Datasets:

Infi-MM

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InfiMM-WebMath-40B

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[ "# If the untyped language is terminating, can we still derive a contradiction from Type : Type?\n\n### Question\n\nIf a pure type system has a terminating proof language, can we have Type : Type at the logic level without causing paradoxes (i.e., without causing ∀ (x : *) -> x to be inhabited)?\n\n### Example\n\nSuppose we take a pure type theory such as the Calculus of Constructions, restrict functions to be affine (at most 1 variable uses) and add constructs for stratified duplications - see here for details. That language - which we could call EACC (Elementary Affine Calculus of Constructions) has a normalizing untyped fragment. That is, the reduction of any term - even ill-typed ones - is guaranteed to terminate due to the reduction rules (and, in particular, the restriction that duplicated terms can not change their levels).\n\n• Perhaps \"Is Type : Type possible in an affine type theory?\" is a better title? – Andrej Bauer Jul 18 '18 at 16:31\n• @AndrejBauer perhaps it needs a better body actually, I used EAL as an example but the actual question is if a terminating proof language makes Type : Type possible. – MaiaVictor Jul 18 '18 at 17:29\n• For basic type:type background I found these helpful: konne.me/2015/08/17/paradoxes.html The coq source is in his github repo github.com/konne88/konne88.github.io . There is also a very nice agda version: gist.github.com/favonia/cc7a504839c5fbb3ebc6 – user833970 Jul 19 '18 at 14:20\n• I'd like to try to look with more detail into your EACC. Are variables classified into affine and non-affine? Do these form separate environments? Why strange syntax with '$'. \"$v x y\" = \"let v = x in y\" right? Is v required to be affine? – Łukasz Lew Jul 19 '18 at 18:48\n• I think that in Coquand's \"An Analysis of Girard's Paradox\" he shows a type system (from Martin Lof) with type:type which is terminating but where every type has an inhabitant. – user833970 Aug 12 '18 at 0:36\n\nThe 'logic' of the contradiction with Type:Type is that you can create a term of any type including 'empty' type by 'cheating' by never returning. This is essentially the only way to cheat because of subject reduction and the subformula properties.\n\nSubject reduction states that evaluation of terms preserves their type. If you have subject reduction and strong normalization (through whatever means), then you can infer that if you have a term of type T, you have a term of type T in a normal form - without any redexes (cuts) in it.\n\nYou have to go over all your type rules and verify that all of them have subformula property except the cut rule (which normalization eliminates). I.e you have to verify that if one of the non-cut rules were to produce a term of type T then you'll have one of its premises produce the term of type T (perhaps somewhere deeper in the derivation tree). AFAIK every good type system requires sub-formula property (I'd like to learn of examples that contradict it).\n\nSince type derivations are supposed to be finite trees, and you have no axiom (0 premises) rule to introduce a term of an empty type, you can prove by induction that no tree can build a term of an empty type.\n\nTo reiterate, the role of normalization is that cut rule does not have sub-formula property, so we have to eliminate it first while preserving the type of the term.\n\nThis is the general sketch of Gentzen's proofs of consistency of logics.\n\n• In order to do that, I would have to perform the work that I described that needs to be done on an unfamiliar set of rules linked by Maia. I'm not confident enough that I would not make a mistake. – Łukasz Lew Jul 19 '18 at 16:05\n• Also, one might notice that I'm quite fuzzy about the subformula property, I would be happy to read some example of it in action. – Łukasz Lew Jul 19 '18 at 16:06\n• Oh I see. Objection withdrawn then :) – Stella Biderman Jul 19 '18 at 16:06\n• I love this answer, thanks for it! If I understand correctly, the cut rule for linear λ-calculus is better in that sense since it has the subformula property, right? – MaiaVictor Sep 27 '18 at 4:10\n• Cut rule (even linear) does not have sub-formula property since the cut-out formula is disappearing. That's why cut elimination (even in linear logic) is so important. – Łukasz Lew Sep 27 '18 at 22:04" ]

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[ "# Black hole entropy from loop quantum gravity in higher dimensions\n\n```@article{Bodendorfer2013BlackHE,\ntitle={Black hole entropy from loop quantum gravity in higher dimensions},\nauthor={Norbert Bodendorfer},\njournal={Physics Letters B},\nyear={2013},\nvolume={726},\npages={887-891}\n}```\nEntropy of higher dimensional nonrotating isolated horizons from loop quantum gravity\n• Physics\n• 2015\nIn this paper, we extend the calculation of the entropy of nonrotating isolated horizons in four-dimensional spacetime to that in a higher-dimensional spacetime. We show that the boundary degrees of\nWald entropy formula and loop quantum gravity\n• Physics\n• 2014\nWe outline how the Wald entropy formula naturally arises in loop quantum gravity based on recently introduced dimension-independent connection variables. The key observation is that in a loop\nThe thermodynamics of isolated horizon in higher dimensional loop quantum gravity\n• Physics\n• 2022\nThe statistical mechanical calculation of the thermodynamical properties of higher-dimensional non-rotating isolated horizons is studied in the loop quantum gravity framework. By employing the\nBTZ Black Hole Entropy and the Turaev–Viro Model\n• Physics\n• 2015\nWe show the explicit agreement between the derivation of the Bekenstein–Hawking entropy of a Euclidean BTZ black hole from the point of view of spin foam models and canonical quantization. This is\nImmirzi parameter and quasinormal modes in four and higher spacetime dimensions\nThere is a one-parameter quantization ambiguity in loop quantum gravity, which is called the Immirzi parameter. In this paper, we fix this free parameter by considering the quasinormal mode spectrum\nA note on entanglement entropy and quantum geometry\nIt has been argued that the entropy computed in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field, and that the\nQuantum theory of charged isolated horizons\n• Physics\n• 2018\nWe describe the quantum theory of isolated horizons with electromagnetic or non-Abelian gauge charges in a setting in which both the gauge and gravitational field are quantized. We consider the\nJ an 2 01 8 Quantum theory of charged isolated horizons\n• Physics\n• 2018\nWe describe the quantum theory of isolated horizons with electromagnetic or non-Abelian gauge charges in a setting in which both gauge and gravitational field are quantized. We consider the distorted\nLoop Quantum Gravity\nThis article presents an \"in-a-nutshell\" yet self-contained introductory review on loop quantum gravity (LQG) -- a background-independent, nonperturbative approach to a consistent quantum theory of\nReformulation of boundary BF theory approach to statistical explanation of the entropy of isolated horizons\n• Physics\n• 2015\nIt is shown in this paper that the symplectic form for the system consisting of D-dimensional bulk Palatini gravity and SO(1, 1) BF theory on an isolated horizon as a boundary just contains the bulk" ]

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[ "# Search result: Catalogue data in Autumn Semester 2018\n\nNumber Title Type ECTS Hours Lecturers Materials Science Bachelor", null, "", null, "3. Semester", null, "", null, "Basic Courses Part 2", null, "", null, "", null, "Examination Block 2 401-0603-00L Stochastics (Probability and Statistics)", null, "O 4 credits 2V + 1U M. H. Maathuis Abstract This class covers the following concepts: random variables, probability, discrete and continuous distributions, joint and conditional probabilities and distributions, the law of large numbers, the central limit theorem, descriptive statistics, statistical inference, inference for normally distributed data, point estimation, and two-sample tests. Objective Knowledge of the basic principles of probability and statistics. Content Introduction to probability theory, some basic principles from mathematical statistics and basic methods for applied statistics. Lecture notes Lecture notes Literature Lecture notes 401-0363-10L Analysis III", null, "O 3 credits 2V + 1U A. Iozzi Abstract Introduction to partial differential equations. Differential equations which are important in applications are classified and solved. Elliptic, parabolic and hyperbolic differential equations are treated. The following mathematical tools are introduced: Laplace transforms, Fourier series, separation of variables, methods of characteristics. Objective Mathematical treatment of problems in science and engineering. To understand the properties of the different types of partial differential equations.The first lecture is on Thursday, September 27 13-15 in HG F 7 and video transmitted into HG F 5.The reference web-page for exercise sheets, solutions and further info is https://metaphor.ethz.ch/x/2018/hs/401-0363-10L/The web-page to enroll for an exercise class ishttps://echo.ethz.chThe coordinator is Stefano D'Alesiohttps://www.math.ethz.ch/the-department/people.html?u=dalesiosStudy Center D-MAVT: 16-18 every Monday from the 3rd week of the semester (first appointment: October the 1st)room HG E22 LinkStudy Center D-MATL: 15-17 every Wednesday from the 5th week of the semester (first appointment: October the 17th)room HCI J 574Ferienpräsenz: Tuesday 15 January 2019, at 12:30-14:00, in room HG G 19.1.Monday 21 January 2019, at 12:30-14:00, in room HG G 19.2.Prüfungseinsicht:Tuesday 26 February 2019, at 17:00-18:30, in room HG 19.1.Monday 4 March 2019, at 18:15-19:45, in room HG 19.1. Content Laplace Transforms:- Laplace Transform, Inverse Laplace Transform, Linearity, s-Shifting - Transforms of Derivatives and Integrals, ODEs- Unit Step Function, t-Shifting- Short Impulses, Dirac's Delta Function, Partial Fractions- Convolution, Integral Equations- Differentiation and Integration of TransformsFourier Series, Integrals and Transforms:- Fourier Series- Functions of Any Period p=2L- Even and Odd Functions, Half-Range Expansions- Forced Oscillations- Approximation by Trigonometric Polynomials- Fourier Integral- Fourier Cosine and Sine TransformPartial Differential Equations:- Basic Concepts- Modeling: Vibrating String, Wave Equation- Solution by separation of variables; use of Fourier series- D'Alembert Solution of Wave Equation, Characteristics- Heat Equation: Solution by Fourier Series- Heat Equation: Solutions by Fourier Integrals and Transforms- Modeling Membrane: Two Dimensional Wave Equation- Laplacian in Polar Coordinates: Circular Membrane, Fourier-Bessel Series- Solution of PDEs by Laplace Transform Lecture notes Lecture notes by Prof. Dr. Alessandra Iozzi:https://polybox.ethz.ch/index.php/s/D3K0TayQXvfpCAA Literature E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 10. Auflage, 2011C. R. Wylie & L. Barrett, Advanced Engineering Mathematics, McGraw-Hill, 6th ed.S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Books on Mathematics, NY.G. Felder, Partielle Differenzialgleichungen für Ingenieurinnen und Ingenieure, hypertextuelle Notizen zur Vorlesung Analysis III im WS 2002/2003.Y. Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005For reference/complement of the Analysis I/II courses:Christian Blatter: Ingenieur-Analysis https://people.math.ethz.ch/~blatter/dlp.html 327-0308-00L Programming Techniques in Materials Science", null, "O 2 credits 2G C. Ederer Abstract This course introduces the general computing and programming skills which are necessary to perform numerical computations and simulations in materials science. This is achieved using the numerical computing environment Matlab and through the use of many practical examples and exercises. Objective On passing this course, the students should be able to develop their own programs for performing numerical computations and simulations, and they should be able to analyse and amend existing code. Content Introduction to Matlab; input/output; structured programming using loops and conditional execution; modular Programming using functions; flow diagrams; numerical accuracy; example: random walk model.\n•", null, "", null, "Page  1  of  1", null, "", null, "" ]

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[ "", null, "", null, "Vincent Selhorst-Jones\n\nInverse Functions\n\nSlide Duration:\n\nSection 1: Introduction\nIntroduction to Precalculus\n\n10m 3s\n\nIntro\n0:00\nTitle of the Course\n0:06\nDifferent Names for the Course\n0:07\nPrecalculus\n0:12\nMath Analysis\n0:14\nTrigonometry\n0:16\nAlgebra III\n0:20\nGeometry II\n0:24\nCollege Algebra\n0:30\nSame Concepts\n0:36\nHow do the Lessons Work?\n0:54\nIntroducing Concepts\n0:56\nApply Concepts\n1:04\nGo through Examples\n1:25\nWho is this Course For?\n1:38\nThose Who Need eExtra Help with Class Work\n1:52\nThose Working on Material but not in Formal Class at School\n1:54\nThose Who Want a Refresher\n2:00\nTry to Watch the Whole Lesson\n2:20\nUnderstanding is So Important\n3:56\nWhat to Watch First\n5:26\nLesson #2: Sets, Elements, and Numbers\n5:30\nLesson #7: Idea of a Function\n5:33\nLesson #6: Word Problems\n6:04\nWhat to Watch First, cont.\n6:46\nLesson #2: Sets, Elements and Numbers\n6:56\nLesson #3: Variables, Equations, and Algebra\n6:58\nLesson #4: Coordinate Systems\n7:00\nLesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope\n7:02\nLesson #6: Word Problems\n7:10\nLesson #7: Idea of a Function\n7:12\nLesson #8: Graphs\n7:14\nGraphing Calculator Appendix\n7:40\nWhat to Watch Last\n8:46\nLet's get Started!\n9:48\nSets, Elements, & Numbers\n\n45m 11s\n\nIntro\n0:00\nIntroduction\n0:05\nSets and Elements\n1:19\nSet\n1:20\nElement\n1:23\nName a Set\n2:20\nOrder The Elements Appear In Has No Effect on the Set\n2:55\nDescribing/ Defining Sets\n3:28\nDirectly Say All the Elements\n3:36\nClearly Describing All the Members of the Set\n3:55\nDescribing the Quality (or Qualities) Each member Of the Set Has In Common\n4:32\nSymbols: 'Element of' and 'Subset of'\n6:01\nSymbol is ∈\n6:03\nSubset Symbol is ⊂\n6:35\nEmpty Set\n8:07\nSymbol is ∅\n8:20\nSince It's Empty, It is a Subset of All Sets\n8:44\nUnion and Intersection\n9:54\nUnion Symbol is ∪\n10:08\nIntersection Symbol is ∩\n10:18\nSets Can Be Weird Stuff\n12:26\nCan Have Elements in a Set\n12:50\nWe Can Have Infinite Sets\n13:09\nExample\n13:22\nConsider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times\n14:08\nThis Set Has Infinitely Many Distinct Elements\n14:40\nNumbers as Sets\n16:03\nNatural Numbers ℕ\n16:16\nIncluding 0 and the Negatives ℤ\n18:13\nRational Numbers ℚ\n19:27\nCan Express Rational Numbers with Decimal Expansions\n22:05\nIrrational Numbers\n23:37\nReal Numbers ℝ: Put the Rational and Irrational Numbers Together\n25:15\nInterval Notation and the Real Numbers\n26:45\nInclude the End Numbers\n27:06\nExclude the End Numbers\n27:33\nExample\n28:28\nInterval Notation: Infinity\n29:09\nUse -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other\n29:14\nAlways Use Parentheses\n29:50\nExamples\n30:27\nExample 1\n31:23\nExample 2\n35:26\nExample 3\n38:02\nExample 4\n42:21\nVariables, Equations, & Algebra\n\n35m 31s\n\nIntro\n0:00\nWhat is a Variable?\n0:05\nA Variable is a Placeholder for a Number\n0:11\nAffects the Output of a Function or a Dependent Variable\n0:24\nNaming Variables\n1:51\nUseful to Use Symbols\n2:21\nWhat is a Constant?\n4:14\nA Constant is a Fixed, Unchanging Number\n4:28\nWe Might Refer to a Symbol Representing a Number as a Constant\n4:51\nWhat is a Coefficient?\n5:33\nA Coefficient is a Multiplicative Factor on a Variable\n5:37\nNot All Coefficients are Constants\n5:51\nExpressions and Equations\n6:42\nAn Expression is a String of Mathematical Symbols That Make Sense Used Together\n7:05\nAn Equation is a Statement That Two Expression Have the Same Value\n8:20\nThe Idea of Algebra\n8:51\nEquality\n8:59\nIf Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same\n9:41\nAlways Do The Exact Same Thing to Both Sides\n12:22\nSolving Equations\n13:23\nWhen You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something\n13:33\nLook For What Values Makes the Equation True\n13:38\nIsolate the Variable by Doing Algebra\n14:37\nOrder of Operations\n16:02\nWhy Certain Operations are Grouped\n17:01\nWhen You Don't Have to Worry About Order\n17:39\nDistributive Property\n18:15\nIt Allows Multiplication to Act Over Addition in Parentheses\n18:23\nWe Can Use the Distributive Property in Reverse to Combine Like Terms\n19:05\nSubstitution\n20:03\nUse Information From One Equation in Another Equation\n20:07\n20:44\nExample 1\n23:17\nExample 2\n25:49\nExample 3\n28:11\nExample 4\n30:02\nCoordinate Systems\n\n35m 2s\n\nIntro\n0:00\nInherent Order in ℝ\n0:05\nReal Numbers Come with an Inherent Order\n0:11\nPositive Numbers\n0:21\nNegative Numbers\n0:58\n'Less Than' and 'Greater Than'\n2:04\n2:56\nInequality\n4:06\nLess Than or Equal and Greater Than or Equal\n4:51\nOne Dimension: The Number Line\n5:36\nGraphically Represent ℝ on a Number Line\n5:43\nNote on Infinities\n5:57\nWith the Number Line, We Can Directly See the Order We Put on ℝ\n6:35\nOrdered Pairs\n7:22\nExample\n7:34\nAllows Us to Talk About Two Numbers at the Same Time\n9:41\nOrdered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ\n10:41\nTwo Dimensions: The Plane\n13:13\nWe Can Represent Ordered Pairs with the Plane\n13:24\nIntersection is known as the Origin\n14:31\nPlotting the Point\n14:32\nPlane = Coordinate Plane = Cartesian Plane = ℝ²\n17:46\n18:50\n19:04\n19:21\n20:04\n20:20\nThree Dimensions: Space\n21:02\nCreate Ordered Triplets\n21:09\nVisually Represent This\n21:19\nThree-Dimension = Space = ℝ³\n21:47\nHigher Dimensions\n22:24\nIf We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power\n22:31\nWe Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers\n22:41\nHard to Visualize Higher Dimensional Spaces\n23:18\nExample 1\n25:07\nExample 2\n26:10\nExample 3\n28:58\nExample 4\n31:05\nMidpoints, Distance, the Pythagorean Theorem, & Slope\n\n48m 43s\n\nIntro\n0:00\nIntroduction\n0:07\nMidpoint: One Dimension\n2:09\nExample of Something More Complex\n2:31\nUse the Idea of a Middle\n3:28\nFind the Midpoint of Arbitrary Values a and b\n4:17\nHow They're Equivalent\n5:05\nOfficial Midpoint Formula\n5:46\nMidpoint: Two Dimensions\n6:19\nThe Midpoint Must Occur at the Horizontal Middle and the Vertical Middle\n6:38\nArbitrary Pair of Points Example\n7:25\nDistance: One Dimension\n9:26\nAbsolute Value\n10:54\nIdea of Forcing Positive\n11:06\nDistance: One Dimension, Formula\n11:47\nDistance Between Arbitrary a and b\n11:48\nAbsolute Value Helps When the Distance is Negative\n12:41\nDistance Formula\n12:58\nThe Pythagorean Theorem\n13:24\na²+b²=c²\n13:50\nDistance: Two Dimensions\n14:59\nBreak Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem\n15:16\nDistance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)\n16:21\nSlope\n19:30\nSlope is the Rate of Change\n19:41\nm = rise over run\n21:27\nSlope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)\n22:31\nInterpreting Slope\n24:12\nPositive Slope and Negative Slope\n25:40\nm=1, m=0, m=-1\n26:48\nExample 1\n28:25\nExample 2\n31:42\nExample 3\n36:40\nExample 4\n42:48\nWord Problems\n\n56m 31s\n\nIntro\n0:00\nIntroduction\n0:05\nWhat is a Word Problem?\n0:45\nDescribes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols\n0:48\nRequires Us to Think\n1:32\nWhy Are They So Hard?\n2:11\nReason 1: No Simple Formula to Solve Them\n2:16\nReason 2: Harder to Teach Word Problems\n2:47\nYou Can Learn How to Do Them!\n3:51\n7:57\n'But I'm Never Going to Use This In Real Life'\n9:46\nSolving Word Problems\n12:58\nFirst: Understand the Problem\n13:37\nSecond: What Are You Looking For?\n14:33\nThird: Set Up Relationships\n16:21\nFourth: Solve It!\n17:48\nSummary of Method\n19:04\nExamples on Things Other Than Math\n20:21\nMath-Specific Method: What You Need Now\n25:30\nUnderstand What the Problem is Talking About\n25:37\nSet Up and Name Any Variables You Need to Know\n25:56\nSet Up Equations Connecting Those Variables to the Information in the Problem Statement\n26:02\nUse the Equations to Solve for an Answer\n26:14\nTip\n26:58\nDraw Pictures\n27:22\nBreaking Into Pieces\n28:28\nTry Out Hypothetical Numbers\n29:52\nStudent Logic\n31:27\nJump In!\n32:40\nExample 1\n34:03\nExample 2\n39:15\nExample 3\n44:22\nExample 4\n50:24\nSection 2: Functions\nIdea of a Function\n\n39m 54s\n\nIntro\n0:00\nIntroduction\n0:04\nWhat is a Function?\n1:06\nA Visual Example and Non-Example\n1:30\nFunction Notation\n3:47\nf(x)\n4:05\nExpress What Sets the Function Acts On\n5:45\nMetaphors for a Function\n6:17\nTransformation\n6:28\nMap\n7:17\nMachine\n8:56\nSame Input Always Gives Same Output\n10:01\nIf We Put the Same Input Into a Function, It Will Always Produce the Same Output\n10:11\nExample of Something That is Not a Function\n11:10\nA Non-Numerical Example\n12:10\nThe Functions We Will Use\n15:05\nUnless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers\n15:11\nUsually Told the Rule of a Given Function\n15:27\nHow To Use a Function\n16:18\nApply the Rule to Whatever Our Input Value Is\n16:28\nMake Sure to Wrap Your Substitutions in Parentheses\n17:09\nFunctions and Tables\n17:36\nTable of Values, Sometimes Called a T-Table\n17:46\nExample\n17:56\nDomain: What Goes In\n18:55\nThe Domain is the Set of all Inputs That the Function Can Accept\n18:56\nExample\n19:40\nRange: What Comes Out\n21:27\nThe Range is the Set of All Possible Outputs a Function Can Assign\n21:34\nExample\n21:49\nAnother Example Would Be Our Initial Function From Earlier in This Lesson\n22:29\nExample 1\n23:45\nExample 2\n25:22\nExample 3\n27:27\nExample 4\n29:23\nExample 5\n33:33\nGraphs\n\n58m 26s\n\nIntro\n0:00\nIntroduction\n0:04\nHow to Interpret Graphs\n1:17\nInput / Independent Variable\n1:47\nOutput / Dependent Variable\n2:00\nGraph as Input ⇒ Output\n2:23\nOne Way to Think of a Graph: See What Happened to Various Inputs\n2:25\nExample\n2:47\nGraph as Location of Solution\n4:20\nA Way to See Solutions\n4:36\nExample\n5:20\nWhich Way Should We Interpret?\n7:13\nEasiest to Think In Terms of How Inputs Are Mapped to Outputs\n7:20\nSometimes It's Easier to Think In Terms of Solutions\n8:39\nPay Attention to Axes\n9:50\nAxes Tell Where the Graph Is and What Scale It Has\n10:09\nOften, The Axes Will Be Square\n10:14\nExample\n12:06\nArrows or No Arrows?\n16:07\nWill Not Use Arrows at the End of Our Graphs\n17:13\nGraph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops\n17:18\nHow to Graph\n19:47\nPlot Points\n20:07\nConnect with Curves\n21:09\nIf You Connect with Straight Lines\n21:44\nGraphs of Functions are Smooth\n22:21\nMore Points ⇒ More Accurate\n23:38\nVertical Line Test\n27:44\nIf a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function\n28:41\nEvery Point on a Graph Tells Us Where the x-Value Below is Mapped\n30:07\nDomain in Graphs\n31:37\nThe Domain is the Set of All Inputs That a Function Can Accept\n31:44\nBe Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'\n33:19\nRange in Graphs\n33:53\nGraphing Calculators: Check the Appendix!\n36:55\nExample 1\n38:37\nExample 2\n45:19\nExample 3\n50:41\nExample 4\n53:28\nExample 5\n55:50\nProperties of Functions\n\n48m 49s\n\nIntro\n0:00\nIntroduction\n0:05\nIncreasing Decreasing Constant\n0:43\nLooking at a Specific Graph\n1:15\nIncreasing Interval\n2:39\nConstant Function\n4:15\nDecreasing Interval\n5:10\nFind Intervals by Looking at the Graph\n5:32\nIntervals Show x-values; Write in Parentheses\n6:39\nMaximum and Minimums\n8:48\nRelative (Local) Max/Min\n10:20\nFormal Definition of Relative Maximum\n12:44\nFormal Definition of Relative Minimum\n13:05\nMax/Min, More Terms\n14:18\nDefinition of Extrema\n15:01\nAverage Rate of Change\n16:11\nDrawing a Line for the Average Rate\n16:48\nUsing the Slope of the Secant Line\n17:36\nSlope in Function Notation\n18:45\nZeros/Roots/x-intercepts\n19:45\nWhat Zeros in a Function Mean\n20:25\nEven Functions\n22:30\nOdd Functions\n24:36\nEven/Odd Functions and Graphs\n26:28\nExample of an Even Function\n27:12\nExample of an Odd Function\n28:03\nExample 1\n29:35\nExample 2\n33:07\nExample 3\n40:32\nExample 4\n42:34\nFunction Petting Zoo\n\n29m 20s\n\nIntro\n0:00\nIntroduction\n0:04\nDon't Forget that Axes Matter!\n1:44\nThe Constant Function\n2:40\nThe Identity Function\n3:44\nThe Square Function\n4:40\nThe Cube Function\n5:44\nThe Square Root Function\n6:51\nThe Reciprocal Function\n8:11\nThe Absolute Value Function\n10:19\nThe Trigonometric Functions\n11:56\nf(x)=sin(x)\n12:12\nf(x)=cos(x)\n12:24\nAlternate Axes\n12:40\nThe Exponential and Logarithmic Functions\n13:35\nExponential Functions\n13:44\nLogarithmic Functions\n14:24\nAlternating Axes\n15:17\nTransformations and Compositions\n16:08\nExample 1\n17:52\nExample 2\n18:33\nExample 3\n20:24\nExample 4\n26:07\nTransformation of Functions\n\n48m 35s\n\nIntro\n0:00\nIntroduction\n0:04\nVertical Shift\n1:12\nGraphical Example\n1:21\nA Further Explanation\n2:16\nVertical Stretch/Shrink\n3:34\nGraph Shrinks\n3:46\nGraph Stretches\n3:51\nA Further Explanation\n5:07\nHorizontal Shift\n6:49\nMoving the Graph to the Right\n7:28\nMoving the Graph to the Left\n8:12\nA Further Explanation\n8:19\nUnderstanding Movement on the x-axis\n8:38\nHorizontal Stretch/Shrink\n12:59\nShrinking the Graph\n13:40\nStretching the Graph\n13:48\nA Further Explanation\n13:55\nUnderstanding Stretches from the x-axis\n14:12\nVertical Flip (aka Mirror)\n16:55\nExample Graph\n17:07\nMultiplying the Vertical Component by -1\n17:18\nHorizontal Flip (aka Mirror)\n18:43\nExample Graph\n19:01\nMultiplying the Horizontal Component by -1\n19:54\nSummary of Transformations\n22:11\nStacking Transformations\n24:46\nOrder Matters\n25:20\nTransformation Example\n25:52\nExample 1\n29:21\nExample 2\n34:44\nExample 3\n38:10\nExample 4\n43:46\nComposite Functions\n\n33m 24s\n\nIntro\n0:00\nIntroduction\n0:04\nArithmetic Combinations\n0:40\nBasic Operations\n1:20\nDefinition of the Four Arithmetic Combinations\n1:40\nComposite Functions\n2:53\nThe Function as a Machine\n3:32\nFunction Compositions as Multiple Machines\n3:59\nNotation for Composite Functions\n4:46\nTwo Formats\n6:02\nAnother Visual Interpretation\n7:17\nHow to Use Composite Functions\n8:21\nExample of on Function acting on Another\n9:17\nExample 1\n11:03\nExample 2\n15:27\nExample 3\n21:11\nExample 4\n27:06\nPiecewise Functions\n\n51m 42s\n\nIntro\n0:00\nIntroduction\n0:04\nAnalogies to a Piecewise Function\n1:16\nDifferent Potatoes\n1:41\nFactory Production\n2:27\nNotations for Piecewise Functions\n3:39\nNotation Examples from Analogies\n6:11\nExample of a Piecewise (with Table)\n7:24\nExample of a Non-Numerical Piecewise\n11:35\nGraphing Piecewise Functions\n14:15\nGraphing Piecewise Functions, Example\n16:26\nContinuous Functions\n16:57\nStatements of Continuity\n19:30\nExample of Continuous and Non-Continuous Graphs\n20:05\nInteresting Functions: the Step Function\n22:00\nNotation for the Step Function\n22:40\nHow the Step Function Works\n22:56\nGraph of the Step Function\n25:30\nExample 1\n26:22\nExample 2\n28:49\nExample 3\n36:50\nExample 4\n46:11\nInverse Functions\n\n49m 37s\n\nIntro\n0:00\nIntroduction\n0:04\nAnalogy by picture\n1:10\nHow to Denote the inverse\n1:40\nWhat Comes out of the Inverse\n1:52\nRequirement for Reversing\n2:02\n2:12\nThe Importance of Information\n2:45\nOne-to-One\n4:04\nRequirement for Reversibility\n4:21\nWhen a Function has an Inverse\n4:43\nOne-to-One\n5:13\nNot One-to-One\n5:50\nNot a Function\n6:19\nHorizontal Line Test\n7:01\nHow to the test Works\n7:12\nOne-to-One\n8:12\nNot One-to-One\n8:45\nDefinition: Inverse Function\n9:12\nFormal Definition\n9:21\nCaution to Students\n10:02\nDomain and Range\n11:12\nFinding the Range of the Function Inverse\n11:56\nFinding the Domain of the Function Inverse\n12:11\nInverse of an Inverse\n13:09\nIts just x!\n13:26\nProof\n14:03\nGraphical Interpretation\n17:07\nHorizontal Line Test\n17:20\nGraph of the Inverse\n18:04\nSwapping Inputs and Outputs to Draw Inverses\n19:02\nHow to Find the Inverse\n21:03\nWhat We Are Looking For\n21:21\nReversing the Function\n21:38\nA Method to Find Inverses\n22:33\nCheck Function is One-to-One\n23:04\nSwap f(x) for y\n23:25\nInterchange x and y\n23:41\nSolve for y\n24:12\nReplace y with the inverse\n24:40\n25:01\nKeeping Step 2 and 3 Straight\n25:44\nSwitching to Inverse\n26:12\nChecking Inverses\n28:52\nHow to Check an Inverse\n29:06\nQuick Example of How to Check\n29:56\nExample 1\n31:48\nExample 2\n34:56\nExample 3\n39:29\nExample 4\n46:19\nVariation Direct and Inverse\n\n28m 49s\n\nIntro\n0:00\nIntroduction\n0:06\nDirect Variation\n1:14\nSame Direction\n1:21\nCommon Example: Groceries\n1:56\nDifferent Ways to Say that Two Things Vary Directly\n2:28\nBasic Equation for Direct Variation\n2:55\nInverse Variation\n3:40\nOpposite Direction\n3:50\nCommon Example: Gravity\n4:53\nDifferent Ways to Say that Two Things Vary Indirectly\n5:48\nBasic Equation for Indirect Variation\n6:33\nJoint Variation\n7:27\nEquation for Joint Variation\n7:53\nExplanation of the Constant\n8:48\nCombined Variation\n9:35\nGas Law as a Combination\n9:44\nSingle Constant\n10:33\nExample 1\n10:49\nExample 2\n13:34\nExample 3\n15:39\nExample 4\n19:48\nSection 3: Polynomials\nIntro to Polynomials\n\n38m 41s\n\nIntro\n0:00\nIntroduction\n0:04\nDefinition of a Polynomial\n1:04\nStarting Integer\n2:06\nStructure of a Polynomial\n2:49\nThe a Constants\n3:34\nPolynomial Function\n5:13\nPolynomial Equation\n5:23\nPolynomials with Different Variables\n5:36\nDegree\n6:23\nInformal Definition\n6:31\nFind the Largest Exponent Variable\n6:44\nQuick Examples\n7:36\nSpecial Names for Polynomials\n8:59\nBased on the Degree\n9:23\nBased on the Number of Terms\n10:12\nDistributive Property (aka 'FOIL')\n11:37\nBasic Distributive Property\n12:21\nDistributing Two Binomials\n12:55\nLonger Parentheses\n15:12\nReverse: Factoring\n17:26\nLong-Term Behavior of Polynomials\n17:48\nExamples\n18:13\nControlling Term--Term with the Largest Exponent\n19:33\nPositive and Negative Coefficients on the Controlling Term\n20:21\n22:07\nEven Degree, Positive Coefficient\n22:13\nEven Degree, Negative Coefficient\n22:39\nOdd Degree, Positive Coefficient\n23:09\nOdd Degree, Negative Coefficient\n23:27\nExample 1\n25:11\nExample 2\n27:16\nExample 3\n31:16\nExample 4\n34:41\nRoots (Zeros) of Polynomials\n\n41m 7s\n\nIntro\n0:00\nIntroduction\n0:05\nRoots in Graphs\n1:17\nThe x-intercepts\n1:33\nHow to Remember What 'Roots' Are\n1:50\nNaïve Attempts\n2:31\nIsolating Variables\n2:45\nFailures of Isolating Variables\n3:30\nMissing Solutions\n4:59\nFactoring: How to Find Roots\n6:28\nHow Factoring Works\n6:36\nWhy Factoring Works\n7:20\nSteps to Finding Polynomial Roots\n9:21\nFactoring: How to Find Roots CAUTION\n10:08\nFactoring is Not Easy\n11:32\n13:08\n13:21\nForm of Factored Binomials\n13:38\nFactoring Examples\n14:40\n16:58\nFactoring Higher Degree Polynomials\n18:19\nFactoring a Cubic\n18:32\n19:04\nFactoring: Roots Imply Factors\n19:54\nWhere a Root is, A Factor Is\n20:01\nHow to Use Known Roots to Make Factoring Easier\n20:35\nNot all Polynomials Can be Factored\n22:30\nIrreducible Polynomials\n23:27\nComplex Numbers Help\n23:55\nMax Number of Roots/Factors\n24:57\nLimit to Number of Roots Equal to the Degree\n25:18\nWhy there is a Limit\n25:25\nMax Number of Peaks/Valleys\n26:39\nShape Information from Degree\n26:46\nExample Graph\n26:54\nMax, But Not Required\n28:00\nExample 1\n28:37\nExample 2\n31:21\nExample 3\n36:12\nExample 4\n38:40\nCompleting the Square and the Quadratic Formula\n\n39m 43s\n\nIntro\n0:00\nIntroduction\n0:05\nSquare Roots and Equations\n0:51\nTaking the Square Root to Find the Value of x\n0:55\nGetting the Positive and Negative Answers\n1:05\nCompleting the Square: Motivation\n2:04\nPolynomials that are Easy to Solve\n2:20\nMaking Complex Polynomials Easy to Solve\n3:03\nSteps to Completing the Square\n4:30\nCompleting the Square: Method\n7:22\nMove C over\n7:35\nDivide by A\n7:44\nFind r\n7:59\nAdd to Both Sides to Complete the Square\n8:49\n9:56\n11:38\nDerivation\n11:43\nFinal Form\n12:23\n13:38\nHow Many Roots?\n14:53\nThe Discriminant\n15:47\nWhat the Discriminant Tells Us: How Many Roots\n15:58\nHow the Discriminant Works\n16:30\nExample 1: Complete the Square\n18:24\n22:00\nExample 3: Solve for Zeroes\n25:28\nExample 4: Using the Quadratic Formula\n30:52\n\n45m 34s\n\nIntro\n0:00\nIntroduction\n0:05\nParabolas\n0:35\nExamples of Different Parabolas\n1:06\nAxis of Symmetry and Vertex\n1:28\nDrawing an Axis of Symmetry\n1:51\nPlacing the Vertex\n2:28\nLooking at the Axis of Symmetry and Vertex for other Parabolas\n3:09\nTransformations\n4:18\nReviewing Transformation Rules\n6:28\nNote the Different Horizontal Shift Form\n7:45\n8:54\nThe Constants: k, h, a\n9:05\nTransformations Formed\n10:01\nAnalyzing Different Parabolas\n10:10\nSwitching Forms by Completing the Square\n11:43\nVertex of a Parabola\n16:30\nVertex at (h, k)\n16:47\nVertex in Terms of a, b, and c Coefficients\n17:28\nMinimum/Maximum at Vertex\n18:19\nWhen a is Positive\n18:25\nWhen a is Negative\n18:52\nAxis of Symmetry\n19:54\nIncredibly Minor Note on Grammar\n20:52\nExample 1\n21:48\nExample 2\n26:35\nExample 3\n28:55\nExample 4\n31:40\nIntermediate Value Theorem and Polynomial Division\n\n46m 8s\n\nIntro\n0:00\nIntroduction\n0:05\nReminder: Roots Imply Factors\n1:32\nThe Intermediate Value Theorem\n3:41\nThe Basis: U between a and b\n4:11\nU is on the Function\n4:52\nIntermediate Value Theorem, Proof Sketch\n5:51\nIf Not True, the Graph Would Have to Jump\n5:58\nBut Graph is Defined as Continuous\n6:43\nFinding Roots with the Intermediate Value Theorem\n7:01\nPicking a and b to be of Different Signs\n7:10\nMust Be at Least One Root\n7:46\nDividing a Polynomial\n8:16\nUsing Roots and Division to Factor\n8:38\nLong Division Refresher\n9:08\nThe Division Algorithm\n12:18\nHow It Works to Divide Polynomials\n12:37\nThe Parts of the Equation\n13:24\nRewriting the Equation\n14:47\nPolynomial Long Division\n16:20\nPolynomial Long Division In Action\n16:29\nOne Step at a Time\n20:51\nSynthetic Division\n22:46\nSetup\n23:11\nSynthetic Division, Example\n24:44\nWhich Method Should We Use\n26:39\n26:49\n27:13\nExample 1\n29:24\nExample 2\n31:27\nExample 3\n36:22\nExample 4\n40:55\nComplex Numbers\n\n45m 36s\n\nIntro\n0:00\nIntroduction\n0:04\nA Wacky Idea\n1:02\nThe Definition of the Imaginary Number\n1:22\nHow it Helps Solve Equations\n2:20\nSquare Roots and Imaginary Numbers\n3:15\nComplex Numbers\n5:00\nReal Part and Imaginary Part\n5:20\nWhen Two Complex Numbers are Equal\n6:10\n6:40\nDeal with Real and Imaginary Parts Separately\n7:36\nTwo Quick Examples\n7:54\nMultiplication\n9:07\nFOIL Expansion\n9:14\nNote What Happens to the Square of the Imaginary Number\n9:41\nTwo Quick Examples\n10:22\nDivision\n11:27\nComplex Conjugates\n13:37\nGetting Rid of i\n14:08\nHow to Denote the Conjugate\n14:48\nDivision through Complex Conjugates\n16:11\nMultiply by the Conjugate of the Denominator\n16:28\nExample\n17:46\n19:24\n20:12\nConjugate Pairs\n20:37\nBut Are the Complex Numbers 'Real'?\n21:27\nWhat Makes a Number Legitimate\n25:38\nWhere Complex Numbers are Used\n27:20\nStill, We Won't See Much of C\n29:05\nExample 1\n30:30\nExample 2\n33:15\nExample 3\n38:12\nExample 4\n42:07\nFundamental Theorem of Algebra\n\n19m 9s\n\nIntro\n0:00\nIntroduction\n0:05\nIdea: Hidden Roots\n1:16\nRoots in Complex Form\n1:42\nAll Polynomials Have Roots\n2:08\nFundamental Theorem of Algebra\n2:21\nWhere Are All the Imaginary Roots, Then?\n3:17\nAll Roots are Complex\n3:45\nReal Numbers are a Subset of Complex Numbers\n3:59\nThe n Roots Theorem\n5:01\nFor Any Polynomial, Its Degree is Equal to the Number of Roots\n5:11\nEquivalent Statement\n5:24\n6:29\nNon-Distinct Roots\n6:59\nDenoting Multiplicity\n7:20\n7:41\n8:55\n9:59\nProof Sketch of n Roots Theorem\n10:45\nFirst Root\n11:36\nSecond Root\n13:23\nContinuation to Find all Roots\n16:00\nSection 4: Rational Functions\nRational Functions and Vertical Asymptotes\n\n33m 22s\n\nIntro\n0:00\nIntroduction\n0:05\nDefinition of a Rational Function\n1:20\nExamples of Rational Functions\n2:30\nWhy They are Called 'Rational'\n2:47\nDomain of a Rational Function\n3:15\nUndefined at Denominator Zeros\n3:25\nOtherwise all Reals\n4:16\nInvestigating a Fundamental Function\n4:50\nThe Domain of the Function\n5:04\nWhat Occurs at the Zeroes of the Denominator\n5:20\nIdea of a Vertical Asymptote\n6:23\nWhat's Going On?\n6:58\nApproaching x=0 from the left\n7:32\nApproaching x=0 from the right\n8:34\nDividing by Very Small Numbers Results in Very Large Numbers\n9:31\nDefinition of a Vertical Asymptote\n10:05\nVertical Asymptotes and Graphs\n11:15\nDrawing Asymptotes by Using a Dashed Line\n11:27\nThe Graph Can Never Touch Its Undefined Point\n12:00\nNot All Zeros Give Asymptotes\n13:02\nSpecial Cases: When Numerator and Denominator Go to Zero at the Same Time\n14:58\nCancel out Common Factors\n15:49\nHow to Find Vertical Asymptotes\n16:10\nFigure out What Values Are Not in the Domain of x\n16:24\nDetermine if the Numerator and Denominator Share Common Factors and Cancel\n16:45\nFind Denominator Roots\n17:33\nNote if Asymptote Approaches Negative or Positive Infinity\n18:06\nExample 1\n18:57\nExample 2\n21:26\nExample 3\n23:04\nExample 4\n30:01\nHorizontal Asymptotes\n\n34m 16s\n\nIntro\n0:00\nIntroduction\n0:05\nInvestigating a Fundamental Function\n0:53\nWhat Happens as x Grows Large\n1:00\nDifferent View\n1:12\nIdea of a Horizontal Asymptote\n1:36\nWhat's Going On?\n2:24\nWhat Happens as x Grows to a Large Negative Number\n2:49\nWhat Happens as x Grows to a Large Number\n3:30\nDividing by Very Large Numbers Results in Very Small Numbers\n3:52\nExample Function\n4:41\nDefinition of a Vertical Asymptote\n8:09\nExpanding the Idea\n9:03\nWhat's Going On?\n9:48\nWhat Happens to the Function in the Long Run?\n9:51\nRewriting the Function\n10:13\nDefinition of a Slant Asymptote\n12:09\nSymbolical Definition\n12:30\nInformal Definition\n12:45\nBeyond Slant Asymptotes\n13:03\nNot Going Beyond Slant Asymptotes\n14:39\nHorizontal/Slant Asymptotes and Graphs\n15:43\nHow to Find Horizontal and Slant Asymptotes\n16:52\nHow to Find Horizontal Asymptotes\n17:12\nExpand the Given Polynomials\n17:18\nCompare the Degrees of the Numerator and Denominator\n17:40\nHow to Find Slant Asymptotes\n20:05\nSlant Asymptotes Exist When n+m=1\n20:08\nUse Polynomial Division\n20:24\nExample 1\n24:32\nExample 2\n25:53\nExample 3\n26:55\nExample 4\n29:22\nGraphing Asymptotes in a Nutshell\n\n49m 7s\n\nIntro\n0:00\nIntroduction\n0:05\nA Process for Graphing\n1:22\n1. Factor Numerator and Denominator\n1:50\n2. Find Domain\n2:53\n3. Simplifying the Function\n3:59\n4. Find Vertical Asymptotes\n4:59\n5. Find Horizontal/Slant Asymptotes\n5:24\n6. Find Intercepts\n7:35\n7. Draw Graph (Find Points as Necessary)\n9:21\nDraw Graph Example\n11:21\nVertical Asymptote\n11:41\nHorizontal Asymptote\n11:50\nOther Graphing\n12:16\nTest Intervals\n15:08\nExample 1\n17:57\nExample 2\n23:01\nExample 3\n29:02\nExample 4\n33:37\nPartial Fractions\n\n44m 56s\n\nIntro\n0:00\nIntroduction: Idea\n0:04\nIntroduction: Prerequisites and Uses\n1:57\nProper vs. Improper Polynomial Fractions\n3:11\nPossible Things in the Denominator\n4:38\nLinear Factors\n6:16\nExample of Linear Factors\n7:03\nMultiple Linear Factors\n7:48\n8:25\n9:26\n9:49\nMixing Factor Types\n10:28\nFiguring Out the Numerator\n11:10\nHow to Solve for the Constants\n11:30\nQuick Example\n11:40\nExample 1\n14:29\nExample 2\n18:35\nExample 3\n20:33\nExample 4\n28:51\nSection 5: Exponential & Logarithmic Functions\nUnderstanding Exponents\n\n35m 17s\n\nIntro\n0:00\nIntroduction\n0:05\nFundamental Idea\n1:46\nExpanding the Idea\n2:28\nMultiplication of the Same Base\n2:40\nExponents acting on Exponents\n3:45\nDifferent Bases with the Same Exponent\n4:31\nTo the Zero\n5:35\nTo the First\n5:45\nFundamental Rule with the Zero Power\n6:35\nTo the Negative\n7:45\nAny Number to a Negative Power\n8:14\nA Fraction to a Negative Power\n9:58\nDivision with Exponential Terms\n10:41\nTo the Fraction\n11:33\nSquare Root\n11:58\nAny Root\n12:59\nSummary of Rules\n14:38\nTo the Irrational\n17:21\nExample 1\n20:34\nExample 2\n23:42\nExample 3\n27:44\nExample 4\n31:44\nExample 5\n33:15\nExponential Functions\n\n47m 4s\n\nIntro\n0:00\nIntroduction\n0:05\nDefinition of an Exponential Function\n0:48\nDefinition of the Base\n1:02\nRestrictions on the Base\n1:16\nComputing Exponential Functions\n2:29\nHarder Computations\n3:10\nWhen to Use a Calculator\n3:21\nGraphing Exponential Functions: a>1\n6:02\nThree Examples\n6:13\nWhat to Notice on the Graph\n7:44\nA Story\n8:27\nStory Diagram\n9:15\nIncreasing Exponentials\n11:29\nStory Morals\n14:40\nApplication: Compound Interest\n15:15\nCompounding Year after Year\n16:01\nFunction for Compounding Interest\n16:51\nA Special Number: e\n20:55\nExpression for e\n21:28\nWhere e stabilizes\n21:55\nApplication: Continuously Compounded Interest\n24:07\nEquation for Continuous Compounding\n24:22\nExponential Decay 0<a<1\n25:50\nThree Examples\n26:11\nWhy they 'lose' value\n26:54\nExample 1\n27:47\nExample 2\n33:11\nExample 3\n36:34\nExample 4\n41:28\nIntroduction to Logarithms\n\n40m 31s\n\nIntro\n0:00\nIntroduction\n0:04\nDefinition of a Logarithm, Base 2\n0:51\nLog 2 Defined\n0:55\nExamples\n2:28\nDefinition of a Logarithm, General\n3:23\nExamples of Logarithms\n5:15\nProblems with Unusual Bases\n7:38\nShorthand Notation: ln and log\n9:44\nbase e as ln\n10:01\nbase 10 as log\n10:34\nCalculating Logarithms\n11:01\nusing a calculator\n11:34\nissues with other bases\n11:58\nGraphs of Logarithms\n13:21\nThree Examples\n13:29\nSlow Growth\n15:19\nLogarithms as Inverse of Exponentiation\n16:02\nUsing Base 2\n16:05\nGeneral Case\n17:10\nLooking More Closely at Logarithm Graphs\n19:16\nThe Domain of Logarithms\n20:41\n21:08\nThe Alternate\n24:00\nExample 1\n25:59\nExample 2\n30:03\nExample 3\n32:49\nExample 4\n37:34\nProperties of Logarithms\n\n42m 33s\n\nIntro\n0:00\nIntroduction\n0:04\nBasic Properties\n1:12\nInverse--log(exp)\n1:43\nA Key Idea\n2:44\nWhat We Get through Exponentiation\n3:18\nB Always Exists\n4:50\nInverse--exp(log)\n5:53\nLogarithm of a Power\n7:44\nLogarithm of a Product\n10:07\nLogarithm of a Quotient\n13:48\nCaution! There Is No Rule for loga(M+N)\n16:12\nSummary of Properties\n17:42\nChange of Base--Motivation\n20:17\nNo Calculator Button\n20:59\nA Specific Example\n21:45\nSimplifying\n23:45\nChange of Base--Formula\n24:14\nExample 1\n25:47\nExample 2\n29:08\nExample 3\n31:14\nExample 4\n34:13\nSolving Exponential and Logarithmic Equations\n\n34m 10s\n\nIntro\n0:00\nIntroduction\n0:05\nOne to One Property\n1:09\nExponential\n1:26\nLogarithmic\n1:44\nSpecific Considerations\n2:02\nOne-to-One Property\n3:30\nSolving by One-to-One\n4:11\nInverse Property\n6:09\nSolving by Inverses\n7:25\nDealing with Equations\n7:50\nExample of Taking an Exponent or Logarithm of an Equation\n9:07\nA Useful Property\n11:57\nBring Down Exponents\n12:01\nTry to Simplify\n13:20\nExtraneous Solutions\n13:45\nExample 1\n16:37\nExample 2\n19:39\nExample 3\n21:37\nExample 4\n26:45\nExample 5\n29:37\nApplication of Exponential and Logarithmic Functions\n\n48m 46s\n\nIntro\n0:00\nIntroduction\n0:06\nApplications of Exponential Functions\n1:07\nA Secret!\n2:17\nNatural Exponential Growth Model\n3:07\nFigure out r\n3:34\nA Secret!--Why Does It Work?\n4:44\ne to the r Morphs\n4:57\nExample\n5:06\nApplications of Logarithmic Functions\n8:32\nExamples\n8:43\nWhat Logarithms are Useful For\n9:53\nExample 1\n11:29\nExample 2\n15:30\nExample 3\n26:22\nExample 4\n32:05\nExample 5\n39:19\nSection 6: Trigonometric Functions\nAngles\n\n39m 5s\n\nIntro\n0:00\nDegrees\n0:22\nCircle is 360 Degrees\n0:48\nSplitting a Circle\n1:13\n2:08\n2:31\n2:52\nHalf-Circle and Right Angle\n4:00\n6:24\n6:52\nCoterminal, Complementary, Supplementary Angles\n7:23\nCoterminal Angles\n7:30\nComplementary Angles\n9:40\nSupplementary Angles\n10:08\nExample 1: Dividing a Circle\n10:38\nExample 2: Converting Between Degrees and Radians\n11:56\nExample 3: Quadrants and Coterminal Angles\n14:18\nExtra Example 1: Common Angle Conversions\n-1\nExtra Example 2: Quadrants and Coterminal Angles\n-2\nSine and Cosine Functions\n\n43m 16s\n\nIntro\n0:00\nSine and Cosine\n0:15\nUnit Circle\n0:22\nCoordinates on Unit Circle\n1:03\nRight Triangles\n1:52\n2:25\nMaster Right Triangle Formula: SOHCAHTOA\n2:48\nOdd Functions, Even Functions\n4:40\nExample: Odd Function\n4:56\nExample: Even Function\n7:30\nExample 1: Sine and Cosine\n10:27\nExample 2: Graphing Sine and Cosine Functions\n14:39\nExample 3: Right Triangle\n21:40\nExample 4: Odd, Even, or Neither\n26:01\nExtra Example 1: Right Triangle\n-1\nExtra Example 2: Graphing Sine and Cosine Functions\n-2\nSine and Cosine Values of Special Angles\n\n33m 5s\n\nIntro\n0:00\n45-45-90 Triangle and 30-60-90 Triangle\n0:08\n45-45-90 Triangle\n0:21\n30-60-90 Triangle\n2:06\nMnemonic: All Students Take Calculus (ASTC)\n5:21\nUsing the Unit Circle\n5:59\nNew Angles\n6:21\n9:43\nMnemonic: All Students Take Calculus\n10:13\n13:11\n16:48\nExample 3: All Angles and Quadrants\n20:21\nExtra Example 1: Convert, Quadrant, Sine/Cosine\n-1\nExtra Example 2: All Angles and Quadrants\n-2\nModified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D\n\n52m 3s\n\nIntro\n0:00\nAmplitude and Period of a Sine Wave\n0:38\nSine Wave Graph\n0:58\nAmplitude: Distance from Middle to Peak\n1:18\nPeak: Distance from Peak to Peak\n2:41\nPhase Shift and Vertical Shift\n4:13\nPhase Shift: Distance Shifted Horizontally\n4:16\nVertical Shift: Distance Shifted Vertically\n6:48\nExample 1: Amplitude/Period/Phase and Vertical Shift\n8:04\nExample 2: Amplitude/Period/Phase and Vertical Shift\n17:39\nExample 3: Find Sine Wave Given Attributes\n25:23\nExtra Example 1: Amplitude/Period/Phase and Vertical Shift\n-1\nExtra Example 2: Find Cosine Wave Given Attributes\n-2\nTangent and Cotangent Functions\n\n36m 4s\n\nIntro\n0:00\nTangent and Cotangent Definitions\n0:21\nTangent Definition\n0:25\nCotangent Definition\n0:47\nMaster Formula: SOHCAHTOA\n1:01\nMnemonic\n1:16\nTangent and Cotangent Values\n2:29\nRemember Common Values of Sine and Cosine\n2:46\n90 Degrees Undefined\n4:36\nSlope and Menmonic: ASTC\n5:47\nUses of Tangent\n5:54\nExample: Tangent of Angle is Slope\n6:09\n7:49\nExample 1: Graph Tangent and Cotangent Functions\n10:42\nExample 2: Tangent and Cotangent of Angles\n16:09\nExample 3: Odd, Even, or Neither\n18:56\nExtra Example 1: Tangent and Cotangent of Angles\n-1\nExtra Example 2: Tangent and Cotangent of Angles\n-2\nSecant and Cosecant Functions\n\n27m 18s\n\nIntro\n0:00\nSecant and Cosecant Definitions\n0:17\nSecant Definition\n0:18\nCosecant Definition\n0:33\nExample 1: Graph Secant Function\n0:48\nExample 2: Values of Secant and Cosecant\n6:49\nExample 3: Odd, Even, or Neither\n12:49\nExtra Example 1: Graph of Cosecant Function\n-1\nExtra Example 2: Values of Secant and Cosecant\n-2\nInverse Trigonometric Functions\n\n32m 58s\n\nIntro\n0:00\nArcsine Function\n0:24\nRestrictions between -1 and 1\n0:43\nArcsine Notation\n1:26\nArccosine Function\n3:07\nRestrictions between -1 and 1\n3:36\nCosine Notation\n3:53\nArctangent Function\n4:30\nBetween -Pi/2 and Pi/2\n4:44\nTangent Notation\n5:02\nExample 1: Domain/Range/Graph of Arcsine\n5:45\nExample 2: Arcsin/Arccos/Arctan Values\n10:46\nExample 3: Domain/Range/Graph of Arctangent\n17:14\nExtra Example 1: Domain/Range/Graph of Arccosine\n-1\nExtra Example 2: Arcsin/Arccos/Arctan Values\n-2\nComputations of Inverse Trigonometric Functions\n\n31m 8s\n\nIntro\n0:00\nInverse Trigonometric Function Domains and Ranges\n0:31\nArcsine\n0:41\nArccosine\n1:14\nArctangent\n1:41\nExample 1: Arcsines of Common Values\n2:44\nExample 2: Odd, Even, or Neither\n5:57\nExample 3: Arccosines of Common Values\n12:24\nExtra Example 1: Arctangents of Common Values\n-1\nExtra Example 2: Arcsin/Arccos/Arctan Values\n-2\nSection 7: Trigonometric Identities\nPythagorean Identity\n\n19m 11s\n\nIntro\n0:00\nPythagorean Identity\n0:17\nPythagorean Triangle\n0:27\nPythagorean Identity\n0:45\nExample 1: Use Pythagorean Theorem to Prove Pythagorean Identity\n1:14\nExample 2: Find Angle Given Cosine and Quadrant\n4:18\nExample 3: Verify Trigonometric Identity\n8:00\nExtra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem\n-1\nExtra Example 2: Find Angle Given Cosine and Quadrant\n-2\nIdentity Tan(squared)x+1=Sec(squared)x\n\n23m 16s\n\nIntro\n0:00\nMain Formulas\n0:19\nCompanion to Pythagorean Identity\n0:27\nFor Cotangents and Cosecants\n0:52\nHow to Remember\n0:58\nExample 1: Prove the Identity\n1:40\nExample 2: Given Tan Find Sec\n3:42\nExample 3: Prove the Identity\n7:45\nExtra Example 1: Prove the Identity\n-1\nExtra Example 2: Given Sec Find Tan\n-2\n\n52m 52s\n\nIntro\n0:00\n0:09\nHow to Remember\n0:48\nCofunction Identities\n1:31\nHow to Remember Graphically\n1:44\nWhere to Use Cofunction Identities\n2:52\nExample 1: Derive the Formula for cos(A-B)\n3:08\nExample 2: Use Addition and Subtraction Formulas\n16:03\nExample 3: Use Addition and Subtraction Formulas to Prove Identity\n25:11\nExtra Example 1: Use cos(A-B) and Cofunction Identities\n-1\nExtra Example 2: Convert to Radians and use Formulas\n-2\nDouble Angle Formulas\n\n29m 5s\n\nIntro\n0:00\nMain Formula\n0:07\nHow to Remember from Addition Formula\n0:18\nTwo Other Forms\n1:35\nExample 1: Find Sine and Cosine of Angle using Double Angle\n3:16\nExample 2: Prove Trigonometric Identity using Double Angle\n9:37\nExample 3: Use Addition and Subtraction Formulas\n12:38\nExtra Example 1: Find Sine and Cosine of Angle using Double Angle\n-1\nExtra Example 2: Prove Trigonometric Identity using Double Angle\n-2\nHalf-Angle Formulas\n\n43m 55s\n\nIntro\n0:00\nMain Formulas\n0:09\nConfusing Part\n0:34\nExample 1: Find Sine and Cosine of Angle using Half-Angle\n0:54\nExample 2: Prove Trigonometric Identity using Half-Angle\n11:51\nExample 3: Prove the Half-Angle Formula for Tangents\n18:39\nExtra Example 1: Find Sine and Cosine of Angle using Half-Angle\n-1\nExtra Example 2: Prove Trigonometric Identity using Half-Angle\n-2\nSection 8: Applications of Trigonometry\nTrigonometry in Right Angles\n\n25m 43s\n\nIntro\n0:00\nMaster Formula for Right Angles\n0:11\nSOHCAHTOA\n0:15\nOnly for Right Triangles\n1:26\nExample 1: Find All Angles in a Triangle\n2:19\nExample 2: Find Lengths of All Sides of Triangle\n7:39\nExample 3: Find All Angles in a Triangle\n11:00\nExtra Example 1: Find All Angles in a Triangle\n-1\nExtra Example 2: Find Lengths of All Sides of Triangle\n-2\nLaw of Sines\n\n56m 40s\n\nIntro\n0:00\nLaw of Sines Formula\n0:18\nSOHCAHTOA\n0:27\nAny Triangle\n0:59\nGraphical Representation\n1:25\nSolving Triangle Completely\n2:37\nWhen to Use Law of Sines\n2:55\nASA, SAA, SSA, AAA\n2:59\nSAS, SSS for Law of Cosines\n7:11\nExample 1: How Many Triangles Satisfy Conditions, Solve Completely\n8:44\nExample 2: How Many Triangles Satisfy Conditions, Solve Completely\n15:30\nExample 3: How Many Triangles Satisfy Conditions, Solve Completely\n28:32\nExtra Example 1: How Many Triangles Satisfy Conditions, Solve Completely\n-1\nExtra Example 2: How Many Triangles Satisfy Conditions, Solve Completely\n-2\nLaw of Cosines\n\n49m 5s\n\nIntro\n0:00\nLaw of Cosines Formula\n0:23\nGraphical Representation\n0:34\nRelates Sides to Angles\n1:00\nAny Triangle\n1:20\nGeneralization of Pythagorean Theorem\n1:32\nWhen to Use Law of Cosines\n2:26\nSAS, SSS\n2:30\nHeron's Formula\n4:49\nSemiperimeter S\n5:11\nExample 1: How Many Triangles Satisfy Conditions, Solve Completely\n5:53\nExample 2: How Many Triangles Satisfy Conditions, Solve Completely\n15:19\nExample 3: Find Area of a Triangle Given All Side Lengths\n26:33\nExtra Example 1: How Many Triangles Satisfy Conditions, Solve Completely\n-1\nExtra Example 2: Length of Third Side and Area of Triangle\n-2\nFinding the Area of a Triangle\n\n27m 37s\n\nIntro\n0:00\nMaster Right Triangle Formula and Law of Cosines\n0:19\nSOHCAHTOA\n0:27\nLaw of Cosines\n1:23\nHeron's Formula\n2:22\nSemiperimeter S\n2:37\nExample 1: Area of Triangle with Two Sides and One Angle\n3:12\nExample 2: Area of Triangle with Three Sides\n6:11\nExample 3: Area of Triangle with Three Sides, No Heron's Formula\n8:50\nExtra Example 1: Area of Triangle with Two Sides and One Angle\n-1\nExtra Example 2: Area of Triangle with Two Sides and One Angle\n-2\nWord Problems and Applications of Trigonometry\n\n34m 25s\n\nIntro\n0:00\nFormulas to Remember\n0:11\nSOHCAHTOA\n0:15\nLaw of Sines\n0:55\nLaw of Cosines\n1:48\nHeron's Formula\n2:46\nExample 1: Telephone Pole Height\n4:01\nExample 2: Bridge Length\n7:48\nExample 3: Area of Triangular Field\n14:20\nExtra Example 1: Kite Height\n-1\nExtra Example 2: Roads to a Town\n-2\nSection 9: Systems of Equations and Inequalities\nSystems of Linear Equations\n\n55m 40s\n\nIntro\n0:00\nIntroduction\n0:04\nGraphs as Location of 'True'\n1:49\nAll Locations that Make the Function True\n2:25\nUnderstand the Relationship Between Solutions and the Graph\n3:43\nSystems as Graphs\n4:07\nEquations as Lines\n4:20\nIntersection Point\n5:19\nThree Possibilities for Solutions\n6:17\nIndependent\n6:24\nInconsistent\n6:36\nDependent\n7:06\nSolving by Substitution\n8:37\nSolve for One Variable\n9:07\nSubstitute into the Second Equation\n9:34\nSolve for Both Variables\n10:12\nWhat If a System is Inconsistent or Dependent?\n11:08\nNo Solutions\n11:25\nInfinite Solutions\n12:30\nSolving by Elimination\n13:56\nExample\n14:22\nDetermining the Number of Solutions\n16:30\nWhy Elimination Makes Sense\n17:25\nSolving by Graphing Calculator\n19:59\nSystems with More than Two Variables\n23:22\nExample 1\n25:49\nExample 2\n30:22\nExample 3\n34:11\nExample 4\n38:55\nExample 5\n46:01\n(Non-) Example 6\n53:37\nSystems of Linear Inequalities\n\n1h 13s\n\nIntro\n0:00\nIntroduction\n0:04\nInequality Refresher-Solutions\n0:46\nEquation Solutions vs. Inequality Solutions\n1:02\nEssentially a Wide Variety of Answers\n1:35\nRefresher--Negative Multiplication Flips\n1:43\nRefresher--Negative Flips: Why?\n3:19\nMultiplication by a Negative\n3:43\nThe Relationship Flips\n3:55\nRefresher--Stick to Basic Operations\n4:34\nLinear Equations in Two Variables\n6:50\nGraphing Linear Inequalities\n8:28\nWhy It Includes a Whole Section\n8:43\nHow to Show The Difference Between Strict and Not Strict Inequalities\n10:08\nDashed Line--Not Solutions\n11:10\nSolid Line--Are Solutions\n11:24\n11:42\nExample of Using a Point\n12:41\n13:14\nGraphing a System\n14:53\nSet of Solutions is the Overlap\n15:17\nExample\n15:22\nSolutions are Best Found Through Graphing\n18:05\nLinear Programming-Idea\n19:52\nUse a Linear Objective Function\n20:15\nVariables in Objective Function have Constraints\n21:24\nLinear Programming-Method\n22:09\nRearrange Equations\n22:21\nGraph\n22:49\nCritical Solution is at the Vertex of the Overlap\n23:40\nTry Each Vertice\n24:35\nExample 1\n24:58\nExample 2\n28:57\nExample 3\n33:48\nExample 4\n43:10\nNonlinear Systems\n\n41m 1s\n\nIntro\n0:00\nIntroduction\n0:06\nSubstitution\n1:12\nExample\n1:22\nElimination\n3:46\nExample\n3:56\nElimination is Less Useful for Nonlinear Systems\n4:56\nGraphing\n5:56\nUsing a Graphing Calculator\n6:44\nNumber of Solutions\n8:44\nSystems of Nonlinear Inequalities\n10:02\nGraph Each Inequality\n10:06\nDashed and/or Solid\n10:18\n11:14\nExample 1\n13:24\nExample 2\n15:50\nExample 3\n22:02\nExample 4\n29:06\nExample 4, cont.\n33:40\nSection 10: Vectors and Matrices\nVectors\n\n1h 9m 31s\n\nIntro\n0:00\nIntroduction\n0:10\nMagnitude of the Force\n0:22\nDirection of the Force\n0:48\nVector\n0:52\nIdea of a Vector\n1:30\nHow Vectors are Denoted\n2:00\nComponent Form\n3:20\nAngle Brackets and Parentheses\n3:50\nMagnitude/Length\n4:26\nDenoting the Magnitude of a Vector\n5:16\nDirection/Angle\n7:52\nAlways Draw a Picture\n8:50\nComponent Form from Magnitude & Angle\n10:10\nScaling by Scalars\n14:06\nUnit Vectors\n16:26\nCombining Vectors - Algebraically\n18:10\nCombining Vectors - Geometrically\n19:54\nResultant Vector\n20:46\nAlternate Component Form: i, j\n21:16\nThe Zero Vector\n23:18\nProperties of Vectors\n24:20\nNo Multiplication (Between Vectors)\n28:30\nDot Product\n29:40\nMotion in a Medium\n30:10\nFish in an Aquarium Example\n31:38\nMore Than Two Dimensions\n33:12\nMore Than Two Dimensions - Magnitude\n34:18\nExample 1\n35:26\nExample 2\n38:10\nExample 3\n45:48\nExample 4\n50:40\nExample 4, cont.\n56:07\nExample 5\n1:01:32\nDot Product & Cross Product\n\n35m 20s\n\nIntro\n0:00\nIntroduction\n0:08\nDot Product - Definition\n0:42\nDot Product Results in a Scalar, Not a Vector\n2:10\nExample in Two Dimensions\n2:34\nAngle and the Dot Product\n2:58\nThe Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors\n2:59\nProof of Dot Product Formula\n4:14\nWon't Directly Help Us Better Understand Vectors\n4:18\nDot Product - Geometric Interpretation\n4:58\nWe Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are\n7:26\nDot Product - Perpendicular Vectors\n8:24\nIf the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other\n8:54\nCross Product - Definition\n11:08\nCross Product Only Works in Three Dimensions\n11:09\nCross Product - A Mnemonic\n12:16\nThe Determinant of a 3 x 3 Matrix and Standard Unit Vectors\n12:17\nCross Product - Geometric Interpretations\n14:30\nThe Right-Hand Rule\n15:17\nCross Product - Geometric Interpretations Cont.\n17:00\nExample 1\n18:40\nExample 2\n22:50\nExample 3\n24:04\nExample 4\n26:20\nBonus Round\n29:18\nProof: Dot Product Formula\n29:24\nProof: Dot Product Formula, cont.\n30:38\nMatrices\n\n54m 7s\n\nIntro\n0:00\nIntroduction\n0:08\nDefinition of a Matrix\n3:02\nSize or Dimension\n3:58\nSquare Matrix\n4:42\nDenoted by Capital Letters\n4:56\nWhen are Two Matrices Equal?\n5:04\nExamples of Matrices\n6:44\nRows x Columns\n6:46\n7:48\nWe Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries\n8:32\nUsing Entries to Talk About Matrices\n10:08\nScalar Multiplication\n11:26\nScalar = Real Number\n11:34\nExample\n12:36\n13:08\nExample\n14:22\nMatrix Multiplication\n15:00\nExample\n18:52\nMatrix Multiplication, cont.\n19:58\nMatrix Multiplication and Order (Size)\n25:26\nMake Sure Their Orders are Compatible\n25:27\nMatrix Multiplication is NOT Commutative\n28:20\nExample\n30:08\nSpecial Matrices - Zero Matrix (0)\n32:48\nZero Matrix Has 0 for All of its Entries\n32:49\nSpecial Matrices - Identity Matrix (I)\n34:14\nIdentity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries\n34:15\nExample 1\n36:16\nExample 2\n40:00\nExample 3\n44:54\nExample 4\n50:08\nDeterminants & Inverses of Matrices\n\n47m 12s\n\nIntro\n0:00\nIntroduction\n0:06\nNot All Matrices Are Invertible\n1:30\nWhat Must a Matrix Have to Be Invertible?\n2:08\nDeterminant\n2:32\nThe Determinant is a Real Number Associated With a Square Matrix\n2:38\nIf the Determinant of a Matrix is Nonzero, the Matrix is Invertible\n3:40\nDeterminant of a 2 x 2 Matrix\n4:34\nThink in Terms of Diagonals\n5:12\nMinors and Cofactors - Minors\n6:24\nExample\n6:46\nMinors and Cofactors - Cofactors\n8:00\nCofactor is Closely Based on the Minor\n8:01\nAlternating Sign Pattern\n9:04\nDeterminant of Larger Matrices\n10:56\nExample\n13:00\nAlternative Method for 3x3 Matrices\n16:46\nNot Recommended\n16:48\nInverse of a 2 x 2 Matrix\n19:02\nInverse of Larger Matrices\n20:00\nUsing Inverse Matrices\n21:06\nWhen Multiplied Together, They Create the Identity Matrix\n21:24\nExample 1\n23:45\nExample 2\n27:21\nExample 3\n32:49\nExample 4\n36:27\nFinding the Inverse of Larger Matrices\n41:59\nGeneral Inverse Method - Step 1\n43:25\nGeneral Inverse Method - Step 2\n43:27\nGeneral Inverse Method - Step 2, cont.\n43:27\nGeneral Inverse Method - Step 3\n45:15\nUsing Matrices to Solve Systems of Linear Equations\n\n58m 34s\n\nIntro\n0:00\nIntroduction\n0:12\nAugmented Matrix\n1:44\nWe Can Represent the Entire Linear System With an Augmented Matrix\n1:50\nRow Operations\n3:22\nInterchange the Locations of Two Rows\n3:50\nMultiply (or Divide) a Row by a Nonzero Number\n3:58\nAdd (or Subtract) a Multiple of One Row to Another\n4:12\nRow Operations - Keep Notes!\n5:50\nSuggested Symbols\n7:08\nGauss-Jordan Elimination - Idea\n8:04\nGauss-Jordan Elimination - Idea, cont.\n9:16\nReduced Row-Echelon Form\n9:18\nGauss-Jordan Elimination - Method\n11:36\nBegin by Writing the System As An Augmented Matrix\n11:38\nGauss-Jordan Elimination - Method, cont.\n13:48\nCramer's Rule - 2 x 2 Matrices\n17:08\nCramer's Rule - n x n Matrices\n19:24\nSolving with Inverse Matrices\n21:10\nSolving Inverse Matrices, cont.\n25:28\nThe Mighty (Graphing) Calculator\n26:38\nExample 1\n29:56\nExample 2\n33:56\nExample 3\n37:00\nExample 3, cont.\n45:04\nExample 4\n51:28\nSection 11: Alternate Ways to Graph\nParametric Equations\n\n53m 33s\n\nIntro\n0:00\nIntroduction\n0:06\nDefinition\n1:10\nPlane Curve\n1:24\nThe Key Idea\n2:00\nGraphing with Parametric Equations\n2:52\nSame Graph, Different Equations\n5:04\nHow Is That Possible?\n5:36\nSame Graph, Different Equations, cont.\n5:42\nHere's Another to Consider\n7:56\nSame Plane Curve, But Still Different\n8:10\nA Metaphor for Parametric Equations\n9:36\nThink of Parametric Equations As a Way to Describe the Motion of An Object\n9:38\nGraph Shows Where It Went, But Not Speed\n10:32\nEliminating Parameters\n12:14\nRectangular Equation\n12:16\nCaution\n13:52\nCreating Parametric Equations\n14:30\nInteresting Graphs\n16:38\nGraphing Calculators, Yay!\n19:18\nExample 1\n22:36\nExample 2\n28:26\nExample 3\n37:36\nExample 4\n41:00\nProjectile Motion\n44:26\nExample 5\n47:00\nPolar Coordinates\n\n48m 7s\n\nIntro\n0:00\nIntroduction\n0:04\nPolar Coordinates Give Us a Way To Describe the Location of a Point\n0:26\nPolar Equations and Functions\n0:50\nPlotting Points with Polar Coordinates\n1:06\nThe Distance of the Point from the Origin\n1:09\nThe Angle of the Point\n1:33\nGive Points as the Ordered Pair (r,θ)\n2:03\nVisualizing Plotting in Polar Coordinates\n2:32\nFirst Way We Can Plot\n2:39\nSecond Way We Can Plot\n2:50\nFirst, We'll Look at Visualizing r, Then θ\n3:09\nRotate the Length Counter-Clockwise by θ\n3:38\nAlternatively, We Can Visualize θ, Then r\n4:06\n'Polar Graph Paper'\n6:17\nHorizontal and Vertical Tick Marks Are Not Useful for Polar\n6:42\nUse Concentric Circles to Helps Up See Distance From the Pole\n7:08\nCan Use Arc Sectors to See Angles\n7:57\nMultiple Ways to Name a Point\n9:17\nExamples\n9:30\nFor Any Angle θ, We Can Make an Equivalent Angle\n10:44\nNegative Values for r\n11:58\nIf r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out\n12:22\nAnother Way to Name the Same Point: Add π to θ and Make r Negative\n13:44\nConverting Between Rectangular and Polar\n14:37\nRectangular Way to Name\n14:43\nPolar Way to Name\n14:52\nThe Rectangular System Must Have a Right Angle Because It's Based on a Rectangle\n15:08\nConnect Both Systems Through Basic Trigonometry\n15:38\nEquation to Convert From Polar to Rectangular Coordinate Systems\n16:55\nEquation to Convert From Rectangular to Polar Coordinate Systems\n17:13\nConverting to Rectangular is Easy\n17:20\nConverting to Polar is a Bit Trickier\n17:21\nDraw Pictures\n18:55\nExample 1\n19:50\nExample 2\n25:17\nExample 3\n31:05\nExample 4\n35:56\nExample 5\n41:49\nPolar Equations & Functions\n\n38m 16s\n\nIntro\n0:00\nIntroduction\n0:04\nEquations and Functions\n1:16\nIndependent Variable\n1:21\nDependent Variable\n1:30\nExamples\n1:46\nAlways Assume That θ Is In Radians\n2:44\nGraphing in Polar Coordinates\n3:29\nGraph is the Same Way We Graph 'Normal' Stuff\n3:32\nExample\n3:52\nGraphing in Polar - Example, Cont.\n6:45\nTips for Graphing\n9:23\nNotice Patterns\n10:19\nRepetition\n13:39\nGraphing Equations of One Variable\n14:39\nConverting Coordinate Types\n16:16\nUse the Same Conversion Formulas From the Previous Lesson\n16:23\nInteresting Graphs\n17:48\nExample 1\n18:03\nExample 2\n18:34\nGraphing Calculators, Yay!\n19:07\nPlot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works\n19:11\nCheck Out the Appendix\n19:26\nExample 1\n21:36\nExample 2\n28:13\nExample 3\n34:24\nExample 4\n35:52\nSection 12: Complex Numbers and Polar Coordinates\nPolar Form of Complex Numbers\n\n40m 43s\n\nIntro\n0:00\nPolar Coordinates\n0:49\nRectangular Form\n0:52\nPolar Form\n1:25\nR and Theta\n1:51\nPolar Form Conversion\n2:27\nR and Theta\n2:35\nOptimal Values\n4:05\nEuler's Formula\n4:25\nMultiplying Two Complex Numbers in Polar Form\n6:10\nMultiply r's Together and Add Exponents\n6:32\nExample 1: Convert Rectangular to Polar Form\n7:17\nExample 2: Convert Polar to Rectangular Form\n13:49\nExample 3: Multiply Two Complex Numbers\n17:28\nExtra Example 1: Convert Between Rectangular and Polar Forms\n-1\nExtra Example 2: Simplify Expression to Polar Form\n-2\nDeMoivre's Theorem\n\n57m 37s\n\nIntro\n0:00\nIntroduction to DeMoivre's Theorem\n0:10\nn nth Roots\n3:06\nDeMoivre's Theorem: Finding nth Roots\n3:52\nRelation to Unit Circle\n6:29\nOne nth Root for Each Value of k\n7:11\nExample 1: Convert to Polar Form and Use DeMoivre's Theorem\n8:24\nExample 2: Find Complex Eighth Roots\n15:27\nExample 3: Find Complex Roots\n27:49\nExtra Example 1: Convert to Polar Form and Use DeMoivre's Theorem\n-1\nExtra Example 2: Find Complex Fourth Roots\n-2\nSection 13: Counting & Probability\nCounting\n\n31m 36s\n\nIntro\n0:00\nIntroduction\n0:08\nCombinatorics\n0:56\nDefinition: Event\n1:24\nExample\n1:50\nVisualizing an Event\n3:02\nBranching line diagram\n3:06\n3:40\nExample\n4:18\nMultiplication Principle\n5:42\nExample\n6:24\nPigeonhole Principle\n8:06\nExample\n10:26\nDraw Pictures\n11:06\nExample 1\n12:02\nExample 2\n14:16\nExample 3\n17:34\nExample 4\n21:26\nExample 5\n25:14\nPermutations & Combinations\n\n44m 3s\n\nIntro\n0:00\nIntroduction\n0:08\nPermutation\n0:42\nCombination\n1:10\nTowards a Permutation Formula\n2:38\nHow Many Ways Can We Arrange the Letters A, B, C, D, and E?\n3:02\nTowards a Permutation Formula, cont.\n3:34\nFactorial Notation\n6:56\nSymbol Is '!'\n6:58\nExamples\n7:32\nPermutation of n Objects\n8:44\nPermutation of r Objects out of n\n9:04\nWhat If We Have More Objects Than We Have Slots to Fit Them Into?\n9:46\nPermutation of r Objects Out of n, cont.\n10:28\nDistinguishable Permutations\n14:46\nWhat If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?\n14:48\nDistinguishable Permutations, cont.\n17:04\nCombinations\n19:04\nCombinations, cont.\n20:56\nExample 1\n23:10\nExample 2\n26:16\nExample 3\n28:28\nExample 4\n31:52\nExample 5\n33:58\nExample 6\n36:34\nProbability\n\n36m 58s\n\nIntro\n0:00\nIntroduction\n0:06\nDefinition: Sample Space\n1:18\nEvent = Something Happening\n1:20\nSample Space\n1:36\nProbability of an Event\n2:12\nLet E Be An Event and S Be The Corresponding Sample Space\n2:14\n'Equally Likely' Is Important\n3:52\nFair and Random\n5:26\nInterpreting Probability\n6:34\nHow Can We Interpret This Value?\n7:24\nWe Can Represent Probability As a Fraction, a Decimal, Or a Percentage\n8:04\nOne of Multiple Events Occurring\n9:52\nMutually Exclusive Events\n10:38\nWhat If The Events Are Not Mutually Exclusive?\n12:20\nTaking the Possibility of Overlap Into Account\n13:24\nAn Event Not Occurring\n17:14\nComplement of E\n17:22\nIndependent Events\n19:36\nIndependent\n19:48\nConditional Events\n21:28\nWhat Is The Events Are Not Independent Though?\n21:30\nConditional Probability\n22:16\nConditional Events, cont.\n23:51\nExample 1\n25:27\nExample 2\n27:09\nExample 3\n28:57\nExample 4\n30:51\nExample 5\n34:15\nSection 14: Conic Sections\nParabolas\n\n41m 27s\n\nIntro\n0:00\nWhat is a Parabola?\n0:20\nDefinition of a Parabola\n0:29\nFocus\n0:59\nDirectrix\n1:15\nAxis of Symmetry\n3:08\nVertex\n3:33\nMinimum or Maximum\n3:44\nStandard Form\n4:59\nHorizontal Parabolas\n5:08\nVertex Form\n5:19\nUpward or Downward\n5:41\nExample: Standard Form\n6:06\nGraphing Parabolas\n8:31\nShifting\n8:51\nExample: Completing the Square\n9:22\nSymmetry and Translation\n12:18\nExample: Graph Parabola\n12:40\nLatus Rectum\n17:13\nLength\n18:15\nExample: Latus Rectum\n18:35\nHorizontal Parabolas\n18:57\nNot Functions\n20:08\nExample: Horizontal Parabola\n21:21\nFocus and Directrix\n24:11\nHorizontal\n24:48\nExample 1: Parabola Standard Form\n25:12\nExample 2: Graph Parabola\n30:00\nExample 3: Graph Parabola\n33:13\nExample 4: Parabola Equation\n37:28\nCircles\n\n21m 3s\n\nIntro\n0:00\nWhat are Circles?\n0:08\nExample: Equidistant\n0:17\n0:32\nEquation of a Circle\n0:44\nExample: Standard Form\n1:11\nGraphing Circles\n1:47\nExample: Circle\n1:56\nCenter Not at Origin\n3:07\nExample: Completing the Square\n3:51\nExample 1: Equation of Circle\n6:44\n11:51\n15:08\nExample 4: Equation of Circle\n16:57\nEllipses\n\n46m 51s\n\nIntro\n0:00\nWhat Are Ellipses?\n0:11\nFoci\n0:23\nProperties of Ellipses\n1:43\nMajor Axis, Minor Axis\n1:47\nCenter\n1:54\nLength of Major Axis and Minor Axis\n3:21\nStandard Form\n5:33\nExample: Standard Form of Ellipse\n6:09\nVertical Major Axis\n9:14\nExample: Vertical Major Axis\n9:46\nGraphing Ellipses\n12:51\nComplete the Square and Symmetry\n13:00\nExample: Graphing Ellipse\n13:16\nEquation with Center at (h, k)\n19:57\nHorizontal and Vertical\n20:14\nDifference\n20:27\nExample: Center at (h, k)\n20:55\nExample 1: Equation of Ellipse\n24:05\nExample 2: Equation of Ellipse\n27:57\nExample 3: Equation of Ellipse\n32:32\nExample 4: Graph Ellipse\n38:27\nHyperbolas\n\n38m 15s\n\nIntro\n0:00\nWhat are Hyperbolas?\n0:12\nTwo Branches\n0:18\nFoci\n0:38\nProperties\n2:00\nTransverse Axis and Conjugate Axis\n2:06\nVertices\n2:46\nLength of Transverse Axis\n3:14\nDistance Between Foci\n3:31\nLength of Conjugate Axis\n3:38\nStandard Form\n5:45\nVertex Location\n6:36\nKnown Points\n6:52\nVertical Transverse Axis\n7:26\nVertex Location\n7:50\nAsymptotes\n8:36\nVertex Location\n8:56\nRectangle\n9:28\nDiagonals\n10:29\nGraphing Hyperbolas\n12:58\nExample: Hyperbola\n13:16\nEquation with Center at (h, k)\n16:32\nExample: Center at (h, k)\n17:21\nExample 1: Equation of Hyperbola\n19:20\nExample 2: Equation of Hyperbola\n22:48\nExample 3: Graph Hyperbola\n26:05\nExample 4: Equation of Hyperbola\n36:29\nConic Sections\n\n18m 43s\n\nIntro\n0:00\nConic Sections\n0:16\nDouble Cone Sections\n0:24\nStandard Form\n1:27\nGeneral Form\n1:37\nIdentify Conic Sections\n2:16\nB = 0\n2:50\nX and Y\n3:22\nIdentify Conic Sections, Cont.\n4:46\nParabola\n5:17\nCircle\n5:51\nEllipse\n6:31\nHyperbola\n7:10\nExample 1: Identify Conic Section\n8:01\nExample 2: Identify Conic Section\n11:03\nExample 3: Identify Conic Section\n11:38\nExample 4: Identify Conic Section\n14:50\nSection 15: Sequences, Series, & Induction\nIntroduction to Sequences\n\n57m 45s\n\nIntro\n0:00\nIntroduction\n0:06\nDefinition: Sequence\n0:28\nInfinite Sequence\n2:08\nFinite Sequence\n2:22\nLength\n2:58\nFormula for the nth Term\n3:22\nDefining a Sequence Recursively\n5:54\nInitial Term\n7:58\nSequences and Patterns\n10:40\nFirst, Identify a Pattern\n12:52\nHow to Get From One Term to the Next\n17:38\nTips for Finding Patterns\n19:52\nMore Tips for Finding Patterns\n24:14\nEven More Tips\n26:50\nExample 1\n30:32\nExample 2\n34:54\nFibonacci Sequence\n34:55\nExample 3\n38:40\nExample 4\n45:02\nExample 5\n49:26\nExample 6\n51:54\nIntroduction to Series\n\n40m 27s\n\nIntro\n0:00\nIntroduction\n0:06\nDefinition: Series\n1:20\nWhy We Need Notation\n2:48\nSimga Notation (AKA Summation Notation)\n4:44\nThing Being Summed\n5:42\nIndex of Summation\n6:21\nLower Limit of Summation\n7:09\nUpper Limit of Summation\n7:23\nSigma Notation, Example\n7:36\nSigma Notation for Infinite Series\n9:08\nHow to Reindex\n10:58\nHow to Reindex, Expanding\n12:56\nHow to Reindex, Substitution\n16:46\nProperties of Sums\n19:42\nExample 1\n23:46\nExample 2\n25:34\nExample 3\n27:12\nExample 4\n29:54\nExample 5\n32:06\nExample 6\n37:16\nArithmetic Sequences & Series\n\n31m 36s\n\nIntro\n0:00\nIntroduction\n0:05\nDefinition: Arithmetic Sequence\n0:47\nCommon Difference\n1:13\nTwo Examples\n1:19\nForm for the nth Term\n2:14\nRecursive Relation\n2:33\nTowards an Arithmetic Series Formula\n5:12\nCreating a General Formula\n10:09\nGeneral Formula for Arithmetic Series\n14:23\nExample 1\n15:46\nExample 2\n17:37\nExample 3\n22:21\nExample 4\n24:09\nExample 5\n27:14\nGeometric Sequences & Series\n\n39m 27s\n\nIntro\n0:00\nIntroduction\n0:06\nDefinition\n0:48\nForm for the nth Term\n2:42\nFormula for Geometric Series\n5:16\nInfinite Geometric Series\n11:48\nDiverges\n13:04\nConverges\n14:48\nFormula for Infinite Geometric Series\n16:32\nExample 1\n20:32\nExample 2\n22:02\nExample 3\n26:00\nExample 4\n30:48\nExample 5\n34:28\nMathematical Induction\n\n49m 53s\n\nIntro\n0:00\nIntroduction\n0:06\nBelief Vs. Proof\n1:22\nA Metaphor for Induction\n6:14\nThe Principle of Mathematical Induction\n11:38\nBase Case\n13:24\nInductive Step\n13:30\nInductive Hypothesis\n13:52\nA Remark on Statements\n14:18\nUsing Mathematical Induction\n16:58\nWorking Example\n19:58\nFinding Patterns\n28:46\nExample 1\n30:17\nExample 2\n37:50\nExample 3\n42:38\nThe Binomial Theorem\n\n1h 13m 13s\n\nIntro\n0:00\nIntroduction\n0:06\nWe've Learned That a Binomial Is An Expression That Has Two Terms\n0:07\nUnderstanding Binomial Coefficients\n1:20\nThings We Notice\n2:24\nWhat Goes In the Blanks?\n5:52\nEach Blank is Called a Binomial Coefficient\n6:18\nThe Binomial Theorem\n6:38\nExample\n8:10\nThe Binomial Theorem, cont.\n10:46\nWe Can Also Write This Expression Compactly Using Sigma Notation\n12:06\nProof of the Binomial Theorem\n13:22\nProving the Binomial Theorem Is Within Our Reach\n13:24\nPascal's Triangle\n15:12\nPascal's Triangle, cont.\n16:12\n16:24\nZeroth Row\n18:04\nFirst Row\n18:12\nWhy Do We Care About Pascal's Triangle?\n18:50\nPascal's Triangle, Example\n19:26\nExample 1\n21:26\nExample 2\n24:34\nExample 3\n28:34\nExample 4\n32:28\nExample 5\n37:12\nTime for the Fireworks!\n43:38\nProof of the Binomial Theorem\n43:44\nWe'll Prove This By Induction\n44:04\nProof (By Induction)\n46:36\nProof, Base Case\n47:00\nProof, Inductive Step - Notation Discussion\n49:22\nInduction Step\n49:24\nProof, Inductive Step - Setting Up\n52:26\nInduction Hypothesis\n52:34\nWhat We What To Show\n52:44\nProof, Inductive Step - Start\n54:18\nProof, Inductive Step - Middle\n55:38\nExpand Sigma Notations\n55:48\nProof, Inductive Step - Middle, cont.\n58:40\nProof, Inductive Step - Checking In\n1:01:08\nLet's Check In With Our Original Goal\n1:01:12\nWant to Show\n1:01:18\nLemma - A Mini Theorem\n1:02:18\nProof, Inductive Step - Lemma\n1:02:52\nProof of Lemma: Let's Investigate the Left Side\n1:03:08\nProof, Inductive Step - Nearly There\n1:07:54\nProof, Inductive Step - End!\n1:09:18\nProof, Inductive Step - End!, cont.\n1:11:01\nSection 16: Preview of Calculus\nIdea of a Limit\n\n40m 22s\n\nIntro\n0:00\nIntroduction\n0:05\nMotivating Example\n1:26\nFuzzy Notion of a Limit\n3:38\nLimit is the Vertical Location a Function is Headed Towards\n3:44\nLimit is What the Function Output is Going to Be\n4:15\nLimit Notation\n4:33\nExploring Limits - 'Ordinary' Function\n5:26\nTest Out\n5:27\nGraphing, We See The Answer Is What We Would Expect\n5:44\nExploring Limits - Piecewise Function\n6:45\nIf We Modify the Function a Bit\n6:49\nExploring Limits - A Visual Conception\n10:08\nDefinition of a Limit\n12:07\nIf f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.\n12:09\nWe Are Not Concerned with f(x) at x=c\n12:49\nWe Are Considering x Approaching From All Directions, Not Just One Side\n13:10\nLimits Do Not Always Exist\n15:47\nFinding Limits\n19:49\nGraphs\n19:52\nTables\n21:48\nPrecise Methods\n24:53\nExample 1\n26:06\nExample 2\n27:39\nExample 3\n30:51\nExample 4\n33:11\nExample 5\n37:07\nFormal Definition of a Limit\n\n57m 11s\n\nIntro\n0:00\nIntroduction\n0:06\nNew Greek Letters\n2:42\nDelta\n3:14\nEpsilon\n3:46\nSometimes Called the Epsilon-Delta Definition of a Limit\n3:56\nFormal Definition of a Limit\n4:22\nWhat does it MEAN!?!?\n5:00\nThe Groundwork\n5:38\nSet Up the Limit\n5:39\nThe Function is Defined Over Some Portion of the Reals\n5:58\nThe Horizontal Location is the Value the Limit Will Approach\n6:28\nThe Vertical Location L is Where the Limit Goes To\n7:00\nThe Epsilon-Delta Part\n7:26\nThe Hard Part is the Second Part of the Definition\n7:30\nSecond Half of Definition\n10:04\nRestrictions on the Allowed x Values\n10:28\nThe Epsilon-Delta Part, cont.\n13:34\nSherlock Holmes and Dr. Watson\n15:08\nThe Adventure of the Delta-Epsilon Limit\n15:16\nSetting\n15:18\nWe Begin By Setting Up the Game As Follows\n15:52\nThe Adventure of the Delta-Epsilon, cont.\n17:24\n17:46\nWhat If I Try Larger?\n19:39\nTechnically, You Haven't Proven the Limit\n20:53\nHere is the Method\n21:18\nWhat We Should Concern Ourselves With\n22:20\nInvestigate the Left Sides of the Expressions\n25:24\nWe Can Create the Following Inequalities\n28:08\nFinally…\n28:50\nNothing Like a Good Proof to Develop the Appetite\n30:42\nExample 1\n31:02\nExample 1, cont.\n36:26\nExample 2\n41:46\nExample 2, cont.\n47:50\nFinding Limits\n\n32m 40s\n\nIntro\n0:00\nIntroduction\n0:08\nMethod - 'Normal' Functions\n2:04\nThe Easiest Limits to Find\n2:06\nIt Does Not 'Break'\n2:18\nIt Is Not Piecewise\n2:26\nMethod - 'Normal' Functions, Example\n3:38\nMethod - 'Normal' Functions, cont.\n4:54\nThe Functions We're Used to Working With Go Where We Expect Them To Go\n5:22\n5:42\nMethod - Canceling Factors\n7:18\nOne Weird Thing That Often Happens is Dividing By 0\n7:26\nMethod - Canceling Factors, cont.\n8:16\nNotice That The Two Functions Are Identical With the Exception of x=0\n8:20\nMethod - Canceling Factors, cont.\n10:00\nExample\n10:52\nMethod - Rationalization\n12:04\nRationalizing a Portion of Some Fraction\n12:05\nConjugate\n12:26\nMethod - Rationalization, cont.\n13:14\nExample\n13:50\nMethod - Piecewise\n16:28\nThe Limits of Piecewise Functions\n16:30\nExample 1\n17:42\nExample 2\n18:44\nExample 3\n20:20\nExample 4\n22:24\nExample 5\n24:24\nExample 6\n27:12\nContinuity & One-Sided Limits\n\n32m 43s\n\nIntro\n0:00\nIntroduction\n0:06\nMotivating Example\n0:56\nContinuity - Idea\n2:14\nContinuous Function\n2:18\nAll Parts of Function Are Connected\n2:28\nFunction's Graph Can Be Drawn Without Lifting Pencil\n2:36\nThere Are No Breaks or Holes in Graph\n2:56\nContinuity - Idea, cont.\n3:38\nWe Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'\n3:52\nContinuity - Definition\n5:16\nA Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does\n5:18\nDiscontinuous\n6:02\nDiscontinuity\n6:10\nContinuity and 'Normal' Functions\n6:48\nReturn of the Motivating Example\n8:14\nOne-Sided Limit\n8:48\nOne-Sided Limit - Definition\n9:16\nOnly Considers One Side\n9:20\nBe Careful to Keep Track of Which Symbol Goes With Which Side\n10:06\nOne-Sided Limit - Example\n10:50\nThere Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits\n11:16\nNormal Limits and One-Sided Limits\n12:08\nLimits of Piecewise Functions\n14:12\n'Breakover' Points\n14:22\nWe Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other\n15:34\nExample 1\n16:40\nExample 2\n18:54\nExample 3\n22:00\nExample 4\n26:36\nLimits at Infinity & Limits of Sequences\n\n32m 49s\n\nIntro\n0:00\nIntroduction\n0:06\nDefinition: Limit of a Function at Infinity\n1:44\nA Limit at Infinity Works Very Similarly to How a Normal Limit Works\n2:38\nEvaluating Limits at Infinity\n4:08\nRational Functions\n4:17\nExamples\n4:30\nFor a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator\n5:22\nThere are Three Possibilities\n6:36\nEvaluating Limits at Infinity, cont.\n8:08\nDoes the Function Grow Without Bound? Will It 'Settle Down' Over Time?\n10:06\n10:26\nLimit of a Sequence\n12:20\nWhat Value Does the Sequence Tend to Do in the Long-Run?\n12:41\nThe Limit of a Sequence is Very Similar to the Limit of a Function at Infinity\n12:52\nNumerical Evaluation\n14:16\nNumerically: Plug in Numbers and See What Comes Out\n14:24\nExample 1\n16:42\nExample 2\n21:00\nExample 3\n22:08\nExample 4\n26:14\nExample 5\n28:10\nExample 6\n31:06\nInstantaneous Slope & Tangents (Derivatives)\n\n51m 13s\n\nIntro\n0:00\nIntroduction\n0:08\nThe Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing\n0:16\nInstantaneous Slop\n0:22\nInstantaneous Rate of Change\n0:28\nSlope\n1:24\nThe Vertical Change Divided by the Horizontal\n1:40\nIdea of Instantaneous Slope\n2:10\nWhat If We Wanted to Apply the Idea of Slope to a Non-Line?\n2:14\nTangent to a Circle\n3:52\nWhat is the Tangent Line for a Circle?\n4:42\nTangent to a Curve\n5:20\nTowards a Derivative - Average Slope\n6:36\nTowards a Derivative - Average Slope, cont.\n8:20\nAn Approximation\n11:24\nTowards a Derivative - General Form\n13:18\nTowards a Derivative - General Form, cont.\n16:46\nAn h Grows Smaller, Our Slope Approximation Becomes Better\n18:44\nTowards a Derivative - Limits!\n20:04\nTowards a Derivative - Limits!, cont.\n22:08\nWe Want to Show the Slope at x=1\n22:34\nTowards a Derivative - Checking Our Slope\n23:12\nDefinition of the Derivative\n23:54\nDerivative: A Way to Find the Instantaneous Slope of a Function at Any Point\n23:58\nDifferentiation\n24:54\nNotation for the Derivative\n25:58\nThe Derivative is a Very Important Idea In Calculus\n26:04\nThe Important Idea\n27:34\nWhy Did We Learn the Formal Definition to Find a Derivative?\n28:18\nExample 1\n30:50\nExample 2\n36:06\nExample 3\n40:24\nThe Power Rule\n44:16\nMakes It Easier to Find the Derivative of a Function\n44:24\nExamples\n45:04\nn Is Any Constant Number\n45:46\nExample 4\n46:26\nArea Under a Curve (Integrals)\n\n45m 26s\n\nIntro\n0:00\nIntroduction\n0:06\nIntegral\n0:12\nIdea of Area Under a Curve\n1:18\nApproximation by Rectangles\n2:12\nThe Easiest Way to Find Area is With a Rectangle\n2:18\nVarious Methods for Choosing Rectangles\n4:30\nRectangle Method - Left-Most Point\n5:12\nThe Left-Most Point\n5:16\nRectangle Method - Right-Most Point\n5:58\nThe Right-Most Point\n6:00\nRectangle Method - Mid-Point\n6:42\nHorizontal Mid-Point\n6:48\nRectangle Method - Maximum (Upper Sum)\n7:34\nMaximum Height\n7:40\nRectangle Method - Minimum\n8:54\nMinimum Height\n9:02\nEvaluating the Area Approximation\n10:08\nSplit the Interval Into n Sub-Intervals\n10:30\nMore Rectangles, Better Approximation\n12:14\nThe More We Us , the Better Our Approximation Becomes\n12:16\nOur Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity\n12:44\nFinding Area with a Limit\n13:08\nIf This Limit Exists, It Is Called the Integral From a to b\n14:08\nThe Process of Finding Integrals is Called Integration\n14:22\nThe Big Reveal\n14:40\nThe Integral is Based on the Antiderivative\n14:46\nThe Big Reveal - Wait, Why?\n16:28\nThe Rate of Change for the Area is Based on the Height of the Function\n16:50\nHeight is the Derivative of Area, So Area is Based on the Antiderivative of Height\n17:50\nExample 1\n19:06\nExample 2\n22:48\nExample 3\n29:06\nExample 3, cont.\n35:14\nExample 4\n40:14\nSection 17: Appendix: Graphing Calculators\n\n10m 41s\n\nIntro\n0:00\n0:06\nShould I Get a Graphing Utility?\n0:20\nFree Graphing Utilities - Web Based\n0:38\nPersonal Favorite: Desmos\n0:58\nFree Graphing Utilities - Offline Programs\n1:18\nGeoGebra\n1:31\nMicrosoft Mathematics\n1:50\nGrapher\n2:18\nOther Graphing Utilities - Tablet/Phone\n2:48\nShould You Buy a Graphing Calculator?\n3:22\nThe Only Real Downside\n4:10\n4:20\nIf You Plan on Continuing in Math and/or Science\n4:26\nIf Money is Not Particularly Tight for You\n4:32\nIf You Don't Plan to Continue in Math and Science\n5:02\nIf You Do Plan to Continue and Money Is Tight\n5:28\n5:44\nWhich Graphing Calculator is Best?\n5:46\nToo Many Factors\n5:54\n6:12\nThe Old Standby\n7:10\nTI-83 (Plus)\n7:16\nTI-84 (Plus)\n7:18\n9:17\n9:19\n9:35\n10:09\nGraphing Calculator Basics\n\n10m 51s\n\nIntro\n0:00\n0:06\nSkim It\n0:20\nPlay Around and Experiment\n0:34\nSyntax\n0:40\nDefinition of Syntax in English and Math\n0:46\nPay Careful Attention to Your Syntax When Working With a Calculator\n2:08\nMake Sure You Use Parentheses to Indicate the Proper Order of Operations\n2:16\n3:54\nSettings\n4:58\nYou'll Almost Never Need to Change the Settings on Your Calculator\n5:00\nTell Calculator In Settings Whether the Angles Are In Radians or Degrees\n5:26\nGraphing Mode\n6:32\nError Messages\n7:10\nDon't Panic\n7:11\nInternet Search\n7:32\nSo Many Things\n8:14\nMore Powerful Than You Realize\n8:18\nOther Things Your Graphing Calculator Can Do\n8:24\nPlaying Around\n9:16\nGraphing Functions, Window Settings, & Table of Values\n\n10m 38s\n\nIntro\n0:00\nGraphing Functions\n0:18\nGraphing Calculator Expects the Variable to Be x\n0:28\nSyntax\n0:58\nThe Syntax We Choose Will Affect How the Function Graphs\n1:00\nUse Parentheses\n1:26\nThe Viewing Window\n2:00\nOne of the Most Important Ideas When Graphing Is To Think About The Viewing Window\n2:01\nFor Example\n2:30\nThe Viewing Window, cont.\n2:36\nWindow Settings\n3:24\nManually Choose Window Settings\n4:20\nx Min\n4:40\nx Max\n4:42\ny Min\n4:44\ny Max\n4:46\nChanging the x Scale or y Scale\n5:08\nWindow Settings, cont.\n5:44\nTable of Values\n7:38\nAllows You to Quickly Churn Out Values for Various Inputs\n7:42\nFor example\n7:44\nChanging the Independent Variable From 'Automatic' to 'Ask'\n8:50\nFinding Points of Interest\n\n9m 45s\n\nIntro\n0:00\nPoints of Interest\n0:06\nInteresting Points on the Graph\n0:11\nRoots/Zeros (Zero)\n0:18\nRelative Minimums (Min)\n0:26\nRelative Maximums (Max)\n0:32\nIntersections (Intersection)\n0:38\nFinding Points of Interest - Process\n1:48\nGraph the Function\n1:49\n2:12\nChoose Point of Interest Type\n2:54\nIdentify Where Search Should Occur\n3:04\nGive a Guess\n3:36\nGet Result\n4:06\n5:10\nFind Out What Input Value Causes a Certain Output\n5:12\nFor Example\n5:24\n7:18\nDerivative\n7:22\nIntegral\n7:30\nBut How Do You Show Work?\n8:20\nParametric & Polar Graphs\n\n7m 8s\n\nIntro\n0:00\nChange Graph Type\n0:08\nLocated in General 'Settings'\n0:16\nGraphing in Parametric\n1:06\nSet Up Both Horizontal Function and Vertical Function\n1:08\nFor Example\n2:04\nGraphing in Polar\n4:00\nFor Example\n4:28\nBookmark & Share Embed\n\n## Copy & Paste this embed code into your website’s HTML\n\nPlease ensure that your website editor is in text mode when you paste the code.\n(In Wordpress, the mode button is on the top right corner.)\n×\n• - Allow users to view the embedded video in full-size.\nSince this lesson is not free, only the preview will appear on your website.\n\n• ## Related Books", null, "1 answer", null, "Last reply by: Professor Selhorst-JonesMon Sep 14, 2020 11:28 AMPost by Chengzu Li on September 12, 2020hi professor, in 47m 26s, I wonder how did you get that green function from?  is there something I missed?", null, "0 answersPost by Li Zeng on April 20, 2019", null, "1 answer", null, "Last reply by: Professor Selhorst-JonesSat Feb 11, 2017 8:14 PMPost by John Lins on February 11, 2017Please, help me with this question:Find Laplace Transform of the following signal:x2 (t)=(1-(1-t)e-3t)u(t)", null, "1 answerLast reply by: John LinsSat Feb 11, 2017 8:06 PMPost by John Lins on February 11, 2017Hello professor vincent. Could you please help me to find this Laplace Transform?x2 (t)=(1-(1-t)e-3t)u(t)", null, "1 answer", null, "Last reply by: Professor Selhorst-JonesFri Mar 25, 2016 5:24 PMPost by Ru Chigoba on November 18, 2015Hi I need help with this problem :Find the inverse of each relation or function, and then determine if the inverse is a function.1 f={1,3), (1,-1), (1,-3), (1,1)}   f-1=", null, "1 answer", null, "Last reply by: Professor Selhorst-JonesTue Mar 17, 2015 11:24 PMPost by thelma clarke on March 17, 2015is there no easier way to solve this it seem very confusing", null, "1 answer", null, "Last reply by: Professor Selhorst-JonesMon Oct 20, 2014 11:27 AMPost by Saadman Elman on October 18, 2014It was a great clarification. Thanks,However, Inverse function is not only has to do with horizontal line test passing but also has to do with vertical test passing. You only stressed on Horizontal test passing. My professor stressed both.", null, "1 answer", null, "Last reply by: Professor Selhorst-JonesMon Jun 16, 2014 9:18 PMPost by Joshua Jacob on June 15, 2014Sorry if this is slightly vague but I'm a little but confused on the last example. Could you explain it in other words please?\n\n### Inverse Functions\n\n• A function does a transformation on an input. But what if there was some way to reverse that transformation? This is the idea of an inverse function: a way to reverse a transformation and get back to our original input.\n• To help us understand this idea, imagine a factory where if you give them a pile of parts, they'll make you a car. Now imagine another factory just down the road, where if you give them that car, they'll give you back the original pile of parts you started with. There is one process, but there is also an inverse process that gets you back to where you started. If you follow one process with the other, nothing happens.\n• It's important to note that not all functions have inverses. Some types of transformation cannot be undone. If the information about what we started with is permanently destroyed by the transformation, it cannot be reversed.\n• A function has an inverse if the function is one-to-one: for any a, b in the domain of f where a ≠ b, then f(a) ≠ f(b). Different inputs produce different outputs.\n• We can see this property in the graph of a function with the Horizontal Line Test. If a function is one-to-one, it is impossible to draw a horizontal line somewhere such that it will intersect the graph twice (or more).\n• Given some function f that is one-to-one, there exists an inverse function, f−1, such that for all x in the domain of f,\n f−1 ⎛⎝ f(x) ⎞⎠ = x.\nIn other words, when f−1 operates on the output of f, it gives the original input that went into f. [Caution: f−1 means the inverse of f, not [1/f]. In general, f−1 ≠ [1/f].]\n• We can figure out the domain and range for f−1 by looking at f. Since the set of all outputs is the range of f, and f−1 can take any output of f, the domain of f−1 is the range of f. Likewise, f−1 can output all possible inputs for f, so the range of f−1 is the domain of f.\n• The inverse of f−1 is simply f. This makes intuitive sense: if you do the opposite of an opposite, you end up doing the original thing.\n• Visually, f−1 is the mirror of f over y=x. This is because f−1 swaps the outputs and inputs from f, which is the same thing as swapping x and y by mirroring over y=x.\n• To find the inverse to a function, we effectively need a way to \"reverse\" the function. This can be a little bit confusing at first, so here is a step-by-step guide for finding inverse functions.\n\n1. Check function is one-to-one;    f(x) = x3+1\n2. Swap f(x) for y;     y = x3+1\n3. Interchange x and y;     x = y3+1\n4. Solve for y;     y = 3√{x−1}\n5. Replace y with f−1(x);     f−1(x) = 3√{x−1}\n• While this method will produce the inverse if followed correctly, it is not perfect. Notice that in steps #2 and #3 above, the equations are completely different, yet they still use the same x and y. Technically, it is not possible for x and y to fulfill both of these equations at the same time. What's really happening is that when we swap in #3, we're actually creating a new, different y. The first one stands in for f(x), but the second stands in for f−1(x). This implicit difference between y's can be confusing, so be careful. Make a note on your paper where you swap x and y so you can see the switch to \"inverse world\".\n• Taking inverses can be difficult: it's easy to make a mistake. This means it's important to check your work. By definition, f−1 ( f(x) ) = x. This means if you know what f−1(x) and f(x) are, you can just compose them! If it really is the inverse, you'll get x. Furthermore, since we know f( f−1 (x) ) = x as well, you can compose them in either order when checking.\n\n### Inverse Functions\n\nThe table below shows how f(x) works. Is f(x) one-to-one?\n x\n f(x)\n apple\n a\n apricot\n a\n a\n banana\n b\n kiwi\n k\n kumquat\n k\n tomato\n t\n• A function is one-to-one when different inputs always produce different outputs. In other words, there are no two inputs that produce the same output.\n• For f(x), we see that there are cases when different inputs produce the same output. For example, f(apple) = f(apricot) = f(avocado) = a. Thus, because it is possible to have two distinctly different inputs result in the same output, the function is NOT one-to-one.\nNo, f(x) is not one-to-one.\ng(x) = 3x. Is g(x) one-to-one?\n• A function is one-to-one when different inputs always produce different outputs. In other words, there are no two inputs that produce the same output.\n• For g(x) to be one-to-one, it must be that if we put in two different values for x, we will never get the same result.\n• If we think about this carefully, we can realize that the only way to get the same output for g(x) is to start with the same x. For example, if x=3, we get g(3) = 9. But there is no other number we could possibly plug in other than x=3 to produce 9 as the output. [If we want to formally prove this, we can do it as follows: Let a and b be two numbers such that g(a) = g(b). Then 3a = 3b. Thus, by algebra, we have a=b. Therefore, if the output is the same, it must be that the inputs were the same, which proves that g(x) is one-to-one. You don't need to formally prove this, though: you can just think about how the function works.]\nYes, g(x) is one-to-one.\nThe graph of h(x) = |x−1| −3 is below. Use the Horizontal Line Test to determine if it is one-to-one.", null, "• The Horizontal Line Test says that if a function is not one-to-one, you can draw a horizontal line somewhere on it that will intersect the graph twice (or more).\n• There are many places a horizontal line can be drawn that will cut the graph twice. Thus, the graph is not one-to-one.\nNo, h(x) is not one-to-one.\nThe graph of f(x) = (x+2)3 −1 is below. Use the Horizontal Line Test to determine if it is one-to-one.", null, "• The Horizontal Line Test says that if a function is not one-to-one, you can draw a horizontal line somewhere on it that will intersect the graph twice (or more).\n• There is nowhere on the graph that a horizontal line can be drawn that will cut the graph twice. Thus, f(x) is one-to-one. [Near the center of the curve, it might look like f(x) is flat enough that you could cut it multiple times with a single horizontal line. This is not true, though. Notice that even when it looks fairly flat, it is still slightly sloped. It is not changing much, but it is still changing enough to pass the horizontal line test.]\nYes, f(x) is one-to-one.\nWhat is f−1 ( f( 4) )? What is f ( f−1 (−27) )? What is f−1 ( f ( f−1 ( f(x)) ) )?\n• f−1 is the inverse function of f: it cancels out whatever f does. In general, f−1 ( f( x) ) = x. By having the inverse operate on a function, it gets us back to where we started. Thus f−1 ( f( 4) ) = 4.\n• The inverse of f−1 is f. Thus, just like f−1 cancels out f, f will cancel out f−1:  f ( f−1 (x) )=x. By having a function operate on its inverse, it gets us back to where we started. Thus f ( f−1 (−27) ) = −27.\n• The above canceling can occur multiple times:\n f−1 ⎛⎝ f ⎛⎝ f−1 ⎛⎝ f(x) ⎞⎠ ⎞⎠ ⎞⎠ = f−1 ⎛⎝ f ⎛⎝ x ⎞⎠ ⎞⎠ = x\nf−1 ( f( 4) ) = 4        f ( f−1 (−27) )=−27        f−1 ( f ( f−1 ( f(x)) ) ) = x\nThe function f(x) = 4x is one-to-one. Find the inverse f−1(x).\n• We already know the function is one-to-one from the problem, so the next step is to replace f(x) with y:\n y = 4x\n• Next, interchange the locations of x and y:\n x = 4y\n• Solve for y from the new equation:\n 1 4 x = y\n• Finally, replace y with f−1(x):\n f−1(x) = 1 4 x\nf−1(x) = [1/4] x\nThe function f(x) = [(x−3)/(x+3)] is one-to-one. Find the inverse f−1(x).\n• We already know the function is one-to-one from the problem, so the next step is to replace f(x) with y:\n y = x−3 x+3\n• Next, interchange the locations of x and y:\n x = y−3 y+3\n• Solve for y from the new equation. This is kind of tricky: use the distributive property in reverse to pull out y once you have everything involving y on one side. xy + 3x = y−3   ⇒  xy−y = −3x −3   ⇒  y(x−1) = −3x−3\n y = 3x+3 1−x\n• Finally, replace y with f−1(x):\n f−1(x) = 3x+3 1−x\nf−1(x) = [(3x+3)/(1−x)]\nLet f(x) = 2x+3 and g(x) = [(x−3)/2]. Show that f and g are inverse functions.\n• Two functions f and f−1 are inverses when f−1( f(x) ) = x or, equivalently, when f( f−1(x) ) = x. This also means we can check to see if two functions are inverses by composing one with the other. Thus, if we can show f(g(x) ) = x or g ( f(x) ) = x, we have shown that they are inverses.\n• This means we have two options. Let us show f(g(x) ) = x is true first:\n f ⎛⎝ g(x) ⎞⎠ = f ⎛⎝ x−3 2 ⎞⎠ = 2 ⎛⎝ x−3 2 ⎞⎠ +3\nSimplifying the above, we find that f(g(x) ) = x, and therefore f and g are inverses.\n• Alternatively, we can show g ( f(x) ) = x is true:\n g ⎛⎝ f(x) ⎞⎠ = g ⎛⎝ 2x+3 ⎞⎠ = (2x+3) −3 2\nSimplifying the above, we find that g ( f(x) ) = x, and therefore f and g are inverses.\n[The problem is answered by showing that f(g(x) ) = x or g ( f(x) ) = x. Look at the steps above to see how this is done if you are uncertain.]\nLet f(x) = 2x3 −12 and g(x) = 3√{[1/2]x+6}. Show that f and g are inverse functions.\n• Two functions f and f−1 are inverses when f−1( f(x) ) = x or, equivalently, when f( f−1(x) ) = x. This also means we can check to see if two functions are inverses by composing one with the other. Thus, if we can show f(g(x) ) = x or g ( f(x) ) = x, we have shown that they are inverses.\n• This means we have two options. Let us show f(g(x) ) = x is true first:\nf\ng(x)\n= f\n\n3\n\n 1 2 x+6\n\n= 2\n\n3\n\n 1 2 x+6\n\n3\n\n−12\nSimplifying the above, we find that f(g(x) ) = x, and therefore f and g are inverses.\n• Alternatively, we can show g ( f(x) ) = x is true:\ng\nf(x)\n= g\n2x3 −12\n=\n3\n\n 1 2 (2x3 −12)+6\n\nSimplifying the above, we find that g ( f(x) ) = x, and therefore f and g are inverses.\n[The problem is answered by showing that f(g(x) ) = x or g ( f(x) ) = x. Look at the steps above to see how this is done if you are uncertain.]\nLet f(x) = √{x−5}. What are the domain and range of f? What is f−1(x)? What are the domain and range of f−1?\n• The domain is the set of all values that the function can accept, while the range is the set of all values the function can possibly output.\n• For f(x), it \"breaks\" when there is a negative in the square root, so the domain of f is x ≥ 5. The range of f is [0, ∞).\n• To find f−1, we follow the same steps we did in previous problems. Working it through, we get f−1 (x) = x2 + 5. [Check it by plugging one function into the other if you're not sure.]\n• At first glance, we might think the domain of f−1(x) is all numbers because x2 +5 never \"breaks\". However, we have to remember that f−1 is the inverse of f: it can only reverse values that could possibly come out of f. Thus, the domain of f−1 is the range of f: [0, ∞). Similarly, the range of f−1 is the domain of f: x ≥ 5 (Alternately, we can use the domain of f−1 to figure out what its range must be).\nDomain of f: [5, ∞)    Range of f: [0, ∞)\nf−1(x) = x2 +5\nDomain of f−1: [0, ∞)    Range of f−1: [5, ∞)\n\n*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.\n\n### Inverse Functions\n\nLecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.\n\n• Intro 0:00\n• Introduction 0:04\n• Analogy by picture 1:10\n• How to Denote the inverse\n• What Comes out of the Inverse\n• Requirement for Reversing 2:02\n• The Importance of Information\n• One-to-One 4:04\n• Requirement for Reversibility\n• When a Function has an Inverse\n• One-to-One\n• Not One-to-One\n• Not a Function\n• Horizontal Line Test 7:01\n• How to the test Works\n• One-to-One\n• Not One-to-One\n• Definition: Inverse Function 9:12\n• Formal Definition\n• Caution to Students\n• Domain and Range 11:12\n• Finding the Range of the Function Inverse\n• Finding the Domain of the Function Inverse\n• Inverse of an Inverse 13:09\n• Its just x!\n• Proof\n• Graphical Interpretation 17:07\n• Horizontal Line Test\n• Graph of the Inverse\n• Swapping Inputs and Outputs to Draw Inverses\n• How to Find the Inverse 21:03\n• What We Are Looking For\n• Reversing the Function\n• A Method to Find Inverses 22:33\n• Check Function is One-to-One\n• Swap f(x) for y\n• Interchange x and y\n• Solve for y\n• Replace y with the inverse\n• Keeping Step 2 and 3 Straight\n• Switching to Inverse\n• Checking Inverses 28:52\n• How to Check an Inverse\n• Quick Example of How to Check\n• Example 1 31:48\n• Example 2 34:56\n• Example 3 39:29\n• Example 4 46:19\n\n### Transcription: Inverse Functions\n\nHi--welcome back to Educator.com.0000\n\nToday, we are going to talk about inverse functions.0002\n\nA function does a transformation on an input; we have talked about functions for a while now.0005\n\nBut what if there was some way to reverse that transformation?0009\n\nThis is the idea behind an inverse function: it is a way to reverse a transformation, reverse the process that another function is doing.0012\n\nIt is a way to get back to our original input.0021\n\nBy way of analogy, let's imagine a factory where, if you give them a pile of parts, they will make you a car.0024\n\nNow, if you take that car down the road to this other factory, you can give them that car,0029\n\nand they will give you back the original pile of parts you started with.0034\n\nThere is one factory where they make cars out of parts, but then there is a second factory0038\n\nwhere they take cars and break them down into the original parts that were used to make them.0042\n\nThere is one process, but there is also an inverse process that gets you back to where you started.0046\n\nIf you follow one process with the other immediately, it ends up as if you haven't done anything.0051\n\nIf you bring the pile of parts to the first factory, and then take that car to the second factory,0055\n\nand they give you back the pile of parts, it is like you just started with a pile of parts and didn't do anything to it.0059\n\nThis is the idea behind an inverse function: it reverses a process--it reverses a transformation and gets you back to where you started.0063\n\nWe have used the analogy of a function as a machine before;0071\n\nand it is a good image for being able to get across what is going on with inverse functions, as well.0073\n\nSo, a function machine takes inputs, and it transforms them to outputs by some rule.0079\n\nSo, what we are used to is: we plug in x into the function f, and it gives out f acting on x, f(x).0084\n\nNow, we could plug it into another one; we could plug it into an inverse machine, an inverse to f;0092\n\nand that would be called \"f inverse,\" the f with the little -1 in the corner.0097\n\nf-1 denotes the inverse of f; we call it \"f inverse.\"0103\n\nWe plug f(x) into f-1, into that machine; we get right back to our original x.0108\n\nIt is as if we hadn't done anything; the first machine does something, but then the second machine0114\n\nreverses that process and gets us back to where we started.0119\n\nSo, is there a requirement for reversing--can we make an inverse out of all functions out there?0123\n\nNo; let's see why by analogy first.0129\n\nImagine a factory where, if you give them a pile of wood or a pile of metal, they give you a basketball in return.0133\n\nThe basketball is the exact same, whether you started with wood or metal; it is always the exact same basketball.0139\n\nIt doesn't matter what you gave them; it is just a basketball.0145\n\nNow, let's say you walk down the road to another factory; you give them that basketball;0148\n\nyou tell them to reverse the process; and then you walk away--you give them no other information.0152\n\nCan that factory take a basketball and transform it back into a pile of wood or a pile of metal, if all they have is a basketball and no other information?0156\n\nNo, they have no idea what you started with.0164\n\nMaybe they have wood; maybe they have metal; but the point is that they have no way0168\n\nto be able to know which one they are supposed to give you at this point.0172\n\nThey don't have the information; the only person who has the information is you, when you brought the original wood pile,0175\n\nor brought the original metal pile--because all they have is the basketball, and the basketball could indicate wood, or it could indicate metal.0180\n\nThey have no way to know what you started with; there is no way to figure it out.0188\n\nThe information about what you originally had has been destroyed (although you would know it, because you brought it to the factory).0191\n\nBut assuming you forgot, then the information has been destroyed--no one has the information anymore.0197\n\nAnother way to think about it would be if you took a piece of paper, and you burned that piece of paper.0202\n\nYou would be left with a pile of ashes.0206\n\nNow, someone could come along and think, \"Oh, a pile of ashes--it used to be a piece of paper.\"0208\n\nBut if you take two pieces of paper, and you write two completely different things on the two pieces of paper,0212\n\nand then you burn the two of them, a person could come along and think, \"Two piles of ashes...\"0217\n\nAnd they would know it was paper, but they wouldn't know what was written on them.0221\n\nThey wouldn't be able to get that information back; the information has been destroyed.0224\n\nThey know it was paper, but they don't know what was written on the paper.0227\n\nThe basketball one...you have given them wood; you have given them metal; you get the same thing.0231\n\nThe information about what you started with has been lost; the information has been destroyed,0235\n\nunless you come along and also say, \"Oh, by the way, that basketball came from ____.\"0239\n\nThe issue, in this scenario, is that we have two inputs providing the same output--whether it is metal or wood, you get a basketball.0245\n\nSo, if we try to have a reverse on that, we have no way to know which one to go back to.0252\n\nWe don't know if we are going to go back to metal; we don't know if we are going to go back to wood,0256\n\nbecause we don't know what the basketball is representing.0259\n\nSo, to be a reversible process--for it to be possible to reverse something--the process has to have a different output for every input.0262\n\nIf you give them metal, they have to give you one color of basketball; and if you give them wood, they give you a totally different color of basketball.0271\n\nThen, the second factory would say, \"Oh, that is a wood basketball\" or \"Oh, that is a metal basketball.\"0277\n\nAnd they would be able to know what to do at that point.0281\n\nSo, for a function f to have an inverse, it has to be that, for any a and b in the domain, any a or b that we could use in f normally,0284\n\nwhere a is not the same thing as b (where a and b are distinct from each other--they are different),0291\n\nthen f(a) is different than f(b); f(a) does not equal f(b).0297\n\nSo, if a and b are different, then the function's outputs on a and b are different, as well.0302\n\nSo, different inputs going into a function have to produce different outputs; we call this property one-to-one.0308\n\nIf this function has a property where whatever you put in, as long as it is different from something else going in,0316\n\nit means the two things will be different, that is called one-to-one: different inputs produce different outputs.0323\n\nYou give them metal; you get one color of basketball; you give them wood--you get a different color of basketball.0328\n\nHere are some examples, so we can see this in a diagram.0334\n\nHere is an example that is one-to-one: a goes to 2; b goes to 1; c goes to 3.0336\n\nThey each go to different things: different inputs each have different places that they end up going.0344\n\nSomething that is not one-to-one: a goes to 1 and b goes to 1.0350\n\nIt doesn't matter that c goes to 3, because a and b have both gone to the same thing, so they have different inputs that are producing the exact same output.0354\n\na and b are different things, but they both produce a 1; so it is not one-to-one.0365\n\nWe have that copy; we are putting in wood, and putting in metal, and we get basketball in both cases.0369\n\nAnd then, finally, just to remind us: this one over here (hopefully you remember this) is not a function.0373\n\nAnd it is not a function, because b is able to produce two outputs at once; and that is something that is not allowed for a function.0378\n\nIf a function takes in one input, it is only allowed to produce one output; it can't produce multiple outputs from a single input.0384\n\nSo, why do we call it one-to-one--why are we using this word, \"one-to-one\"?0391\n\nWell, we can think of it as being because a has one partner, and b has one partner, and c has one partner.0395\n\nEverybody gets a partner, and nobody has shared partners; everybody gets their partner, and that partner is theirs.0402\n\nThey don't have to share it with anybody else.0408\n\nIt is one-to-one: this thing is matched to this thing, and there is nobody else who is going to match up to that one: one-to-one.0410\n\nAll right, how can we test for this?0417\n\nOne way to test for this, to test if a function is one-to-one: we know, if we are going to be one-to-one,0420\n\nthat every input must have a unique output; that was what it meant to be one-to-one.0425\n\nIf we have different inputs, we have different outputs.0429\n\nSo, if we draw a horizontal line on the graph, it can intersect the graph only once, or not at all.0432\n\nRemember, if we have some picture on a graph--like if we have this point--then what that means0438\n\nis that this is the input, and this over here is the output.0443\n\nWe make it a point: (input,output); that is why it is (x,f(x)).0449\n\nIf it is f(x) = x2, then we plug in 3, and we get (3,9), (3,32).0458\n\nThe input is our horizontal location; the output is our vertical location.0465\n\nThe horizontal line test is a way to test if the function has the same output for multiple inputs.0469\n\nWe draw a horizontal line across, because wherever an output hits the graph, we know that there must be an input directly below it.0477\n\nIf a function is not one-to-one, you will be able to draw a horizontal line that will intersect it twice, or maybe even more.0485\n\nLet's look at some examples: first, here is one that is one-to-one, because whenever we draw0490\n\nany sort of horizontal line, it is only going to cut it once.0496\n\nThe only place that might seem a little confusing is if we draw it near here.0499\n\nIt might make you think, \"Well, doesn't it look like those are stacked?\"0503\n\nWell, remember, we can't draw perfectly what is being represented by the mathematics.0506\n\nWe have to give our lines some thickness; in reality, the line is infinitely thin.0511\n\nSo, while it looks like they are kind of getting stacked, it is actually still moving through that zone; it is not constant.0517\n\nIt is increasing just a little bit, but it isn't constant.0522\n\nLet's look at one that is not one-to-one: over here, this horizontal line (or many horizontal lines that we could make)--0525\n\nit cuts it in two places; so we know that, here and here, there are two inputs.0532\n\nWe can produce the same output from two different locations.0537\n\nWe have two inputs making one output; so that means we are not one-to-one,0541\n\nbecause this one is partner to that height, but this one is also partner to this height.0545\n\nSo, we are not one-to-one, because we have to share an output.0550\n\nNow that we have all of these ideas in mind, we are finally ready to define an inverse function; we can really talk about them and sink our teeth into them.0554\n\nGiven some function f that is one-to-one (it has to be one-to-one for this to happen),0561\n\nthere exists and inverse function, which we denote as f-1 such that, for all x in the domain of f,0566\n\nany x that could normally go into f, any value that could normally be input,0574\n\nf-1 acting on f(x) becomes just x.0578\n\nSo, we have f acting on x like normal; and then, f-1 acts on that whole thing.0583\n\nAnd it breaks the action that was done by f and returns us back to our original input.0591\n\nIn other words, when f-1 operates on the output of f, it gives the original input that went into f.0596\n\nCaution: I want to warn you about something: f-1 means the inverse of f, not 1/f.0604\n\nThis can be confusing, because, if you have taken algebra and remember your exponents0612\n\n(you might have forgotten them, but we will talk about them later in this course), -1 can mean a reciprocal for numbers.0617\n\nSo, 7-1 becomes 1/7; x-1 becomes 1/x.0622\n\nBut f is not \"to the -1\"; it is just a symbol that says inverse--\"This is a function inverse.\"0630\n\nSo, in general, for the most part, f-1, \"f inverse,\" is not the same thing as 1/f.0636\n\nThe inverse of f is normally not the reciprocal of f, 1/f.0643\n\nThis exponent thing, where 7-1 is 1/7, is not the case for functions.0649\n\nOn a function, the -1 does not represent an exponent; it is not an exponent.0656\n\nBut it instead tells us function inverse; it is a way of saying, \"This is an inverse function\"; that is what it is telling us, not \"flip it to the reciprocal.\"0661\n\nHow can we get domain and range for f-1?0673\n\nWe can figure these out by looking at f; remember, the set of all outputs from f is its range.0675\n\nThe things that x can get mapped to by f, what f is able to map x to, is the range of f.0687\n\nThe domain of f is everything that x can be, everything that we can plug into f.0695\n\nAnd then, the range of f is everything that can come out of f.0701\n\nNow, f-1 has to be able to take any output of f.0704\n\nIt is not very good at reversing if there are some numbers that it is not allowed to reverse.0709\n\nSo, it has to be able to reverse anything from f.0713\n\nIf it is able to reverse anything from f, then that means the range of f has to be everything that we can put into f-1.0716\n\nSo, the domain of f-1, the things we can put into f-1, is the range of f.0723\n\nThe domain of f-1 is the range of f.0728\n\nLikewise, because f-1 then breaks that f(x) and turns it back into original inputs,0732\n\nwe must be able to turn it back into all of the original inputs, because all of the original outputs have to be over here.0738\n\nSo, anything that we can make it to, we have to be able to make it back from.0745\n\nSo, since we are able to get back everywhere, that means that we can output all the possible inputs for f.0748\n\nSince we can output all of the possible inputs, because we can reverse any of the processes,0755\n\nthen it must mean that we are able to get the range of f-1 from the domain of f.0760\n\nSo, the domain of f tells us the range of f-1, what we are allowed to output with f-1.0767\n\nAnd the domain of f-1 tells us the range of f.0772\n\nSo, the domain of f-1 is the things that f is able to output; and the range of f-1,0776\n\nthe things that f-1 is able to output, is what we can put into f, the domain of f.0784\n\nThis idea is going to let us prove something later on.0790\n\nNow we can get to that proof--the inverse of an inverse: what is the inverse of an inverse?0793\n\nIn symbols, what is (f-1)-1--what do we do if we are going to take the inverse of something that is already doing inverses?0797\n\nNow, it might seem a little surprising, but it turns out that the inverse of f-1 is just f.0806\n\nThe inverse of f-1 is f; it seems a little surprising, maybe, but it makes a sort of intuitive sense.0812\n\nIf you do the opposite of an opposite, you get to the original thing.0820\n\nIf you do an action, but then you are going to do the opposite of that action,0824\n\nbut then you do the opposite of the opposite of the action, then you must be back at your original action.0828\n\nSo, we might be able to believe this on an intuitive level; it makes sense, intuitively, that two opposites gets us back to where we started.0833\n\nBut let's see a proof of this fact, formally; let's see it in formal mathematics.0840\n\nSo, how do we get this started? Well, by definition, f-1 is the function where,0845\n\nfor any x, f-1 acting on f(x) is going to just give us our original x.0849\n\nIf f acts on an input, and then f-1 comes and acts on that, we get back to our original input.0856\n\nNow, consider (f-1)-1: by this definition of the way inverses work, it must be that f inverse, inverse,0862\n\nwhen it acts on the thing that it is an inverse of...f inverse, inverse, is an inverse of f inverse...0869\n\nI know it is complicated to say...but this one right here is going to be the opposite action of f-1.0874\n\nSo, if we take any y (don't get too worried about x and y; remember, they are just placeholders for inputs),0881\n\nsimilarly, for any y, (f-1)-1, acting on f-1(y),0887\n\nis going to just get us right back to our original y.0892\n\nIt is the same structure as what is going on here with f-1(f(x)) = x: we are just reversing a process.0895\n\nSo, it doesn't matter that one of the processes is already a reversed process, because we are reversing this other reversed process.0901\n\nSo, we get back to our original thing.0907\n\nNow, we know that y is in the domain of f-1, because we are allowed to put it into f-1.0910\n\nIt is allowed to go into f-1; now, we know, from our thing that we were just talking about,0917\n\nthat the domain of f-1 is the range of f; so there has to be...0922\n\nIf f-1 is the range of f (the domain of f-1 is the range of f),0930\n\nif you are in the range, then that means that there is something out there that can produce this.0940\n\nThat means that, if you are in the range of f, there must be some x in the domain of f;0944\n\nthere has to be some way to get to that place in the range, so that f(x) is equal to y.0949\n\nThere is some x out there in the domain of our original f, that f(x) is equal to y.0954\n\nSo now, we have what we need: we can use this f(x) = y, and we can just plug it in right here and here.0960\n\nWe will plug it in for the two y's up here, and we will see what happens.0967\n\nThus, f inverse, inverse, acting on f-1(f(x))...0970\n\nbecause remember, we know that there has to be some way to get an x such that f(x) = y,0974\n\nbecause of this business about domain and range; so we plug that in here, and we plug that in here.0979\n\nAnd we have that f inverse, inverse, on f inverse of f of x, must be equal to this over here on the right, as well.0985\n\nWe are just doing substitution.0993\n\nBut we know, by the definition of f-1(f(x)), that this just turns out to be x.0995\n\nThis whole thing right here just comes down to x--it simplifies right out to x.1002\n\nSo, it must be the case that f inverse, inverse, of x is the same thing as f(x).1007\n\nIf f(x) is the exact same thing as f inverse, inverse of x, it must be the case that (f-1)-1 is just the same thing as f.1012\n\nAnd our proof is finished; great.1023\n\nHow can we interpret this graphically?--there is a great way to interpret inverses through graphs.1028\n\nFirst, let's consider f(x) = x3 + 1.1032\n\nNow, we know that this one has to be one-to-one, because it passes the horizontal line test.1035\n\nWe come along and try to cut this with any horizontal line; they are only going to be able to cut in one place.1040\n\nEven here, where we have sort of seemed to flatten out, it is still moving, because we know it is x3 + 1.1045\n\nAnd it never actually stops going up; it just slows down how fast it is going up.1051\n\nAnd our lines have to have thickness, so while it kind of looks like they are stacked, they are not really.1055\n\nSo, we see that it passed the horizontal line test; so it must be one-to-one.1059\n\nIf it is one-to-one, we know it has to have an inverse; that is how we talked about this, right from the beginning.1064\n\nNow, notice that the graph, any graph, is made up of the points (x,f(x)).1069\n\nWe talked about this before: 0 gets mapped to 1 when we plug it in as f(0) = 1; so that gives us the point (0,1).1073\n\nThat is how we make up our original graph for f.1082\n\nNow, the graph of f-1 will swap these coordinates.1084\n\nIt takes in outputs and gives out inputs, in a way; so its input will swap these two things.1088\n\nIt takes in f(x), and it gives out x; so the points of f-1 will be the reverse of what we had for the points of the other one.1096\n\nSo, (f(x),x) is what we get for f-1.1105\n\nNow, visually, what that means is that f-1 is going to be the mirror of y = x; and that is our line right here, y = x.1109\n\nWhy is that the case? Well, look: we swap x and y coordinates if we go across this,1117\n\nbecause (-3,0) swaps to (0,-3); if we are going over y = x, if we are mirroring across this, we will swap the locations.1122\n\nAnd so, if we mirror over y = x, we are going to swap x and y, y and x; we will swap the order of our points,1138\n\nbecause y = x is sort of a way of saying, \"Let's pretend for today that I am you and you are me.\"1145\n\ny is going to pretend that it is x, and x is going to pretend that it is y.1153\n\nThey are sort of swapping places when we do a mirror over it.1156\n\nSo, that means our picture, mirroring f over y = x...we get the graph of f-1.1159\n\nSo, we look at this; we mirror over it; and we get places like this.1164\n\nAll right, we see how we are just sort of bouncing across it.1171\n\nAnd this is going to happen with any of our inverses graphically.1179\n\nSo, any time f-1 is being looked at, we know it is going to be a bounce, a reflection through, a mirror over;1183\n\nit is going to be symmetric to f with respect to the line y = x.1190\n\nSince f-1 is swapping outputs and inputs, it is going to be sort of reversing the placements of these.1193\n\nSo, the graph of f-1 will always be symmetric to f, with respect to the line y = x.1201\n\nIt will bounce across, because when you bounce across y = x, you swap your coordinates.1206\n\nNow, there are many ways to say this; I am saying \"bounce across\"; that is not really formal.1211\n\nBut we can formally say that it is symmetric to f, with respect to the line y = x.1215\n\nYou could also say that it reflects through y = x, or it reflects over y = x.1220\n\nYou could also say that it mirrors over, or it mirrors through, y = x.1224\n\nThere are many ways to say it; but in all of these things, the same idea is that we are going to bounce across,1227\n\nand that that point will now show up at that same distance here.1232\n\nSo, let's see what it looks like: we bring them in, and indeed, they pop into those places.1235\n\nThey pop into being a nice, symmetric-to-the-line, y = x; and that makes sense.1240\n\nWe replaced the inputs with outputs and outputs with inputs; they have swapped locations.1247\n\nWe look at this one here, and the point (3,0) on the inverse is connected to (0,3) on the original function--the same sort of thing on both of them.1252\n\nAll right, so we have talked a lot about what is going on; we have a really great understanding of the mechanics behind an inverse.1264\n\nBut how do we actually find an inverse?1270\n\nNow that we understand them, we are ready to actually go and find them.1273\n\nHow do we turn an algebraic function like f(x) = x3 + 1 into a formula?1276\n\nBefore we do this formula for f-1, consider that f-1 is taking the output of f(x); and it is transforming that into x.1282\n\nTo find a formula for f-1, we want a formula that gives x, if we know f(x).1290\n\nNormally, f(x) = x3 + 1, for example--normally we have x.1298\n\nWe know x, and from that, we get our f(x); you plug in an x into a function, and it gives out f(x).1303\n\nSo, f-1 is the reverse of that; we know f(x), and we want to get x out of it.1312\n\nSo, to be able to do this, we are solving f(x) = x3 + 1 in reverse.1318\n\nf(x) = x3 + 1; well, we move that over: f(x) - 1 = x3; so now we have 3√(f(x) - 1) = x.1323\n\nIf we know what f(x) is, we can figure out that that is what the original x that did it is.1338\n\nWe are solving it in reverse; we have reversed the function.1342\n\nAs opposed to solving f(x) in terms of x, we are solving x in terms of f(x).1345\n\nNow, this is a little bit of a confusing idea; so instead, I am going to show you a method to do this.1351\n\nThe idea of reversing is really what is behind inverses; it can be a little hard to understand what to do on a step-by-step basis.1357\n\nWe are normally used to solving for f(x) in terms of x, having f(x) just on its own on one side, and having a bunch of stuff involving x on the other side.1365\n\nSo, at this point, it might be a little bit confusing for you to try to do it the other way.1372\n\nAnd it would work; but let's learn a method that makes some of that confusion go away, and do things we are more used to doing.1376\n\nHere is one step-by-step guide for finding inverse functions.1382\n\nThe very first thing we have to do: we have to check that the function is one-to-one.1384\n\nIt has to be one-to-one for us to be able to find an inverse at all.1388\n\nNow, f(x) = x3 + 1...we just saw its graph; remember, it looked something like this.1391\n\nSo, we already know that it passes the horizontal line test; it does a great job; it is a great function.1397\n\nIt is one-to-one; great--we have already passed that part for this.1401\n\nNext, we swap f(x) for y; this is going to be a little bit easier for us in solving.1406\n\nWe are used to solving for y's; we are not really used to solving for f(x)'s; so this will make it a little bit less confusing.1410\n\nWe switch out f(x) for y; great; in the next step, we interchange the x and the y.1415\n\nIn this one, we have x in its normal place and y in its normal place.1423\n\nWhat we do on step 3 is swap their places: y takes the place of all of the x's, and x takes the place of y.1427\n\nWe swap x and y, interchange x and y; every time you had an x in step 2, you are now going to have a y;1436\n\nevery time you had a y (which is probably just the one time, since it was from a function), you are going to now have an x.1443\n\nThat is how we are doing this step that is the reversing step.1448\n\nSolve for y: at this point, we have x = y3 + 1; so if x = y3 + 1, we solve for it.1453\n\nWe just move that 1 over: x - 1 = y3; and we have 3√(x - 1) = y.1460\n\nAnd you will notice that this actually looks pretty much the exact same as what we just did on the previous slide--1469\n\nbut perhaps a little less confusing, because it is what we are used to seeing.1473\n\nSo, we have y = 3√(x - 1); and finally, just like we replaced f(x) with y, we now do a reverse replacement.1476\n\nBut we are now going to f-1; so y now becomes f-1(x).1486\n\nf-1(x) is equal to 3√(x - 1); f-1(input) = 3√(input - 1).1492\n\nGreat; now, while this method will produce the inverse if followed correctly, it is not perfect.1500\n\nNow, remember steps #2 and #3; in that, we had to swap f(x) for y, and then we were told to interchange x and y.1507\n\nRemember, they swapped places; now notice, these equations are completely different.1515\n\nThey are totally, totally different from one another; yet they are still using the same x and y.1520\n\nTechnically, it is not possible to have both of these equations be true with the same x and y.1530\n\nx and y can't possibly fulfill both of these equations at the same time, because they are completely different equations.1534\n\nSo, what is going on here? When we swap in step #3, we are really creating a new, different y.1541\n\nWhen we have \"swap f(x) for y,\" it is really red y or something here.1548\n\nBut then, when we do the interchange, it becomes a totally different color of y; it becomes like blue y here.1553\n\nSo, we are creating a new, different y; the y when we first swap is different than the meaning of the second y.1560\n\nThe swapping y is a different y from our first time that we replaced f(x).1567\n\nThe first one is standing in for f(x); that was our red y.1572\n\nAnd then, the second one stands in for f-1(x); that is really taking the place there.1578\n\nThis implicit difference between y's can be confusing; so be careful.1584\n\nI would recommend making a note on your paper; make a note when you are working that says where you swap.1588\n\nUse a note to see that swap of x and y, so that you can see the switch over to this inverse world,1593\n\nwhere you are now in an inverse world, and you can solve for an answer.1599\n\nThis is a bit confusing; so why are we learning this method, if it has this hidden, confusing1603\n\nimplicit difference, when we really think about what is going on?1609\n\nIn short, the reason we are doing this is because everybody else does.1612\n\nThat is not because it is perfect; it is because everyone else out there pretty much learns this method for solving inverses.1616\n\nMost textbooks, and almost all of the teachers out there, teach this method.1623\n\nSo, it is important to learn, not because it is absolutely, perfectly correct, but because it is standard--1627\n\nso that you can talk to other people, and talk about inverses, and they will understand1632\n\nwhat you are talking about, because they are doing the same method that you are doing.1635\n\nIf you do something different, they might get confused.1639\n\nIf they are really clever, or they really understand what is going on, they will think, \"Oh, yes, that makes perfect sense.\"1641\n\nBut we want to go with the standard method, so that other people will understand what we are doing.1645\n\nAnd if we are taking a course at the same time as we are watching this course,1649\n\nthe teacher will think that is correct, as opposed to being confused by what you are doing and marking your grade down.1653\n\nBut the important idea here, the really important idea inside of this thing, that is confusing, is the reversal.1657\n\nThat is what the moment is all about--that moment between #2 and #3--the #3 step where we reverse, and we create this new y.1663\n\nWe reverse the places; instead of solving for an output, we are solving for input.1671\n\nWe are reversing the places, so we can do this directly; I did that with f(x), where I did f(x) = y;1678\n\nand I solved directly for if we know what f(x) is.1685\n\nI'm sorry, f(x) equals stuff involving x; I solved it directly for f(x)...1687\n\nWe had f(x) = x3 + 1, and we figured out that it also is the same thing that the cube root of f(x) - 1 is equal to x.1694\n\nWe figured that out; so there is this direct way of being able to do this.1706\n\nWe can do this directly; but lots of students find this difficult or confusing, so we have this method of swapping x and y.1709\n\nAnd also, it has just become the standard way to do things; so it is good to practice this way, even though it is not absolutely perfect.1716\n\nIt is not a perfect method, but it does the job.1723\n\nAs long as you are careful and you pay attention to what you are doing--you closely follow its steps--1725\n\nyou will be able to get the answer, and you will be able to find the inverse function.1729\n\nTaking inverses can be difficult; it seemed a little bit confusing from what I have been saying so far.1733\n\nAnd it is an easy one to make a mistake on; this means it is really important to check your work.1737\n\nYou really want to make sure that you check your work on this.1743\n\nHow do you do this? Well, remember: by definition, f-1(f(x)) is equal to x.1746\n\nThat means, if we know what f-1 is (we have figured out its formula), and we know what f(x) is1752\n\n(we were probably told f(x), we can just compose them.1756\n\nWe know how to compose them from our lesson Composite Functions.1760\n\nIf you didn't check out Composite Functions, you will have to watch that before you are able to compose them and do this check.1763\n\nBut if it is really the inverse, you will get x; if you compose f-1 with f(x), it has to come out to be x,1768\n\nbecause that is the definition of how we are creating this stuff, right from the beginning.1775\n\nFurthermore, we also know that f-1, inverse, was just f;1779\n\nso it also must be the case that f acting on f-1(x) will give us x, as well.1785\n\nYou can compose them in either order when you are doing a check; and you will end up being able to get it correct.1790\n\nLet's see a quick example: for example, if f(x) = x3 + 1, and f-1(x) = 3√(x - 1)1796\n\n(the ones we have been working with), how do we check this?1803\n\nWell, let's start with f-1(f(x)); we compose this: we plug in f(x) = x3 + 1.1805\n\nSo, f-1 acting on x3 + 1...now, remember, we are going to plug that into f-1(x).1814\n\nBut it is f-1(input); whatever is in the box just goes to the box over here.1821\n\nSo, it is going to be that f-1 will become cube root...where does the box go?1825\n\nx3 + 1...that is our box...minus 1; so the cube root of x3 + 1 - 1...1832\n\n+ 1 - 1 cancels; the cube root of x3 equals x; great--that checks out.1841\n\nWhat about if we did it the other way--if we did it as f(f-1(x))?1847\n\nHopefully, this will work out, as well (and it will).1852\n\nSo, what is f-1(x)? f-1(x) is the cube root of x - 1, so f(3√(x - 1))...1854\n\nwhat is going to happen over here?--we know that you plug in the box; you plug in the box.1864\n\nSo, f(3√(x - 1))...we are going to take that, and we are going to plug it in right here.1869\n\nIt is going to be 3√(x - 1), the quantity cubed, because it has to go in as the box; plus 1--finish out that function.1875\n\nThe cube root, cubed...those are going to cancel each other; we will get x - 1 + 1, which is just equal to x; and it checks out.1885\n\nSo, we can check it as f-1(f(x)) or f(f-1(x)); sometimes it might be easier for us to do it one way or the other.1895\n\nWe could also do both ways, if you want to check and be absolutely, doubly sure that we really got our work correct.1902\n\nAll right, let's move on to some examples.1907\n\nUsing these graphs for assistance, which of the following functions are one-to-one?1909\n\nThe first one is f(x) = 1/x; we do the horizontal line test--it is going to pass any high horizontal lines.1914\n\nWhat about as we get lower? Well, we know that 1/x continues to move--it never freezes and becomes constant.1921\n\nDoes it ever cross this x-axis, though? No, it doesn't.1928\n\nWe haven't formally talked about asymptotes yet; we will talk about asymptotes in a later lesson.1932\n\nBut 1/x...as we go positive (f of a positive), 1 over a positive is going to also have to be positive.1936\n\nSo, it never crosses the x-axis; the same thing goes with the negatives--f of a negative is going to be a negative.1945\n\nSo, when it goes to the left, it never manages to cross this x-axis; as it goes to the right, it never manages to cross this x-axis.1951\n\nAnd it keeps changing; so the two things never cross over each other.1958\n\nSo, yes, this is one-to-one.1961\n\nWhat about the blue one, g(x) = x3 - 2x2 - x + 1?1968\n\nIt is easy to say it fails: we cross lots of places in the middle here, and it is able to have multiple points at the same time.1973\n\nSo, any one of these hits here and here and here; there are three points that all give the same output of 0; so it fails the horizontal line test.1982\n\nIt is not one-to-one.1992\n\nFinally, (2x - 1) and (x2 + 1); 2x - 1 is just a line that is going to keep going on this way forever and ever and ever.1998\n\n2x - 1, when x is less than or equal to 1...this is from piecewise functions; if you haven't checked out piecewise functions, this might be a little confusing.2007\n\nBut hopefully, you have watched that lesson already.2013\n\n2x - 1 is x ≤ -1; it is just going to keep going on down and down and down, to the left and left and left.2015\n\nAnd x2 + 1 is the right side of the parabola; if we plug in higher and higher numbers, it just keeps curving up and up and up to the right.2020\n\nSo, that means that we are never going to cross; the parabola is never going to double back and manage to touch itself again.2027\n\nThe parabola might eventually do this, but that part isn't on it.2033\n\nAnd the line is never going to be able to go down to have itself crossed horizontally.2037\n\nSo, if we do any horizontal line crossing on this, it is never going to hit twice; so it is one-to-one.2042\n\nOne thing I would like to make a special comment on: notice that right here there is an empty space.2055\n\nThere is this gap where it jumps; is that a problem for a horizontal line test?2060\n\nNo, it is not a problem at all, because the horizontal line test is allowed to hit no points, as well.2064\n\nIt is allowed to hit one point or zero points; in this case, if it goes through that gap, it hits no points; but that is OK.2070\n\nWe are only worried about having multiple inputs for the same output.2076\n\nIt is OK if there are no inputs to make an output; the important thing2080\n\nis that there are no double sets of inputs that all make the same output.2082\n\nLike, in the blue one, where we had multiple different places where we could plug in some number--2087\n\nplug in different numbers, but they would all produce zeroes.2092\n\nAll right, let's actually find an f-1: f(x) = -3x/(x + 3).2096\n\nThey told us, right from the beginning, that it is one-to-one; so we can jump right to figuring it out: what is f-1(x)?2102\n\nAnd then, after it, we need to check our answer.2108\n\nOK, so what is f-1(x)? Remember all of our steps, one by one.2111\n\nf(x) = -3x/(x + 3): they told us, right from the beginning, that it is one-to-one, so we are already checked out.2115\n\nWe have already checked out the first one.2122\n\nThe next step: we swap y for f(x): y = -3x/(x + 3).2124\n\nNow, that is not the important part of when we reverse, though; we reverse into inverse world.2131\n\nSo, here is when we go into inverse world; we reverse the place of x and y.2136\n\nSo now, it is x where y was, and it is -3y/(y + 3).2144\n\nMultiply both sides by y; we get x times (y + 3) equals -3y; let's distribute this out: xy + 3x...let's also move the 3y over, so + 3y = 0.2154\n\nOK, at this point, we will pull out the y's from these two things; we will move them together, so we can see it a little bit easier at first.2170\n\nxy + 3y + 3x = 0; let's subtract that 3x to move it over; -3x, -3x here.2176\n\nSo, then we will pull out the y's to the right; so we have x + 3, times quantity y, equals -3x.2186\n\nFinally, we divide by that x + 3, and we get y = -3x/(x + 3).2195\n\nAnd now, finally, we can plug in f-1 for this y; so we plug it in, and we get f-1(x) = -3x/(x + 3).2202\n\nGreat; now, let's check and make sure that we got this right.2216\n\nWe check this in red; here is our check--let's check it by plugging f into f-1.2219\n\nSo, we want this to come out to be x; it should be x, if we got everything right.2232\n\nSo, f-1(f(x)); what is f(x)? f(x) is this; and here is something funny to notice.2239\n\nNotice -3x/(x + 3); amazingly, it just so happens that for -3x/(x + 3), f(x) and f-1(x) are the exact same thing--kind of impressive.2245\n\nWe plug this in; we have f-1(f(x)); f(x) is -3x/(x + 3); now, over here, we plug it in; what is in the box?2260\n\nThe box shows up here; the box shows up here; it shows up twice, so it is f-1 on -3x/(x + 3).2274\n\nIt is going to be -3...what is in the box?...-3x/(x + 3), over (-3x/(x + 3)) + 3.2281\n\nGreat; so the first thing that is going to be confusing is that we have this x + 3, and we have this x + 3 here.2301\n\nSo, let's take that out by multiplying the whole thing by (x + 3)/(x + 3).2307\n\nWe can get away with that, because it is just the same thing as 1: (x + 3)/(x + 3) is just 1.2312\n\nSo, (x + 3)/(x + 3)...multiply that here; the (x + 3) will cancel out here and cancel out here.2317\n\nBut remember, it also has to distribute to the other part, because they are not connected through multiplication on that; they are connected through addition.2324\n\nSo, we have -3, -3x, over -3x plus 3 times x plus 3.2330\n\nThese two negatives cancel out; so we have 3 times 3x on the top, -3x plus 3(x + 3)...so we have 9x on the top,2341\n\ndivided by -3x plus 3x plus 9; -3x plus 3x...they cancel each other out; we have 9x/9.2350\n\n9 over 9...those cancel out, and we have just x.2362\n\nSo, that checks out--great, we have the answer.2367\n\nAll right, the next one: we have, this time, a piecewise function.2370\n\nThis is a little confusing: we didn't talk about this formula, but we will see how to do it.2375\n\nf(x) = -x + 1 when x < 0, and -√x when x ≥ 0.2378\n\nThis is confusing; we don't know what to do about the different pieces of the piecewise function.2387\n\nWe don't know what to do about these two different categories: we have x < 0 and x ≥ 0.2393\n\nWe didn't learn that when we learned how to do inverses; but we could still figure out these two.2397\n\nWe could figure out what is the inverse of -x + 1 and what is the inverse of -√x.2402\n\nWe were told, explicitly, that this is one-to-one; so we can go ahead and do this, and then we will think about it.2407\n\nFirst, we will do inverses on these two rules; and then we will figure out how they fit together--what are the categories for these two rules?2413\n\nSo, first, -x + 1; we will have y = -x + 1; we swap them, so we now get into our inverse world.2422\n\nSwap their locations; we interchange them, and we are now at negative...sorry, not -x; the negative does not swap.2439\n\nWe are at x = -y + 1; we move the y over and move the x over; we get -x + 1; so y = -x + 1,2446\n\nwhich is going to give us f-1 for at least the first rule here.2458\n\nNow, what about the other one?--let's do that, as well.2464\n\nSo, y = -√x; we go into inverse mode; we reverse their locations; and we are now at x = -√y.2467\n\nSo, how do we solve for y? Well, we move this negative over: -x = √y.2482\n\nSquare both sides; we get (-x)2 = y; and then (-x)2...the negatives will cancel out, so we get just x2 = y.2487\n\nAnd so, this is the inverse rule for this part.2498\n\nNow, here is the part where we start thinking: we know that f-1 is going to break into a piecewise function using these two different things.2502\n\ny = -x + 1...so it will be -x + 1 for the first rule, and then x2 for the second rule.2513\n\nBut the question is that we don't know what the categories are.2520\n\nHow do we figure out what the categories are?2524\n\nWell, remember: if f goes from its domain to its range, let's call that a to b, then f-1 does the reverse of that.2526\n\nf-1 goes from b to a; it does the reverse.2543\n\nWhat that means is that the domain...the thing that determined which rule we used...we need to do the range to determine which way to get back.2548\n\nThe range on these two rules...now we are back to using f, so range on f...for -x + 1:2557\n\nwell, -x + 1 was x < 0; that was the category, so it has to be within those.2571\n\nSo, what can it go to? Well, if we plug in a really big negative number, like, say, -100, we will get -(-100) + 1; so we get 101.2577\n\nSo, as long as we keep plugging in more and more negative numbers, we get bigger and bigger numbers.2584\n\nWe are able to get all the way out to positive infinity, as we are really far in negative numbers.2587\n\nWhat is the lowest that we can get to? Well, we could get really close to 1, as we plug in -0.00000001.2592\n\nWe are really close to being to 1, so we can get right up to 1; but we can't actually touch it.2601\n\nWe have to exclude it, so we use parentheses.2605\n\nSo, the range for the first rule is this: -x + 1 becomes this.2607\n\nSo, I will put a red dot on that, because that matches to this rule here.2614\n\nNow, what about the range for the other rule?2619\n\nThe range on this rule is -√x; it has x ≥ 0 as its domain; what are the numbers we can get out of this?2622\n\nWhat is the largest number we can get out of it?2630\n\nThe largest number we can get out of it is actually 0; why?--because, when we plug in any reasonably large positive number,2632\n\nlike, say, 100, then -√100 is -10; so as we get bigger and bigger positive numbers that we plug in,2639\n\nwe actually get more and more negative.2649\n\nSo, we can actually go to any negative number we want; we can go all the way down to negative infinity.2651\n\nCan we actually reach 0? Yes, we actually can reach 0, because it is greater than or equal to.2655\n\nSo, if we plug in x = 0, we get -√0, which is just 0; and we put a bracket to indicate that we are actually allowed to do it.2660\n\nThis one is the range for -√x, that rule; it is going to get a green dot on it, because it matches to the green rule.2668\n\nThat means that -x + 1 is allowed to take in...what values? It is allowed to take in the range values.2677\n\nIt is allowed to do a reverse on anything that shows up in the range (1,∞).2684\n\nAlso notice: these two ranges, (1,∞) and (-∞,0], don't have any intersection.2689\n\nThey don't overlap at all, so we don't have any worries about pulling from one versus pulling from the other.2695\n\nThey will never get in each other's way.2700\n\nSo, for this one, -x + 1, if it is going to be allowed to go from 1 to infinity, then that means we can plug in anything into f-1,2703\n\nwhere x is greater than 1, which is to say input; it is not the same x that was up here.2711\n\nIt is now just saying \"placeholder--anything that we are plugging in.\"2719\n\nWhat about x2? Well, that was the green dot--that was allowed to go from negative infinity up to 0.2722\n\nSo, it is allowed to have x ≤ 0; it is allowed to go all the way up to negative infinity, but it can only just get to touching 0.2727\n\nIt is allowed to actually have 0, though; x > 1 is not actually allowed to touch 1, but it is able to get as close as it wants.2735\n\nAnd there is our piecewise inverse function.2741\n\nIt is a little bit difficult, but if you think about it, you do each of the inverses, and then you think about2747\n\n\"How do I get the domain for the inverse? I get it from the domain of f becoming the range of our inverse,2750\n\nand the range of our f becoming the domain of our inverse.\"2760\n\nSo, what the original function was able to output to is what the inverse is allowed to take in.2766\n\nAnd that is how we figured out these rules, these categories--what the categories were for these two different transformations.2773\n\nAll right, the final example: f and g are one-to-one functions; now, prove that f composed with g, inverse, is equal to g inverse composed with f inverse.2779\n\nThis might be a little daunting at first; these are weird symbols; we are not used to using these sorts of things.2791\n\nSo, if that is the case, let's remind ourselves: from composition, f composed with g, acting on x, is equal to f(g(x)).2795\n\nNow, I said before: it makes things always, always, always easier to see it in that format.2806\n\nWhat we want to show is that g-1 composed with f-1 (which would be g-1(f-1(x)))...2811\n\nwe want to show that this one here is an inverse to that one over there.2823\n\nThat is what we are trying to prove, that f composed with g inverse...2829\n\nWe know, by the definition of how this symbol works, by how inverses work...2833\n\nf composed with g-1, acting on f composed with g, on x, is going to just leave us as if we had done nothing,2837\n\nbecause we are putting an inverse on something.2845\n\nSo, we want to show that this means the exact same thing as this right here.2848\n\nSo, let's just try it out: we will set it up like this: f composed with g-1, acting on f composed with g, acting on x.2854\n\nOK, so what does that become? Well, we know that f composed with g, acting on x, is the same thing as f(g(x)).2871\n\nAll right, what is f composed with g-1? Well, we know (from what we did over here)2881\n\nthat we can bring that into g-1 acting on f-1, acting on whatever is going into it.2887\n\nWhat is going into it here is this whole thing; so, it is going to be g-1, acting on f-1, acting on f, acting on g, acting on x.2891\n\nAnd then, we close up all of those parentheses.2908\n\nThat is a little bit confusing; but we are seeing inverses right next to functions: f-1 acting on f, acting on whatever is in there.2912\n\nIt just cancels out and gets us right back to what we originally had in there.2922\n\nSo, f-1 acting on f...that cancels out, and we get g-1, acting on whatever was in there, which was g(x).2926\n\nSo, g-1 acting on g(x)...the exact same thing: we get down to x; so we have proved it.2933\n\ng-1 composed with f-1 is how we create f composed with g, inverse.2940\n\nGreat; we have proved it.2948\n\nAll right, I hope you have a much better idea of how inverses work at this point.2949\n\nThey can be a little bit confusing, but you have that method to be able to guide you through it.2952\n\nJust follow it really carefully, step-by-step.2956\n\nThe danger is if you break from those steps and do something else; that is where you can make mistakes.2958\n\nIf you really understand what is going on, you don't even have to use that method.2963\n\nBut it really is the standard method, so it is a good idea to stick with it, just because it is what a lot of other people are used to using.2966\n\nAnd you can find it in a lot of textbooks.2972\n\nAll right, we will see you at Educator.com later--goodbye!2974" ]

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[ "# Department of Mathematics and Statistics\n\n## Past Exams (Archive) for Practice\n\nLink to an archive for past M.A., M.S., and Ph.D. exams\n\n## M.A. Comprehensive Examination\n\nThe M.A. Comprehensive Examination is a written examination consisting of three parts. Each part is based on topics normally treated in the courses indicated.\n\n1. Real and Complex Analysis (5820, 5830, 5880)\n2. Abstract and Linear Algebra (5330, 5340, 5300, 5310)\n3. One of:\n1. Topology (5450, 5460)\n2. Differential Equations (5800, 5810)\n3. Probability and Statistics (5680, 5690)\n\nParts 1 and 2 are each three hours in length and Part 3 is a two hour exam. The parts will normally be written on three consecutive Saturdays in the Fall or in the Spring.\n\nThe following rules apply to this examination:\n\n1. If a student's score on any part of the written exam is only marginally below the passing level, the examining committee for that part may defer its final decision pending the outcome of a one-hour follow-up oral examination of the student, to be administered within one month of the date of the written exam. This option will not be exercised by the committee if the margin of failure on the entire written exam is deemed to be decisive.\n2. A student who passes two of the three parts of the exam is credited with those parts and is required to retake only the part failed; a student who fails two or more parts will receive no credit for any part.\n3. A student is allowed two attempts on the full three-part examination. A third (and final) attempt on one part of the exam will be permitted if credit has previously been earned for the other two parts.\n4. A student's first attempt at the examination must occur at one of the two regularly scheduled periods (Fall or Spring). Following an unsuccessful attempt made during one of these periods, a student who is eligible under rule (3) may elect to reattempt the exam (or part thereof as governed by rule (2)) within two months, at a time agreeable to both the student and the exam committee. Otherwise, the retake must take place at the next regularly scheduled examination.\n5. A student who intends the M.A. degree to be her or his terminal degree in this department may elect to satisfy the degree requirements by submitting and defending a thesis instead of writing the M.S. Comprehensive Examination, but a student may not switch from the exam option to the thesis option after an unsuccessful attempt at the examination.\n\n### Syllabus for M.A. Exam\n\n1. Real and Complex Analysis\n\nReal Analysis:\n\nThe real number system\n\nElementary metric space theory\n\nSequences and series\n\nDifferential calculus\n\nIntegral calculus (the Riemann integral)\n\nSequences and sums of functions (Weierstrass Approximation theorem, uniform convergence, Arzela-Ascoli theorem)\n\nThe Lebesgue integral (on the real line)\n\nComplex Analysis:\n\nComplex functions: limits, continuity, properties of elementary functions including branches\n\nDifferentiability: derivatives Cauchy-Riemann equations, analyticity, harmonic functions\n\nIntegration: Cauchy theorems, Residue theorem, Morera's theorem, Maximum-Modulus theorem\n\nSeries: Taylor's theorem, Laurent series expansions\n\nMappings: elementary functions, properties of conformal maps\n\nFurther properties of analytic functions: singular points, zeros, analytic continuation, residues, evaluation of real and complex integrals using residues\n\nSome references:\nReal Analysis:\n\nRudin, Principals of Mathematical Analysis, McGraw-Hill\n\nGoldberg, Methods of Real Analysis, Blaisdell\n\nComplex Analysis:\n\nChurchill, Complex Variables and Applications, McGraw-Hill\n\nSaff & Smider, Fundamentals of Complex Analysis, Prentice-Hall\n\n2. Abstract and Linear Algebra\n\nAbstract Algebra:\n\nBasic concepts of groups and rings including Lagrange's theorem, normal subgroups, factor groups, homomorphisms and isomorphisms, permutations, the field of quotients of an integral domain, polynomial rings, factorization in integral domains.\n\nLinear Algebra:\n\nVector spaces, linear transformations and matrices, determinants, canonical forms.\n\nSome references:\nAbstract Algebra:\n\nHerstein, Topics in Algebra, Wiley\n\nFraleigh, A First Course in Abstract Algebra, Addison-Wesley\n\nLinear Algebra:\n\nHoffman and Kunze, Linear Algebra, Prentice-Hall\n\nCurtis, Linear Algebra, Springer-Verlag\n\nFriedberg, Insel, Spence, Linear Algebra, Prentice-Hall\n\n3. One of:\n\n1. Topology\n\nAxiomatization of topological spaces\n\nDifferent ways of introducing a topological structure.\n\nContinuous maps\n\nCharacterizations of continuity, initial sources and final sinks, discrete and indiscrete spaces.\n\nFundamental constructions\n\nBasis for open (or closed) sets, subbase, subspaces, products, quotients, sums. Lattice of topologies on a set.\n\nConvergence\n\nSequences; filters and ultrafilters.\n\nCountability\n\nFirst and second axiom. Lindelof spaces.\n\nSeparation\n\nHausdorff, regular, completely regular, normal spaces; $T_i$-spaces, $i=0,1,2,3\\frac{1}{2},4$. Urysohn's Lemma.\n\nCompactness\n\nIn Euclidean spaces. Tychonoff Theorem, Stone-Cech compactification. Local compactness in $T$-spaces, Alexandroff compactification.\n\nConnectedness\n\nComponents, local connectedness, path connectedness.\n\nMetric spaces\n\nCauchy sequences, completeness. Uniform continuity. Baire's theorem.\n\nMetrization theorems and paracompactness\n\nThe classical theorems of Urysohn and of Nagata-Smirnov-Bing. Stone's theorem.\n\nFunction spaces\n\nPointwise and compact convergence. The compact-open topology, Ascoli's theorem. Uniform convergence for metric spaces.\n\nApproximation\n\nStone-Weierstrass theorem.\n\nSome References\n\nJ. Dugundji, Topology, Allyn and Bacon, Inc., 1966\n\nR. Engelking, General Topology, PWN-Polish Scientific Publishers, 1977\n\nJames R. Munkres, Topology, Prentice-Hall, Inc., 1975\n\nStephen Willard, General Topology, Addison-Wesley, 1968\n\n2. Differential Equations\n\nOrdinary Differential Equations:\n\nGeneralities and soluble classes of first order ODE's.\n\nSecond order linear ODE's (General theory and explicit solutions in case of constant coefficients).\n\nSystems of linear ODE's with constant coefficients.\n\nSeries solutions of second order linear ODE's with analytic coefficients near an ordinary point and near a regular singularity.\n\nNon-linear autonomous ODE's of second order. Phase plane analysis; in particular, equilibrium solutions, their classifications and their stability.\n\nExistence and uniqueness theorems. Nature of dependence on initial conditions.\n\nPartial Differential Equations:\n\nFirst order linear (and quasi-linear) equations.\n\nClassification of second order equations and their canonical forms.\n\nMethod of separation of variables to solve boundary value problems, in particular the heat equation, the wave equation and the Laplace equation. In this connection: the Convergence theorem for Fourier series. Also Fourier integrals.\n\nWave equation (or other hyperbolic equations). The Initial (Boundary) value problem. D'Alembert's Principle. (Huygens' principle.)\n\nHeat equation (or other parabolic equations). The Initial (Boundary) value problem. Existence and uniqueness theorem in one-dimensional heat equation.\n\nLaplace equation (or other elliptic equation). Basic properties of harmonic functions, in particular, the Maximum Principle. Boundary value problems. The Dirichlet problem, Green's function and Poisson's formula.\n\nSome References:\nODE's:\n\nBoyce and DePrima, Elementary Differential Equations, Wiley.\n\nSimmons, Ordinary Differential Equations, McGraw-Hill.\n\nBirkhoff and Rota, Ordinary Differential Equations.\n\nPDE's:\n\nZachmano glu and Thoe, Introduction to Partial Differential Equations, William & Willkins Comp., Baltimore.\n\nColton, Partial Differential Equations: An Introduction, Random House Birkhaeuser Math Series.\n\nBerg and McGregor, Elementary Partial Differential Equations, Holden-Day, San Francisco.\n\n3. Probability and Statistics\n\nThis will be based upon a knowledge of the material in Mathematics 5680 and 5690.\n\n## M.S. (Applied Math) Comprehensive Exam\n\nThe comprehensive examination for the M.S. in Applied Mathematics will be a written examination consisting of two parts. Each part is based on topics normally treated in the courses indicated.\n\n1. Real and Complex Analysis (5820, 5830, 5880)\n\nThe examination will test the student's knowledge of elementary real and complex analysis.\n\n2. Differential Equations (6500, 6510)\n\nThe examination will test students' knowledge of both ordinary and partial differential equations. The exam will be based upon the more computational aspects of the material in the above courses.\n\nThe two parts of the exam will each be three hours in length and will normally be taken on consecutive Saturdays in the Fall or in the Spring.\n\nThe following rules apply to this examination:\n\n1. The entire (two part) examination can be taken only twice.\n2. If a student fails only one part of the examination, only that part need be retaken. A third (and final) attempt on one part only will be allowed.\n3. If a student needs to retake one or both parts, this may be done either within two months or at the next regularly scheduled examination session.\n4. If a student's score on one or both parts of the examination is only marginally below the passing level, in the examining committee's judgment, the committee may defer its final decision pending the outcome of a one-hour oral examination on that part. Such oral examination(s) are to be given within one month of the date of the written examination and at the mutual convenience of the student and the examining committee.\n\n### Syllabus for M.S. (Applied Math) Exam\n\n1. Real and Complex Analysis\n\nReal Analysis:\n\nCompleteness of $\\mathbb{R}^n$, Sequences and Series.\n\nCompactness of $\\mathbb{R}^n$, Connectedness.\n\nUniform Convergence.\n\nRiemann integral, existence of the integral, uniform convergence and the integral.\n\nImproper integrals.\n\nComplex Analysis:\n\nAnalytic functions and the Cauchy-Riemann Equations.\n\nElementary conformal mappings.\n\nCauchy-Goursat theorem, Cauchy integral formula, residue calculus.\n\nTaylor and Laurent series.\n\n2. Differential Equations\n\nOrdinary Differential Equations:\n\nLinear systems, calculation of fundamental matrices.\n\nVariation of parameters for systems.\n\nBoundary value problems, eigenvalue problems, Sturm-Liouville theory.\n\nPlane autonomous systems, Liapunov stability.\n\nPartial Differential Equations:\n\nMethod of characteristic for first order equations.\n\nBoundary value problems for the Laplace, Wave and Heat equations, separation of variables.\n\nGreens functions for the Laplace, Wave and Heat equations, Poisson's kernel, Dirichlet problem, method of images.\n\n## M.S. (Statistics) Comprehensive Exam\n\nThe purpose of this examination is to ensure that the M.S. graduate has acquired statistical knowledge and skills adequate for a practicing statistician in engineering, science, management, pharmacy, medicine, and other areas.\n\nThe examination will be a written examination, including a take-home project. It will consist of two parts:\n\n1. Probability and Statistical Theory.\n\nThis part will normally be based upon a detailed knowledge of the material in Mathematics 5680, 5690 and 6680. In addition, on a more general level, and with some choice given between questions, material from Mathematics 5660, 5700, 6630, 6690, and 6710 will also be tested. If material from any other courses will be tested, you will be notified of that fact in writing at least one month in advance of the examination.\n\n2. Applied Statistics.\n\nThis part will normally be based upon a detailed knowledge of the material in Mathematics 5620, 6630, 6640, and 6690. In addition, on a more general level, and with some choice given between questions, material from Mathematics 5610, 5640, 5660, 5700, 6610, and 6710 will also be tested. If material from any other courses will be tested, you will be notified of that fact in writing at least one month in advance of the examination.\n\nThe two parts of the exam will each be three hours in length and will normally be given on consecutive Saturdays in the Fall and/or in the Spring.\n\nThe take-home project will be given at least one week before the Applied Statistics portion of the exam, and will be due at the time that the Applied Statistics portion is given. In addition to material from the courses listed above in the description for the Applied Statistics exam, material from Mathematics 5670 may also be covered on the take-home portion of the exam.\n\nThe rules applying to this examination are the same as rules (1) through (4) of the M.S. Applied Mathematics exam as delineated in Appendix A2 of this Guide with one additional rule:\n\n1. Books and notes (as approved by the examining committee) and calculators may be used during this examination. Your course notes and course textbooks will be approved. Please obtain approval in advance for the use of other written materials during the examination.\n\nAny questions regarding the content of this examination should be directed to Professor Donald White, Head of the Statistics Group.\n\n## Ph.D. Examinations\n\nThe following regulations apply to students entering the Ph.D. program in or after September 1994.\n\n## Ph.D. Qualifying Exam\n\nThe Ph.D. qualifying examination is a preliminary examination for the Ph.D. program. It consists of two three-hour parts, each on a separate topic. The two topics on the examination are to be chosen from among the following general areas: algebra, real analysis,topology, differential equations (for students intending to write a dissertation in an applied area) or statistics (for students intending to write a dissertation in the area of statistics). The content of each part will be based on the material presented in the respective required first year-long sequences in algebra (6300, 6310), real analysis (6800, 6810), topology (6400, 6410), differential equations (6500, 6510). See below for more details on the exams syllabus. (Students intending to take the Statistics exam should consult with the statistics program adviser.)\n\nThe following rules apply:\n\n1. The Ph.D. preliminary exam will generally be offered twice a year, in the Fall and Spring semesters, and will be administered over a period of two weeks.\n2. Each part of the examination will be graded on a pass/fail basis. Students who fail are required to pass the failed exams by the end of their second year.\n3. In order to continue in the program both exams must be passed by the end of the student's second year. There will be two opportunities to pass the exam. However, should the student elect to take the exams at the end of their first year this will not count as one of the two opportunities.\n\n### Syllabus for Ph.D. Qualifying Exams\n\n1. Differential Equations\n\nOrdinary Differential Equations\nGeneral theory for first order equations\n\nExistence and uniqueness of solutions\n\nContinuous dependence on parameter and initial conditions\n\nInfinite series solutions and method of majorants\n\nLinear Systems\n\nGeneral theory of linear systems\n\nLinear periodic systems\n\nSecond Order linear equations\n\nBoundary value problems, Green's functions, Sturm-Liouville theory\n\nComparison theorems\n\nQualitative theory of ordinary differential equations in the plane\n\nLimit cycles, Poincaré-Bendixson Theorem\n\nStability, Liapunov's method\n\nPartial Differential Equations\n\nCauchy-Kowalevski Theorem\n\nHyperbolic systems\nExistence and uniqueness of solutions\n\nMethod of characteristics\n\nEnergy estimates\n\nFourier transforms, Green's functions\n\nSecond order elliptic equations\n\nApplication of the Maximum Principle\n\nElementary Sobolev space theory:\n\n$H^1(\\mathbb{R}^n)$ and $H^2(\\mathbb{R}^n)$\n\nExistence and uniqueness of solutions\n\nDirichelet Principle\n\nPerron's Method\n\nEigenvalues of the Laplace operator\n\nHeat equation\nExistence and uniqueness of solutions\n\nFundamental solutions\n\nEnergy estimates\n\nMaximum principle\n\n2. Algebra\n\nBackground material\nLinear Algebra\nVector spaces and linear transformations\n\nDeterminants\n\nCanonical forms\n\nGeneral\n\nHomomorphism theorems\n\nJordan-Hölder Theorem\n\nFittings's Lemma\n\nKrull-Schmidt Theorem\n\nGroups\nGroup actions\n\nFundamental counting theorem\n\nPermutation groups\n\nTransitivity and primitivity\n\nSimplicity of $A_n$ for $n \\ge 5$\n\nClass equation\n\nFrattini Argument\n\nSylow's theorems and $p$-groups\n\nGroup constructions\n\nDirect products\n\nSemidirect products\n\nStructure of finitely generated Abelian groups\n\nDerived series and central series\n\nSolvable groups and nilpotent groups\n\nFields\n\nSimple extensions (algebraic and transcendental)\n\nGalois Group of an extension\n\nAlgebraic closure\n\nSeparable and inseparable extensions\n\nNormal extensions\n\nFundamental theorem of Galois Theory\n\nFinite fields\n\nRings\n\nProjective and Injective modules\n\nCommutative Rings\n\nFactorization\n\nLocalization with respect to multiplicatively closed sets\n\nSimple modules and primitive rings\n\nJacobson Density Theorem\n\nArtinian Rings\n\nWedderburn-Artin Theorems\n\n3. Real Analysis\n\nBackground material\n\nInfinite series and products\n\nPower series\n\nElementary theory of the derivative\n\nMonotone function and functions of bounded variation\n\nMetric spaces\nTopology\n\nCompleteness\n\nConnectedness\n\nCompactness and totally boundedness\n\nUniform convergence\n\nBaire Category Theorem\n\nAscoli-Arzela Theorem\n\nStone-Weierstrass Theorem\n\nIntegration\nMeasure Theory on the real line\n\nOuter measure\n\nRelation between measure and content\n\nMeasurable function, Egoroff and Lusin Theorems\n\nLebesgue integral on the real line\n\nMonotone and bounded convergence theorems, Fatou's Lemma\n\nRiemann integral and relation to Lebesgue integral\n\nImproper Riemann and Lebesgue integrals\n\nSome References:\n\nT. Apostol, Mathematical Analysis\n\nR. Goldberg, Methods of Real Analysis\n\nH.L. Royden, Real Analysis\n\nW. Rudin, Principles of Mathematical Analysis\n\n4. Topology\n\nBackground material\n\nCardinality and countability\n\nAxiom of Choice, Well Ordering, Maximum Principle\n\nGeneral topology\n\nOpen set, closed set, closure, interior, and neighborhood systems\n\nConvergence of filters and nets\n\nSeparation and countability axioms\n\nContinuous functions\n\nConnectedness and path connectedness\n\nCompactness\n\nProduct spaces and Tychonov Theorem\n\nUrysohn Lemma and Tietze Extension Theorem\n\nLocal compactness and paracompactness\n\nQuotient spaces\n\nAlgebraic topology\n\nHomotopy of maps and homotopy equivalence\n\nFundamental group\n\nCovering spaces and classification\n\nSeifert-van Kampen Theorem\n\nSome references:\n\nW.S. Massey, A Basic Course in Algebraic Topology\n\nJ.R. Munkres, Topology, A First Course\n\nI.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry\n\nS. Willard, General Topology\n\n## Ph.D. Oral Examination\n\nA student shall take the Ph.D. oral examination upon successful completion of the Ph.D. qualifying examination, or at the request of a faculty member whom the student has asked to be his/her thesis supervisor and who has not yet accepted the student as an advisee. The oral examination will be administered by a committee of three faculty members. The exam will be concerned with the general area of specialization of the student. It is the student's responsibility to ask a faculty member to agree to serve as the chair of the examination committee. The examination will consist of two parts. Part I shall be a talk by the student on a topic at a level sufficient to demonstrate the student's ability to engage in mathematical research. The topic will be chosen in consultation with and approval of the committee chair. Part II shall be an examination of the student by the committee to ascertain the level of the student's understanding of the topic and related background material. Parts I and II will be administered at the same session.\n\nThe following rules apply:\n\n1. The oral exam will be graded on a pass/fail basis.\n2. The student must pass the oral examination within one year of passing the Ph.D. qualifying exam or by the end of the second year, which ever is later.\nLast Updated: 6/27/22" ]

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[ "## Oct 27 What is \"Delta\"?\n\nDelta is a term originally coined by MarketDelta back in 2002 when we created the Footprint Chart. We needed a way to describe order flow and nothing existed because it was new territory for everyone at the time.\n\n### Delta is the net difference between the buying and selling at each price (footprint delta), each bar (bar delta), or the the entire day (cumulative delta).\n\nDelta is calculated by subtracting the volume transacted at the bid price from the volume transacted at the ask price. MarketDelta considers trades that occur at the ask price to be trades initiated by aggressive buyers. Trades that occur at the bid price are considered to be initiated by aggressive sellers.\n\nThus a positive delta would reflect more aggressive buying as the result of motivated buyers trading more at the ask. A negative delta would reflect more aggressive selling as the result of motivated sellers trading more at the bid.\n\nCalculation: Ask Traded Volume – Bid Traded Volume = Delta.\n\nThe components required to calculate delta are: bid price, ask price, last trade price, last trade volume, time.\n\nExample: If the bid price for WTI Crude Oil is 63.50 and the ask price is 63.51 and 25 contracts trade at 63.51, then the delta would be counted as +25. If a second trade occurred but this time the trade occurs at the bid price of 63.50 for 10 contracts the delta for that trade would be -10 and the total bar delta would now be +15.\n\n### Footprint Delta\n\nThis is a close up view of delta because it looks at a single price or group of prices (if the price scale is compressed).\n\nThe value is positive or negative. A positive delta will have a green background and reflect \"positive\" order flow as the result of buyers being more aggressive at that price. A negative delta will have a red background and reflect \"negative\" order flow as the result of sellers being more aggressive at that price.\n\nThere is a high correlation between price direction and order flow, therefore having the ability to see the footprint when trading is a valuable tool for the astute trader.\n\n### Bar Delta\n\nThis is the the sum of delta for each bar. So no matter the time frame or interval of chart you are viewing, the software totals the delta for each bar and displays it at the bottom of the chart as a histogram or numerical value. Below are 2 different ways of displaying the bar delta.\n\nThe bar delta is positive or negative. A positive delta will have a green/blue background and reflect \"positive\" order flow as the result of buyers being more aggressive for a particular bar. A negative delta will have a red background and reflect \"negative\" order flow as the result of sellers being more aggressive for a particular bar.\n\nOne benefit of bar delta is it takes a step back from the the price by price footprint delta and allows you to focus on the order flow for the entire bar. A good way to use it is to compare the bar delta with the price direction of the bar. Is there a divergence? Another way is to look at a series of bars and look for patterns.\n\n### Volume Study\n\nThis shows the bar delta as a histogram.\n\n### Footprint Bar Statistics Study\n\nThis shows the bar delta as a numerical value in the row labeled \"Delta\".\n\n### Cumulative Delta\n\nThis provides the biggest picture of delta because it spans the entire session or day. So no matter the time frame or interval of chart you are viewing, the software totals the ALL the delta's and displays a running total. Below are 2 different ways of displaying the bar delta. However, it can also be displayed using the Footprint Bar Statistic study and adding the \"Delta Day\" row.\n\nCumulative Delta is called Delta Day in the software and is either positive or negative. A positive delta will have a green/blue background and reflect \"positive\" order flow for the session as the result of buyers being more aggressive for the current day. A negative delta will have a red background and reflect \"negative\" order flow as the result of sellers being more aggressive for the current day.\n\nOne strategy of applying cumulative delta is to use it to help determine which side to trade from. When cumulative delta is positive it reflects buyers have been more aggressive over the course of the session. When cumulative delta is negative it reflects sellers have been more aggressive over the course of the session. It is a very useful tool to confirm price direction.\n\n### Delta Bar Study\n\nThis shows the Delta Bar study set to accumulate. It is unique because it shows the range cumulative delta had for each bar and provides unique insight.\n\n### Volume Study\n\nThis shows the cumulative delta as a histogram.\n\n### Next Step\n\nYour next step in learning should be to watch the video below. It will introduce you to Footprint charts and explain how they are constructed and help you grasp a better idea of delta." ]

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[ "", null, "", null, "", null, "Trending ▼   ResFinder", null, "", null, "# ICSE Class IX Prelims 2022 : Mathematics (Saint Lawrence School (SLS), Tentoloi, Balaramprasad, Angul)\n\n3 pages, 34 questions, 2 questions with responses, 2 total responses,", null, "2", null, "0", null, "", null, "", null, "", null, "Anil Kumar Dash Saint Lawrence School (SLS), Tentoloi, Balaramprasad, Angul\n+Fave Message\n Home > 9938480846 >   F Also featured on: School Page icse9", null, "Formatting page ...\n\nClass 9 Saint Lawrence School, Tentoloi Half Yearly Examination 2021 Sub- Mathematics Date- 05.10.2021 Time 8am to 10am F.M - 80 -:General Instructions:1. Attempt all questions from section A and any four questions from section B. 2. All working including rough work must be clearly shown and must be done on the same sheet as the rest of the answer. 3. Omission of essential working will result in the loss of marks. 4. The intended marks for questions as parts of questions are given in brackets [ ] Section-A (40 marks) Q1). Q2). Q3). Q4). (a) Solve for x & y by substitution method. 3x - 7= & x + = 1 (b). The area of circular ring enclosed between two concentric is 286 cm2. Find the radii of the two circles given that their difference is 7 cm. (c). Factories the following. (i) (x2 - x) (4x2 - 4x - 5) - 6 (ii) a3 - a - 120 1 (a) If x=2+ , find the value of + x (b). Calculate the interest Rs 6000 in 3 years at 5% per annum compounded annually. (Without using formula) (c). In an isosceles ABC with AB=AC, the bisectors of B & C intersect each other at O. Show that : (i) OB = OC (ii) AO bisects A 1 1 1 2 2 (a) If ab + bc + ca = 0, Then find the value of :2 a bc b ac c ab (b). The ratio of incomes of two persons is 9:7 & the ratio of their expenditures is 4:3. If each of them manages to save Rs 2000 per month. Find their monthly incomes. (c). If O is any point in the interior of ABC prove that :- OA+OB+OC < AB+BC+CA 1 3 1 (a) If x2 + = 8 , Find x3 + 2 5 25 x 125 x 3 (b). The base of a triangular field is 3 times its height. If the cost of cultivating the field at the rate of Rs25 per 100m2 is Rs60,000. Find its base& the height. (c). ABCD is a parallogram.. E & F are the mid points of the sides AB & CD. Prove that AF & CE trisect the diagonal BD.", null, "", null, "Formatting page ...\n\nTop Contributors\nto this ResPaper", null, "Shashwat Shaw 9H 35(1)", null, "ResPaper Admins(1)", null, "Bob Bird(1)", null, "", null, "Formatting page ...", null, "", null, "", null, "" ]

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[ "Book I.  Propositions 33 and 34\n\nProblems\n\nBack to Propositions 33, 34.\n\n11.   a)  State the hypothesis of Proposition 33.\n\nTo see the answer, pass your mouse over the colored area.\nTo cover the answer again, click \"Refresh\" (\"Reload\").\nDo the problem yourself first!\n\nStraight lines join the extremities on the same side of two equal and parallel straight lines.\n\n12.  b)  State the conclusion.\n\nThose straight lines are themselves equal and parallel.\n\n12.  c)  Practice Proposition 33.\n\n12.   State the definition of a \"parallelogram.\"\n\nA parallelogram is a quadrilateral whose opposite sides are parallel.  (Definition 14)\n\n13.   a)  State the hypothesis of Proposition 34.\n\nA figure is a parallelogram.\n\n13.  b)  State the conclusion.\n\nThe opposite sides and angles are equal, and the diagonal bisects the area.\n\n13.  c)  Practice Proposition 34.\n\n14.   The straight line ADE is parallel to the straight line BC, and AB is\n14.   parallel to DC.", null, "Prove that AB is equal to DC, and angle EDC is equal to angle DAB.\n\nSince AD is parallel to BC, and AB is parallel to DC,\nABCD is a parallelogram.   (Definition 14)\nTherefore the opposite sides AB, DC are equal.   (I. 34)\nAlso, because AE meets the two parallel lines AB, DC,\nthe exterior angle EDC is equal to the opposite interior angle DAB.   (I. 29)\n\n15.   By the distance from a point to a line, we mean the length of the\n15.   perpendicular from the point to the line.\n\n15.   Prove that two parallel lines are everywhere the same distance apart.", null, "Let the straight lines AB, CD be parallel, and let E and G be any two points on AB;\nthen the distance from E to CD is equal to the distance from G from CD.\n\nDraw EF and GH perpendicular to CD;\nthen EF will be equal to GH.\n\nFor, the straight line AB meets the two straight lines\nEF, GH, and the exterior angle AEF is equal to opposite interior angle EGH,\nbecause they are right angles;\ntherefore EF and GH are parallel.   (I. 28)\nAnd by hypothesis, AB, CD are parallel.\nTherefore EFHG is a parallelogram,   (Def. 14)\nhence the opposite sides EF, GH are equal.   (I. 34)\nAnd E and G were any two points on AB.\nThis implies that the parallel lines AB and CD are everywhere the same distance apart.\n\nThis could also be proved by showing that EG, FH are equal, and\n15.  then citing I. 33.\n\n16.   Straight lines EH, BCG are parallel; EB and HC are straight lines;\n16.   BC is equal to FG; and EFGH is a parallelogram.", null, "Prove that EBCH is a parallelogram.\n\nSince EFGH is a parallelogram, then EH is equal to FG;   (I. 34)\nbut BC is equal to FG;   (Hypothesis)\ntherefore BC is equal to EH.   (Axiom 2)\nAnd since EH, BC are parallel,  (Hypothesis)\nand EB, HC join their extremities on the same side,\nthen EB, HC are parallel.   (I. 33)\nTherefore EBCH is a parallelogram.   (Definition 14)\n\n17.   Prove that a square is a certain kind of parallelogram.\n\n18.   A rhombus is a quadrilateral which is equilateral but not\n18.   right-angled.", null, "Prove that a rhombus is a certain kind of parallelogram.\n\n19.   A rectangle is a quadrilateral in which all the angles are right angles.", null, "Prove that a rectangle is a certain kind of parallelogram.\n\n10.   Prove:  Equal squares have equal sides.\n\n11.   ABCD is a square; ACFE is a square drawn on the diagonal AC;\n10.   and ED is a straight line.", null, "10.   a)   Prove that ED is equal to DC, and is in a straight line with DC;\n10.   a)   that is, EC is the diagonal of that square.\n\nThe diagonal AC bisects ABCD into two congruent and isosceles right triangles.   (I. 34)\nSince ABCD is a square, then angle BAD is a right angle;\ntherefore angle CAD is half a right angle.\nAnd since CAEF is a square, angle DAE is the other half of the right angle CAE;\ntherefore angle DAE is equal to angle CAD.", null, "Next, side EA is equal to side AC;\nand we have shown that angle EAD is equal to angle CAD.\nTherefore (S.A.S) the remaining side is equal to the remaining side:\nED is equal to DC;\nand the remaining angles are equal:\nangle EDA is equal to angle CDA.\nBut angle CDA is a right angle.\nTherefore angle EDA is also a right angle,\nand therefore ED is is a straight line with DC.   (I. 14)", null, "10.   b)   Prove that the square drawn on the diagonal AC is twice\n10.   a)   as large as the square on the side BC.\n\nIf we draw DF, then AF is also a diagonal of that square.\nAnd each diagonal bisects that square;\ntherefore AF and EC divide the square into four equal and isosceles right triangles.\nBut the square whose side is BC is made up of two of those triangles.\nTherefore the square drawn on the diagonal AC is twice the square on the side BC.\nThe square drawn on the diagonal of a square is twice\nthe square on the side.\n\nThis theorem asserts an essential knowledge of a square figure. Moreover, it led Pythagoras to realize that the diagonal and side are incommensurable. See Topic 11 of The Evolution of the Real Numbers.\n\nNext proposition" ]

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[ "# Electrical Engineering - Parallel Circuits - Discussion\n\n### Discussion :: Parallel Circuits - General Questions (Q.No.5)\n\n5.\n\nWhen a 1.6 k", null, "resistor and a 120", null, "resistor are connected in parallel, the total resistance is\n\n [A]. greater than 1.6 k", null, "[B]. greater than 120", null, "but less than 1.6 k", null, "[C]. less than 120", null, "but greater than 100", null, "[D]. less than 100", null, "Explanation:\n\nNo answer description available for this question.\n\n Anand said: (Jul 26, 2011) Can anybody explain about the answer ?\n\n Jinal said: (Sep 19, 2011) 1/R=1/R1+1/R2 =1/1600+1/120 =(120+1600)/(120*1600) so that, R=111.63 That is less than 120 ohm and greater than 100 ohm.\n\n Saravanan said: (Mar 20, 2013) R = R1R2/ (R1+R2). R = 1600*120/ (1600+120). R = 111.62.\n\n Madhusudana B R said: (Jan 17, 2016) In parallel circuit, if 10 no of different rating resistance connected parallel in that total resistance should be less than smaller value resistance of the parallel circuit. So you can consider less than 120 ohms as your answer." ]

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[ "# Interval\n\nAn interval is the subset of all the real numbers lying between a lower and an upper bound. If the interval includes a bound it is written with a square bracket, if not with a round bracket.\nFor example\n\\begin{align*} x \\in (l, u) &\\rightarrow l < x < u\\\\ x \\in [l, u) &\\rightarrow l \\leqslant x < u\\\\ x \\in (l, u] &\\rightarrow l < x \\leqslant u\\\\ x \\in [l, u] &\\rightarrow l \\leqslant x \\leqslant u \\end{align*}\nwhere $$\\in$$ means within and $$\\rightarrow$$ means implies.\n\n### Gallimaufry", null, "", null, "AKCalc ECMA", null, "", null, "Endarkenment Turning Sixteen\n\n•", null, "Subscribe\n•", null, "Contact\n•", null, "Downloads\n•", null, "GitHub\n\nThis site requires HTML5, CSS 2.1 and JavaScript 5 and has been tested with", null, "Chrome 26+", null, "Firefox 20+", null, "Internet Explorer 9+" ]

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[ "gemseo / mlearning / regression\n\n# polyreg module¶\n\nPolynomial regression model.\n\nPolynomial regression is a particular case of the linear regression, where the input data is transformed before the regression is applied. This transform consists of creating a matrix of monomials by raising the input data to different powers up to a certain degree $$D$$. In the case where there is only one input variable, the input data $$(x_i)_{i=1, \\dots, n}\\in\\mathbb{R}^n$$ is transformed into the Vandermonde matrix:\n\n$\\begin{split}\\begin{pmatrix} x_1^1 & x_1^2 & \\cdots & x_1^D\\\\ x_2^1 & x_2^2 & \\cdots & x_2^D\\\\ \\vdots & \\vdots & \\ddots & \\vdots\\\\ x_n^1 & x_n^2 & \\cdots & x_n^D\\\\ \\end{pmatrix} = (x_i^d)_{i=1, \\dots, n;\\ d=1, \\dots, D}.\\end{split}$\n\nThe output variable is expressed as a weighted sum of monomials:\n\n$y = w_0 + w_1 x^1 + w_2 x^2 + ... + w_D x^D,$\n\nwhere the coefficients $$w_1, w_2, ..., w_d$$ and the intercept $$w_0$$ are estimated by least square regression.\n\nIn the case of a multidimensional input, i.e. $$X = (x_{ij})_{i=1,\\dots,n; j=1,\\dots,m}$$, where $$n$$ is the number of samples and $$m$$ is the number of input variables, the Vandermonde matrix is expressed through different combinations of monomials of degree $$d, (1 \\leq d \\leq D)$$; e.g. for three variables $$(x, y, z)$$ and degree $$D=3$$, the different terms are $$x$$, $$y$$, $$z$$, $$x^2$$, $$xy$$, $$xz$$, $$y^2$$, $$yz$$, $$z^2$$, $$x^3$$, $$x^2y$$ etc. More generally, for $$m$$ input variables, the total number of monomials of degree $$1 \\leq d \\leq D$$ is given by $$P = \\binom{m+D}{m} = \\frac{(m+D)!}{m!D!}$$. In the case of 3 input variables given above, the total number of monomial combinations of degree lesser than or equal to three is thus $$P = \\binom{6}{3} = 20$$. The linear regression has to identify the coefficients $$w_1, \\dots, w_P$$, in addition to the intercept $$w_0$$.\n\n## Dependence¶\n\nThe polynomial regression model relies on the LinearRegression and PolynomialFeatures classes of the scikit-learn library.\n\nclass gemseo.mlearning.regression.polyreg.PolynomialRegressor(data, degree, transformer=mappingproxy({}), input_names=None, output_names=None, fit_intercept=True, penalty_level=0.0, l2_penalty_ratio=1.0, **parameters)[source]\n\nPolynomial regression model.\n\nParameters:\n• data (IODataset) – The learning dataset.\n\n• degree (int) – The polynomial degree.\n\n• transformer (TransformerType) –\n\nThe strategies to transform the variables. The values are instances of Transformer while the keys are the names of either the variables or the groups of variables, e.g. \"inputs\" or \"outputs\" in the case of the regression algorithms. If a group is specified, the Transformer will be applied to all the variables of this group. If IDENTITY, do not transform the variables.\n\nBy default it is set to {}.\n\n• input_names (Iterable[str] | None) – The names of the input variables. If None, consider all the input variables of the learning dataset.\n\n• output_names (Iterable[str] | None) – The names of the output variables. If None, consider all the output variables of the learning dataset.\n\n• fit_intercept (bool) –\n\nWhether to fit the intercept.\n\nBy default it is set to True.\n\n• penalty_level (float) –\n\nThe penalty level greater or equal to 0. If 0, there is no penalty.\n\nBy default it is set to 0.0.\n\n• l2_penalty_ratio (float) –\n\nThe penalty ratio related to the l2 regularization. If 1, the penalty is the Ridge penalty. If 0, this is the Lasso penalty. Between 0 and 1, the penalty is the ElasticNet penalty.\n\nBy default it is set to 1.0.\n\n• **parameters (float | int | str | bool | None) – The parameters of the machine learning algorithm.\n\nRaises:\n\nValueError – If the degree is lower than one.\n\nclass DataFormatters\n\nBases: DataFormatters\n\nMachine learning regression model decorators.\n\nclassmethod format_dict(predict)\n\nMake an array-based function be called with a dictionary of NumPy arrays.\n\nParameters:\n\npredict (Callable[[ndarray], ndarray]) – The function to be called; it takes a NumPy array in input and returns a NumPy array.\n\nReturns:\n\nA function making the function ‘predict’ work with either a NumPy data array or a dictionary of NumPy data arrays indexed by variables names. The evaluation will have the same type as the input data.\n\nReturn type:\nclassmethod format_dict_jacobian(predict_jac)\n\nWrap an array-based function to make it callable with a dictionary of NumPy arrays.\n\nParameters:\n\npredict_jac (Callable[[ndarray], ndarray]) – The function to be called; it takes a NumPy array in input and returns a NumPy array.\n\nReturns:\n\nThe wrapped ‘predict_jac’ function, callable with either a NumPy data array or a dictionary of numpy data arrays indexed by variables names. The return value will have the same type as the input data.\n\nReturn type:\nclassmethod format_input_output(predict)\n\nMake a function robust to type, array shape and data transformation.\n\nParameters:\n\npredict (Callable[[ndarray], ndarray]) – The function of interest to be called.\n\nReturns:\n\nA function calling the function of interest ‘predict’, while guaranteeing consistency in terms of data type and array shape, and applying input and/or output data transformation if required.\n\nReturn type:\nclassmethod format_samples(predict)\n\nMake a 2D NumPy array-based function work with 1D NumPy array.\n\nParameters:\n\npredict (Callable[[ndarray], ndarray]) – The function to be called; it takes a 2D NumPy array in input and returns a 2D NumPy array. The first dimension represents the samples while the second one represents the components of the variables.\n\nReturns:\n\nA function making the function ‘predict’ work with either a 1D NumPy array or a 2D NumPy array. The evaluation will have the same dimension as the input data.\n\nReturn type:\nclassmethod format_transform(transform_inputs=True, transform_outputs=True)\n\nForce a function to transform its input and/or output variables.\n\nParameters:\n• transform_inputs (bool) –\n\nWhether to transform the input variables.\n\nBy default it is set to True.\n\n• transform_outputs (bool) –\n\nWhether to transform the output variables.\n\nBy default it is set to True.\n\nReturns:\n\nA function evaluating a function of interest, after transforming its input data and/or before transforming its output data.\n\nReturn type:\nclassmethod transform_jacobian(predict_jac)\n\nApply transformation to inputs and inverse transformation to outputs.\n\nParameters:\n\npredict_jac (Callable[[ndarray], ndarray]) – The function of interest to be called.\n\nReturns:\n\nA function evaluating the function ‘predict_jac’, after transforming its input data and/or before transforming its output data.\n\nReturn type:\nget_coefficients(as_dict=False)[source]\n\nReturn the regression coefficients of the linear model.\n\nParameters:\n\nas_dict (bool) –\n\nIf True, return the coefficients as a dictionary of Numpy arrays indexed by the names of the coefficients. Otherwise, return the coefficients as a Numpy array. For now the only valid value is False.\n\nBy default it is set to False.\n\nReturns:\n\nThe regression coefficients of the linear model.\n\nRaises:\n\nNotImplementedError – If the coefficients are required as a dictionary.\n\nReturn type:\nget_intercept(as_dict=True)\n\nReturn the regression intercepts of the linear model.\n\nParameters:\n\nas_dict (bool) –\n\nIf True, return the intercepts as a dictionary. Otherwise, return the intercepts as a numpy.array\n\nBy default it is set to True.\n\nReturns:\n\nThe regression intercepts of the linear model.\n\nRaises:\n\nValueError – If the coefficients are required as a dictionary even though the transformers change the variables dimensions.\n\nReturn type:\nlearn(samples=None, fit_transformers=True)\n\nTrain the machine learning algorithm from the learning dataset.\n\nParameters:\n• samples (Sequence[int] | None) – The indices of the learning samples. If None, use the whole learning dataset.\n\n• fit_transformers (bool) –\n\nWhether to fit the variable transformers.\n\nBy default it is set to True.\n\nReturn type:\n\nNone\n\nLoad a machine learning algorithm from a directory.\n\nParameters:\n\ndirectory (str | Path) – The path to the directory where the machine learning algorithm is saved.\n\nReturn type:\n\nNone\n\npredict(input_data, *args, **kwargs)\n\nEvaluate ‘predict’ with either array or dictionary-based input data.\n\nFirstly, the pre-processing stage converts the input data to a NumPy data array, if these data are expressed as a dictionary of NumPy data arrays.\n\nThen, the processing evaluates the function ‘predict’ from this NumPy input data array.\n\nLastly, the post-processing transforms the output data to a dictionary of output NumPy data array if the input data were passed as a dictionary of NumPy data arrays.\n\nParameters:\n• input_data (ndarray | Mapping[str, ndarray]) – The input data.\n\n• *args – The positional arguments of the function ‘predict’.\n\n• **kwargs – The keyword arguments of the function ‘predict’.\n\nReturns:\n\nThe output data with the same type as the input one.\n\nReturn type:\npredict_jacobian(input_data, *args, **kwargs)\n\nEvaluate ‘predict_jac’ with either array or dictionary-based data.\n\nFirstly, the pre-processing stage converts the input data to a NumPy data array, if these data are expressed as a dictionary of NumPy data arrays.\n\nThen, the processing evaluates the function ‘predict_jac’ from this NumPy input data array.\n\nLastly, the post-processing transforms the output data to a dictionary of output NumPy data array if the input data were passed as a dictionary of NumPy data arrays.\n\nParameters:\n• input_data – The input data.\n\n• *args – The positional arguments of the function ‘predict_jac’.\n\n• **kwargs – The keyword arguments of the function ‘predict_jac’.\n\nReturns:\n\nThe output data with the same type as the input one.\n\npredict_raw(input_data)\n\nPredict output data from input data.\n\nParameters:\n\ninput_data (ndarray) – The input data with shape (n_samples, n_inputs).\n\nReturns:\n\nThe predicted output data with shape (n_samples, n_outputs).\n\nReturn type:\n\nndarray\n\nto_pickle(directory=None, path='.', save_learning_set=False)\n\nSave the machine learning algorithm.\n\nParameters:\n• directory (str | None) – The name of the directory to save the algorithm.\n\n• path (str | Path) –\n\nThe path to parent directory where to create the directory.\n\nBy default it is set to “.”.\n\n• save_learning_set (bool) –\n\nWhether to save the learning set or get rid of it to lighten the saved files.\n\nBy default it is set to False.\n\nReturns:\n\nThe path to the directory where the algorithm is saved.\n\nReturn type:\n\nstr\n\nDEFAULT_TRANSFORMER: DefaultTransformerType = mappingproxy({'inputs': <gemseo.mlearning.transformers.scaler.min_max_scaler.MinMaxScaler object>, 'outputs': <gemseo.mlearning.transformers.scaler.min_max_scaler.MinMaxScaler object>})\n\nThe default transformer for the input and output data, if any.\n\nFILENAME: ClassVar[str] = 'ml_algo.pkl'\nIDENTITY: Final[DefaultTransformerType] = mappingproxy({})\n\nA transformer leaving the input and output variables as they are.\n\nLIBRARY: Final[str] = 'scikit-learn'\n\nThe name of the library of the wrapped machine learning algorithm.\n\nSHORT_ALGO_NAME: ClassVar[str] = 'PolyReg'\n\nThe short name of the machine learning algorithm, often an acronym.\n\nTypically used for composite names, e.g. f\"{algo.SHORT_ALGO_NAME}_{dataset.name}\" or f\"{algo.SHORT_ALGO_NAME}_{discipline.name}\".\n\nalgo: Any\n\nThe interfaced machine learning algorithm.\n\nproperty coefficients: ndarray\n\nThe regression coefficients of the linear model.\n\nproperty input_data: ndarray\n\nThe input data matrix.\n\nproperty input_dimension: int\n\nThe input space dimension.\n\ninput_names: list[str]\n\nThe names of the input variables.\n\ninput_space_center: dict[str, ndarray]\n\nThe center of the input space.\n\nproperty intercept: ndarray\n\nThe regression intercepts of the linear model.\n\nproperty is_trained: bool\n\nReturn whether the algorithm is trained.\n\nproperty learning_samples_indices: Sequence[int]\n\nThe indices of the learning samples used for the training.\n\nlearning_set: Dataset\n\nThe learning dataset.\n\nproperty output_data: ndarray\n\nThe output data matrix.\n\nproperty output_dimension: int\n\nThe output space dimension.\n\noutput_names: list[str]\n\nThe names of the output variables.\n\nparameters: dict[str, MLAlgoParameterType]\n\nThe parameters of the machine learning algorithm.\n\ntransformer: dict[str, Transformer]\n\nThe strategies to transform the variables, if any.\n\nThe values are instances of Transformer while the keys are the names of either the variables or the groups of variables, e.g. “inputs” or “outputs” in the case of the regression algorithms. If a group is specified, the Transformer will be applied to all the variables of this group.\n\n## Examples using PolynomialRegressor¶", null, "MSE example - test-train split\n\nMSE example - test-train split", null, "Polynomial regression\n\nPolynomial regression" ]

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[ "# flambe.metric¶\n\n## Package Contents¶\n\nclass flambe.metric.Metric[source]\n\nBase Metric interface.\n\nObjects implementing this interface should take in a sequence of examples and provide as output a processd list of the same size.\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nComputes the metric over the given prediction and target.\n\nParameters: pred (torch.Tensor) – The model predictions target (torch.Tensor) – The ground truth targets The computed metric torch.Tensor\naggregate(self, state: dict, *args, **kwargs)\n\nAggregates by simply storing preds and targets\n\nParameters: state (dict) – the metric state args (the pred, target tuple) – the state dict dict\nfinalize(self, state: Dict)\n\nFinalizes the metric computation\n\nParameters: state (dict) – the metric state The final score. float\n__call__(self, *args, **kwargs)\n\nMakes Featurizer a callable.\n\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\nclass flambe.metric.MultiLabelCrossEntropy(weight: Optional[torch.Tensor] = None, ignore_index: Optional[int] = None, reduction: str = 'mean')[source]\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nComputes the multilabel cross entropy loss.\n\nParameters: pred (torch.Tensor) – input logits of shape (B x N) target (torch.LontTensor) – target tensor of shape (B x N) loss – Multi label cross-entropy loss, of shape (B) torch.Tensor\nclass flambe.metric.MultiLabelNLLLoss(weight: Optional[torch.Tensor] = None, ignore_index: Optional[int] = None, reduction: str = 'mean')[source]\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nComputes the Negative log likelihood loss for multilabel.\n\nParameters: pred (torch.Tensor) – input logits of shape (B x N) target (torch.LontTensor) – target tensor of shape (B x N) loss – Multi label negative log likelihood loss, of shape (B) torch.float\nclass flambe.metric.Accuracy[source]\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nComputes the loss.\n\nParameters: pred (Tensor) – input logits of shape (B x N) target (LontTensor) – target tensor of shape (B) or (B x N) accuracy – single label accuracy, of shape (B) torch.Tensor\nclass flambe.metric.Perplexity[source]\n\nToken level perplexity, computed a exp(cross_entropy).\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute the preplexity given the input and target.\n\nParameters: pred (torch.Tensor) – input logits of shape (B x N) target (torch.LontTensor) – target tensor of shape (B) Output perplexity torch.float\naggregate(self, state: dict, *args, **kwargs)\n\nAggregates by only storing entropy per sample\n\nParameters: state (dict) – the metric state args (the pred, target tuple) – the state dict dict\nfinalize(self, state: Dict)\n\nFinalizes the metric computation\n\nParameters: state (dict) – the metric state The final score. float\nclass flambe.metric.BPC[source]\n\nBits per character. Computed as log_2(perplexity)\n\nInherits from Perplexity to share aggregate functionality.\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute the bits per character given the input and target.\n\nParameters: pred (torch.Tensor) – input logits of shape (B x N) target (torch.LontTensor) – target tensor of shape (B) Output perplexity torch.float\nfinalize(self, state: Dict)\n\nFinalizes the metric computation\n\nParameters: state (dict) – the metric state The final score. float\nclass flambe.metric.AUC(max_fpr=1.0)[source]\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute AUC at the given max false positive rate.\n\nParameters: pred (torch.Tensor) – The model predictions of shape numsamples target (torch.Tensor) – The binary targets of shape numsamples The computed AUC torch.Tensor\nclass flambe.metric.MultiClassAUC[source]\n\nN-Ary (Multiclass) AUC for k-way classification\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute multiclass AUC at the given max false positive rate.\n\nParameters: pred (torch.Tensor) – The model predictions of shape numsamples x numclasses target (torch.Tensor) – The binary targets of shape: numsamples. In this case the elements index into the different classes numsamples x numclasses. This implementation only considers the indices of the max values as positive labels The computed AUC torch.Tensor\nclass flambe.metric.BinaryPrecision(threshold: float = 0.5, positive_label: int = 1)[source]\n\nCompute Binary Precision.\n\nAn example is considered negative when its score is below the specified threshold. Binary precition is computed as follows:\n\n |True positives| / |True Positives| + |False Positives| \n\ncompute_binary(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute binary precision.\n\nParameters: pred (torch.Tensor) – Predictions made by the model. It should be a probability 0 <= p <= 1 for each sample, 1 being the positive class. target (torch.Tensor) – Ground truth. Each label should be either 0 or 1. The computed binary metric torch.float\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\nclass flambe.metric.BinaryRecall(threshold: float = 0.5, positive_label: int = 1)[source]\n\nCompute binary recall.\n\nAn example is considered negative when its score is below the specified threshold. Binary precition is computed as follows:\n\n |True positives| / |True Positives| + |False Negatives| \n\ncompute_binary(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute binary recall.\n\nParameters: pred (torch.Tensor) – Predictions made by the model. It should be a probability 0 <= p <= 1 for each sample, 1 being the positive class. target (torch.Tensor) – Ground truth. Each label should be either 0 or 1. The computed binary metric torch.float\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\nclass flambe.metric.BinaryAccuracy[source]\n\nCompute binary accuracy.\n\n |True positives + True negatives| / N \n\ncompute_binary(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute binary accuracy.\n\nParameters: pred (torch.Tensor) – Predictions made by the model. It should be a probability 0 <= p <= 1 for each sample, 1 being the positive class. target (torch.Tensor) – Ground truth. Each label should be either 0 or 1. The computed binary metric torch.float\nclass flambe.metric.F1(threshold: float = 0.5, positive_label: int = 1, eps: float = 1e-08)[source]\ncompute_binary(self, pred: torch.Tensor, target: torch.Tensor)\n\nCompute F1. Score, the harmonic mean between precision and recall.\n\nParameters: pred (torch.Tensor) – Predictions made by the model. It should be a probability 0 <= p <= 1 for each sample, 1 being the positive class. target (torch.Tensor) – Ground truth. Each label should be either 0 or 1. The computed binary metric torch.float\nclass flambe.metric.Recall(top_k: int = 1)[source]\n__str__(self)\n\nReturn the name of the Metric (for use in logging).\n\ncompute(self, pred: torch.Tensor, target: torch.Tensor)\n\nComputes the recall @ k.\n\nParameters: pred (Tensor) – input logits of shape (B x N) target (LongTensor) – target tensor of shape (B) or (B x N) recall – single label recall, of shape (B) torch.Tensor" ]

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[ "# Electrons Accelerating#\n\nAn electron accelerated (with acceleration $$a_1$$) from rest for a time interval $$t_1$$ and travels a distance of 20 $$m$$.\n\n## Part 1#\n\nA second electron given the acceleration $${a_1}$$/6 for a time interval 6$$t_1$$, after starting from rest, will travel a distance of:\n\n• 20 $$m$$\n\n• 120.0 $$m$$\n\n• 720.0 $$m$$\n\n• 3.3 $$m$$\n\n• 0.3 $$m$$", null, "" ]

[ null, "https://raw.githubusercontent.com/firasm/bits/master/by-nc-sa.png", null ]

[ "Home » Blog » Prove a property of the integral of the product of continuous functions\n\n# Prove a property of the integral of the product of continuous functions\n\nLet", null, "be a continuous function on the interval", null, "and assume", null, "for every function", null, "which is continuous on the interval", null, ". Prove that", null, "for all", null, ".\n\nProof. Since", null, "must hold for every function", null, "that is continuous on", null, ", it must hold for", null, "itself (since", null, "is continuous on", null, "by hypothesis). Therefore we must have,", null, "However,", null, "for all", null, ". We know from the previous exercise (Section 3.20, #7) that a non-negative function whose integral is zero on an interval must be zero at every point at which it is continuous. By hypothesis", null, "is continuous at every point of", null, "; hence,", null, "is also continuous at every point of", null, "(since the product of continuous functions is continuous). Therefore,", null, "Which implies,", null, "1.", null, "•", null, "" ]

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[ "# 一个标明 1kΩ,10W 的电阻.一个标明 1kΩ,10W 的电阻,允许通过的最大电流为_____,允许加在这个电阻两端的最大电压为______.当这个电阻两端的电压为40V时,它消耗的实际功率为______.（设电阻\n\n1.I=根号P/R=根号10/1000=0.1安\n2.U=IR=0.1*1000=100伏\n3.P=U^2/R=40^2/1000=16wa\n\n`0.1A,100V,1.6W`\n`楼上前两个对的1.I=根号P/R=根号10/1000=0.1安 2.U=IR=0.1*1000=100伏第三个是3.P=U^2/R=40^2/1000=1.6W`\n\n2020 及时作业本 webmaster#jeepfushi.net\n10 q. 0.010 s." ]

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[ "## Elementary Algebra\n\nConverting the expression $5\\times5\\times5\\times a\\times b\\times b$ to exponential notation, we must count the number of times each number is being multiplied by itself and make that the exponent for the number being multiplied. Since $5$ is multiplied by itself three times, $a$ is multiplied by itself once, and $b$ is multiplied by itself twice, the expression in exponential form should read $5^3ab^2$, not $5^3ab$." ]

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[ "# What is 111 inches in feet?\n\n111 inches = 9.25 feet\n\nSaid another way, 111 inches = 9 feet 3 inches\n\nConvert another measurement\n\n## Formula for converting inches to feet\n\nThe formula for converting inches to feet is inches / 12. So for a length of 111 inches, the formula would be 111 / 12, with a result of 9.25 feet.\n\nThe process for converting inches to feet and inches is inches / 12, keeping the whole number, and then multiplying whatever is after the decimal by 12. So for a length of 111 inches, you would divide 111 by 12, with a result of 9.25. Set the 9 aside for now and multiply 0.25 by 12 to find the remaining number of inches, which in this case would be 3, for a final result of 9 feet 3 inches.\n\n## Convert 111.01 - 111.99 feet\n\n222222222222222222222222222222222222222222222222222222222222222222222222222\n Feet\n222222222222222222222222222222222222222222222222222222222222222222222222222\n Feet\n222222222222222222222222222222222222222222222222222222222222222222222222222\n Feet\n222222222222222222222222222222222222222222222222222222222222222222222222\n Feet\n\n## Look up numbers near 111\n\n← Prev num Next num →\n110 112" ]

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[ "Solutions by everydaycalculation.com\n\n1st number: 2 15/30, 2nd number: 2 2/8\n\n75/30 + 18/8 is 19/4.\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 30 and 8 is 120\n2. For the 1st fraction, since 30 × 4 = 120,\n75/30 = 75 × 4/30 × 4 = 300/120\n3. Likewise, for the 2nd fraction, since 8 × 15 = 120,\n18/8 = 18 × 15/8 × 15 = 270/120\n300/120 + 270/120 = 300 + 270/120 = 570/120\n5. 570/120 simplified gives 19/4\n6. So, 75/30 + 18/8 = 19/4\nIn mixed form: 43/4\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "# std::operator+(std::basic_string) (3) - Linux Man Pages\n\n## NAME\n\nstd::operator+(std::basic_string) - std::operator+(std::basic_string)\n\n## Synopsis\n\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (1)\noperator+( const basic_string<CharT,Traits,Alloc>& lhs,\nconst basic_string<CharT,Traits,Alloc>& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (2)\noperator+( const basic_string<CharT,Traits,Alloc>& lhs,\nconst CharT* rhs );\ntemplate<class CharT, class Traits, class Alloc>\nbasic_string<CharT,Traits,Alloc> (3)\noperator+( const basic_string<CharT,Traits,Alloc>& lhs,\nCharT rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (4)\noperator+( const CharT* lhs,\nconst basic_string<CharT,Traits,Alloc>& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (5)\noperator+( CharT lhs,\nconst basic_string<CharT,Traits,Alloc>& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (6) (since C++11)\noperator+( basic_string<CharT,Traits,Alloc>&& lhs,\nbasic_string<CharT,Traits,Alloc>&& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (7) (since C++11)\noperator+( basic_string<CharT,Traits,Alloc>&& lhs,\nconst basic_string<CharT,Traits,Alloc>& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (8) (since C++11)\noperator+( basic_string<CharT,Traits,Alloc>&& lhs,\nconst CharT* rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (9) (since C++11)\noperator+( basic_string<CharT,Traits,Alloc>&& lhs,\nCharT rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (10) (since C++11)\noperator+( const basic_string<CharT,Traits,Alloc>& lhs,\nbasic_string<CharT,Traits,Alloc>&& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (11) (since C++11)\noperator+(const CharT* lhs,\nbasic_string<CharT,Traits,Alloc>&& rhs );\ntemplate< class CharT, class Traits, class Alloc >\nbasic_string<CharT,Traits,Alloc> (12) (since C++11)\noperator+( CharT lhs,\nbasic_string<CharT,Traits,Alloc>&& rhs );\n\nReturns a string containing characters from lhs followed by the characters from rhs.\n\nThe allocator used for the result is:\n1-3) std::allocator_traits<Alloc>::select_on_container_copy_construction(lhs.get_allocator())\n4-5) std::allocator_traits<Alloc>::select_on_container_copy_construction(rhs.get_allocator())\n6-9) lhs.get_allocator() (since C++11)\n10-12) rhs.get_allocator()\nIn other words, if one operand is a basic_string rvalue, its allocator is used; otherwise, select_on_container_copy_construction is used on the allocator of the lvalue basic_string operand. In each case, the left operand is preferred when both are basic_strings of the same value category.\nFor (6-12), all rvalue basic_string operands are left in valid but unspecified states.\n\n## Parameters\n\nlhs - string, character, or pointer to the first character in a null-terminated array\nrhs - string, character, or pointer to the first character in a null-terminated array\n\n## Return value\n\nA string containing characters from lhs followed by the characters from rhs\n, using the allocator determined as above\n(since C++11).\n\n## Notes\n\noperator+ should be used with great caution when stateful allocators are involved\n(such as when std::pmr::string is used)\n(since C++17). Prior to P1165R1, the allocator used for the result was determined by historical accident and can vary from overload to overload for no apparent reason. Moreover, for (1-5), the allocator propagation behavior varies across major standard library implementations and differs from the behavior depicted in the standard.\nBecause the allocator used by the result of operator+ is sensitive to value category, operator+ is not associative with respect to allocator propagation:\n\nusing my_string = std::basic_string<char, std::char_traits<char>, my_allocator<char>>;\nmy_string cat();\nconst my_string& dog();\n\nmy_string meow = /* ... */, woof = /* ... */; (since C++11)\nmeow + cat() + /*...*/; // uses SOCCC on meow's allocator\nwoof + dog() + /*...*/; // uses allocator of dog()'s return value instead\n\nmeow + woof + meow; // uses SOCCC on meow's allocator\nmeow + (woof + meow); // uses SOCCC on woof's allocator instead\n\nFor a chain of operator+ invocations, the allocator used for the ultimate result may be controlled by prepending an rvalue basic_string with the desired allocator:\n\n// use my_favorite_allocator for the final result\nmy_string(my_favorite_allocator) + meow + woof + cat() + dog();\n\nFor better and portable control over allocators, member functions like append(), insert(), and operator+=() should be used on a result string constructed with the desired allocator.\n\n## Example\n\n// Run this code\n\n#include <iostream>\n#include <string>\n\nint main()\n{\nstd::string s1 = \"Hello\";\nstd::string s2 = \"world\";\nstd::cout << s1 + ' ' + s2 + \"!\\n\";\n}\n\n## Output:\n\nHello world!\n\nDefect reports\n\nThe following behavior-changing defect reports were applied retroactively to previously published C++ standards.\n\nDR Applied to Behavior as published Correct behavior\nP1165R1 C++11 allocator propagation is haphazard and inconsistent made more consistent" ]

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[ "# Exponentially weighted moving average\n\nThe exponentially weighted moving average, or in short the exponential moving average, is a moving average technical indicator which uses an exponential weighting scheme of past prices. Compared to the simple moving average indicator, this metric puts more weight on recent prices.\n\n## Dual exponentially weighted moving average\n\nSimilar like the dual moving average trading system based on the simple moving average, the dual exponentially weighted moving average uses 2 indicators to generate buy and sell signals. The first moving average uses a smaller lag parameter and imposes an exponential weighting scheme. Therefore, this indicator follows closely the price series but also evolves more erratic over time. The second indicator uses a larger lag parameter and thus evolves more smoothly over time but follows less recent price changes. Technical analysts argue that the crossovers generates buy signals whenever the short moving average crosses the long one from below upwards. Sell signals are generated when the short moving average crosses the longer one from above downwards.", null, "## Exponentially weighted moving average formula\n\nHow to calculate an exponentially weighted moving average? This is somewhat more tricky than the calculation of a simple moving average as a result of a more complicated weighting scheme. First a memory parameter, α, need to be determined.  This parameter needs to have a value between 0 and 1. The larger this parameter is, the longer its memory and the more time it will take to incorporate new changes. This parameter can be directly imposed, or calculated based on a required amount of look back days as shown in the formula below.", null, "", null, "## Summary\n\nThe EWMA is a specific type of moving average that puts more weight on recent price. As such, this indicator will follow more closely the price series than a simple moving average." ]

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[ "The score in specifying heat capacity is to relate transforms in the internal power to measured changes in the variables that characterize the states of the system. Because that a mechanism consisting that a solitary pure substance, the only kind of job-related it can do is atmospheric work, and also so the an initial law reduces come dU = dQP dV. (28)\n\nSuppose currently that U is related to as gift a duty U(T, V) of the live independence pair the variables T and V. The differential quantity dU can constantly be expanded in terms of its partial derivatives according to", null, "(29) wherein the subscripts represent the quantity being held constant when calculating derivatives. Substituting this equation into dU = dQP dV then yields the general expression", null, "(30) because that the path-dependent heat. The path have the right to now be specified in regards to the live independence variables T and also V. For a temperature adjust at continuous volume, dV = 0 and, by an interpretation of heat capacity, dQV = CV dT. (31) The above equation climate gives instantly", null, "(32) for the heat capacity at consistent volume, showing that the change in internal power at continuous volume is due entirely to the warmth absorbed.\n\nYou are watching: What equation represents the energy it takes to heat a substance?\n\nTo uncover a matching expression for CP, one need only readjust the live independence variables come T and P and also substitute the development", null, "(33) for dV in equation (28) and correspondingly for dU to attain", null, "(34)\n\nFor a temperature change at continuous pressure, dP = 0, and, by definition of warm capacity, dQ = CP dT, causing", null, "(35)\n\nThe two additional terms beyond CV have actually a direct physical meaning. The term", null, "to represent the additional atmospheric occupational that the system does as it undergoes thermal growth at consistent pressure, and also the second term involving represents the interior work that should be excellent to pull the device apart versus the pressures of attraction in between the molecule of the substance (internal stickiness). Due to the fact that there is no inner stickiness for perfect gas, this ax is zero, and, indigenous the best gas law, the continuing to be partial derivative is", null, "(36) through these substitutions the equation for CP becomes merely CP = CV + nR (37) or cP = cV + R (38) for the molar particular heats. Because that example, because that a monatomic ideal gas (such as helium), cV = 3R/2 and cP = 5R/2 to a great approximation. cVT represents the quantity of translational kinetic energy possessed by the atoms of an ideal gas together they bounce approximately randomly inside your container. Diatomic molecules (such together oxygen) and also polyatomic molecule (such together water) have added rotational activities that additionally store thermal energy in their kinetic power of rotation. Each extr degree of flexibility contributes second amount R to cV. Since diatomic molecules have the right to rotate around two axes and polyatomic molecules have the right to rotate around three axes, the worths of cV rise to 5R/2 and 3R respectively, and cP correspondingly rises to 7R/2 and 4R. (cV and cP rise still additional at high temperatures due to the fact that of vibrational levels of freedom.) for a genuine gas such as water vapour, these values are only approximate, yet they offer the exactly order that magnitude. Because that example, the correct values room cP = 37.468 joules every K (i.e., 4.5R) and also cPcV = 9.443 joules every K (i.e., 1.14R) for water vapour in ~ 100 °C and 1 setting pressure.\n\n## Entropy as specific differential\n\nBecause the amount dS = dQmax/T is specific differential, countless other vital relationships connecting the thermodynamic nature of substances deserve to be derived. Because that example, v the substitutions dQ = T dS and dW = P dV, the differential kind (dU = dQdW) that the an initial law the thermodynamics becomes (for a solitary pure substance) dU = T dSP dV. (39)\n\nThe benefit gained by the above formula is that dU is currently expressed completely in regards to state features in place of the path-dependent amounts dQ and also dW. This adjust has the very important mathematical implication the the ideal independent variables space S and V in location of T and also V, respectively, for interior energy.\n\nThis instead of of T by S as the most appropriate independent variable for the internal energy of building materials is the solitary most an important insight provided by the combined an initial and 2nd laws the thermodynamics. Through U related to as a role U(S, V), its differential dU is", null, "(40)\n\nA comparison through the preceding equation shows instantly that the partial derivatives room", null, "(41) Furthermore, the overcome partial derivatives,", null, "(42) need to be equal due to the fact that the bespeak of differentiation in calculating the second derivatives the U does not matter. Equating the right-hand political parties of the above pair of equations then yields", null, "(43)\n\nThis is one of four Maxwell relations (the others will follow shortly). They room all extremely valuable in that the amount on the right-hand side is virtually impossible to measure directly, if the quantity on the left-hand side is easily measured in the laboratory. Because that the present situation one simply actions the adiabatic sport of temperature through volume in an insulated cylinder so that there is no heat flow (constant S).\n\nSee more: How Long Egg Salad Keep In Refrigerator, How Long Is Egg Salad Good In The Fridge\n\nThe other three Maxwell relationships follow by an in similar way considering the differential expressions for the thermodynamic potentials F(T, V), H(S, P), and also G(T, P), v independent variables together indicated. The results are", null, "(44)\n\nAs an instance of the use of this equations, equation (35) because that CPCV includes the partial derivative which vanishes for an ideal gas and is difficult to evaluate directly from speculative data for genuine substances. The basic properties that partial derivatives can an initial be used to write it in the form", null, "(45)\n\nCombining this with equation (41) for the partial derivatives together with the very first of the Maxwell equations from equation (44) then yields the desired an outcome", null, "(46)\n\nThe amount", null, "comes directly from differentiating the equation the state. For perfect gas", null, "(47) and also so is zero together expected. The leave of indigenous zero reveals straight the impacts of interior forces in between the molecule of the substance and the work-related that must be done against them as the substance increases at consistent temperature." ]

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[ "", null, "### Mike Trienis\n\nAll about data product and services that scale; from design to implementation\n\n# Python concepts for interviews\n\nPreparing for a technical interview with Python means that you should have a decent understanding of the following concepts.\n\n## Iterating over iterators\n\nLists, dictionaries, generators and strings are all example of iterators. Each one of these constructs support `for` statements.\n\n``````# looping through a list\n>>> for i in [1,2,3]:\n... print i\n...\n1\n2\n3\n\n# looping through a dictionary\n>>> for i in {\"key\":\"value\"}:\n... print i\n...\nkey\n\n# looping through a string\n>>> for i in \"hi\":\n... print i\n...\nh\ni``````\n\n## Generator functions\n\nGenerators are used to produce iterators with minimum memory consumption.\n\n``````# Using the generator pattern (an iterable)\nclass firstn(object):\ndef __init__(self, n):\nself.n = n\nself.num, self.nums = 0, []\n\ndef __iter__(self):\nreturn self\n\n# Python 3 compatibility\ndef __next__(self):\nreturn self.next()\n\ndef next(self):\nif self.num < self.n:\ncur, self.num = self.num, self.num+1\nreturn cur\nelse:\nraise StopIteration()\n\nsum_of_first_n = sum(firstn(1000000))``````\n\n## Anonymous (lambda) functions\n\nThe next snippet of code defines a lambda function which is essentially a function that is not bounded to a name.\n\n``````>>> print lambda x : i * x\n<function <lambda> at 0x109ae6938>``````\n\n## List comprehensions\n\nA list comprehension a way to generate lists using a natural syntax. If we build on the previous example, we can generate a list of lambda functions by:\n\n``````>>> print [lambda x : i * x for i in range(4)]\n[<function <lambda> at 0x109ae6938>, <function <lambda> at 0x109ae6758>, <function <lambda> at 0x109ae69b0>, <function <lambda> at 0x109ae6a28>]``````\n\n## Dynamic arguments\n\nIt’s possible to pass a dynamic number of arguments. Either a sequence of values or a sequence of key-values.\n\n``````>>> def dynamicArguments(*arg, **kwargs):\n... print arg\n... print kwargs\n...\n>>> dynamicArguments(1,2,3, first=4, second=5, third=6)\n(1, 2, 3)\n{'second': 5, 'third': 6, 'first': 4}``````\n\n## Linked-list implementation in python\n\nA node in a linked list can represented as a class with a storage variable and another variable that points to the next node in the list. If the next pointer is `None` then it’s the last element of the list.\n\n``````class Node:\ndef __init__(self, item=None, next=None):\nself.item = cargo\nself.next = next\ndef __str__(self):\nreturn str(self.item)\n\nnodeA = Node(\"A\")\nnodeB = Node(\"B\")\nnodeC = Node(\"C\")\n\nnodeA.next = nodeB\nnodeB.next = nodeC``````\n\n## Private members in class\n\nA member variable in a class can be set to private by prefixing with `__`.\n\n``````>>> class Foo:\n... myPublicVar = \"a\"\n... __myPrivateVar = \"b\"\n...\n>>> foo = Foo()\n>>> print foo.myPublicVar\na\n>>> print foo.__myPrivateVar``````\n\n## Class variable and instance variables\n\nClass variables:\n\n``````MyController(Controller):\n\npath = \"something/\"\nchildren = [AController, BController]\n\ndef action(request):\npass``````\n\nInstance variables:\n\n``````MyController(Controller):\n\ndef __init__(self):\nself.path = \"something/\"\nself.children = [AController, BController]\n\ndef action(self, request):\npass``````\n\n## Command-line arguments (sys.argv)\n\nPassing command line arguments to a python script is as simple as importing `sys` module and using the `sys.argv` for retrieving the arguments.\n\n``````import sys\nprint sys.argv``````\n\nThe arguments are constructed as a list when executing the python script.\n\n``````Mikes-MacBook-Pro-3:code miketrienis\\$ python example1_4.py a b\n['example1_4.py', 'a', 'b']``````\n\n## Pass statement\n\nThe pass statement does nothing. It can be used when a statement is required syntactically but the program requires no action. For example:\n\n``````>>> def initlog(*args):\n... pass # Remember to implement this!\n...``````" ]

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[ "# isHolonomic -- determines whether a D-module (or ideal in Weyl algebra) is holonomic\n\n## Synopsis\n\n• Usage:\nisHolonomic M\nisHolonomic I\n• Inputs:\n• M, , over the Weyl algebra D\n• I, an ideal, which represents the module M = D/I\n• Outputs:\n\n## Description\n\nLet $D$ be the Weyl algebra with generators $x_1,\\dots,x_n$ and $\\partial_1,\\dots,\\partial_n$. over a field. A $D$-module is holonomic if it has dimension $n$. For more details see [SST, Section 1.4].\n\n i1 : D = makeWA(QQ[x_1..x_3]) o1 = D o1 : PolynomialRing, 3 differential variables i2 : A = matrix{{1,1,1},{0,1,2}} o2 = | 1 1 1 | | 0 1 2 | 2 3 o2 : Matrix ZZ <--- ZZ i3 : b = {3,4} o3 = {3, 4} o3 : List i4 : I = gkz(A,b,D) 2 o4 = ideal (x dx + x dx + x dx - 3, x dx + 2x dx - 4, - dx + dx dx ) 1 1 2 2 3 3 2 2 3 3 2 1 3 o4 : Ideal of D i5 : isHolonomic I o5 = true" ]

[ null ]

[ "## Creating arrays of wave formulas with BWise\n\nTheo Verelst\n\nGranted, it sounds a bit scienfic, but it's actually quite generally useful and quite followable. Like with the fairly quickly programmed experiments on Creating wave formulas with BWise and for window users a major part can run also via the code refered to on Wave player for formulas from BWise and Maxima on Windows XP the idea is to create\n\n``` bwise graph --> tcl list --> maxima formula --> fortran --> linked C program --> sound program on Linux with jack and Alsa midi\nof wave of complete |---> latex --> gif prettyprinted formula\nformulas formula```\n\nwhere most of these tools are automatically called by tcl, under Tk button control.\n\nIn this case it is fun to play with the BWise blocks creating the signal-determining formula, or instance sine waves as connected blocks, for instance I created this bwise graph automatically:", null, "You need the startup sources of the above pages, and bwise first, because they contain the necessary proc definitions !\n\n``` #\n# Make a ten input maxima formula addition block\n#\nproc_toblock add10 {} {} 700 100\n#\n# The output block which multiplies the final formula with the final volume, which has a name linked with the Run button\n#\nnewproc {set mult11.out \"((\\${mult11.a})*(\\${mult11.b}))\"} mult11 {a b} out 40 {} {} 800 100\nset mult11.b (1/10)\nconnect {} add10 out mult11 a\n#\n# First column\n#\nfor {set i 1} {\\$i < 11} {incr i} { proc_toblock sinea [list {f 440*\\$i} {m 0} {a (1/\\$i)}] sine\\${i}_1 [expr 400] [expr 50+75*\\$i] ; connect {} sine\\${i}_1 out add10 i[expr \\$i] }\n#\n# Second column\n#\nfor {set i 1} {\\$i < 11} {incr i} { proc_toblock sinea [list {f 440*\\$i} {m 0} {a (1/(1000*\\$i))}] sine\\${i}_2 [expr 300] [expr 50+75*\\$i] ; connect {} sine\\${i}_2 out sine\\${i}_1 m }\n\n# Create entries with modulation formulas of first column (where v is the velocity of the keypress, [0,1])\n#\ntoplevel .if1\nfor {set i 1} {\\$i < 11} {incr i} { pack [entry .if1.e1\\$i -textvar sine\\${i}_2.a -width 16] -side top -expand 1 -fill x }\n#\n# Set the variables\n#\nfor {set i 1} {\\$i < 11} {incr i} { set sine\\${i}_2.a \"(v/(2000*\\$i))\" }```\n\nFor Fortran interested people, this is the fortran file generated by the tcl routine after applying the Run button to the BWise graph:\n\n``` subroutine sayhello(x,r,v)\ndouble precision x,r,v\nr = (sin(2.764601535159018d+4*(v*sin(2.764601535159018d+4*x)/2.d+4+x))\n1 /1.d+1+sin(2.4881413816431164d+4*(v*sin(2.4881413816431164d+4*x\n2 )/1.8d+4+x))/9.d+0+sin(2.2116812281272147d+4*(v*sin(2.211681228\n3 1272147d+4*x)/1.6d+4+x))/8.d+0+sin(1.9352210746113127d+4*(v*sin\n4 (1.9352210746113127d+4*x)/1.3999999999999999d+4+x))/7.d+0+sin(1\n5 .6587609210954105d+4*(v*sin(1.6587609210954105d+4*x)/1.2d+4+x))\n6 /6.d+0+sin(1.382300767579509d+4*(v*sin(1.382300767579509d+4*x)/\n7 1.d+4+x))/5.d+0+sin(1.1058406140636072d+4*(v*sin(1.105840614063\n8 6072d+4*x)/8.d+3+x))/4.d+0+sin(8.293804605477053d+3*(v*sin(8.29\n9 3804605477053d+3*x)/6.d+3+x))/3.d+0+sin(5.529203070318036d+3*(v```\n*sin(5.529203070318036d+3*x)/4.d+3+x))/2.d+0+sin(2.764601535159\n``` ; 018d+3*(v*sin(2.764601535159018d+3*x)/8.d+3+x)))/1.d+1\nreturn\nend```\n\nWhen you have Latex installed, use from bwise:\n\n` newimage /dev/shm/formula.gif`\n\nto see the formula.\n\nDuring trying the column creation or otherwise it may be handy to delete per column, to do this for the second one:\n\n``` for {set i 1} {\\$i < 11} {incr i} { cbbox sine\\${i}_2}\ndelete_selblocks```\n\nTo also delete the associated variables:\n\n` foreach i [info var *sine*_2.*] {unset \\$i}`\n\nYesterday evening I also tried a real big graph, how about a hundred sine wave FM algorithm?", null, "That gives this [1 ] fortran function, and this http://www.theover.org/Bwise/Arr1/formula.gif formula (watch out, very big gif file!) and the algorithm has been turned into a succesfull executable using the aforementioned tcl scripts, except on a not so slow machine I could only have monophonic sound (1 100-oscilator voice at the time).\n\nTV (sept 30 10)\n\nFor a musical experiment, see this wiki page ( http://www.theover.org/wiki/index.php/Chords_from_math_waves ) which will also contain pointers to code for making pop-up menus for creating math blocks automatically, and on a recent page on bwise I´ve made a \"select all left\" option to delete parts of graphs like above with only popup menus.\n\n Category Bwise" ]

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[ "", null, "", null, "", null, "", null, "When one unleashes the full power of a 'forcing net', then contradictions are found to occur very commonly. The following contradictions are discovered:\n\n• Two same numbers in the one unit are both inferred to be true;\n• Two different numbers in one cell are both inferred to be true;\n• All numbers in one cell are inferred to be false;\n• All instances of one number in a row, column or box are false\n\nBasically, if a network of strong and weak links is created based on the false assuption that a number is true, then contradictions may or may not occur. If the network of chains doesn't get off the ground because of few strong links being present, then no contradiction may be revealed. Alternatively, when the network is generated based on the correct assumption that a number is true, then there are no contradictions. Thus, the lack of contradiction does not allow one to make any conclusions (unless every number is assigned true or false without contradiction, delivering the solution to the puzzle).\n\nFinding a contradiction based on the assumption that a number is false allows one to asign this number true. This can be very rewarding if the cell being assigned a single number (ie being solved) had three or more candidates to begin. However, contradictions based on a number being true only yield a single elimination of that number.\n\nIn order to clarify the process, a utility is provided which lets you enter the number and its address, and then press the 'Show True/False' button, and the end result of all the strong and weak chains that can be generated is displayed. Red numbers are 'true', blue are 'false', and orange are ones that have been assigned both true and false. These latter are the result of a number at an address which was first assigned true and added to an array of true number+address, was later assigned false, and added to the array of false number+address.\n\nWhen you try your luck with hard puzzles like the Easter Monster, even the above techniques are not enough. Postulating two different numbers are true and seeing if there is a contradiction doesn't help, because if there is one, one doesn't know which one is the culprit, or if both are culprits.\n\nA solution is to take all occurances of pairs, ie where a number appears only twice in a unit. If each of the pair is taken one at a time to be true, and a second number is then taken to be true and all chains possible are generated, then if a contradiction occurs both times then we are in business. Since one or other of the pair must be true, but combining with the other number causes a contradiction each time, then the other number has to be false and can be removed.\n\nIn a partially solved sudoku, naked sets are sets of n cells containing n candidates, within a single unit. The final placement of the n candidates must be one of the possible combinations of the digits. For example, with the naked triplet of 12, 123, 123 the 3 cells could end up with one of the following 4 combinations: 1-2-3, 1-3-2, 2-1-3, 2-3-1. Each of these can be tested to see if the derived forcing net creates contradictions, and those that do removed. For more detail, refer to the appropriate help page.", null, "", null, "", null, "", null, "" ]

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[ "# Indole-3-carbinol\n\nCatalog No. T2947   CAS 700-06-1\nSynonyms: 吲哚-3-甲醇, 3-Indolemethanol, Indole-3-Methanol, I3C, 3-吲哚甲醇\n\nIndole-3-carbinol, a naturally occurring, orally available cleavage product of the glucosinolate glucobrassicanin, inhibits NF-κB and IκBα kinase activation.\n\nAll products from TargetMol are for Research Use Only. Not for Human or Veterinary or Therapeutic Use.", null, "Indole-3-carbinol, CAS 700-06-1\nProduct consultation\nGet quote\nPurity: 98%\nBiological Description\nChemical Properties\nStorage & Solubility Information\n Description Indole-3-carbinol, a naturally occurring, orally available cleavage product of the glucosinolate glucobrassicanin, inhibits NF-κB and IκBα kinase activation.\n Synonyms 吲哚-3-甲醇, 3-Indolemethanol, Indole-3-Methanol, I3C, 3-吲哚甲醇 Molecular Weight 147.177 Formula C9H9NO CAS No. 700-06-1\n\n#### Storage\n\nPowder: -20°C for 3 years\n\nIn solvent: -80°C for 2 years\n\n#### Solubility Information\n\nDMSO: 28 mg/mL (190.3 mM)\n\nEthanol: 28 mg/mL (190.3 mM)\n\nH2O: 7 mg/mL (47.56 mM)\n\n( < 1 mg/ml refers to the product slightly soluble or insoluble )\n\n## Related compound libraries\n\nThis product is contained In the following compound libraries：\n\n## Related Products\n\nRelated compounds with same targets\n\n##", null, "Dose Conversion\n\nYou can also refer to dose conversion for different animals. More\n\n##", null, "In vivo Formulation Calculator (Clear solution)\n\nStep One: Enter information below\nDosage\nmg/kg\nAverage weight of animals\ng\nDosing volume per animal\nul\nNumber of animals\nStep Two: Enter the in vivo formulation\n% DMSO\n%\n% Tween 80\n% ddH2O\n\n##", null, "Calculator\n\nMolarity Calculator\nDilution Calculator\nReconstitution Calculation\nMolecular Weight Calculator\n=\nX\nX\n\n### Molarity Calculator allows you to calculate the\n\n• Mass of a compound required to prepare a solution of known volume and concentration\n• Volume of solution required to dissolve a compound of known mass to a desired concentration\n• Concentration of a solution resulting from a known mass of compound in a specific volume\nSee Example\n\nAn example of a molarity calculation using the molarity calculator\nWhat is the mass of compound required to make a 10 mM stock solution in 10 ml of water given that the molecular weight of the compound is 197.13 g/mol?\nEnter 197.13 into the Molecular Weight (MW) box\nEnter 10 into the Concentration box and select the correct unit (millimolar)\nEnter 10 into the Volume box and select the correct unit (milliliter)\nPress calculate\nThe answer of 19.713 mg appears in the Mass box\n\nX\n=\nX\n\n### Calculator the dilution required to prepare a stock solution\n\nCalculate the dilution required to prepare a stock solution\nThe dilution calculator is a useful tool which allows you to calculate how to dilute a stock solution of known concentration. Enter C1, C2 & V2 to calculate V1.\n\nSee Example\n\nAn example of a dilution calculation using the Tocris dilution calculator\nWhat volume of a given 10 mM stock solution is required to make 20ml of a 50 μM solution?\nUsing the equation C1V1 = C2V2, where C1=10 mM, C2=50 μM, V2=20 ml and V1 is the unknown:\nEnter 10 into the Concentration (start) box and select the correct unit (millimolar)\nEnter 50 into the Concentration (final) box and select the correct unit (micromolar)\nEnter 20 into the Volume (final) box and select the correct unit (milliliter)\nPress calculate\nThe answer of 100 microliter (0.1 ml) appears in the Volume (start) box\n\n=\n/\n\n### Calculate the volume of solvent required to reconstitute your vial.\n\nThe reconstitution calculator allows you to quickly calculate the volume of a reagent to reconstitute your vial.\nSimply enter the mass of reagent and the target concentration and the calculator will determine the rest.\n\ng/mol\n\n### Enter the chemical formula of a compound to calculate its molar mass and elemental composition\n\nTip: Chemical formula is case sensitive: C10H16N2O2 c10h16n2o2\n\nInstructions to calculate molar mass (molecular weight) of a chemical compound:\nTo calculate molar mass of a chemical compound, please enter its chemical formula and click 'Calculate'.\nDefinitions of molecular mass, molecular weight, molar mass and molar weight:\nMolecular mass (molecular weight) is the mass of one molecule of a substance and is expressed n the unified atomic mass units (u). (1 u is equal to 1/12 the mass of one atom of carbon-12)\nMolar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol.\n\nbottom\n\n## Tech Support\n\nPlease see Inhibitor Handling Instructions for more frequently ask questions. Topics include: how to prepare stock solutions, how to store products, and cautions on cell-based assays & animal experiments, etc." ]

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[ "Weak Gravity Conjecture and Holographic Dark Energy Model with Interaction and Spatial Curvature\n\n# Weak Gravity Conjecture and Holographic Dark Energy Model with Interaction and Spatial Curvature\n\nCheng-Yi [email protected]; [email protected]\n\nInstitute of Modern Physics, Northwest University,\nXian 710069, P.R. China.\n###### Abstract\n\nIn the paper, we apply the weak gravity conjecture to the holographic quintessence model of dark energy. Three different holographic dark energy models are considered: without the interaction in the non-flat universe; with interaction in the flat universe; with interaction in the non-flat universe. We find that only in the models with the spatial curvature and interaction term proportional to the energy density of matter, it is possible for the weak gravity conjecture to be satisfied. And it seems that the weak gravity conjecture favors an open universe and the decaying of matter into dark energy.\n\nPACS: 98.80.Cq; 98.80.-k; 11.25.-w\n\nKey words: weak gravity conjecture, holographic, quintessence\n\n## 1 Introduction\n\nIncreasing evidence suggests that the expansion of our universe is being accelerated [1, 2, 3]. Within the framework of the general relativity, the acceleration can be phenomenally attributed to the existence of a mysterious exotic component with negative pressure, namely the dark energy [4, 5], which dominates the present evolution of the universe. However, we know little about the nature of dark energy. The dark energy problem has become one of the focuses in the fields of both cosmology and theoretical physics today.\n\nThe most nature, simple and important candidate for dark energy is the Einstein’s cosmological constant, which can fit the observations well so far. But the cosmological constant is plagued with the well-known fine-tuning and cosmic coincidence problems [4, 5]. Another candidate for dark energy is scalar-field dark energy model. So far, a wide variety of scalar-filed dark energy models have been proposed, such as quintessence , phantom , -essence , tachkyon , quintom , hessence , etc. Usually, the scalar-field models are regarded as an low-energy effective description of the underlying theory of dark energy. Other dynamical dark energy models include Chaplygin gas models , braneworld models , etc.\n\nA lot of efforts have been made to solve the dark energy problem, but no effort seems to be successful so far. Actually, it is generally believed that the dark energy problem is in essence an issue of quantum gravity. However a complete theory of quantum gravity is still unknown. Then it becomes natural for physicists to explore the nature of dark energy just in light of some fundamental principles of quantum gravity. The holographic principle is commonly believed to be such a principle . Based on the principle, the holographic dark energy models has been suggested . The model has been studied widely, and supported by various observations (see citations of Ref.). And even, it is found that the holographic dark energy model is favored by the anthropic principle .\n\nOn the other hand, it is generally believed that string theory is the most promising theory of quantum gravity. Recent progress suggests that there exist a vast number of semi-classical consistent vacua in string theory, named Landscape . However, not all semi-classical consistent vacua are actually consistent on the quantum level, and these actually inconsistent vacua are called Swampland . Self-consistent landscape is surrounded by the swampland. The weak gravity conjecture (WGC) is suggested to be a new criterion to distinguish the landscape from the swampland [20, 21]. The conjecture can be most simply stated as gravity is the weakest force. For a four-dimensional U(1) gauge theory, WGC implies that there is an intrinsic UV cutoff \n\n Λ≤gMp,\n\nwhere is the gauge coupling constant and is the Planck scale. Furthermore, in , it is argued that WGC also indicates an intrinsic UV cutoff for the scalar field theories with gravity, e.g.\n\n Λ≤λ1/2Mp\n\nfor theory. In the slow-roll inflation model with the potential , Hubble constant can be taken as the IR cutoff for the field theory. Then the requirement that the IR cutoff should be lower than the UV cutoff indicates \n\n λ1/2ϕ2Mp∼H≤Λ≤λ1/2Mp,or,ϕ≤Mp (1)\n\nThis leads the author in to conjecture that the variation of the inflaton during the period of inflation should be less than ,\n\n |Δϕ|≤Mp. (2)\n\nAnd it is found that this can make stringent constraint on the spectral index of the inflation model .\n\nRecently, the criterion (2) has been used in to explore the quintessence model of dark energy, in to study Chaplygin gas models , and in to study the agegraphic dark energy model . Then if the holographic dark energy scenario is assumed to be the underlying theory of dark energy, and the low-energy scalar field can be used to describe it effectively , can the weak gravity conjecture, i.e., , be satisfied? In the direction, some work has been done [29, 30]. It is found that the holographic quintessence model does not satisfied the conjecture . However, in [29, 30], only the non-interacting holographic quintessence in the flat universe is discussed. In , it is found that when simultaneously considering the interaction and spatial curvature in the holographic dark energy model, the parameter space is amplified much more. Then it may be possible for the weak gravity conjecture to be satisfied in the interacting holographic quintessence model with spatial curvature. Here we will discuss the problem.\n\nIn the paper, we will first recall the interacting holographic dark energy model in the non-flat universe. Then we will discuss the possible theoretical constraints on the holographic quintessence model from the weak gravity conjecture and try to find the possibility for the conjecture to be satisfied within the parameter space displayed in . Finally, conclusion will be given.\n\n## 2 Holographic Dark Energy Model with Interaction and Space Curvature\n\nThe Friedmann-Robertson-Walker (FRW) universe is described by the line element\n\n ds2=−dt2+a2(t)(dr21−kr2+r2dΩ2), (3)\n\nwhere is the scale factor, and is the curvature parameter with corresponding to a spatially open, flat and closed universe, respectively. The Friedmann equation is\n\n 3M2p(H2+ka2)=ρm+ρD, (4)\n\nwhere , is the energy density of matter and is the energy density of dark energy. By defining\n\n Ωk=ρkρc=kH2a2,ΩD=ρDρc,Ωm=ρmρc, (5)\n\nwhere and , we can rewrite the Friedmann equation as\n\n 1+Ωk=ΩD+Ωm. (6)\n\nIn the holographic dark energy model, the energy density is assumed to be \n\n ρD=3d2M2pR−2h, (7)\n\nwhere is a constant parameter, and satisfies \n\n ∫r(t)0dr√1−kr2=∫+∞tdta(t). (8)\n\nFor a closed universe, , from the equation above we have\n\n arcsin(√kr)=√k∫+∞tdta(t). (9)\n\nTogether with Eq.(7), we have\n\n arcsin(√k3d2M2pa2ρD)=√k∫+∞tdta(t). (10)\n\nThe derivative of the equation with respect to gives\n\n ˙ρD=−3HρD×23(1−√ΩDd2−Ωk). (11)\n\nThe equation is obtained by using . But it can be easily checked that Eq.(11) holds in the cases of and , too. We can define an effective equation of state parameter by\n\n ˙ρD+3H(1+weffD)ρD=0. (12)\n\nThen comparing Eqs.(11) and (12), we have\n\n weffD=−13(1+2√ΩDd2−Ωk). (13)\n\nUsing Eqs.(5), we may rewrite Eq.(11) as \n\n √ΩDH2d2−ka2=˙ΩD2ΩD+H+˙HH. (14)\n\nNow consider some interaction between holographic dark energy and matter \n\n ˙ρm+3Hρm =Q, (15) ˙ρD+3H(1+wD)ρD =−Q, (16)\n\nwhere denotes the phenomenological interaction term. In Ref. three types of interaction are considered,\n\n Q1 =−3bHρD (17) Q2 =−3bH(ρD+ρm) (18) Q3 =−3bHρm. (19)\n\nFollowing Ref., for convenience, we uniformly express the interaction term as , where and , for and , respectively.\n\nFrom Eqs.(4), (15) and (16), we can obtain\n\n wD=−13HρD(2˙HHρc−2Hρk)−ρc−ρk. (20)\n\nSubstituting the equation into Eq.(16) and using Eq.(5), we can have \n\n 2(ΩD−1)˙HH+˙ΩD+H(3ΩD−3−Ωk)=3bHΩi, (21)\n\nwhere and . From Eqs.(14) and (21), finally we get the following two equations governing the evolution of the interacting holographic dark energy in the non-flat universe\n\n d˜Hdz =−˜H1+zΩD(3ΩD−Ωk0(1+z)2˜H2−3−3bΩi2ΩD−1+ ⎷ΩDd2−Ωk0(1+z)2˜H2), (22) dΩDdz =−2ΩD(1−ΩD)1+z( ⎷ΩDd2−Ωk0(1+z)2˜H2−1−3ΩD−Ωk0(1+z)2˜H2−3−3bΩi2(1−ΩD)), (23)\n\nwhere , and hereafter the subscript denotes the present value of the corresponding parameter. And we have used and .\n\n## 3 Holographic Quintessence Model and Weak Gravity Conjecture\n\nFor a single-scalar-field quintessence model with potential , the energy density and pressure of the quintessence scalar field are\n\n ρϕ =12˙ϕ2+V(ϕ), (24) pϕ =12˙ϕ2−V(ϕ). (25)\n\nSo the equation of state is\n\n wϕ=ρϕpϕ=12˙ϕ2+V12˙ϕ2−V (26)\n\nFrom Eqs.(24) and (26), we can obtain easily\n\n ρϕ=˙ϕ21+wϕ (27)\n\nWithout loss of generality, we may assume and . Thus from Eq.(27), we may have\n\n ˙ϕ=√(1+wϕ)ρϕ (28)\n\nThen the weak gravity conjecture tells us [22, 24, 29, 30]\n\n 1≥|Δϕ(z)|Mp=∫˙ϕMpdt=∫z0√3[1+wϕ(z′)]Ωϕ(z′)dz′1+z′, (29)\n\nwhere .\n\nIn the holographic quintessence model, the holographic dark energy is assumed to be described by the effective scalar field. Then naturally we have\n\n ρϕ=ρD⇒Ωϕ=ΩD (30)\n\nHere following Ref., we identify with , instead of ,\n\n wϕ=weffD=−13(1+2√ΩDd2−Ωk). (31)\n\nWe must require so that . Substituting Eq.(31) into Eq.(29), then the weak gravity conjecture for the holographic quintessence with interaction and spatial curvature reads\n\n 1≥|Δϕ(z)|Mp=∫z0 ⎷2(1− ⎷ΩD(z′)d2−Ωk0(1+z)2˜H2)ΩD(z′)dz′1+z′. (32)\n\nwhere and can obtained by numerically solving Eqs.(22) and (23) if the initial values and , and the values of the constant parameters and are given.\n\nThe case of and has been discussed in Ref.[29, 30], and it is found the weak gravity conjecture cannot be satisfied for a holographic quintessence model. Here we expect the weak gravity conjecture may be satisfied when both the interaction and spatial curvature are considered.\n\nSince\n\n ΩD=d2H2R2h, (33)\n\nthen larger indicates bigger and more difficult for Eq.(32) to be satisfied. On the other hand, we must require so that . So in the paper, we will focus on the case of . Then Eq.(32) becomes\n\n 1≥|Δϕ(z)|Mp=∫z0 ⎷2(1− ⎷ΩD(z′)−Ωk0(1+z)2˜H2)ΩD(z′)dz′1+z′. (34)\n\n### 3.1 the non-interacting holographic quintessence model with spatial curvature\n\nWith , we can rewrite Eq.(34) as\n\n 1≥|Δϕ(z)|Mp=∫z0 ⎷2(1−Ωm0(1+z)α˜H2)ΩD(z′)dz′1+z′, (35)\n\nwhere , and we have used and Eq.(6). Then the equation above tells us that larger will make it easier for Eq.(35) to be satisfied. However, in Ref., it is shown that Eq.(35) can not be satisfied in the flat universe even for .\n\nNow let us consider the effect of the spatial curvature. From Eq.(6), we know that larger indicates larger and then makes it more difficult for Eq.(35) to be satisfied. So negative will make it easier for Eq.(35) to be satisfied than that in the flat universe. We plot the result of versus the redshift in Fig.1. In the case, we find that it is impossible for the weak gravity conjecture to be satisfied within the parameter space displayed in FIG.4 in . Actually, in order to match the weak gravity conjecture, we should take which has been far outside the range of displayed in FIG. 4 in . In Fig.1, we have fixed which is slightly bigger than the maximum value of displayed in FIG.4 in Ref., since smaller would make it more difficult for Eq.(35) to be satisfied.", null, "Fig. 1: Δϕ(z)/Mp versus the redshift z in the non-interacting holographic quintessence model with spatial curvature for fixed Ωm0=0.34 and different Ωk0.\n\n### 3.2 the interacting holographic quintessence model without spatial curvature\n\nWith the interaction between the dark energy and matter, we can rewrite Eq.(34) as\n\n 1≥|Δϕ(z)|Mp=∫z0 ⎷2(1−Ωm0(1+z)α(z)˜H2)ΩD(z′)dz′1+z′, (36)\n\nwhere is defined by\n\n lnρmρm0=α(z)ln(1+z). (37)\n\n 1=Ωm+ΩD.\n\nSince the form of Eq.(36) is similar to that of Eq.(35), we can get a similar conclusion that larger will make it easier for Eq.(36) to be satisfied.\n\nFor the three types of interaction, we can uniformly express the conservation law Eq.(15) as\n\n ˙ρm=−3H(1+bΩiΩm)ρm, (38)\n\nwhere and . From Eqs.(37) and (38), we have\n\n α(z)=3ln(1+z)∫z0(1+bΩi(z′)Ωm(z′))dz′1+z′. (39)\n\nObviously, for , for and for . Or, in the other words, smaller indicates smaller , and then will make it more difficult for Eq.(36) to be satisfied.\n\nOn the other hand, it has been shown in that the weak gravity conjecture cannot be satisfied in the holographic quintessence models in the flat universe with even for . Since smaller or would make it more difficult for the weak gravity conjecture to be satisfied, and has been far beyond the range of displayed in FIG.1 in Ref., we can conclude that the weak gravity conjecture cannot be satisfied in the flat universe with non-positive . Our conclusion is illustrated by the results displayed in Fig.2 and Fig.3.", null, "Fig. 2: Δϕ(z)/Mp versus the redshift z in the holographic quintessence model with the interaction term Q=Q1 in the flat universe for fixed Ωm0=0.34 and different b.", null, "Fig. 3: Δϕ(z)/Mp versus the redshift z in the holographic quintessence model with the interaction term Q=Q2 in the flat universe for fixed Ωm0=0.34 and different b.", null, "Fig. 4: Δϕ(z)/Mp versus the redshift z in the holographic quintessence model with the interaction term Q=Q3 in the flat universe for fixed Ωm0=0.34 and different b.\n\nFor the models with the interaction term or , only the case of is regarded as the realistic physical situation. The reason is that in the two types of models, positive will lead to become negative in the far future. Then we know that, in the holographic quintessence models with interaction term or in the flat universe, the weak gravity conjecture can not be satisfied, as shown in Fig.2 and Fig.3.\n\nIf , the conservation law of matter is\n\n ˙ρm=−3H(1+b)ρm⇒ρm∝a−3(1+b). (40)\n\nThen is also in the realistic physical region since will never become negative in the case. But we find even for positive , in the range of given in FIG.2 in Ref., Eq.(36) can not be satisfied yet. Our results are shown in Fig.4. Naively, in order to match the weak gravity conjecture, we should take which is far beyond the range of given in FIG.2 in Ref..\n\nIn the subsection, we still fix which is slightly bigger than the maximum value of displayed in FIG.2 in Ref., since smaller would make it more difficult for Eq.(36) to be satisfied.\n\n### 3.3 the holographic quintessence model with interaction and spatial curvature\n\nWhen both the interaction and spatial curvature are considered, we still have Eqs.(36) and (39). Of course, now the Friedmann equation includes the spatial curvature:\n\n 1+Ωk=Ωm+ΩD.\n\nThe analysis of the effects of and still works in the models with interaction and spatial curvature: larger , smaller or larger will make it easier for the weak gravity conjecture to be satisfied. From FIG.5 in , we know and is anti-correlated: larger corresponds to smaller . Then there might exit the combinations of , and that satisfies the weak gravity conjecture Eq.(36).\n\nHowever, unfortunately, for the models with the interaction term or in the non-flat universe, we can not find any combination of , and within the parameter space given in FIG.5 in Ref. so that Eq.(36) is satisfied. We display our results in Fig.5 and Fig.6. In the two figures, we fix and , since smaller or negative would make it more difficult for Eq.(36) to be satisfied, and larger or positive are not physical.", null, "Fig. 5: Δϕ(z)/Mp versus the redshift z in the holographic quintessence model with the interaction term Q=Q1 and spatial curvature for Ωm0=0.35, b=0 and different Ωk0.", null, "Fig. 6: Δϕ(z)/Mp versus the redshift z in the holographic quintessence model with the interaction term Q=Q2 and spatial curvature for Ωm0=0.35, b=0 and different Ωk0.", null, "Fig. 7: Δϕ(z)/Mp versus the redshift z in the holographic quintessence model with the interaction term Q=Q1 and spatial curvature for different Ωm0, b and Ωk0.\n\nFor the models with the interaction term , We find that the weak gravity conjecture can not be satisfied within the parameter space displayed in FIG.5 in if . However, if and , the combinations of and within the parameter space in FIG.5 of Ref. which satisfy Eq.(36) can be found. Our results are shown in Fig.7.\n\n## 4 Conclusion\n\nIn the paper, we have discussed the theoretical limits on the holographic quintessence model of dark energy from the weak gravity conjecture. Since the non-interacting holographic quintessence model without spatial curvature has been investigated in [29, 30], here we consider the other three cases separately: without interaction in the non-flat universe; with interaction in the flat universe; with interaction in the non-flat universe. Here, we use the observational constraints given in . We find that the the weak gravity conjecture can not be satisfied even in the the holographic quintessence models with interaction in the flat universe or the models without interaction in the non-flat universe. In , it is shown that the parameter space is amplified when simultaneously considering the interaction and spatial curvature. Then we might expect that it should be possible for the weak gravity conjecture to be satisfied in the models with interaction and spatial curvature.\n\nHowever, we find that the models with the interaction term or in the non-flat universe are still inconsistent with the weak gravity conjecture within the parameter space in . Fortunately, we find that, in the models with the spatial curvature and interaction term , it is possible for the weak gravity conjecture to be satisfied. A roughly necessary condition for the weak gravity conjecture to be satisfied is shown: and .\n\nThen our results indicate that only the holographic dark energy models with the spatial curvature and interaction term may be described by a consistent low-energy effective scalar field theory. And it seems that the weak gravity conjecture favors an open universe and the decaying of matter into dark energy.\n\n## Acknowledgments\n\nThis work is supported by the Natural Science Foundation of the Northwest University of China under Grant No. NS0927.\n\n## References\n\n• A. G. Riess et al., Astron. J. 116, 1009 (1998), [astro-ph/9805201]; S. Perlmutter et al., Astrophys. J. 517, 565 (1999), [astro-ph/9812133].\n• D. N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003), [astro-ph/0302209]; D. N. Spergel et al., Astrophys. J. Suppl. 170, 377 (2007), [astro-ph/0603449].\n• M. Tegmark et al., Phys. Rev. D 69, 103501 (2004), [astro-ph/0310723]; K. Abazajian et al., Astron. J. 128, 502 (2004), [astro-ph/0403325]; K. Abazajian et al., Astron. J. 129, 1755 (2005), [astro-ph/0410239].\n• S. Weinberg, Rev. Mod. Phys. 61, 1 (1989);\n• V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), [astro-ph/9904398]; S. M. Carroll, Living Rev. 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Lett. B 632, 605 (2006), [arXiv:gr-qc/0509040].\nYou are adding the first comment!\nHow to quickly get a good reply:\n• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.\n• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.\n• Your comment should inspire ideas to flow and help the author improves the paper.\n\nThe better we are at sharing our knowledge with each other, the faster we move forward.\nThe feedback must be of minimum 40 characters and the title a minimum of 5 characters", null, "", null, "", null, "" ]

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[ "Isothermal and isentropic process are identical at\n\nThis question was previously asked in\nUPRVUNL JE ME 2016 Official Paper Shift 1\nView all UPRVUNL JE Papers >\n1. Saturation temperature\n2. Critical temperature\n3. Absolute zero temperature\n4. Below 0°C temperature\n\nOption 3 : Absolute zero temperature\n\nDetailed Solution\n\nExplanation:\n\nFor isentropic process:\n\n$$\\frac{T_2}{T_1}=\\left(\\frac{P_2}{P_1}\\right)^{\\frac{γ -1}{γ}}=\\left(\\frac{V_1}{V_2}\\right)^{{γ -1}}$$\n\nFor isothermal process T1 = T2\n\n$$\\frac{T_2}{T_1}=\\left(\\frac{P_2}{P_1}\\right)^{\\frac{γ -1}{γ}}$$\n\n$$1=\\left(\\frac{P_2}{P_1}\\right)^{\\frac{γ -1}{γ}}$$\n\n$$\\frac{γ -1}{γ}=0$$\n\nγ = 1\n\nγ is the ratio of specific heat at constant pressure Cp and constant volume Cv.\n\nWe know that for the difference in specific heat at constant pressure Cp and constant volume Cv is:\n\n$$C_p-C_v=-T\\left(\\frac{\\partial S}{\\partial V}\\right)_T\\left(\\frac{\\partial S}{\\partial P}\\right)_T$$\n\nAt Absolute zero temperature dS = 0\n\n∴ Cp - Cv = 0\n\nCp = Cv\n\n$$\\frac{C_p}{C_v}=1$$" ]

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[ "# Using GPU ARRAYFUN for Monte-Carlo Simulations\n\nThis example shows how prices for financial options can be calculated on a GPU using Monte-Carlo methods. Three simple types of exotic option are used as examples, but more complex options can be priced in a similar way.\n\nThis example is a function so that the helpers can be nested inside it.\n\nfunction paralleldemo_gpu_optionpricing \n\nThis example uses long-running kernels, so cannot run if kernel execution on the GPU can time-out. A time-out is usually only active if the selected GPU is also driving a display.\n\ndev = gpuDevice(); if dev.KernelExecutionTimeout error( 'pctexample:gpuoptionpricing:KernelTimeout', ... ['This example cannot run if kernel execution on the GPU can ', ... 'time-out.'] ); end \n\n### Stock Price Evolution\n\nWe assume that prices evolve according to a log-normal distribution related to the risk-free interest rate, the dividend yield (if any), and the volatility in the market. All of these quantities are assumed fixed over the lifetime of the option. This gives the following stochastic differential equation for the price:", null, "where", null, "is the stock price,", null, "is the risk-free interest rate,", null, "is the stock's annual dividend yield,", null, "is the volatility of the price and", null, "represents a Gaussian white-noise process. Assuming that", null, "is log-normally distributed, this can be discretized to:", null, "As an example let's use $100 of stock that yields a 1% dividend each year. The central government interest rate is assumed to be 0.5%. We examine a two-year time window sampled roughly daily. The market volatility is assumed to be 20% per annum. stockPrice = 100; % Stock price starts at$100. dividend = 0.01; % 1% annual dividend yield. riskFreeRate = 0.005; % 0.5 percent. timeToExpiry = 2; % Lifetime of the option in years. sampleRate = 1/250; % Assume 250 working days per year. volatility = 0.20; % 20% volatility. \n\nWe reset the random number generators to ensure repeatable results.\n\nseed = 1234; rng( seed ); % Reset the CPU random number generator. gpurng( seed ); % Reset the GPU random number generator. \n\nWe can now loop over time to simulate the path of the stock price:\n\nprice = stockPrice; time = 0; hold on; while time < timeToExpiry time = time + sampleRate; drift = (riskFreeRate - dividend - volatility*volatility/2)*sampleRate; perturbation = volatility*sqrt( sampleRate )*randn(); price = price*exp(drift + perturbation); plot( time, price, '.' ); end axis tight; grid on; xlabel( 'Time (years)' ); ylabel( 'Stock price ($)' );", null, "### Running on the GPU To run stock price simulations on the GPU we first need to put the simulation loop inside a helper function: function finalStockPrice = simulateStockPrice(S,r,d,v,T,dT) t = 0; while t < T t = t + dT; dr = (r - d - v*v/2)*dT; pert = v*sqrt( dT )*randn(); S = S*exp(dr + pert); end finalStockPrice = S; end We can then call it thousands of times using arrayfun. To ensure the calculations happen on the GPU we make the input prices a GPU vector with one element per simulation. To accurately measure the calculation time on the GPU we use the gputimeit function. % Create the input data. N = 1000000; startStockPrices = stockPrice*ones(N,1,'gpuArray'); % Run the simulations. finalStockPrices = arrayfun( @simulateStockPrice, ... startStockPrices, riskFreeRate, dividend, volatility, ... timeToExpiry, sampleRate ); meanFinalPrice = mean(finalStockPrices); % Measure the execution time of the function on the GPU using gputimeit. % This requires us to store the |arrayfun| call in a function handle. functionToTime = @() arrayfun(@simulateStockPrice, ... startStockPrices, riskFreeRate, dividend, volatility, ... timeToExpiry, sampleRate ); timeTaken = gputimeit(functionToTime); fprintf( 'Calculated average price of$%1.4f in %1.3f secs.\\n', ... meanFinalPrice, timeTaken ); clf; hist( finalStockPrices, 100 ); xlabel( 'Stock price ($)' ) ylabel( 'Frequency' ) grid on; Calculated average price of$98.9563 in 0.283 secs.", null, "### Pricing an Asian Option\n\nAs an example, let's use a European Asian Option based on the arithmetic mean of the price of the stock during the lifetime of the option. We can calculate the mean price by accumulating the price during the simulation. For a call option, the option is exercised if the average price is above the strike, and the payout is the difference between the two:\n\n function optionPrice = asianCallOption(S,r,d,v,x,T,dT) t = 0; cumulativePrice = 0; while t < T t = t + dT; dr = (r - d - v*v/2)*dT; pert = v*sqrt( dT )*randn(); S = S*exp(dr + pert); cumulativePrice = cumulativePrice + S; end numSteps = (T/dT); meanPrice = cumulativePrice / numSteps; % Express the final price in today's money. optionPrice = exp(-r*T) * max(0, meanPrice - x); end \n\nAgain we use the GPU to run thousands of simulation paths using arrayfun. Each simulation path gives an independent estimate of the option price, and we therefore take the mean as our result.\n\nstrike = 95; % Strike price for the option ($). optionPrices = arrayfun( @asianCallOption, ... startStockPrices, riskFreeRate, dividend, volatility, strike, ... timeToExpiry, sampleRate ); meanOptionPrice = mean(optionPrices); % Measure the execution time on the GPU and show the results. functionToTime = @() arrayfun( @asianCallOption, ... startStockPrices, riskFreeRate, dividend, volatility, strike, ... timeToExpiry, sampleRate ); timeTaken = gputimeit(functionToTime); fprintf( 'Calculated average price of$%1.4f in %1.3f secs.\\n', ... meanOptionPrice, timeTaken ); \nCalculated average price of $19.2711 in 0.286 secs. ### Pricing a Barrier Option This final example uses an \"up and out\" barrier option which becomes invalid if the stock price ever reaches the barrier level. If the stock price stays below the barrier level then the final stock price is used in a normal European call option calculation. function optionPrice = upAndOutCallOption(S,r,d,v,x,b,T,dT) t = 0; while (t < T) && (S < b) t = t + dT; dr = (r - d - v*v/2)*dT; pert = v*sqrt( dT )*randn(); S = S*exp(dr + pert); end if S<b % Within barrier, so price as for a European option. optionPrice = exp(-r*T) * max(0, S - x); else % Hit the barrier, so the option is withdrawn. optionPrice = 0; end end Note that we must now supply both a strike price for the option and the barrier price at which it becomes invalid: strike = 95; % Strike price for the option ($). barrier = 150; % Barrier price for the option ($). optionPrices = arrayfun( @upAndOutCallOption, ... startStockPrices, riskFreeRate, dividend, volatility, ... strike, barrier, ... timeToExpiry, sampleRate ); meanOptionPrice = mean(optionPrices); % Measure the execution time on the GPU and show the results. functionToTime = @() arrayfun( @upAndOutCallOption, ... startStockPrices, riskFreeRate, dividend, volatility, ... strike, barrier, ... timeToExpiry, sampleRate ); timeTaken = gputimeit(functionToTime); fprintf( 'Calculated average price of$%1.4f in %1.3f secs.\\n', ... meanOptionPrice, timeTaken ); \nCalculated average price of \\$6.8166 in 0.289 secs. \nend \n\n## Support", null, "" ]

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[ "# Answers\n\nSolutions by everydaycalculation.com\n\n## Compare 42/9 and 4/50\n\n1st number: 4 6/9, 2nd number: 4/50\n\n42/9 is greater than 4/50\n\n#### Steps for comparing fractions\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 9 and 50 is 450\n\nNext, find the equivalent fraction of both fractional numbers with denominator 450\n2. For the 1st fraction, since 9 × 50 = 450,\n42/9 = 42 × 50/9 × 50 = 2100/450\n3. Likewise, for the 2nd fraction, since 50 × 9 = 450,\n4/50 = 4 × 9/50 × 9 = 36/450\n4. Since the denominators are now the same, the fraction with the bigger numerator is the greater fraction\n5. 2100/450 > 36/450 or 42/9 > 4/50\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:\nAndroid and iPhone/ iPad\n\n#### Compare Fractions Calculator\n\nand\n\n© everydaycalculation.com" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "### These are solutions for NPTEL Introduction To Machine Learning IITKGP ASSIGNMENT 5 Week 5\n\nCourse Name: Introduction To Machine Learning IITKGP\n\nQ1) What would be the ideal complexity of the curve which can be used for separating the two classes shown in the image below?\nA) Linear\nC) Cubic\nD) insufficient data to draw conclusion\n\nQ2) Which of the following option is true?\nA) Linear regression error values have to normally distributed but not in the caseof the logistic regression\nB) Logistic regression values have to be normally distributed but not in the case of the linear regression\nC) Both linear and logistic regression error values have to be normally distributed\nD) Both linear and logistic regression error values need not to be normally distributed\n\nAnswer: A) Linear regression error values have to normally distributed but not in the caseof the logistic regression\n\nThese are solutions for NPTEL Introduction To Machine Learning IITKGP ASSIGNMENT 5 Week 5\n\nQ3) Which of the following methods do we use to best fit the data in Logistic Regression?\nA) Manhattan distance\nB) Maximum Likelihood\nC) Jaccard distance\nD) Both A and B\n\nQ4) Imagine, you have given the below graph of logistic regression which shows the relationships between cost function and number of iterations for 3 different learning rate values (different colors are showing different curves at different learning rates).\n\nSuppose, you save the graph for future reference but you forgot to save the value of different learning rates for this graph. Now, you want to find out the relation between the leaning rate values of these curve. Which of the following will be the true relation?\nNote : 1. The learning rate for blue is L1.\n2. The learning rate for red is L2.\n3. The learning rate for green is L3.\n\nA) L1>L2>L3\nB) L1=L2-L3\nC) L1 L2 L3\nD) None of these\n\nThese are solutions for NPTEL Introduction To Machine Learning IITKGP ASSIGNMENT 5 Week 5\n\nQ5) State whether True or False.\nAfter training an SVM, we can discard all examples which are not support vectors and can still classify new examples.\n\nA) TRUE\nB) FALSE\n\nQ6) Suppose you are dealing with 3 class classification problem and you want to train a SVM model on the data for that you are using One-vs-all method. How many times we need to train our SVM model in such case?\nA) 1\nB) 2\nC) 3\nD) 4\n\nThese are solutions for NPTEL Introduction To Machine Learning IITKGP ASSIGNMENT 5 Week 5\n\nQ7) What is/are true about kernel in SVM?\n1. Kernel function map low dimensional data to high dimensional space\n2. It’s a similarity function\n\nA) 1\nB) 2\nC) 1 and 2\nD) None of these.\n\nQ8) Suppose you are using RBF kernel in SVM with high Gamma value. What does this signify?\nA) The model would consider even far away points from hyperplane for modelling.\nB) The model would consider only the points close to the hyperplane for modelling.\nC) The model would not be affected by distance of points from hyperplane for modelling.\nD) None of the above\n\nAnswer: B) The model would consider only the points close to the hyperplane for modelling.\n\nQ9) Below are the labelled instances of 2 classes and hand drawn decision boundaries for logistic regression. Which of the following figure demonstrates overfitting of the training data?\nA) A\nB) B\nC) C\nD) None of these\n\nQ10) What do you conclude after seeing the visualization in previous question?\nC1. The training error in first plot is higher as compared to the second and third plot.\nC2. The best model for this regression problem is the last (third) plot because it has minimum training error (zero).\nC3. Out of the 3 models, the second model is expected to perform best on unseen data.\nC4. All will perform similarly because we have not seen the test data.\n\nA) C1 and C2\nB) C1 and C3\nC) C2 and C3\nD) C4\n\nThese are solutions for NPTEL Introduction To Machine Learning IITKGP ASSIGNMENT 5 Week 5\n\nMore weeks solution of this course: https://progies.in/answers/nptel/introduction-to-machine-learning\n\n* The material and content uploaded on this website are for general information and reference purposes only. Please do it by your own first. COPYING MATERIALS IS STRICTLY PROHIBITED." ]

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[ "# #0b4828 Color Information\n\nIn a RGB color space, hex #0b4828 is composed of 4.3% red, 28.2% green and 15.7% blue. Whereas in a CMYK color space, it is composed of 84.7% cyan, 0% magenta, 44.4% yellow and 71.8% black. It has a hue angle of 148.5 degrees, a saturation of 73.5% and a lightness of 16.3%. #0b4828 color hex could be obtained by blending #169050 with #000000. Closest websafe color is: #003333.\n\n• R 4\n• G 28\n• B 16\nRGB color chart\n• C 85\n• M 0\n• Y 44\n• K 72\nCMYK color chart\n\n#0b4828 color description : Very dark cyan - lime green.\n\n# #0b4828 Color Conversion\n\nThe hexadecimal color #0b4828 has RGB values of R:11, G:72, B:40 and CMYK values of C:0.85, M:0, Y:0.44, K:0.72. Its decimal value is 739368.\n\nHex triplet RGB Decimal 0b4828 `#0b4828` 11, 72, 40 `rgb(11,72,40)` 4.3, 28.2, 15.7 `rgb(4.3%,28.2%,15.7%)` 85, 0, 44, 72 148.5°, 73.5, 16.3 `hsl(148.5,73.5%,16.3%)` 148.5°, 84.7, 28.2 003333 `#003333`\nCIE-LAB 26.329, -27.33, 13.978 2.838, 4.859, 2.796 0.27, 0.463, 4.859 26.329, 30.697, 152.913 26.329, -21.515, 17.657 22.043, -15.591, 7.91 00001011, 01001000, 00101000\n\n# Color Schemes with #0b4828\n\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #480b2b\n``#480b2b` `rgb(72,11,43)``\nComplementary Color\n• #0d480b\n``#0d480b` `rgb(13,72,11)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #0b4846\n``#0b4846` `rgb(11,72,70)``\nAnalogous Color\n• #480b0d\n``#480b0d` `rgb(72,11,13)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #460b48\n``#460b48` `rgb(70,11,72)``\nSplit Complementary Color\n• #48280b\n``#48280b` `rgb(72,40,11)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #280b48\n``#280b48` `rgb(40,11,72)``\n• #2b480b\n``#2b480b` `rgb(43,72,11)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #280b48\n``#280b48` `rgb(40,11,72)``\n• #480b2b\n``#480b2b` `rgb(72,11,43)``\n• #010603\n``#010603` `rgb(1,6,3)``\n• #041c0f\n``#041c0f` `rgb(4,28,15)``\n• #08321c\n``#08321c` `rgb(8,50,28)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #0e5e34\n``#0e5e34` `rgb(14,94,52)``\n• #127441\n``#127441` `rgb(18,116,65)``\n• #158a4d\n``#158a4d` `rgb(21,138,77)``\nMonochromatic Color\n\n# Alternatives to #0b4828\n\nBelow, you can see some colors close to #0b4828. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #0b4819\n``#0b4819` `rgb(11,72,25)``\n• #0b481e\n``#0b481e` `rgb(11,72,30)``\n• #0b4823\n``#0b4823` `rgb(11,72,35)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #0b482d\n``#0b482d` `rgb(11,72,45)``\n• #0b4832\n``#0b4832` `rgb(11,72,50)``\n• #0b4837\n``#0b4837` `rgb(11,72,55)``\nSimilar Colors\n\n# #0b4828 Preview\n\nText with hexadecimal color #0b4828\n\nThis text has a font color of #0b4828.\n\n``<span style=\"color:#0b4828;\">Text here</span>``\n#0b4828 background color\n\nThis paragraph has a background color of #0b4828.\n\n``<p style=\"background-color:#0b4828;\">Content here</p>``\n#0b4828 border color\n\nThis element has a border color of #0b4828.\n\n``<div style=\"border:1px solid #0b4828;\">Content here</div>``\nCSS codes\n``.text {color:#0b4828;}``\n``.background {background-color:#0b4828;}``\n``.border {border:1px solid #0b4828;}``\n\n# Shades and Tints of #0b4828\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #010402 is the darkest color, while #f2fdf7 is the lightest one.\n\n• #010402\n``#010402` `rgb(1,4,2)``\n• #03150c\n``#03150c` `rgb(3,21,12)``\n• #062615\n``#062615` `rgb(6,38,21)``\n• #08371f\n``#08371f` `rgb(8,55,31)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #0e5931\n``#0e5931` `rgb(14,89,49)``\n• #106a3b\n``#106a3b` `rgb(16,106,59)``\n• #137b44\n``#137b44` `rgb(19,123,68)``\n• #158c4e\n``#158c4e` `rgb(21,140,78)``\n• #189d57\n``#189d57` `rgb(24,157,87)``\n• #1bae61\n``#1bae61` `rgb(27,174,97)``\n• #1dbf6a\n``#1dbf6a` `rgb(29,191,106)``\n• #20d074\n``#20d074` `rgb(32,208,116)``\n• #26de7d\n``#26de7d` `rgb(38,222,125)``\n• #37e087\n``#37e087` `rgb(55,224,135)``\n• #48e392\n``#48e392` `rgb(72,227,146)``\n• #59e69c\n``#59e69c` `rgb(89,230,156)``\n• #6ae8a6\n``#6ae8a6` `rgb(106,232,166)``\n• #7bebb0\n``#7bebb0` `rgb(123,235,176)``\n• #8cedba\n``#8cedba` `rgb(140,237,186)``\n• #9df0c4\n``#9df0c4` `rgb(157,240,196)``\n• #aef3cf\n``#aef3cf` `rgb(174,243,207)``\n• #bff5d9\n``#bff5d9` `rgb(191,245,217)``\n• #d0f8e3\n``#d0f8e3` `rgb(208,248,227)``\n• #e1faed\n``#e1faed` `rgb(225,250,237)``\n• #f2fdf7\n``#f2fdf7` `rgb(242,253,247)``\nTint Color Variation\n\n# Tones of #0b4828\n\nA tone is produced by adding gray to any pure hue. In this case, #282b29 is the less saturated color, while #015228 is the most saturated one.\n\n• #282b29\n``#282b29` `rgb(40,43,41)``\n• #252e29\n``#252e29` `rgb(37,46,41)``\n• #213229\n``#213229` `rgb(33,50,41)``\n• #1e3529\n``#1e3529` `rgb(30,53,41)``\n• #1b3829\n``#1b3829` `rgb(27,56,41)``\n• #183b29\n``#183b29` `rgb(24,59,41)``\n• #153e28\n``#153e28` `rgb(21,62,40)``\n• #114228\n``#114228` `rgb(17,66,40)``\n• #0e4528\n``#0e4528` `rgb(14,69,40)``\n• #0b4828\n``#0b4828` `rgb(11,72,40)``\n• #084b28\n``#084b28` `rgb(8,75,40)``\n• #054e28\n``#054e28` `rgb(5,78,40)``\n• #015228\n``#015228` `rgb(1,82,40)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #0b4828 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "Calculus Of One Real Variable – By Pheng Kim Ving Chapter 1: Limits And Continuity – Section 1.1.1: Limits 1.1.1 Limits\n\n 1. Origin Of The Concept Of Limits\n\nTangents\n\nA tangent to a circle is a straight line that intersects the circle at a single point. See Fig. 1.1. So can we define a tangent\nto an arbitrary curve as a straight line that intersects the curve at a single point? To find out let's look at Fig. 1.2. Clearly", null, "Fig. 1.1   Tangent to a circle intersects circle at a single point.", null, "Fig. 1.2   T1: a tangent, T2: a tangent, S: not a tangent.\n\nline T1 intersects curve C at a single point and is a tangent to C, line T2 intersects C at more than 1 point and is tangent\nto\nC, and line S intersects C at a single point and isn't a tangent to C. Thus we can't define a tangent to a curve as a\nstraight line that intersects the curve at a single point.\n\nTo define a tangent to a curve we proceed as follow. Let C be a curve and P a point on it. Refer to Fig. 1.3. Let's define\nthe tangent to\nC at P. Let S be a secant to C thru P and intersecting C at another point Q. Let Q get closer and closer\nto, or approach or tend to,\nP. See Fig. 1.4, where Q is on one side of P, and Fig. 1.5, where Q is on the other side of P.\nPoints\nQ1, Q2, and Q3 and secants S1, S2, and S3 represent 3 positions of Q and the 3 corresponding positions of S\nrespectively as\nQ approaches P. For this curve C and this point P, as Q approaches P from one side, S approaches a", null, "Fig. 1.3   Using Secant S To Define Tangent T.", null, "Fig. 1.4   As Q approaches P from one side, S approaches T.", null, "Fig. 1.5   And as Q approaches P from the other side, S also approaches T.\n\nline T, and as Q approaches P from the other side, S also approaches the same line T. The tangent to C at P is defined\nto be\nT. We can make S arbitrarily close to T (meaning as close to T as we like) by making Q sufficiently close to P. This\nsituation was a part of the origin of the concept of limits. We say that\nT is the limit  of S or that S approaches T as Q\napproaches\nP. The tangent to C at P is defined to be the limit of secant PQ as Q approaches P. Here “approaches”\nmeans approaches from both sides. When there's no specifying of side, it's understood that both sides must be\nconsidered. The limit is a unique line that\nPQ approaches as Q approaches P from both sides of P.\n\nTo see why we require that both sides must be considered, look at Fig. 1.6. Curve C changes direction smoothly at every\npoint of it except at point\nP, where it changes direction abruptly. Now look at Fig. 1.7. As Q labelled Q1 approaches P", null, "Fig. 1.6   C changes direction abruptly at P.", null, "Fig. 1.7   L1 and L2 are different. No tangent to C at P.\n\nfrom one side of P, secant PQ labelled PQ1 approaches line L1, and as Q now labelled Q2 approaches P from the other\nside of\nP, secant PQ now labelled PQ2 approaches line L2, which is different from L1. Now if a tangent exists then it must\nbe unique. Consequently\nL1 and L2 can't be tangents to C at P. If we consider only 1 side of P then we'll be led to the\nwrong conclusion that either\nL1 or L2 is a tangent to C at P. That's why we require that both sides must be considered.\n\nSlopes\n\nRefer to Fig. 1.3. As Q approaches P, secant PQ approaches tangent T, so naturally the slope of PQ approaches the\nslope of\nT. The slope of the tangent at P is defined to be the limit of the slope of secant PQ as Q approaches P.\n\nRates Of Change\n\nImagine there's a road having a portion that's a straight line measuring a little more than 300 km. A car travelling on it\npasses point\nA at 1:00 pm, point B, which is east of A, at 3:00 pm on the same day, and point C, which is east of B, at\n5:00 pm on the same day. See Fig. 1.8. The distances\nAB and AC are 100 km and 300 km respectively. The portion\nAC is a straight line.\n\nTake the direction from west to east of the road as positive. Choose A as the initial position s = 0 of the car. So B is at\nposition\ns = 100 and C at position s = 300, where s is in km. Since AC is a straight line, any position to the right of A up\nto\nC can be determined by a single number, the positive distance from A to that position (positive because of being east\nof\nA). Select 1:00 pm when the car passes A as the initial time t = 0. Thus 3:00 pm corresponds to t = 2 and 5:00 pm to\nt = 4, where t is in h (hours). Position s is a function of time t: s = s(t). We have s(0) = 0, s(2) = 100, and s(4) = 300.\n\nIf the car travels from B to C at a constant velocity, then its average velocity over time interval [2, 4] is:", null, "and its velocity at any instant or point in that time interval, which exists as evidenced by the speedometer, is also\n100 km/h. Now suppose it travels from\nB to C at a velocity that sometimes changes. Its average velocity over [2, 4] is its", null, "Fig. 1.8   Velocity And Rate Of change.\n\nvelocity that it would have if it travelled at a constant velocity over that time interval. As a consequence that average\nvelocity is still (\ns(4) – s(2))/(4 – 2) = (300 – 100)/2 = 100 km/h. However its velocity at any particular instant in that time\ninterval may be different from its average velocity over that time interval.\n\nThe car changes position from B to C by 200 km in 2 h. Hence in average it changes position by 200/2 = 100 km per h,\nwritten as 100 km/h. As the hour is a unit of time, this is the average change of position per 1-unit change of time. It's\ncalled the average rate of change of position with respect to time. The minute, second, or day are also units of time.\nPosition is a function of time. In general the average rate of change of a function\ny = f(x) over [x1, x2] (with respect to x)\nis the average change of the function\nf per 1-unit change of the variable x in [x1, x2], given by the ratio ( f(x2) – f(x1))/\n(\nx2x1). At any point or instant x = a in dom( f ), f also has a rate of change, called instantaneous rate of change of f at\na, or simply rate of change  of f at a, because a is a point or instant and therefore the rate of change of f at it can't be\nanything other than instantaneous.\n\nConsider the function y = f(x) = x2. See Figs. 1.9 and 1.10. In dom( f ) let a be a given point and x an arbitrary point\ndifferent from\na. It may be that x > a as in Fig. 1.9 or x < a as in Fig. 1.10. The rate of change of f (with respect to x)\nis its change corresponding to a 1-unit change of\nx. The average rate of change rax of f over the interval [a, x] is defined\nas follows:", null, "We also want to define the rate of change ra of f at just point a, not over an interval. It's called the instantaneous rate of\nchange\nof f at a, to emphasize that it's the rate at an instant or a point, not over an interval. It' also simply called the", null, "Fig. 1.9   Instantaneous rate of change of f at a is equal to limit, which is a unique number, of average rate of change of f over [a, x] as x approaches a from the right ... (Continued to Fig. 1.10)", null, "Fig. 1.10   (Continued from Fig. 1.9) ... and from the left.\n\nrate of change of f at a, for the reason seen above. Now how can we define ra? If there's an interval, say [a1, a2], that\ncontains\na and such that at each point of it the rate of change is constant, then ra is equal to the average rate of change\nover [\na1, a2]. Otherwise ra generally isn't equal to the average rate of change over any interval that contains a. In this\ncase, should\nra be defined as ( f(a) _ f(a))/(a _ a), imitating the definition of the average rate of change over an interval?\nNo, because we can't divide anything by 0, not even 0 by 0. However we can choose an interval, say [\na3, a4], that\ncontains\na and that is small enough so that the rate of change at each point of it is approximately the same. Then ra is\napproximately equal to the average rate of change over [\na3, a4]. Better still, we can proceed as follows.\n\nLet x get closer and closer to a, from the right of a then from the left, but remain different from it, and see what\nhappens. As\nx gets closer and closer to a, the interval [a, x] becomes smaller and smaller toward almost being point a,\nrax gets closer and closer to ra , and the value a + x of rax gets closer and closer to a + a = 2a. So, it's reasonable to\naccept that the value of\nra is 2a. Thus, let's define ra to be the quantity that rax gets closer to, or approaches or tends to,\nas\nx gets closer to, or approaches or tends to, a, and we obtain ra = 2a, as desired. Note that in this case the\ninstantaneous rate of change is exactly, not just approximately, equal to 2\na. This situation was another part of the origin\nof the concept of limits. The limit  of\nrax as x approaches a is defined to be the quantity that rax approaches as x\napproaches\na. This yields:\n\nra = ( limit of rax as x approaches a) = 2a.\n\nNote that we require that the limits from both sides of a exist and are equal in order for the limit at a to exist, as it's\ndefined to be their common value, and the rate of change of\nf at a is defined to be it. Similarly to the case of the tangent\nwhich must be unique if it exists, the rate of change must also be unique if it exists. For example a car on a road can't\nhave more than 1 different velocity relative to that road at the same time.\n\nSlopes And Rates Of Change\n\nLet f be a function. See Fig. 1.11. Let a and u be 2 distinct points in dom( f ), P = (a, f(a)), Q = (u, f(u)), and T the\ntangent to the graph of\nf at P. We have:", null, "So slope of secant equals average rate of change and slope of tangent equals instantaneous rate of change.", null, "Fig. 1.11   Slope Of Secant = Average Rate Of Change, Slope Of Tangent = Instantaneous Rate Of Change.\n\n 2. Limits\n\nThe limit of 1/(x – 2) as x approaches 0 is –1/2. Now consider the limit of 1/(x – 2) as x approaches 2. As x approaches\n2 from the left, which means\nx approaches 2 and is < 2, x – 2 approaches 0 and is < 0, so 1/(x – 2) is < 0 and gets\nlarger and larger without bound. As\nx approaches 2 from the right, which means x approaches 2 and is > 2, x – 2\napproaches 0 and is > 0, so 1/(\nx – 2) is > 0 and gets larger and larger without bound. Thus, the limit of 1/(x – 2) as x\napproaches 2 doesn't exist. In general, the limit of\nf(x) as x approaches a may or may not exist.\n\nFor the tangent to the graph of f(x) and the instantaneous rate of change of f at x = a to exist and be unique, both the\nlimits of (\nf(x) – f(a))/(xa) as x approaches a from the right and from the left must exist and be equal, so that the\n2-sided limit as\nx approaches a from both sides exists and is unique, as it's defined to be their common value, and the\nslope of the tangent and the instantaneous rate of change will be equal to this 2-sided limit. When we talk of a limit\nwithout specifying from the left or from the right, we refer to the 2-sided limit that must be obtained equally from both\nsides. It's this type of limit that we discuss in this section.\n\nNow let's define limits formally.", null, "", null, "Fig. 2.1   Limit of f(x) as x approaches a is L.\n\napproaches 5 is 6. We remark that we can make h(x) as close to 6 as we like by making x sufficiently close to 5. In\ngeneral we say that the limit of\nf(x) as x approaches a is L if we can make the value f(x) as close to L as we like by\nmaking\nx sufficiently close to a. This means that we can make the distance | f(x) _ L| from f(x) to L as small as we like\nby making the distance\n|x _ a| from x to a sufficiently small.\n\nDefinition 2.1", null, "In the last line in the above definition, it's clear that x must also be in dom( f ) because f(x) must exist. The line is a\nsimplified form of this statement:", null, "###### Remarks 2.1", null, "", null, "Fig. 2.2", null, "", null, "", null, "#", null, "L = f(a); graph of f“touches” point (a, L) and contains it.", null, "# Fig. 2.4", null, "graph of f   touches” point (a, L) but doesn't contain it.", null, "# Fig. 2.5", null, "f(a) doesn't exist;\ngraph of f   touches” point (a, L) but doesn't contain it.\n\ngraph of f near the point (a, L) that satisfies these 2 properties: (1) it extends from the point (a, L) to both sides of\nthat point (or one side if the limit is one-sided, as we'll see in a later section), and (2) it's solid (ie, it has no holes or\njumps inside it) except it may possibly have a hole at the point (a, L). The graph of f may or may not have holes or\njumps in it, small or large. But it must have a piece of it that satisfies those 2 properties. It doesn't matter how small\nthe piece is, as long as it satisfies the 2 properties. We can intuitively think of the graph of f like this: it “ touches” the\npoint (a, L) but may or may not contain it.", null, "3. Proving A Limit Using Its Definition\n\n###### Example 3.1", null, "### Solution", null, "### EOS\n\n###### Example 3.2\n\nProve that:", null, "### Solution", null, "### EOS\n\nRemarks 3.1\n\ni.  When you're asked to prove a limit using the definition of the limit, you must just do that: use the definition of the\n\nlimit, even if the limit is intuitively “ obvious”.", null, "vi.  The definition of the limit provides a means of proving whether or not a particular number is the limit of  a\nparticular function at a particular point. It provides no means of finding an unknown limit.\n\n 4. Why Bother With The Definition Of The Limit?", null, "when x gets closer and closer to a, f(x) gets closer and closer to L.” ?”,\n\nyou propose.\n\nOK. Let's see. Suppose we take the phrase gets closer and closer to  to mean approaches  or tends to. In the definition,\nwe define what approaches\nor tends to  means. If we replace the last line of the definition with the “ proposed” line, then\nthe definition will be incomplete, because\ngets closer and closer to  simply means approaches  or tends to, and thus\nwon't have been defined yet.\n\nNow suppose we don't take gets closer and closer to  to explicitly mean approaches  or tends to, but we simply employ it\nin its ordinary everyday meaning without defining what it means mathematically. Well, in this case, trouble may be\nwaiting for us. For example, let:", null, "See Fig. 4.1. When\nx gets closer and closer to 0 from the right, f(x) gets closer and closer to 0.001 from above; since\n0.001 is close to 0, we can say that\nf(x) “ gets closer and closer to 0” from above (from above because 0.001 > 0) (if this", null, "# Fig. 4.1\n\nThe limit of f(x) as x approaches 0 isn't 0. After studying a later section\nwe'll know that such a limit doesn't exist.\n\nfails to convince you, replace 0.001 with 0.000,001). When x gets closer and closer to 0 from the left, f(x) gets closer\nand closer to\n_ 0.001 from below; since _ 0.001 is close to 0, we can say that f(x) “ gets closer and closer to 0” from\nbelow (from below because\n_ 0.001 < 0) (if this fails to convince you, replace _ 0.001 with _ 0.000,000,001).\nConsequently, we can say that when\nx gets closer and closer to 0, f(x) “ gets closer and closer to 0”. This leads us to\nconclude that\nf(x) “approaches 0” as x approaches 0. Well, clearly that conclusion is wrong, because f(x) never gets\ninto the interval (\n_ 0.001, 0.001), which contains 0, and it follows that f(x) can't approach 0. The word limit  is the name\ngiven to the number that the function approaches (or tends to or gets closer and closer to).\n\nWe can use Definition 2.1 to prove formally that:", null, "Problems & Solutions", null, "Solution", null, "", null, "", null, "Solution", null, "", null, "3.  Prove that:", null, "utilizing the definition of the limit.\n\nSolution", null, "That completes the proof.", null, "4.  Prove that:", null, "employing the definition of the limit.\n\nSolution", null, "", null, "", null, "Solution", null, "" ]

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[ "top of page\nSearch\n\n# 28. The Relationship between the 0.729166667\" digit and the Egyptian Royal Cubit of 20.625\"\n\nUpdated: Feb 23, 2022\n\nThe 0.729166667\" digit is an important unit. It goes 1,575,000,000 times into the circumference of the earth (the polar circumference of 24,857.95454545 miles to be exact), and it fits well with the remen of 14.583333\", this remen consisting of 20 of these digits. But the 20.625\" Egyptian Royal cubit seems not to fit, unless of course by means of the square root of 2. And this itself has to be taken as 99/70. So we have 20.625 x 7 / (2 x 99) = 0.729166667. The relationship between the digit and the Egyptian Royal Cubit, as 0.729166667\" and 20.625\" respectively, is of course as the side of a square is to it's diagonal.\n\nHowever, the relationship between the 0.729166667\" digit and the Egyptian Royal Cubit of 20.625\" could also be though of as this: the digit goes in 1,400,000 x 4,374 / 4,375 times into a hypothetical 25,920 ancient metres unit of length (using the 39.375\" conversion to metres). And the 20.625\" cubit goes into this 25,920 mm unit 49,500 x 4374 / 4375 x 9,801 / 9,800 times. Royal Egyptian cubit and digit, as 20.625\" and 0.729166667\", are linked like this: cubit x 49,500 / 1,400,000 x 9,800/9,801 = digit.\n\nThis also has the advantage of linking up with the Neal / Michell value for the Egyptian Royal Cubit of 20.618181818\", as 25,920 x 39.375 / 49,500 = 20.6181818181. This is because the link between these two cubits, the 20.6181818\" one and the 20.625\" one, is 9,801/9,800 x 4,375/4,374.\n\nSo we could define the digit of 0.7291666667\" as simply 25,920 ancient metres divided by 49,500, and multiplied by 4,375/4,374 (ragisma) and 9,801/9,800.\n\nThis offers an intriguing connection between the digit, the inch and the ancient metre.\n\nA hypothetical unit of 25,920 ancient metres in fact lends itself to many other possible connections. For example, 25,920 / 49,500 = 0.523636363..., and this is the value in metres of the Neal / Michell Royal Egyptian Cubit, otherwise known as 1,134/55 = 20.618181818 inches. The 1,134 part of this fraction is in fact 25,920 x 4,375 / 100,000. The 55 part of the fraction is 49,500 / 9, and also links up with an approximation of Phi or Phi squares using the Fibonacci numbers. For example 55/34 = 1.617647, and 144/55 = 2.618181818.\n\nThere is good reason to think of these values in metres, of the ancient kind at least, using the 39.375\" conversion. For example, 25,920 / 49,500 = 0.523636363... is 2.618181818 + 3.141818181818, these last two numbers being values for Phi squared (144/55) and pi (144/55 x 6/5).\n\nTo convert from the ancient metre to the modern one, the 8,001/8,000 ratio works perfectly.\n\nAnd so the digit of 0.7291666667\" is 25,920 / 140 x 4,375 / 4,374 ancient mm, or 25,920 / 140 x 4,375 / 4,374 x 39.375 / 10,000 ancient mm, multiplied by 8,000/8,001 to obtain the modern metre value. (39.375 = 10,000 / 254 x 8,001/8,000)\n\nThe 39.375 conversion between ancient metre and inch is in fact 9 x 4,375.\n\nThe 0.729166667\" digit is itself already connected to the ragisma 4,375/4,374, as it can also be written 9 x 9 x 9 / 1,000 x 4,375 / 4,374 . Curiously, 25,920 / 140 x 4,375 = 81 = 9 x 9. This means then that the digit expressed in ancient centimetres, 1.851851851851, is 9 x 9 / 4,374.\n\nAnd expressed in inches, that's simply 9 x 9 x 9 x 4375/ 4,374 x 1/1,000.\n\nThe remen of 14.5833333\", in relation to 25,920 ancient mm, is 25920 / 1,800 x 7,875 / 7,776, or 14.4 x 7,875/7,776. The remen is an Egyptian Royal Cubit of 20.625\" divided by 99/70. This approximation of root 2 is also 25,920 / (2.618181818 x 7,000).\n\n7,875/7,776 is an interesting ratio that comes up from time to time in metrology, and here it seems to bridge the values in inches and metres between the 25,920 mm unit and the 14.583333\" remen. 7,875 inches are in fact 200 ancient metres. And 14.4 / 7,776 is in fact the digit in ancient metres: 0.0185185185....\n\nThe 0.7291666667\" or 1.851851851 (ancient) cm digit is 2 (ancient) metres / 1,080.\n\nThe moon's radius is 1,080 miles.\n\nI thought I might try and see if the 0.729166667\" digit might connect to other units.\n\nI went back to a text I was looking at a few months ago by Mauss, to see how the measures he gave fitted in with a 0.72916666\" digit. The first unit he mentions is the Assyrian and Persian Royal Cubit, also referred to as the \"grande Hachémique\", which he assigns 658.285 mm to. The conversion rate for all his measures is 39.375\" to the metre. We know this because he gives the value of the English yard in millimetres as 914.2857. This article is from 1892, so before the 1897 Weights and Measures Act, and long before 1930 when the 25.4 mm inch was adopted. The conversion rate is curious however as I've not been able to find any mentrion anywhere of this rate ever having been in existence. Still, it ties in perfectly with the 0.7291666667\" digit.\n\n914.2857, which you could take as 6.4/7, multplied by 39.375 = 36, so 36 inches.\n\nInterestingly, 6.4 m = 252\", so 7 yards, is quite compatible with the 0.729166667\" digit.\n\n6.4 metres are obviously 54 x 0.72916667 x 6.4, or perhaps also 21.6 x 16 digits, 14.4 x 24 digits, 12.8 x 27 digits, etc.\n\nThis makes the yard 14.4 x 24 x 0.72916667 / 7 inches = 12 x 12 x 12 x 2 x 0.729166667 / 70\", and the English foot = 115.2 / 7 x 0.729166667\"\n\nTo go back to the Assyrian and Persian royal cubit that Mauss writes about, 658.285 mm, this is also a seventh division of something. 4608 / 7 = 658.2857142857 mm = 25,920 / 1,000 inches, which is 2.16 English feet, and also 248.832 / 7 digits, or 12⁵ / 7,000.\n\n25.92\" for an Assyrian cubit is an interesting number. In the article on weights and measures here, an Assyrian cubit is mentioned, \"a royal cubit of 7/6 the U cubit, or 25.20, and four monuments show a cubit averaging 25.28 (...) we may take 25.24 as the nearest approach to the ancient Persian unit\". (p 4 on the webpage)\n\nThe 25.2\" value would work well with the 0.729166667\" digit, being 2 x 12³ / 100 digits.\n\nThe same article gives the size of the \"U\" as 10.806 inches. If it were in fact 10.8 inches, that would also go well with the digit, as 12³ x 6/700 digits. (In fact in the little table below this sentence in the article the \"U\" is down as 10.80, 6 of these are a qanu, and there are more multiples of this, worth 129.6\", 648\", 7776\" and 233,280\", whose names are hard to read.)", null, "The article then says there is a 2\"U\" unit of 21.6\", and this ties in well with the Mauss article.\n\nMauss has a unit which is 5/6 of the royal Assyrian and Persian cubit, which he calls the worker's cubit, or Coudée ouvriere, worth, as he states it, 548.571 mm, which is also 3,840 / 7 mm, and which is also 21.6\" exactly.\n\nThe 21.6\" unit doesn't seem to work with the digit at first, but bring in the number 7 again and it does: 21.6\" x 7 = 151.2\" = 12⁴ x 0.729166667 / 100\n\nRoyal assyrian cubit = 4,608 / 7 mm = 12⁵ / 7,000 digits = 25.92\"\n\nWorker's cubit = 3,840 / 7 mm = 12⁴ / 700 digits = 21.6\"\n\nThis worker's unit of 548.57142 mm is also 3 digits x 144/55 x 6/5 x 22/7, so the two pis, 3.142857 and 3.141818181. Three digits are perhaps a tenth of Ezekiel's cubit, as Jim Alison says in his article: \"Ezekiel’s cubit of 30 digits, which is contained 72,000,000 times in the polar circumference, or 18,000,000 times in the quarter circumference, or 200,000 times in one degree of latitude\"\n\nRoyal foot = 2,304 / 7 mm = `12² x 16 / 7 mm = 6 x 12⁴ / 700 digits = 12.96\"\n\n`Royal foot x 7/8 = 288 mm = 11.34\" = 55/100 x 20.6181818 (Michell / Neal Egyptian Egyptian cubit)\n\nSo the 20.6181818\" cubit = 6 x 12⁴ / 440 digits = 28.8 / 55 = 0.52363636 metres = 2 x Phi squared / 10 = metres (with Phi squared as 144 / 55). This would make the metre an integral part of Neal and Michell's Egyptian cubit, via Phi squared. (And of course, as pointed out earlier, this unit of 0.52363636.. = 3.1418181818... - 2.618181818... metres, or (144/55 x 6/5) - (144/55) metres.)\n\nOther units mentioned in the Mauss article include:\n\n• Dieulafoy's worker's cubit = 550 mm = 20.625 x 24/25 x 3/2 digits = 21.65625\"\n\n• Dieulafoy's worker's foot = 330 mm = 20.625 x 24/25 x 18/20 digits = 12.99375\"\n\n• Dieulafoy's royal cubit = 660 mm = 20.625 x 24/25 x 18/10 digits = 25.9875\"\n\n• Watering cubit (\"coudée de l'arroseur\") = 576 mm = 126 x 12³ / 7,000 digits = 22.68\"\n\n• Hand cubit (\"coudée de la main\") = 480 mm = 105 x 12³ / 7,000 = 18.9\". This is also the quim cubit and divides into 24 digits of 0.7875\".\n\n• Iron cubit / Black cubit (\"coudée de fer / coudée noire\") = 540 mm = 105 x 12³ x 9 / (8 x 7,000) digits = 21.2625\". This is also the same as the \"coudée des étoffes\", the fabric cubit, which divides up into 27 digits. These digits would be of 0.7875\", which is 0.729166667 x 1.08 inches, same as the hand cubit digits. After all, a digit of 0.729166667\" or 1.851851851 ancient cm, is 2 / 108 ancient metres.\n\n32 of these digits of 0.7875\", or 2 ancient cm, make up the Hachemi cubit of 0.64 metres = 25.2\"\n\nThese 0.7875\" digits connect back to the English foot as 320 / 21 digits of 0.7875\". The Egyptian Royal cubit of 20.625\" contains 550/21 of these digits. The Neal / Michell Egyptian cubit contains 10 x Phi squared of these cubits, or 10 x 144/55. The 14.58333\" remen would contain 2000 / (3 x 36) of these 0.7875\" digits. This remen is also 4/9 x 2.618181818 / 3.1418181818 metres. If you take 1,000 of these digits of 0.7875\" that's also 20 metres.\n\nIt's actually surprisingly helpful to think of the digit and Egyptian / Royal Cubit in terms of metres, however much certain researchers might like to disagree. The idea of a hypothetical unit of 25,920 ancient metres is intriguing.\n\nThe 39.375\" 'metre' is just another subdivision of the 1,575,000,000\" meridian circumference, and a third of it is Jim Alison's Northern foot of 13.125\", which is Jim Wakefield's 13.2\" inch Saxon foot x 175/176, 13.2 x 100,000 x 10/3 x 360 = 1,584,000,000. The question is always going to be what was the value for the meridian circumference in the first place? the 12\" English foot relates to the 1,575,000,000\" circumference, as 1,575,000,000 x 9 x 176 / (360 x 12 x 250,000 x 1.1 x 175) = 12, and the 0.729166667\" digit is the 12\" foot multiplied by 1.1, 175/176, and divided by 18.\n\nAre all units of measure linked? The idea has come up in the work of various researchers recently. It was also a common idea in 18th century France. Bailly, Letronne, and Gosselin found in their research that many units of measure could be traced back to an accurate survey of the size of the planet, in particular to the length of the polar meridian. Paucton, also from the same period, wrote about various types of feet being already integer parts of a degree. He also says if you're designing a new system, you should really take one particular length for a degree and stick to it, or else it will be too confusing when people travel abroad. And he says that the project of post-revolutionary France in implementing a universal system of measure based on an exact division of the earth's meridian arc is actually the very same project that was undertaken in the \"remotest antiquity\", when the units designed were based precisely on a meridian degree.\n\nQuote Paucton Voilà préciſément quel étoit le ſyſtême métrique des peuples dans l'antiquité la plus reculée . Cette partie de la légiſlation leur avoit paru mériter une attention particuliere . Ils fixerent d'une maniere irrévocable leurs meſures en les rendant dépendantes de ta grandeur d'un degré du Méridien . Ils en prirent préciſément la quatre cent millieme partie qu'ils appellerent , tantôt pied & tantôt coudée (...)\nThis is precisely the measurement system of the peoples of the remotest antiquity. This part of their legislation seemed to warrant particular attention. They irrevocably fixed their measures making them dependant on the length of a meridian degree. They tool precisely the 400,000th part, which they sometimes called foor, sometimes cubit (...)\n\nJim Alison recently wrote on GHMB:\n\nThe heart of Washington DC is defined by the NS sections 3-6 and the EW sections B-D. This forms a perfect square of 3600 x 3600 meters, or 10,800 x 10,800 Northern feet, with an area of 116,640,000 square Northern feet. I wonder where L'enfant was coming from with that?\n\nJim has done some amazing work on the city of Washington D.C., see here. He has found that the metre, albeit in its ancient form of 39.375\", comes up in surprising places, for example in the US, where the metre was never actually adopted, despite the closeness of the founders of the country with the French revolutionaries who implemented the metre in France as part of their design for a new country. L'Enfant, who was responsible for designing the city of Washington, was of course French, and both a revolutionary anda trained painter at the French Academy, and trained by his father who himself wa a court painter. Jim Alison has shown that L'Enfant used a grid defined in metres to place his design for the city in.\n\nIn the same post, Jim Alison also wrote:\n\nIf, instead of saying the proposed metric system was based on the most recent, most extensive and most accurate global survey in the history of the world, they had said their proposed metric system was the same as the oldest system of measurement in the history of the world, would the rest of the world, or even the French, have considered adopting it?\n\nOne of my webpages about the global alignment of Teotihuacan, Washington DC, Stonehenge, Troy, etc., has to do with the street plan of Washington DC that was designed by Pierre L'enfant. Although the U.S. rejected the metric system, Washington DC, despite the diagonal avenues, is based on a due NS-EW street grid, and a metric grid, with NS lengths of 900 meters, and EW lengths of 1200 meters, defines the locations of the main buildings and monuments and the slopes, angles and distances of the diagonal avenues.\n\nSee his webpage here.\n\nI wondered if you could extend Jim's grid slightly to the east to include the hospital / prison site, which is at the apex of on of the huge triangles in the city's design. In another post, I wrote about how if you superimpose a picture of Orion's Belt onto L'Enfant's plan, it forms a nice little trio with the other two more important sites: the White House and the Capitol. In the mind of a revolutionary, perhaps the place where 'broken' citizens go to get 'fixed' should be placed in an almost equally important part of the city as the Congress and President's House, one which is directly linked to them.You can then get a nice 2:1 rectangle, with an area of 259,200 square km. Again this number: 25,920, that I was imagining as a possible unit of measure that might make sense of the relationship between the digit of 0.72916666\" and the Royal cubit of 20.625\".\n\nHere is my variation on Jim Alison's grid.", null, "", null, "" ]

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[ "# Codomain\n\nIn mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: XY. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.", null, "A function f from X to Y. The blue oval Y is the codomain of f. The yellow oval inside Y is the image of f, and the red oval X is the domain of f.\n\nA codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.\n\nA codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: XY.\n\n## Examples\n\nFor a function\n\n$f\\colon \\mathbb {R} \\rightarrow \\mathbb {R}$\n\ndefined by\n\n$f\\colon \\,x\\mapsto x^{2},$  or equivalently $f(x)\\ =\\ x^{2},$\n\nthe codomain of f is $\\textstyle \\mathbb {R}$ , but f does not map to any negative number. Thus the image of f is the set $\\textstyle \\mathbb {R} _{0}^{+}$ ; i.e., the interval [0, ∞).\n\nAn alternative function g is defined thus:\n\n$g\\colon \\mathbb {R} \\rightarrow \\mathbb {R} _{0}^{+}$\n$g\\colon \\,x\\mapsto x^{2}.$\n\nWhile f and g map a given x to the same number, they are not, in this view, the same function because they have different codomains. A third function h can be defined to demonstrate why:\n\n$h\\colon \\,x\\mapsto {\\sqrt {x}}.$\n\nThe domain of h cannot be $\\textstyle \\mathbb {R}$  but can be defined to be $\\textstyle \\mathbb {R} _{0}^{+}$ :\n\n$h\\colon \\mathbb {R} _{0}^{+}\\rightarrow \\mathbb {R} .$\n\nThe compositions are denoted\n\n$h\\circ f,$\n$h\\circ g.$\n\nOn inspection, hf is not useful. It is true, unless defined otherwise, that the image of f is not known; it is only known that it is a subset of $\\textstyle \\mathbb {R}$ . For this reason, it is possible that h, when composed with f, might receive an argument for which no output is defined – negative numbers are not elements of the domain of h, which is the square root function.\n\nFunction composition therefore is a useful notion only when the codomain of the function on the right side of a composition (not its image, which is a consequence of the function and could be unknown at the level of the composition) is a subset of the domain of the function on the left side.\n\nThe codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection.\n\nA second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from $\\textstyle \\mathbb {R} ^{2}$  to itself, which can be represented by the 2×2 matrices with real coefficients. Each matrix represents a map with the domain $\\textstyle \\mathbb {R} ^{2}$  and codomain $\\textstyle \\mathbb {R} ^{2}$ . However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or 0). Take for example the matrix T given by\n\n$T={\\begin{pmatrix}1&0\\\\1&0\\end{pmatrix}}$\n\nwhich represents a linear transformation that maps the point (x, y) to (x, x). The point (2, 3) is not in the image of T, but is still in the codomain since linear transformations from $\\textstyle \\mathbb {R} ^{2}$  to $\\textstyle \\mathbb {R} ^{2}$  are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that T does not have full rank since its image is smaller than the whole codomain." ]

[ null, "https://upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/250px-Codomain2.SVG.png", null ]

[ "Computing the Sum of All Numbers from F to L\n\nIf you want to add all the numbers from F (First) to L (Last), here is an easy way to do it, and many times can be performed in your head.\n\n$\\sum_{F}^{L} = (L^{2} - F^{2} + F + L)/2$\n\nWhere F is the first number in the number line and L is the Last.\n\nExample:\n\nTo add all the numbers from 1 to 10, plug in 1 for F and 10 for L.\n\n$$(10^{2} - 1^{2} + 10 + 1)/2 =$$\n\n$$(100 - 1 + 11)/2 =$$\n\n$$110/2 = 55$$\n\nThe result is the same as adding all the numbers from 1 to 10 like this:\n\n$$1+2+3+4+5+6+7+8+9+10=55$$\n\nThis works for all nonnegative integers where 0 < F < L. You don't have to start with 1.\n\nSubmitted by former curiousmath.com member \"deud\"." ]

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[ "author Bodo Möller Fri, 25 Feb 2000 07:40:53 +0000 (07:40 +0000) committer Bodo Möller Fri, 25 Feb 2000 07:40:53 +0000 (07:40 +0000)\n\nindex ce90175..6b158f0 100644 (file)\n@@ -282,13 +282,13 @@ static void ssleay_rand_add(const void *buf, int num, double add)\n{\nmd[k] ^= local_md[k];\n}\n+       if (entropy < ENTROPY_NEEDED) /* stop counting when we have enough */\n+           entropy += add;\nCRYPTO_w_unlock(CRYPTO_LOCK_RAND);\n\n#ifndef THREADS\nassert(md_c == md_count);\n#endif\n-       if (entropy < ENTROPY_NEEDED) /* stop counting when we have enough */\n-           entropy += add;\n}\n\nstatic void ssleay_rand_seed(const void *buf, int num)\n@@ -318,8 +318,8 @@ static void ssleay_rand_initialize(void)\nRAND_add(&l,sizeof(l),0);\n\n#ifdef DEVRANDOM\n-       /* Use a random entropy pool device. Linux and FreeBSD have\n-        * this. Use /dev/urandom if you can as /dev/random will block\n+       /* Use a random entropy pool device. Linux, FreeBSD and OpenBSD\n+        * have this. Use /dev/urandom if you can as /dev/random may block\n* if it runs out of random entries.  */\n\nif ((fh = fopen(DEVRANDOM, \"r\")) != NULL)\n@@ -388,6 +388,19 @@ static int ssleay_rand_bytes(unsigned char *buf, int num)\nssleay_rand_initialize();\n\nok = (entropy >= ENTROPY_NEEDED);\n+       if (!ok)\n+               {\n+               /* If the PRNG state is not yet unpredictable, then seeing\n+                * the PRNG output may help attackers to determine the new\n+                * state; thus we have to decrease the entropy estimate.\n+                * Once we've had enough initial seeding we don't bother to\n+                * adjust the entropy count, though, because we're not ambitious\n+                * to provide *information-theoretic* randomness.\n+                */\n+               entropy -= num;\n+               if (entropy < 0)\n+                       entropy = 0;\n+               }\n\nst_idx=state_index;\nst_num=state_num;" ]

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[ "## Dividend discount model in stock market\n\nPerhaps more importantly, valuing stocks enables you to take a deeper look at factors that drive stock price. Characteristics such as growth and fundamental  24 Jul 2019 The Gordon growth model is a simple and convenient way of valuing stocks but it is extremely sensitive to the inputs for the growth rate. Used  In finance theory, the value of an asset is defined as the present value of future cash flows expected from owning that asset. In case of stocks, the only visible cash\n\nKEY WORDS: dividend discount model, Gordon growth model, stock valuation, European equity market. Introduction. Valuation as a strategy is especially crucial   What is Dividend Discount Model? Meaning of Dividend Discount Model as a finance term. What does Dividend Discount Model mean in finance? Multi Stage Dividend Discount Model Calculator - Value of a Stock Based on Certain industries are very sensitive to market events and stocks will rise/fall  30 Jul 2016 You can compare Total Equity Value to MasterCard's Market Cap. . To get to a per share Fair Value that we can compare to the current stock price  10 Feb 2016 The dividend discount model is one of the prized possessions in the field of finance. It was developed by Myron Gordon in 1959, and simplifies\n\n## 24 Jul 2019 The Gordon growth model is a simple and convenient way of valuing stocks but it is extremely sensitive to the inputs for the growth rate. Used\n\n28 Feb 2018 Keywords: Common stocks; Constant growth; Dividend discount model (DDM) intrinsic value; Philippine stock exchange (PSE). Introduction. Perhaps more importantly, valuing stocks enables you to take a deeper look at factors that drive stock price. Characteristics such as growth and fundamental  24 Jul 2019 The Gordon growth model is a simple and convenient way of valuing stocks but it is extremely sensitive to the inputs for the growth rate. Used  In finance theory, the value of an asset is defined as the present value of future cash flows expected from owning that asset. In case of stocks, the only visible cash  14 Aug 2019 In the case of listed companies, an investor can sell his stake in the market. Multi period model of Equity Valuation. How to Calculate Stock Price  Learn how to identify stocks with the best dividend payout ratio using dividend discount model & discounted cash flow model. Visit our Knowledge Bank section\n\n### In finance theory, the value of an asset is defined as the present value of future cash flows expected from owning that asset. In case of stocks, the only visible cash\n\n17 Jan 2020 The Dividend Discount Model is a simplified version of discounted cash flow analysis that is specifically tailored for stocks that pay fairly high  14 Nov 2019 As a bonus, it additionally includes dividend lookups for 2,000+ publicly listed US stocks and can automatically populate dividend history. Primer. The dividend discount model provides a means of developing an explicit expected return for the stock market. By comparing this return with the expected  8 Jan 2020 He explains how to Value Stocks Using Dividend Discount Models (DDM), like the Gordon Growth Model and Multi-Stage DDMs. 3 Oct 2019 If you are investing into stocks or equity, then you need to know how to forecast earnings growth. That's exactly what the Gordon Growth model\n\n### Multi Stage Dividend Discount Model Calculator - Value of a Stock Based on Certain industries are very sensitive to market events and stocks will rise/fall\n\n22 Nov 2019 The dividend discount model, or DDM, is a method used to value stocks that uses the theory that a stock is worth the sum of all of its future  discount model -- the value of a stock is the present value of expected The Gordon growth model is a simple and convenient way of valuing stocks but it is. In an efficient market, the market price of a stock is considered equal to the intrinsic value of the stock, where the capitalization rate is equal to the market  1+r. < 1, i.e. as long as r (the expected return on the stock market) is greater than g (the growth rate of dividends). We will assume this holds. Thus, we have. Pt. =.\n\n## In an efficient market, the market price of a stock is considered equal to the intrinsic value of the stock, where the capitalization rate is equal to the market\n\nGenerally, the dividend discount model is best used for larger blue-chip stocks because the growth rate of dividends tends to be predictable and consistent.\n\n24 Jul 2019 The Gordon growth model is a simple and convenient way of valuing stocks but it is extremely sensitive to the inputs for the growth rate. Used  In finance theory, the value of an asset is defined as the present value of future cash flows expected from owning that asset. In case of stocks, the only visible cash  14 Aug 2019 In the case of listed companies, an investor can sell his stake in the market. Multi period model of Equity Valuation. How to Calculate Stock Price  Learn how to identify stocks with the best dividend payout ratio using dividend discount model & discounted cash flow model. Visit our Knowledge Bank section   Gordon growth model (Constant growth dividend discount model): assumes that Calculate the value of a stock that paid a \\$10 dividend last year, if dividends are Concept 80: Mechanisms Available for Issuing Bonds in Primary Markets" ]

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[ "September 10, 2021\n\n# Poor Lil Jessi Sadly Sleep Sit Huq Mom Behind", null, "here is an image excerpted from the video below.", null, "​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​\n\nAwww look how sweet Jessie looks. Hey mom we need to see Bella haven’t seen her in awhile.😘😘", null, "​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​\n\nPrecious little one. So sweet❤️", null, "​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​\n\nAwesome Jessi🍼very sweet she like a human baby love you💕🙏", null, "​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​\n\nHi! Little Jessi ❤sweet dreams baby hugs and kisses sweetie love you guy’s 💓😍😘💗💖💕", null, "​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​\n\n​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​\n\nPoor Lil Jessi Sadly Sleep Sit Huq Mom Behind\n\n#### You may have missed", null, "#### BiBi takes care of Ody cat so sweet", null, "", null, "", null, "" ]

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[ "# Glossary\n\n## Algo: Directed acyclic graph\n\nA cycle in a directed graph is a circular path $v_0 \\rightarrow v_1 \\rightarrow v_2 \\rightarrow \\dots \\rightarrow v_k \\rightarrow v_0$. The graph below has quite a few of them, for example, $B \\rightarrow E \\rightarrow F \\rightarrow B$.", null, "A graph without cycles is acyclic. It turns out we can test for acyclicity in linear time, with a single depth-first search.\n\nProperty A directed graph has a cycle if and only if its depth-first search reveals a back edge.\n\nProof. One direction is quite easy: if $(u, v)$ is a back edge, then there is a cycle consisting of this edge together with the path from $v$ to $u$ in the search tree.\n\nConversely, if the graph has a cycle $v_0 \\rightarrow v_1 \\rightarrow v_2 \\rightarrow \\dots \\rightarrow v_k \\rightarrow v_0$, look at the first node on this cycle to be discovered (the node with the lowest pre number). Suppose it is $v_i$. All the other $v_j$ on the cycle are reachable from it and will therefore be its descendants in the search tree. In particular, the edge $v_{i−1} \\rightarrow v_i$ (or $v_k \\rightarrow v_0$ if $i = 0$) leads from a node to its ancestor and is thus by definition a back edge. ∎\n\nDirected acyclic graphs (DAG), or dags for short, come up all the time. They are good for modeling relations like causalities, hierarchies, and temporal dependencies." ]

[ null, "https://rosalind.info/media/directedgraph.png", null ]

[ "# Blog\n\n## Geek Challenge Results: Fractal Snowflake", null, "The winner of the Fractal Snowflake Geek Challenge is John Jacobsma of Dickson. Tim Jager of DMC also responded with a correct value of C, 10830.\n\nThe Fractal Snowflake Challenge was a study in recursive programming. Recursive programming is a technique where a function calls itself, usually many times, until eventually instead of calling itself, it just does something. The program had two layers of recursive programming in it:\n\nThe function `MakeTriangle` was what actually did something in the end. It made one of the little blue triangles we were counting.\n\nThe function `Sierpinski` made the triangle shaped fractal. It is called with an Order parameter, and if the order is zero, then it calls `MakeTriangle`. If the order parameter is > 0, it just calls itself 3 times with an order decremented by 1.\n\nThe function `KochSide` is responsible for jagged looking sides of the snowflake. It also takes an Order parameter; At order 0, it does nothing, but at higher orders it calls itself 4 times, and `Sierpinski` once, all with order decremented by 1.\n\nThe function `CombinedFractal` calls `Sierpinski` once to fill in the largest fractal triangle, then calls `KochSide `3 times to fill out the jagged edge pattern on all sides of the triangle.\n\nTo solve the puzzle of counting the little triangles, the easiest way to do it was to modify `MakeTriangle` to simply count instead of actually making triangles. Then run the program, and it will give the answer: 10830.\n\nJohn Jacobsma simplified and re-wrote the program in Java. Without any of the geometry, this program shows just the recursive nature of the problem:\n\n```package counttriangles;\n\npublic class CountTriangles implements Runnable\n{\npublic static void main(String[] args)\n{\nnew CountTriangles().run();\n}\n\n@Override\npublic void run()\n{\nSystem.out.println(combined(6));\n}\n\nint combined(int order)\n{\nreturn sierpinski(order) + 3 * koch(order);\n}\n\nint koch(int order)\n{\nif (order <= 0)\nreturn 0;\nreturn sierpinski(order - 1) + 4 * koch(order - 1);\n}\n\nint sierpinski(int order)\n{\nif (order <= 0)\nreturn 1;\nreturn 3 * sierpinski(order - 1);\n}\n}```\n\nThe problem can also be solved without delving into the programming, but by analyzing the pattern.", null, "Order 0 has just 1 triangle. Each additional order spits each triangle into 3, so it has 3 times more triangles than the previous. So, the triangle counts of Sierpinski triangles are 1, 3, 9, 27, 81, 243 and 729. Also note that the order of a Sierpinski triangle can be determined by counting the interior white triangles down the middle.\n\nThree white triangles down the middle indicates an order 3 Sierpinski triangle:", null, "Next, the Koch pattern can be analyzed. A level 2 Koch snowflake is shown here with the edges marked. The level pattern in Red is repeated 4 times by pattern in Green. Each additional level continues to multiply by 4.  But the first Red pattern is performed only 3 times, for the 3 sides of the center triangle.", null, "So, the number of triangles that make up solid Koch snowflake is 1, 3, 12, 48, 192, 768, 3072.\n\nSince the fractal snowflake pattern is made of Sierpinski triangles instead of solid triangles, the pattern of the order of the Sierpinski triangles has to be determined. It can be observed that the central triangle is order 6, and each smaller triangle on the perimeter is seen to be 1 level smaller, down to level 0 triangles on the fringes:", null, "", null, "Now, the two numeric sequences can be combined and summed to determine the total number of triangles:\n\n Sierpinski Koch Total Triangles Triangles Triangles 1 3072 3072 3 768 2304 9 192 1728 27 48 1296 81 12 972 243 3 729 729 1 729 Sum 10830\n\nThere are currently no comments, be the first to post one.\n\nName (required)\n\nEmail (required)", null, "Enter the code shown above:\n\n## Popular Articles\n\nDMC to Open Five New Offices in 2022\n08/31/2021" ]

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[ "", null, "", null, "# Résolution de problèmes algorithmiques\n\nGiven a grid with two openings and two sided mirrors in some grid cells, find an orientation for the mirrors at 45 or -45 degrees, such that a laser beam entering one opening would be reflected all the way to the second opening.\n\n## An $O(n^2)$ algorithm\n\nWe use the following graph representation for this problem. Every mirror a generates up to 4 vertices, one for each direction up, left, down, right. If some mirror b is reachable from a in direction d then there is a vertex (a,d) and by symmetry a vertex (b,inv(d)) — where the inv function inverts the direction — and these vertices are connected by an edge. In addition vertices (a,d) and (a,e) are connected by an edge if d and e are adjacent directions. This construction is completed with a vertex for each of two openings. Each opening vertex is connected to the reachable mirror (if any). The opening vertices are connected together by an edge, or — variant of the graph model — connected each to an additional vertex. We distinguish two kind of edges: (i) horizontal or vertical ones and (ii) diagonal or anti-diagonal ones.", null, "We claim that the graph has a perfect matching if and only if the mirror maze instance has a solution. The interpretation of a matching is the following. A vertical or horizontal edge in the matching means that the laser does not follow this trajectory. A diagonal or anti diagonal edge in the matching, say ((a,d),(a,e)), means that the laser reflects on the mirror a without however distinguishing the case that the laser enters a from direction d and leaves in direction e or the opposite case. Note that if there are two reflections happening on a mirror then these are not conflicting and there is a unique corresponding mirror position.\n\nThe algorithm consists of\n\n• building the graph model,\n• creating a near perfect matching, consisting of all horizontal and vertical edges. This corresponds to the complete absence of the laser beam.\n• tries to build a perfect matching, by searching for a single alternating augmenting path,\n\nThis last step is hard to implement as it needs Edmond’s blossom algorithm.\n\n## Example", null, "## Note\n\nThe mirror maze problem can be solved with backtracking. Such a solution is of course not running in polynomial time in the worst case, but seems to pass all the tests and is quite easy to implement. But such a solution is cheating, wouldn’t you agree ?\n\nMost available implementations of Edmond’s blossom algorithm are a bit long (who can blame?), but the following by David Eppstein is quite elegant:" ]

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[ "# Interactive pixel information of an image\n\nYou can monkey-patch `ax.format_coord`, similar to this official example. I’m going to use a slightly more “pythonic” approach here that doesn’t rely on global variables. (Note that I’m assuming no `extent` kwarg was specified, similar to the matplotlib example. To be fully general, you need to do a touch more work.)\n\n``````import numpy as np\nimport matplotlib.pyplot as plt\n\nclass Formatter(object):\ndef __init__(self, im):\nself.im = im\ndef __call__(self, x, y):\nz = self.im.get_array()[int(y), int(x)]\nreturn 'x={:.01f}, y={:.01f}, z={:.01f}'.format(x, y, z)\n\ndata = np.random.random((10,10))\n\nfig, ax = plt.subplots()\nim = ax.imshow(data, interpolation='none')\nax.format_coord = Formatter(im)\nplt.show()\n``````", null, "Alternatively, just to plug one of my own projects, you can use `mpldatacursor` for this. If you specify `hover=True`, the box will pop up whenever you hover over an enabled artist. (By default it only pops up when clicked.) Note that `mpldatacursor` does handle the `extent` and `origin` kwargs to `imshow` correctly.\n\n``````import numpy as np\nimport matplotlib.pyplot as plt\nimport mpldatacursor\n\ndata = np.random.random((10,10))\n\nfig, ax = plt.subplots()\nax.imshow(data, interpolation='none')\n\nmpldatacursor.datacursor(hover=True, bbox=dict(alpha=1, fc=\"w\"))\nplt.show()\n``````", null, "Also, I forgot to mention how to show the pixel indices. In the first example, it’s just assuming that `i, j = int(y), int(x)`. You can add those in place of `x` and `y`, if you’d prefer.\n\nWith `mpldatacursor`, you can specify them with a custom formatter. The `i` and `j` arguments are the correct pixel indices, regardless of the `extent` and `origin` of the image plotted.\n\nFor example (note the `extent` of the image vs. the `i,j` coordinates displayed):\n\n``````import numpy as np\nimport matplotlib.pyplot as plt\nimport mpldatacursor\n\ndata = np.random.random((10,10))\n\nfig, ax = plt.subplots()\nax.imshow(data, interpolation='none', extent=[0, 1.5*np.pi, 0, np.pi])\n\nmpldatacursor.datacursor(hover=True, bbox=dict(alpha=1, fc=\"w\"),\nformatter=\"i, j = {i}, {j}\\nz = {z:.02g}\".format)\nplt.show()\n``````", null, "" ]

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[ "Community Profile", null, "# Philippe Lebel\n\nLast seen: 7 months ago Active since 2018\n\n#### Statistics\n\nAll\n•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "#### Content Feed\n\nView by\n\nMatch numbers with letters\nim late to the show, but here we go anyway... clear clc name = 'david'; a = {'a','b','c','d','e','f','g','h','i','j','k','...\n\n1 year ago | 0\n\nHow do I add a column vector or matrix to a 6X6 original matrix?\nhere is an example: clear all zeros_col = zeros(142,1); c11 = (1:142)'; fantastic_matrix = [c11 zeros_col zeros_col zeros_co...\n\n2 years ago | 0\n\n| accepted\n\nthe loop does not display results\nYou are not looping over the values of \"i\" nor \"j\" i=1:length(X(:,1)) i = 1 2 3 4 5 6 7 ...\n\n2 years ago | 0\n\nBetter parallelization than parfor?\nmaybe take a look at arrayfun(). https://www.mathworks.com/help/matlab/ref/arrayfun.html\n\n2 years ago | 0\n\nElements of table equal to elements of another table\nHere you go: I loop over all station names and check where in the data table i find these names. I generate a boolean index in ...\n\n2 years ago | 0\n\nHow to set the function to be able to use all variables in the script without need to be named in the function def?\ni would suggest creating a struct containing the required variables. %my script params.a = 1; params.b = \"very string\" param...\n\n2 years ago | 0\n\nHow can i speed up my function?\nIn order to help you in a significant way, it would be useful to have a sample data set on which we could run the function. On ...\n\n2 years ago | 1\n\n| accepted\n\nDifferential equation and problem with minmax\nThis error message tells you that you don't have the required package installed in order to use the minmax() function. Matlab c...\n\n2 years ago | 0\n\nDecrease time to upload a figure\nIn addition to @Looky 's answer, try to remove the set(gca, 'YDir','reverse') from the while loop, I'm pretty sure you only n...\n\n3 years ago | 1\n\nHow to plot curve with one color without precising which color it is?\naloa for i = 1:10 if i == 1 h = plot(1:10,1:2*i:20*i) v = get(h,'Color'); else plot(1:10,1...\n\n3 years ago | 0\n\n| accepted\n\nDeleting plotted image from a figure if conditions are met\nIf you get the handle of the plot like so: h = plot(1,1,'Xr') you can just delete the handle and the data disapears delete(h)...\n\n3 years ago | 0\n\n| accepted\n\nDoes anyone know why it is also taking point 9 as convex hull point eventhough it shouldn't?\nBy observation, points 8, 9 and 10 are on the same line. You have to add a condition to decimate redundant points. (points that...\n\n3 years ago | 0\n\n| accepted\n\nHow to crop detected face part?\nRGB Images are 3D matrices with dimenions = (resolution_x, resolution_y, 3) I suppose the bounding box contains the face. Try ...\n\n3 years ago | 1\n\nHow to extract matrix from a bigger matrix based on threshold of each column?\nhere is my try based on what i understood. given a matrix a nx3: a = [5,6,8;1,2,3;2,3,4;3,4,5;4,5,6] a = 5 6 ...\n\n3 years ago | 0\n\n| accepted\n\nBlock diagonal matrix of identity times scalar.\nhere is my take. a=[1,2]; c = round(a(1):1/(length(b(:,1))-1):a(2)); matrix = diag(c);\n\n3 years ago | 0\n\n| accepted\n\nadd vertical line on a plot for Confidence intervalle.\nHave you tried this function? https://www.mathworks.com/help/matlab/ref/errorbar.html\n\n3 years ago | 0\n\n| accepted\n\nIs it possible to perform additional operation on job cancellation?\nCheckout this function: https://www.mathworks.com/help/matlab/ref/oncleanup.html\n\n3 years ago | 1\n\nHow can I fill my cell array?\nhere is my try: U{1} = [1,2]; U{2} = [2,3]; T = [1,2,3]; S = {}; for i = 1:length(T) temp_cell = {}; k=1; f...\n\n3 years ago | 0\n\n| accepted\n\nHow do i create a projectile?\nas example for a non-accelerating projectile (which is not your case) this could look like time = 0:1:100; % time goes from 0 t...\n\n3 years ago | 0\n\nCreating Sine wave with random values\nhere is my try: random_phase_offset = rand(1,1)*2*pi; max_amplitude = 20; random_amplitude = rand(1,1)*max_amplitude; t = ...\n\n3 years ago | 1\n\nSaving each data from for loop\nIf you want to store elements (like strings) that are not the same size you can use cells. A = {'abc', 'defg', 'hijklmnop', 1...\n\n3 years ago | 0\n\nbasic question about matrices\nAs i think this is an homework i'd suggest reading: https://www.mathworks.com/help/matlab/ref/ones.html https://www.mathworks....\n\n3 years ago | 0\n\n| accepted\n\nhow can i change an indice in Matrix as vector?\nNow i understand. Here is a solution that you can easily expand. clear protein(1).name = 'A'; protain(1).bool_value = [1...\n\n3 years ago | 1\n\nhow can i change an indice in Matrix as vector?\nI am not sure what you are trying to do as a whole, but if you want to quickly find where there are occurences of a certain stri...\n\n3 years ago | 1\n\nproblem in optimization by MATLAB using genetic algorithm\nTry to see if the problem is not the same as in this question: https://www.mathworks.com/matlabcentral/answers/76730-genetic-al...\n\n3 years ago | 1\n\nCreate a string array (MxN) where each element is a repeated character based off a numeric array (MxN)\ntry this: B = arrayfun(@(x) repmat('#',1,x),A, 'UniformOutput', false) B = '####' [1x0 char] '#' '####...\n\n3 years ago | 2\n\nDelete - Variance vs standard deviation when using var() and std() with decimal values from CSV file?\nIsn't this normal? As the variance is the square of the standard deviation? 0.1^2 = 0.01\n\n3 years ago | 0\n\n| accepted\n\nUnknown operation performed.\nThis is called logical (or boolean) indexing. https://www.mathworks.com/company/newsletters/articles/matrix-indexing-in-matlab....\n\n3 years ago | 0\n\n| accepted\n\nSolved\n\nExtract leading non-zero digit\n<http://en.wikipedia.org/wiki/Benford%27s_law Benford's Law> states that the distribution of leading digits is not random. This...\n\n3 years ago\n\nSolved\n\nDetermine if input is odd\nGiven the input n, return true if n is odd or false if n is even.\n\n3 years ago" ]

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[ "# How Many Kilograms in 38.7 Pounds?\n\n38.7 lbs to kg converter. How many kg in 38.7 pounds?\n\n38.7 lbs equal to 17.5540247 kg or there are 17.5540247 kg in 38.7 pounds.\n\n←→\nstep\nRound:\nEnter Pound\nEnter Kilogram\n\n## How to convert 38.7 lbs to kg?\n\nThe conversion factor from lbs to kg is 0.45359237. To convert any value of lbs to kg, multiply the pound value by the conversion factor.\n\nTo convert 38.7 lbs to kg, multiply 38.7 by 0.45359237 (or divide by 2.2046226218), that makes 38.7 lbs equal to 17.5540247 kg.\n\n38.7 lbs to kg formula\n\nkg = lbs value * 0.45359237\n\nkg = 38.7 * 0.45359237\n\nkg = 17.5540247\n\nCommon conversions from 38.7x lbs to kg:\n(rounded to 3 decimals)\n\n• 38.7 lbs = 17.554 kg\n• 38.71 lbs = 17.559 kg\n• 38.72 lbs = 17.563 kg\n• 38.73 lbs = 17.568 kg\n• 38.74 lbs = 17.572 kg\n• 38.75 lbs = 17.577 kg\n• 38.76 lbs = 17.581 kg\n• 38.77 lbs = 17.586 kg\n• 38.78 lbs = 17.59 kg\n• 38.79 lbs = 17.595 kg\n• 38.8 lbs = 17.599 kg\n\nWhat is a Kilogram?\n\nKilogram (kilo) is the metric system base unit of mass. 1 Kilogram = 2.2046226218 Pounds. The symbol is \"kg\".\n\nWhat is a Pound?\n\nPound is an imperial system mass unit. 1 Pound = 0.45359237 Kilogram. The symbol is \"lb\".\n\nCreate Conversion Table\nClick \"Create Table\". Enter a \"Start\" value (5, 100 etc). Select an \"Increment\" value (0.01, 5 etc) and select \"Accuracy\" to round the result." ]

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[ "# Pattern Problems in C++ Part-1\n\n## Simple Patterns Problems using Nested Loops\n\n### How to make patterns ?\n\nFirstly take pencil paper and draw a matrix with dots(.) and then joint the points and make the pattern you want on this matrix and find out a specific pattern that diagram follows and make a calculation that you would apply in the loops for forming the pattern. Initially, you find it difficult to make every pattern on paper but this will increase your way of thinking and after practicing these patterns you can solve the loops formula in your mind.\n\n### Topics Covered\n\n1. Print a Solid Rectangle using nested loops\n2. Print a Hollow Rectangle using nested loops\n3. Print a Solid Rhombus using nested loops\n4. Print a Hollow Rhombus using nested loops\n\n## Print a solid rectangle using nested loop\n\n```* * * * * *\n* * * * * *\n* * * * * *\n* * * * * *\n* * * * * *```\n\nCODE:\n\n``````//program in cpp\n#include<iostream>\nusing namespace std;\nint main()\n{\nint rows, columns;\ncout<<\"Enter the number of rows\"<<endl;\ncin>>rows;\ncout<<\"Enter the number of columns\"<<endl;\ncin>>columns;\n\n//nested loops applied here\nfor(int i=1; i<=rows; i++)\n{\nfor(int j=1; j<=columns; j++)\n{\ncout<<\"*\";\n}\ncout<<endl; //change the line\n}\nreturn 0;\n}``````\n\n## Print a Hollow rectangle using nested loops\n\n``````* * * * * *\n* *\n* *\n* *\n* * * * * *``````\n\nCODE:\n\n``````//program in cpp\n#include<iostream>\nusing namespace std;\nint main()\n{\nint rows, columns;\ncout<<\"Enter the number of rows\"<<endl;\ncin>>rows;\ncout<<\"Enter the number of columns\"<<endl;\ncin>>columns;\n\n//nested loops applied here\nfor(int i=1; i<=rows; i++)\n{\nfor(int j=1; j<=columns; j++)\n{\nif (i==1 || i==rows || j==1 || j==columns)\n{\ncout<<\"* \";\n}\nelse\n{\ncout<<\" \";\n}\n}\ncout<<endl;\n}\nreturn 0;\n}``````\n\n## Print a Solid Rhombus using nested loops\n\n`````` * * * * * *\n* * * * * *\n* * * * * *\n* * * * * *\n* * * * * *\n* * * * * *``````\n\nCODE:\n\n``````//program in cpp\n#include<iostream>\nusing namespace std;\nint main()\n{\nint s;\ncout<<\"Enter the value of side \"<<endl;\ncin>>s;\n//nested loops starts here\nfor(int i=1; i<=s; i++)\n{\nfor(int j=1; j<=s-i; j++)\n{\ncout<<\" \";\n}\nfor(int j=1; j<=s; j++)\n{\ncout<<\"* \";\n}\ncout<<endl;\n}\nreturn 0;\n}``````\n\n## Print a Hollow Rhombus using nested loops\n\n`````` * * * * * *\n* *\n* *\n* *\n* *\n* * * * * *``````\n\nCODE:\n\n``````//program in cpp\n#include<iostream>\nusing namespace std;\nint main()\n{\nint s;\ncout<<\"Enter the value of side \"<<endl;\ncin>>s;\n//nested loops starts here\nfor(int i=1; i<=s; i++)\n{\nfor(int j=1; j<=s-i; j++)\n{\ncout<<\" \";\n}\nfor(int j=1; j<=s; j++)\n{\nif(i==1 || i==s ||j==1 || j==s)\n{\ncout<<\"* \";\n}\nelse\n{\ncout<<\" \";\n}\n}\ncout<<endl;\n}\nreturn 0;\n}``````\n\n#### OUTPUT:\n\nThese are some of the basic and easy pattern problems that we solved above you can copy these codes and try at your own machines and do some experiments in these patterns and comment out the different patterns that you can make from these patterns, modifications like giving space after star(*) in the code (especially in the hollow patterns ) and see the amazing patterns.\n\nIn Part-2 of this we will discuss about some of the pyramid patterns using patterns.\n\nWe will be happy to hear your thoughts", null, "Hello!👋" ]

[ null, "https://edusera.org/wp-content/uploads/2021/02/cropped-Logopit_1610967121036.png", null ]

[ "# ISSA Proceedings 2014 ~ A Formal Model Of Conductive Reasoning\n\nAbstract: I propose a formal model of representation and numerical evaluation of conductive arguments. Such arguments consist not only of pro-premises supporting a claim, but also of contra-premises denying this claim. Offering a simple and intuitive alternative to accounts developed in the area of computational models of argument, the proposed model recognizes internal structure of arguments, allows infinitely many degrees of acceptability, reflects the cumulative nature of convergent reasoning, and enables to interpret attack relation.\n\nKeywords: argument evaluation, argument structure, attack relation, conductive reasoning, logical force of argument, rebuttal.\n\n1. Introduction\nAccording to Wellman’s original definition (1971) the conclusion of any conductive argument is drawn inconclusively from its premises. Moreover, the premises and the conclusion are about one and the same individual case, i.e. the conclusion is drawn without appeal to any other case. Wellman also gave three leading examples of conductive arguments, which determine three patterns of conduction:\n\n(1) You ought to help him for he has been very kind to you.\n(2) You ought to take your son to the movie because you promised, and you have nothing better to do this afternoon.\n(3) Although your lawn needs cutting, you want to take your son to the movies because the picture is ideal for children and will be gone by tomorrow.\n\nWellman’s definition was an object of many interesting views, opinions and interpretations, mostly surveyed in (Blair & Johnson 2011). However, we do not discuss this issue here, but we simply follow these authors who, as Walton & Gordon (2013), focus on the third pattern and propose to take conductive arguments to be the same as pro-contra arguments. Such arguments, except of a normal pro-premise or premises (The picture is ideal for children; It will be gone by tomorrow), have also a con-premise or premises (Your lawn needs cutting).\n\nIn the next two chapters we analyze conductive arguments from the logical point of view. The conduction is regarded here as one act of reasoning, in which a conclusion is drawn by the same time from both types of premises. In Chapter 2 we describe the structure and in Chapter 3 – a method of evaluation of conductive arguments. This method is based on the model of argument proposed in (Selinger 2014). In Chapter 4 we introduce a dialectical component of the analysis. Namely, by means of our model, we discuss definition of attack relation holding between arguments.\n\n2. Structure of conductive arguments\nThere are many ways of expressing conductive arguments in natural language. Some of them are the following:\n\nSince A, even though B, therefore C.\nA, therefore C, although B.\nAlthough B, C because A.\nB, but (on the other hand) A, therefore C.\nDespite B, (we know that) A, therefore C.\n\nIn the above schemes the letter A represents a pro-premise (or pro-premises), B – a con-premise (or con-premises) and C – a conclusion. It is worth to note that pro-premises are presented as overcoming con-premises, so that an argument can be accepted if they really do. There are two types of inference in conductive arguments: pro-premises support and con-premises deny (contradict, attack) conclusions. They can be represented using the standard diagramming method. Figure 1 shows the diagram of Wellman’s third example.\n\nRelation of support is represented by the solid and relation of contradiction – by the dashed line.[i] In order to reflect this duality in our formal model we follow Walton & Gordon’s idea involving the assignment of Boolean values to these two types of inference, however, we propose to use simpler formal structures than the so-called argument graphs (cf. Walton & Gordon 2013).\n\nLet L be a language, i.e. a set of sentences. Sequents are all the tuples of the form <P, c, d>, where P ⊆ L is a non-empty, finite set of sentences (premises), c ∈ L is a single sentence (conclusion), and d is a Boolean value (1 in pro-sequents and 0 in con-sequents). An argument is simply any finite, non-empty set of sequents. If an argument consists of only one sequent then it will be called an atomic argument.\n\nThe premises of an argument are all the premises of all its sequents. The conclusions of an argument are all the conclusions of all its sequents. The first premises are those premises, which are not the conclusions, and the final conclusions are those conclusions, which are not the premises. Finally, the intermediate conclusions are those sentences, which are both the conclusions and the premises. A typical (abstract) argument structure is presented in Figure 2 by the diagram corresponding to the set: {<{α1}, α5, 1>, <{α2}, α5, 0>, <{α3}, α5, 0>, <{α4}, α9, 1>, <{α5}, α13, 1>, <{α6}, α15, 1>, <{α7}, α15, 1>, <{α8}, α15, 0>, <{α9}, α16, 1>, <{α10}, α18, 1>, <{α11}, α18, 0>, <{α12, α13, α14}, α20, 1>, <{α15, α16}, α, 1>, <{α17}, α, 1>, <{α18, α19}, α, 0>, <{α20}, α, 0>}. This argument consists of 16 different sequents (10 of them are pro- and 6 are con-sequents), so it is the sum of the same number of atomic arguments. The premises are all the sentences in the diagram except of α, which is the final conclusion; the conclusions are: α5, α9, α13, α15, α16, α18, α20, α; the first premises: α1, α2, α3, α4, α6, α7, α8, α10, α11, α12, α14, α17, α19; the intermediate conclusions: α5, α9, α13, α15, α16, α18, α20.\n\nBy the means of our formalism also atypical structures can be distinguished (cf. Selinger 2014). Some of them are illustrated by Figure 3. Circular arguments can have no first premises and/or no final conclusion (two examples in Figure 3 have neither the first premises nor the final conclusion). They are interesting argument structures, e.g. for those who deal with antinomies, however, we do not discuss them, since they are mostly regarded as faulty. On the other hand, divergent arguments and incoherent arguments can have more than one final conclusion. They are not faulty (unless from some purely pragmatic point of view), but they can be represented as the sums of non-divergent and coherent arguments. Therefore, when discussing evaluation of conductive arguments in the next chapter, we focus on typical argument structures like that shown in Figure 2.\n\n3. Evaluation of conductive arguments\nThe central question to be considered in this section is: how to transform the values of first premises into the value of final conclusion? We answer this question in three steps concerning evaluation of atomic, convergent and, finally, conductive arguments.\n\nFirst we introduce some basic notions. Each partial function v: L’→[0, 1], where L’ ⊆ L, is an evaluation function. The value v(p) is the (degree of) acceptability of p. We consider also a predefined function w: LχL→[0, 1]. The value w(c/p) is the acceptability of c under the condition that v(p) = 1, so that the function w will be called conditional acceptability.\n\nWe assume that L contains the negation connective. If the premises of some sequent deny its conclusion c then evaluation of c will be based on evaluation of the sentence ¬c in the corresponding pro-sequent, in which the same premises support ¬c. Let us note that for a perfectly rational agent the condition v(¬c) = 1 – v(c) should be satisfied. This postulate will be useful to evaluate con-sequents.\n\nLet v be a given evaluation function (we assume that v is fixed in the following part of our exposition). By ∧P we denote the conjunction of all the sentences belonging to a finite, non-empty set P (if P is a singleton then ∧P is the sole element of P). We assume that L contains the conjunction connective, and if P⊆ dom(v) then ∧P ∈dom(v).[ii] The value w(c/∧P) will be called the internal strength of a pro-sequent <P, c, 1>, and the value w(¬c/∧P) – the internal strength of a con-sequent <P, c, 0>.\n\nLet A = {<P, c, d>} be an atomic argument, where P ∈ dom(v), c∉ dom(v), and d is a Boolean value. The function vA is the following extension of v to the set dom(v) ∪ {c}:\n\n(4) If d = 1 then vA(c) = v(∧P)⋅w(c/∧P);\n(5) If d = 0 then vA(c) = 1 – v(∧P)⋅w(¬c/∧P).\n\nThus the acceptability of the conclusion of an atomic argument under condition that its premises are fully acceptable is reduced proportionally to the actual acceptability of the premises. The value vA(c) will be called the (logical) strength (or force) of an argument A. We will say that a pro-argument is acceptable iff its strength is greater than ½, and a con-argument is acceptable iff its strength is smaller than ½.\n\nIn the next step we consider evaluation of convergent reasoning. Since convergent argumentation is used to cumulate the forces of different reasons supporting (or denying) a claim we have to add these forces in a way adapted to our scale. Strengths of pro- and con-components will be added separately in each of both groups, independently of the other. Let A = A1 ∪ A2 , where both A1 and A2 are acceptable arguments and they either consist of only pro- or of only con-sequents having the same conclusion c. Let vA1(c) = a1 and vA2(c) = a2.\n\n(6) If A1 and A2 are independent pro-arguments, and a1, a2 > ½, then vA(c) = a1 ⊕a2;\n(7) If A1 and A2 are independent con-arguments, and a1, a2 < ½, then vA(c) = 1 – (1–a1)⊕ (1–a2), where x ⊕ y = 2•x + 2•y – 2•x•y ¬– 1.\n\nIn (Selinger 2014) we provide a justification of this algorithm, deriving it from the principle (satisfied also by the algorithms given in (4) and (5)) that can be called the principle of proportionality, according to which the strength of argument should vary proportionally to the values assigned to its components. We also discuss properties of the operation ⊕ (here let us only mention that it is both commutative and associative, therefore the strengths of any number of converging, independent arguments can be added in any order).\n\nFinally, we consider conductive reasoning. In order to compute the final value of a conductive argument we will subtract the strength of its con- from the strength of its pro-components in a way adapted to our scale. Let A = Apro  Acon, where Apro consists only of pro-sequents and Acon only of con-sequents having the same conclusion c. We assume that both groups of arguments are acceptable, i.e. vApro(c) > ½ and vAcon(c) < ½.\n\n(8) If vApro(c) < 1, and vAcon(c) > 0, then vA(c) = vApro(c) + vAcon(c) ¬– ½;\n\nThe idea of this algorithm is illustrated by Figure 4. Since we want to know how much pro-arguments outweigh con-arguments (or vice versa), we subtract the value ½¬ –vAcon(c) represented by the interval [vAcon(c), ½] in this figure from the value vApro(c) – ½ represented by the interval [½, vApro(c)]. In order to finally receive the acceptability of c we add this differential to ½. Let us note that the considered value is directly proportional to the acceptability of pro- and reversely proportional to the acceptability of con-arguments, so that the algorithm satisfies the principle of proportionality.\n\nThe algorithm given by (8) assumes that both pro- and con-arguments are, as defined by Wellman, inconclusive. However in real-life argumentation it happens, for example in mathematical practice, that initial considerations concerning some hypothesis, which are based on subjective premonitions, analogies, incomplete calculations etc., are finally overcame by a mathematical proof. Then all the objections raised originally are no longer significant, and the hypothesis becomes a theorem. Therefore, if either pro- or con-arguments are conclusive, then so the whole conductive argument is.\n\n(9) If vApro(c) = 1, and vAcon(c) ≠ 0, then vA(c) = 1;\n(10) If vApro(c) ≠ 1, and vAcon(c) = 0, then vA(c) = 0.\n\nIf both pro- and con-arguments happen to be conclusive then it is an evidence of a contradiction in underlying knowledge, and the initial evaluation function requires revision. Therefore we claim that the values of such strongly antinomian arguments cannot be found.\n\n(11) If vApro(c) = 1, and vAcon(c) = 0, then vA(c) is not computable.\n\nOtherwise, the strength of weakly antinomian arguments, which consist of equal inconclusive components, can be computed as ½ using the algorithm given by (8).\n\nIn order to complete this section let us add that the acceptability of the conclusions of complex, multilevel argument structures, as the one represented by Figure 3, can be calculated level by level using the algorithms (4) – (10). An analogous process concerning only pro-arguments is described in (Selinger 2014).\n\n4. Attack relation\nOur goal is to define attack relation, which holds between arguments. For the sake of simplicity we consider only attack relation restricted to the set of atomic arguments. There are three components of atomic arguments that can be an object of a possible attack: premises, inferences and conclusions. The latter is the case of conduction. If we take into account a pro- and a con-argument, which have the same conclusion, then the stronger of them attacks the weaker one (in the case of an antinomy both arguments attack each other, so that it can be called the mutual attack case).\n\n(12) An argument A attacks (the conclusion of) an argument B iff A = {<P1, c, d>}, B = {<P2, c, 1 – d >}, and either d = 0 and 1 – vA(c) ≤ vB(c), or d = 1 and 1 – vA(c) ≤ vB(c).\n\nThe second kind of attack is the attack on a premise. Obviously, it is effective if (i) some premise of an attacked argument is shown to be not acceptable on the basis of the remaining knowledge.\n\n(13) An argument A attacks (a premise of) an argument B iff A = {<P1, c1, 0>}, B = {<P2, c2, d>}, c1 < P2, and v’A(c1) ≤ ½, where v’ is the function obtained from v by deleting c1 from its domain, i.e. dom(v’) = dom(v) – {c1}.\n\nHowever, with respect to the proposed method of evaluation, two further situations are possible: (ii) the premises of an attacked argument considered separately are acceptable, however their conjunction is not; (iii) the conjunction of the premises of an attacked argument is acceptable and the internal strength of its constituent (pro- or con-) sequent is greater than ½, but the product of these values is not. Thus, in view of the evaluation method proposed here, merely weakening a premise can cause an effective attack, and the definition (13) should be replaced by the following broader one.\n\n(13’) An argument A attacks (a premise of) an argument B iff A = {<P1, c1, 0>}, B={<P2, c2, d>}, c1 ∉P2, v’A(c1) ≤v(c1), and either d = 1 and v’A(∧P2)∧w(c2/∧P2)∧ ½, or d = 0 and v’A(∧P2) w(~c2/∧P2) ⊆ ½, where v’ is the function obtained from v by deleting c1 from its domain.\n\nIn order to consider attack on the relationship between the premises and the conclusion of an attacked argument, let us take into account the following Pollock’s example of an undercutting defeater:\n\n(14) The object looks red, thus it is red unless it is illuminated by a red light.\n\nFollowing Toulmin’s terminology, the sentence The object is illuminated by a red light will be called rebuttal. Let us note, that rebuttals are not con-premises, since they do not entail the negation of the conclusion (the fact that the object is illuminated by a red light does not imply that the object is not red). Thus Pollock’s example cannot be diagrammed like conductive arguments. Since it is an arrow that represents the inference, which is denied by the rebuttal, rather the diagram shown by Figure 5 seems to be relevant here.\n\nHowever, structures such as the one in Figure 5 have no direct representation within the formalism introduced in this paper to examine conductive reasoning. In order to fill this gap we propose to add the fourth element, namely the set of rebuttals, to the sequents considered so far. Such extended sequents will have the form <P, c, d, R>, where R is the set of (linked) rebuttals.\n\nSince our goal is to define attack relation as holding between arguments, we propose to take an argument without rebuttals (i.e. with the empty set of rebuttals) as being attacked by the argument with the same premises and conclusion, but with a rebuttal added. For example (14) can be regarded as an attacker of the simple argument\n\n(15) The object looks red, thus it is red.\n\nThis argument (15) has the following representation: {<{The object looks red}, The object is red, 1, ∅>}, and its attacker (14): {<{The object looks red}, The object is red, 1, {The object is illuminated by a red light}>}. In general, an argument of the form {<P, c, d,∅>} can be attacked by any argument of the form {<P, c, d, R>}. Effectiveness of this sort of attack depends on evaluation of such arguments. It is not the aim of this paper to develop an evaluation method for arguments with rebuttals systematically, however, let us note that the strength of an argument {<P, c, d, R>}, where R ≠∅, seems to be strictly connected with the strength of the corresponding argument {<P∪{~∧R}, c, d, ∅>}, which has an empty set of rebuttals. For example, the strength of (14) depends on the strength of the argument:\n\n(16) The object looks red, and it is not illuminated by a red light, thus it is red.\n\nIf this argument is acceptable then so is its second premise (The object is not illuminated by a red light), which is the negation of the rebuttal in (14). By the same the rebuttal is not acceptable so that the attack on (15) cannot be effective. Thus (16) cannot be acceptable if (14) attacks the inference of (15). In general, if A = {<P, c, d, R>} attacks (the inference of) B = {<P, c, d,∅>}, then R≠∅ and A’ = {<P∪{~∧R}, c, d, ∅>} is not acceptable. Obviously, the converse does not hold, because not any acceptable set of sentences can be a good rebuttal. If the attack is to be effective the set R must be relevant to deny the inference in B. A test of relevance that we propose is based on an observation concerning (15) and (16). Intuitively, the inference in (16) is stronger than the inference in (15), i.e. the internal strength of the sequent in (16) is greater than the internal strength of the sequent in (15). This is because (16) assumes that a possible objection against the inference in (15) has been overcome. Thus, the condition w(c/∧P∧~∧R) > w(c/∧P)  can be proposed to determine the relevance of the rebuttal in A. Following these intuitions we recognize arguments overcoming rebuttals as hybrid arguments in the sense defined by Vorobej (1995). Such arguments contain a premise that strengthens them, but this premise does not work alone so that it cannot be taken as the premise of a separate convergent reasoning (in (16) such a premise is the sentence The object is not illuminated by a red light).\n\nSumming up, we claim that (a) non-acceptability of the hybrid counterparts corresponding to arguments having rebuttals and (b) relevance of rebuttals are necessary for attack on inference to be effective. However, we leave open the question whether they are sufficient.\n\n5. Conclusion\nWe showed how the model of representation and evaluation of arguments elaborated in (Selinger 2014) can be enriched in order to cover the case of conductive reasoning. The extended model allowed us to define in formal terms two kinds of attack relation, namely attack on conclusion and attack on premise. However, the definition of attack on inference requires further extension of the model. In order to initiate more profound studies, we outlined a possible direction of making such an extension.\n\nAcknowledgements\nI would like to thank Professor David Hitchcock for his inspiring remarks concerning my ideas, and for his helpful terminological suggestions.\n\nNOTES\ni. Let us note that Walton & Gordon (2013) interpret both pro-premises as supporting the claim independently of each other, and they draw separate arrows connecting each pro-premise with the conclusion, which represent convergent reasoning. However, it seems to be problematic whether the premise The picture will be gone tomorrow alone (i.e. without any further information about the movie) actually supports the conclusion.\nii. In order to avoid this assumption the acceptability of an independent set of sentences can be calculated as the product of the values of its elements. Thus the acceptability of a conjunction can be smaller than the acceptability of its components considered separately (cf. Selinger 2014).\nReferences\nBlair, J. A., & Johnson, R. H. (2011). Conductive argument: an overlooked type of defeasible reasoning. London: King’s College Publications.\nSelinger, M. (2014). Towards formal representation and evaluation of arguments. Argumentation, 26(3), 379-393. (K. Budzynska, & M. Koszowy (Eds.), The Polish School of Argumentation, the special the issue of the journal.)\nVorobej, M. (1995). Hybrid arguments. Informal Logic, 17(2), 289-296.\nWellman, C. (1971). Challenge and response: justification in ethics. Carbondale: Southern Illinois University Press.\nWalton, D., & Gordon, T. F. (2013). How to formalize informal logic. In M. Lewiński, & D. Mohammed (Eds.), Proceedings of the 10th OSSA Conference at the University of Windsor, May 2013 (pp. 1-13). Windsor: Centre for Research in Reasoning, and the University of Windsor.\n\n## You May Also Like\n\nWhat is 14 + 15 ?\nIMPORTANT! To be able to proceed, you need to solve the following simple math (so we know that you are a human) :-)\n\nRozenberg Quarterly aims to be a platform for academics, scientists, journalists, authors and artists, in order to offer background information and scholarly reflections that contribute to mutual understanding and dialogue in a seemingly divided world. By offering this platform, the Quarterly wants to be part of the public debate because we believe mutual understanding and the acceptance of diversity are vital conditions for universal progress. Read more...\n• ## Support\n\nRozenberg Quarterly does not receive subsidies or grants of any kind, which is why your financial support in maintaining, expanding and keeping the site running is always welcome. 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[ "## boiler Agent capacity unit sq ft to tons Related Information", null, "### Boiler Capacity - Engineering ToolBox\n\nA is usually expressed as kBtu/hour (1000 Btu/hour) and can be calculated as W = (hg - hf) m (1) where W = (Btu/h kW) hg = enthalpy steam (Btu/lb kJ/kg) hf = enthalpy condensate (Btu/lb kJ/kg) m = steam evaporated (lb/h kg/s)", null, "### Steam Capacity Conversion Table | Industrial Controls\n\n· Other Useful Information: 1 EDR = 240 BTUH @215 ° F 1 HP = 33493 BTU/Hr Where SpGr = Specific Gravity (Water is 10) Condensate Pump Sizing: A: Pump = 2 x Actual GPM Condensate B: Minimum Receiver Size in Gallons = 1 minute x GPM found in part A C: Pump Head (In ) = Static Lift in + Friction Losses in + Pressure In Vessel Thats Being", null, "### How to Calculate Furnace Tonnage | Hunker\n\nDivide the result from Step 4 by 12000 to determine the requirement of the In the example you would divide 35000 by 12000 and get 292 This home would require a that has a of approximately 3", null, "· Multiple /cooling systems: Another important new feature is calculating cost of multiple / cooling systems being installed in large homes (over 3000 ) and specifying largest possible BTU lead HVAC system (s) and then the smallest size system for the remainder of the total BTU load For example if your heat load is 150K BTUs and maximum residential Central AC size is 60K BTUs (5 ) then you need two 60K BTU compressors and a 30K (25", null, "### Rule of Thumb Sizing - The McDermott Group\n\nRule of Thumb HVAC Sizing Frequently designers and contractors \"guestimate\" the size of HVAC units by figuring (12000 Btuh) of air conditioning will cover 400 square feet (Sq-Ft) of building area This ratio is vastly overused and often leads to undersized HVAC units in", null, "### Heat Pump Sizing Guide\n\nThe heating and cooling capacity of the heat pump is expressed in where one ton equals 12000 Btu/h 2 - 1000 sq ft; 3 - 1500 sq ft; 4 - 2000 sq ft; 5 - 2500 sq ft; 6 - 3000 sq ft; Depends on the region the supplemental heating might be needed (usually when the balance point is" ]

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[ "Giáo trình\n\n# College Physics\n\nScience and Technology\n\n## Production of Electromagnetic Waves\n\nTác giả: OpenStaxCollege\n\nWe can get a good understanding of electromagnetic waves (EM) by considering how they are produced. Whenever a current varies, associated electric and magnetic fields vary, moving out from the source like waves. Perhaps the easiest situation to visualize is a varying current in a long straight wire, produced by an AC generator at its center, as illustrated in [link].", null, "This long straight gray wire with an AC generator at its center becomes a broadcast antenna for electromagnetic waves. Shown here are the charge distributions at four different times. The electric field (E size 12{E} {}) propagates away from the antenna at the speed of light, forming part of an electromagnetic wave.\n\nThe electric field ($\\mathbf{E}$) shown surrounding the wire is produced by the charge distribution on the wire. Both the $\\mathbf{E}$ and the charge distribution vary as the current changes. The changing field propagates outward at the speed of light.\n\nThere is an associated magnetic field ($\\mathbf{B}$) which propagates outward as well (see [link]). The electric and magnetic fields are closely related and propagate as an electromagnetic wave. This is what happens in broadcast antennae such as those in radio and TV stations.\n\nCloser examination of the one complete cycle shown in [link] reveals the periodic nature of the generator-driven charges oscillating up and down in the antenna and the electric field produced. At time $t=0$, there is the maximum separation of charge, with negative charges at the top and positive charges at the bottom, producing the maximum magnitude of the electric field (or $E$-field) in the upward direction. One-fourth of a cycle later, there is no charge separation and the field next to the antenna is zero, while the maximum $E$-field has moved away at speed $c$.\n\nAs the process continues, the charge separation reverses and the field reaches its maximum downward value, returns to zero, and rises to its maximum upward value at the end of one complete cycle. The outgoing wave has an amplitude proportional to the maximum separation of charge. Its wavelength$\\left(\\lambda \\right)$ is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. (As usual, wavelength and frequency$\\left(f\\right)$ are inversely proportional.)\n\n# Electric and Magnetic Waves: Moving Together\n\nFollowing Ampere’s law, current in the antenna produces a magnetic field, as shown in [link]. The relationship between $\\mathbf{E}$ and $\\mathbf{B}$ is shown at one instant in [link] (a). As the current varies, the magnetic field varies in magnitude and direction.", null, "(a) The current in the antenna produces the circular magnetic field lines. The current (I size 12{I} {}) produces the separation of charge along the wire, which in turn creates the electric field as shown. (b) The electric and magnetic fields (E size 12{E} {} and B size 12{B} {}) near the wire are perpendicular; they are shown here for one point in space. (c) The magnetic field varies with current and propagates away from the antenna at the speed of light.\n\nThe magnetic field lines also propagate away from the antenna at the speed of light, forming the other part of the electromagnetic wave, as seen in [link] (b). The magnetic part of the wave has the same period and wavelength as the electric part, since they are both produced by the same movement and separation of charges in the antenna.\n\nThe electric and magnetic waves are shown together at one instant in time in [link]. The electric and magnetic fields produced by a long straight wire antenna are exactly in phase. Note that they are perpendicular to one another and to the direction of propagation, making this a transverse wave.", null, "A part of the electromagnetic wave sent out from the antenna at one instant in time. The electric and magnetic fields (E size 12{E} {} and B size 12{B} {}) are in phase, and they are perpendicular to one another and the direction of propagation. For clarity, the waves are shown only along one direction, but they propagate out in other directions too.\n\nElectromagnetic waves generally propagate out from a source in all directions, sometimes forming a complex radiation pattern. A linear antenna like this one will not radiate parallel to its length, for example. The wave is shown in one direction from the antenna in [link] to illustrate its basic characteristics.\n\nInstead of the AC generator, the antenna can also be driven by an AC circuit. In fact, charges radiate whenever they are accelerated. But while a current in a circuit needs a complete path, an antenna has a varying charge distribution forming a standing wave, driven by the AC. The dimensions of the antenna are critical for determining the frequency of the radiated electromagnetic waves. This is a resonant phenomenon and when we tune radios or TV, we vary electrical properties to achieve appropriate resonant conditions in the antenna.\n\n# Receiving Electromagnetic Waves\n\nElectromagnetic waves carry energy away from their source, similar to a sound wave carrying energy away from a standing wave on a guitar string. An antenna for receiving EM signals works in reverse. And like antennas that produce EM waves, receiver antennas are specially designed to resonate at particular frequencies.\n\nAn incoming electromagnetic wave accelerates electrons in the antenna, setting up a standing wave. If the radio or TV is switched on, electrical components pick up and amplify the signal formed by the accelerating electrons. The signal is then converted to audio and/or video format. Sometimes big receiver dishes are used to focus the signal onto an antenna.\n\nIn fact, charges radiate whenever they are accelerated. When designing circuits, we often assume that energy does not quickly escape AC circuits, and mostly this is true. A broadcast antenna is specially designed to enhance the rate of electromagnetic radiation, and shielding is necessary to keep the radiation close to zero. Some familiar phenomena are based on the production of electromagnetic waves by varying currents. Your microwave oven, for example, sends electromagnetic waves, called microwaves, from a concealed antenna that has an oscillating current imposed on it.\n\n# Relating $E$-Field and $B$-Field Strengths\n\nThere is a relationship between the $E$- and $B$-field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the $E$-field created by a separation of charge, the greater the current and, hence, the greater the $B$-field created.\n\nSince current is directly proportional to voltage (Ohm’s law) and voltage is directly proportional to $E$-field strength, the two should be directly proportional. It can be shown that the magnitudes of the fields do have a constant ratio, equal to the speed of light. That is,\n\n$\\frac{E}{B}=c$\n\nis the ratio of $E$-field strength to $B$-field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and elegant result.\n\nCalculating $B$-Field Strength in an Electromagnetic Wave\n\nWhat is the maximum strength of the $B$-field in an electromagnetic wave that has a maximum $E$-field strength of $\\text{1000 V/m}$?\n\nStrategy\n\nTo find the $B$-field strength, we rearrange the above equation to solve for $B$, yielding\n\n$B=\\frac{E}{c}.$\n\nSolution\n\nWe are given $E$, and $c$ is the speed of light. Entering these into the expression for $B$ yields\n\n$B=\\frac{\\text{1000 V/m}}{3\\text{.}\\text{00}×{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m/s}}=\\text{3}\\text{.}\\text{33}×{\\text{10}}^{-6}\\phantom{\\rule{0.25em}{0ex}}\\text{T},$\n\nWhere T stands for Tesla, a measure of magnetic field strength.\n\nDiscussion\n\nThe $B$-field strength is less than a tenth of the Earth’s admittedly weak magnetic field. This means that a relatively strong electric field of 1000 V/m is accompanied by a relatively weak magnetic field. Note that as this wave spreads out, say with distance from an antenna, its field strengths become progressively weaker.\n\nThe result of this example is consistent with the statement made in the module Maxwell’s Equations: Electromagnetic Waves Predicted and Observed that changing electric fields create relatively weak magnetic fields. They can be detected in electromagnetic waves, however, by taking advantage of the phenomenon of resonance, as Hertz did. A system with the same natural frequency as the electromagnetic wave can be made to oscillate. All radio and TV receivers use this principle to pick up and then amplify weak electromagnetic waves, while rejecting all others not at their resonant frequency.\n\n# Section Summary\n\n• Electromagnetic waves are created by oscillating charges (which radiate whenever accelerated) and have the same frequency as the oscillation.\n• Since the electric and magnetic fields in most electromagnetic waves are perpendicular to the direction in which the wave moves, it is ordinarily a transverse wave.\n• The strengths of the electric and magnetic parts of the wave are related by\n$\\frac{E}{B}=\\text{c},$\n\nwhich implies that the magnetic field $B$ is very weak relative to the electric field $E$.\n\n# Conceptual Questions\n\nThe direction of the electric field shown in each part of [link] is that produced by the charge distribution in the wire. Justify the direction shown in each part, using the Coulomb force law and the definition of $\\mathbf{E}=\\mathbf{F}/q$, where $q$ is a positive test charge.\n\nIs the direction of the magnetic field shown in [link] (a) consistent with the right-hand rule for current (RHR-2) in the direction shown in the figure?\n\nWhy is the direction of the current shown in each part of [link] opposite to the electric field produced by the wire’s charge separation?\n\nIn which situation shown in [link] will the electromagnetic wave be more successful in inducing a current in the wire? Explain.\n\nIn which situation shown in [link] will the electromagnetic wave be more successful in inducing a current in the loop? Explain.\n\nShould the straight wire antenna of a radio be vertical or horizontal to best receive radio waves broadcast by a vertical transmitter antenna? How should a loop antenna be aligned to best receive the signals? (Note that the direction of the loop that produces the best reception can be used to determine the location of the source. It is used for that purpose in tracking tagged animals in nature studies, for example.)\n\nUnder what conditions might wires in a DC circuit emit electromagnetic waves?\n\nGive an example of interference of electromagnetic waves.\n\n[link] shows the interference pattern of two radio antennas broadcasting the same signal. Explain how this is analogous to the interference pattern for sound produced by two speakers. Could this be used to make a directional antenna system that broadcasts preferentially in certain directions? Explain.\n\nWhat is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of $5\\text{.}\\text{00}×{\\text{10}}^{-4}\\phantom{\\rule{0.25em}{0ex}}\\text{T}$ (about 10 times the Earth’s)?\nThe maximum magnetic field strength of an electromagnetic field is $5×{\\text{10}}^{-6}\\phantom{\\rule{0.25em}{0ex}}\\text{T}$. Calculate the maximum electric field strength if the wave is traveling in a medium in which the speed of the wave is $\\text{0.75}c$.\nVerify the units obtained for magnetic field strength $B$ in [link] (using the equation $B=\\frac{E}{c}$) are in fact teslas (T)." ]

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[ "# A scalar quantity is one that: 1. is conserved in a process. 2. will never accept negative values. 3. must be dimensionless. 4. has the same value for observers with different orientations of axes.\n\nSubtopic:  Scalars & Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nThe position of a particle in a rectangular co-ordinate system is (3, 2, 5). Then its position vector will be:\n\n1. $5\\stackrel{^}{\\mathrm{i}}+6\\stackrel{^}{\\mathrm{j}}+2\\stackrel{^}{\\mathrm{k}}$\n\n2. $3\\stackrel{^}{\\mathrm{i}}+2\\stackrel{^}{\\mathrm{j}}+5\\stackrel{^}{\\mathrm{k}}$\n\n3. $5\\stackrel{^}{\\mathrm{i}}+3\\stackrel{^}{\\mathrm{j}}+2\\stackrel{^}{\\mathrm{k}}$\n\n4. None of these\n\nSubtopic:  Scalars & Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\n$\\stackrel{\\to }{\\mathrm{A}}$ is a vector with magnitude A, then the unit vector $\\stackrel{^}{\\mathrm{A}}$ in the direction of $\\stackrel{\\to }{\\mathrm{A}}$ is\n\n1. $\\mathrm{A}\\stackrel{\\to }{\\mathrm{A}}$\n\n2. $\\stackrel{\\to }{\\mathrm{A}}.\\stackrel{\\to }{\\mathrm{A}}$\n\n3. $\\stackrel{\\to }{\\mathrm{A}}×\\stackrel{\\to }{\\mathrm{A}}$\n\n4. $\\stackrel{\\to }{\\mathrm{A}}/\\mathrm{A}$\n\nSubtopic:  Resolution of Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nIf $\\stackrel{\\to }{\\mathrm{A}}=2\\stackrel{^}{\\mathrm{i}}+4\\stackrel{^}{\\mathrm{j}}-5\\stackrel{^}{\\mathrm{k}}$, then the direction cosines of the vector are:\n\n(direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three +ve coordinate axes.)\n\n1.\n\n2.\n\n3.\n\n4.\n\nSubtopic:  Resolution of Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nA force F applied at a 30° angle to the x-axis has the following X and Y components:\n\n1.\n\n2.\n\n3.\n\n4.\n\nSubtopic:  Resolution of Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nIf $\\stackrel{\\to }{\\mathrm{P}}=\\stackrel{\\to }{\\mathrm{Q}}$ , then which of the following is NOT correct?\n\n1. $\\stackrel{^}{\\mathrm{P}}=\\stackrel{^}{\\mathrm{Q}}$\n\n2. $\\left|\\stackrel{\\to }{\\mathrm{P}}\\right|=\\left|\\stackrel{\\to }{\\mathrm{Q}}\\right|$\n\n3. $\\mathrm{P}\\stackrel{^}{\\mathrm{Q}}=\\mathrm{Q}\\stackrel{^}{\\mathrm{P}}$\n\n4. $\\stackrel{\\to }{\\mathrm{P}}+\\stackrel{\\to }{\\mathrm{Q}}=\\stackrel{^}{\\mathrm{P}}+\\stackrel{^}{\\mathrm{Q}}$\n\nSubtopic:  Resultant of Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nThere are two force vectors, one of 5N and the other of 12N. At what angle should the two vectors be added to get the resultant vector of 17N, 7N, and 13N, respectively:\n\n1.\n\n2.\n\n3.\n\n4.\n\nSubtopic:  Resultant of Vectors |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nA particle moves from position null to $\\left(11\\stackrel{^}{\\mathrm{i}}+11\\stackrel{^}{\\mathrm{j}}+15\\stackrel{^}{\\mathrm{k}}\\right)$ due to a uniform force of $\\left(4\\stackrel{^}{\\mathrm{i}}+\\stackrel{^}{\\mathrm{j}}+3\\stackrel{^}{\\mathrm{k}}\\right)$N. If the displacement is in m, then the work done will be: (Given: $$W=\\vec{F}.\\vec{S}$$)\n\n1. 100 J\n\n2. 200 J\n\n3. 300 J\n\n4. 250 J\n\nSubtopic:  Scalar Product |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nIf for two vectors $\\stackrel{\\to }{\\mathrm{A}}$ and $\\stackrel{\\to }{\\mathrm{B}}$$\\stackrel{\\to }{\\mathrm{A}}×\\stackrel{\\to }{\\mathrm{B}}=0$, then the vectors:\n\n1. are perpendicular to each other.\n\n2. are parallel to each other.\n\n3. act at an angle of $60°.$\n\n4. act at an angle of $30°.$\n\nSubtopic:  Vector Product |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nLaunched MCQ Practice Books\n\nPrefer Books for Question Practice? Get NEETprep's Unique MCQ Books with Online Audio/Video/Text Solutions via Telegram Bot\n\nThe angle between vectors $\\left(\\stackrel{\\to }{\\mathrm{A}}×\\stackrel{\\to }{\\mathrm{B}}\\right)$ and $\\left(\\stackrel{\\to }{\\mathrm{B}}×\\stackrel{\\to }{\\mathrm{A}}\\right)$ is\n\n1. Zero\n\n2. $\\mathrm{\\pi }$\n\n3. $\\mathrm{\\pi }/4$\n\n4. $\\mathrm{\\pi }/2$\n\nSubtopic:  Vector Product |\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh\nTo view explanation, please take trial in the course below.\nNEET 2023 - Target Batch - Aryan Raj Singh" ]

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[ "# Kotlin Lambda Expressions\n\nHaving seen through functions, we have seen a variety of them having varied flavours. In this article here we would be going through functions which are of anonymous type but can be treated as values. By this we mean that we can pass them as arguments to fetch a value, return them directly from a function or in simple words, and treat them as a normal object.\n\nLambdas are not unique programming concept and have been there for quite some time having their implementations in different languages.\n\n## Kotlin Lambdas\n\nDefinition\n\nLambdas are function literals which denotes the functions that are not declared but passed on immediately as expressions. We use Lambdas to make our code simpler and compact.\n\nSyntax\n\n`val lamdbaFun : data_type = { arguments -> action }`\n\nAs per the above syntax, lamdbaFun is the name which we giving to our lambda function. The naming convention for lambda remains similar to any function. We also mention the data type of the return value to be returned by the function. The data type of return can be left blank as well, since the kotlin compiler can derive it out during compilation time of the function.\n\nLet’s have a look at the most basic lambda expression which neither accepts an input nor does return any value as shown in below example:\n\n```fun main(args: Array<String>) {\n// Declaring Lambda Function with name : printMessage\nval printMessage = { println(\"Hello World! \")}\n// Calling function : Print Message\nprintMessage()\n}```\n\nOutput:\n\nHello World!\n\nIn the above example “printMessage” is a lambda expression which is neither accepting an argument nor returning any value.\n\nLet’s see how we can have a function which accepts two input of type integer and returns the sum of two number.\n\n```fun main(args: Array<String>) {\n// Calling function: Calculate SUM\nprint (\"The sum of number is \\$sum (10,20)\")\n}\n\n// declaring Lambda Function with name: sum which accepts integer a & b as input\n// and returns the sum as output.\nval sum = { a: Int , b : Int -> a + b }```\n\nIn the example above we have created a function to calculate the sum, but if you look carefully there’s no return type of the function mentioned. The compiler automatically picks up the return type.\n\nNow let’s have a look at the example below which prints the square of a number as a String.\n\n```fun main(args: Array<String>) {\nprint(\"The square of number is: \"+ square(2))\n\n}\n\nval square = { num : Int ->\nval output = num * num\noutput.toString()\n}```\n\nIn the above example we have created the function to return a string and the same is passed back. The output of the above code is:\n\nThe square of number is: 4\n\nTill now we have covered the basic lambda expressions, let’s have a look at tweaked versions of lambdas which can be used in different ways:\n\n### Lambdas with when\n\nYes, we agree lambdas are single expression based functions and so are when which can be used in case of switch in kotlin. We can use when in the lambdas, since when also form a single expression:\n\n```fun main(args: Array<String>) {\n\n}\nval compileGrade = { marks : Int ->\nwhen(marks) {\nin 0..40 -> \"C\"\nin 41..70 -> \"B\"\nin 71..80 -> \"A\"\nin 81..90 -> \"A+\"\nin 91..100 -> \"A++\"\nelse -> false\n}\n}```\n\nIn the above example the entire when block is treated as a single expression and the output of the above code is as below:\n\nGrade for student with marks 95 is: A++\n\n### Lambdas Using the ‘it’ keyword\n\nIn cases when lambda functions are accepting multiple elements of the same type and there is an operation to be performed on each one of them, we can do it by using ‘it’ operator. It operator would reduce the need to pass the input argument and can be directly used with ‘it’ operator.\n\nLet’s see how we can implement it in coding:\n\n```fun main(args: Array<String>) {\n\nval arr = arrayOf(1,2,3,4,5)\n\nprint(\"Employee Code when using Regular Lambda Expressions\")\n// Regular/ long hand lambda expressions\narr.forEach { num -> println(\"EMP000\"+num) }\n\nprint(\"Employee Code when using Short Hand Lambda Expressions\")\n// Short Hand Lambda Expressions\narr.forEach { println(\"EMP000\"+it) }\n\n}```\n\nThe output of the above code is like:\n\nEmployee Code when using Regular Lambda Expressions\nEMP0001\nEMP0002\nEMP0003\nEMP0004\nEMP0005\nEmployee Code when using Short Hand Lambda Expressions\nEMP0001\nEMP0002\nEMP0003\nEMP0004\nEMP0005\n\nIn the above example, ‘it’ represents the each element of the array.\n\n### High-Order Functions & Lambdas\n\nKotlin supports functional programming. Kotlin provides the ability to pass functions as arguments to other functions. Also, kotlin functions have ability to return functions from other functions, these functions are referred as higher order functions. Kotlin has provided us with several built in functions which are available in standard library and accept arguments as lambdas.\n\nLets see an example of these functions by using lambdas in functions while working over collection. In the example below we are taking an example of function to find the employee with the highest salary.\n\n```data class Employee(val name: String, val sal: Int)\n\nfun main(args: Array<String>) {\n\nval employeeList = listOf(\nEmployee(\"Ram Mohan\", 5000),\nEmployee(\"Rohit Aryan\", 6200),\nEmployee(\"Hello World\", 5100),\nEmployee(\"New Employee\", 5150),\nEmployee(\"Sachin\", 5950),\nEmployee(\"Ramesh\", 4950)\n)\n\nval maxSalEmp = employeeList.maxBy({ emp -> emp.sal })\nprintln(maxSalEmp)\nprintln(\"Name Of Employee With Maximum Salary: \\${maxSalEmp?.name}\" )\nprintln(\"Maximum Salary od Employee : \\${maxSalEmp?.sal}\" )\n}```\n\nThe output of the above code would be like:\n\nEmployee(name=Rohit Aryan, sal=6200)\nName Of Employee With Maximum Salary: Rohit Aryan\nMaximum Salary od Employee : 6200\n\nSo, how it exactly happened?\n\nWe created a data class as Employee which has 2 properties i.e. name and salary, for the sake of simplicity we have taken salary type as int. Now, we make a list of employees and store it inside a variable as “employeeList”.  On this list we are calling a function “maxBy” which accepts the condition to be tested as lambda expression. That’s it, the work is done. We got the employee with maximum salary in the variable “maxSalEmp”.\n\nThe variable maxSalEmp is like just another Employee object class having the details of employee with the maximum salary.\n\nWe can even use the multiple functions in one case like shown in the example below:\n\n```data class Employee(val name: String, val sal: Int)\n\nfun main(args: Array<String>) {\n\nval employeeList = listOf(\nEmployee(\"Ram Mohan\", 5000),\nEmployee(\"Rohit Aryan\", 6200),\nEmployee(\"Hello World\", 5100),\nEmployee(\"New Employee\", 5150),\nEmployee(\"Sachin\", 5950),\nEmployee(\"Ramesh\", 4950)\n)\n\nval maxSalEmp = employeeList\n.filter { it.name.startsWith(\"R\")}\n.maxBy { it.sal }\n\nprintln(maxSalEmp)\nprintln(\"Name Of Employee starting with R and having Maximum Salary: \\${maxSalEmp?.name}\" )\nprintln(\"Name Of Employee starting with R with Maximum Salary od Employee : \\${maxSalEmp?.sal}\" )\n}```\n\nIn the above example, we have first used filter and the based on the filtered value we are checking the maximum salary.\n\nThe output of the above code is as below:\n\nEmployee(name=Rohit Aryan, sal=6200)\nName Of Employee starting with R and having Maximum Salary: Rohit Aryan\nName Of Employee starting with R with Maximum Salary od Employee: 6200\n\nSo, that was all about the Kotlin lambdas we just referred above, one of another major application of Lambdas function is the onClick() which we use in Android. Now, its time to put on your thinking cap and use lambdas there. Till then Happy Coding!" ]

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[ "# When was Hubble tension first noticed? When was this term first used?\n\nWhen was Hubble tension first noticed? When was this term first used? Who used this term for the first time?\n\n• There have always been disagreements about the value of $H_0$ using different techniques. All that has changed is that people are becoming more confident that the statistical and systematic errors are well enough understood that real discrepancies might be there. – ProfRob May 13 '20 at 11:04\n\nHubble tension refers to the incompatibility between different measurements of the value of the Hubble constant. These measurements are incompatible up to more than $$5 \\sigma$$. This incompatibility arises between what we measure \"nearby\" and what we measure further away, and indicates that there might be some physics we don't understand yet.\n\nNow there has always been some disagreement between the different values for $$H_0$$ -- in the early 20th century, estimates ranges from 50 to 550 km/s/Mpc -- but the error bars on those measurements were very large. So in the early days of the Hubble constant, although the proposed values varied wildly, there was no tension (well, there were quite a few arguments, but no Hubble tension yet), since it was accepted that the constant was the same on all scales.\n\nThe most likely value for $$H_0$$ mostly shrank in the middle of the 20th century, as bias in previous measurements were discovered. By the 60's, most agreed that $$60 < H_0 < 130$$. But around 1975, there started to be a divide between those who though that the constant was around $$55$$ km/s/Mpc and those who thought it was around $$100$$ km/s/Mpc. For instance, de Vaucouleurs found $$H_0 = 50$$ km/s/Mpc in 1970 and $$H_0 = 100\\pm 10$$ km/s/Mpc in 1977. But error bars were still large, and it was expected that the values for $$H_0$$ would converge.\n\nUp until very recently, the difference between measurements could reasonably be explained by the size of the error bars. But as measurement errors decreased, the range of possible values did not. This is what led to what we call the Hubble tension.\n\nWhat really made the Hubble tension clear were the measurements of the Cosmic Microwave Background (CMB) done by Planck in 2013. These were the best observations of the CMB that indicated that $$H_0 = 67.74±0.46$$ km/s/Mpc, in complete disagreement with the value around $$73$$ or $$74$$ km/s/Mpc derived from type Ia supernova. The Planck observations made the discrepancy between these values statistically significant. Around the same time, measurements of the Baryonic Acoustic Oscillations (BAO) confirmed the tension between the values obtained at low redshift (with the standard distance ladder) and the values obtained at high redshift.\n\nThe Hubble tension was noticed before it was formally called that way. See the following articles that describe this tension without using the expression \"Hubble tension\":\n\nIn fact, a search on arxiv, ads or google scholar for \"Hubble tension\", or \"Hubble-parameter tension\" before 2014 yields no results. It seems that in 2015, this exact phrase started to really be used (for instance here).\n\nIn the last five years, more measurements of $$H_0$$ have been made, confirming the tension between the \"high-redshift\" values and the \"low-redshift\" values, and solving the Hubble tension is a hot topic of research.\n\n• This is a very nicely written answer; thank you very much! – uhoh May 18 '20 at 10:58\n• The connection made in this answer between historical disagreements on the size of the Hubble constant, and the current tension between “local” and “cosmological” measurements seems unfounded. – mmeent May 19 '20 at 8:18\n• I tried to underline the fact that the disagreement about the value of $H_0$ is not sufficient to qualify as the Hubble tension. – usernumber May 19 '20 at 12:20" ]

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[ "# There should be universal LaTeX/MathJax guide for sites supporting it\n\nCurrently, stats, math, cstheory and physics have MathJax support which turns LaTeX code into equation.\n\nAt least on math, occasionally there are users that do not know this feature or don't know it's possible to use LaTeX (MathJax), resulting a badly formatted post.\n\nI suggest that there should be a common \"LaTeX/MathJax typesetting help\" page for those sites that enabled MathJax support, like the \"Markdown editing help\" that is present on all sites. Preferably, there should also be a line\n\n• LaTeX equations $\\sin^2 \\theta$\n\nin the \"How to Format\" on the right hand side.\n\n### See also: The MathJax help link should point to a more specific guide\n\n• Wait. Geeks aren't born knowing LaTeX? Oct 23, 2010 at 15:36\n• You mean a MathJaX help. MathJaX =/= LaTeX. Oct 23, 2010 at 15:41\n• @AndrewStacey: well, MathJax ≈ LaTeX-surrounded-by-dollar-signs\n– Kip\nNov 17, 2010 at 14:16\n• @Kip: *shakes head sadly at the ignorance of youth*. I \\emph{guess} it depends on what \\textbf{you} mean by \\LaTeX\\ really. Nov 17, 2010 at 14:24\n\nThis is basically completed, but I can't find a good help / demos page to link to for MathJaX.\n\nThere is this:\n\nhttp://www.mathjax.org/demos/\n\nbut oddly it requires users to \"view source\" before showing them the markup required, which is ... annoying.\n\nPer suggestion, I am now changing it to\n\nhttp://www.math.harvard.edu/texman/\n\n• Maybe you can pop a question on meta.math.se and ask the good people there to write one? Not sure how much work it would take though. Nov 19, 2010 at 6:50\n• @YiJiang: meta.stackexchange.com/questions/68388/…. (I'm putting it here instead of meta.math.SE as this is not just a math.SE feature.) Nov 19, 2010 at 19:26\n• I found the Wikipedia formula-writing page very helpful: en.wikipedia.org/wiki/Help:Displaying_a_formula\n– Kip\nNov 20, 2010 at 3:17\n• Jeff, this was added as a comment to the meta.math post, but I'll repeat it here so you're more likely to see it- the help box is almost perfect, but MathJax will render the $\\sin^2 \\theta$ in it, so what the user actually sees is \"MathJax equation sin² θ\", which isn't correct. It has to be wrapped between <code/> tags or <span class=\"tex2jax_ignore\"/> tags to avoid MathJax parsing it.\n– Kip\nNov 20, 2010 at 3:19\n• I just spent 15 minutes trying to figure out how to do a simple subscript. Sending me off to mathjax.org/demos frankly wasted my time as I vainly clicked around looking for help. Please don't provide that link any more as there's no obvious way to get any syntax help! I wrote the MathJax people directly and they suggested math.harvard.edu/texman. Would you please consider changing your link on the question-asking page to point there instead? It will save your users much wasted time and frustration. Jan 21, 2011 at 20:19\n• @emt sure I can change it to that, checking that change in now. Jan 22, 2011 at 0:51\n• Thanks a lot. There may be better resources out there (in fact, the MathJax folks asked me to keep them posted if a better one is found) but for now that one is at least adequate and did have syntax help like I was expecting. Jan 22, 2011 at 2:22\n\nLet's write our own help page for our use of MathJax. Here is a start. It is based on Stack Overflow's own Markdown editing help, the FAQ for typing math on math.SE and Math Overflow, and \"Using LaTeX\" on ask NRICH.\n\nCharacters in bold italics indicate highlighting.\n\n# MathJax turns LaTeX markup into beautiful formulae\n\nThis site supports typesetting mathematical formula with AMS-LaTeX markup, powered by the MathJax rendering engine.\n\n## Entering math mode\n\nSurround the TeX code with dollar signs to insert an inline equation\n\nThe integers $x,y,z$ form a Pythagorean triplet when $x^2 + y^2 = z^2$.\n\nand use double dollar signs to insert an equation in its own line\n\nThe Bessel functions $J_n(x)$ and $Y_n(x)$ are\nsolutions to the Bessel equation\n$$x^2 y'' + x y' + (x^2 - n^2) y = 0$$\nwhere $n$ is a constant.\n\n## Basic LaTeX markup\n\n• Superscript and subscript — x^2, a_n, a_{n+1}, H_n^{(2)}\n\n• Spacing — a\\ b (text space). Other kinds of spacing.\n\n• Square root and radicals — \\sqrt{x}, \\sqrt{x}\n\n• Fraction — \\frac{a}{b}\n\n• Sum and integral — \\sum_{k=0}^n k^2, \\int_0^1 x^3 dx\n\n• Greek letters — \\alpha (α) to \\omega (ω); \\Gamma (Γ) to \\Omega (Ω)\n\n• Symbols — \\ne (≠), \\ge (≥), \\le (≤), \\sim (∼), \\pm (±), \\to (→), \\infty (∞), etc.\n\n• Function names — \\sin, \\cos, \\log, \\lim, etc.\n\nVisit Detexify2 to lookup command for a symbol.\n\nCheck the MathJax documentation for the complete list of commands supported.\n\n## Show source\n\nRight-clicking on any equations should reveal a context menu. Clicking \"Show source\" will open up a new window showing the LaTeX markup that generates it.\n\n## Need More Detail?\n\nThe Not So Short Introduction to LaTeX2e is a good beginner's guide on the LaTeX system.\n\n• Let's get this up! Right now the link provided is not helping anybody. Jan 21, 2011 at 20:20\n• Lack of proper docs & examples is actually getting worse, as even more fantasy-themed sites such as Worldbuilding are supporting it, but only link to mathjax.org, which only links to docs.mathjax.org which has LaTeX links to the glossary of docs.mathjax.org which links to wikipedia.org/wiki/LaTeX... 15 clicks and 3 deliberate searches later to find something that resembles respectable docs just to type $\\Delta$. This may be the opportunity for SO Docs to step up as the de-facto resource. Feb 14, 2017 at 20:47\n\nI brought this up on meta.math and was directed to this question. I thought I'd share the mockup here too, showing what I'd like:", null, "• Heyo, and guess what happened! At least on math.SE Oct 8, 2017 at 20:31\n\nA much much more detailed MathJax tutorial/reference has been created on Math.SE: MathJax basic tutorial and quick reference" ]

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[ "# The Raw data structure: continuous data¶\n\nContinuous data is stored in objects of type Raw. The core data structure is simply a 2D numpy array (channels × samples) (in memory or loaded on demand) combined with an Info object (.info attribute) (see The Info data structure).\n\nThe most common way to load continuous data is from a .fif file. For more information on loading data from other formats, or creating it from scratch.\n\nimport mne\nimport os.path as op\nfrom matplotlib import pyplot as plt\n\n\nLoad an example dataset, the preload flag loads the data into memory now:\n\ndata_path = op.join(mne.datasets.sample.data_path(), 'MEG',\n'sample', 'sample_audvis_raw.fif')\nraw.set_eeg_reference('average', projection=True) # set EEG average reference\n\n# Give the sample rate\nprint('sample rate:', raw.info['sfreq'], 'Hz')\n# Give the size of the data matrix\nprint('%s channels x %s samples' % (len(raw), len(raw.times)))\n\n\nOut:\n\nOpening raw data file /home/circleci/mne_data/MNE-sample-data/MEG/sample/sample_audvis_raw.fif...\nRead a total of 3 projection items:\nPCA-v1 (1 x 102) idle\nPCA-v2 (1 x 102) idle\nPCA-v3 (1 x 102) idle\nRange : 25800 ... 192599 = 42.956 ... 320.670 secs\nCurrent compensation grade : 0\nReading 0 ... 166799 = 0.000 ... 277.714 secs...\nAdding average EEG reference projection.\n1 projection items deactivated\nAverage reference projection was added, but has not been applied yet. Use the apply_proj method to apply it.\nsample rate: 600.614990234375 Hz\n166800 channels x 166800 samples\n\n\nNote\n\nThis size can also be obtained by examining raw._data.shape. However this is a private attribute as its name starts with an _. This suggests that you should not access this variable directly but rely on indexing syntax detailed just below.\n\nInformation about the channels contained in the Raw object is contained in the Info attribute. This is essentially a dictionary with a number of relevant fields (see The Info data structure).\n\n## Indexing data¶\n\nTo access the data stored within Raw objects, it is possible to index the Raw object.\n\nIndexing a Raw object will return two arrays: an array of times, as well as the data representing those timepoints. This works even if the data is not preloaded, in which case the data will be read from disk when indexing. The syntax is as follows:\n\n# Extract data from the first 5 channels, from 1 s to 3 s.\nsfreq = raw.info['sfreq']\ndata, times = raw[:5, int(sfreq * 1):int(sfreq * 3)]\n_ = plt.plot(times, data.T)\n_ = plt.title('Sample channels')", null, "### Selecting subsets of channels and samples¶\n\nIt is possible to use more intelligent indexing to extract data, using channel names, types or time ranges.\n\n# Pull all MEG gradiometer channels:\n# Make sure to use .copy() or it will overwrite the data\nmeg_only = raw.copy().pick_types(meg=True)\neeg_only = raw.copy().pick_types(meg=False, eeg=True)\n\n# The MEG flag in particular lets you specify a string for more specificity\n\n# Or you can use custom channel names\npick_chans = ['MEG 0112', 'MEG 0111', 'MEG 0122', 'MEG 0123']\nspecific_chans = raw.copy().pick_channels(pick_chans)\nprint(meg_only)\nprint(eeg_only)\nprint(specific_chans)\n\n\nOut:\n\n<Raw | sample_audvis_raw.fif, n_channels x n_times : 305 x 166800 (277.7 sec), ~391.7 MB, data loaded>\n<Raw | sample_audvis_raw.fif, n_channels x n_times : 59 x 166800 (277.7 sec), ~78.2 MB, data loaded>\n<Raw | sample_audvis_raw.fif, n_channels x n_times : 203 x 166800 (277.7 sec), ~261.7 MB, data loaded>\n<Raw | sample_audvis_raw.fif, n_channels x n_times : 4 x 166800 (277.7 sec), ~8.1 MB, data loaded>\n\n\nNotice the different scalings of these types\n\nf, (a1, a2) = plt.subplots(2, 1)\neeg, times = eeg_only[0, :int(sfreq * 2)]\nmeg, times = meg_only[0, :int(sfreq * 2)]\na1.plot(times, meg)\na2.plot(times, eeg)\ndel eeg, meg, meg_only, grad_only, eeg_only, data, specific_chans", null, "You can restrict the data to a specific time range\n\nraw = raw.crop(0, 50) # in seconds\nprint('New time range from', raw.times.min(), 's to', raw.times.max(), 's')\n\n\nOut:\n\nNew time range from 0.0 s to 50.00041705299622 s\n\n\nAnd drop channels by name\n\nnchan = raw.info['nchan']\nraw = raw.drop_channels(['MEG 0241', 'EEG 001'])\nprint('Number of channels reduced from', nchan, 'to', raw.info['nchan'])\n\n\nOut:\n\nNumber of channels reduced from 376 to 374\n\n\n### Concatenating Raw objects¶\n\nRaw objects can be concatenated in time by using the append function. For this to work, they must have the same number of channels and their Info structures should be compatible.\n\n# Create multiple :class:Raw <mne.io.RawFIF> objects\nraw1 = raw.copy().crop(0, 10)\nraw2 = raw.copy().crop(10, 20)\nraw3 = raw.copy().crop(20, 40)\n\n# Concatenate in time (also works without preloading)\nraw1.append([raw2, raw3])\nprint('Time extends from', raw1.times.min(), 's to', raw1.times.max(), 's')\n\n\nOut:\n\nTime extends from 0.0 s to 40.00399655463821 s\n\n\nTotal running time of the script: ( 0 minutes 6.486 seconds)\n\nEstimated memory usage: 1657 MB\n\nGallery generated by Sphinx-Gallery" ]

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[ "# Sudoku\n\nSudoku puzzle consists of 81 squares (some of them are prefilled) and the objective is to fill all of them such that all rows, columns and boxes (groups of 3x3 squares) contain digits 1 to 9. Let $$x_{sd}$$ be an indicator variable for the square $$s$$ and the digit $$d$$. $$\\sum_{s \\in u} x_{sd} = 1 \\quad \\forall u \\ \\forall d$$\n\nSince the number of digits and the number of squares in a unit is the same we can define the following (redundent?) constraints. $$\\sum_d x_{sd} = 1 \\quad \\forall s$$\n\nWhen a digit is assigned to a square then we can set zero that digit in neigbor squares. $$x_{sd}=1 \\Rightarrow x_{s'd}=0 \\quad \\forall s'\\in N(s)$$\n\nIf we can find two digits elligible to only two squares of a unit then we can safely remove the other possible digits from these squares. (Hall's theorem? pigoenhole principle?) \\begin{align} \\exists s_1, s_2 \\in u, \\ \\exists d_1, d_2 \\quad x_{s_1 d_1} + x_{s_1 d_2} + x_{s_2 d_1} + x_{s_2 d_2} = 2 \\quad \\\\ \\Rightarrow x_{s_1d} = 0, x_{s_2d}=0 \\quad \\forall d \\notin (d_1, d_2) \\end{align}\n\nIf possible squares in a box aligns to a row (resp. column) then we can remove that digit the rest of the row (resp. column). $$x_{s_1d} + x_{s_2d} = 1 \\Rightarrow x_{s'd} = 0 \\quad \\forall s' \\in N(s_1, s_2)$$\n\nIf for a square, all the digits are eliminated except one then we can assign that digit to that square. $$\\exists d,s \\quad x_{sd'}=0 \\quad \\forall d' \\neq d \\Rightarrow x_{sd} = 1$$\n\nIf in a unit there is only one elligible square remained for a digit then we can assign that digit to that square. $$\\exists u,d,s \\sum_{s'\\in u, \\ s' \\neq s} x_{s'd} = 0 \\Rightarrow x_{sd} = 1$$" ]

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[ "# Solve the following pair of linear equations\n\n## Question :\n\nSolve the following pair of linear equations :\n\n(i)   px + qy = p – q       and    qx – py = p + q\n\n(ii)  ax + by = c     and    bx + ay = 1 + c\n\n(iii)  $$x\\over a$$ – $$y\\over b$$ = 0      and      ax + by = $$a^2 + b^2$$\n\n(iv)  Solve for x and y :\n\n(a – b)x + (a + b)y = $$a^2 – 2ab – b^2$$\n\n(a + b)(x + y) = $$a^2 + b^2$$\n\n(v)  152x – 378y = -74        and        -378x + 152y = -604\n\n## Solution :\n\n(i)  The given linear equations are\n\npx + qy = p – q     $$\\implies$$  px + qy – (p – q) = 0             ………….(1)\n\nqx – py = p + q       $$\\implies$$   qx – py – (p + q) = 0           ………….(2)\n\nSolving it by cross multiplication method, we get\n\n$$x\\over -q(p + q) – p(p – q)$$ = $$y\\over -q(p – q) + p(p + q)$$ = $$1\\over -p^2 – q^2$$\n\n$$\\implies$$  $$x\\over -pq – q^2 – p^2 + pq$$ = $$y\\over – pq + q^2 + p^2 + pq$$ = $$1\\over -(p^2 + q^2)$$\n\n$$\\implies$$  $$x\\over -(p^2 + q^2)$$ = $$y\\over p^2 + q^2$$ = $$1\\over -(p^2 + q^2)$$\n\n$$\\implies$$  x = 1 and y = 1.\n\n(ii)  The given linear equations are\n\nax + by – c = 0             ………….(1)\n\nbx + ay – (1 + c) = 0           ………….(2)\n\nSolving it by cross multiplication method, we get\n\n$$x\\over -b(1 + c) + ac$$ = $$y\\over -bc + a(1 + c)$$ = $$1\\over a^2 – b^2$$\n\n$$\\implies$$  $$x\\over -b – bc + ac$$ = $$y\\over -bc + a + ac$$ = $$1\\over a^2 – b^2)$$\n\n$$\\implies$$  $$x\\over c(a – b) – b$$ = $$y\\over c(a – b) + a$$ = $$1\\over (a – b)(a + b)$$\n\n$$\\implies$$  x = $$c\\over a + b$$ – $$b\\over (a – b)(a + b)$$\n\ny = $$c\\over a + b$$ + $$b\\over (a – b)(a + b)$$\n\n(iii)  The given linear equations are\n\n$$x\\over a$$ – $$y\\over b$$ = 0    $$\\implies$$    bx – ay = 0               ………..(1)\n\nax + by -$$(a^2 + b^2)$$ = 0           ………(2)\n\nSolving it by cross multiplication method, we get\n\n$$x\\over a(a^2 + b^2) – 0$$ = $$y\\over 0 + b(a^2 + b^2)$$ = $$1\\over b^2 + a^2$$\n\n$$\\implies$$   $$x\\over a(a^2 + b^2)$$ = $$y\\over b(a^2 + b^2)$$ = $$1\\over a^2 + b^2$$\n\n$$\\implies$$  x = a,  y = b.\n\n(iv)  The given linear equations are\n\n(a – b)x + (a + b)y = $$a^2 – 2ab – b^2$$              ………..(1)\n\n(a + b)(x + y) = $$a^2 + b^2$$          ………(2)\n\nSolving it by cross multiplication method, we get\n\n$$x\\over -(a + b)(a^2 + b^2) + (a + b)(a^2 – 2ab – b^2)$$ = $$y\\over -(a + b)(a^2 – 2ab – b^2) + (a – b)(a^2 + b^2)$$ = $$1\\over (a – b)(a + b) – {(a + b)}^2$$\n\n$$\\implies$$   $$x\\over (a + b)(-a^2 – b^2 + a^2 – 2ab – b^2)$$ = $$y\\over -a^3 + 2a^2b + ab^2 – a^2b + 2ab^2 – b^3 + a^3 + ab^2 – a^2b – b^3$$ = $$1\\over a^2 – b^2 – a^2 – b^2 – 2ab$$\n\n$$\\implies$$  $$x\\over (a + b)(-2ab – 2b^2)$$ = $$y\\over 4ab^2$$ = $$1\\over -2b^2 – 2ab$$\n\n$$\\implies$$  $$x\\over (a + b)(-2b)(a + b)$$ = $$y\\over 4ab^2$$ = $$1\\over -2b(a + b)$$\n\n$$\\implies$$  x = a + b,  y = $$-2ab\\over a + b$$\n\n(v)  The given linear equations are\n\n152x – 378y = -74            ………(1)\n\n-378x + 152y = -604         ……….(2)\n\nAdding equation (1) and (2), we get\n\n-226x – 226y = -678       $$\\implies$$     x + y = 3           ……(3)\n\nSubtracting equation (1) from (2),  we get\n\n-530x + 530y = -530      $$\\implies$$     x – y = 1             …….(4)\n\nAdding equation (3) and (4), we get\n\n2x = 4   $$\\implies$$   x = 2\n\nPut the value of x = 2 in equation (4), we get\n\ny = 1" ]

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[ "# trimesh.inertia\n\n## inertia.py\n\nFunctions for dealing with inertia tensors.\n\nResults validated against known geometries and checked for internal consistency.\n\nReturn the inertia tensor of a cylinder.\n\nParameters\n• mass (float) – Mass of cylinder\n\n• height (float) – Height of cylinder\n\n• transform ((4, 4) float) – Transformation of cylinder\n\nReturns\n\ninertia – Inertia tensor\n\nReturn type\n\n(3, 3) float\n\ntrimesh.inertia.principal_axis(inertia)\n\nFind the principal components and principal axis of inertia from the inertia tensor.\n\nParameters\n\ninertia ((3, 3) float) – Inertia tensor\n\nReturns\n\n• components ((3,) float) – Principal components of inertia\n\n• vectors ((3, 3) float) – Row vectors pointing along the principal axes of inertia\n\nCheck whether a mesh has radial symmetry.\n\nReturns\n\n• symmetry (None or str) – None No rotational symmetry ‘radial’ Symmetric around an axis ‘spherical’ Symmetric around a point\n\n• axis (None or (3,) float) – Rotation axis or point\n\n• section (None or (3, 2) float) – If radial symmetry provide vectors to get cross section\n\nReturn the inertia tensor of a sphere.\n\nParameters\n• mass (float) – Mass of sphere\n\nReturns\n\ninertia – Inertia tensor\n\nReturn type\n\n(3, 3) float\n\ntrimesh.inertia.transform_inertia(transform, inertia_tensor)\n\nTransform an inertia tensor to a new frame.\n\nMore details in OCW PDF: MIT16_07F09_Lec26.pdf\n\nParameters\n• transform ((3, 3) or (4, 4) float) – Transformation matrix\n\n• inertia_tensor ((3, 3) float) – Inertia tensor\n\nReturns\n\ntransformed – Inertia tensor in new frame\n\nReturn type\n\n(3, 3) float" ]

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[ "If assets are valued at net book value, ROI and residual income figures generally improve as assets get older. The bigger investment would give a net return of \\$50,000. Hence, senior managers need to introduce systems of performance measurement to ensure that decisions made by junior managers are in the best interests of the company as a whole. In this formula, the monthly net income is the sum of all passive income earned which can be from royalties, rental income, interest earning on saving, subscription or service fee for a service rendered. Residual Income Equation Components. controllable (traceable) profit - an imputed interest charge on controllable (traceable) investment. For the first company division listed, insert formula commands to calculate ROI and Residual income (assume the company has a 25% target rate of return). Viewing 10 posts - 1 through 10 (of 10 total) Residual income (RI) can mean different things depending on the context. Evaluation of RI as a performance measure . Compared to using return on investment (ROI) as a measure of performance, RI has several advantages and disadvantages:. Since the ROI (ROA) for ABC, Inc. is below the industry average, you want to find out why. Residual income also features in corporate finance and valuation where it equals the difference between a company's net income and the product of the company's equity capital and its cost of equity. C. only the gross book value of assets needs to be calculated. OK References Return On Investment (ROI) And Residual Income 1. When looking at corporate finance, residual income is any excess that an investment earns relative to the opportunity cost Opportunity Cost Opportunity cost is one of the key concepts in the study of economics and is prevalent throughout various decision-making processes. r = Cost of equity. It is quite possible that some departments may be able to accept a lower ROI project while a higher ROI project at another department may not get the required investment.eval(ez_write_tag([[580,400],'xplaind_com-box-4','ezslot_2',134,'0','0'])); CP Inc. is a company engaged in production and distribution of computers and printers. Internal Rate of Return (IRR) The Internal Rate of Return (IRR) is the discount rate that makes the net present value (NPV) of a project zero. The formula for residual income (RI) is: If department managers are evaluated based on the residual income that their departments generate, they have an incentive to accept all such projects which earn a return greater than the minimum required rate of return. Under ROI the basic objective is to maximize the rate of return percentage. Department C's average operating assets are \\$1.05 billion on which the minimum required return is \\$157.5 million (=\\$1,050 million × 15%). Residual Income is the money remaining after paying the necessary expenses and costs. Explain the meaning of, and calculate, Return on Investment (ROI) and Residual Income (RI), and discuss their shortcomings. ADVERTISEMENTS: Return on Investment (ROI): Advantages and Disadvantages! Return on investment (ROI) calculates total return in percentage terms and is a better measure of relative performance. This method (RI) is an alternative approach to calculate the performance of the investment center. It is based on the company's cost of capital and the risk of the project. Residual Income (RI) Residual income is a measure used as part of divisional performance management for investment centres. In this example, Department C has a return on investment (ROI) of 28.6% (\\$300 million/\\$1,050 million) while Department P has return on investment (ROI) of 21.67% (\\$130 million/\\$600 million).eval(ez_write_tag([[300,250],'xplaind_com-leader-1','ezslot_4',109,'0','0'])); by Obaidullah Jan, ACA, CFA and last modified on Apr 7, 2019Studying for CFA® Program? Investment center. Better Measure of Profitability: It relates net income to investments made in a division giving a better measure of divisional profitability. Residual income (RI) is defined as the amount of income a segment has in excess of the segment’s investment base times its cost of capital percentage. All divisional managers know that their performance will be judged in terms of how they have utilized […] Target rate of return X total assets (target rate of return is the same as ROI, but it is set as a desired goal by management) Gross book value. For example, assume you could borrow unlimited amounts of money from the bank at a cost of 10% per annum. Advantages of ROI: ROI has the following advantages: 1. Residual Income and EVA (Economic Value Added) are two methods that assess how much funds in excess of the business’ cost of capital the investment is projected to generate. What is Residual Income? Click OK To Begin. Return on Investment (ROI) Vs Residual Income (RI): RI is favoured for reasons of goal congruence and managerial effort. The following table lists the operating income and assets of the departments: The company's weighted average cost of capital is 12% and the highest return available on new investment opportunities foregone equals 15%.eval(ez_write_tag([[250,250],'xplaind_com-banner-1','ezslot_3',135,'0','0'])); Since the company can earn 15% on alternate projects, it is treated as the minimum required return. Since the residual income in both cases is positive, we conclude that both have met the minimum return requirements. It encourages investment centre managers to make new investments if they add to RI. In management accounting, the following formula works out the return on investment of a department: Department's net operating income (also called segment margin) equals the department's revenue minus all controllable expenses. Residual Income is the money remaining after paying the necessary expenses and costs. Understanding Return on Investment (ROI) ROI is a popular metric because of its versatility and simplicity. The formula to calculate Residual Income is the following: RI = Net income – (Equity * Cost of Equity). Return on investment (ROI) is a ratio which measures gain/income generated by an investment per dollar of capital invested. 2. We use cookies to help make our website better. Advantages . --------------------------------------                                            Residual Income Calculator - calculate the residual income. Return on investment (ROI) measures the rate of profitability of a given investment. Yes, a leasing Company, Inc. (YCI), is a mid-size company in terms of market capitalization, and as per public records, the firm has reported total assets of US\\$4 million and the capital structure of the firm is Fifty % with equity capital and Fifty % with debt. Its residual income is hence \\$142.5 million: Residual Income (Department C) = \\$300 million - \\$1,050 million × 15% = \\$142.5 millioneval(ez_write_tag([[250,250],'xplaind_com-large-leaderboard-2','ezslot_5',136,'0','0'])); Department P's average operating assets are \\$0.6 billion on which the minimum required return is \\$90 million. The minimum required return is 15% and the department manager is considering a project that will earn \\$50,000 and require additional capital of \\$300,000. Both measures require an estimate of the cost of capital, a figure which can be difficult to calculate. Use the DuPont formula to compute the rate of return on investment. Starting with operating profit, then deducting the adjusted tax charge (because tax charge includes the tax benefit of interest). In fact, the residual income is the performance indicator for the companies just like return on investment for portfolio managers. Two measures of divisional performance are commonly used: Return on investment (ROI) All Answers Must Be Entered As A Formula. V 0 = Market Value of the Firm. Return on investment (ROI) is very similar to return on capital employed (ROCE) except the focus is on controllable and traceable revenues, expenses and assets. This clearly shows that assessing the performance of the investment center with residual income (RI) is a better option since it provides a better analysis, and it … This video discusses the difference between ROI and Residual Income. It has two main operating departments: Department C specializes in design, production and marketing of computers and Department P deals in printers. Let us take the example of an investment center that had an operating income of \\$1,000,000 during the year by using operating assets worth \\$5,000,000. Since the manager’s goal is to continually increase RI, the proposed investment would be accepted resulting in an increase of \\$3,500 in RI (= \\$13,500 − \\$10,000). RI = Operating Income - (Operating Assets x Target Rate of Return) ROI % = Operating Income / Operating Assets. Subtract that from your income, and you’ve got what’s left over — the residual income. Net Income: Net earnings after deducing all costs, expenses, depreciation, amortization, interest charges and taxes from the business revenues. There is another measuring tool for the assessment i-e Return on investment (ROI). Both techniques attempt to measure divisional performance in a single figure. Re… This video discusses the difference between ROI and Residual Income.                                      The formula for residual income can be derived by deducting the product of the minimum required rate of return and average operating assets from the operating income. The averageof the operating assets is used when possible. In fact, they require some ongoing effort, too (to various degrees). In this formula, the monthly net income is the sum of all passive income earned which can be from royalties, rental income, interest earning on saving, subscription or service fee for a service rendered. Residual income, being an absolute measure, would lead you to select the project that maximises your wealth. Residual income does, however, experience problems in comparing managerial performance in divisions of different sizes. Average operating assets of the department represents the total capital employed by the department. Even though ROI is the most popular measure, it suffers from a serious drawback. One key consideration for this item is the adjustment of the cost of interest. Compute Return On Investment (ROI) 2. This method is used in comparison to the return on investment (ROI) method. Definitions and meanings: Return on investment: Return on investment (ROI) is a measure which calculates the efficiency of an investment by calculating percentage of return earned by that investment. Under ROI the basic objective is to maximize the rate of return percentage. In case of an investment in capital markets, ROI can be calc… In this regard, the residual income formula becomes: Residual Income = Monthly Net Income – Monthly Debts. The ROI is one of the most widely used performance measurement tool in evaluating an investment center. Residual income is a better measure for performance evaluation of an investment center manager than return on investment because: desirable investment decisions will not be rejected by divisions that already have a high ROI. Business residual income is the net operating income of a department whereas personal residual income is your monthly income after paying off your debt. \\$ Profit Margin, Investment Turnover, and ROI Campbell Company has income from operations of \\$29,400, invested assets of \\$140,000, and sales of \\$294,000. Residual Income = Net Income of the firm – Equity Charge = 123765.00 – 110000.00; Example #2. To do that, you can use the Dupont Model and break down the ROI into its component parts. It encourages investment centre managers to make new investments if they add to RI. However, with residual income is not particularly useful in comparing performance. Yes, traditional investing is a residual income idea. Residual income approach is useful in allocating resources among projects or investments. The result in both cases would be a certain amount of money. A positive residual income means that the department has met the minimum return requirement while a negative residual income means that the department has failed to meet it. It is calculated by subtracting the product of a department's average operating assets and the minimum required rate of return from its controllable margin. Residual income: Residual income (RI) is the amount of income an investment opportunity generates above the minimum level of rate of return. residual income. Residual Income (RI) Residual income is a measure used as part of divisional performance management for investment centres. A return on investment (ROI) for real estate can vary greatly depending on how the property is financed, the rental income, and the costs involved. Compute Residual Income. We hope you like the work that has been done, and if you have any suggestions, your feedback is highly valuable. It is calculated by dividing the sum of the opening and closing operating assets balances by 2. In technical terms, it is the income that one generates in excess of the minimum rate of return or the opportunity cost of capital.There is a residual income formula that helps in ascertaining residual income. What is the residual income for the division? An investment center is a subunit of an organization that has control over its own sources of revenues, the costs incurred, and assets (investments) employed. Return on Investment (ROI) Vs Residual Income (RI): RI is favoured for reasons of goal congruence and managerial effort. Residual Income = Net Income of the firm – Equity Charge = 123765.00 – 110000.00; Example #2. In technical terms, it is the income that one generates in excess of the minimum rate of return or the opportunity cost of capital.There is a residual income formula that helps in ascertaining residual income. The formula in computing for the residual income is: where: Desired income = Minimum required rate of return x Operating assets Note: In most cases, the minimum required rate of return is equal to the cost of capital. The residual income formula is calculated by subtracting the product of the minimum required return on capital and the average cost of the department’s capital from the department’s operating income. It creates an incentive for managers to not invest in projects which reduce their composite ROI even though those projects generate a return greater than the minimum required return.eval(ez_write_tag([[468,60],'xplaind_com-medrectangle-4','ezslot_1',133,'0','0'])); Let us consider a department whose current operating income is \\$200,000 and its asset base is \\$1,000,000. Thus, managers of highly profitable […] Mathematically, Residual Income is represented as, This approach is used when opposed to the approach of return on investment (ROI). Yes, a leasing Company, Inc. (YCI), is a mid-size company in terms of market capitalization, and as per public records, the firm has reported total assets of US\\$4 million and the capital structure of the firm is Fifty % with equity capital and Fifty % with debt. There is a second definition for residual income that’s an accounting term used to help businesses calculate net income. Return on investment (ROI) is another performance evaluation tool which equals the operating income earned by a department divided by its asset base. The cost of interest is included in the finance charge (WACC*capital) that is deducted from NOPAT in the EVA calculation and can be approached in two ways: 1. controllable (traceable) profit %  In this regard, the residual income formula becomes: Residual Income = Monthly Net Income – Monthly Debts. Formula. Compute Return On Investment (ROI) 2. Return on investment (ROI) is very similar to return on capital employed (ROCE) except the focus is on controllable and traceable revenues, expenses and assets. The formula of ROI is: ROI % = Operating Income / Operating Assets . When there is a positive RI, it indicates that the company has met the minimum return cost. Residual income also ties in with net present value, theoretically the best way to make investment decisions. Investment projects with positive net present value can show poor ROI and residual income figures in early years leading to rejection of projects by managers. My rich dad taught me to focus on passive income streams and spend my time acquiring the assets that provide passive and long-term residual income …income from capital gains, dividends, residual income from business, rental income from real estate, and royalties.. The formula for residual income is the same, whether the metric is used for personal or business finance Additionally, residual income serves as a way to track the flow of your earnings. Let's connect. OK References Return On Investment (ROI) And Residual Income 1. If assets are valued at net book value, ROI and residual income figures generally improve as assets get older. Equity: The total equity as stated in the Balance Sheet. In the long run, companies that maximise residual income will also maximise net present value and in turn shareholder wealth. When companies use the residual income method, management is evaluated based on the growth in the residual income from year to year instead of the growth in the rate of return. However, from the company’s perspective, accepting the project is the right thing to do because the project's return of 16.67% is higher than the minimum required return. Or, as one of our financial hero’s Robert Kiyosaki puts it:. The basic formula is: Net Income – (Expenses + Debts) = Residual Income. Compute Residual Income. The most detailed measure of return is known as the Internal Rate of Return (IRR). RI = Residual Income. Advantages . The formula of ROI is: Compared to using return on investment (ROI) as a measure of performance, RI has several advantages and disadvantages:. You are welcome to learn a range of topics from accounting, economics, finance and more. Compute Return On Investment (ROI) 2. If possible, the averageamount for the period is used. Although the smaller investment has the higher percentage rate of return, it would only give you an absolute net return (residual income) of \\$15 per annum after borrowing costs. The company will accept a project whose residual income is a positive number, because it shows that the project is earning more than the minimum as expected by the company. Better Measure of Profitability: It relates net income to investments made in a division giving a better measure of divisional profitability. Both residual income and EVA are based on the same principle … ROI is composed of two parts, the company's profit margin and the asset turnover—the firm's ability to generate profit and make sales based on its asset base. Residual Income (Department P) = \\$130 million - \\$600 million × 15% = \\$40 million. Compute the return on investment (ROI) for each division, using the formula stated in terms of margin and turnover. It is calculated by dividing the sum of income and capital gain of an investment by the cost of investment. Solution 13.2: Compare and contrast the return on investment and residual income measures of divisional performance. Of students, roi and residual income formula students, and for students and in turn shareholder.. To maximize the rate of return is known as the Internal rate of is. In this regard, the calculation for ROI would be: ROI % = \\$ 40 million period used.: department C specializes in design, production and marketing of computers and department deals! Can use the DuPont formula to compute the return on investment for portfolio managers at AlphaBetaPrep.com 130 million \\$., with residual income will also maximise net present value, ROI be! In case of an investment 's performance balances by 2, theoretically the best way to new! ; of students, and you ’ ve got What ’ s an accounting term to. When opposed to the use the residual income formula is: net after! Hence \\$ 40 million, the residual income figures generally improve as assets get.... It indicates that the residual income is the following: RI = net income – Debts... Are good candidates for expansion divisional performance help businesses calculate net income – ( expenses + Debts =! Online merchandise, take a lot of upfront time and effort equity as stated in terms margin! The period is used in comparison to the very generic return on and... As assets get older generally improve as assets get older can mean different things depending on the principle. Both have met the minimum return cost balances by 2 ) investment What... About the difference between return on investment is the performance indicator for the?. Debt costs establishes its cost of 10 % per annum – Monthly.! The money remaining after paying off your debt s an accounting term used to help make our website.! Assets needs to be calculated \\$ 40 million ( \\$ 130 million - \\$ million. Calculates total return in percentage terms and is a free educational website ; of students and. Of equity ) be calculated net earnings after deducing all costs, expenses roi and residual income formula depreciation amortization. Encourage managers to retain outdated plant and machinery be calculated of modern,... Better measure of performance are necessary, working capital, a company may not be able to arrange money all! Met the minimum return cost management for investment centres, ROI can be calc… investment center performed... If possible, the residual income is the net operating income - ( operating.. Range of topics from accounting, residual income approach is useful in allocating resources among or... Make new investments if they add to RI of profitability: it relates net income – Monthly Debts calculate income! The complex nature of modern businesses, multi-faceted measures of performance, roi and residual income formula has advantages... Measure of an investment by the cost of equity ) income also ties in with net present value, the... Some ongoing effort, too ( to various degrees ) can be difficult and turnover \\$ \\$! ’ s left over — the residual income is the residual income encourage to... Working capital, stockholders ' equity, or initial cash outlay alternatives the... Formula is: ROI ( ROA ) for ABC, Inc. is below the industry average you! Definition for residual income figures generally improve as assets get older identifying controllable ( traceable profits... Roi the basic objective is to maximize the rate of return percentage, they some! Depreciation, amortization, interest charges and taxes from the business is Vt = Bt PV. A serious drawback this if you continue book value of assets needs to be calculated tax! Used in comparison to the return on investment ( ROI ) as a measure used part. Same principle … What is the adjustment of the operating assets of the operating assets balances by.. Ri is favoured for reasons of goal congruence and managerial effort RI is favoured for of! Serious drawback has been done, and you ’ ve got What ’ s stock value: it net!\nPuppy Essentials Reddit, Dental Surgery Course, Noah's Ark Book Pdf, Saxophone Intro Pop Song, Elios My Deepest Secret Last Name, 5 Tier Greenstalk Vertical Planter With Mover, Atlantic Technology Company, Ultra Modern Gas Fires, Canon Camera Rebates 2020, Impact Investing Companies, Used Volvos Under \\$5,000," ]

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[ "", null, "Example Worksheet - Maple Help\n\nHome : Support : Online Help : Programming : Domains : Example Worksheet\n\nThe Domains Package\n\nThis example worksheet demonstrates commands from the Domains package. The Domains package can be used to create domains of computation and then develop code for complicated algorithms.\n\n > $\\mathrm{restart}$\n > $\\mathrm{with}\\left(\\mathrm{Domains}\\right):$\n ---------------------- Domains version 1.0 --------------------- Initially defined domains are Z and Q the integers and rationals Abbreviations, e.g. DUP for DenseUnivariatePolynomial, also made", null, "Basic Examples Using Domains Z and Q\n\nThe domains Z (the integers) and Q (the rationals) have been defined. Let's do some operations.\n\n > $Z\\left[\\mathrm{Gcd}\\right]\\left(8,12\\right)$\n ${4}$ (1.1)\n > $Q\\left[\\mathrm{+}\\right]\\left(\\frac{1}{2},\\frac{1}{3},\\frac{1}{4}\\right)$\n $\\frac{{13}}{{12}}$ (1.2)\n\nWhat are the objects Z and Q? Z and Q are Maple tables\n\n > $\\mathrm{type}\\left(Z,\\mathrm{table}\\right)$\n ${\\mathrm{true}}$ (1.3)\n\nThe table contains operations (Maple procedures) for computing in Z and Q. What operations are available?\n\n > $\\mathrm{show}\\left(Z,\\mathrm{operations}\\right)$\n     Signatures for constructor Z     note: operations prefixed by  --  are not available      * : (Z,Z*) -> Z      * : (Integers,Z) -> Z      + : (Z,Z*) -> Z      - : Z -> Z      - : (Z,Z) -> Z      0 : Z      1 : Z      < : (Z,Z) -> Boolean      <= : (Z,Z) -> Boolean      <> : (Z,Z) -> Boolean      = : (Z,Z) -> Boolean      > : (Z,Z) -> Boolean      >= : (Z,Z) -> Boolean      Abs : Z -> Z      Characteristic : Integers      Coerce : Integers -> Z      Div : (Z,Z) -> Union(Z,FAIL)      EuclideanNorm : Z -> Integers      Factor : Z -> [Z,[[Z,Integers]*]]      Gcd : Z* -> Z      Gcdex : (Z,Z,Name) -> Z      Gcdex : (Z,Z,Name,Name) -> Z      Input : Expression -> Union(Z,FAIL)      Inv : Z -> Union(Z,FAIL)      Lcm : Z* -> Z      Max : (Z,Z*) -> Z      Min : (Z,Z*) -> Z      Modp : (Z,Z) -> Z      Mods : (Z,Z) -> Z      ModularHomomorphism : () -> (Z -> Z,Z)      Normal : Z -> Z      Output : Z -> Expression      Powmod : (Z,Integers,Z) -> Z      Prime : Z -> Boolean      Quo : (Z,Z) -> Z      Quo : (Z,Z,Name) -> Z      Random : () -> Z      RelativelyPrime : (Z,Z) -> Boolean      Rem : (Z,Z) -> Z      Rem : (Z,Z,Name) -> Z      Sign : Z -> UNION(1,-1,0)      SmallerEuclideanNorm : (Z,Z) -> Boolean      Sqrfree : Z -> [Z,[[Z,Integers]*]]      Type : Expression -> Boolean      Unit : Z -> Z      UnitNormal : Z -> [Z,Z,Z]      Zero : Z -> Boolean      ^ : (Z,Integers) -> Z" ]

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[ "# Dynamic Regret of Convex and Smooth Functions\n\n7 Jul 2020Peng ZhaoYu-Jie ZhangLijun ZhangZhi-Hua Zhou\n\nWe investigate online convex optimization in non-stationary environments and choose the dynamic regret as the performance measure, defined as the difference between cumulative loss incurred by the online algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path-length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamic regret is $\\mathcal{O}(\\sqrt{T(1+P_T)})$... (read more)\n\nPDF Abstract\n\nNo code implementations yet. Submit your code now" ]

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[ "# Questions tagged [lapack]\n\nLAPACK (Linear Algebra PACKage) is a commonly used library of subroutines for numerical linear algebra tasks, including solutions of linear sets of equations, linear least squares, eigenvalue problems, and singular value decomposition. LAPACK routines may be used with fortran, C and relatives and a variety of other languages.\n\n120 questions\nFilter by\nSorted by\nTagged with\n4k views\n\n### What is the relationship of BLAS, LAPACK, and other linear algebra libraries?\n\nI have been looking into C++ linear algebra libraries for a project I've been working on. Something that I still don't have any grasp on is the connection of BLAS and LAPACK to other linear algebra ...\n3k views\n\n### Why isn't my Matrix-Vector Multiplication Scaling?\n\nSorry for the long post but I wanted to include everything that I thought was relevant in the first go. What I want I am implementing a parallel version of Krylov Subspace Methods for Dense Matrices. ...\n8k views\n\n### Understanding how Numpy does SVD\n\nI have been using different methods to calculate both the rank of a matrix and the solution of a matrix system of equations. I came across the function linalg.svd. Comparing this to my own effort of ...\n458 views\n\n### Rapidly determining whether or not a dense matrix is of low rank\n\nIn a software project that I'm working on, certain computations are vastly easier for dense low-rank matrices. Some problem instances involve dense low-rank matrices, but they're given to me in full, ...\n1k views\n\n### Is there any benefit to compiling LAPACK from source versus installing the prebuilt package from Ubuntu?\n\nI know that ATLAS is able to optimize itself for the machine it is compiled on and thus maximum benefits are found by compiling from source. Is there any benefit to compiling LAPACK from source? It ...\n20k views\n\n### How to start using LAPACK in c++?\n\nI'm new to computational science and I already have learned basic methods for integration, interpolation, methods like RK4, Numerov etc on c++ but recently my professor asked me to learn how to use ...\n3k views\n\n### What are the fastest available implementations of BLAS/LAPACK or other linear algebra routines on GPU systems?\n\nnVidia, for example, has CUBLAS, which promises 7-14x speedup. Naively, this is nowhere near the theoretical throughput of any of nVidia's GPU cards. What are the challenges in speeding up linear ...\n3k views\n\n### What is the corresponding LAPACK function behind Matlab [Q,R,E]=qr(A)?\n\nI currently trying to cheaply compute a good rank estimate for a matrix $A$. Therefore I compute a columnt pivoting QR decompostion using [Q,R,E]=qr(A) in ...\n7k views\n\n### solve $xA=b$ for $x$ using LAPACK and BLAS\n\nI am porting an existing code from MATLAB to C++ and have a linear system to solve $xA=b$ (rather than the more typical form $Ax=b$) The matrix $A$ is dense, and of general form, but is no larger ...\n3k views\n\n### Optimized open source BLAS / LAPACK package\n\nI was wondering what is a more optimized open source BLAS/LAPACK package with respect to modern multi-core processors (Haswell and beyond). Is there any distribution that can attain performance close ...\n2k views\n\n### Matrix exponential of a skew-Hermitian matrix with fortran 95 and LAPACK\n\nI'm just getting tucked into fortran 95 for some quantum mechanics simulations. Honestly, I've been spoiled by Octave so I've taken matrix exponentiation for granted. Given a (small, $n\\leq 36$) skew-...\n758 views\n\n### What is the reason that LAPACK uses $\\tau$ in QR decomposition (instead of normalizing the reflection vector)?\n\nLAPACK's QR routine stores Q as Householder reflectors. It scales the reflection vector $v$ with $1/v_1$, so the first element of the result becomes $1$, so it doesn't have to be stored. And it stores ...\n232 views\n\n### Benchmark problems for eigenvalue reordering algorithms sought\n\nEvery real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...\n3k views\n\n### Matrix exponential of a real asymmetric matrix with Fortran 95 and LAPACK\n\nI recently asked a question along the same lines for skew-Hermitian matrices. Inspired by the success of that question, and after banging my head against a wall for a couple of hours, I'm looking at ...\n878 views\n\n### Does PETSc ever make use of LAPACK libraries for sparse matrix math?\n\nDoes compiling PETSc with an external BLAS/LAPACK library significantly affect performance on sparse matrices, or does it only use those libraries for dense matrix math?\n5k views\n\n### Solving a sparse and highly ill-conditioned system\n\nI intend to solve Ax = b where A is complex, sparse, unsymmetric and highly ill-conditioned (condition number ~ 1E+20) square or rectangular matrix. I have been able to solve the system with ZGELSS in ...\n647 views\n\n### Matrix Balancing Algorithm\n\nI have been writing a control system toolbox from scratch and purely in Python3 (shameless plug : harold ). From my past research, I have always complaints about ...\n4k views\n\n### How does LAPACK solve tridiagonal systems and why?\n\nIn my project I have to solve a couple of tridiagonal matrices at every time step, so it is crucial to have a good solver for those. I did my own implementation, just the classical way to do it ...\n155 views\n\n3k views\n\n### Fast vector - \"diagonal\" matrix multiplication\n\nLet $\\mathbf{1}\\in\\mathbb{R}^d$ be a vector with all elements equal to $1$. Define: \\mathbf{D} = \\mathrm{diag}(\\mathbf{1}^\\top,\\mathbf{1}^\\top,\\ldots,\\mathbf{1}^\\top) = \\begin{bmatrix} 1 \\cdots 1 &...\n10k views\n\n### Fast c++ library to solve very big sparse systems\n\nI am working on a project with electrical circuits, where I am trying to compute the voltages at all the nodes of an electrical circuit. I know that the electrical circuit is a perfect grid, so each ...\n4k views\n\n### BLAS, LAPACK or ATLAS for Matrix Multiplication in C\n\nI am trying to find the most optimized way to perform Matrix Multiplication of very large sizes in C language and under Windows 7 or Ubuntu 14.04. And searching led me to BLAS, LAPACK and ATLAS. ..." ]

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[ "", null, "# Why Work And Heat Are Not Properties?\n\n## What are the 3 types of heat?\n\nThe three types of heat transfer Heat is transfered via solid material (conduction), liquids and gases (convection), and electromagnetical waves (radiation).\n\nHeat is usually transfered in a combination of these three types and seldomly occurs on its own..\n\n## Is pressure a form of energy?\n\nPressure energy is the ‘can do work’ created in a closed system with elastic particles being under pressure. So the pressure energy is the energy of the particles elasticity. Pressure is defined as force per unit area.\n\n## Why work is not a property?\n\nWork is not a property of a system. Work is a process done by or on a system, but a system contains no work. This distinction between the forms of energy that are properties of a system and the forms of energy that are transferred to and from a system is important to the understanding of energy transfer systems.\n\n## Is work dependent on path?\n\nThe work a conservative force does on an object is path-independent; the actual path taken by the object makes no difference. … Conservative forces are easier to work with in physics because they don’t “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy.\n\n## Is heat a state variable?\n\nIn thermodynamics, a state variable is an independent variable of a state function like internal energy, enthalpy, and entropy. Examples include temperature, pressure, and volume. Heat and work are not state functions, but process functions.\n\n## Is Gibbs free energy a path function?\n\nGibbs free energy (G) is a state function since it depends on enthalpy (H), absolute temperature (T) and entropy (S), all of which are state…\n\n## Why heat and work are not state functions?\n\nTemperature is a state function. … Heat and work are not state functions. Work can’t be a state function because it is proportional to the distance an object is moved, which depends on the path used to go from the initial to the final state.\n\n## Why are heat and work path functions?\n\nHeat and work are path functions because they depend on the actual path traversed to move from initial to final state of the system. … Hence the work and heat involved in the individual processes are different.\n\n## Is heat a property?\n\nHeat energy is the internal energy of a substance. … It is impossible to accurately measure all the kinetic energy of the moving particles in a substance. The more particles you have, the more heat energy you have, thus heat energy is an extensive property.\n\n## Why is Q used for heat?\n\nClapeyron,a french engineer first used the symbol “Q” to describe the thermal energy. As the thermodynamics was in it’s premature form,he used the symbol ‘Q’ to describe the quantity of heat. Later on Horstmann used ‘Q’ to describe the amount of thermal energy required to decompose a mole of compound.\n\n## Is heat a point function?\n\nWork (W), heat (Q) are path functions. Point Function: They depend on the state only, and not on how a system reaches that state. All properties are point functions.\n\n## Is work a process function?\n\nIn thermodynamics, a quantity that is well defined so as to describe the path of a process through the equilibrium state space of a thermodynamic system is termed a process function, or, alternatively, a process quantity, or a path function. … Examples of path functions include work, heat and arc length.\n\n## Is entropy a path function?\n\nEntropy is a Point function which doesn’t depend upon path history, but Entropy generate due to irreversibility and always have a positive value Path function.\n\n## Is electric force a conservative force?\n\nForce is conservative if work done along a path that starts and ends at the same point is 0. We have shown that the work done by the electric force along a path starting and ending at the same point is 0. Hence, the electric force is conservative.\n\n## What is the difference between work and heat?\n\nHeat and work are two different ways of transferring energy from one system to another. Heat is the transfer of thermal energy between systems, while work is the transfer of mechanical energy between two systems. …\n\n## Which quantities are state functions?\n\nThe thermodynamic state of a system refers to the temperature, pressure and quantity of substance present. State functions only depend on these parameters and not on how they were reached. Examples of state functions include density, internal energy, enthalpy, entropy.\n\n## Is free energy a state function?\n\nThe Gibbs free energy of a system at any moment in time is defined as the enthalpy of the system minus the product of the temperature times the entropy of the system. The Gibbs free energy of the system is a state function because it is defined in terms of thermodynamic properties that are state functions.\n\n## Is heat a physical or chemical property?\n\nA physical property is a characteristic of matter that is not associated with a change in its chemical composition. Familiar examples of physical properties include density, color, hardness, melting and boiling points, and electrical conductivity." ]

[ null, "https://mc.yandex.ru/watch/68554720", null ]

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[ "# Nonlinear analysis of braking delay dynamics for the progressive gears in variable operating conditions\n\n1The State School of Higher Education, The Institute of Technical Sciences, Chełm, Poland\n\n2Lublin University of Technology, Institute of Technological Systems of Information, Lublin, Poland\n\n1Corresponding author\n\nJournal of Vibroengineering, Vol. 18, Issue 7, 2016, p. 4401-4408. https://doi.org/10.21595/jve.2016.17000\nReceived 21 March 2016; received in revised form 18 August 2016; accepted 22 August 2016; published 15 November 2016\n\nCopyright © 2016 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.\nViews 44\nAbstract.\n\nThe article presents the impact of variable operating conditions on the value of braking process delay for the progressive gears of PP16 type and the newly constructed CHP2000 gears. Tests were conducted in the regular working conditions with the application of mineral oil as lubricating agent. Values of delays were recorded for the loading of 400 and 1000 kg. This research study presents the recurrence plot analysis to investigate variability of received test runs. The results may be useful for testing new solutions brakes in varying conditions of service and may set new directions of research in the discussed topics.\n\nKeywords: safety gears, nonlinear dynamics, recurrence analysis, elevators.\n\n#### 1. Introduction\n\nThe brake systems belong to the most important systems of all technical constructions. Friction gears are equipped with two types of brakes (gears): instantaneous and progressive ones. Many authors [1-7] touch the aspects related to the friction gears construction and operation. In most cases, the papers described in the literature refer to instantaneous brake systems. With respect to the characteristics of their operation they require to apply the safety coefficients of 3rd order. The topic of the progressive gears construction and operation is brought up by the authors in the papers [8, 9]. The authors used the numerical modeling and analyzed the impact of a gear construction on the length of braking distance. The wavelet analysis to investigate the pressures received in the engine chamber is described by the authors in publication . Nevertheless, the world literature providing descriptions, analyses and mathematical models of progressive gears represents deficiency of information. Thus, it is justified to perform research studies within the impact of variable loading and operating conditions on the progressive gears braking process.\n\n#### 2. Experimental tests results\n\nBelow presented research study illustrates the possibility to apply the recurrence analysis to evaluate the delay value for the progressive gears. It refers to the gears subject to variable loadings and operating conditions. Free fall method, in dry conditions (without lubrication) and with RENOLIN B20 oil applied, was used in the tests. A test bench, the scheme of which is presented in Fig. 1 and 2, was used to perform the tests.\n\nFree fall of the friction elevator was simulated during the tests. In order to that the carrying ties were broken off. In the frame of the system No. 6 the bodies No. 1 were installed in turn together with PP16 and CHP 2000 gears respectively. The guides No. 7 were mounted in the test bench construction. The frame loading the gears was moving along the guides. The gears and the frame were coupled with the system to control the free fall speed. The system had a task to initiate the braking process once the nominal value of speed was exceeded by 0.3 m/s. The measurement of braking parameters was recorded by an accelerometer of MMA7341LC type. It was manufactured by Pololu 5 company. This device was coupled with PC class computer. Apart from „${a}_{op}$” delay parameter also the “$s$” free fall distance was recorded. The free fall distance was recorded with an optical sensor of CFR-22 type. The sensor was manufactured by Sensor company. The sensor possessed a graduation that was scaled every 10 mm. It was placed on the slat No. 4. Once the free fall was initiated, a lever of the gear No. 2 hit a clamp of the line No. 3. In consequence the braking process was started. The height of free fall, at which the frame system should have been placed, was defined via an empirical dependence. That was the empirical dependence between potential and kinetic energy. The calculated value of the free fall height was 250 mm – This dependence is presented in Eq. (1):\n\n(1)\n$h=\\frac{{v}_{1}^{2}}{2g}+0.1+0.03,$\n\nwhere: $h$ – the free fall height [m], ${v}_{1}$ – the speed of the gears release, 1.25 [m/s] value accepted for tests, $g$ –the acceleration of gravity – 9.81 [m/s2], 0.1 – the delay coefficient of the gears activation [-], 0.03 – the excessive clearance coefficient in the brake system according to [-].\n\nFig. 1. The test bench scheme", null, "Fig. 2. The test bench scheme", null, "All measuring tracks were coupled with LABVIEW 9215 measuring card. The card was connected with PC class computer. Data was recorded on the computer.\n\nBraking distance dependencies for tested gears received via the experiments are presented in Fig. 3.\n\nWhile analyzing the characteristics obtained in the experiment the influence of operating conditions on the value of braking process delay can be noticed. The maximum values reaching 130 m/s2 with lubrication applied were received for the gears of CHP 2000 type and with 1000 kg loading. For the same gears type but without lubrication the values of delay reached 110 m/s2. The values of delay 90 and 105 m/s2 werereached respectively for the loading of 400 kg. During tests the gears of PP16 type demonstrated more fixed characteristic. It is reflected on the plots. The maximum values reaching 130 m/s2 with lubrication and 1000 kg loading were obtained for the gears of CHP 2000 type. For the same gears type but without lubrication the values of delay reached 128 m/s2. The values of delay 125 and 120 m/s2 werereached respectively for the loading of 400 kg.\n\n#### 3. The analysis of recurrence\n\nOriginally, the analysis of recurrence was proposed by Eckmann in 1987 . It was presented as a new graphic tool to identify time dependencies in dynamic systems. It looks for the presence of the same dynamical system conditions in successive time periods. It means that, if two points on trajectory in phase space are close enough to each other they are marked as “the recurrence points” [13, 14]. If dynamics is periodic the number of recurrent points increases. If it is non-periodic or stochastic dynamics, then the number of recurrent states decreases or disappears. This method has been widely used in many fields, especially in physiology, technical science or economics where periodicity can be observed and identified in short time scales or/and with presence of noise [13-20]. In nonlinear time series analysis, all phase space coordinates can be used if available or the phase space can be reconstructed from one-dimensional time series with the use of embedding theorem by Takens . Let us have $y\\left(t\\right)$ time series of $n$ length. On its basis the $m$ dimensional reconstruction of phase space can be done:\n\n(2)\n$y=\\left[y\\left(t\\right),y\\left(t-\\tau \\right),\\dots ,y\\left(t-\\left(m-1\\right)\\tau \\right)\\right],$\n\nwhere $\\tau$ means time delay, and $m$ stands for the phase space dimension. Generally, any time delay can be selected. Nevertheless, due to too low value the respective constituents are strongly correlated with each other. On the other hand, too high value causes weaker correlation. Thus, it can lead to the loss of information about the system in short time intervals.\n\nFig. 3. The results of measurements for acceleration of the gears braking process depending on the operating conditions and loading: a) CHP 2000 progressive gear with 1000 kg loading – operated with lubrication, b) CHP 2000 progressive gear with 1000 kg loading – operated without lubrication c) CHP 2000 progressive gear with 400 kg loading – operated with lubrication, d) CHP 2000 progressive gear with 400 kg loading – operated without lubrication, e) PP16 progressive gear with 1000 kg loading – operated with lubrication, f) PP16 progressive gear with 1000 kg loading – operated without lubrication, g) PP16 progressive gear with 400 kg loading – operated with lubrication, h) PP16 progressive gear with 400 kg loading - operated without lubrication", null, "a)", null, "b)", null, "c)", null, "d)", null, "e)", null, "f)", null, "g)", null, "h)\n\nMutual Information method ($MI$) is one of the methods to select optimal value of $\\tau$. The method takes into account nonlinear dependencies:\n\n(3)\n$MI=-{\\sum }_{ij}{p}_{ij}\\left(\\tau \\right)\\mathrm{l}\\mathrm{o}\\mathrm{g}\\frac{{p}_{ij}\\left(\\tau \\right)}{{p}_{i}{p}_{j}},$\n\nwhere ${p}_{j}$ is the probability of finding the value from time series in $i$th interval and ${p}_{ij}\\left(\\tau \\right)$ is the probability after the time $\\tau$ of the value from time series in $j$th interval (the number of intervals is 16). The value $\\tau$ is selected as the first minimum of MI function. In this case 4. Then, the identified value of delay is used to determine a dimension of reconstructed phase space with False Nearest Neighbors Fraction (FNNF) [23, 24] method. It is an iterative comparison of the distance relation between neighboring points when dimension of reconstructed space increases:\n\n(4)\n$\\mathrm{F}\\mathrm{N}\\mathrm{N}\\mathrm{F}=\\frac{{‖{y}_{j}-{y}_{i}‖}_{m+1}}{{‖{y}_{j}-{y}_{i}‖}_{m}},$\n\nwhere ${y}_{i}$ means a given point, and ${y}_{j}$ its closest neighbour. If the relation received with the Eq. (4) is higher than a given value (in our case it is the value 2) then the number of false neighbors goes up. One of the means to select the $m$ embedding dimension is zero of FNNF function, in our case $m$ equals 5. It should be noticed that the embedded space dimension can be higher. In the space with higher dimension, the false neighbors also disappear. Periodicity of analyzed dynamics can be tested with the use of the recurrence analysis. This analysis, in opposition to the majority of statistical methods, can be used for short and non-stationary time series. If the distance between two points on the trajectory is small enough they are marked as neighboring ones. It can be expressed with a distance matrix, ${R}_{ij}^{\\epsilon }$ elements of which are calculated with the following formula:\n\n(5)\n${R}_{ij}^{\\epsilon }=\\mathrm{\\Theta }\\left(\\epsilon -‖{y}_{i}-{y}_{j}‖\\right),$\n\nwhere $\\epsilon$ means the highest value of distance, and $\\mathrm{\\Theta }$ means Heaviside function. The number of recurrence points depends both on $\\epsilon$ value and dynamics of the system. Usually, $\\epsilon$ is selected in such a way so as the number of recurrence points equals to several percent. Fig. 4 presents the comparison between (RP) recurrence plots including 10 % of recurrence points received in the time series (Fig. 3).\n\nThe recurrence plot can consist of diagonal lines, isolated points and vertical lines. Presence of different length diagonal lines gives evidence of periodic dynamics. Additional isolated points indicate non-periodic dynamics or noise. The vertical lines belong to the features distinguished for intermittent states. Comparing the recurrence plots presented in Fig. 4 the similarity between several examples but subject to different operating conditions (it is also visible in the chart in Fig. 3) can be noticed. White areas represent the fact that tested time series are non-stationary. The concentration of recurrence points in the lower left corner in the plots in Fig. 4(a), (b) and (e), (f) indicates more regular dynamics of braking process at the initial stage. Analogously, the clusters of recurrence points located in the upper right corner give evidence of higher regularity of the phenomenon at the final stage. They are clearly visible in the plot in Fig. 4(c), (d) but also (g), (h). It should be added that analyzed data comes from the test bench and it is influenced by noise.\n\nVisual analysis of recurrence plots can be helpful to identify the dynamics type. Nevertheless, more precise comparison can be received when the statistics of isolated points, diagonal and vertical lines are used.\n\nFig. 4. The recurrence plots prepared for all analyzed accelerations: a) CHP 2000 progressive gear with 1000 kg loading – operated with lubrication b) CHP 2000 progressive gear with 1000 kg loading – operated without lubrication c) CHP 2000 progressive gear with 400 kg loading – operated with lubrication, d) CHP 2000 progressive gear with 400 kg loading – operated without lubrication, e) PP16 progressive gear with 1000 kg loading – operated with lubrication f) PP16 progressive gear with 1000 kg loading – operated without lubrication g) PP16 progressive gear with 400 kg loading – operated with lubrication h) PP16 progressive gear with 400 kg loading - operated without lubrication", null, "a)", null, "b)", null, "c)", null, "d)", null, "e)", null, "f)", null, "g)", null, "h)\n\n– RR (Recurrence Rate) – the ratio between the number of recurrence points and the total number of points:\n\n(6)\n$RR=\\frac{1}{{N}^{2}}{\\sum }_{ij\\ne i}^{N}{R}_{ij}^{\\epsilon }.$\n\n– DET (Determinism) – the ratio between the number of recurrence points creating diagonal lines and the total number of points:\n\n(7)\n$DET=\\frac{{\\sum }_{l=lmin}^{N}{lP}^{\\epsilon }\\left(l\\right)}{{\\sum }_{l=1}^{N}{lP}^{\\epsilon }\\left(l\\right)}.$\n\n– LAM (Laminarity) – the ratio between the number of recurrence points creating vertical lines and the total number of points:\n\n(8)\n$LAM=\\frac{{\\sum }_{v=vmin}^{N}{vP}^{\\epsilon }\\left(v\\right)}{{\\sum }_{v=1}^{N}{vP}^{\\epsilon }\\left(v\\right)}.$\n\n– LMAX – the longest diagonal line.\n\n– VMAX – the longest vertical line.\n\n$L$ – the average length of diagonal lines:\n\n(9)\n$L=\\frac{{\\sum }_{l=lmin}^{N}{lP}^{\\epsilon }\\left(l\\right)}{{\\sum }_{l=1}^{N}{P}^{\\epsilon }\\left(l\\right)}.$\n\n– TT – the average length of vertical lines:\n\n(10)\n$TT=\\frac{{\\sum }_{v=vmin}^{N}{vP}^{\\epsilon }\\left(v\\right)}{{\\sum }_{v=1}^{N}{P}^{\\epsilon }\\left(v\\right)},$\n\nwhere $p\\left(z\\right)$ means probability of the diagonal lines arrangement for $p\\left(l\\right)$ or vertical lines for $p\\left(v\\right)$. The values $lmin$ and $vmin$ mean a minimum length of diagonal or vertical lines (the following was selected: 2).\n\nTable 1. presents the recurrence indicators for respective cases.\n\nTable 1. The analysis of recurrence quantification\n\n Case $\\epsilon$ RR DET LAM LMAX VMAX L TT CHP2000-1000 OL 1.2 10 % 0.55 0.68 88 48 5 4 CHP2000-1000 S 1.3 10 % 0.46 0.55 74 18 4 2 CHP2000-400 OL 0.9 10 % 0.74 0.55 131 48 6 2 CHP2000-400 S 1.1 10 % 0.84 0.77 165 109 13 14 PP16-1000 OL 1.2 10 % 0.65 0.69 72 31 5 3 PP16-1000 S 1.5 10 % 0.56 0.64 75 83 6 4 PP16-400 OL 1.5 10 % 0.58 0.61 83 44 5 3 PP16-400 S 1.3 10 % 0.61 0.44 47 23 4 2\n\nComparing the values of recurrence indicators, quite high values of DET can be noticed. They reflect predictability of the system and give evidence of not big impact of measuring noise. It is worth mentioning that the highest level of determinism was reached for the experiment conducted with the use of CHP2000 type gears. Moreover, for lower loading (400 kg) determinism is lower with lubrication in comparison to the runs without lubrication. On the other hand, for the greater loading (1000 kg) the runs of acceleration are more predictable while operated with lubrication. Presence of vertical lines can be also noticed. Relatively high values of laminarity (LAM) indicate the presence of vertical lines. At the same time, average lengths of diagonal and vertical lines are short. And, quite high LMAX and VMAX values indicate the determinism of analyzed process. Furthermore, they confirm the highest regularity of braking process for CHP2000-400 experiment.\n\n#### 4. Conclusions\n\nThis paper presents nonlinear methods used to analyze dynamics of the elevator braking process. Two types of the gears (CHP2000 and PP16) were investigated. Different loadings (1000 kg and 400 kg) were applied during tests. Moreover, they were working in various operating conditions (with and without lubrication). Time series of accelerations vibrations (400 measuring points) were used to reconstruct the space where the process periodicity was compared. It can be assumed that higher periodicity of braking process increases durability of the brake elements and provides better comfort for the passengers. Due to the recurrence analysis it was discovered that at the initial stage of braking process (the first 200 ms) the higher periodicity exists for the lower loading (400 kg). On the other hand, at the final stage of braking process (the last 200 ms) the higher regularity was observed for the higher loading (1000 kg). While analyzing the values of recurrence indicators, similar periodicity of braking process for PP16 gears was observed regardless of operating conditions. Yet, dynamics of the braking process for CHP2000 gears changes considerably together with changing operating conditions. Summing up, it seems that higher loading increases periodicity of the process. To confirm the conclusions, more systematic research studies with more experiment repetitions should be performed.\n\n1. Feng L., Bao Y., Zhou X., Wang Y. High speed elevator car frame’s finite elements analysis. Advanced Materials Research, Vol. 510, 2012, p. 298-303. [Publisher]\n2. Filas J., Mudro M. The dynamic equation of motion of driving mechanism of a freight elevator. Procedia Engineering, Vol. 48, 2012, p. 149-152. [Publisher]\n3. Jong de J. Understanding the natural behavior of elevator safety gears and their triggering. The International Congress on Vertical Transportation Technologies, Istambul, 2004. [Search CrossRef]\n4. Kayaoglu E., Salman O., Candas A. Study on stress and deformation of an elevator safety gear brake block using experimental and FEA methods. Advanced Materials Research, Vols. 308-310, 2011, p. 1513-1518. [Publisher]\n5. Onur Y. A., Imrak C. E. Reliability analysis of elevator car frame using analytical and finite element methods. Building Services Engineering Research and Technology, Vol. 33, 2012, p. 293-305. [Publisher]\n6. Taplak H., Erkaya S., Yildirim S., Uzmay I. The use of neural network predictors for analyzing the elevator vibrations. Mechanical Engineering, Vol. 39, 2014, p. 1157-1170. [Publisher]\n7. Zhu W. D., Ren H. A linear model of stationary elevator traveling and compensation cables. Journal of Sound and Vibration, Vol. 332, 2012, p. 3086-3097. [Publisher]\n8. Lonkwic P., Różyło P., Dębski H. Numerical and experimental analysis of the progressive gear body with the use of finite-element method. Maintenance and Reliability, Vol. 17, 2015, p. 542-548. [Publisher]\n9. Lonkwic P. Influence of friction drive lift gears construction on the length of braking distance. Chinese Journal of Mechanical Engineering, Vol. 28, 2015, p. 363-368. [Publisher]\n10. Longwic R., Litak G., Asok K. Sen. Recurrence plots for diesel engine variability tests. Verlag der Zeitschrift fur Naturforschung, Vol. 64, 2009, p. 96-102. [Publisher]\n11. Polish Standard PN EN 81.1+A3, Safety Regulations Concerning the Structure and Installation of Lifts, Part I. Electric Lifts. [Search CrossRef]\n12. Eckmann J. P., Kamphorst S. O., Ruelle D. Recurrence plots of dynamical systems. Europhysics Letters, Vol. 5, 1987, p. 973-977. [Publisher]\n13. Webber Jr., C. L., Zbilut, J. P. Dynamical assessment of physiological systems and states using recurrence plot strategies. Journal of Applied Physiology, Vol. 76, 1994, p. 965-973. [Search CrossRef]\n14. Zbilut J. P., Webber Jr. C. L. Embeddings and delays as derived from quantification of recurrence plots. Physics Letters A, Vol. 171, 1992, p. 199-203. [Publisher]\n15. Zbilut J. P., Giuliani A., Webber Jr. C. L. Recurrence quantification analysis and principal components in the detection of short complex signals. Physics Letters A, Vol. 237, 1998, p. 131-135. [Publisher]\n16. Marwan N., Romano M. C., Thiel M., Kurths J. Recurrence plots for the analysis of complex systems. Physics Reports, Vol. 438, 2007, p. 237-329. [Publisher]\n17. Casdagli M. C. Recurrence plots revisited. Physica D, Vol. 108, 1997, p. 12-44. [Publisher]\n18. Syta A., Jonak J., Jedlinski L., Litak G. Failure diagnosis of a gear box by recurrences. Journal of Vibration and Acoustics – Transactions of the ASME, Vol. 134, 2012, p. 041006. [Publisher]\n19. Litak G., Gajewski J., Syta A., Jonak J. Quantitative estimation of the tool wear effects in a ripping head by recurrence plots. Journal of Theoretical and Applied Mechanics, Vol. 46, 2008, p. 521-530. [Search CrossRef]\n20. Litak G., Syta A., Gajewski J., Jonak J. Nonlinear analysis of the ripping head power time series. Journal of Vibroengineering, Vol. 13, 2011, p. 39-51. [Search CrossRef]\n21. Takens F. Detecting strange attractors in turbulence. Lecture Notes in Mathematics, Vol. 898, 1981, p. 366-381. [Publisher]\n22. Fraser A. M., Swinney H. L. Independent coordinates for strange attractors from mutual information Physical Review A, Vol. 33, 1986, p. 1134-1140. [Publisher]\n23. Hegger R., Kantz H., Schreiber T. Practical implementation of non-linear time series methods: the TISEAN package. Chaos, Vol. 9, 1999, p. 413-435. [Publisher]\n24. Kantz H., Schreiber T. Non-linear Time Series Analysis. Cambridge University Press, Cambridge, 1997. [Search CrossRef]\n\n#### Cited By\n\n Transport and Telecommunication Journal Krzysztof Przystupa, Zhang Qin, Serhii Zabolotnii, Volodymyr Pohrebennyk, Sergii Mogilei, Chen Zhongju, Leszek Gil 2021 Energies Peter Girovský, Jaroslava Žilková, Ján Kaňuch 2020 Metals Poul Lonkwic, Tomasz Krakowski, Hubert Ruta 2020 Meccanica Piotr Wolszczak, Pawel Lonkwic, Americo Cunha, Grzegorz Litak, Szymon Molski 2019 Energy Efficiency Tomasz Krakowski, Hubert Ruta 2019 Measurement Paweł Lonkwic, Krystian Łygas, Piotr Wolszczak, Szymon Molski, Grzegorz Litak 2017" ]

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[ "# Normal products and radicals in finite groups\n\nIf $G$ is a finite group with normal subgroups $M$ and $N$, then $MN$ is a subgroup, called the normal product of $M$ and $N$.\n\nIf $\\mathcal{F}$ is a set of finite groups closed under isomorphism and normal products, then there is a subgroup $O_\\mathcal{F}(G)$ called the $\\mathcal{F}$-radical which is the normal product of all normal subgroups of $G$ that are in $\\mathcal{F}$. If $\\mathcal{F}$ additionally is closed under normal subgroups, $O_\\mathcal{F}(G)$ has the property that a subgroup of $G$ is a subnormal subgroup from $\\mathcal{F}$ if and only if it is a subnormal subgroup of $O_\\mathcal{F}(G)$. This is standard material on Fitting classes, covered in say 6.3 of Kurzweil–Stellmacher's textbook.\n\nI'd like to conclude quite generally that $O_\\mathcal{F}(MN) = O_\\mathcal{F}(M) O_\\mathcal{F}(N)$. This is definitely true of direct products, and I'm having a hard time seeing any relevant differences to the normal product. However, I am also having trouble proving it.\n\nI get that $O_\\mathcal{F}(M) O_\\mathcal{F}(N) \\leq O_\\mathcal{F}(MN)$ in general, simply because $\\mathcal{F}$ is closed under normal products.\n\nHow do I show the reverse containment? If it is not true, is there some extra hypothesis on $\\mathcal{F}$ that does it (like subgroup or quotient closure)?\n\nThe particular case of concern is $\\mathcal{F}$ consisting of all $\\pi$-groups. I can brute-force it for $\\mathcal{F}$ consisting of all solvable $\\pi$-groups, and technically all groups my audience was considering were solvable, but I'd prefer a technique that worked in general, or some counterexamples to show what extra hypotheses are actually being used.\n\nFor instance, in my application $\\mathcal{F}$ is closed under quotients and all subgroups too (a subgroup closed (saturated by Bryce-Cossey) Fitting formation), but I doubt much if any of that extra hypothesis is needed.\n\nEdit: Assume $\\mathcal{F}$ is quotient closed for the positive answer. I don't current have a counterexample for the general $\\mathcal{F}$, but they are apparently plentiful and well known.\n\nApparently direct products don't work this way for general $\\mathcal{F}$ (for any normal Fitting class properly contain in the class of all solvable groups, for instance). Lockett figured out how to fix this in a fairly low impact way. For any Fitting class $\\mathcal{F}$, he associated $\\mathcal{F}^*$ with the property that $\\mathcal{F} \\subseteq \\mathcal{F}^* = \\mathcal{F}^{**} \\subseteq \\mathcal{F} \\mathcal{A}$ and that $O_{\\mathcal{F}^*}(G \\times H) = O_{\\mathcal{F}^*}(G) \\times O_{\\mathcal{F}^*}(H)$ and $O_{\\mathcal{F}^*}(G) = \\{ g \\in G : (g,h) \\in O_\\mathcal{F}(G\\times G) \\}$.\n\nTheorem 2.2d shows that if $\\mathcal{F}$ is closed under quotients or residual products (the other aspect of being a formation), then $\\mathcal{F}=\\mathcal{F}^*$, so all Fitting formations work the way I thought.\n\n... Still checking on normal products not directly addressed in the article ...\n\n• Lockett, F. Peter. “The Fitting class $\\mathfrak{F}^*$” Math. Z. 137 (1974), 131–136. MR364435 DOI:10.1007/BF01214854\n• There is some small possibility that this is not true, and the class I'm looking for are called Lockett classes, but that is a few hundred pages of reading ahead of me, so I'm very happy for any summaries. – Jack Schmidt Jun 8 '13 at 22:53\n• Normal Product? You should have answered my question here! – user1729 Aug 1 '13 at 19:01\n\nLet $\\mathcal{F}$ be the class of all $2$-groups. Consider $G = S_3 \\times C_2$. For our normal subgroups $M$ and $N$, let $M = S_3 \\times 1$ and $N = \\{(\\sigma, \\operatorname{sgn}(\\sigma)): \\sigma \\in S_3\\}$. Now $M \\cong N \\cong S_3$, so $O_{\\mathcal{F}}(M)O_{\\mathcal{F}}(N)$ is trivial. But $O_{\\mathcal{F}}(MN) = O_{\\mathcal{F}}(G) = 1 \\times C_2$.\nImitating the above example, I think the following should also work. Let $\\mathcal{F}$ be a class of finite groups closed under isomorphism and normal products, and such that $O_{\\mathcal{F}}(S_n)$ is trivial and $O_{\\mathcal{F}}(C_2) = C_2$ (ie. the cyclic group of order $2$ is a $\\mathcal{F}$-group). Like in the previous example, let $G = S_n \\times C_2$ and $M = S_n \\times 1$ and $N = \\{(\\sigma, \\operatorname{sgn}(\\sigma)): \\sigma \\in S_n\\}$. Then $M$ and $N$ are normal in $G$ and $M \\cong N \\cong S_n$ so $O_{\\mathcal{F}}(M)O_{\\mathcal{F}}(N)$ is trivial. Now $1 \\times C_2$ is a nontrivial normal $\\mathcal{F}$-subgroup of $G = MN$, so $O_{\\mathcal{F}}(MN)$ is nontrivial.\n• Thanks, $O_\\mathcal{F}$ in your first example is called $O_2$ and is of fundamental importance. I am disturbed to have not seen this before. Thanks! – Jack Schmidt Aug 1 '13 at 18:03\n• Same group works for $\\mathcal{F}$ the class of 3-nilpotent groups. Fixed in the 3-length 1 proof. – Jack Schmidt Aug 1 '13 at 18:22" ]

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[ "## Quick Conceptual question\n\nJennifer Torres 2L\nPosts: 62\nJoined: Tue Nov 14, 2017 3:01 am\n\n### Quick Conceptual question\n\nCould someone explain the difference between enthalpy and entropy? And what equations would be used in each case?\n\nVicky Lu 1L\nPosts: 60\nJoined: Fri Sep 28, 2018 12:18 am\n\n### Re: Quick Conceptual question\n\nEnthalpy (H) refers to the heat in the system while entropy (S) can be thought of as the disorder of the system. They are connected by delta G = delta H - T*delta S. For enthalpy, use the 3 methods: Standard Enthalpy of Formation, Bond Enthalpy, and Hess's Law. In Entropy, changes in volume would use delta s = nRln(Vf/Vi), changes in pressure would use delta s = nRln(Pi/Pf), and changes in temperature uses delta s = nCspln(Tf/Ti).\n\nSophie Roberts 1E\nPosts: 61\nJoined: Fri Sep 28, 2018 12:17 am\n\n### Re: Quick Conceptual question\n\nIn addition to what was said above, entropy can determine if a reaction is spontaneous (favorable) or not but Enthalpy cannot determine this.\n\nTotal entropy >>0 - Spontaneous\nTotal entropy <<0 - Non spontaneous\n\nPhilipp_V_Dis1K\nPosts: 32\nJoined: Fri Sep 28, 2018 12:20 am\n\n### Re: Quick Conceptual question\n\nEntropy is the amount of chaos in the system, while enthalpy is the heat released\n\nCynthia Aragon 1B\nPosts: 47\nJoined: Mon Apr 09, 2018 1:38 pm\n\n### Re: Quick Conceptual question\n\nEnthalpy refers to the measure of total heat content in a thermodynamic system under constant pressure and is denoted by the symbol H. Entropy refers to the measure of the level of disorder and is denoted by the symbol S. Entropy is calculated in terms of change Delta S = q/T where q is the heat content and T is the temperature.\n\nLedaKnowles2E\nPosts: 62\nJoined: Fri Sep 28, 2018 12:27 am\n\n### Re: Quick Conceptual question\n\nEnthalpy is the heat transferred in/out of the system, while entropy is the disorder of the system (how many possible positions, vibrations, etc).\n\nLaurenJuul_1B\nPosts: 65\nJoined: Fri Sep 28, 2018 12:17 am\n\n### Re: Quick Conceptual question\n\nenthalpy is the heat released from the system and entropy is the amount of disorder of the system\n\n904914909\nPosts: 60\nJoined: Fri Sep 28, 2018 12:26 am\n\n### Re: Quick Conceptual question\n\nEnthalpy is a measure of heat transfer while entropy is a measure of disorder in a system" ]

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[ "Wednesday, February 10, 2010\n\nOVERCOMING \"SCIENTIFIC\" SUPERSTITION\n\nOVERCOMING “SCIENTIFIC” SUPERSTITION:\n\n“It is easy to explain something to a layman. It is easier to explain the same thing to an expert. But even the most knowledgeable person cannot explain something to one who has limited half-baked knowledge.” ------------- (Hitopadesha).\n\n“To my mind there must be, at the bottom of it all, not an equation, but an utterly simple idea. And to me that idea, when we finally discover it, will be so compelling, so inevitable, that we will say to one another: ‘Oh, How wonderful! How could it have been otherwise.” -----------(John Wheeler).\n\n“All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken”. --------------- Einstein, 1954\n\nTwentieth century was a marvel in technological advancement. But except for the first quarter, the advancement of theoretical physics has nothing much to be written about. The principle of mass-energy equivalence, which is treated as the corner-stone principle of all nuclear interactions, binding energies of atoms and nucleons, etc., enters physics only as a corollary of the transformation equations between frames of references in relative motion. Quantum Mechanics (QM) cannot justify this equivalence principle on its own, even though it is the theory concerned about the energy exchanges and interactions of fundamental particles. Quantum Field Theory (QFT) is the extension of QM (dealing with particles) over to fields. In spite of the reported advancements in QFT, there is very little back up experimental proof to validate many of its postulates including Higgs mechanism, bare mass/charge, infinite charge etc. It seems almost impossible to think of QFT without thinking of particles which are accelerated and scattered in colliders. But interestingly, the particle interpretation has the best arguments against QFT. Till recently, the Big Bang hypothesis held the center stage in cosmology. Now Loop Quantum Cosmology (LQC) with its postulates of the “Big Bounce” is taking over. Yet there are two distinctly divergent streams of thought on this subject also. The confusion surrounding interpretation of quantum physics is further compounded by the modern proponents, who often search historical documents of discarded theories and come up with new meanings to back up their own theories. For example, the cosmological constant, first proposed and subsequently rejected as the greatest blunder of his life by Einstein; has made a come back in cosmology. Bohr’s complementarity principle, originally central to his vision of quantum particles, has been reduced to a corollary and is often identified with the frameworks in Consistent Histories.\n\nThere are a large number of different approaches or formulations to the foundations of Quantum Mechanics. There is the Heisenberg’s Matrix Formulation, Schrödinger’s Wave-function Formulation, Feynman’s Path Integral Formulation, Second Quantization Formulation, Wigner’s Phase Space Formulation, Density Matrix Formulation, Schwinger’s Variational Formulation, de Broglie-Bohm’s Pilot Wave Formulation, Hamilton-Jacobi Formulation etc. There are several Quantum Mechanical pictures based on placement of time-dependence. There is the Schrödinger Picture: time-dependent Wave-functions, the Heisenberg Picture: time-dependent operators and the Interaction Picture: time-dependence split. The different approaches are in fact, modifications of the theory. Each one introduces some prominent new theoretical aspect with new equations, which needs to be interpreted or explained. Thus, there are many different interpretations of Quantum Mechanics, which are very difficult to characterize. Prominent among them are; the Realistic Interpretation: wave-function describes reality, the Positivistic Interpretation: wave-function contains only the information about reality, the famous Copenhagen Interpretation: which is the orthodox Interpretation. Then there is Bohm’s Causal Interpretation, Everett’s Many World’s Interpretation, Mermin’s Ithaca Interpretation, etc. With so many contradictory views, quantum physics is not a coherent theory, but truly weird.\n\nGeneral relativity breaks down when gravity is very strong: for example when describing the big bang or the heart of a black hole. And the standard model has to be stretched to the breaking point to account for the masses of the universe’s fundamental particles. The two main theories; quantum theory and relativity, are also incompatible, having entirely different notions: such as for the concept of time. The incompatibility of quantum theory and relativity has made it difficult to unite the two in a single “Theory of everything”. There are almost infinite numbers of the “Theory of Everything” or the “Grand Unified Theory”. But none of them are free from contradictions. There is a vertical split between those pursuing the superstrings route and others, who follow the little Higgs route.\n\nString theory, which was developed with a view to harmonize General Relativity with Quantum theory, is said to be a high order theory where other models, such as supergravity and quantum gravity appear as approximations. Unlike super-gravity, string theory is said to be a consistent and well-defined theory of quantum gravity, and therefore calculating the value of the cosmological constant from it should, at least in principle, be possible. On the other hand, the number of vacuum states associated with it seems to be quite large, and none of these features three large spatial dimensions, broken super-symmetry, and a small cosmological constant. The features of string theory which are at least potentially testable - such as the existence of super-symmetry and cosmic strings - are not specific to string theory. In addition, the features that are specific to string theory - the existence of strings - either do not lead to precise predictions or lead to predictions that are impossible to test with current levels of technology.\n\nThere are many unexplained questions relating to the strings. For example, given the measurement problem of quantum mechanics, what happens when a string is measured? Does the uncertainty principle apply to the whole string? Or does it apply only to some section of the string being measured? Does string theory modify the uncertainty principle? If we measure its position, do we get only the average position of the string? If the position of a string is measured with arbitrarily high accuracy, what happens to the momentum of the string? Does the momentum become undefined as opposed to simply unknown? What about the location of an end-point? If the measurement returns an end-point, then which end-point? Does the measurement return the position of some point along the string? (The string is said to be a Two dimensional object extended in space. Hence its position cannot be described by a finite set of numbers and thus, cannot be described by a finite set of measurements.) How do the Bell’s inequalities apply to string theory? We must get answers to these questions first before we probe more and spend (waste!) more money in such research. These questions should not be put under the carpet as inconvenient or on the ground that some day we will find the answers. That someday has been a very long period indeed!\n\nThe energy “uncertainty” introduced in quantum theory combines with the mass-energy equivalence of special relativity to allow the creation of particle/anti-particle pairs by quantum fluctuations when the theories are merged. As a result there is no self-consistent theory which generalizes the simple, one-particle Schrödinger equation into a relativistic quantum wave equation. Quantum Electro-Dynamics began not with a single relativistic particle, but with a relativistic classical field theory, such as Maxwell’s theory of electromagnetism. This classical field theory was then “quantized” in the usual way and the resulting quantum field theory is claimed to be a combination of quantum mechanics and relativity. However, this theory is inherently a many-body theory with the quanta of the normal modes of the classical field having all the properties of physical particles. The resulting many-particle theory can be relatively easily handled if the particles are heavy on the energy scale of interest or if the underlying field theory is essentially linear. Such is the case for atomic physics where the electron-volt energy scale for atomic binding is about a million times smaller than the energy required to create an electron positron pair and where the Maxwell theory of the photon field is essentially linear.\n\nHowever, the situation is completely reversed for the theory of the quarks and gluons that compose the strongly interacting particles in the atomic nucleus. While the natural energy scale of these particles, the proton, r meson, etc. is on the order of hundreds of millions of electron volts, the quark masses are about one hundred times smaller. Likewise, the gluons are quanta of a Yang-Mills field which obeys highly non-linear field equations. As a result, strong interaction physics has no known analytical approach and numerical methods is said to be the only possibility for making predictions from first principles and developing a fundamental understanding of the theory. This theory of the strongly interacting particles is called quantum chromodynamics or QCD, where the non-linearities in the theory have dramatic physical effects. One coherent, non-linear effect of the gluons is to “confine” both the quarks and gluons so that none of these particles can be found directly as excitations of the vacuum. Likewise, a continuous “chiral symmetry”, normally exhibited by a theory of light quarks, is broken by the condensation of chirally oriented quark/anti-quark pairs in the vacuum. The resulting physics of QCD is thus entirely different from what one would expect from the underlying theory, with the interaction effects having a dominant influence.\n\nIt is known that the much celebrated Standard Model of Particle Physics is incomplete as it relies on certain arbitrarily determined constants as inputs - as “givens”. The new formulations of physics such as the Super String Theory and M-theory do allow mechanisms where these constants can arise from the underlying model. However, the problem with these theories is that they postulate the existence of extra dimensions that are said to be either “extra-large” or “compactified” down to the Planck length, where they have no impact on the visible world we live in. In other words, we are told to blindly believe that extra dimensions must exist, but on a scale that we cannot observe. The existence of these extra dimensions has not been proved. However, they are postulated to be not fixed in size. Thus, the ratio between the compactified dimensions and our normal four space-time dimensions could cause some of the fundamental constants to change! If this could happen then it might lead to physics that are in contradiction to the universe we observe.\n\nThe concept of “absolute simultaneity” – an off-shoot of quantum entanglement and non-locality, poses the gravest challenge to Special Relativity. But here also, a different interpretation is possible for the double-slit experiment, Bell’s inequality, entanglement and decoherence, which can rub them off of their mystic character. The Ives - Stilwell experiment conducted by Herbert E. Ives and G. R. Stilwell in 1938 is considered to be one of the fundamental tests of the special theory of relativity. The experiment was intended to use a primarily longitudinal test of light wave propagation to detect and quantify the effect of time dilation on the relativistic Doppler effect of light waves received from a moving source. Also it intended to indirectly verify and quantify the more difficult to detect transverse Doppler effect associated with detection at a substantial angle to the path of motion of the source - specifically the effect associated with detection at a 90° angle to the path of motion of the source. In both respects it is believed that, a longitudinal test can be used to indirectly verify an effect that actually occurs at a 90° transverse angle to the path of motion of the source.\n\nBased on recent theoretical findings of the relativistic transverse Doppler effect, some scientists have shown that such comparison between longitudinal and transverse effects is fundamentally flawed and thus invalid; because it assumes compatibility between two different mathematical treatments. The experiment was designed to detect the predicted time dilation related red-shift effect (increase in wave-length with corresponding decrease in frequency) of special relativity at the fundamentally longitudinal angles at or near 00 and 1800, even though the time dilation effect is based on the transverse angle of 900. Thus, the results of the said experiment do not prove anything. More specifically, it can be shown that the mathematical treatment of special relativity to the transverse Doppler effect is invalid and thus incompatible with the longitudinal mathematical treatment at distances close to the moving source. Any direct comparisons between the longitudinal and transverse mathematical predictions under the specified conditions of the experiment are invalid.\n\nCosmic rays are particles - mostly protons but sometimes heavy atomic nuclei - that travel through the universe at close to the speed of light. Some cosmic rays detected on Earth are produced in violent events such as supernovae, but physicists still don’t know the origins of the highest-energy particles, which are the most energetic particles ever seen in nature. As cosmic-ray particles travel through space, they lose energy in collisions with the low-energy photons that pervade the universe, such as those of the cosmic microwave background radiation. Special theory of relativity dictates that any cosmic rays reaching Earth from a source outside our galaxy will have suffered so many energy-shedding collisions that their maximum possible energy cannot exceed 5 × 1019 electron-volts. This is known as the Greisen-Zatsepin-Kuzmin limit. Over the past decade, University of Tokyo’s Akeno Giant Air Shower Array - 111 particle detectors have detected several cosmic rays above the GZK limit. In theory, they could only have come from within our galaxy, avoiding an energy-sapping journey across the cosmos. However, astronomers cannot find any source for these cosmic rays in our galaxy. One possibility is that there is something wrong with the observed results. Another possibility is that Einstein was wrong. His special theory of relativity says that space is the same in all directions, but what if particles found it easier to move in certain directions? Then the cosmic rays could retain more of their energy, allowing them to beat the GZK limit. A recent report (Physical Letters B, Vol. 668, p-253) suggests that the fabric of space-time is not as smooth as Einstein and others have predicted.\n\nDuring 1919, Eddington started his much publicised eclipse expedition to observe the bending of light by a massive object (here the Sun) to verify the correctness of General Relativity. The experiment in question concerned the problem of whether light rays are deflected by gravitational forces, and took the form of astrometric observations of the positions of stars near the Sun during a total solar eclipse. The consequence of Eddington’s theory-led attitude to the experiment, along with alleged data fudging, was claimed to favor Einstein’s theory over Newton’s when in fact the data supported no such strong construction. In reality, both the predictions were based on Einstein’s calculations in 1908 and again in 1911 using Newton’s theory of gravitation. In 1911, Einstein wrote: “A ray of light going past the Sun would accordingly undergo deflection to an amount of 4’10-6 = 0.83 seconds of arc”. He did not clearly explain which fundamental principle of physics used in his paper and giving the value of 0.83 seconds of arc (dubbed half deflection) was wrong. He revised his calculation in 1916 to hold that light coming from a star far away from the Earth and passing near the Sun will be deflected by the Sun’s gravitational field by an amount that is inversely proportional to the star’s radial distance from the Sun (1.745” at the Sun’s limb - dubbed full deflection). Einstein never explained why he revised his earlier figures. Eddington was experimenting which of the above two values calculated by Einstein is correct.\n\nSpecifically it has been alleged that a sort of data fudging took place when Eddington decided to reject the plates taken by the one instrument (the Greenwich Observatory’s Astrographic lens, used at Sobral), whose results tended to support the alternative “Newtonian” prediction of light bending (as calculated by Einstein). Instead the data from the inferior (because of cloud cover) plates taken by Eddington himself at Principe and from the inferior (because of a reduced field of view) 4-inch lens used at Sobral were promoted as confirming the theory. While he claimed that the result proved Einstein right and Newton wrong, an objective analysis of the actual photographs shows no such clear cut result. Both theories are consistent with the data obtained. It may be recalled that when someone said that there are only two persons in the world besides Einstein who understood relativity, Eddington had replied that he does not know who the other person was. This arrogance clouded his scientific acumen, as was confirmed by his distaste for the theories of Dr. S Chandrasekhar, which subsequently won the Nobel Prize.\n\nHeisenberg’s Uncertainty relation is still a postulate, though many of its predictions have been verified and found to be correct. Heisenberg never called it a principle. Eddington was the first to call it a principle and others followed him. But as Karl Popper pointed out, uncertainty relations cannot be granted the status of a principle because theories are derived from principles, but uncertainty relation does not lead to any theory. We can never derive an equation like the Schrödinger equation or the commutation relation from the uncertainty relation, which is an inequality. Einstein’s distinction between “constructive theories” and “principle theories” does not help, because this classification is not a scientific classification. Serious attempts to build up quantum theory as a full fledged Theory of Principle on the basis of the uncertainty relation have never been carried out. At best it can be said that Heisenberg created “room” or “freedom” for the introduction of some non-classical mode of description of experimental data. But these do not uniquely lead to the formalism of quantum mechanics.\n\nThere are a plethora of other postulates in Quantum Mechanics; such as: the Operator postulate, the Hermitian property postulate, Basis set postulate, Expectation value postulate, Time evolution postulate, etc. The list goes on and on and includes such undiscovered entities as strings and such exotic particles as the Higg’s particle (which is dubbed as the “God particle”) and graviton; not to speak of squarks et all. Yet, till now it is not clear what quantum mechanics is about? What does it describe? It is said that quantum mechanical systems are completely described by its wave function? From this it would appear that quantum mechanics is fundamentally about the behavior of wave-functions. But do the scientists really believe that wave-functions describe reality? Even Schrödinger, the founder of the wave-function, found this impossible to believe! He writes (Schrödinger 1935): “That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message”. Rather, he was worried about the “blurring” suggested by the spread-out character of the wave-function, which he describes as, “affects macroscopically tangible and visible things, for which the term ‘blurring’ seems simply wrong”.\n\nSchrödinger goes on to note that it may happen in radioactive decay that “the emerging particle is described … as a spherical wave … that impinges continuously on a surrounding luminescent screen over its full expanse. The screen however, does not show a more or less constant uniform surface glow, but rather lights up at one instant at one spot …”. He observed further that one can easily arrange, for example by including a cat in the system, “quite ridiculous cases” with the ψ-function of the entire system having in it the living and the dead cat mixed or smeared out in equal parts. Resorting to epistemology cannot save such doctrines.\n\nThe situation was further made complicated by Bohr with interpretation of quantum mechanics. But how many scientists truly believe in his interpretation? Apart from the issues relating to the observer and observation, it usually is believed to address the measurement problem. Quantum mechanics is fundamentally about the micro-particles such as quarks and strings etc, and not the macroscopic regularities associated with measurement of their various properties. But if these entities are somehow not to be identified with the wave-function itself and if the description is not about measurements, then where is their place in the quantum description? Where is the quantum description of the objects that quantum mechanics should be describing? This question has led to the issues raised in the EPR argument. As we will see, this question has not been settled satisfactorily.\n\nThe formulations of quantum mechanics describe the deterministic unitary evolution of a wave-function. This wave-function is never observed experimentally. The wave-function allows computation of the probability of certain macroscopic events of being observed. However, there are no events and no mechanism for creating events in the mathematical model. It is this dichotomy between the wave-function model and observed macroscopic events that is the source of the various interpretations in quantum mechanics. In classical physics, the mathematical model relates to the objects we observe. In quantum mechanics, the mathematical model by itself never produces observation. We must interpret the wave-function in order to relate it to experimental observation. Often these interpretations are related to the personal and socio-cultural bias of the scientist, which gets weightage based on his standing in the community. Thus, the arguments of Einstein against Bohr’s position has roots in Lockean notions of perception, which opposes the Kantian metaphor of the “veil of perception” that pictures the apparatus of observation as like a pair of spectacles through which a highly mediated sight of the world can be glimpsed. According to Kant, “appearances” simply do not reflect an independently existing reality. They are constituted through the act of perception in such a way that conform them to the fundamental categories of sensible intuitions. Bohr maintained that “measurement has an essential influence on the conditions on which the very definition of physical quantities in question rests” (Bohr 1935, 1025).\n\nIn modern science, there is no unambiguous and precise definition of the words time, space, dimension, numbers, zero, infinity, charge, quantum particle, wave-function etc. The operational definitions have been changed from time to time to take into account newer facts that facilitate justification of the new “theory”. For example, the fundamental concept of the quantum mechanical theory is the concept of “state”, which is supposed to be completely characterized by the wave-function. However, till now it is not certain “what” a wave-function is. Is the wave-function real - a concrete physical object or is it something like a law of motion or an internal property of particles or a relation among spatial points? Or is it merely our current information about the particles? Quantum mechanical wave-functions cannot be represented mathematically in anything smaller than a 10 or 11 dimensional space called configuration space. This is contrary to experience and the existence of higher dimensions is still in the realm of speculation. If we accept the views of modern physicists, then we have to accept that the universe’s history plays itself out not in the three dimensional space of our everyday experience or the four-dimensional space-time of Special Relativity, but rather in this gigantic configuration space, out of which the illusion of three-dimensionality somehow emerges. Thus, what we see and experience is illusory! Maya?\n\nThe measurement problem in quantum mechanics is the unresolved problem of how (or if) wave-function collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer. If it is postulated that a particle does not have a value before measurement, there has to be conclusive evidence to support this view. The wave-function in quantum mechanics evolves according to the Schrödinger equation into a linear superposition of different states, but actual measurements always find the physical system in a definite state. Any future evolution is based on the state the system was “discovered” to be in when the measurement was made, implying that the measurement “did something” to the process under examination. Whatever that “something” may be does not appear to be explained by the basic theory. Further, quantum systems described by linear wave-functions should be incapable of non-linear behavior. But chaotic quantum systems have been observed. Though chaos appears to be probabilistic, it is actually deterministic. Further, if the collapse causes the quantum state to jump from superposition of states to a fixed state, it must be either an illusion or an approximation to the reality at quantum level. We can rule out illusion as it is contrary to experience. In that case, there is nothing to suggest that events in quantum level are not deterministic. We may very well be able to determine the outcome of a quantum measurement provided we set up an appropriate measuring device!\n\nThe operational definitions and the treatment of the term wave-function used by researchers in quantum theory progressed through intermediate stages. Schrödinger viewed the wave-function associated with the electron as the charge density of an object smeared out over an extended (possibly infinite) volume of space. He did not regard the waveform as real nor did he make any comment on the waveform collapse. Max Born interpreted it as the probability distribution in the space of the electron’s position. He differed from Bohr in describing quantum systems as being in a state described by a wave-function which lives longer than any specific experiment. He considered the waveform as an element of reality. According to this view, also known as State Vector Interpretation, measurement implied the collapse of the wave function. Once a measurement is made, the wave-function ceases to be smeared out over an extended volume of space and the range of possibilities collapse to the known value. However, the nature of the waveform collapse is problematic and the equations of Quantum Mechanics do not cover the collapse itself.\n\nThe view known as “Consciousness Causes Collapse” regards measuring devices also as quantum systems for consistency. The measuring device changes state when a measurement is made, but its wave-function does not collapse. The collapse of the wave-function can be traced back to its interaction with a conscious observer. Let us take the example of measurement of the position of an electron. The waveform does not collapse when the measuring device initially measures the position of the electron. Human eye can also be considered a quantum system. Thus, the waveform does not collapse when the photon from the electron interacts with the eye. The resulting chemical signals to the brain can also be treated as a quantum system. Hence it is not responsible for the collapse of the wave-form. However, a conscious observer always sees a particular outcome. The wave-form collapse can be traced back to its first interaction with the consciousness of the observer. This begs the question: what is consciousness? At which stage in the above sequence of events did the wave-form collapse? Did the universe behave differently before life evolved? If so, how and what is the proof for that assumption? No answers.\n\nMany-worlds Interpretation tries to overcome the measurement problem in a different way. It regards all possible outcomes of measurement as “really happening”, but holds that somehow we select only one of those realities (or in their words - universes). But this view clashes with the second law of thermodynamics. The direction of the thermodynamic arrow of time is defined by the special initial conditions of the universe which provides a natural solution to the question of why entropy increases in the forward direction of time. But what is the cause of the time asymmetry in the Many-worlds Interpretation? Why do universes split in the forward time direction? It is said that entropy increases after each universe-branching operation – the resultant universes are slightly more disordered. But some interpretations of decoherence contradict this view. This is called macroscopic quantum coherence. If particles can be isolated from the environment, we can view multiple interference superposition terms as a physical reality in this universe. For example, let us consider the case of the electric current being made to flow in opposite directions. If the interference terms had really escaped to a parallel universe, then we should never be able to observe them both as physical reality in this universe. Thus, this view is questionable.\n\nTransactional Interpretation accepts the statistical nature of waveform, but breaks it into an “offer” wave and an “acceptance” wave, both of which are treated as real. Probabilities are assigned to the likelihood of interaction of the offer waves with other particles. If a particle interacts with the offer wave, then it “returns” a confirmation wave to complete the transaction. Once the transaction is complete, energy, momentum, etc., are transferred in quanta as per the normal probabilistic quantum mechanics. Since Nature always takes the shortest and the simplest path, the transaction is expected to be completed at the first opportunity. But once that happens, classical probability and not quantum probability will apply. Further, it cannot explain how virtual particles interact. Thus, some people defer the waveform collapse to some unknown time. Since the confirmation wave in this theory is smeared all over space, it cannot explain when the transaction begins or is completed and how the confirmation wave determines which offer wave it matches up to.\n\nQuantum decoherence, which was proposed in the context of the many-worlds interpretation, but has also become an important part of some modern updates of the Copenhagen interpretation based on consistent histories, allows physicists to identify the fuzzy boundary between the quantum micro-world and the world where the classical intuition is applicable. But it does not describe the actual process of the wave-function collapse. It only explains the conversion of the quantum probabilities (that are able to interfere) to the ordinary classical probabilities. Some people have tried to reformulate quantum mechanics as probability or logic theories. In some theories, the requirements for probability values to be real numbers have been relaxed. The resulting non-real probabilities correspond to quantum waveform. But till now a fully developed theory is missing.\n\nHidden Variables Theories treat Quantum mechanics as incomplete. Until a more sophisticated theory underlying Quantum mechanics is discovered, it is not possible to make any definitive statement. It views quantum objects as having properties with well-defined values that exist separately from any measuring devices. According to this view, chance plays no roll at all and everything is fully deterministic. Every material object invariably does occupy some particular region of space. This theory takes the form of a single set of basic physical laws that apply in exactly the same way to every physical object that exists. The waveform may be a purely statistical creation or it may have some physical role. The Causal Interpretation of Bohm and its latter development, the Ontological Interpretation, emphasize “beables” rather than the “observables” in contradistinction to the predominantly epistemological approach of the standard model. This interpretation is causal, but non-local and non-relativistic, while being capable of being extended beyond the domain of the current quantum theory in several ways.\n\nThere are divergent views on the nature of reality and the role of science in dealing with reality. Measuring a quantum object was supposed to force it to collapse from a waveform into one position. According to quantum mechanical dogma, this collapse makes objects “real”. But new verifications of “collapse reversal” suggest that we can no longer assume that measurements alone create reality. It is possible to take a “weak” measurement of a quantum particle continuously partially collapsing the quantum state, and then “unmeasure” it altering certain properties of the particle and perform the same weak measurement again. In one such experiment reported in Nature News, the particle was found to have returned to its original quantum state just as if no measurement had ever been taken. This implies that, we cannot assume that measurements create reality because; it is possible to erase the effects of a measurement and start again.\n\nNewton gave his laws of motion in the second chapter, entitled “Axioms, or Laws of motion” of his book Principles of Natural Philosophy published in 1687 in Latin language. The second law says that the change of motion is proportional to the motive force impressed. Newton relates the force to the change of momentum (not to the acceleration as most textbooks do). Momentum is accepted as one of two quantities that, taken together, yield the complete information about a dynamic system at any instant. The other quantity is position, which is said to determine the strength and direction of the force. Since then the earlier ideas have changed considerably. The pairing of momentum and position is no longer viewed in the Euclidean space of three dimensions. Instead, it is viewed in phase space, which is said to have six dimensions, three for position and three for momentum. But here the term dimension has actually been used for direction, which is not a scientific description.\n\nIn fact most of the terms used by modern scientists have not been precisely defined - they have only an operational definition, which is not only incomplete, but also does not stand scientific scrutiny, though it is often declared as “reasonable”. This has been done not by chance, but by design, as modern science is replete with such instances. For example, we quote from the paper of Einstein and his colleagues Boris Podolsky and Nathan Rosen, which is known as the EPR argument (Phys. Rev. 47, 777 (1935):\n\n“A comprehensive definition of reality is, however, unnecessary for our purpose. We shall be satisfied with the following criterion, which, we regard as reasonable. If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. It seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one such way, whenever the conditions set down in it occur. Regarded not as necessary, but merely as a sufficient, condition of reality, this criterion is in agreement with classical as well as quantum-mechanical ideas of reality.”\n\nPrima facie, what Einstein and his colleagues argued was that under ideal conditions, observation (includes measurement) functions like a mirror reflecting an independently existing, external reality. The specific criterion for describing reality characterizes it in terms of objectivity understood as independence from any direct measurement. This implies that, when a direct measurement of physical reality occurs, it merely passively reflects rather than actively constitutes the object under observation. It further implies that ideal observations not only reflect the state of the object during observation, but also before and after observation just like a photograph taken. It has a separate and fixed identity than the object whose photograph has been taken. While the object may be evolving in time, the photograph depicts a time invariant state. Bohr and Heisenberg opposed this notion based on the Kantian view by describing acts of observation and measurement more generally as constitutive of phenomena. More on this will be discussed later.\n\nThe fact that our raw sense impressions and experiences are compatible with widely differing concepts of the world has led some philosophers to suggest that we should dispense with the idea of an “objective world” altogether and base our physical theories on nothing but direct sense impressions only. Berkeley expressed the positivist identification of sense impressions with objective existence by the famous phrase “esse est percipi” (to be is to be perceived). This has led to the changing idea of “objective reality”. However, if we can predict with certainty “the value of a physical quantity”, it only means that we have partial and not complete “knowledge” – which is the “total” result of “all” measurements - of the system. It has not been shown that knowledge is synonymous with reality. We may have the “knowledge” of mirage, but it is not real. Based on the result of our measurement, we may have knowledge that something is not real, but only apparent.\n\nThe partial definition of reality is not correct as it talks about “the value of a physical quantity” and not “the value of all physical quantities”. We can predict with certainty “the value of a physical quantity” such as position or momentum, which are classical concepts, without in any way disturbing the system. This has been accepted for past events by Heisenberg himself, which has been discussed in latter pages. Further, measurement is a process of comparison between similars and not bouncing light off something to disturb it. This has been discussed in detail while discussing the measurement problem. We cannot classify an object being measured (observed) separately from the apparatus performing the measurement (though there is lot of confusion in this area). They must belong to the same class. This is clearly shown in the quantum world where it is accepted that we cannot divorce the property we are trying to measure from the type of observation we make: the property is dependent on the type of measurement and the measuring instrument must be designed to use that particular property. However, this interpretation can be misleading and may not have anything to do with reality as described below. Such limited treatment of the definition of “reality” has given the authors the freedom to manipulate the facts to suit their convenience. Needless to say; the conclusions arrived at in that paper has been successively proved wrong by John S. Bell, Alain Aspect, etc, though for a different reason.\n\nIn the double slit experiment, it is often said that whether the electron has gone through the hole No.1 or No. 2 is meaningless. The electron, till we observe which hole it goes through, exists in a superposition state of equal measure of probability wave for going through the hole 1 and through the hole 2. This is a highly misleading notion as after it went through, we can always see its imprint on the photographic plate at a particular position and that is real. Before such observation we do not know which hole it went through, but there is no reason to presume that it went through a mixed state of both holes. Our inability to measure or know cannot change physical reality. It can only limit our knowledge of such physical reality. This aspect and the interference phenomenon have been discussed elaborately in later pages.\n\nIf, we accept the modern view of superposition of states, we land in many complex situations. Suppose the Schrödinger’s cat is somewhere in deep space and a team of astronauts were sent to measure its state According to the Copenhagen interpretation, the astronauts by opening the box and performing the observation have now put the cat into a definite quantum state; say find it alive. For them, the cat is no longer in a superposition state of equal measure of probability of living or dead. But for their Earth bound colleagues, the cat and the astronauts on board the space shuttle who know the state of the cat (did they change to a quantum state?), are still in a probability wave superposition state of live cat and dead cat. Finally, when the astronauts communicate with a computer down on earth, they pass on the information that is stored in the magnetic memory of the computer. After the computer receives the information, but before its memory is read by the earth-bound scientists, the computer is part of the superposition state for the earth-bound scientists. Finally, in reading the computer output, the earth-bound scientists reduce the superposition state to a definite one. Reality springs into being or rather from being to becoming only after we observe it. Is the above description sensible?\n\nWhat really happens is that the cat interacts with the particles around it – protons, electrons, air molecules, dust particles, radiation, etc, which has the effect of “observing” it. The state is accessed by each of the conscious observers (as well as the other particles) by intercepting on its/our retina a small fraction of the light that has interacted with the cat. Thus, in reality, the field set up by his retina is perturbed and the impulse is carried out to the brain, where it is compared with previous similar impressions. If the impression matches with any previous impressions, we cognize it to be like that. Thereafter only we cognize the result of the measurement: the cat is alive or dead at the moment of observation. Thus, the process of measurement is carried out constantly without disturbing the system and evolution of the observed has nothing to do with the observation. This has been elaborated while discussing the measurement problem.\n\nFurther someone has put the cat and the deadly apparatus in the box. Thus according to the generally accepted theory, the wave-function has collapsed for him at that time. The information is available to us. Only afterwards, the evolutionary state of the cat – whether living or dead – is not known to us including the person who put the cat in the box in the first place. But according to the above description, the cat, whose wave-function has collapsed for the person who put the cat in the box, again goes into a “superposition of states of both alive and dead” and needs another observation – directly or indirectly through a set of apparatus - to describe its proper state at any subsequent time. This implies that after the second observation, the cat again goes into a “superposition of states of both alive and dead” till it is again observed and so on ad infinitum till it is found dead. But then the same story repeats for the dead cat – this time about his state of decomposition!\n\nThe cat example shows three distinct aspects: the state of the cat, i.e., dead or alive at the moment of observation (which information is time invariant as it is fixed), the state of the cat prior to and after the moment of observation (which information is time variant as the cat will die at some unspecified time due to unspecified reasons), and the cognition of these information by a conscious observer, which is time invariant but about the time evolution of the states of the cat. In his book “Popular Astronomy”, Prof. Bigelow says; Force, Mass, Surface, Electricity, Magnetism, etc., “are apprehended only during instantaneous transfer of energy”. He further adds; “Energy is the great unknown quantity, and its existence is recognized only during its state of change”. This is an eternal truth. We endorse the above view. It is well-known that the Universe is so called because everything in it is ever moving. Thus the view that observation not only describes the state of the object during observation, but also the state before and after it, is misleading. The result of measurement is the description of a state frozen in time, thus a fixed quantity. Its time evolution is not self-evident in the result of measurement. It has any meaning only after it is cognized by a conscious agent, as consciousness is time invariant. Thus, the observable, observation and observer depict three aspects of confined mass, displacing energy and revealing radiation of a single phenomenon depicting reality. Quantum physics has to explain these phenomena scientifically. We will discus it later.\n\nWhen one talks about what an electron is “doing”, one implies what sort of a wave function is associated with it. But the wave function is not a physical object in the sense a proton or an electron or a billiard ball. In fact, the rules of quantum theory do not even allot a unique wave function to a given state of motion, since multiplying the wave function by a factor of modulus unity does not change any physical consequence. Thus, Heisenberg opined that “the atoms or elementary particles are not as real; they form a world of potentialities or possibilities rather than one of things or facts”. This shows the helplessness of the physicists to explain the quantum phenomena in terms of the macro world. The activities of the elementary particles appear essential as long as we believe in the independent existence of fundamental laws that we can hope to understand better.\n\nReality cannot differ from person to person or from measurement to measurement because it has existence independent of these factors. The elements of our “knowledge” are actually derived from our raw sense impressions, by automatically interpreting them in conventional terms based on our earlier impressions. Since these impressions vary, our responses to the same data also vary. Yet, unless an event is observed, it has no meaning by itself. Thus, it can be said that while observables have a time evolution independent of observation, it depends upon observation for any meaningful description in relation to others. For this reason the individual responses/readings to the same object may differ based on their earlier (at a different time and may be space) experience/environment. As the earlier example of the cat shows, it requires a definite link between the observer and the observed – a split (from time evolution), and a link (between the measurement representing its state and the consciousness of the observer for describing such state in communicable language). This link varies from person to person. At every interaction, the reality is not “created”, but the “presently evolved state” of the same reality gets described and communicated. Based on our earlier experiences/experimental set-up, it may return different responses/readings.\n\nThere is no proof to show that a particle does not have a value before measurement. The static attributes of a proton or an electron such as its charge or its mass have well defined properties and will remain so even before and after observation even though it may change its position or composition due to the effect of the forces acting on it – spatial translation. The dynamical attributes will continue to evolve – temporal translation. The life cycles of stars and galaxies will continue till we notice their extinction in a supernova explosion. The moon will exist even when we are not observing it. The proof for this is their observed position after a given time matches our theoretical calculation. Before measurement, we do not know the “present” state. Since present is a dynamical entity describing time evolution of the particle, it evolves continuously from past to future. This does not mean that the observer creates reality – after observation at a given instant he only discovers the spatial and temporal state of its static and dynamical aspects.\n\nThe prevailing notion of superposition (an unobserved proposition) only means that we do not know how the actual fixed value after measurement has been arrived at (described elaborately in later pages), as the same value could be arrived at by infinite numbers of ways. We superimpose our ignorance on the particle and claim that the value of that particular aspect is undetermined whereas in reality the value might already have been fixed (the cat might have died). The observer cannot influence the state of the observed (moment of death of the cat) before or after observation. He can only report the “present state”. Quantum mechanics has failed to describe the collapse mechanism satisfactorily. In fact many models (such as the Copenhagen interpretation) treat the concept of collapse as non-sense. The few models that accept collapse as real are incomplete and fail to come up with a satisfactory mechanism to explain it. In 1932, John von Neumann showed that if electrons are ordinary objects with inherent properties (which would include hidden variables) then the behavior of those objects must contradict the predictions of quantum theory. Because of his stature in those days, no one contradicted him. But in 1952, David Bohm showed that hidden variables theories were plausible if super-luminal velocities are possible. Bohm’s mechanics has returned predictions equivalent to other interpretations of quantum mechanics. Thus, it cannot be discarded lightly. If Bohm is right, then Copenhagen interpretation and its extensions are wrong.\n\nThere is no proof to show that the characteristics of particle states are randomly chosen instantaneously at the time of observation/measurement. Since the value remains fixed after measurement, it is reasonable to assume that it remained so before measurement also. For example, if we measure the temperature of a particle by a thermometer, it is generally assumed that a little heat is transferred from the particle to the thermometer thereby changing the state of the particle. This is an absolutely wrong assumption. No particle in the Universe is perfectly isolated. A particle inevitably interacts with its environment. The environment might very well be a man-made measuring device.\n\nIntroduction of the thermometer does not change the environment as all objects in the environment are either isothermic or heat is flowing from higher concentration to lower concentration. In the former case there is no effect. In the latter case also it does not change anything as the thermometer is isothermic with the environment. Thus the rate of heat flow from the particle to the thermometer remains constant – same as that of the particle to its environment. When exposed to heat, the expansion of mercury shows a uniform gradient in proportion to the temperature of its environment. This is sub-divided over a randomly chosen range and taken as the unit. The expansion of mercury when exposed to the heat flow from a particle till both become isothermic is compared with this unit and we get a scalar quantity, which we call the result of measurement at that instant. Similarly, the heat flow to the thermometer does not affect the object as it was in any case continuing with the heat flow at a steady rate and continued to do so even after measurement. This is proved from the fact that the thermometer reading does not change after sometime (all other conditions being unchanged). This is common to all measurements. Since the scalar quantity returned as the result of measurement is a number, it is sometimes said that numbers are everything.\n\nWhile there is no proof that measurement determines reality, there is proof to the contrary. Suppose we have a random group of people and we measure three of their properties: sex, height and skin-color. They can be male or female, tall or short and their skin-color could be fair or brown. If we take at random 30 people and measure the sex and height first (male and tall), and then the skin-color (fair) for the same sample, we will get one result (how many tall men are fair). If we measure the sex and the skin-color first (male and fair), and then the height (tall), we will get a different result (how many fare males are tall). If we measure the skin-color and the height first (fair and tall), and then the sex (male), we will get a yet different result (how many fare and tall persons are male). Order of measurement apparently changes result of measurement. But the result of measurement really does not change anything. The tall will continue to be tall and the fair will continue to be fair. The male and female will not change sex either. This proves that measurement does not determine reality, but only exposes selected aspects of reality in a desired manner – depending upon the nature of measurement. It is also wrong to say that whenever any property of a microscopic object affects a macroscopic object, that property is observed and becomes physical reality. We have experienced situations when an insect bite is not really felt (measure of pain) by us immediately even though it affects us. A viral infection does not affect us immediately.\n\nWe measure position, which is the distance from a fixed reference point in different coordinates, by a tape of unit distance from one end point to the other end point or its sub-divisions. We measure mass by comparing it with another unit mass. We measure time, which is the interval between events by a clock, whose ticks are repetitive events of equal duration (interval) which we take as the unit, etc. There is no proof to show that this principle is not applicable to the quantum world. These measurements are possible when both the observer with the measuring instrument and the object to be measured are in the same frame of reference (state of motion); thus without disturbing anything. For this reason results of measurement are always scalar quantities – multiples of the unit. Light is only an accessory for knowing the result of measurement and not a pre-condition for measurement. Simultaneous measurement of both position and momentum is not possible, which is correct, though due to different reasons explained in later pages. Incidentally, both position and momentum are regarded as classical concepts.\n\nIn classical mechanics and electromagnetism, properties of a point mass or properties of a field are described by real numbers or functions defined on two or three dimensional sets. These have direct, spatial meaning, and in these theories there seems to be less need to provide a special interpretation for those numbers or functions. The accepted mathematical structure of quantum mechanics, on the other hand, is based on fairly abstract mathematics (?), such as Hilbert spaces, (which is the quantum mechanical counterpart of the classical phase-space) and operators on those Hilbert spaces. Here again, there is no precise definition of space. The proof for the existence and justification of the different classification of “space” and “vacuum” are left unexplained.\n\nWhen developing new theories, physicists tend to assume that quantities such as the strength of gravity, the speed of light in vacuum or the charge on the electron are all constant. The so-called universal constants are neither self-evident in Nature nor have been derived from fundamental principles (though there are some claims to the contrary, each has some problem). They have been deduced mathematically and their value has been determined by actual measurement. For example, the fine structure constant has been postulated in QED, but its value has been derived only experimentally (We have derived the measured value from fundamental principles). Yet, the regularity with which such constants of Nature have been discovered points to some important principle underlying it. But are these quantities really constant?\n\nThe velocity of light varies according to the density of the medium. The acceleration due to gravity “g” varies from place to place. We have measured the value of “G” from earth. But we do not know whether the value is the same beyond the solar system. The current value of the distance between the Sun and the Earth has been pegged at 149,597,870.696 kilometers. A recent (2004) study shows that the Earth is moving away from the Sun @ 15 cm per annum. Since this value is 100 times greater than the measurement error, something must really be pushing Earth outwards. While one possible explanation for this phenomenon is that the Sun is losing enough mass via fusion and the solar wind, alternative explanations include the influence of dark matter and changing value of G. We will explain it later.\n\nEinstein proposed the Cosmological Constant to allow static homogeneous solutions to his equations of General Relativity in the presence of matter. When the expansion of the Universe was discovered, it was thought to be unnecessary forcing Einstein to declare was it was his greatest blunder. There have been a number of subsequent episodes in which a non-zero cosmological constant was put forward as an explanation for a set of observations and later withdrawn when the observational case evaporated. Meanwhile, the particle theorists are postulating that the cosmological constant can be interpreted as a measure of the energy density of the vacuum. This energy density is the sum of a number of apparently unrelated contributions: potential energies from scalar fields and zero-point fluctuations of each field theory degree of freedom as well as a bare cosmological constant λ0, each of magnitude much larger than the upper limits on the cosmological constant as measured now. However, the observed vacuum energy is very very small in comparison to the theoretical prediction: a discrepancy of 120 orders of magnitude between the theoretical and observational values of the cosmological constant. This has led some people to postulate an unknown mechanism which would set it precisely to zero. Others postulate the mechanism to suppress the cosmological constant by just the right amount to yield an observationally accessible quantity. However, all agree that this illusive quantity does play an important dynamical role in the Universe. The confusion can be settled if we accept the changing value of G, which can be related to the energy density of the vacuum. Thus, the so-called constants of Nature could also be thought of as the equilibrium points, where different forces acting on a system in different proportions balance each other.\n\nFor example, let us consider the Libration points called L4 and L5, which are said to be places that gravity forgot. They are vast regions of space, sometimes millions of kilometers across, in which celestial forces cancel out gravity and trap anything that falls into them. The Libration points, known as ¨ÉxnùÉäSSÉ and {ÉÉiÉ in earlier times, were rediscovered in 1772 by the mathematician Joseph-Louis Lagrange. He calculated that the Earth’s gravitational field neutralizes the gravitational pull of the sun at five regions in space, making them the only places near our planet where an object is truly weightless. Astronomers call them Libration points; also Lagrangian points, or L1, L2, L3, L4 and L5 for short. Of the five Libration points, L4 and L5 are the most intriguing.\n\nTwo such Libration points sit in the Earth’s orbit also, one marching ahead of our planet, the other trailing along behind. They are the only ones that are stable. While a satellite parked at L1 or L2 will wander off after a few months unless it is nudged back into place (like the American satellite SOHO), any object at L4 or L5 will stay put due to a complex web of forces (like the asteroids). Evidence for such gravitational potholes appears around other planets too. In 1906, Max Wolf discovered an asteroid outside of the main belt between Mars and Jupiter, and recognized that it was sitting at Jupiter’s L4 point. The mathematics for L4 uses the “brute force approach” making it approximate. Lying 150 million kilometers away along the line of Earth’s orbit, L4 circles the sun about 60 degrees (slightly more, according to our calculation) in front of the planet while L5 lies at the same angle behind. Wolf named it Achilles, leading to the tradition of naming these asteroids after characters from the Trojan wars.\n\nThe realization that Achilles would be trapped in its place and forced to orbit with Jupiter, never getting much closer or further away, started a flurry of telescopic searches for more examples. There are now more than 1000 asteroids known to reside at each of Jupiter’s L4 and L5 points. Of these, about ⅔ reside at L4 while the rest ⅓ are at L5. Perturbations by the other planets (primarily Saturn) causes these asteroids to oscillate around L4 and L5 by about 15-200 and at inclinations of up to 400 to the orbital plane. These oscillations generally take between 150 years and 200 years to complete. Such planetary perturbations may also be the reason why there have been so few Trojans found around other planets. Searches for “Trojan” asteroids around other planets have met with mixed results. Mars has 5 of them at L5 only. Saturn seemingly has none. Neptune has two.\n\nThe asteroid belt surrounds the inner Solar system like a rocky, ring-shaped moat, extending out from the orbit of Mars to that of Jupiter. But there are voids in that moat in distinct locations called Kirkwood gaps that are associated with orbital resonances with the giant planets - where the orbital influence of Jupiter is especially potent. Any asteroid unlucky enough to venture into one of these locations will follow chaotic orbits and will be perturbed and ejected from the cozy confines of the belt, often winding up on a collision course with one of the inner, rocky planets (such as Earth) or the moon. But Jupiter’s pull cannot account for the extent of the belt’s depletion seen at present or for the spotty distribution of asteroids across the belt - unless there was a migration of planets early in the history of the solar system. According to a report (Nature 457, 1109-1111 dated 26 February 2009), the observed distribution of main belt asteroids does not fill uniformly even those regions that are dynamically stable over the age of the Solar System. There is a pattern of excess depletion of asteroids, particularly just outward of the Kirkwood gaps associated with the 5:2, the 7:3 and the 2:1 Jovian resonances. These features are not accounted for by planetary perturbations in the current structure of the Solar System, but are consistent with dynamical ejection of asteroids by the sweeping of gravitational resonances during the migration of Jupiter and Saturn.\n\nSome researchers designed a computer model of the asteroid belt under the influence of the outer “gas giant” planets, allowing them to test the distribution that would result from changes in the planets’ orbits over time. A simulation wherein the orbits remained static, did not agree with observational evidence. There were places where there should have been a lot more asteroids than we saw. On the other hand, a simulation with an early migration of Jupiter inward and Saturn outward - the result of interactions with lingering planetesimals (small bodies) from the creation of the solar system - fit the observed layout of the belt much better. The uneven spacing of asteroids is readily explained by this planet-migration process that other people have also worked on. In particular, if Jupiter had started somewhat farther from the sun and then migrated inward toward its current location, the gaps it carved into the belt would also have inched inward, leaving the belt looking much like it does now. The good agreement between the simulated and observed asteroid distributions is quite remarkable.\n\nOne significant question not addressed in this paper is the pattern of migration - whether the asteroid belt can be used to rule out one of the presently competing theories of migratory patterns. The new study deals with the speed at which the planets’ orbits have changed. The simulation presumes a rather rapid migration of a million or two million years, but other models of Neptune’s early orbital evolution tend to show that migration proceeds much more slowly, over millions of years. We hold this period as 4.32 million years for the Solar system. This example shows that the orbits of planets, which are stabilized due to balancing of the centripetal force and gravity, might be changing from time to time. This implies that either the masses of the Sun and the planets or their distance from each other or both are changing over long periods of time (which is true). It can also mean that G is changing. Thus, the so-called constants of Nature may not be so constants after all.\n\nEarlier, a cosmology with changing physical values for the gravitational constant G was proposed by P.A.M. Dirac in 1937. Field theories applying this principle have been proposed by P. Jordan and D.W. Sciama and in 1961 by C. Brans and R.H. Dicke. According to these theories the value of G is diminishing. Brans and Dicke suggested a change of about 0.00000000002 per year. This theory has not been accepted on the ground that it would have profound effect on the phenomena ranging from the evolution of the Universe to the evolution of the Earth. For instance, stars evolve faster if G is greater. Thus, the stellar evolutionary ages computed with constant G at its present value would be too great. The Earth compressed by gravitation would expand having a profound effect on surface features. The Sun would have been hotter than it is now and the Earth’s orbit would have been smaller. No one bothered to check whether such a scenario existed or is possible. Our studies in this regard show that the above scenario did happen. We have data to prove the above point.\n\nPrecise measurements in 1999 gave so divergent values of G from the currently accepted value that the result had to be pushed under the carpet, as otherwise most theories of physics would have tumbled. Presently, physicists are measuring gravity by bouncing atoms up and down off a laser beam (arXiv:0902.0109). The experiments have been modified to perform atom interferometry, whereby quantum interference between atoms can be used to measure tiny accelerations. Those still using the earlier value of G in their calculations, land in trajectories much different from their theoretical calculations. Thus, modern science is based on a value of G that has been proved to be wrong. The Pioneer and Fly-by anomalies and the change of direction of Voyager 2 after it passed the orbit of Saturn have cast a shadow on the authenticity of the theory of gravitation. Till now these have not been satisfactorily explained. We have discussed these problems and explained a different theory of gravitation in later pages.\n\nAccording to reports published in several scientific journals, precise measurements of the light from distant quasars and the only known natural nuclear reactor, which was active nearly 2 billion years ago at what is now Oklo in Gabon suggest that the value of the fine-structure constant may have changed over the history of the universe (Physical Review D, vol 69, p 121701). If confirmed, the results will be of enormous significance for the foundations of physics. Alpha is an extremely important constant that determines how light interacts with matter - and it shouldn’t be able to change. Its value depends on, among other things, the charge on the electron, the speed of light and Planck’s constant. Could one of these really have changed?\n\nIf the fine-structure constant changes over time, it allows postulating that the velocity of light might not be constant. This would explain the flatness, horizon and monopole problems in cosmology. Recent work has shown that the universe appears to be expanding at an ever faster rate, and there may well be a non-zero cosmological constant. There is a class of theories where the speed of light is determined by a scalar field (the force making the cosmos expand, the cosmological constant) that couples to the gravitational effect of pressure. Changes in the speed of light convert the energy density of this field into energy. One off-shoot of this view is that in a young and hot universe during the radiation epoch, this prevents the scalar field dominating the universe. As the universe expands, pressure-less matter dominates and variations in c decreases making α (alpha) fixed and stable. The scalar field begins to dominate, driving a faster expansion of the universe. Whether the variation of the fine-structure constant claimed exists or not, putting bounds on the rate of change puts tight constraints on new theories of physics.\n\nOne of the most mysterious objects in the universe is what is known as the black hole – a derivative of the general theory of relativity. It is said to be the ultimate fate of a super-massive star that has exhausted its fuel that sustained it for millions of years. In such a star, gravity overwhelms all other forces and the star collapses under its own gravity to the size of a pinprick. It is called a black hole as nothing – not even light – can escape it. A black hole has two parts. At its core is a singularity, the infinitesimal point into which all the matter of the star gets crushed. Surrounding the singularity is the region of space from which escape is impossible - the perimeter of which is called the event horizon. Once something enters the event horizon, it loses all hope of exiting. It is generally believed that a large star eventually collapses to a black hole. Roger Penrose conjectured that the formation of a singularity during stellar collapse necessarily entails the formation of an event horizon. According to him, Nature forbids us from ever seeing a singularity because a horizon always cloaks it. Penrose’s conjecture is termed the cosmic censorship hypothesis. It is only a conjecture. But some theoretical models suggest that instead of a black hole, a collapsing star might become a naked singularity.\n\nMost physicists operate under the assumption that a horizon must indeed form around a black hole. What exactly happens at a singularity - what becomes of the matter after it is infinitely crushed into oblivion - is not known. By hiding the singularity, the event horizon isolates this gap in our knowledge. General relativity does not account for the quantum effects that become important for microscopic objects, and those effects presumably intervene to prevent the strength of gravity from becoming truly infinite. Whatever happens in a black hole stays in a black hole. Yet Researchers have found a wide variety of stellar collapse scenarios in which an event horizon does not form, so that the singularity remains exposed to our view. Physicists call it a naked singularity. In such a case, Matter and radiation can both fall in and come out, whereas matter falling into the singularity inside a black hole would land in a one-way trip.\n\nIn principle, we can come as close as we like to a naked singularity and return back. Naked singularities might account for unexplained high-energy phenomena that astronomers have seen, and they might offer a laboratory to explore the fabric of the so-called space-time on its finest scales. The results of simulations by different scientists show that most naked singularities are stable to small variations of the initial setup. Thus, these situations appear to be generic and not contrived. These counterexamples to Penrose’s conjecture suggest that cosmic censorship is not a general rule.\n\nThe discovery of naked singularities would transform the search for a unified theory of physics, not the least by providing direct observational tests of such a theory. It has taken so long for physicists to accept the possibility of naked singularities because they raise a number of conceptual puzzles. A commonly cited concern is that such singularities would make nature inherently unpredictable. Unpredictability is actually common in general relativity and not always directly related to cosmic censorship violation described above. The theory permits time travel, which could produce causal loops with unforeseeable outcomes, and even ordinary black holes can become unpredictable. For example, if we drop an electric charge into an uncharged black hole, the shape of space-time around the hole radically changes and is no longer predictable. A similar situation holds when the black hole is rotating.\nSpecifically, what happens is that space-time no longer neatly separates into space and time, so that physicists cannot consider how the black hole evolves from some initial time into the future. Only the purest of pure black holes, with no charge or rotation at all, is fully predictable. The loss of predictability and other problems with black holes actually stem from the occurrence of singularities; it does not matter whether they are hidden or not. Cosmologists dread the singularity because at this point gravity becomes infinite, along with the temperature and density of the universe. As its equations cannot cope with such infinities, general relativity fails to describe what happens at the big bang.\nIn the mid 1980s, Abhay Ashtekar rewrote the equations of general relativity in a quantum-mechanical framework to show that the fabric of space-time is woven from loops of gravitational field lines. The theory is called the loop quantum gravity. If we zoom out far enough, the space appears smooth and unbroken, but a closer look reveals that space comes in indivisible chunks, or quanta, 10-35 square meters in size. In 2000, some scientists used loop quantum gravity to create a simple model of the universe. This is known as the LQC. Unlike general relativity, the physics of LQC did not break down at the big bang. Some others developed computer simulations of the universe according to LQC. Early versions of the theory described the evolution of the universe in terms of quanta of area, but a closer look revealed a subtle error. After this mistake was corrected it was found that the calculations now involved tiny volumes of space. It made a crucial difference. Now the universe according to LQC agreed brilliantly with general relativity when expansion was well advanced, while still eliminating the singularity at the big bang. When they ran time backwards, instead of becoming infinitely dense at the big bang, the universe stopped collapsing and reversed direction. The big bang singularity had disappeared (Physical Review Letters, vol.96, p-141301). The era of the Big Bounce has arrived. But the scientists are far from explaining all the conundrums.\n\nOften it is said that the language of physics is mathematics. In a famous essay, Wigner wrote about the “unreasonable effectiveness of mathematics”. Most physicists resonate with the perplexity expressed by Wigner and Einstein’s dictum that “the most incomprehensible thing about the universe is that it is comprehensible”. They marvel at the fact that the universe is not anarchic - that atoms obey the same laws in distant galaxies as in the lab. Yet, Gödel’s Theorem implies that we can never be certain that mathematics is consistent: it leaves open the possibility that a proof exists demonstrating that 0=1. The quantum theory tells that, on the atomic scale, nature is intrinsically fuzzy. Nonetheless, atoms behave in precise mathematical ways when they emit and absorb light, or link together to make molecules. Yet, is Nature mathematical?\n\nLanguage is a means of communication. Mathematics cannot communicate in the same manner like a language. Mathematics on its own does not lead to a sensible universe. The mathematical formula has to be interpreted in communicable language to acquire some meaning. Thus, mathematics is only a tool for describing some and not all ideas. For example, “observer” has an important place in quantum physics. Everett addressed the measurement problem by making the observer an integral part of the system observed: introducing a universal wave function that links observers and objects as parts of a single quantum system. But there is no equation for the “observer”.\nWe have not come across any precise and scientific definition of mathematics. Concise Oxford Dictionary defines mathematics as: “the abstract science of numbers, quantity, and space studied in its own right”, or “as applied to other disciplines such as physics, engineering, etc”. This is not a scientific description as the definition of number itself leads to circular reasoning. Even the mathematicians do not have a common opinion on the content of mathematics. There are at least four views among mathematicians on what mathematics is. John D Barrow calls these views as:\nPlatonism: It is the view that concepts like groups, sets, points, infinities, etc., are “out there” independent of us – “the pie is in the sky”. Mathematicians discover them and use them to explain Nature in mathematical terms. There is an offshoot of this view called “neo-Platonism”, which likens mathematics to the composition of a cosmic symphony by independent contributors, each moving it towards some grand final synthesis. Proof: completely independent mathematical discoveries by different mathematicians working in different cultures so often turn out to be identical.\n\nConceptualism: It is the anti-thesis of Platonism. According to this view, scientists create an array of mathematical structures, symmetries and patterns and force the world into this mould, as they find it so compelling. The so-called constants of Nature, which arise as theoretically undetermined constants of proportionality in the mathematical equations, are solely artifacts of the peculiar mathematical representation they have chosen to use for different purposes.\n\nFormalism: This was developed during the last century, when a number of embarrassing logical paradoxes were discovered. There was proof which established the existence of particular objects, but offered no way of constructing them explicitly in a finite number of steps. Hilbert’s formalism belongs to this category, which defines mathematics as nothing more than the manipulation of symbols according to specified rules (not natural, but sometimes un-physical man-made rules). The resultant paper edifice has no special meaning at all. If the manipulations are done correctly, it should result in a vast collection of tautological statements: an embroidery of logical connections.\n\nIntuitionism: Prior to Cantor’s work on infinite sets, mathematicians had not made use of actual infinities, but only exploited the existence of quantities that could be made arbitrarily large or small – the concept of limit. To avoid founding whole areas of mathematics upon the assumption that infinite sets share the “obvious” properties possessed by finite one’s, it was proposed that only quantities that can be constructed from the natural numbers 1,2,3,…, in a finite number of logical steps, should be regarded as proven true.\n\nNone of the above views is complete because it neither is a description derived from fundamental principles nor conforms to a proper definition of mathematics, whose foundation is built upon logical consistency. The Platonic view arose from the fact that mathematical quantities transcend human minds and manifests the intrinsic character of reality. A number, say three or five codes some information differently in various languages, but conveys the same concept in all civilizations. They are abstract entities and mathematical truth means correspondence between the properties of these abstract objects and our system of symbols. We associate the transitory physical objects such as three worlds or five sense organs to these immutable abstract quantities as a secondary realization. These ideas are somewhat misplaced. Numbers are a property of all objects by which we distinguish between similars. If there is nothing similar to an object, it is one. If there are similars, the number is decided by the number of times we perceive such similars (we may call it a set). Since perception is universal, the concept of numbers is also universal.\n\nBelievers in eternal truth often point to mathematics as a model of a realm with timeless truths. Mathematicians explore this realm with their minds and discover truths that exist outside of time, in the same way that we discover the laws of physics by experiment. But mathematics is not only self-consistent, but also plays a central role in formulating fundamental laws of physics, which the physics Nobel laureate Eugene Wigner once referred to as the “unreasonable success of mathematics in physics”. One way to explain this “success” within the dominant metaphysical paradigm of the timeless multiverse is to suppose that physical reality is mathematical, i.e. we are creatures within the timeless Platonic realm. The cosmologist Max Tegmark calls this the mathematical universe hypothesis. A slightly less provocative approach is to posit that since the laws of physics can be represented mathematically, not only is their essential truth outside of time, but there is in the Platonic realm a mathematical object, a solution to the equations of the final theory, that is “isomorphic” in every respect to the history of the universe. That is, any truth about the universe can be mapped into a theorem about the corresponding mathematical object. If nothing exists or is true outside of time, then this description is void. However, if mathematics is not the description of a different timeless realm of reality, what is it? What are the theorems of mathematics about if numbers, formulas and curves do not exist outside of our world?\n\nLet us consider a game of chess. It was invented at a particular time, before which there is no reason to speak of any truths of chess. But once the game was invented, a long list of facts became demonstrable. These are provable from the rules and can be called the theorems of chess. These facts are objective in that any two minds that reason logically from the same rules will reach the same conclusions about whether a conjectured theorem is true or not. Platonists would say that chess always existed timelessly in an infinite space of mathematically describable games. By such an assertion, we do not achieve anything except a feeling of doing something elevated. Further, we have to explain how we finite beings embedded in time can gain knowledge about this timeless realm. It is much simpler to think that at the moment the game was invented, a large set of facts become objectively demonstrable, as a consequence of the invention of the game. There is no need to think of the facts as eternally existing truths, which are suddenly discoverable. Instead we can say they are objective facts that are evoked into existence by the invention of the game of chess. The bulk of mathematics can be treated the same way, even if the subjects of mathematics such as numbers and geometry are inspired by our most fundamental observations of nature. Mathematics is no less objective, useful or true for being evoked by and dependent on discoveries of living minds in the process of exploring the time-bound universe.\n\nThe Mandelbrot Set is often cited as a mathematical object with an independent existence of its own. Mandelbrot Set is produced by a remarkably simple mathematical formula – a few lines of code (f(z) = z2+c) describing a recursive feed-back loop – but can be used to produce beautiful colored computer plots. It is possible to endlessly zoom in to the set revealing ever more beautiful structures which never seem to repeat themselves. Penrose called it “not an invention of the human mind: it was a discovery”. It was just out there. On the other hand, fractals – geometrical shapes found through out Nature – are self-similar because how far you zoom into them; they still resemble the original structure. Some people use these factors to plead that mathematics and not evolution is the sole factor in designing Nature. They miss the deep inner meaning of these, which will be described later while describing the structure of the Universe.\n\nThe opposing view reflects the ideas of Kant regarding the innate categories of thought whereby all our experience is ordered by our minds. Kant pointed out the difference between the internal mental models we build of the external world and the real objects that we know through our sense organs. The views of Kant have many similarities with that of Bohr. The Consciousness of Kant is described as intelligence by Bohr. The sense organs of Kant are described as measuring devices by Bohr. Kant’s mental models are Bohr’s quantum mechanical models. This view of mathematics stresses more on “mathematical modeling” than mathematical rules or axioms. In this view, the so-called constants of Nature that arise as theoretically determined constants of proportionality in our mathematical equations, are solely artifacts of the particular mathematical representation we have chosen to use for explaining different natural phenomena. For example, we use G as the Gravitational constant because of our inclination to express the gravitational interaction in a particular way. This view is misleading as the large number of the so-called constants of Nature points to some underlying reality behind it. We will discuss this point later.\n\nThe debate over the definition of “physical reality” led to the notion that it should be external to the observer – an observer-independent objective reality. The statistical formulation of the laws of atomic and sub-atomic physics has added a new dimension to the problem. In quantum mechanics, the experimental arrangements are treated in classical terms, whereas the observed objects are treated in probabilistic terms. In this way, the measuring apparatus and the observer are effectively joined into one complex system which has no distinct, well defined parts, and the measuring apparatus does not have to be described as an isolated physical entity.\n\nAs Max Tegmark in his External Reality Hypothesis puts it: If we assume that reality exists independently of humans, then for a description to be complete, it must also be well-defined according to non-human entities that lack any understanding of human concepts like “particle”, “observation”, etc. A description of objects in this external reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings what-so-ever. To understand the concept, you have to distinguish between two ways of viewing reality. The first is from outside, like the overview of a physicist studying its mathematical structure – a bird’s eye view. The second way is the inside view of an observer living in the structure – the view of a frog in the well.\n\nThough Tegmark’s view is nearer the truth (it will be discussed later), it has been contested by others on the ground of contradicting logical consistency. Tegmark relies on a quote of David Hilbert: “Mathematical existence is merely freedom from contradiction”. This implies that mathematical structures simply do not exist unless they are logically consistent. They cite the Russell’s paradox (discussed in detail in later pages) and other paradoxes - such as the Zermelo-Frankel set theory that avoids the Russell’s paradox - to point out that mathematics on its own does not lead to a sensible universe. We seem to need to apply constraints in order to obtain consistent physical reality from mathematics. Unrestricted axioms lead to Russell’s paradox.\n\nConventional bivalent logic is assumed to be based on the principle that every proposition takes exactly one of two truth values: “true” or “false”. This is a wrong conclusion based on European tradition as in the ancient times students were advised to: observe, listen (to teachings of others), analyze and test with practical experiments before accepting anything as true. Till it is conclusively proved or disproved, it was “undecided”. The so-called discovery of multi-valued logic is nothing new. If we extend the modern logic then why stop at ternary truth values: it could be four or more-valued logic. But then what are they? We will discuss later.\n\nThough Euclid with his Axioms appears to be a Formalist, his Axioms were abstracted from the real physical world. But the focus of attention of modern Formalists is upon the relations between entities and the rules governing them, rather than the question of whether the objects being manipulated have any intrinsic meaning. The connection between the Natural world and the structure of mathematics is totally irrelevant to them. Thus, when they thought that the Euclidean geometry is not applicable to curved surfaces, they had no hesitation in accepting the view that the sum of the three angles of a triangle need not be equal to 1800. It could be more or less depending upon the curvature. This is a wholly misguided view. The lines or the sides drawn on a curved surface are not straight lines. Hence the Axioms of Euclid are not violated, but are wrongly applied. Riemannian geometry, which led to the chain of non-Euclidean geometry, was developed out of his interest in trying to solve the problems of distortion of metal sheets when they were heated. Einstein used this idea to suggest curvature of space-time without precisely defining space or time or spece-time. But such curvature is a temporary phenomenon due to the application of heat energy. The moment the external heat energy is removed, the metal plate is restored to its original position and Euclidean geometry is applicable. If gravity changes the curvature of space, then it should be like the external energy that distorts the metal plate. Then who applies gravity to mass or what is the mechanism by which gravity is applied to mass. If no external agency is needed and it acts perpetually, then all mass should be changing perpetually, which is contrary to observation. This has been discussed elaborately in latter pages.\n\nOnce the notion of the minimum distance scale was firmly established, questions were raised about infinity and irrational numbers. Feynman raised doubts about the relevance of infinitely small scales as follows: “It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space”. Paul Davies asserted: “the use of differential equations assumes the continuity of space-time on arbitrarily small scales.\n\nThe frequent appearance of π implies that their numerical values may be computed to arbitrary precision by an infinite sequence of operations. Many physicists tacitly accept these mathematical idealizations and treat the laws of physics as implementable in some abstract and perfect Platonic realm. Another school of thought, represented most notably by Wheeler and Landauer, stresses that real calculations involve physical objects, such as computers, and take place in the real physical universe, with its specific available resources. In short, information is physical. That being so, it follows that there will be fundamental physical limitations to what may be calculated in the real world”. Thus, Intuitionism or Constructivism divides mathematical structures into “physically relevant” and “physically irrelevant”. It says that mathematics should only include statements which can be deduced by a finite sequence of step-by-step constructions starting from the natural numbers. Thus, according to this view, infinity and irrational numbers cannot be part of mathematics.\n\nInfinity is qualitatively different from even the largest number. Finite numbers, however large, obey the laws of arithmetic. We can add, multiply and divide them, and put different numbers unambiguously in order of size. But infinity is the same as a part of itself, and the mathematics of other numbers is not applicable to it. Often the term “Hilbert’s hotel” is used as a metaphor to describe infinity. Suppose a hotel is full and each guest wants to bring a colleague who would need another room. This would be a nightmare for the management, who could not double the size of the hotel instantly. In an infinite hotel, though, there is no problem. The guest from room 1 goes into room 2, the guest in room 2 into room 4, and so on. All the odd-numbered rooms are then free for new guests. This is a wrong analogy. The numbers are divided into two categories based on whether there is similar perception or not. If after the perception of one object there is further similar perception, they are many, which can range from 2,3,4,…..n depending upon the sequence of perceptions? If there is no similar perception after the perception of one object, then it is one. In the case of Infinity, neither of the above conditions applies. However, Infinity is more like the number ‘one’ – without a similar – except for one characteristic. While one object has a finite dimension, infinity has infinite dimensions. The perception of higher numbers is generated by repetition of ‘one’ that many number of times, but the perception of infinity is ever incomplete.\n\nSince interaction requires a perceptible change anywhere in the system under examination or measurement, normal interactions are not applicable in the case of infinity. For example, space and time in their absolute terms are infinite. Space and time cannot be measured, as they are not directly perceptible through our sense organs, but are deemed to be perceived. Actually what we measure as space is the interval between objects or points on objects. These intervals are mental constructs and have no physical existence other than the objects, which are used to describe space through alternative symbolism. Similarly, what we measure as time is the interval between events. Space and time do not and cannot interact with each other or with other objects or events as no mathematics is possible between infinities. Our measurements of an arbitrary segment of space or time (which are really the intervals) do not affect space or time in any way. We have explained the quantum phenomena with real numbers derived from fundamental principles and correlated them to the macro world. The quantities like π and φ etc have other significances, which will be discussed later.\n\nThe fundamental “stuff” of the Universe is the same and the differences arise only due to the manner of their accumulation and reduction – magnitude and sequential arrangement. Since number is a property of all particles, physical phenomena have some associated mathematical basis. However, the perceptible structures and processes of the physical world are not the same as their mathematical formulations, many of which are neither perceptible nor feasible. Thus the relationship between physics and mathematics is that of the map and the territory. Map facilitates study of territory, but it does not tell all about territory. Knowing all about the territory from the map is impossible. This creates the difficulty. Science is increasingly becoming less objective. The scientists are presenting data as if it is absolute truth merely liberated by their able hands for the benefit of lesser mortals. Thus, it has to be presented to the lesser mortals in a language that they do not understand – thus do not question. This leads to misinterpretations to the extent that some classic experiments become dogma even when they are fatally flawed. One example is the Olber’s paradox.\n\nIn order to understand our environment and interact effectively with it, we engage in the activities of counting the total effect of each of the systems. Such counting is called mathematics. It covers all aspects of life. We are central to everything in a mathematical way. As Barrow points out; “While Copernicus’s idea that our position in the universe should not be special in every sense is sound, it is not true that it cannot be special in any sense”. If we consider our positioning as opposed to our position in the Universe, we will find our special place. For example, if we plot a graph with mass of the star relative to the Sun (with Sun at 1) and radius of orbit relative to Earth (with Earth at 1) and consider scale of the planets, its distance from the Sun, its surface conditions, the positioning of the neighboring planets etc; and consider these variables in a mathematical space, we will find that the Earth’s positioning is very special indeed. It is in a narrow band called the Habitable zone (For details, please refer to Wikipedia on planetary habitability hypothesis).\n\nIf we imagine the complex structure of the Mandelbrot Set as representative of the Universe (since it is self similar), then we could say that we are right in the border region of the fractal structure. If we consider the relationship between different dimensions of space or a (bubble), then we find their exponential nature. If we consider the center of the bubble as 0 and the edge as 1 and map it in a logarithmic scale, we will find an interesting zone at 0.5. Starting for the Galaxy, to the Sun to Earth to the atoms, everything comes in this zone. For example, we can consider the galactic core as the equivalent of the S orbital of the atom, the bars as equivalent of the P orbital, the spiral arms as equivalent of the D orbital and apply the logarithmic scale, we will find the Sun at 0.5 position. The same is true for Earth. It is known that both fusion and fission push atoms towards iron. The element finds itself in the middle group of the middle period of the periodic table; again 0.5. Thus, there can be no doubt that Nature is mathematical. But the structures and the processes of the world are not the same as mathematical formulations. The map is not the territory. Hence there are various ways of representing Nature. Mathematics is one of them. However, only mathematics cannot describe Nature in any meaningful way.\n\nEven the modern mathematician and physicists do not agree on many concepts. Mathematicians insist that zero has existence, but no dimension, whereas the physicists insist that since the minimum possible length is the Planck scale; the concept of zero has vanished! The Lie algebra corresponding to SU (n) is a real and not a complex Lie algebra. The physicists introduce the imaginary unit i, to make it complex. This is different from the convention of the mathematicians. Mathematicians treat any operation involving infinity is void as it does not change by addition or subtraction of or multiplication or division by any number. History of development of science shows that whenever infinity appears in an equation, it points to some novel phenomenon or some missing parameters. Yet, physicists use renormalization by manipulation to generate another infinity in the other side of the equation and then cancel both! Certainly it is not mathematics!\n\nOften the physicists apply the “brute force approach”, in which many parameters are arbitrarily reduced to zero or unity to get the desired result. One example is the mathematics for solving the equations for the libration points. But such arbitrary reduction changes the nature of the system under examination (The modern values are slightly different from our computation). This aspect is overlooked by the physicists. We can cite many such instances, where the conventions of mathematicians are different from those of physicists. The famous Cambridge coconut puzzle is a clear representation of the differences between physics and mathematics. Yet, the physicists insist that unless a theory is presented in a mathematical form, they will not even look at it. We do not accept that the laws of physics break down at singularity. At singularity only the rules of the game change and the mathematics of infinities takes over.\n\nModern scientists claim to depend solely on mathematics. But most of what is called as “mathematics” in modern science fails the test of logical consistency that is a corner stone for judging the truth content of a mathematical statement. For example, mathematics for a multi-body system like a lithium or higher atom is done by treating the atom as a number of two body systems. Similarly, the Schrödinger equation in so-called one dimension (it is a second order equation as it contains a term x2, which is in two dimensions and mathematically implies area) is converted to three dimensional by addition of two similar factors for y and z axis. Three dimensions mathematically imply volume. Addition of three areas does not generate volume and x2+y2+z2 ≠ (x.y.z). Similarly, mathematically all operations involving infinity is void. Hence renormalization is not mathematical. Thus, the so called mathematics of modern physicists is not mathematical at all!\n\nIn fact, some recent studies appear to hint that perception is mathematically impossible. Imagine a black-and-white line drawing of a cube on a sheet of paper. Although this drawing looks to us like a picture of a cube, there are actually infinite numbers of other three-dimensional objects that could have produced the same set of lines when collapsed on the page. But we don’t notice any of these alternatives. The reason for the same is that, our visual systems have more to go on than just bare perceptual input. They are said to use heuristics and short cuts, based on the physics and statistics of the natural world, to make the “best guesses” about the nature of reality. Just as we interpret a two-dimensional drawing as representing a three-dimensional object, we interpret the two-dimensional visual input of a real scene as indicating a three-dimensional world. Our perceptual system makes this inference automatically, using educated guesses to fill in the gaps and make perception possible. Our brains use the same intelligent guessing process to reconstruct the past and help in perceiving the world.\n\nMemory functions differently than a video-recording with a moment-by-moment sensory image. In fact, it’s more like a puzzle: we piece together our memories, based on both what we actually remember and what seems most likely given our knowledge of the world. Just as we make educated guesses – inferences - in perception, our minds’ best inferences help “fill in the gaps” of memory, reconstructing the most plausible picture of what happened in our past. The most striking demonstration of the minds’ guessing game occurs when we find ways to fool the system into guessing wrong. When we trick the visual system, we see a “visual illusion” - a static image might appear as if it’s moving, or a concave surface will look convex. When we fool the memory system, we form a false memory - a phenomenon made famous by researcher Elizabeth Loftus, who showed that it is relatively easy to make people remember events that never occurred. As long as the falsely remembered event could plausibly have occurred, all it takes is a bit of suggestion or even exposure to a related idea to create a false memory.\n\nEarlier, visual illusions and false memories were studied separately. After all, they seem qualitatively different: visual illusions are immediate, whereas false memories seemed to develop over an extended period of time. A recent study blurs the line between these two phenomena. The study reveals an example of false memory occurring within 42 milliseconds - about half the amount of time it takes to blink your eye. It relied upon a phenomenon known as “boundary extension”, an example of false memory found when recalling pictures. When we see a picture of a location - say, a yard with a garbage can in front of a fence - we tend to remember the scene as though more of the fence were visible surrounding the garbage can. In other words, we extend the boundaries of the image, believing that we saw more fence than was actually present. This phenomenon is usually interpreted as a constructive memory error - our memory system extrapolates the view of the scene to a wider angle than was actually present. The new study, published in the November 2008 issue of the journal Psychological Science, asked how quickly this boundary extension happens.\n\nThe researchers showed subjects a picture, erased it for a very short period of time by overlaying a new image, and then showed a new picture that was either the same as the first image or a slightly zoomed-out view of the same place. They found that when people saw the exact same picture again, they thought the second picture was more zoomed-in than the first one they had seen. When they saw a slightly zoomed-out version of the picture they had seen before, however, they thought this picture matched the first one. This experience is the classic boundary extension effect. However, the gap between the first and second picture was less than 1/20th of a second. In less than the blink of an eye, people remembered a systematically modified version of pictures they had seen. This modification is, by far, the fastest false memory ever found.\n\nAlthough it is still possible that boundary extension is purely a result of our memory system, the incredible speed of this phenomenon suggests a more parsimonious explanation: that boundary extension may in part be caused by the guesses of our visual system itself. The new dataset thus blurs the boundaries between the initial representation of a picture (via the visual system) and the storage of that picture in memory. This raises the question: is boundary extension a visual illusion or a false memory? Perhaps these two phenomena are not as different as previously thought. False memories and visual illusions both occur quickly and easily, and both seem to rely on the same cognitive mechanism: the fundamental property of perception and memory to fill in gaps with educated guesses, information that seems most plausible given the context. The work adds to a growing movement that suggests that memory and perception may be simply two sides of the same coin. This, in turn, implies that mathematics, which is based on perception of numbers and other visual imagery, could be misleading for developing theories of physics.\nThe essence of creation is accumulation and reduction of the number of particles in each system in various combinations. Thus, Nature has to be mathematical. But then physics should obey the laws of mathematics, just as mathematics should comply with the laws of physics. We have shown elsewhere that all of mathematics cannot be physics. We may have a mathematical equation without a corresponding physical explanation. Accumulation or reduction can be linear or non-linear. If they are linear, the mathematics is addition and subtraction. If they are non-linear, the mathematics is multiplication and division. Yet, this principle is violated in a large number of equations. For an example, the Schrödinger’s equation in one dimension has been discussed earlier. Then there are unphysical combinations. For example, certain combinations of protons and neutrons are prohibited physically, though there is no restriction on devising one such mathematical formula. There is no equation for the observer. Thus, sole dependence on mathematics for discussing physics is neither desirable nor warranted.\n\nWe accept “proof” – mathematical or otherwise - to validate the reality of any physical phenomena. We depend on proof to validate a theory as long as it corresponds to reality. The modern system of proof takes five stages: observation/experiment, developing hypothesis, testing the hypothesis, acceptance or rejection or modification of hypothesis based on the additional information and lastly, reconstruction of the hypothesis if it was not accepted. We also adopt a five stage approach to proof. First we observe/experiment and hypothesize. Then we look for corroborative evidence. In the third stage we try to prove that the opposite of the hypothesis is wrong. In the fourth stage we try to prove whether the hypothesis is universally valid or has any limitations. In the last stage we try to prove that any theory other than this is wrong.\n\nMathematics is one of the tools of “proof” because of its logical constancy. It is a universal law that the tools are selected based on the nature of operations and not vice-versa. The tools can only restrict the choice of operations. Hence mathematics by itself does not provide proof, but the proof may use mathematics as a tool. We also depend on symmetry, as it is a fundamental property of Nature. In our theory, different infinities co-exist and do not interact with each other. Thus, we agree that the evolutionary process of the Universe could be explained mathematically, as basically it is a process of non-linear accumulation and corresponding reduction of particles and energies in different combinations. But we differ on the interpretation of the equation. For us, the left hand side of the equation represents the cause and the right hand side the effect, which is reversible only in the same order. If the magnitudes of the parameters of one side are changed, the effect on the other side also correspondingly changes. But such changes must be according to natural laws and not arbitrary changes. For example, we agree that e/m = c2 or m/e = 1/c2, which we derive from fundamental principles. But we do not agree that e = mc2. This is because we treat mass and energy as inseparable conjugates with variable magnitude and not interchangeable, as each has characteristics not found in the other. Thus, they are not fit to be used in an equation as cause and effect. Simultaneously, we agree with c2 as energy flow is perceived in fields, which are represented by second order quantities.\n\nIf we accept the equation e = mc2, according to modern principles, it will lead to m = e/c2. In that case, we will land in many self contradicting situations. For example, if photon has zero rest mass, then m0 = 0/c2 (at rest, external energy that moves a particle has to be zero. Internal energy is not relevant, as a stable system has zero net energy). This implies that m0c2 = 0, or e = 0, which makes c2 = 0/0, which is meaningless. But if we accept e/m = c2 and both sides of the equation as cause and effect, then there is no such contradiction. As we have proved in our book “Vaidic Theory of Numbers”, all operations involving zero except multiplication are meaningless. Hence if either e or m becomes zero, the equation becomes meaningless and in all other cases, it matches the modern values. Here we may point out that the statement that the rest mass of matter is determined by its total energy content is not susceptible of a simple test since there is no independent measure of the later quantity. This proves our view that mass and energy are inseparable conjugates.\n\nThe domain that astronomers call “the universe” - the space, extending more than 10 billion light years around us and containing billions of galaxies, each with billions of stars, billions of planets (and maybe billions of biospheres) - could be an infinitesimal part of the totality. There is a definite horizon to direct observations: a spherical shell around us, such that no light from beyond it has had time to reach us since the big bang. However, there is nothing physical about this horizon. If we were in the middle of an ocean, it is conceivable that the water ends just beyond your horizon - except that we know it doesn’t. Likewise, there are reasons to suspect that our universe - the aftermath of our big bang - extends hugely further than we can see.\n\nAn idea called eternal inflation suggested by some cosmologists envisages big bangs popping off, endlessly, in an ever-expanding substratum. Or there could be other space-times alongside ours - all embedded in a higher-dimensional space. Ours could be but one universe in a multiverse. Other branches of mathematics then may become relevant. This has encouraged the use of exotic mathematics such as the transfinite numbers. It may require a rigorous language to describe the number of possible states that a universe could possess and to compare the probability of different configurations. It may just be too hard for human brains to grasp. A fish may be barely aware of the medium in which it lives and swims; certainly it has no intellectual powers to comprehend that water consists of interlinked atoms of hydrogen and oxygen. The microstructure of empty space could, likewise, be far too complex for unaided human brains to grasp. Can we guarantee that with the present mathematics we can overcome all obstacles and explain all complexities of Nature? Should we not resort to the so-called exotic mathematics? But let us see where it lands us.\n\nThe manipulative mathematical nature of the descriptions of quantum physics has created difficulties in its interpretation. For example, the mathematical formalism used to describe the time evolution of a non-relativistic system proposes two somewhat different kinds of transformations:\n\n· Reversible transformations described by unitary operators on the state space. These transformations are determined by solutions to the Schrödinger equation.\n· Non-reversible and unpredictable transformations described by mathematically more complicated transformations. Examples of these transformations are those that are undergone by a system as a result of measurement.\n\nThe truth content of a mathematical statement is judged from its logical consistency. We agree that mathematics is a way of representing and explaining the Universe in a symbolic way because evolution is logically consistent. This is because everything is made up of the same “stuff”. Only the quantities (number or magnitude) and their ordered placement or configuration create the variation. Since numbers are a property by which we differentiate between similar objects and all natural phenomena are essentially accumulation and reduction of the fundamental “stuff” in different permissible combinations, physics has to be mathematical. But then mathematics must conform to Natural laws: not un-physical manipulations or the brute force approach of arbitrarily reducing some parameters to zero to get a result that goes in the name of mathematics. We suspect that the over-dependence on mathematics is not due to the fact that it is unexceptionable, but due to some other reason described below.\n\nIn his book “The Myth of the Framework”, Karl R Popper, acknowledged as the major influence in modern philosophy and political thought, has said: “Many years ago, I used to warn my students against the wide-spread idea that one goes to college in order to learn how to talk and write “impressively” and incomprehensibly. At that time many students came to college with this ridiculous aim in mind, especially in Germany …………. They unconsciously learn that highly obscure and difficult language is the intellectual value par excellence……………Thus arose the cult of incomprehensibility, of “impressive” and high sounding language. This was intensified by the impenetrable and impressive formalism of mathematics…………….” It is unfortunate that even now many Professors, not to speak of their students, are still devotees of the above cult.\n\nThe modern Scientists justify the cult of incomprehensibility in the garb of research methodology - how “big science” is really done. “Big science” presents a big opportunity for methodologists. With their constant meetings and exchanges of e-mail, collaboration scientists routinely put their reasoning on public display (not the general public, but only those who subscribe to similar views), long before they write up their results for publication in a journal. In reality, it is done to test the reaction of others as often bitter debate takes place on such ideas. Further, when particle physicists try to find a particular set of events among the trillions of collisions that occur in a particle accelerator, they focus their search by ignoring data outside a certain range. Clearly, there is a danger in admitting a non-conformist to such raw material, since a lack of acceptance of their reasoning and conventions can easily lead to very different conclusions, which may contradict their theories. Thus, they offer their own theory of “error-statistical evidence” such as in the statement, “The distinction between the epistemic and causal relevance of epistemic states of experimenters may also help to clarify the debate over the meaning of the likelihood principle”. Frequently they refer to ceteris paribus (other things being equal), without specifying which other things are equal (and then face a challenge to justify their statement).\n\nThe cult of incomprehensibility has been used even the most famous scientists with devastating effect. Even the obvious mistakes in their papers have been blindly accepted by the scientific community and remained un-noticed for hundreds of years. Here we quote from an article written by W.H. Furry of Department of Physics, Harvard University, published in March 1, 1936 issue of Physical Review, Volume 49. The paper “Note on the Quantum-Mechanical Theory of Measurement” was written in response to the famous EPR Argument and its counter by Bohr. The quote relates to the differentiation between “pure state” and “mixture state”.\n\n“2. POSSIBLE TYPES OF STATISTICAL INFORMATION ABOUT A SYSTEM.\n\nOur statistical information about a system may always be expressed by giving the expectation values of all observables. Now the expectation value of an arbitrary observable F, for a state whose wave function is φ, is\nIf we do not know the state of the system, but know that wi\n\nare the respective probabilities of its being in states whose wave functions are φi, then we must assign as the expectation value of F the weighted average of its expectation values for the states φi. Thus,\nThis formula for is the appropriate one when our system is one of an ensemble of systems of which numbers proportional to wi are in the states φi. It must not be confused with any such formula as\nwhich corresponds to the system’s having a wave function which is a linear combination of the φi. This last formula is of the type of (1), while (2) is an altogether different type.\n\nAn alternative way of expressing our statistical information is to give the probability that measurement of an arbitrary observable F will give as result an arbitrary one of its eigenvalues, say δ. When the system is in the state φ, this probability is\nwhere xδ is the eigenfunction of F corresponding to the eigenvalues δ. When we know only that wi are the probabilities of the system’s being in the states φi, the probability in question is\nFormula (2’) is not the same as any special case of (1’) such as\nIt differs generically from (1’) as (2) does from (1).\nWhen such equations as (1), (1’) hold, we say that the system is in the “pure state” whose wave function is φ. The situation represented by Eqs. (2), (2’) is called a “mixture” of the states φi with the weights wi. It can be shown that the most general type of statistical information about a system is represented by a mixture. A pure state is a special case, with only one non-vanishing wi. The term mixture is usually reserved for cases in which there is more than one non-vanishing wi.It must again be emphasized that a mixture in this sense is essentially different from any pure state whatever.”\n\nNow we quote from a recent Quantum Reality Web site the same description of “pure state” and “quantum state”:\n\n“The statistical properties of both systems before measurement, however, could be described by a density matrix. So for an ensemble system such as this the density matrix is a better representation of the state of the system than the vector.\n\nSo how do we calculate the density matrix? The density matrix is defined as the weighted sum of the tensor products over all the different states:\n\nWhere p and q refer to the relative probability of each state. For the example of particles in a box, p would represent the number of particles in state│ψ>, and q would represent the number of particles in state │φ>.\n\nLet’s imagine we have a number of qubits in a box (these can take the value │0> or│1>.\n\nLet’s say all the qubits are in the following superposition state: 0.6│0> +0.8i│1>.\nIn other words, the ensemble system is in a pure state, with all of the particles in an identical quantum superposition of states │0> and│1>. As we are dealing with a single, pure state, the construction of the density matrix is particularly simple: we have a single probability p, which is equal to 1.0 (certainty), while q (and all the other probabilities) are equal to zero. The density matrix then simplifies to: │ψ><ψ│\n\nThis state can be written as a column (“ket”) vector. Note the imaginary component (the expansion coefficients are in general complex numbers):\nIn order to generate the density matrix we need to use the Hermitian conjugate (or adjoint) of this column vector (the transpose of the complex conjugate│ψ>. So in this case the adjoint is the following row (“bra”) vector:\nWhat does this density matrix tell us about the statistical properties of our pure state ensemble quantum system? For a start, the diagonal elements tell us the probabilities of finding the particle in the│0> or│1> eigenstate. For example, the 0.36 component informs us that there will be a 36% probability of the particle being found in the │0> state after measurement. Of course, that leaves a 64% chance that the particle will be found in the │1> state (the 0.64% component).\nThe way the density matrix is calculated, the diagonal elements can never have imaginary components (this is similar to the way the eigenvalues are always real). However, the off-diagonal terms can have imaginary components (as shown in the above example). These imaginary components have a associated phase (complex numbers can be written in polar form). It is the phase differences of these off-diagonal elements which produces interference (for more details, see the book Quantum Mechanics Demystified). The off-diagonal elements are characteristic of a pure state. A mixed state is a classical statistical mixture and therefore has no off-diagonal terms and no interference.\n\nSo how do the off-diagonal elements (and related interference effects) vanish during decoherence?\n\nThe off-diagonal (imaginary) terms have a completely unknown relative phase factor which must be averaged over during any calculation since it is different for each separate measurement (each particle in the ensemble). As the phase of these terms is not correlated (not coherent) the sums cancel out to zero. The matrix becomes diagonalised (all off-diagonal terms become zero. Interference effects vanish. The quantum state of the ensemble system is then apparently “forced” into one of the diagonal eigenstates (the overall state of the system becomes a mixture state) with the probability of a particular eigenstate selection predicted by the value of the corresponding diagonal element of the density matrix.\n\nConsider the following density matrix for a pure state ensemble in which the off-diagonal terms have a phase factor of θ:\n\nThe above statement can be written in a simplified manner as follows: Selection of a particular eigenstate is governed by a purely probabilistic process. This requires a large number of readings. For this purpose, we must consider an ensemble – a large number of quantum particles in a similar state and treat them as a single quantum system. Then we measure each particle to ascertain a particular value; say color. We tabulate the results in a statement called the density matrix. Before measurement, each of the particles is in the same state with the same state vector. In other words, they are all in the same superposition state. Hence this is called a pure state. After measurement, all particles are in different classical states – the state (color) of each particle is known. Hence it is called a mixed state.\n\nIn common-sense language, what it means is that: if we take a box of billiard balls of say 100 numbers of random colors - say blue and green, before counting balls of each color, we could not say what percentage of balls are blue and what percentage green. But after we count the balls of each color and tabulate the results, we know that (in the above example) 36% of the balls belong to one color and 64% belong to another color. If we have to describe the balls after counting, we will give the above percentage or say that 36 numbers of balls are blue and 64 numbers of balls are green. That will be a pure statement. But before such measurement, we can describe the balls as 100 balls of blue and green color. This will be a mixed state.\n\nAs can be seen, our common-sense description is opposite of the quantum mechanical classification, which are written by two scientists about 75 years apart and which is accepted by all scientists unquestioningly. Thus, it is no wonder that one scientist jokingly said that: “A good working definition of quantum mechanics is that things are the exact opposite of what you thought they were. Empty space is full, particles are waves, and cats can be both alive and dead at the same time.”\n\nWe quote another example from the famous EPR argument of Einstein and others (Phys. Rev. 47, 777 (1935): “To illustrate the ideas involved, let us consider the quantum-mechanical description of the behavior of a particle having a single degree of freedom. The fundamental concept of the theory is the concept of state, which is supposed to be completely characterized by the wave function ψ, which is a function of the variables chosen to describe the particle’s behavior. Corresponding to each physically observable quantity A there is an operator, which may be designated by the same letter.\n\nIf ψ is an eigenfunction of the operator A, that is, if ψ’ ≡ Aψ = aψ (1)\n\nwhere a is a number, then the physical quantity A has with certainty the value a whenever the particle is in the state given by ψ. In accordance with our criterion of reality, for a particle in the state given by ψ for which Eq. (1) holds, there is an element of physical reality corresponding to the physical quantity A”.\n\nWe can write the above statement and the concept behind it in various ways that will be far easier to understand by the common man. We can also give various examples to demonstrate the physical content of the above statement. However, such statements and examples will be difficult to twist and interpret differently when necessary. Putting the concept in an ambiguous format helps in its subsequent manipulation, as is explained below citing from the same example:\n\n“In accordance with quantum mechanics we can only say that the relative probability that a measurement of the coordinate will give a result lying between a and b is\nSince this probability is independent of a, but depends only upon the difference b - a, we see that all values of the coordinate are equally probable”.\n\nThe above conclusion has been arrived at based on the following logic: “More generally, it is shown in quantum mechanics that, if the operators corresponding to two physical quantities, say A and B, do not commute, that is, if AB ≠ BA, then the precise knowledge of one of them precludes such a knowledge of the other. Furthermore, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first”.\n\nThe above statement is highly misleading. The law of commutation is a special case of non-linear accumulation as explained below. All interactions involve application of force which leads to accumulation and corresponding reduction. Where such accumulation is between similars, it is linear accumulation and its mathematics is called addition. If such accumulation is not fully between similars, but partially similars (and partially dissimilar) it is non-linear accumulation and its mathematics is called multiplication. For example, 10 cars and another 10 cars are twenty cars through addition. But if there are 10 cars in a row and there are two rows of cars, then rows of cars is common to both statements, but one statement shows the number of cars in a row while the other shows the number of rows of cars. Because of this partial dissimilarity, the mathematics has to be multiplication of 10 x 2 or 2 x 10. We are free to use any of the two sequences and the result will be the same. This is the law of commutation. However, no multiplication is possible if the two factors are not partially similar. In such cases, the two factors are said to be non-commutable. If the two terms are mutually exclusive, i.e., one of the terms will always be zero, the result of their multiplication will always be zero. Hence they may be said to be not commutable though in reality they are commutable, but the result of their multiplication is always zero. This implies that the knowledge of one precludes the knowledge of the other. The commutability or otherwise depend on the nature of the quantities – whether they are partially related and partially non-related to each other or not.\n\nPosition is a fixed co-ordinate in a specific frame of reference. Momentum is a mobile co-ordinate in the same frame of reference. Both fixedity and mobility are mutually exclusive. If a particle has a fixed position, its momentum is zero. If it has momentum, it does not have a fixed position. Since “particle” is similar in both the above statements, i.e., since both are related to the particle, they can be multiplied, hence commutable. But since one or the other factors is always zero, the result will always be zero and the equation AB ≠ BA does not hold. In other words, while uncertainty is established due to other reasons, the equation Δx. Δp ≥ h is a mathematically wrong statement, as mathematically the answer will always be zero. The validity of a physical statement is judged by its correspondence to reality or as Einstein and others put it: “by the degree of agreement between the conclusions of the theory and human experience”. Since in this case the degree of agreement between the conclusions of the theory and human experience is zero, it cannot be a valid physical statement either. Hence, it is no wonder that the Heisenberg’s Uncertainty relation is still a hypothesis and not proven. In latter pages we have discussed this issue elaborately.\n\nIn modern science there is a tendency of generalization or extension of one principle to others. For example; the Schrödinger equation in the so-called one dimension (actually it contains a second order term; hence cannot be an equation in one dimension) is generalized (?) to three dimensions by adding two more terms for y and z dimensions (mathematically and physically it is a wrong procedure). We have discussed it in latter pages. While position and momentum are specific quantities, the generalizations are done by replacing these quantities with A and B. When a particular statement is changed to a general statement by following algebraic principles, the relationship between the quantities of the particular statement is not changed. However, physicists often bypass or over-look this mathematical rule. A and B could be any set of two quantities. Since they are not specified, it is easy to use them in any way one wants. Even if the two quantities are commutable, since they are not precisely described, it gives one the freedom to manipulate by claiming that they are not commutable and vice-versa. Modern science is full of such manipulations.\n\nHere we give another example to prove that physics and modern mathematics are not always compatible. Bell’s Inequality is one of the important equations used by all quantum physicists. We will discuss it repeatedly for different purposes. Briefly the theory holds that if a system consists of an ensemble of particles having three Boolean properties A, B and C, and there is a reciprocal relationship between the values of measurement of A on two particles, the same type relationship exists between the particles with respect to the quantity B, the value of one particle measured and found to be a, and the value of another particle measured and found to be b, then the first particle must have started with state (A = a, B = b). In that event, the Theorem says that P (A, C) ≤ P (A, B) + P (B, C). In the case of classical particles, the theorem appears to be correct.\n\nQuantum mechanically: P(A, C) = ½ sin2 (θ), where θ is the angle between the analyzers. Let an apparatus emit entangled photons that pass through separate polarization analysers. Let A, B and C be the events that a single photon will pass through analyzers with axis set at 00, 22.50, and 450 to vertical respectively. It can be proved that C → C.\n\nThus, according to Bell’s theorem: P(A, C) ≤ P(A, B) + P(B, C),\nOr ½ sin2 (450) ≤ ½ sin2 (22.50) + ½ sin2 (22.50),\nOr 0.25 ≤0.1464, which is clearly absurd.\n\nThis inequality has been used by quantum physicists to prove entanglement and distinguish quantum phenomena from classical phenomena. We will discuss it in detail to show that the above interpretation is wrong and the same set of mathematics is applicable to both macro and the micro world. The real reason for such deviation from common sense is that because of the nature of measurement, measuring one quantity affects the measurement of another. The order of measurement becomes important in such cases. Even in the macro world, the order of measurement leads to different results. However, the real implication of Bell’s original mathematics is much deeper and points to one underlying truth that will be discussed later.\n\nA wave function is said to describe all possible states in which a particle may be found. To describe probability, some people give the example of a large, irregular thundercloud that fills up the sky. The darker the thundercloud, the greater the concentration of water vapor and dust at that point. Thus by simply looking at a thundercloud, we can rapidly estimate the probability of finding large concentrations of water and dust in certain parts of the sky. The thundercloud may be compared to a single electron's wave function. Like a thundercloud, it fills up all space. Likewise, the greater its value at a point, the greater the probability of finding the electron there! Similarly, wave functions can be associated with large objects, like people. As one sits in his chair, he has a Schrödinger probability wave function. If we could somehow see his wave function, it would resemble a cloud very much in the shape of his body. However, some of the cloud would spread out all over space, out to Mars and even beyond the solar system, although it would be vanishingly small there. This means that there is a very large likelihood that his, in fact, sitting here in his chair and not on the planet Mars. Although part of his wave function has spread even beyond the Milky Way galaxy, there is only an infinitesimal chance that he is sitting in another galaxy. This description is highly misleading.\n\nThe mathematics for the above assumption is funny. Suppose we choose a fixed point A and walked in the north-eastern direction by 5 steps. We mark that point as B. There are an infinite number of ways of reaching the point B from A. For example, we can walk 4 steps to the north of A and then walk 3 steps to the east. We will reach at B. Similarly, we can walk 6 steps in the northern direction, 3 steps in the eastern direction and 2 steps in the Southern direction. We will reach at B. Alternatively; we can walk 8 steps in the northern direction, 6 steps in the eastern direction and 5 steps in the South-eastern direction. We will reach at B. It is presumed that since the vector addition or “superposition” of these paths, which are different sorts from the straight path, lead to the same point, the point B could be thought of as a superposition of paths of different sort from A. Since we are free to choose any of these paths, at any instant, we could be “here” or “there”. This description is highly misleading.\n\nTo put the above statement mathematically, we take a vector V which can be resolved into two vectors V1 and V2 along the directions 1 and 2, we can write: V = V1 + V2. If a unit of displacement along the direction 1 is represented by 1, then V1 = V11, wherein V1 denotes the magnitude of the displacement V1. Similarly, V2 = V22. Therefore:\n\nV = V1 + V2 = V11 + V22. [1 and 2 are also denoted as (1,0) and (0,1) respectively].\n\nThis equation is also written as: V = λ1 + λ2, where λ is treated as the magnitude of the displacement. Here V is treated as a superposition of any standard vectors (1,0) and (0,1) with coefficients given by the numbers (ordered pair) (V1 , V2). This is the concept of a vector space. Here the vector has been represented in two dimensions. For three dimensions, this equation is written as V = λ1 + λ2 + λ3. For an n-tuple in n dimensions, the equation is written as V = λ1 + λ2 + λ3 +…… λn.\n\nIt is said that the choice of dimensions appropriate to a quantum mechanical problem depends on the number of independent possibilities the system possesses. In the case of polarization of light, there are only two possibilities. The same is true for electrons. But in the case of electrons, it is not dimensions, but spin. If we choose a direction and look at the electron’s spin in relation to that direction, then either its axis of rotation points along that direction or it is wholly in the reverse direction. Thus, electron spin is described as “up” and “down”. Scientists describe the spin of electron as something like that of a top, but different from it. In reality, it is something like the nodes of the Moon. At one node, Moon appear to be always going in the northern direction and at the other node, it always appears to be going in the southern direction. It is said that the value of “up” and “down” for an electron spin is always valid irrespective of the directions we may choose. There is no contradiction here, as direction is not important in the case of nodes. It is only the lay out of the two intersecting planes that is relevant. In many problems, the number of possibilities is said to be unbounded. Thus, scientists use infinite dimensional spaces to represent them. For this they use something called the Hilbert space. We will discuss about these later.\n\nAny intelligent reader would have seen through the fallacy of the vector space. Still we are describing it again. Firstly, as we have described in the wave phenomena in later pages, superposition is a merger of two waves, which lose their own identity to create something different. What we see is the net effect, which is different from the individual effects. There are many ways in which it could occur at one point. But all waves do not stay in superposition. Similarly, the superposition is momentary, as the waves submit themselves to the local dynamics. Thus, only because there is a probability of two waves joining to cancel the effect of each other and merge to give a different picture, we cannot formulate a general principle such as the equation: V = λ1 + λ2 to cover all cases, because the resultant wave or flat surface is also transitory.\n\nSecondly, the generalization of the equation V = λ1 + λ2 to V = λ1 + λ2 + λ3 +…… λn is mathematically wrong as explained below. Even though initially we mentioned 1 and 2 as directions, they are essentially dimensions, because they are perpendicular to each other. Direction is the information contained in the relative position of one point with respect to another point without the distance information. Directions may be either relative to some indicated reference (the violins in a full orchestra are typically seated to the left of the conductor), or absolute according to some previously agreed upon frame of reference (Kolkata lies due north-east of Puri). Direction is often indicated manually by an extended index finger or written as an arrow. On a vertically oriented sign representing a horizontal plane, such as a road sign, “forward” is usually indicated by an upward arrow. Mathematically, direction may be uniquely specified by a unit vector in a given basis, or equivalently by the angles made by the most direct path with respect to a specified set of axes. These angles can have any value and their inter-relationship can take an infinite number of values. But in the case of dimensions, they have to be at right angles to each other which remain invariant under mutual transformation.\n\nAccording to Vishwakaema the perception that arises from length is the same that arises from the perception of breadth and height – thus they belong to the same class, so that the shape of the particle remains invariant under directional transformations. There is no fixed rule as to which of the three spreads constitutes either length or breadth or height. They are exchangeable in re-arrangement. Hence, they are treated as belonging to one class. These three directions have to be mutually perpendicular on the consideration of equilibrium of forces (for example, electric field and the corresponding magnetic field) and symmetry. Thus, these three directions are equated with “forward-backward”, “right-left”, and “up-down”, which remain invariant under mutual exchange of position. Thus, dimension is defined as the spread of an object in mutually perpendicular directions, which remains invariant under directional transformations. This definition leads to only three spatial dimensions with ten variants. For this reason, the general equation in three dimensions uses x, y, and z (and/or c) co-ordinates or at least third order terms (such as a3+3a2b+3ab2+b3), which implies that with regard to any frame of reference, they are not arbitrary directions, but fixed frames at right angles to one another, making them dimensions. A one dimensional geometric shape is impossible. A point has imperceptible dimension, but not zero dimensions. The modern definition of a one dimensional sphere or “one sphere” is not in conformity with this view. It cannot be exhibited physically, as anything other than a point or a straight line has a minimum of two dimensions.\n\nWhile the mathematicians insist that a point has existence, but no dimensions, the Theoretical Physicists insist that the minimum perceptible dimension is the Planck length. Thus, they differ over the dimension of a point from the mathematicians. For a straight line, the modern mathematician uses the first order equation, ax + by + c = 0, which uses two co-ordinates, besides a constant. A second order equation always implies area in two dimensions. A three dimensional structure has volume, which can be expressed only by an equation of the third order. This is the reason why Born had to use the term “d3r” to describe the differential volume element in his equations.\n\nThe Schrödinger equation was devised to find the probability of finding the particle in the narrow region between x and x+dx, which is denoted by P(x) dx. The function P(x) is the probability distribution function or probability density, which is found from the wave function ψ(x) in the equation P(x) = [ψ(x)]2. The wave function is determined by solving the Schrödinger’s differential equation: d2ψ/dx2 + 8π2m/h2 [E-V(x)]ψ = 0, where E is the total energy of the system and V(x) is the potential energy of the system. By using a suitable energy operator term, the equation is written as Hψ = Eψ. The equation is also written as iħ ∂/∂tψ› = Hψ›, where the left hand side represents iħ times the rate of change with time of a state vector. The right hand side equates this with the effect of an operator, the Hamiltonian, which is the observable corresponding to the energy of the system under consideration. The symbol ψ indicates that it is a generalization of Schrödinger’s wave-function. The equation appears to be an equation in one dimension, but in reality it is a second order equation signifying a two dimensional field, as the original equation and the energy operator contain a term x2. A third order equation implies volume. Three areas cannot be added to create volume. Thus, the Schrödinger equation described above is an equation not in one, but in two dimensions. The method of the generalization of the said Schrödinger equation to the three spatial dimensions does not stand mathematical scrutiny.\n\nThree areas cannot be added to create volume. Any simple mathematical model will prove this. Hence, the Schrödinger equation could not be solved for other than hydrogen atoms. For many electron atoms, the so called solutions simply consider them as many one-electron atoms, ignoring the electrostatic energy of repulsion between the electrons and treating them as point charges frozen to some instantaneous position. Even then, the problem remains to be solved. The first ionization potential of helium is theorized to be 20.42 eV, against the experimental value of 24.58 eV. Further, the atomic spectra show that for every series of lines (Lyman, Balmer, etc) found for hydrogen, there is a corresponding series found at shorter wavelengths for helium, as predicted by theory. But in the spectrum of helium, there are two series of lines observed for every single series of lines observed for hydrogen. Not only does helium possess the normal Balmer series, but also it has a second “Balmer” series starting at λ = 3889 Å. This shows that, for the helium atom, the whole series repeats at shorter wavelengths.\n\nFor the lithium atom, it is even worse, as the total energy of repulsion between the electrons is more complex. Here, it is assumed that as in the case of hydrogen and helium, the most stable energy of lithium atom will be obtained when all three electrons are placed in the 1s atomic orbital giving the electronic configuration of 1s3, even though it is contradicted by experimental observation. Following the same basis as for helium, the first ionization potential of lithium is theorized to be 20.4 eV, against the experimental value of 202.5 eV to remove all three electrons and only 5.4 eV to remove one electron from lithium. Experimentally, it requires less energy to ionize lithium than it does to ionize hydrogen, but the theory predicts ionization energy one and half times larger. More serious than this is the fact that, the theory should never predict the system to be more stable than it actually is. The method should always predict energy less negative than is actually observed. If this is not found to be the case, then it means that an incorrect assumption has been made or that some physical principle has been ignored.\n\nFurther, it contradicts the principle of periodicity, as the calculation places each succeeding electron in the 1s orbital as it increases nuclear charge by unity. It must be remembered that, with every increase in n, all the preceding values of l are repeated, and a new l value is introduced. The reasons why more than two electrons could not be placed in the 1s orbit has not been explained. Thus, the mathematical formulations are contrary to the physical conditions based on observation. To overcome this problem, scientists take the help of operators. An operator is something which turns one vector into another. Often scientists describe robbery as an operator that transforms a state of wealth to a state of penury for the robbed and vice versa for the robber. Another example of an operator often given is the operation that rotates a frame clockwise or anticlockwise changing motion in northern direction to that in eastern or western directions. The act of passing light through a polarizer is called an operator as it changes the physical state of the photons polarization. Thus, the use of a polarizer is described as measurement of polarization, since the transmitted beam has to have its polarization in the direction perpendicular to it. We will come back to operators later.\n\nThe probability does not refer to (as is commonly believed) whether the particle will be observed at any specific position at a specific time or not. Similarly the description of different probability of finding the particle at any point of space is misleading. A particle will be observed only at a particular position at a particular time and no where else. Since a mobile particle does not have a fixed position, the probability actually refers to the state in which the particle is likely to be observed. This is because all the forces acting on it and their dynamics, which influence the state of the particle, may not be known to us. Hence we cannot predict with certainty whether the particle will be found here or elsewhere. After measurement, the particle is said to acquire a time invariant “fixed state” by “wave-function collapse”. This is referred to as the result of measurement, which is an arbitrarily frozen time invariant non-real (since in reality, it continues to change) state. This is because; the actual state with all influences on the particle has been measured at “here-now”, which is a perpetually changing state. Since all mechanical devices are subject to time variance in their operational capacities, they have to be “operated” by a “conscious agent” – directly or indirectly - because, as will be shown later, only consciousness is time invariant. This transition from a time variant initial state to a time invariant hypothetical “fixed state” through “now” or “here-now” is the dividing line between quantum physics and the classical physics, as well as conscious actions and mechanical actions. To prove the above statement, we have examined what is “information” in latter pages, because only conscious agents can cognize information and use it to achieve the desired objects. However, before that we will briefly discuss the chaos prevailing in this area among the scientists.\n\nModern science fails to answer the question “why” on many occasions. In fact it avoids such inconvenient questions. Here we may quote an interesting anecdote from the lives of two prominent persons. Once, Arthur Eddington was explaining the theory of the expanding universe to Bertrand Russell. Eddington told Russell that the expansion was so rapid and powerful that even a most powerful dictator would not be able to control the entire universe. He explained that even if the orders were sent with the speed of light, they would not reach the farthest parts of the universe. Bertrand Russell asked, “If that is so, how does God supervise what is going on in those parts?” Eddington looked keenly at Russell and replied, “That, dear Bertrand does not lie in the province of the physicists.” This begs the question: What is physics? We cannot take the stand that the role of physics is not to explain, but to describe reality. Description is also an explanation. Otherwise, why and to whom do you describe? If the validity of a physical statement is judged by its correspondence to reality, we cannot hide behind the veil of reductionism, but explain scientifically the theory behind the seemingly “acts of God”.\n\nThere is a general belief that we can understand all physical phenomenon if we can relate it to the interactions of atoms and molecules. After all, the Universe is made up of these particles only. Their interactions – in different combinations – create everything in the Universe. This is called a reductionist approach because it is claimed that everything else can be reduced to this supposedly more fundamental level. But this approach runs into problem with thermodynamics and its arrow of time. In the microscopic world, no such arrow of time is apparent, irrespective of whether it is being described by Newtonian mechanics, relativistic or quantum mechanics. One consequence of this description is that there can be no state of microscopic equilibrium. Time-symmetric laws do not single out a special end-state where all potential for change is reduced to zero, since all instants in time are treated as equivalent.\n\nThe apparent time reversibility of motion within the atomic and molecular regimes, in direct contradiction to the irreversibility of thermodynamic processes constitutes the celebrated irreversibility paradox put forward by in 1876 by Loschmidt among others (L. Boltzmann: Lectures on Gas Theory – University of California Press, 1964, page 9). The paradox suggests that the two great edifices – thermodynamics and mechanics – are at best incomplete. It represents a very clear problem in need of an explanation which should not be swept under carpet. As Lord Kelvin says: If the motion of every particle of matter in the Universe were precisely reversed at any instant, the course of Nature would be simply reversed for ever after. The bursting bubble of foam at the foot of a waterfall would reunite and descend into water. The thermal motions would reconcentrate energy and throw the mass up the fall in drops reforming in a close column of ascending water. Living creatures would grow backwards – from old age to infancy till they are unborn again – with conscious knowledge of the future but no memory of the past. We will solve this paradox in later pages.\n\nThe modern view on reductionism is faulty. Reductionism is based on the concept of differentiation. When an object is perceived as a composite that can be reduced to different components having perceptibly different properties which can be differentiated from one another and from the composite as a whole, the process of such differentiation is called reductionism. Some objects may generate similar perception of some properties or the opposite of some properties from a group of substances. In such cases the objects with similar properties are grouped together and the objects with opposite properties are grouped together. The only universally perceived aspect that is common to all objects is physical existence in space and time, as the radiation emitted by or the field set up by all objects create a perturbation in our sense organs always in identical ways. Since intermediate particles exhibit some properties similar with other particles and are similarly perceived with other such objects and not differentiated from others, reductionism applies only to the fundamental particles. This principle is violated in most modern classifications.\n\nTo give one example, x-rays and γ-rays exhibit exclusive characteristics that are not shared by other rays of the electromagnetic spectrum or between themselves – such as the place of their origin. Yet, they are clubbed under one category. If wave nature of propagation is the criterion for such categorisation, then sound waves that travel through a medium such as air or other gases in addition to liquids and solids of all kinds should also have been added to the classification. Then there are mechanical waves, such as the waves that travel though a vibrating string or other mechanical object or surface, waves that travel through a fluid or along the surface of a fluid, such as water waves. If electromagnetic properties are the criteria for such categorisation, then it is not scientific, as these rays do not interact with electromagnetic fields. If they have been clubbed together on the ground that theoretically they do not require any medium for their propagation, then firstly there is no true vacuum and secondly, they are known to travel through various mediums such as glass. There are many such examples of wrong classification due to reductionism and developmental history.\n\nThe cults of incomprehensibility and reductionism have led to another deficiency. Both cosmology and elementary particle physics share the same theory of the plasma and radiation. They have independent existence that is seemingly eternal and may be cyclic. Their combinations lead to the sub-atomic particles that belong to the micro world of quantum physics. The atoms are a class by itself, whose different combinations lead to the perceivable particles and bodies that belong to the macro world of the so-called classical physics. The two worlds merge in the stars, which contain plasma of the micro world and the planetary system of the macro world. Thus, the study of the evolution of stars can reveal the transition from the micro world to the macro world. For example, the internal structures of planet Jupiter and protons are identical and like protons, Jupiter-like stars are abundant in the stars. Yet, in stead of unification of all branches of science, Cosmology and nuclear physics have been fragmented into several “specialized” branches.\n\nHere we are reminded of an anecdote related to Lord Chaitanya. While in his southern sojourn, a debate was arranged between him and a great scholar of yore. The scholar went off explaining many complex doctrines while Lord Chaitanya sat quietly and listened with rapt attention without any response. Finally the scholar told Lord Chaitanya that he was not responding at all to his discourse. Was it too complex for him? The Scholar was sure from the look on Lord Chaitanya’s face that he did not understand anything. To this, Lord Chaitanya replied; “I fully understand what you are talking about. But I was wondering why you are making the simple things look so complicated?” Then he explained the same theories in plain language after which the scholar fell at his feet.\n\nThere has been very few attempts to list out the essence of all branches and develop “one” science. Each branch has its huge data bank with its specialized technical terms glorifying some person at the cost of a scientific nomenclature thereby enhancing incomprehensibility. Even if we read the descriptions of all six proverbial blind men repeatedly, one who has not seen an elephant cannot visualize it. This leaves the students with little opportunity to get a macro view of all theories and evaluate their inter-relationship. The educational system with its examination method of emphasizing the aspects of “memorization and reproduction at a specific instant” compounds the problem. Thus, the students have to accept many statements and theories as “given” without questioning it even on the face of ambiguities. Further, we have never come across any book on science, which does not glorify the discoveries in superlative terms, while leaving out the uncomfortable and ambiguous aspects, often with an assurance that they are correct and should be accepted as such. This creates an impression on the minds of young students to accept the theories unquestioningly making them superstitious. Thus, whenever some deficiencies have been noticed in any theory, there is an attempt at patch work within the broad parameters of the same theories. There have been few attempts to review the theories ab initio. Thus, the scientists cannot relate the tempest at a distant land to the flapping of the wings of the butterfly elsewhere.\n\nTill now scientists do not know “what” are electrons, photons, and other subatomic objects that have made the amazing technological revolution possible? Even the modern description of the nucleus and the nucleons leave many aspects unexplained. Photo-electric effect, for which Einstein got his Noble Prize, deals with electrons and photons. But it does not clarify “what” are these particles. The scientists, who framed the current theories, were not gifted with the benefit of the presently available data. Thus, without undermining their efforts, it is necessary to ab initio re-formulate the theories based on the presently available data. Only this way we can develop a theory whose correspondence resembles to reality. Here is an attempt in this regard from a different perspective. Like the child revealing the secret of the Emperor’s clothes, we, a novice in this field, are attempting to point the lamp in the direction of the Sun.\n\nThousands of papers are read every year in various forums on as yet undiscovered particles. This reminds us of the saying which means: after taking bath in the water of the mirage, wearing the flower of the sky in the head, holding the bow made of the horns of a rabbit, here goes the son of the barren woman! Modern scientists are precisely making similar statements. This is a sheer waste of not only valuable time but also public money worth trillions for the pleasure of a few. In addition, this amounts to misguiding general public for generations. This is unacceptable because a scientific theory must stand up to experimental scrutiny within a reasonable time period. Till it is proved or disproved, it cannot be accepted, though not rejected either. We cannot continue for three quarters and more of a century to develop “theories” based on such unproven postulates in the hope that we may succeed someday – may be after a couple of centuries! We cannot continue research on the properties of the “flowers of the sky” on the ground that someday it may be discovered.\n\nExperiments with the subatomic phenomena show effects that have not been reconciled with our normal view of an objective world. Yet, they cannot be treated separately. This implies the existence of two different states – classical and quantum – with different dynamics, but linked to each other in some fundamentally similar manner. Since the validity of a physical statement is judged by its correspondence to reality, there is a big question mark on the direction in which theoretical physics is moving. Technology has acquired a pre-eminent position in the global epistemic order. However, Engineers and Technologists, who progress by trial and error methods, have projected themselves as experimental scientists. Their search for new technology has been touted as the progress of science, questioning whose legitimacy is projected as deserving a sacrament. Thus, everything that exposes the hollowness or deficiencies of science is consigned to defenestration. The time has come to seriously consider the role, the ends and the methods of scientific research. If we are to believe that the sole objective of the scientists is to make their impressions mutually consistent, then we lose all motivation in theoretical physics. These impressions are not of a kind that occurs in our daily life. They are extremely special, are produced at great cost, time and effort. Hence it is doubtful whether the mere pleasure their harmony gives to a selected few can justify the huge public spending on such “scientific research”.\n\nA report published in the Notices of the American Mathematical Society, October 2005 issue shows that the Theory of Dynamical Systems that is used for calculating the trajectories of space flights and the Theory of Transition States for chemical reactions share the same mathematics. This is the proof of a universally true statement that both microcosm and the macrocosm replicate each other. The only problem is to find the exact correlations. For example, as we have repeated pointed out, the internal structure of a proton and that of planet Jupiter are identical. We will frequently use this and other similarities between the microcosm and the macrocosm (from astrophysics) in this presentation to prove the above statement. Also we will frequently refer to the definitions of technical terms as defined precisely in our book “Vaidic Theory of Numbers”." ]

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[ "## Genetic Algorithm Options\n\n### Options for Genetic Algorithm\n\nSet options for `ga` by using `optimoptions`.\n\n`options = optimoptions('ga','Option1','value1','Option2','value2');`\n• Some options are listed in `italics`. These options do not appear in the listing that `optimoptions` returns. To see why '`optimoptions` hides these option values, see Options that optimoptions Hides.\n\n• Ensure that you pass options to the solver. Otherwise, `patternsearch` uses the default option values.\n\n`[x,fval] = ga(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)`\n\n### Plot Options\n\n`PlotFcn` specifies the plot function or functions called at each iteration by `ga` or `gamultiobj`. Set the `PlotFcn` option to be a built-in plot function name or a handle to the plot function. You can stop the algorithm at any time by clicking the button on the plot window. For example, to display the best function value, set `options` as follows:\n\n`options = optimoptions('ga','PlotFcn','gaplotbestf');`\n\nTo display multiple plots, use a cell array of built-in plot function names or a cell array of function handles:\n\n`options = optimoptions('patternsearch','PlotFcn', {@plotfun1, @plotfun2, ...});`\n\nwhere `@plotfun1`, `@plotfun2`, and so on are function handles to the plot functions. If you specify more than one plot function, all plots appear as subplots in the same window. Right-click any subplot to obtain a larger version in a separate figure window.\n\nAvailable plot functions for `ga` or for `gamultiobj`:\n\n• `'gaplotscorediversity'` plots a histogram of the scores at each generation.\n\n• `'gaplotstopping'` plots stopping criteria levels.\n\n• `'gaplotgenealogy'` plots the genealogy of individuals. Lines from one generation to the next are color-coded as follows:\n\n• Red lines indicate mutation children.\n\n• Blue lines indicate crossover children.\n\n• Black lines indicate elite individuals.\n\n• `'gaplotscores'` plots the scores of the individuals at each generation.\n\n• `'gaplotdistance'` plots the average distance between individuals at each generation.\n\n• `'gaplotselection'` plots a histogram of the parents.\n\n• `'gaplotmaxconstr'` plots the maximum nonlinear constraint violation at each generation. For `ga`, available only when the `NonlinearConstraintAlgorithm` option is `'auglag'` (default for non-integer problems). Therefore, not available for integer-constrained problems, as they use the `'penalty'` nonlinear constraint algorithm.\n\n• You can also create and use your own plot function. Structure of the Plot Functions describes the structure of a custom plot function. Pass any custom function as a function handle.\n\nThe following plot functions are available for `ga` only:\n\n• `'gaplotbestf'` plots the best score value and mean score versus generation.\n\n• `'gaplotbestindiv'` plots the vector entries of the individual with the best fitness function value in each generation.\n\n• `'gaplotexpectation'` plots the expected number of children versus the raw scores at each generation.\n\n• `'gaplotrange'` plots the minimum, maximum, and mean score values in each generation.\n\nThe following plot functions are available for `gamultiobj` only:\n\n• `'gaplotpareto'` plots the Pareto front for the first two objective functions.\n\n• `'gaplotparetodistance'` plots a bar chart of the distance of each individual from its neighbors.\n\n• `'gaplotrankhist'` plots a histogram of the ranks of the individuals. Individuals of rank 1 are on the Pareto frontier. Individuals of rank 2 are lower than at least one rank 1 individual, but are not lower than any individuals from other ranks, etc.\n\n• `'gaplotspread'` plots the average spread as a function of iteration number.\n\n#### Structure of the Plot Functions\n\nThe first line of a plot function has this form:\n\n`function state = plotfun(options,state,flag)`\n\nThe input arguments to the function are\n\n• `options` — Structure containing all the current options settings.\n\n• `state` — Structure containing information about the current generation. The State Structure describes the fields of `state`.\n\n• `flag` — Description of the stage the algorithm is currently in. For details, see Output Function Options.\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\nThe output argument `state` is a state structure as well. Pass the input argument, modified if you like; see Changing the State Structure. To stop the iterations, set `state.StopFlag` to a nonempty character vector, such as `'y'`.\n\n#### The State Structure\n\nga.  The state structure for `ga`, which is an input argument to plot, mutation, and output functions, contains the following fields:\n\n• `Generation` — Current generation number.\n\n• `StartTime` — Time when genetic algorithm started, returned by `tic`.\n\n• `StopFlag` — Reason for stopping, a character vector.\n\n• `LastImprovement` — Generation at which the last improvement in fitness value occurred.\n\n• `LastImprovementTime` — Time at which last improvement occurred.\n\n• `Best` — Vector containing the best score in each generation.\n\n• `how` — The `'augLag'` nonlinear constraint algorithm reports one of the following actions: `'Infeasible point'`, ```'Update multipliers'```, or ```'Increase penalty'```; see Augmented Lagrangian Genetic Algorithm.\n\n• `FunEval` — Cumulative number of function evaluations.\n\n• `Expectation` — Expectation for selection of individuals.\n\n• `Selection` — Indices of individuals selected for elite, crossover, and mutation.\n\n• `Population` — Population in the current generation.\n\n• `Score` — Scores of the current population.\n\n• `NonlinIneq` — Nonlinear inequality constraints at current point, present only when a nonlinear constraint function is specified, there are no integer variables, `flag` is not `'interrupt'`, and `NonlinearConstraintAlgorithm` is `'auglag'`.\n\n• `NonlinEq` — Nonlinear equality constraints at current point, present only when a nonlinear constraint function is specified, there are no integer variables, `flag` is not `'interrupt'`, and `NonlinearConstraintAlgorithm` is `'auglag'`.\n\n• `EvalElites` — Logical value indicating whether `ga` evaluates the fitness function of elite individuals. Initially, this value is `true`. In the first generation, if the elite individuals evaluate to their previous values (which indicates that the fitness function is deterministic), then this value becomes `false` by default for subsequent iterations. When `EvalElites` is `false`, `ga` does not reevaluate the fitness function of elite individuals. You can override this behavior in a custom plot function or custom output function by changing the output `state.EvalElites`.\n\n• `HaveDuplicates` — Logical value indicating whether `ga` adds duplicate individuals for the initial population. `ga` uses a small relative tolerance to determine whether an individual is duplicated or unique. If `HaveDuplicates` is `true`, then `ga` locates the unique individuals and evaluates the fitness function only once for each unique individual. `ga` copies the fitness and constraint function values to duplicate individuals. `ga` repeats the test in each generation until all individuals are unique. The test takes order `n*m*log(m)` operations, where `m` is the population size and `n` is `nvars`. To override this test in a custom plot function or custom output function, set the output `state.HaveDuplicates` to `false`.\n\ngamultiobj.  The state structure for `gamultiobj`, which is an input argument to plot, mutation, and output functions, contains the following fields:\n\n• `Population` — Population in the current generation\n\n• `Score` — Scores of the current population, a `Population`-by-`nObjectives` matrix, where `nObjectives` is the number of objectives\n\n• `Generation` — Current generation number\n\n• `StartTime` — Time when genetic algorithm started, returned by `tic`\n\n• `StopFlag` — Reason for stopping, a character vector\n\n• `FunEval` — Cumulative number of function evaluations\n\n• `Selection` — Indices of individuals selected for elite, crossover, and mutation\n\n• `Rank` — Vector of the ranks of members in the population\n\n• `Distance` — Vector of distances of each member of the population to the nearest neighboring member\n\n• `AverageDistance` — Standard deviation (not average) of `Distance`\n\n• `Spread` — Vector where the entries are the spread in each generation\n\n• `mIneq` — Number of nonlinear inequality constraints\n\n• `mEq` — Number of nonlinear equality constraints\n\n• `mAll` — Total number of nonlinear constraints, `mAll` = `mIneq` + `mEq`\n\n• `C` — Nonlinear inequality constraints at current point, a `PopulationSize`-by-`mIneq` matrix\n\n• `Ceq` — Nonlinear equality constraints at current point, a `PopulationSize`-by-`mEq` matrix\n\n• `isFeas` — Feasibility of population, a logical vector with `PopulationSize` elements\n\n• `maxLinInfeas` — Maximum infeasibility with respect to linear constraints for the population\n\n### Population Options\n\nPopulation options let you specify the parameters of the population that the genetic algorithm uses.\n\n`PopulationType` specifies the type of input to the fitness function. Types and their restrictions are:\n\n• `'doubleVector'` — Use this option if the individuals in the population have type `double`. Use this option for mixed integer programming. This is the default.\n\n• `'bitstring'` — Use this option if the individuals in the population have components that are `0` or `1`.\n\nCaution\n\nThe individuals in a `Bit string` population are vectors of type `double`, not strings or characters.\n\nFor `CreationFcn` and `MutationFcn`, use `'gacreationuniform'` and `'mutationuniform'` or handles to custom functions. For `CrossoverFcn`, use `'crossoverscattered'`, `'crossoversinglepoint'`, `'crossovertwopoint'`, or a handle to a custom function. You cannot use a `HybridFcn`, and `ga` ignores all constraints, including bounds, linear constraints, and nonlinear constraints.\n\n• `'custom'` — Indicates a custom population type. In this case, you must also use a custom `CrossoverFcn` and `MutationFcn`. You must provide either a custom creation function or an `InitialPopulationMatrix`. You cannot use a `HybridFcn`, and `ga` ignores all constraints, including bounds, linear constraints, and nonlinear constraints.\n\n`PopulationSize` specifies how many individuals there are in each generation. With a large population size, the genetic algorithm searches the solution space more thoroughly, thereby reducing the chance that the algorithm returns a local minimum that is not a global minimum. However, a large population size also causes the algorithm to run more slowly. The default is ```'50 when numberOfVariables <= 5, else 200'```.\n\nIf you set `PopulationSize` to a vector, the genetic algorithm creates multiple subpopulations, the number of which is the length of the vector. The size of each subpopulation is the corresponding entry of the vector. Note that this option is not useful. See Migration Options.\n\n`CreationFcn` specifies the function that creates the initial population for `ga`. Do not specify a creation function with integer problems because `ga` overrides any choice you make. Choose from:\n\n• `[]` uses the default creation function for your problem type.\n\n• `'gacreationuniform'` creates a random initial population with a uniform distribution. This is the default when there are no linear constraints, or when there are integer constraints. The uniform distribution is in the initial population range (`InitialPopulationRange`). The default values for `InitialPopulationRange` are `[-10;10]` for every component, or `[-9999;10001]` when there are integer constraints. These bounds are shifted and scaled to match any existing bounds `lb` and `ub`.\n\nCaution\n\nDo not use `'gacreationuniform'` when you have linear constraints. Otherwise, your population might not satisfy the linear constraints.\n\n• `'gacreationlinearfeasible'` is the default when there are linear constraints and no integer constraints. This choice creates a random initial population that satisfies all bounds and linear constraints. If there are linear constraints, `'gacreationlinearfeasible'` creates many individuals on the boundaries of the constraint region, and creates a well-dispersed population. `'gacreationlinearfeasible'` ignores `InitialPopulationRange`. `'gacreationlinearfeasible'` calls `linprog` to create a feasible population with respect to bounds and linear constraints.\n\nFor an example showing its behavior, see Custom Plot Function and Linear Constraints in ga.\n\n• `'gacreationnonlinearfeasible'` is the default creation function for the `'penalty'` nonlinear constraint algorithm. For details, see Constraint Parameters.\n\n• A function handle lets you write your own creation function, which must generate data of the type that you specify in `PopulationType`. For example,\n\n`options = optimoptions('ga','CreationFcn',@myfun);`\n\nYour creation function must have the following calling syntax.\n\n`function Population = myfun(GenomeLength, FitnessFcn, options)`\n\nThe input arguments to the function are:\n\n• `Genomelength` — Number of independent variables for the fitness function\n\n• `FitnessFcn` — Fitness function\n\n• `options` — Options\n\nThe function returns `Population`, the initial population for the genetic algorithm.\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\nCaution\n\nWhen you have bounds or linear constraints, ensure that your creation function creates individuals that satisfy these constraints. Otherwise, your population might not satisfy the constraints.\n\n`InitialPopulationMatrix` specifies an initial population for the genetic algorithm. The default value is `[]`, in which case `ga` uses the default `CreationFcn` to create an initial population. If you enter a nonempty array in the `InitialPopulationMatrix`, the array must have no more than `PopulationSize` rows, and exactly `nvars` columns, where `nvars` is the number of variables, the second input to `ga` or `gamultiobj`. If you have a partial initial population, meaning fewer than `PopulationSize` rows, then the genetic algorithm calls `CreationFcn` to generate the remaining individuals.\n\n`InitialScoreMatrix` specifies initial scores for the initial population. The initial scores can also be partial. Do not specify initial scores with integer problems because `ga` overrides any choice you make.\n\n`InitialPopulationRange` specifies the range of the vectors in the initial population that is generated by the `gacreationuniform` creation function. You can set `InitialPopulationRange` to be a matrix with two rows and `nvars` columns, each column of which has the form `[lb;ub]`, where `lb` is the lower bound and `ub` is the upper bound for the entries in that coordinate. If you specify `InitialPopulationRange` to be a 2-by-1 vector, each entry is expanded to a constant row of length `nvars`. If you do not specify an `InitialPopulationRange`, the default is `[-10;10]` (`[-1e4+1;1e4+1]` for integer-constrained problems), modified to match any existing bounds. `'gacreationlinearfeasible'` ignores `InitialPopulationRange`. See Set Initial Range for an example.\n\n### Fitness Scaling Options\n\nFitness scaling converts the raw fitness scores that are returned by the fitness function to values in a range that is suitable for the selection function.\n\n`FitnessScalingFcn` specifies the function that performs the scaling. The options are\n\n• `'fitscalingrank'` — The default fitness scaling function, `'fitscalingrank'`, scales the raw scores based on the rank of each individual instead of its score. The rank of an individual is its position in the sorted scores. An individual with rank r has scaled score proportional to $1/\\sqrt{r}$. So the scaled score of the most fit individual is proportional to 1, the scaled score of the next most fit is proportional to $1/\\sqrt{2}$, and so on. Rank fitness scaling removes the effect of the spread of the raw scores. The square root makes poorly ranked individuals more nearly equal in score, compared to rank scoring. For more information, see Fitness Scaling.\n\n• `'fitscalingprop'` — Proportional scaling makes the scaled value of an individual proportional to its raw fitness score.\n\n• `'fitscalingtop'` — Top scaling scales the top individuals equally. You can modify the top scaling using an additional parameter:\n\n`options = optimoptions('ga','FitnessScalingFcn',{@fitscalingtop,quantity})`\n\n`quantity` specifies the number of individuals that are assigned positive scaled values. `quantity` can be an integer from 1 through the population size or a fraction from 0 through 1 specifying a fraction of the population size. The default value is `0.4`. Each of the individuals that produce offspring is assigned an equal scaled value, while the rest are assigned the value 0. The scaled values have the form [01/n 1/n 0 0 1/n 0 0 1/n ...].\n\n• `'fitscalingshiftlinear'` — Shift linear scaling scales the raw scores so that the expectation of the fittest individual is equal to a constant called `rate` multiplied by the average score. You can modify the `rate` parameter:\n\n```options = optimoptions('ga','FitnessScalingFcn',... {@fitscalingshiftlinear, rate})```\n\nThe default value of `rate` is `2`.\n\n• A function handle lets you write your own scaling function.\n\n`options = optimoptions('ga','FitnessScalingFcn',@myfun);`\n\nYour scaling function must have the following calling syntax:\n\n`function expectation = myfun(scores, nParents)`\n\nThe input arguments to the function are:\n\n• `scores` — A vector of scalars, one for each member of the population\n\n• `nParents` — The number of parents needed from this population\n\nThe function returns `expectation`, a column vector of scalars of the same length as `scores`, giving the scaled values of each member of the population. The sum of the entries of `expectation` must equal `nParents`.\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\n### Selection Options\n\nSelection options specify how the genetic algorithm chooses parents for the next generation.\n\nThe `SelectionFcn` option specifies the selection function. Do not use with integer problems.\n\n`gamultiobj` uses only the `'selectiontournament'` selection function.\n\nFor `ga` the options are:\n\n• `'selectionstochunif'` — The `ga` default selection function, `'selectionstochunif'`, lays out a line in which each parent corresponds to a section of the line of length proportional to its scaled value. The algorithm moves along the line in steps of equal size. At each step, the algorithm allocates a parent from the section it lands on. The first step is a uniform random number less than the step size.\n\n• `'selectionremainder'` — Remainder selection assigns parents deterministically from the integer part of each individual's scaled value and then uses roulette selection on the remaining fractional part. For example, if the scaled value of an individual is 2.3, that individual is listed twice as a parent because the integer part is 2. After parents have been assigned according to the integer parts of the scaled values, the rest of the parents are chosen stochastically. The probability that a parent is chosen in this step is proportional to the fractional part of its scaled value.\n\n• `'selectionuniform'` — Uniform selection chooses parents using the expectations and number of parents. Uniform selection is useful for debugging and testing, but is not a very effective search strategy.\n\n• `'selectionroulette'` — Roulette selection chooses parents by simulating a roulette wheel, in which the area of the section of the wheel corresponding to an individual is proportional to the individual's expectation. The algorithm uses a random number to select one of the sections with a probability equal to its area.\n\n• `'selectiontournament'` — Tournament selection chooses each parent by choosing `size` players at random and then choosing the best individual out of that set to be a parent. `size` must be at least 2. The default value of `size` is `4`. Set `size` to a different value as follows:\n\n```options = optimoptions('ga','SelectionFcn',... {@selectiontournament,size})```\n\nWhen `NonlinearConstraintAlgorithm` is `Penalty`, `ga` uses `'selectiontournament'` with size `2`.\n\n• A function handle enables you to write your own selection function.\n\n`options = optimoptions('ga','SelectionFcn',@myfun);`\n\nYour selection function must have the following calling syntax:\n\n`function parents = myfun(expectation, nParents, options)`\n\n`ga` provides the input arguments `expectation`, `nParents`, and `options`. Your function returns the indices of the parents.\n\nThe input arguments to the function are:\n\n• `expectation`\n\n• For `ga`, `expectation` is a column vector of the scaled fitness of each member of the population. The scaling comes from the Fitness Scaling Options.\n\nTip\n\nYou can ensure that you have a column vector by using `expectation(:,1)`. For example, `edit selectionstochunif` or any of the other built-in selection functions.\n\n• For `gamultiobj`, `expectation` is a matrix whose first column is the negative of the rank of the individuals, and whose second column is the distance measure of the individuals. See Multiobjective Options.\n\n• `nParents`— Number of parents to select.\n\n• `options` — Genetic algorithm `options`.\n\nThe function returns `parents`, a row vector of length `nParents` containing the indices of the parents that you select.\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\n### Reproduction Options\n\nReproduction options specify how the genetic algorithm creates children for the next generation.\n\n`EliteCount` specifies the number of individuals that are guaranteed to survive to the next generation. Set `EliteCount` to be a positive integer less than or equal to the population size. The default value is `ceil(0.05*PopulationSize)` for continuous problems, and `0.05*(default PopulationSize)` for mixed-integer problems.\n\n`CrossoverFraction` specifies the fraction of the next generation, other than elite children, that are produced by crossover. Set `CrossoverFraction` to be a fraction between `0` and `1`. The default value is `0.8`.\n\nSee Setting the Crossover Fraction for an example.\n\n### Mutation Options\n\nMutation options specify how the genetic algorithm makes small random changes in the individuals in the population to create mutation children. Mutation provides genetic diversity and enables the genetic algorithm to search a broader space. Specify the mutation function in the `MutationFcn` option. Do not use with integer problems.\n\n`MutationFcn` options:\n\n• `'mutationgaussian'` — The default mutation function for unconstrained problems, `'mutationgaussian'`, adds a random number taken from a Gaussian distribution with mean 0 to each entry of the parent vector. The standard deviation of this distribution is determined by the parameters `scale` and `shrink`, and by the `InitialPopulationRange` option. Set `scale` and `shrink` as follows:\n\n```options = optimoptions('ga','MutationFcn', ... {@mutationgaussian, scale, shrink})```\n• The `scale` parameter determines the standard deviation at the first generation. If you set `InitialPopulationRange` to be a 2-by-1 vector `v`, the initial standard deviation is the same at all coordinates of the parent vector, and is given by `scale``*(v(2)-v(1))`.\n\nIf you set `InitialPopulationRange` to be a vector `v` with two rows and `nvars` columns, the initial standard deviation at coordinate `i` of the parent vector is given by `scale````*(v(i,2) - v(i,1))```.\n\n• The `shrink` parameter controls how the standard deviation shrinks as generations go by. If you set `InitialPopulationRange` to be a 2-by-1 vector, the standard deviation at the kth generation, σk, is the same at all coordinates of the parent vector, and is given by the recursive formula\n\n`${\\sigma }_{k}={\\sigma }_{k-1}\\left(1-\\text{Shrink}\\frac{k}{\\text{Generations}}\\right).$`\n\nIf you set `InitialPopulationRange` to be a vector with two rows and `nvars` columns, the standard deviation at coordinate i of the parent vector at the kth generation, σi,k, is given by the recursive formula\n\n`${\\sigma }_{i,k}={\\sigma }_{i,k-1}\\left(1-\\text{Shrink}\\frac{k}{\\text{Generations}}\\right).$`\n\nIf you set `shrink` to `1`, the algorithm shrinks the standard deviation in each coordinate linearly until it reaches 0 at the last generation is reached. A negative value of `shrink` causes the standard deviation to grow.\n\nThe default value of both `scale` and `shrink` is 1.\n\nCaution\n\nDo not use `mutationgaussian` when you have bounds or linear constraints. Otherwise, your population will not necessarily satisfy the constraints. Instead, use `'mutationadaptfeasible'` or a custom mutation function that satisfies linear constraints.\n\n• `'mutationuniform'` — Uniform mutation is a two-step process. First, the algorithm selects a fraction of the vector entries of an individual for mutation, where each entry has a probability `rate` of being mutated. The default value of `rate` is `0.01`. In the second step, the algorithm replaces each selected entry by a random number selected uniformly from the range for that entry.\n\nTo change the default value of `rate`,\n\n`options = optimoptions('ga','MutationFcn', {@mutationuniform, rate})`\n\nCaution\n\nDo not use `mutationuniform` when you have bounds or linear constraints. Otherwise, your population will not necessarily satisfy the constraints. Instead, use `'mutationadaptfeasible'` or a custom mutation function that satisfies linear constraints.\n\n• `'mutationadaptfeasible'`, the default mutation function when there are constraints, randomly generates directions that are adaptive with respect to the last successful or unsuccessful generation. The mutation chooses a direction and step length that satisfies bounds and linear constraints.\n\n• A function handle enables you to write your own mutation function.\n\n`options = optimoptions('ga','MutationFcn',@myfun);`\n\nYour mutation function must have this calling syntax:\n\n```function mutationChildren = myfun(parents, options, nvars, FitnessFcn, state, thisScore, thisPopulation)```\n\nThe arguments to the function are\n\n• `parents` — Row vector of parents chosen by the selection function\n\n• `options` — Options\n\n• `nvars` — Number of variables\n\n• `FitnessFcn` — Fitness function\n\n• `state` — Structure containing information about the current generation. The State Structure describes the fields of `state`.\n\n• `thisScore` — Vector of scores of the current population\n\n• `thisPopulation` — Matrix of individuals in the current population\n\nThe function returns `mutationChildren`—the mutated offspring—as a matrix where rows correspond to the children. The number of columns of the matrix is `nvars`.\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\nCaution\n\nWhen you have bounds or linear constraints, ensure that your mutation function creates individuals that satisfy these constraints. Otherwise, your population will not necessarily satisfy the constraints.\n\n### Crossover Options\n\nCrossover options specify how the genetic algorithm combines two individuals, or parents, to form a crossover child for the next generation.\n\n`CrossoverFcn` specifies the function that performs the crossover. Do not use with integer problems. You can choose from the following functions:\n\n• `'crossoverscattered'`, the default crossover function for problems without linear constraints, creates a random binary vector and selects the genes where the vector is a 1 from the first parent, and the genes where the vector is a 0 from the second parent, and combines the genes to form the child. For example, if `p1` and `p2` are the parents\n\n```p1 = [a b c d e f g h] p2 = [1 2 3 4 5 6 7 8]```\n\nand the binary vector is [1 1 0 0 1 0 0 0], the function returns the following child:\n\n`child1 = [a b 3 4 e 6 7 8]`\n\nCaution\n\nWhen your problem has linear constraints, `'crossoverscattered'` can give a poorly distributed population. In this case, use a different crossover function, such as `'crossoverintermediate'`.\n\n• `'crossoversinglepoint'` chooses a random integer n between 1 and `nvars` and then\n\n• Selects vector entries numbered less than or equal to n from the first parent.\n\n• Selects vector entries numbered greater than n from the second parent.\n\n• Concatenates these entries to form a child vector.\n\nFor example, if `p1` and `p2` are the parents\n\n```p1 = [a b c d e f g h] p2 = [1 2 3 4 5 6 7 8]```\n\nand the crossover point is 3, the function returns the following child.\n\n`child = [a b c 4 5 6 7 8]`\n\nCaution\n\nWhen your problem has linear constraints, `'crossoversinglepoint'` can give a poorly distributed population. In this case, use a different crossover function, such as `'crossoverintermediate'`.\n\n• `'crossovertwopoint'` selects two random integers `m` and `n` between `1` and `nvars`. The function selects\n\n• Vector entries numbered less than or equal to `m` from the first parent\n\n• Vector entries numbered from `m+1` to `n`, inclusive, from the second parent\n\n• Vector entries numbered greater than `n` from the first parent.\n\nThe algorithm then concatenates these genes to form a single gene. For example, if `p1` and `p2` are the parents\n\n```p1 = [a b c d e f g h] p2 = [1 2 3 4 5 6 7 8]```\n\nand the crossover points are 3 and 6, the function returns the following child.\n\n`child = [a b c 4 5 6 g h]`\n\nCaution\n\nWhen your problem has linear constraints, `'crossovertwopoint'` can give a poorly distributed population. In this case, use a different crossover function, such as `'crossoverintermediate'`.\n\n• `'crossoverintermediate'`, the default crossover function when there are linear constraints, creates children by taking a weighted average of the parents. You can specify the weights by a single parameter, `ratio`, which can be a scalar or a row vector of length `nvars`. The default value of `ratio` is a vector of all 1's. Set the `ratio` parameter as follows.\n\n```options = optimoptions('ga','CrossoverFcn', ... {@crossoverintermediate, ratio});```\n\n`'crossoverintermediate'` creates the child from `parent1` and `parent2` using the following formula.\n\n`child = parent1 + rand * Ratio * ( parent2 - parent1)`\n\nIf all the entries of `ratio` lie in the range [0, 1], the children produced are within the hypercube defined by placing the parents at opposite vertices. If `ratio` is not in that range, the children might lie outside the hypercube. If `ratio` is a scalar, then all the children lie on the line between the parents.\n\n• `'crossoverheuristic'` returns a child that lies on the line containing the two parents, a small distance away from the parent with the better fitness value in the direction away from the parent with the worse fitness value. You can specify how far the child is from the better parent by the parameter `ratio`. The default value of `ratio`is 1.2. Set the `ratio` parameter as follows.\n\n```options = optimoptions('ga','CrossoverFcn',... {@crossoverheuristic,ratio});```\n\nIf `parent1` and `parent2` are the parents, and `parent1` has the better fitness value, the function returns the child\n\n`child = parent2 + ratio * (parent1 - parent2);`\n\nCaution\n\nWhen your problem has linear constraints, `'crossoverheuristic'` can give a poorly distributed population. In this case, use a different crossover function, such as `'crossoverintermediate'`.\n\n• `'crossoverarithmetic'` creates children that are the weighted arithmetic mean of two parents. Children are always feasible with respect to linear constraints and bounds.\n\n• A function handle enables you to write your own crossover function.\n\n`options = optimoptions('ga','CrossoverFcn',@myfun);`\n\nYour crossover function must have the following calling syntax.\n\n```xoverKids = myfun(parents, options, nvars, FitnessFcn, ... unused,thisPopulation)```\n\nThe arguments to the function are\n\n• `parents` — Row vector of parents chosen by the selection function\n\n• `options` — options\n\n• `nvars` — Number of variables\n\n• `FitnessFcn` — Fitness function\n\n• `unused` — Placeholder not used\n\n• `thisPopulation` — Matrix representing the current population. The number of rows of the matrix is `PopulationSize` and the number of columns is `nvars`.\n\nThe function returns `xoverKids`—the crossover offspring—as a matrix where rows correspond to the children. The number of columns of the matrix is `nvars`.\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\nCaution\n\nWhen you have bounds or linear constraints, ensure that your crossover function creates individuals that satisfy these constraints. Otherwise, your population will not necessarily satisfy the constraints.\n\n### Migration Options\n\nNote\n\nSubpopulations refer to a form of parallel processing for the genetic algorithm. `ga` currently does not support this form. In subpopulations, each worker hosts a number of individuals. These individuals are a subpopulation. The worker evolves the subpopulation independently of other workers, except when migration causes some individuals to travel between workers.\n\nBecause `ga` does not currently support this form of parallel processing, there is no benefit to setting `PopulationSize` to a vector, or to setting the `MigrationDirection`, `MigrationInterval`, or `MigrationFraction` options.\n\nMigration options specify how individuals move between subpopulations. Migration occurs if you set `PopulationSize` to be a vector of length greater than 1. When migration occurs, the best individuals from one subpopulation replace the worst individuals in another subpopulation. Individuals that migrate from one subpopulation to another are copied. They are not removed from the source subpopulation.\n\nYou can control how migration occurs by the following three options:\n\n• `MigrationDirection` — Migration can take place in one or both directions.\n\n• If you set `MigrationDirection` to `'forward'`, migration takes place toward the last subpopulation. That is, the nth subpopulation migrates into the (n+1)th subpopulation.\n\n• If you set `MigrationDirection` to `'both'`, the nth subpopulation migrates into both the (n–1)th and the (n+1)th subpopulation.\n\nMigration wraps at the ends of the subpopulations. That is, the last subpopulation migrates into the first, and the first may migrate into the last.\n\n• `MigrationInterval` — Specifies how many generation pass between migrations. For example, if you set `MigrationInterval` to `20`, migration takes place every 20 generations.\n\n• `MigrationFraction` — Specifies how many individuals move between subpopulations. `MigrationFraction` specifies the fraction of the smaller of the two subpopulations that moves. For example, if individuals migrate from a subpopulation of 50 individuals into a subpopulation of 100 individuals and you set `MigrationFraction` to `0.1`, the number of individuals that migrate is 0.1*50=5.\n\n### Constraint Parameters\n\nConstraint parameters refer to the nonlinear constraint solver. For details on the algorithm, see Nonlinear Constraint Solver Algorithms.\n\nChoose between the nonlinear constraint algorithms by setting the `NonlinearConstraintAlgorithm` option to `'auglag'` (Augmented Lagrangian) or `'penalty'` (Penalty algorithm).\n\n#### Augmented Lagrangian Genetic Algorithm\n\n• `InitialPenalty` — Specifies an initial value of the penalty parameter that is used by the nonlinear constraint algorithm. `InitialPenalty` must be greater than or equal to `1`, and has a default of `10`.\n\n• `PenaltyFactor` — Increases the penalty parameter when the problem is not solved to required accuracy and constraints are not satisfied. `PenaltyFactor` must be greater than `1`, and has a default of `100`.\n\n#### Penalty Algorithm\n\nThe penalty algorithm uses the `'gacreationnonlinearfeasible'` creation function by default. This creation function uses `fmincon` to find feasible individuals. `'gacreationnonlinearfeasible'` starts `fmincon` from a variety of initial points within the bounds from the `InitialPopulationRange` option. Optionally, `'gacreationnonlinearfeasible'` can run `fmincon` in parallel on the initial points.\n\nYou can specify tuning parameters for `'gacreationnonlinearfeasible'` using the following name-value pairs.\n\nNameValue\n`SolverOpts``fmincon` options, created using `optimoptions` or `optimset`.\n`UseParallel`When `true`, run `fmincon` in parallel on initial points; default is `false`.\n`NumStartPts`Number of start points, a positive integer up to `sum(PopulationSize)` in value.\n\nInclude the name-value pairs in a cell array along with `@gacreationnonlinearfeasible`.\n\n```options = optimoptions('ga','CreationFcn',{`@gacreationnonlinearfeasible`,... 'UseParallel',true,'NumStartPts',20});```\n\n### Multiobjective Options\n\nMultiobjective options define parameters characteristic of the `gamultiobj` algorithm. You can specify the following parameters:\n\n• `ParetoFraction` — Sets the fraction of individuals to keep on the first Pareto front while the solver selects individuals from higher fronts. This option is a scalar between 0 and 1.\n\nNote\n\nThe fraction of individuals on the first Pareto front can exceed `ParetoFraction`. This occurs when there are too few individuals of other ranks in step 6 of Iterations.\n\n• `DistanceMeasureFcn` — Defines a handle to the function that computes distance measure of individuals, computed in decision variable space (genotype, also termed design variable space) or in function space (phenotype). For example, the default distance measure function is `'distancecrowding'` in function space, which is the same as `{@distancecrowding,'phenotype'}`.\n\n“Distance” measures a crowding of each individual in a population. Choose between the following:\n\n• `'distancecrowding'`, or the equivalent `{@distancecrowding,'phenotype'}` — Measure the distance in fitness function space.\n\n• `{@distancecrowding,'genotype'}` — Measure the distance in decision variable space.\n\n• `@distancefunction` — Write a custom distance function using the following template.\n\n```function distance = distancefunction(pop,score,options) % Uncomment one of the following two lines, or use a combination of both % y = score; % phenotype % y = pop; % genotype popSize = size(y,1); % number of individuals numData = size(y,2); % number of dimensions or fitness functions distance = zeros(popSize,1); % allocate the output % Compute distance here```\n\n`gamultiobj` passes the population in `pop`, the computed scores for the population in `scores`, and the options in `options`. Your distance function returns the distance from each member of the population to a reference, such as the nearest neighbor in some sense. For an example, edit the built-in file `distancecrowding.m`.\n\n### Hybrid Function Options\n\n#### `ga` Hybrid Function\n\nA hybrid function is another minimization function that runs after the genetic algorithm terminates. You can specify a hybrid function in the `HybridFcn` option. Do not use with integer problems. The choices are\n\n• `[]` — No hybrid function.\n\n• `'fminsearch'` — Uses the MATLAB® function `fminsearch` to perform unconstrained minimization.\n\n• `'patternsearch'` — Uses a pattern search to perform constrained or unconstrained minimization.\n\n• `'fminunc'` — Uses the Optimization Toolbox™ function `fminunc` to perform unconstrained minimization.\n\n• `'fmincon'` — Uses the Optimization Toolbox function `fmincon` to perform constrained minimization.\n\nNote\n\nEnsure that your hybrid function accepts your problem constraints. Otherwise, `ga` throws an error.\n\nYou can set separate options for the hybrid function. Use `optimset` for `fminsearch`, or `optimoptions` for `fmincon`, `patternsearch`, or `fminunc`. For example:\n\n`hybridopts = optimoptions('fminunc','Display','iter','Algorithm','quasi-newton');`\nInclude the hybrid options in the Genetic Algorithm `options` as follows:\n`options = optimoptions('ga',options,'HybridFcn',{@fminunc,hybridopts}); `\n`hybridopts` must exist before you set `options`.\n\nSee Hybrid Scheme in the Genetic Algorithm for an example. See When to Use a Hybrid Function.\n\n#### `gamultiobj` Hybrid Function\n\nA hybrid function is another minimization function that runs after the multiobjective genetic algorithm terminates. You can specify the hybrid function `'fgoalattain'` in the `HybridFcn` option.\n\nIn use as a multiobjective hybrid function, the solver does the following:\n\n1. Compute the maximum and minimum of each objective function at the solutions. For objective j at solution k, let\n\n`$\\begin{array}{c}{F}_{\\mathrm{max}}\\left(j\\right)=\\underset{k}{\\mathrm{max}}{F}_{k}\\left(j\\right)\\\\ {F}_{\\mathrm{min}}\\left(j\\right)=\\underset{k}{\\mathrm{min}}{F}_{k}\\left(j\\right).\\end{array}$`\n2. Compute the total weight at each solution k,\n\n`$w\\left(k\\right)=\\sum _{j}\\frac{{F}_{\\mathrm{max}}\\left(j\\right)-{F}_{k}\\left(j\\right)}{1+{F}_{\\mathrm{max}}\\left(j\\right)-{F}_{\\mathrm{min}}\\left(j\\right)}.$`\n3. Compute the weight for each objective function j at each solution k,\n\n`$p\\left(j,k\\right)=w\\left(k\\right)\\frac{{F}_{\\mathrm{max}}\\left(j\\right)-{F}_{k}\\left(j\\right)}{1+{F}_{\\mathrm{max}}\\left(j\\right)-{F}_{\\mathrm{min}}\\left(j\\right)}.$`\n4. For each solution k, perform the goal attainment problem with goal vector Fk(j) and weight vector p(j,k).\n\n### Stopping Criteria Options\n\nStopping criteria determine what causes the algorithm to terminate. You can specify the following options:\n\n• `MaxGenerations` — Specifies the maximum number of iterations for the genetic algorithm to perform. The default is `100*numberOfVariables`.\n\n• `MaxTime` — Specifies the maximum time in seconds the genetic algorithm runs before stopping, as measured by `tic` and `toc`. This limit is enforced after each iteration, so `ga` can exceed the limit when an iteration takes substantial time.\n\n• `FitnessLimit` — The algorithm stops if the best fitness value is less than or equal to the value of `FitnessLimit`. Does not apply to `gamultiobj`.\n\n• `MaxStallGenerations` — The algorithm stops if the average relative change in the best fitness function value over `MaxStallGenerations` is less than or equal to `FunctionTolerance`. (If the `StallTest` option is `'geometricWeighted'`, then the test is for a geometric weighted average relative change.) For a problem with nonlinear constraints, `MaxStallGenerations` applies to the subproblem (see Nonlinear Constraint Solver Algorithms).\n\nFor `gamultiobj`, if the geometric average of the relative change in the spread of the Pareto solutions over `MaxStallGenerations` is less than `FunctionTolerance`, and the final spread is smaller than the average spread over the last `MaxStallGenerations`, then the algorithm stops. The geometric average coefficient is ½. The spread is a measure of the movement of the Pareto front. See gamultiobj Algorithm.\n\n• `MaxStallTime` — The algorithm stops if there is no improvement in the best fitness value for an interval of time in seconds specified by `MaxStallTime`, as measured by `tic` and `toc`.\n\n• `FunctionTolerance` — The algorithm stops if the average relative change in the best fitness function value over `MaxStallGenerations` is less than or equal to `FunctionTolerance`. (If the `StallTest` option is `'geometricWeighted'`, then the test is for a geometric weighted average relative change.)\n\nFor `gamultiobj`, if the geometric average of the relative change in the spread of the Pareto solutions over `MaxStallGenerations` is less than `FunctionTolerance`, and the final spread is smaller than the average spread over the last `MaxStallGenerations`, then the algorithm stops. The geometric average coefficient is ½. The spread is a measure of the movement of the Pareto front. See gamultiobj Algorithm.\n\n• `ConstraintTolerance` — The `ConstraintTolerance` is not used as stopping criterion. It is used to determine the feasibility with respect to nonlinear constraints. Also, `max(sqrt(eps),ConstraintTolerance)` determines feasibility with respect to linear constraints.\n\nSee Set Maximum Number of Generations and Stall Generations for an example.\n\n### Output Function Options\n\nOutput functions are functions that the genetic algorithm calls at each generation. Unlike other solvers, a `ga` output function can not only read the values of the state of the algorithm, but also modify those values. An output function can also halt the solver according to conditions you set.\n\n`options = optimoptions('ga','OutputFcn',@myfun);`\n\nFor multiple output functions, enter a cell array of function handles:\n\n`options = optimoptions('ga','OutputFcn',{@myfun1,@myfun2,...});`\n\nTo see a template that you can use to write your own output functions, enter\n\n`edit gaoutputfcntemplate`\n\nat the MATLAB command line.\n\nFor an example, see Custom Output Function for Genetic Algorithm.\n\n#### Structure of the Output Function\n\nYour output function must have the following calling syntax:\n\n`[state,options,optchanged] = myfun(options,state,flag)`\n\nMATLAB passes the `options`, `state`, and `flag` data to your output function, and the output function returns `state`, `options`, and `optchanged` data.\n\nNote\n\nTo stop the iterations, set `state.StopFlag` to a nonempty character vector, such as `'y'`.\n\nThe output function has the following input arguments:\n\n• `options` — Options\n\n• `state` — Structure containing information about the current generation. The State Structure describes the fields of `state`.\n\n• `flag` — Current status of the algorithm:\n\n• `'init'` — Initialization state\n\n• `'iter'` — Iteration state\n\n• `'interrupt'` — Iteration of a subproblem of a nonlinearly constrained problem for the `'auglag'` nonlinear constraint algorithm. When `flag` is `'interrupt'`:\n\n• The values of `state` fields apply to the subproblem iterations.\n\n• `ga` does not accept changes in `options`, and ignores `optchanged`.\n\n• The `state.NonlinIneq` and `state.NonlinEq` fields are not available.\n\n• `'done'` — Final state\n\nPassing Extra Parameters explains how to provide additional parameters to the function.\n\nThe output function returns the following arguments to `ga`:\n\n• `state` — Structure containing information about the current generation. The State Structure describes the fields of `state`. To stop the iterations, set `state.StopFlag` to a nonempty character vector, such as `'y'`.\n\n• `options` — Options as modified by the output function. This argument is optional.\n\n• `optchanged` — Boolean flag indicating changes to `options`. To change `options` for subsequent iterations, set `optchanged` to `true`.\n\n#### Changing the State Structure\n\nCaution\n\nChanging the state structure carelessly can lead to inconsistent or erroneous results. Usually, you can achieve the same or better state modifications by using mutation or crossover functions, instead of changing the state structure in a plot function or output function.\n\n`ga` output functions can change the `state` structure (see The State Structure). Be careful when changing values in this structure, as you can pass inconsistent data back to `ga`.\n\nTip\n\nIf your output structure changes the `Population` field, then be sure to update the `Score` field, and possibly the `Best`, `NonlinIneq`, or `NonlinEq` fields, so that they contain consistent information.\n\nTo update the `Score` field after changing the `Population` field, first calculate the fitness function values of the population, then calculate the fitness scaling for the population. See Fitness Scaling Options.\n\n### Display to Command Window Options\n\n`'Display'` specifies how much information is displayed at the command line while the genetic algorithm is running. The available options are\n\n• `'final'` (default) — The reason for stopping is displayed.\n\n• `'off'` or the equivalent `'none'` — No output is displayed.\n\n• `'iter'` — Information is displayed at each iteration.\n\n• `'diagnose'` — Information is displayed at each iteration. In addition, the diagnostic lists some problem information and the options that have been changed from the defaults.\n\nBoth `'iter'` and `'diagnose'` display the following information:\n\n• `Generation` — Generation number\n\n• `f-count` — Cumulative number of fitness function evaluations\n\n• `Best f(x) `— Best fitness function value\n\n• `Mean f(x) `— Mean fitness function value\n\n• `Stall generations `— Number of generations since the last improvement of the fitness function\n\nWhen a nonlinear constraint function has been specified, `'iter'` and `'diagnose'` do not display the `Mean f(x)`, but will additionally display:\n\n• `Max Constraint` — Maximum nonlinear constraint violation\n\n### Vectorize and Parallel Options (User Function Evaluation)\n\nYou can choose to have your fitness and constraint functions evaluated in serial, parallel, or in a vectorized fashion. Set the `'UseVectorized'` and `'UseParallel'` options with `optimoptions`.\n\n• When `'UseVectorized'` is `false` (default), `ga` calls the fitness function on one individual at a time as it loops through the population. (This assumes `'UseParallel'` is at its default value of `false`.)\n\n• When `'UseVectorized'` is `true`, `ga` calls the fitness function on the entire population at once, in a single call to the fitness function.\n\nIf there are nonlinear constraints, the fitness function and the nonlinear constraints all need to be vectorized in order for the algorithm to compute in a vectorized manner.\n\nSee Vectorize the Fitness Function for an example.\n\n• When `UseParallel` is `true`, `ga` calls the fitness function in parallel, using the parallel environment you established (see How to Use Parallel Processing in Global Optimization Toolbox). Set `UseParallel` to `false` (default) to compute serially.\n\nNote\n\nYou cannot simultaneously use vectorized and parallel computations. If you set `'UseParallel'` to `true` and `'UseVectorized'` to `true`, `ga` evaluates your fitness and constraint functions in a vectorized manner, not in parallel.\n\nHow Fitness and Constraint Functions Are Evaluated\n\n`UseVectorized` = `false``UseVectorized` = `true`\n`UseParallel` = `false`SerialVectorized\n`UseParallel` = `true`ParallelVectorized\n\n## Support", null, "Get trial now" ]

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[ "# week 06: the trouble with p-values\n\n## it all started so well\n\nGiven some observed data, it would be useful to know how surprising they are. Are these data consistent with what I'd expect by chance? If not, something more interesting might be going on.\n\nTake Laplace's question of the birth rate of boy vs. girls in Paris, for example. He observed 251,527 boys in 493,472 births. Is this 0.51 frequency surprisingly different from what we'd expect by chance?\n\nTo be quantitative, we have to specify exactly what chance means. We formulate a hypothesis called the \"null hypothesis\", $$H_0$$, so that we can calculate $$P(D \\mid H_0)$$, the probability distribution over different data outcomes, given the null hypothesis.\n\nIn the Laplace example, the obvious null hypothesis is that boys and girls are equiprobable: $$p = 0.5$$. The possible data outcomes, in $$n=493472$$ total births, are $$c = 0...493472$$ boys. The probability of any given count of boys $$c$$ is a binomial distribution $$P(c \\mid p, n) = {n \\choose c} p^c (1-p)^{n-c}$$.\n\nAs Laplace was aware, the probability of any specific outcome might be absurdly small, especially as $$n$$ gets large. A specific outcome can be unlikely (in the sense that you wouldn't have bet on exactly that outcome beforehand), but unsurprising (in the sense that it's one of many outcomes that are consistent with the null hypothesis). If $$p = 0.5$$, the probability of getting exactly 50% boys ($$c = 246736$$) is tiny, 0.001. But the probability of getting a number within $$\\pm$$ 1000 of 246736 is more than 99%. I calculated that using the binomial's cumulative distribution function in Python, which you're about to see.\n\nIndeed, we may be able to find a range of data that are consistent with the null hypothesis, even if any particular one outcome is unlikely, and then ask if our observed data is outside that plausible range. This requires that the data can be arranged in some sort of linear order, so that it makes sense to talk about a range, and about data outside that range. That's true for our counts of boys $$c$$, and it's also true of a wide range of \"statistics\" that we might calculate to summarize a dataset (for example, the mean $$\\bar{x}$$ of a bunch of observations $$x_1..x_n$$).\n\nFor example, what's the probability that we would observe $$c$$ boys or more in Laplace's problem, if p=0.5? For this, we (and Laplace) use a cumulative probability function (CDF), the probability of getting a result of $$x$$ or less:\n\n$$P(X \\leq x \\mid \\theta) = \\sum_{-\\infty}^{x} P(X = x \\mid \\theta)$$\n\nOur boy count $$c$$ is discrete (and only defined on $$c \\geq 0$$) so $$P(C \\geq 251527) = 1 - P(C \\leq 251526 \\mid p)$$, which of course we can get in Python's SciPy stats.binom module:\n\nimport scipy.stats as stats\n\nc = 251527\nn = 493472\np = 0.5\n1 - stats.binom.cdf(c-1,n,p)\n\n\nwhich gives us 1.1e-16.\n\nUh, wait a second. That's not what Laplace got.\n\n### computers are annoying\n\nHa! This number is totally wrong! The math is right, but computers are annoying. The number is so close to zero, we get garbage, from an immense floating-point rounding error. When numbers get very small, we have to worry about pesky details. So let's take a detour for a second into how machines do arithmetic, and where it can go wrong if you're not paying enough attention.\n\nOn a machine, in floating point math, $$1+\\epsilon = 1$$ for some small threshold $$\\epsilon$$. In double-precision floating-point math (what Python uses internally), the machine $$\\epsilon$$ is 1.1e-16. This is the smallest relative unit of magnitude that two floating point numbers can differ by. The result of stats.binom.cdf() is so close to 1 that the machine can't keep track of the precision; it just left its return value at one epsilon less than 1, and $$1 - (1-\\epsilon)$$ gives us $$\\epsilon$$.\n\nWe have to make sure that we never try to represent $$1 \\pm x$$ if we know $$x$$ might be small; we need to use $$x$$ instead. Here that means we want SciPy to tell us 1 - CDF instead of the CDF. That's got a name: the survival function, .sf() in SciPy. Let's try again:\n\n c = 251527\nn = 493472\np = 0.5\nstats.binom.sf(c-1,n,p)\n\n\nNow we get 1.2e-42, which is right. There's a tiny probability that we'd observe 251,527 boys or more, if the boy-girl ratio is 50:50.\n\n## definition of a p-value\n\nA p-value is the probability that we would have gotten a result at least this extreme, if the null hypothesis is true.\n\nWe get the p-value from a cumulative probability function $$P(X \\leq x)$$, so it has to make sense to calculate a CDF. There has to be an order to the data, so that \"more extreme\" is meaningful. Usually this means we're representing the data as a single number: either the data is itself a number ($$c$$, in the Laplace example), or a summary statistic like a mean.\n\nFor example, it wouldn't make sense to talk about the p-value of the result of rolling a die $$n$$ times. The observed data are six values $$c_1..c_6$$, and it's not obvious how to order them. We could calculate the p-value of observing $$c_6$$ sixes or more out of $$n$$ rolls, though. Similarly, it wouldn't make sense to talk about the p-value of a specific poker hand, but you could talk about the p-value of drawing a pair or better, because the value of a poker hand is orderable.\n\n### a p-value is a false positive rate\n\nRecall that a false positive rate is the fraction of false positives out of all negatives: $$\\frac{\\mathrm{FP}}{\\mathrm{FP} + \\mathrm{TN}}$$. If we consider our test statistic $$x$$ to be the threshold for defining positives, i.e. everything that scores at least $$x$$ is called positive, then the p-value and the false positive rate are the same thing: for data samples generated by the null hypothesis (negatives), what fraction of the time do they nonetheless score $$x$$ or greater?\n\nThis idea leads to a simple way of calculating p-values called order statistics. Generate $$N$$ synthetic negative datasets, calculate the score (test statistic) for each of them, and count the fraction of times that you get $$x$$ or more; that's the p-value for score $$x$$.\n\nAny biologist is familiar with this idea. Do negative controls. Simulate negative datasets and count how frequently a negative dataset gets a score of your threshold $$x$$ or more.\n\n### p-values are uniformly distributed on (0,1)\n\nIf the data were actually generated by the null hypothesis, and you did repeated experiments, calculating a p-value for each observed data sample, you would see that the p-value is uniformly distributed. By construction -- simply because it's a cumulative distribution! 5% of the time, if the null hypothesis is true, we'll get a p-value of $$< 0.05$$; 50% of the time, we'll get a p-value $$< 0.5$$.\n\nUnderstanding this uniform distribution of p-values is important. Sometimes people say that a result with a p-value of 0.7 is \"less significant\" than a result with a p-value of 0.3, but in repeated samples from the null hypothesis, you expect to obtain the full range of possible p-values from 0..1 in a uniform distribution. Seeing a p-value of 0.7 is literally equally probable as seeing a p-value of 0.3, or 0.999, or 0.001, under the null hypothesis. Indeed, seeing a uniform distribution is a good check that you're calculating p-values correctly.\n\n## null hypothesis significance testing\n\nP-values were introduced in the 1920's by the biologist and statistician Ronald Fisher. He intended them to be used as a tool for detecting unusual results:\n\n\"Personally, the writer prefers to set a low standard of significance at the 5 per cent point, and ignore entirely all results which fail to reach this level. A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.\"\n\nThere are three important things in this passage. First, it introduced $$P < 0.05$$ as a standard of scientific evidence. Second, Fisher recognized that this was a \"low standard\". Third, by saying \"rarely fails\", Fisher meant it to be used in the context of repeated experiments, not a single experiment: a true effect should reproducibly and repeatedly be distinguishable from chance.\n\nMany fields of science promptly forgot about the second two points and adopted $$P<0.05$$ as a hard standard of scientific evidence. A result is said to be \"statistically significant\" if it achieves $$P < 0.05$$. Sometimes, contrary to both logic and what Fisher intended, a single result with $$P<0.05$$ is publishable in some fields (cough cough, TED talks, cough cough). How this travesty happened, nobody quite seems to know.\n\nNowadays there's a backlash. Some people want to change the 0.05 threshold to 0.005, which rather misses the point. Some people want to ban P-values altogether.\n\nP-values are useful, if you're using them the way Fisher intended. It is useful to know when the observed data aren't matching well to an expected null hypothesis, alerting you to the possibility that something else may be going on. But 5% is a low standard -- even if the null hypothesis is true, 5% of the time you're going to get results with $$P<0.05$$. You need to see your unusual result reproduce consistently before you're going to believe in it.\n\nWhere you get into trouble is when you try to use a p-value as more than just a rule-of-thumb filter for potentially interesting results:\n\n• when you say that you've rejected the null hypothesis $$H_0$$, and therefore your hypothesis $$H_1$$ is true. A tiny p-value doesn't necessarily mean the data support some other hypothesis of yours, just because the data don't agree with the null hypothesis. Nothing about a p-value calculation tests any other hypothesis, other than the null hypothesis.\n\n• when you equate \"statistical significance\" with effect size. A miniscule difference can become statistically significant, given large sample sizes. The p-value is a function of both the sample size and the effect size. In a sufficiently large dataset, it is easy to get small p-values, because real data always depart from simple null hypotheses. This is often the case in large, complex biological datasets.\n\n• when you do multiple tests but you don't correct for it. Remember that the p-value is the probability that your test statistic would be at least this extreme if the null hypothesis is true. If you chose $$\\alpha = 0.05$$ (the \"standard\" significance threshold), you're going to get values that small 5% of the time, even if the null hypothesis is true: that is, you are setting your expected false positive rate to 5%. Suppose there's nothing going on and your samples are generated by the null hypothesis. If you test one sample, you have a 5% chance of erroneously rejecting the null. But if you test a million samples, 50,000 of them will be \"statistically significant\".\n\nMost importantly, using a p-value to test whether your favorite hypothesis $$H_1$$ is supported by the data is fundamentally illogical. A p-value test never even considers $$H_1$$; it only considers the null hypothesis $$H_0$$. \"Your model is unlikely; therefore my model is right!\" is just not the way logic works.\n\n### multiple testing correction\n\nSuppose you do test $$n=$$ one million things. What do you need your p-value to be (per test), to decide that any positive result you get in $$N$$ tests is statistically significant?\n\nWell, you expect $$np$$ false positives. The probability of obtaining one or more false positives is (by Poisson) $$1 - e^{-np}$$. This is still a p-value, but with a different meaning, conditioned on the fact that we did $$n$$ tests: now we're asking, what is the probability that we get result at least this extreme (at least one positive prediction), given the null hypothesis, when we do $$n$$ independent experiments? For small $$x$$, $$1-e^{-x} \\simeq x$$, so the multiple-test-corrected p-value is approximately $$np$$. That is, multiply your per-test p-value by the number of tests you did to get a \"corrected p-value\". Like many simple ideas, this simple idea has a fancy name: it's called a Bonferroni correction. It's considered to be a very conservative correction; I'll explain one of the main reasons why in a bit.\n\n### probability of what now?\n\nRather than the names of things, I'd rather that you remembered the principles. A lot of confusion stems from not knowing what probability we're talking about. Someone will say \"with a p of < 0.05\". Someone like me will say, yeah? probability of what? It's actually sort of crazy that we just write down \"p < 0.05\" like that's enough information to know what it means. It's important to understand what your null hypothesis actually is, for one thing! Also, there's at least two major different probabilities in play, because of the multiple testing issue.\n\nThe per-test p value is (something like) $$P(s > x \\mid H_0)$$: the probability that your test statistic would have been at least as extreme as $$x$$, if the null hypothesis were true.\n\nThe \"multiply corrected\" p-value is (something like) $$P( c(s > x) > 0 \\mid n, H_0)$$: the probability that, if you ran $$n$$ independent samples from the null hypothesis and counted the number of times a sample gave a score exceeding threshold $$x$$, you get at least one. Now the \"test statistic\" that we're testing for extreme-ness is the count of tests being declared positives.\n\nIf you don't explain which one you're using in your work, your reader doesn't know what you're talking about when you say \"p < 0.05\".\n\nOne way to make it clear without a lot of jargon is just to explain where your expected false positives are. If you say you ran a screen on $$n$$ samples at a threshold of $$p<0.05$$, I'm looking for where in your paper you put the 5% false positive predictions that you expected: there should've been about $$np$$ of them. I'll feel better that you know what you're doing if you acknowledge that they're in your results somewhere. That is, don't talk only in terms of \"statistical significance\"; in large dataset analysis, you can talk in terms of the number of expected false positives. If you made 100 positive predictions, you can say \"under a null hypothesis that (whatever it is), we would expect xxx of these to be false.\"\n\n### the false discovery rate (FDR)\n\nOne reason that the Bonferroni correction is conservative is the following. Suppose you run a genome-wide screen and you make 80,000 predictions. Do you really need all of them to be \"statistically significant\" on their own? That is, do you really need to know that the probability of even one false positive in that search is $$< 0.05$$ or whatever? More reasonably, you might say you'd like to know that 99% of your 80,000 results are true positives, and 1% or less of them are false positives.\n\nSuppose you tested a million samples to get your 80,000 positives, at a per-test p-value threshold of $$< 0.05$$. By the definition of the p-value you expected up to 50,000 false positives, because in the worst case, all million samples are in fact from the null hypothesis, and at a significance threshold $$\\alpha = 0.05$$, you expect 5% of them to be called as (false) positives. So if you trust your numbers, at least 30,000 of your 80,000 predictions (80000 positives - 50000 false positives) are expected to be true positives. You could say that the expected fraction of false positives in your 80,000 positives is 50000/80000 = 62.5%.\n\nThis is called a false discovery rate calculation -- specifically, (fancy names for simple ideas again) it is called the Benjamini-Hochberg FDR.\n\nThe false discovery rate (FDR) is the proportion of your called \"positives\" that are expected to be false positives, given your p-value threshold, the number of samples you tested, and the number of positives that were \"statistically significant\".\n\n## what Bayes says about p-values\n\nA good way to see the issues with using p-values for hypothesis testing is to look at a Bayesian posterior probability calculation. Suppose we're testing our favorite hypothesis $$H_1$$ against a null hypothesis $$H_0$$, and we've collected some data $$D$$. What's the probability that $$H_1$$ is true? That's its posterior:\n\n$$P(H_1 \\mid D) = \\frac{ P(D \\mid H_1) P(H_1) } { P(D \\mid H_1) P(H_1) + P(D \\mid H_0) P(H_0) }$$\n\nTo put numbers into this, we need to be able to calculate the likelihoods $$P(D \\mid H_1)$$ and $$P(D \\mid H_0)$$, and we need to know how the priors $$P(H_0)$$ and $$P(H_1)$$ -- how likely $$H_0$$ and $$H_1$$ were before the data arrived.\n\nWhat does p-value testing give us? It gives us $$P( s(D) \\geq x \\mid H_0)$$: the cumulative probability that some statistic of the data $$s(D)$$ has a value at least as extreme as $$x$$, under the null hypothesis.\n\nWe don't know anything about how likely the data are under our hypothesis $$H_1$$. We don't know how likely $$H_0$$ or $$H_1$$ were in the first place. And we don't even know $$P(D \\mid H_0)$$, really, because all we know is a related cumulative probability function of $$H_0$$ and the data.\n\nTherefore it is utterly impossible (in general) to calculate a Bayesian posterior probability, given a p-value -- which means, a p-value simply cannot tell you how much evidence your data give in support of your hypothesis $$H_1$$.\n\n(This was the fancy way of saying that just because the data are unlikely given $$H_0$$ does not logically mean that $$H_1$$ must be true.)\n\n### what Nature (2014) said about p-values\n\nThe journal Nature ran a commentary article in 2014 called \"Statistical errors\", about the fallacies of p-value testing. The article showed one figure, reproduced to the right. The figure shows how a p-value corresponds to a Bayesian posterior probability, under three different assumptions of the prior odds, for $$P=0.05$$ or $$P=0.01$$. It shows, for example, a result of $$P=0.01$$ might be \"very significant\", but the posterior probability of the null hypothesis might still be quite high: 30%, if the null hypothesis was pretty likely to be true to begin with. The commentary was trying to illustrate the point that the p-value is not a posterior probability, and that a \"significant\" p-value does not move the evidence as much as you might guess.\n\nFigure from Nuzzo, \"Statistical errors\", Nature (2014), showing how a P-value affects a posterior probability.", null, "But hang on, what? I just finished telling you that it is utterly impossible (in general) to calculate a posterior probability from a p-value, and here we have Nature doing exactly that. What's going on? The p-value literature is often like this: many righteous voices declaiming what's wrong with p-values, while making things even more confusing by leaving out important details and/or making subtle errors. Which means I'm probably doing the same thing, even as I try not to!\n\nThe key detail in the Nature commentary flashes by in a phrase -- \"According to one widely used calculation...\", and references a 2001 paper from statistican Steven Goodman. Let's look at that calculation and see if we can understand it, and if we agree with its premises.\n\nFirst we need some background information on statistical tests involving Gaussian distributions.\n\n### differences of means of Gaussian distributions\n\nSuppose I've collected a dataset with a mean value of $$\\bar{x}$$, and suppose I have reason to expect, in replicate experiments, that the mean $$\\bar{x}$$ is normally (Gaussian) distributed with a standard error SE$$_{\\bar{x}}$$. (I'll explain \"standard error\" sometime soon - for now, it's just the standard deviation of our observed means, in repeated experiments.) My null hypothesis is that the true mean is $$\\mu$$. I want to know if my observed $$\\bar{x}$$ is surprisingly far from $$\\mu$$ -- that is: what is the probability that I would have observed an absolute difference $$|\\bar{x} - \\mu|$$ at least this large, if I were sampling $$\\bar{x}$$ from a Gaussian of mean $$\\mu$$ and standard deviation $$\\sigma = \\mathrm{SE}_{\\bar{x}}$$?\n\nA Gaussian probability density function is defined as: And where's this come from? Magic? Turns out that we can derive the Gaussian from first principles with a little calculus, if we assume we seek the least informative -- i.e. maximum entropy -- distribution that is constrained to have some mean $$\\mu$$ and variance $$\\sigma^2$$. If instead we only constrain to a mean $$\\mu$$, we derive the exponential distribution. We'll leave this aside, and maybe get back to it someday, since it's fun to see.\n\n$$P(x \\mid \\mu, \\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} e^{- \\frac{(x - \\mu)^2}{2 \\sigma^2}}$$\n\nA useful thing to notice about Gaussian distributions is that they're identical under a translation of $$x$$ and $$\\mu$$, and under a multiplicative rescaling of $$\\sigma$$. The probability only depends on the ratio $$(x - \\mu) / \\sigma$$: that is, on the number of standard deviations away from the parametric mean. So, if I calculate a so-called Z-score:\n\n$$Z = \\frac{x - \\mu} {\\sigma}$$\n\nthen I can talk in terms of a simplified standard normal distribution:\n\n$$P(Z) = \\frac{1}{\\sqrt{2 \\pi}} e^{- \\frac{Z^2}{2}}$$\n\nThis is a very useful simplification - among other things, it'll be easier to remember things in units of \"number of standard deviations away from the mean\", and will help you develop general, quantitative intuition for Gaussian-distributed quantities.\n\nThe probability that we get a Z score at least as extreme as $$z$$ is an example of a p-value:\n\n$$P(Z \\geq z) = \\int_z^{\\infty} P(Z)$$\n\nWe might be interested not just in whether our mean $$\\bar{x}$$ is surprisingly larger than $$\\mu$$, but also if it's surprisingly smaller. That's the difference between what statisticians call a \"one-tailed\" test versus a \"two-tailed\" test. In a one-tailed test, I'm specifically testing whether $$P(Z \\geq z)$$, for example; in a two-tailed test, I'm testing the absolute value, $$P(|Z| \\geq z)$$. The Gaussian is symmetric, so $$P(|Z| \\geq z) = P(Z \\leq -z) + P(Z \\geq z) = 2 P(Z \\geq z)$$.\n\nWe get $$P(Z \\geq z)$$ from the Gaussian cumulative distribution function (CDF):\n\n$$P(Z \\geq z) = 1 - P(Z < z) = 1 - \\mathrm{CDF}(z)$$\n\n(Because this is a continuous function, $$P(Z < z) = P(Z \\leq z)$$ asymptotically; there's asymptotically zero mass exactly at $$P(Z = z)$$.)\n\nThere's no analytical expression for a Gaussian CDF, but it can be computed numerically. In Python, the scipy.stats.norm module includes a CDF method and more:\n\n from scipy.stats import norm\n\nz = 1.96\n1 - norm.cdf(z) # gives one-tailed p-value P(Z >= 1.96) = 0.025\nnorm.sf(z) # 1-CDF(x) is the \"survival function\"\n\np = 0.05 # you can invert the survival function with .isf():\nnorm.isf(p) # given a 1-tailed p-value P(Z > z), what Z do you need? (gives 1.64)\nnorm.isf(p/2) # or for a two-tailed P(|Z| > z) (gives 1.96)\n\n\nNow we've got enough background (and Python) to get back to the calculation that Goodman makes, that the Nature commentary is based on.\n\n### back to Goodman's calculation\n\nCrucially, for a Gaussian-distributed statistic, if I tell you the p-value, you can calculate the Z-score (by inverting the CDF); and given the Z-score, you can calculate the likelihood $$P(Z)$$. Thus in this case we can convert a P-value to a likelihood of the null hypothesis for a $$Z$$-score (that we got from our mean $$\\bar{x}$$):\n\n$$P(Z \\mid H_0) = \\frac{1}{\\sqrt{2 \\pi}} e^{- \\frac{Z^2}{2}}$$\n\nFor example, a p-value of $$P = 0.05$$ for a two tailed test $$P(|Z| > z)$$ implies Z = 1.96, by inverting the CDF. Roughly speaking, 5% of the time, we expect a result more than 2 standard deviations away from the mean on either side.\n\nSo now we've got $$P(Z \\mid H_0)$$. That's one of our missing terms dealt with, in the Bayesian posterior probability calculation. How about $$P(Z \\mid H_1)$$? Well, that's the tricksy one.\n\nObserve that the best possible hypothesis $$H_1$$ is that its mean $$\\mu$$ happens to be exactly the observed mean of the data $$\\bar{x}$$. If so, then $$(\\bar{x} - \\mu) = 0$$, thus $$Z = 0$$, so:\n\n$$P(\\bar{x} \\mid H_1) < \\frac{1}{\\sqrt{2 \\pi}}$$\n\n(In part it's an upper bound because it's cheating to choose this as our $$H_1$$ after we've looked at the data; we'd actually have to look at a range of possible $$\\mu_1$$ values for $$H_1$$, with a prior distribution. We'll come back to this.)\n\nNow we can calculate a bound on the likelihood odds ratio, the so-called \"Bayes factor\" for the support for the null hypothesis:\n\n$$\\frac{P(\\bar{x} \\mid H_0)} {P(\\bar{x} \\mid H_1)} > e^{-\\frac{Z^2}{2}}$$\n\nThe \"Bayes factor\" represents how much the relative odds in favor of the null hypothesis changes after we observed the data.\n\nWe still need to deal with the prior probabilities $$P(H_0)$$ and $$P(H_1)$$. Recall that we can rearrange the posterior in terms of the Bayes factor and the prior odds:\n\n$$P(H_0 \\mid \\bar{x}) = \\frac{P(\\bar{x} \\mid H_0) P(H_0)} {P(\\bar{x} \\mid H_0) P(H_0) + P(\\bar{x} \\mid H_1) P(H_1)} = \\frac { \\frac{P(\\bar{x} \\mid H_0) P(H_0)} {P(\\bar{x} \\mid H_1) P(H_1)} } { \\frac{P(\\bar{x} \\mid H_0) P(H_0)} {P(\\bar{x} \\mid H_1) P(H_1)} + 1} = \\frac { \\frac{P(\\bar{x} \\mid H_0) } { P(\\bar{x} \\mid H_1)} } { \\frac{P(\\bar{x} \\mid H_0) } { P(\\bar{x} \\mid H_1)} + \\frac{P(H_1)}{P(H_0)}}$$\n\nSince we have a lower bound on the Bayes factor, we're also going to get a lower bound on the posterior probability of the null hypothesis.\n\nNow we're ready to plug numbers in, to see what Goodman is doing.\n\nSuppose $$H_1$$ and $$H_0$$ are 50:50 equiprobable a priori, and you observe a mean $$\\bar{x}$$ that is $$Z=1.96$$ standard deviations away from the null hypothesis' $$\\mu$$. The two-tailed P-value is 0.05. The minimum Bayes factor is $$e^{-\\frac{Z^2}{2}} = 0.15$$. The posterior probability of the null hypothesis is no less than $$\\frac{0.15} {0.15 + 1} = 13$$%. Obtaining our \"significant\" P-value of 0.05 only moved our confidence in the null hypothesis from 50% to 13%.\n\nNow suppose that the null hypothesis is more likely a priori, with prior probability 95%. (In many biological experiments, we're usually going to observe unsurprising, expected results.) Now the posterior probability of the null is $$\\frac{0.15} {0.15 + \\frac{0.05}{0.95}} = 74$$%. Our \"significant\" P-value hardly means a thing -- it's still 74% probable that the null hypothesis is true. If this is how we did science -- if we published all our \"statistically significant\" results, with only the p-value as evidence -- 74% of our papers would be wrong.\n\nThis is the gist of it, though the numbers in Nuzzo's 2014 Nature commentary are actually based on a second calculation that's a step more sophisticated than this. Instead of saying that $$H_1$$ has exactly $$\\mu = \\bar{x}$$ (which, as we said, is somewhat bogus, choosing your hypothesis to test after looking at the data), you can do a version of the calculation where you say that any $$\\mu_1$$ is possible for $$H_1$$, with a prior distribution symmetrically decreasing around $$\\mu_0$$. Under this calculation, the probability $$P(\\bar{x} \\mid H_1)$$ is lower, so the bound on the Bayes factor is higher -- so the posterior probability of the null hypothesis is decreased even less, for a given p-value. Thus Nuzzo's figure says \"29% chance of no real effect\" for the p=0.05/50:50 case and \"89% chance of no real effect\" for the p=0.05/95:5 case, instead of the 13% and 74% I just calculated. Table 1 in the Goodman paper shows both variants of the calculation, for a range of p-values.\n\n### summary\n\nThus the exact numbers in Nuzzo's 2014 Nature commentary turn out to depend on very specific assumptions -- primarily, that the p-value comes from a hypothesis test where we're asking if a Gaussian-distributed mean is different from an expected value. There are many other situations in which we would use p-values and hypothesis testing. It is indeed true that in general you cannot convert a p-value to a Bayesian posterior probability.\n\nBut the general direction of Goodman's and Nuzzo's arguments is correct, and it is useful to see a worked example. A \"significant\" or even a \"highly significant\" p-value does does not mean that the null hypothesis has been disproven -- it can easily remain more probable than the alternative hypothesis." ]

[ null, "http://mcb112.org/w06/fig-nuzzo.png", null ]

[ "# Cellular Automata variant\n\nThis is a simple note on Turing completeness of 2-neighbourhood 1-dimensional Cellular Automata.\n\nA Cellular Automaton (pl. Cellular Automata) is a model of computation based on a grid of cells that evolve according to a simple set of rules. The grid can be multi-dimensional, the most famous and studied cellular automata are 1-dimensional and 2-dimensional. Each cell can be in a particular finite state ($k$-states CA), at each step of the evolution (generation) the cell changes its state according to its current state and the state of its neighboring cells. The number of neighbours considered is finite; for a 1-dimensional CA, the usual choice is the adjacent neighbours. A m-neighbourhood 1-dimensional CA is a CA in which the next state of cell $c_i$ depends on the state of $c_i$ and the state of the $m-1$ adjacent cells; usually $c_i$ is the central cell.\n\nFormally, if $t_n(c_i)$ is the state of cell $c_i$ at time $t_n$:\n\n$$t_{n+1}(c_i) = f( t_n(c_{i-\\ell}), …, t_n(c_{i-2}), t_n(c_{i-1}), t_n(c_i), t_n(c_{i+1}), t_n(c_{i+2}), …, t_n(c_{i+\\ell}) )$$\n\nand $m = 2\\ell+1$.\n\nFor more details and basic references see: Das D. (2012) A Survey on Cellular Automata and Its Applications.\n\nEven basic cellular automata can be Turing Complete, for example see the (controversial) proof of Turing completeness of the 3-neighbourhood Cellular Autaton identified by Rule 110 .\n\nIf only 1 neigbour is considered (i.e. the next state of a cell only depend on its current state), then we cannot achieve Turing completeness. But with 2-neighbourhood and enough states it’s possible to emulate every CA." ]

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[ "", null, "Download Presentation", null, "Evaluation of Relational Operations\n\n# Evaluation of Relational Operations - PowerPoint PPT Presentation", null, "Download Presentation", null, "## Evaluation of Relational Operations\n\n- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -\n##### Presentation Transcript\n\n1. Evaluation of Relational Operations Chapter 14, Part B (Joins)\n\n2. Schema for Examples Sailors (sid: integer, sname: string, rating: integer, age: real) Reserves (sid: integer, bid: integer, day: dates, rname: string) • Similar to old schema; rname added for variations. • Reserves: • Each tuple is 40 bytes long, 100 tuples per page, 1000 pages. • Sailors: • Each tuple is 50 bytes long, 80 tuples per page, 500 pages.\n\n3. Equality Joins With One Join Column SELECT * FROM Reserves R1, Sailors S1 WHERE R1.sid=S1.sid • In algebra: R S. Common! Must be carefully optimized. R S is large; so, R S followed by a selection is inefficient. • Assume: M pages in R, pR tuples per page, N pages in S, pS tuples per page. • In our examples, R is Reserves and S is Sailors. • We will consider more complex join conditions later. • Cost measure: # of I/Os. We will ignore output costs.\n\n4. Simple Nested Loops Join foreach tuple r in R do foreach tuple s in S do if ri == sj then add <r, s> to result • For each tuple in the outer relation R, we scan the entire inner relation S. • Cost: M + pR * M * N = 1000 + 100*1000*500 I/Os. • Page-oriented Nested Loops join: For each page of R, get each page of S, and write out matching pairs of tuples <r, s>, where r is in R-page and S is in S-page. - Cost: M + M*N = 1000 + 1000*500 • If smaller relation (S) is outer, cost = 500 + 500*1000\n\n5. Index Nested Loops Join foreach tuple r in R do foreach tuple s in S where ri == sj do add <r, s> to result • If there is an index on the join column of one relation (say S), can make it the inner and exploit the index. • Cost: M + ( (M*pR) * cost of finding matching S tuples) • For each R tuple, cost of probing S index is about 1.2 for hash index, 2-4 for B+ tree. Cost of then finding S tuples (assuming Alt. (2) or (3) for data entries) depends on clustering. • Clustered index: 1 I/O (typical), unclustered: up to 1 I/O per matching S tuple.\n\n6. Examples of Index Nested Loops • Hash-index (Alt. 2) on sid of Sailors (as inner): • Scan Reserves: 1000 page I/Os, 100*1000 tuples. • For each Reserves tuple: 1.2 I/Os to get data entry in index, plus 1 I/O to get (the exactly one) matching Sailors tuple. Total: 220,000 I/Os. • Hash-index (Alt. 2) on sid of Reserves (as inner): • Scan Sailors: 500 page I/Os, 80*500 tuples. • For each Sailors tuple: 1.2 I/Os to find index page with data entries, plus cost of retrieving matching Reserves tuples. Assuming uniform distribution, 2.5 reservations per sailor (100,000 / 40,000). Cost of retrieving them is 1 or 2.5 I/Os depending on whether the index is clustered.\n\n7. . . . Block Nested Loops Join • Use one page as an input buffer for scanning the inner S, one page as the output buffer, and use all remaining pages to hold ``block’’ of outer R. • For each matching tuple r in R-block, s in S-page, add <r, s> to result. Then read next R-block, scan S, etc. R & S Join Result Hash table for block of R (k < B-1 pages) . . . . . . Output buffer Input buffer for S\n\n8. Exercise Simulate a block-nested loop join of these two tables with Sailors are the outer. Assume that you read just one tuple at a time (i.e., 1 tuple per page), and that you have 5 buffers. How many I/Os do you need?\n\n9. Examples of Block Nested Loops • Cost: Scan of outer + #outer blocks * scan of inner • #outer blocks = • With Reserves (R) as outer, and 100 pages of R in block: • Cost of scanning R is 1000 I/Os; a total of 10 blocks. • Per block of R, we scan Sailors (S); 10*500 I/Os. • If space for just 90 pages of R, we would scan S 12 times. • With 100-page block of Sailors as outer: • Cost of scanning S is 500 I/Os; a total of 5 blocks. • Per block of S, we scan Reserves; 5*1000 I/Os.\n\n10. Sort-Merge Join (R S) i=j • Sort R and S on the join column, then scan them to do a ``merge’’ (on join col.), and output result tuples. • Advance scan of R until current R-tuple >= current S tuple, then advance scan of S until current S-tuple >= current R tuple; do this until current R tuple = current S tuple. • At this point, all R tuples with same value in Ri (current R group) and all S tuples with same value in Sj (current S group) match; output <r, s> for all pairs of such tuples. • Then resume scanning R and S. • R is scanned once; each S group is scanned once per matching R tuple. (Multiple scans of an S group are likely to find needed pages in buffer.)\n\n11. Example of Sort-Merge Join • Cost: M log M + N log N + (M+N) • The cost of scanning, M+N, could be M*N (very unlikely!) • With 35, 100 or 300 buffer pages, both Reserves and Sailors can be sorted in 2 passes; total join cost: 7500. (BNL cost: 2500 to 15000 I/Os)\n\n12. Exercise Simulate a sort-merge join of these two tables. Assuming that you read just one tuple at a time (i.e., 1 tuple per page), how many I/Os do you need?\n\n13. Original Relation Partitions OUTPUT 1 1 2 INPUT 2 hash function h . . . B-1 B-1 B main memory buffers Disk Disk Partitions of R & S Join Result Hash table for partition Ri (k < B-1 pages) hash fn h2 h2 Output buffer Input buffer for Si B main memory buffers Disk Disk Hash-Join • Partition both relations using hash fn h: R tuples in partition i will only match S tuples in partition i. • Read in a partition of R, hash it using h2 (<> h!). Scan matching partition of S, search for matches.\n\n14. Exercise Simulate a hash join of these two tables with Sailor as the outer. Assuming 4 buffers and that you read just one tuple at a time (i.e., 1 tuple per page), how many I/Os do you need?\n\n15. Observations on Hash-Join • #partitions k < B-1, and B-2 > size of largest partition to be held in memory. Assuming uniformly sized partitions, and maximizing k, we get: • k= B-1, and M/(B-1) < B-2, i.e., B must be > • If we build an in-memory hash table to speed up the matching of tuples, a little more memory is needed (“fudge factor”). • If the hash function does not partition uniformly, one or more R partitions may not fit in memory. Can apply hash-join technique recursively to do the join of this R-partition with corresponding S-partition.\n\n16. Cost of Hash-Join • In partitioning phase, read+write both relns; 2(M+N). In matching phase, read both relns; M+N I/Os. • In our running example, this is a total of 4500 I/Os. • Sort-Merge Join vs. Hash Join: • Given a minimum amount of memory both have a cost of 3(M+N) I/Os. Hash Join superior on this count if relation sizes differ greatly. Also, Hash Join shown to be highly parallelizable. • Sort-Merge less sensitive to data skew; result is sorted.\n\n17. General Join Conditions • Equalities over several attributes (e.g., R.sid=S.sid ANDR.rname=S.sname): • For Index NL, build index on <sid, sname> (if S is inner); or use existing indexes on sid or sname. • For Sort-Merge and Hash Join, sort/partition on combination of the two join columns. • Inequality conditions (e.g., R.rname < S.sname): • For Index NL, need (clustered!) B+ tree index. • Range probes on inner; # matches likely to be much higher than for equality joins. • Hash Join, Sort Merge Join not applicable. • Block NL quite likely to be the best join method here." ]

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[ "# Parameters: Simulation\n\nModel ElementParam_Simulation defines the solution control parameters for simulations that are associated with more than one analysis method. Exceptions are noted.\n\n## Format\n\n<Param_Simulation\n[ constr_tol = \"real\" ]\n[ implicit_diff_tol = \"real\" ]\n[ nr_converge_ratio = \"real\" ]\n[ linsolver = \"HARWELL\" ]\n[ usrsub_param_string = \"USER([par1,par2,…,parN])\"\n{ usrsub_dll_name = \"string\"\n|\ninterpreter = \"PYTHON | MATLAB\"\nscript_name = \"string\"\n}\nusrsub_fnc_name = \"string\"\n]\n[ acusolve_cosim = \"TRUE | FALSE\" ]\n/> \n\n## Attributes\n\nconstr_tol\nDefines the accuracy to which the system configuration and motion constraints are to be satisfied at each step. This is used by kinematics and the transient solver in the ODE formulation (ABAM, VSTIFF, or MSTIFF), but not the DAE form, which has its own tolerance for this (Param_Transient::dae_constr_tol). This should be a small positive number.\nThe default value for constr_tol = 1.0E-10.\nimplicit_diff_tol\nDefines the accuracy to which implicit differential equations, such as Control_Diff equations with the is_implicit = \"TRUE\", are to be satisfied. The default value for implicit_diff_tol = 1.0E-5.\nnr_converg_ratio\nDefines a measure of the rate of convergence in the Newton-Raphson method for ODE solvers. If the maximum entry in the constraint vector is larger than nr_converg_ratio times the maximum entry from the previous iteration, the Newton-Raphson iterations are converging slowly and the generalized coordinates will be re-partitioned in the next integration step to select a new set of independent coordinates. This attribute is applicable only if an ODE integrator (ABAM, VSTIFF, or MSTIFF) is selected.\n\nThe default value is 0.09.\n\nlinsolver\nThe type of linear solver used, which currently is set to HARWELL for all analysis.\nusrsub_param_string\nThe list of parameters that are passed from the data file to the user defined subroutine, defined by usrsub_fnc_name. This attribute is common to all types of user subroutines and scripts.\nusrsub_dll_name\nSpecifies the path and name of the shared library containing the user subroutine. MotionSolve uses this information to load the user subroutine at run time.\nusrsub_fnc_name\nSpecifies an alternative name for the user subroutine TUNSUB. For details on the definition and syntax of the TUNSUB, please refer to the TUNSUB documentation.\nscript_name\nSpecifies the path and name of the user written script that contains the routine specified by usrsub_fnc_name.\ninterpreter\nSpecifies the interpreted language that the user script is written in. Valid choices are MATLAB or PYTHON.\nacusolve_cosim\nA Boolean flag that determines whether the simulation will be coupled with AcuSolve or not.\n\"TRUE\" means that the MBD model is to be coupled with the CFD model in a co-simulation between MotionSolve and AcuSolve.\n\"FALSE\" means that the MBD model is not to be coupled with the CFD model. The default value is FALSE.\n\n## Example\n\nThe example below shows the default settings for the Param_Simulation modeling element.\n\n<Param_Simulation\nconstr_tol = \"1E-10\"\nimplicit_diff_tol = \"1E-5\"\nnr_converg_ratio = \"0.09\"\nlinsolver = \"HARWELL\"\n/>\n\n1. Linear Solvers: At the core of most analyses in MotionSolve is solving a set of linear equations (A x = b). For example, the Newton-Raphson method solves a set of linear equations as part of the iteration process to find the solution to a set of non-linear algebraic equations. A brief explanation of how the linear solver works follows.\n\nThe Harwell Linear Solver is a tool for solving linear algebraic equations. It is especially suited for linear systems characterized by non-singular, unsymmetrical and sparse coefficient matrices A that have a fixed non-zero entry pattern. The latter implies that while matrix entries are allowed to change with time, the pattern of non-zeroes is not. The matrix entries must be real.\n\nThis methodology is suitable for solving small to medium sets of equations (< 10,000 equations) and is therefore quite suitable for multi-body systems simulation. The Harwell software solves the equations in three major steps:\n1. Symbolic LU factorization.\n2. Numeric LU factorization.\n3. Forward-Backward Substitution.\nSymbolic LU Factorization\nGiven a pattern of non-zero entries in A and their representative values, this function computes the symbolic lower- and upper-triangular (LU) factors of A. A partial pivoting scheme is used. It tries to maximize the stability of the LU factors while still maintaining sparsity of the factors. This operation is typically done once or only a few times during the simulation.\nNumeric LU Factorization\nGiven the current values of the non-zero entries in A and the symbolic LU factors, this utility returns the numeric LU factors of A. The symbolic LU factorization must therefore precede the numeric LU factorization. This operation is done whenever a new Jacobian is needed.\nForward-Backward Substitution\nGiven the numeric LU factors of a sparse matrix A and an appropriately sized right-hand-side vector b, this utility returns the solution x for the linear problem. The Numeric LU factorization must precede the forward-backward substitution operation. This operation is performed at each iteration." ]

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[ "", null, "## May 9, 2019 Role Of Statistics in Data Science\n\nData Science and Statistics definitions from the Data Science Association’s , “personal code of conduct”:\n\n“Data Scientists” means a professional who uses scientific methods to liberate and create meaning from raw data.\n\nTo a statistician this sounds like a lot similar what applied statistician do : use methodology to make inferences from data\n\n“Statistics” means the practice or science of collecting and analyzing numerical data in large quantities.\n\n#### Relation Between the two fields, “Statistics & Data Science”:\n\nDoes statistics plays a very crucial role in the field of Data Science and the answers is yes in most cases . Statistics is foundational to Data Science , there is strong relationship between these two fields.Statistics is one of the most important discipline that provide tools and methods to find structure and give deeper insights into data.\n\nMachine Learning is rapidly growing field at the intersection of computer science and statistics concerned with finding patterns in data. It is responsible for various advancements in technology,  product recommendations to Speech recognition to Autonomous driving ; machine learning is having presence in almost the fields.\n\n#### Is Data Science all about statistics?\n\nNo, Data science is not only statistics, it is the field which is comprised of Statistics, Probability, Mathematics(mainly Linear Algebra and Calculus) and Programming.\n\n#### How does Data Scientists takes advantage of  statistics knowledge?\n\nBuilding models using popular statistical methods such as Regression, Classification, Time Series Analysis and Hypothesis Testing. Data Scientists run suitable experiments and interpret the results with the help of these statistical methods.\n\nStatistics is also used for summarizing the data fairly quickly.\n\n#### How one can applies its knowledge in Statistics and Data Science?\n\nThere are numerous challenges posted on Topcoder of various level and domains are being posted on https://www.topcoder.com/challenges?filter[tracks][data_science]=true&tab=details . Anyone can choose the appropriate challenge as per individual interest and skills and work on those challenges. Apart from getting the experience of working on real data science challenges participant is rewarded with the money prizes.\n\nExample :\n\nLet’s discuss about the role of statistics in Machine Learning (Machine Learning is itself an integral part of Data Science) with an example of Classification Problem.\n\nBefore diving right into how we can solve classification problem using popular statistical model let’s first have some brief about what exactly are classification problem.\n\nClassification problems are essential part of Machine Learning . Around 70% problems in Data Science are Classification problems. Classification problems are the ones which have a qualitative response such an email is a “Spam” or  “Not Spam” or a Cancer is of type “Malignant” or “Benign”. The method used for classification first predict the probability of each of the categories of a qualitative variable as a basis for making the classification.\n\nThere are number of classification techniques that might one can use to predict the qualitative response. Most widely used classification techniques are:\n\n• Logistic Regression\n• Linear Discriminant Analysis\n• K-nearest neighbors\n\nLet’s talk about more on Logistic regression in this blog post.\n\nLogistic Regression is one of the most popular classification method for predicting the qualitative response for ex: predicting that patient has cancer or not or predicting whether a particular customer will churn or not.\n\nLogistic Regression does not gives a straight line fit for predicting the response but uses the Logistic function for the prediction.", null, "", null, "The Logistic function described above will always produce an S-shaped curve and can take any real value between 0 and 1.  This function is also known as Sigmoid function.\n\n#### Sigmoid Function\n\nIn sigmoid function if the curve goes to positive infinity the predicted value will be close to 1 and if the curve goes to negative infinity the predicted value  will be close to 0.\n\nThe coefficients (", null, ") mentioned in the logistic function needs to be estimated based on the available training data. Maximum Likelihood Estimation (MSE) is the preferred method for estimating the coefficients since it has better statistical properties. The basic idea behind the Maximum Likelihood to fit a logistic regression model is it seeks estimates of coefficients", null, "such that predicted probability is as close as the actual observation.\n\nLet’s build logistic regression model for diabetes prediction\n\nWe will be taking the dataset from  kaggle website. Link for the dataset is :https://www.kaggle.com/uciml/pima-indians-diabetes-database\n\nImport pandas as pd\ncolumn_names = ['pregnant', 'glucose', 'bp', 'skin', 'insulin', 'bmi', 'pedigree', 'age', 'label']\n\n#split dataset in features and target variable\nfeature_cols = ['pregnant', 'insulin', 'bmi', 'age','glucose','bp','pedigree']\nX = pima[feature_cols] # Features\ny = pima.label # Target variable\n\n\nSplitting the dataset into train and test data is good strategy to analyze model performance.\n\n# split X and y into training and testing sets\nfrom sklearn.cross_validation import train_test_split\nX_train,X_test,y_train,y_test=train_test_split(X,y,test_size=0.25,random_state=0)\n\n\n#### Model Development and prediction\n\nFor Model development and prediction we will be using Scikit-learn library logistic regression function\n\nWe will use fit() to fit the model and predict() for prediction on test data\n\n# import the class\nfrom sklearn.linear_model import LogisticRegression\n# instantiate the model (using the default parameters)\nlogreg = LogisticRegression()\n# fit the model with data\nlogreg.fit(X_train,y_train)\n#\ny_pred=logreg.predict(X_test)\n\n\nPerformance Evaluation using confusion Matrix\n\n# import the metrics class\nfrom sklearn import metrics\ncnf_matrix = metrics.confusion_matrix(y_test, y_pred)\ncnf_matrix\n\n\nConfusion matrix is an array object. The dimension of this matrix is 2*2 matrix since it is a binary classification. In the output as shown below the 119 and 36 are actual predictions and 26 and 11 are incorrect predictions.\n\narray([[119, 11],\n[ 26, 36]])\n\nprint(\"Accuracy:\",metrics.accuracy_score(y_test, y_pred))\nprint(\"Precision:\",metrics.precision_score(y_test, y_pred))\nprint(\"Recall:\",metrics.recall_score(y_test, y_pred))\n\nAccuracy: 0.8072916666666666\nPrecision: 0.7659574468085106\nRecall: 0.5806451612903226\n\n\nAccuracy of the prediction model is 80% which is good accuracy rate.\n\nPrecision: precision is about being precise i.e how accurate the model is.In the above case the logistic regression model predicted patients who are going to suffer from diabetes , 76% of time patients have diabetes.\n\nRecall: Logistic regression model predicted 58% of time about the patients who have diabetes in test set.\n\n#### Conclusion:\n\nIn this post we learned that statistics  can therefore go a long way for data scientist to make solid and dependable business insights. We also learned about the very popular statistical model “Logistic regression” and how logistic regression can be used to make predictions about the classification problems.\n\nHopefully, you can now utilize the Logistic Regression technique to analyze your own datasets. Thanks for reading this blog.\n\nrkhemka\n\nGuest Blogger\n\ncategories & Tags\n\nUNLEASH THE GIG ECONOMY. START A PROJECT OR TALK TO SALES\nClose" ]

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[ "", null, "# Working with Even More Functions in Excel\n\nRelated Articles\n\nAs I showed in my first Excel functions article, the serious meat in any spreadsheet is the formulas and functions that perform the calculations. Excel offers functions that you can use to make the process of computing the results simpler for many standard calculations. In this article I'll show you some useful examples; I'll tell you how to find functions to perform specific tasks and what to consider when building your own formulas.\n\n### Practical Formulas\n\nProbably the first function you’ll have learned to use is =SUM, which calculates the sum of a range of numbers. This saves you from having to build simple additions one cell at a time. However this function will fail spectacularly if you have a filtered list.", null, "The SUBTOTAL function is required when totaling only visible values in a filtered list. (Click for larger image).\n\nIf you filter a list to show only a small range of numbers and use =SUM to total the visible numbers –you’ll be disappointed as the function sums all values in the selected range – visible and not. Instead, when you total a filtered list you must use the SUBTOTAL function and, in fact, if you select the AutoSum button to calculate the sum automatically Excel will apply the SUBTOTAL function where the area to sum is a filtered list.\n\nOnce you know the limitations you can write your own SUBTOTAL function – it's essentially the same as the SUM function but with an extra argument: =SUBTOTAL(9,A2:A10) – the number nine is the trigger that makes the SUBTOTAL function sum only the visible numbers in the filtered range A2:A10.\n\n### Working with Days\n\nWhen planning projects and calculating timeframes, you often need to know the number of workdays between two dates. The NETWORKDAYS function can do this ‑‑ it calculates the number of days between two dates ignoring Saturdays, Sundays and any holidays that you specify. You'll find the function in the Excel Analysis Toolpak. To check if the Toolpak is installed, choose Tools > Add-ins and then select the Analysis Toolpak checkbox.", null, "Use the Analysis Toolpak’s NETWORKDAYS function to calculate the workdays between two dates. (Click for larger image).\n\nTo use the NETWORKDAYS function, enter the holiday days as a series of dates. Select the cells containing the dates and name them by choosing Insert > Name > Define; type in the holidays and press OK. To calculate the NETWORKDAYS, type the start and end dates in cells A1 and A2. The formula =NETWORKDAYS (A1,A2,holidays) will return the number of workdays between the two dates excluding weekends and the dates in the range you've named holidays.\n\nTwo other useful date functions are =NOW() and =TODAY(). The function =NOW inserts the current date and time into the cell and the =TODAY function enters the current date. You may need to format the cells to a date or time format using Format > Cell > Number Format for the date and time data to show correctly.\n\n### Referring to a Cell\n\nTo refer to the contents of one cell in another cell, type the cell reference with an = symbol. So, to refer to the contents of cell C2 in another cell, type =C2. If the value in cell C2 changes in the future, this cell’s value will change, too. You can also refer to a cell on another worksheet by including the sheet name – for example, to refer to the contents of C2 in Sheet 3 of the workbook use =Sheet3!C2.\n\nPage 1 of 2", null, "" ]

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[ "CTS  Placement Paper   General - Other   -\n\nCTS  Placement Paper   General - Other   -\n\n• Posted by  FreshersWorld\n7 Jan, 2012\n\nThe  written  tests are  based  on  critical   reasoning  type questions. Word -based  problems,  verbal  ability, pattern recognition  and  pattern  matching, series  type, arithematic-based (including  functions  and  permutations) are usually  asked. This is only a sample paper. We are not providing you with all the questions - just some questions to give you a general idea of the test pattern.\n\nSECTION I - 8 questions based on series.\n\n1.\nThese questions involve interchange of letters in a word at particular locations and also interchanging letters adjacent to those particular locations.Certain other conditions may also be given For eg.Let the word be ABBAABA\nIf we apply 25 on this, it means we have to interchange the letters at positions 2 and 5, also we have to change the letters adjacent to positions 2 and 5 i.e.from A to B and B to A. A B B A A B after Step 1 i.e interchange of 2 and 5 becomes AABABB Now change adjacent elements of 2 and 5...finally answer becomes\nAns: B A A B B A\nQuestions 1-5 are based on the pattern with changed numbers as described above\n\nQuestions 6-8 are of the following type\n\nTo get AAABBD from BBBAAA what number should be applied:-\na) 25\nb) 34\nc) 25 & 34\nd) none\n\nSECTION II\n\n1. Given the following functions\n(1) f(n a b c ) = ac if n=1\n(2) f(n a b c) = f( n-1 a c b) + f( 1 a b c) + f( n-1 b a c ) if n > 1 Then what is the value   f( 2 a b c ) = ?\nAns: f( 2 a c b)= ab+ac+ bc\n\n2. Similar question on functions.\n\n3. [ Based on the function in the first question] For the function f( 4 a b c ) the number of terms is...?\nHint f( 4 a b c ) = f( 3 a c b ) + f( 1 a b c ) + f( 3 b a c ) etc.\n\n4. What is the value of the function  f( 5 a b c ) = ?\n\nSECTION III\n\nPermutations and Combinations. 8 Questions.\n\n1. r = number of flags;n = number of poles;Any number of flags can be accommodated on any single pole.\n\n1)r=5,n=5 The no. of ways the flags can be arranged ? Questions 2-5 are based on the above pattern\n\n6. r = 5 n = 3 . If first pole has 2 flags, third pole has 1 flag How many ways can the remaining be arranged?\n\nQuestions 7.& 8. are similar to Question 6.\n\nSECTION IV\nQuestion consisting of figures - Pattern-matching type.\n\nRefer R.S Agarwal's book on Analytical Reasoning & TMHs Quantitative ability book by Edgar Thorpe.\n\nSECTION V\n\nIn this section first part of compound word is given. Select meaning of the second part from the choice given:\n1. Swan\n2. Swans\n3. Fool\n4. Fools\n5. Stare\nFor all above 4 choices are given....\n\ng. Swan ---> Swansong (compound word)\na) category b) music c) television d) none\n\nAns: Swansong is compound word. But song is not given as an option. so (b) music is the answer\n\nAnalogies\n\n1. slur : speech : : smudge :?\nAns. writing\n\n2. epaulet : shoulder : : ring :?\nAns.finger\n\n3. vernacular : place : :  fingerprint : ?\nAns.identical\n\nOpposites\n\nQ. corpulent\nAns: emaciated\n\nQ. officious\nAns: pragmate\n\nQ. dextrous\nAns: clumsy\n\nThe following sentences are broken into 4 sections-\n\nA, B, C, D\nChoose the part which has a mistake\nMark (E) if you find no mistake.\n\nQ.A) psychologists point out that  B)  there are human processes C)   which does not involve D) the use of words\n\nAns. (C) which does not involve (do)\n\nQ.A)jack ordered for B)two plates of chicken C)and a glass D)of water\n\nAns. (A)jack ordered for\n\nThe following is a group of questions is based on a passage or a set of conditions for each question.\nSelect the best answer choice given.\n\n(i). If it is fobidden by law if the object of agreement is the doing of an act, that is forbidden by law the agreement is\n\nvoid.\n\n(ii). If it is of the nature that,it would defeat the provision of any law is the agreement is void.if the object of agreement\n\nis such that thing got directly forbidden by law it would defeat the provision of statuary law.\n\n(iii). If the object of agreement is fraddulent it is void.\n\n(iv). An object of agreement is void if it involves or implies to the personnal property of another.\n\n(v). An object of agreement is void where the constant regards as ignored.\n\n(vi). An object of agreement is void where the constant regards is as opposed to public policy.\n\nQ. An algorithm follws a six step process za,zb,zc,zd,ze,zf, it is governed by the following\n\n(ii) the first may be za,zd or zf\n(iii) zb and zc have to be performed after zd\n(iv) zc must be immediately after zb\n\nQ. If za is the first set zd must be\n\na) 3rd\nb) 5th\nc) 2nd\nd) 4th\n\nQ. If zb must follow za then za can be\n\na) third or fourth\nb) first or second\nc) can not be third\nd) fouth or fifth\ne) none\n\nQ. If ze is third term the number of different operations possible are\n\nThe following questions are based on the given statements\nRavi plants six seperate saplings -- x,y,z,w,u,v in rows no 1 to 6 ,according to the follwing conditions\nHe must plant x before y and u\nHe must plant y and w\nThe third has to be z\n\nQ. Which of the following is acceptable\n\na) xuywzv\nb) xvzyuw\nc) zuyxwv\nd) zvxuwy\ne) wyzuvx\n\nQ. Which of the following is true\n\na) z before v\nb) z before x\nc) w before u\nd) y before u\ne) x before w\n\nQ. If he plants v first, then which can be planted second\n\na) x\nb) y\nc) z\nd) w\ne) u\n\nQ. Which of the following describes a correct combination of sapling and row?\n\na)  x,3\nb) y,6\nc) z,1\nd) w,2\ne) u,6\n\nQ. If he plants b 6th which would be planted first and second\n\na) x and w\nb) x and y\nc)y and x\nd)w and z\ne) w and u\n\nQ. If he plants w before u and after v he should plant w at\n\na) first\nb) second\nc) fourth\nd) fifth\ne) sixth\n\nQ. At a certain moment a watch shows 2 min lag although it is running fast If it showed a 3 min lag at that moment, but\n\nalso gains by 1/2 min more a day than its current speed it would show the true time one day sooner than it usually\n\ndoes.\nHow many mins does the watch gain per day.\n\na).2\nb).5\nc).6\nd).4\ne).75\n\nQ. In 400m race A gives B a start of 7 sec and beats him by 24 sec. In another race A beats B by 10 sec.the speeds\n\nare in the ratio\n\na)8:7\nb)7:6\nc)10:8\n\nd)6:8\ne)12:10\n\nQ.   3x+4y=10\nx3 + y3=6    What is the minimum value of 3x+11y=?\n\nQ. There are 600 tennis players 4% wear wrist band on one wrist Of the remaining, 25% wear wrist bands on both\n\nhands How many players don't wear a wrist band?\n\nAns. 432\nQ. Three types of tea the a,b,c costs Rs. 95/kg,100/kg and70/kg respectively.How many kgs of each should be blended\n\nto produce 100 kg of mixture worth Rs.90/kg, given that the quntities of band c are equal\n\na)70,15,15\nb)50,25,25\nc)60,20,20\nd)40,30,30\n\nAns. (b)\nQ. Two distinct no's are taken from 1,2,3,4......28 .Find the probability that their sum is less than 13" ]

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[ "", null, "Manfred Weber Metra Mess- und Frequenztechnik in Radebeul e.K. PC Vibration Measurement System VibroMetra", null, "VM-OCT (+)\n\nIn addition to the calculation of overall values within a certain frequency range and frequency analysis of amplitudes at discrete frequencies vibrations can also be analyzed in defined frequency intervals, so called octave bands or fractions of them. Octave analysis is typically performed with logarithmic frequency scale. The corner frequencies of an octave band have always a ratio of 2:1. Hence with a logarithmic frequency axis octave bands appear with fixed width.\n\nOctave band analysis is a mix of overall values and frequency analysis. It shows the vibration values of different frequency intervals at a glance.", null, "There are mainly two fields of application for this type of measurement:\n\n•  Acoustic measurements: The frequency range of 16 .. 16000 Hz is used. Octaves and fractions of 1/3 and 1/6 octaves are analyzed.\n\n• Vibration monitoring of very sensitive equipment: Mainly 1/3 octave bands are measured. The frequency range of 1 (4) .. 80 Hz is commonly used in the semiconductor industry for monitoring the production environment.\n\nVM-OCT can be used for the measurement of so called Vibration Criteria (VC lines), Nano lines and for acoustic measurements with octave, 1/3 octave and 1/6 octave bands.\n\nVibroMetra uses octave filters in compliance with EN 61260." ]

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[ "Purchase Solution\n\n# Marginal probability\n\nNot what you're looking for?\n\n2.94 The joint probability function for the random variables X and Y is given in Table 2-9.... (see attached)\n\n##### Solution Summary\n\nThis shows how to find marginal probability functions.\n\n##### Geometry - Real Life Application Problems\n\nUnderstanding of how geometry applies to in real-world contexts\n\n##### Exponential Expressions\n\nIn this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.\n\n##### Graphs and Functions\n\nThis quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.\n\n##### Probability Quiz\n\nSome questions on probability\n\n##### Multiplying Complex Numbers\n\nThis is a short quiz to check your understanding of multiplication of complex numbers in rectangular form." ]

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[ "# Functional field score: the effect of using a Goldmann V-4e isopter instead of a Goldmann III-4e isopter\n\nMaaike Langelaan, Bill Wouters, Annette C Moll, Michiel R de Boer, Ger H M B van Rens\n\n### Abstract\n\nPURPOSE: To investigate the underestimation of field loss in functional field score (FFS) between the Goldmann isopters III-4e and V-4e in visually impaired patients, in order to develop a predictive model for the FFS(III-4e) based on FFS(v-4e) that adjusts for possible confounders. Although the visual field is generally evaluated using Goldmann isopter III-4e, it has the disadvantage that not all low-vision patients are able to see the stimulus corresponding to this isopter.\n\nMETHODS: Goldmann visual fields were obtained from 58 patients with a variety of eye diseases. Eligibility criteria were age of 18 years or older and valid results of a Goldmann III-4e and V-4e visual field test in at least one eye. Linear regression was used to develop the model, setting FFS(III-4e) as the dependent variable and FFS(V-4e) as the independent one.\n\nRESULTS: The FFS(V-4e) was higher than the FFS(III-4e), the mean difference being 14.56 points (95% CI, 12.48 -16.64). Multiple linear regression analysis showed that age, functional acuity score, primary eye disease, and central-peripheral loss were not confounders for the prediction of FFS(III-4e). FFS(III-4e) was estimated with the following equation: FFS(III-4e) = -19.25 + 1.063 x FFS(V-4e).\n\nCONCLUSIONS: The relationship between FFS(III-4e) and FFS(V-4e) is linear, and the FFS(V-4e) can be used to estimate the FFS(III-4e). In practice, just subtracting 19.25 points of the value of FFS(V-4e) will be sufficient to estimate the value of FFS(III-4e). This model should give confidence about using the bigger isopter for determining the visual impairment of a person by the FFS.\n\nOriginal language English 1817-23 7 Investigative Ophthalmology and Visual Science 47 5 https://doi.org/10.1167/iovs.04-1345 Published - May 2006\n\n### Fingerprint\n\nEye Diseases\nVisual Fields\nLinear Models\nLow Vision\nVisual Field Tests\nVision Disorders\nRegression Analysis\n\n### Keywords\n\n• Female\n• Humans\n• Linear Models\n• Male\n• Quality of Life\n• Retrospective Studies\n• Vision, Low\n• Visual Field Tests\n• Visual Fields\n• Visually Impaired Persons\n• Journal Article\n• Research Support, Non-U.S. Gov't\n\n### Cite this\n\nLangelaan, Maaike ; Wouters, Bill ; Moll, Annette C ; de Boer, Michiel R ; van Rens, Ger H M B. / Functional field score : the effect of using a Goldmann V-4e isopter instead of a Goldmann III-4e isopter. In: Investigative Ophthalmology and Visual Science. 2006 ; Vol. 47, No. 5. pp. 1817-23.\n@article{2579dc3e69014ae69ceee6129fa6c461,\ntitle = \"Functional field score: the effect of using a Goldmann V-4e isopter instead of a Goldmann III-4e isopter\",\nabstract = \"PURPOSE: To investigate the underestimation of field loss in functional field score (FFS) between the Goldmann isopters III-4e and V-4e in visually impaired patients, in order to develop a predictive model for the FFS(III-4e) based on FFS(v-4e) that adjusts for possible confounders. Although the visual field is generally evaluated using Goldmann isopter III-4e, it has the disadvantage that not all low-vision patients are able to see the stimulus corresponding to this isopter.METHODS: Goldmann visual fields were obtained from 58 patients with a variety of eye diseases. Eligibility criteria were age of 18 years or older and valid results of a Goldmann III-4e and V-4e visual field test in at least one eye. Linear regression was used to develop the model, setting FFS(III-4e) as the dependent variable and FFS(V-4e) as the independent one.RESULTS: The FFS(V-4e) was higher than the FFS(III-4e), the mean difference being 14.56 points (95{\\%} CI, 12.48 -16.64). Multiple linear regression analysis showed that age, functional acuity score, primary eye disease, and central-peripheral loss were not confounders for the prediction of FFS(III-4e). FFS(III-4e) was estimated with the following equation: FFS(III-4e) = -19.25 + 1.063 x FFS(V-4e).CONCLUSIONS: The relationship between FFS(III-4e) and FFS(V-4e) is linear, and the FFS(V-4e) can be used to estimate the FFS(III-4e). In practice, just subtracting 19.25 points of the value of FFS(V-4e) will be sufficient to estimate the value of FFS(III-4e). This model should give confidence about using the bigger isopter for determining the visual impairment of a person by the FFS.\",\nkeywords = \"Adult, Female, Humans, Linear Models, Male, Quality of Life, Retrospective Studies, Vision, Low, Visual Field Tests, Visual Fields, Visually Impaired Persons, Journal Article, Research Support, Non-U.S. Gov't\",\nauthor = \"Maaike Langelaan and Bill Wouters and Moll, {Annette C} and {de Boer}, {Michiel R} and {van Rens}, {Ger H M B}\",\nyear = \"2006\",\nmonth = \"5\",\ndoi = \"10.1167/iovs.04-1345\",\nlanguage = \"English\",\nvolume = \"47\",\npages = \"1817--23\",\njournal = \"Investigative ophthalmology & visual science\",\nissn = \"0146-0404\",\npublisher = \"Association for Research in Vision and Ophthalmology\",\nnumber = \"5\",\n\n}\n\nFunctional field score : the effect of using a Goldmann V-4e isopter instead of a Goldmann III-4e isopter. / Langelaan, Maaike; Wouters, Bill; Moll, Annette C; de Boer, Michiel R; van Rens, Ger H M B.\n\nIn: Investigative Ophthalmology and Visual Science, Vol. 47, No. 5, 05.2006, p. 1817-23.\n\nTY - JOUR\n\nT1 - Functional field score\n\nT2 - the effect of using a Goldmann V-4e isopter instead of a Goldmann III-4e isopter\n\nAU - Langelaan, Maaike\n\nAU - Wouters, Bill\n\nAU - Moll, Annette C\n\nAU - de Boer, Michiel R\n\nAU - van Rens, Ger H M B\n\nPY - 2006/5\n\nY1 - 2006/5\n\nN2 - PURPOSE: To investigate the underestimation of field loss in functional field score (FFS) between the Goldmann isopters III-4e and V-4e in visually impaired patients, in order to develop a predictive model for the FFS(III-4e) based on FFS(v-4e) that adjusts for possible confounders. Although the visual field is generally evaluated using Goldmann isopter III-4e, it has the disadvantage that not all low-vision patients are able to see the stimulus corresponding to this isopter.METHODS: Goldmann visual fields were obtained from 58 patients with a variety of eye diseases. Eligibility criteria were age of 18 years or older and valid results of a Goldmann III-4e and V-4e visual field test in at least one eye. Linear regression was used to develop the model, setting FFS(III-4e) as the dependent variable and FFS(V-4e) as the independent one.RESULTS: The FFS(V-4e) was higher than the FFS(III-4e), the mean difference being 14.56 points (95% CI, 12.48 -16.64). Multiple linear regression analysis showed that age, functional acuity score, primary eye disease, and central-peripheral loss were not confounders for the prediction of FFS(III-4e). FFS(III-4e) was estimated with the following equation: FFS(III-4e) = -19.25 + 1.063 x FFS(V-4e).CONCLUSIONS: The relationship between FFS(III-4e) and FFS(V-4e) is linear, and the FFS(V-4e) can be used to estimate the FFS(III-4e). In practice, just subtracting 19.25 points of the value of FFS(V-4e) will be sufficient to estimate the value of FFS(III-4e). This model should give confidence about using the bigger isopter for determining the visual impairment of a person by the FFS.\n\nAB - PURPOSE: To investigate the underestimation of field loss in functional field score (FFS) between the Goldmann isopters III-4e and V-4e in visually impaired patients, in order to develop a predictive model for the FFS(III-4e) based on FFS(v-4e) that adjusts for possible confounders. Although the visual field is generally evaluated using Goldmann isopter III-4e, it has the disadvantage that not all low-vision patients are able to see the stimulus corresponding to this isopter.METHODS: Goldmann visual fields were obtained from 58 patients with a variety of eye diseases. Eligibility criteria were age of 18 years or older and valid results of a Goldmann III-4e and V-4e visual field test in at least one eye. Linear regression was used to develop the model, setting FFS(III-4e) as the dependent variable and FFS(V-4e) as the independent one.RESULTS: The FFS(V-4e) was higher than the FFS(III-4e), the mean difference being 14.56 points (95% CI, 12.48 -16.64). Multiple linear regression analysis showed that age, functional acuity score, primary eye disease, and central-peripheral loss were not confounders for the prediction of FFS(III-4e). FFS(III-4e) was estimated with the following equation: FFS(III-4e) = -19.25 + 1.063 x FFS(V-4e).CONCLUSIONS: The relationship between FFS(III-4e) and FFS(V-4e) is linear, and the FFS(V-4e) can be used to estimate the FFS(III-4e). In practice, just subtracting 19.25 points of the value of FFS(V-4e) will be sufficient to estimate the value of FFS(III-4e). This model should give confidence about using the bigger isopter for determining the visual impairment of a person by the FFS.\n\nKW - Female\n\nKW - Humans\n\nKW - Linear Models\n\nKW - Male\n\nKW - Quality of Life\n\nKW - Retrospective Studies\n\nKW - Vision, Low\n\nKW - Visual Field Tests\n\nKW - Visual Fields\n\nKW - Visually Impaired Persons\n\nKW - Journal Article\n\nKW - Research Support, Non-U.S. Gov't\n\nU2 - 10.1167/iovs.04-1345\n\nDO - 10.1167/iovs.04-1345\n\nM3 - Article\n\nVL - 47\n\nSP - 1817\n\nEP - 1823\n\nJO - Investigative ophthalmology & visual science\n\nJF - Investigative ophthalmology & visual science\n\nSN - 0146-0404\n\nIS - 5\n\nER -" ]

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[ "# Factors of 17\n\nFactors of 17 are the numbers obtained when you divide the original number 17 completely without leaving the remainder. These factors of a number can be positive or negative but cannot be in decimal or fraction form. Hence, the factors of 17 are 1 and 17, where 17 is the prime number. Introduce factors of 17 to kids in the form of activities for better retention of numbers they are learning.\n\nContents\n\nBesides learning factors of 17, you can teach kids pair and prime factors of 17 to help them solve mathematical problems easily. The pair factors of 17 are (1, 17), whereas the negative pair factors of 17 are (-1, -17). The prime factors of 17 are 17. What are the best ways to teach factors of a number to kids? It would be great if you could teach them to calculate and memorize the factors of numbers by conducting math activities. These activities create a fun learning environment for children.\n\nCheck Math Related Articles:\n\n## What are the Factors of 17?\n\nFactors of 17 are the numbers that divide the original number 17 evenly with zero remainders. Though number 17 is composite, it has no factors other than one and the number itself. The factors of 17 are 1 and 17, where 17 is the prime number. You can find the factors of 17 using the division method. Besides this, there are pair and prime factors of 17. The pair factors of 17 are (1, 17), whereas the negative pair factors of 17 are (-1, -17). The prime factors of 17 are 17. Children can find the pair and prime factors of 17 using the multiplication and division methods, respectively. This helps them find the factors of a number efficiently. The total sum of factors of 17 are 1 + 17 = 18.\n\n## How to Find the Factors of 17?\n\nTo calculate the factors of 17, you need to divide the original number 17 with the integers using the division method. Kids must be familiar with tables 11 to 20 to solve multiplication and division related problems easily. Check out the method to find the factors of 17 as given below.\n\n17/1 = 17\n\n17/17 = 1\n\nTherefore, factors of 17 are 1 and 17.\n\n### Pair Factors of 17\n\nThe pair factors of numbers are obtained when you multiply the integers in pairs to get the original number. These numbers can be positive or negative. Check out the multiplication method to calculate the pair factors of 17 as given below.\n\n1 x 17 = 17\n\n17 x 1 = 17\n\nTherefore, the positive pair factors of 17 are (1, 17). When two negative integers multiply together, the result you obtain is positive. You can find the negative pair factors of 17 similar to the positive pair factors of 17.\n\n-1 x -17 = 17\n\n-17 x -1 = 17\n\nTherefore, the negative pair factors of 17 are (-1, -17).\n\n### Prime Factors of 17\n\nTo find prime factors of 17, you must divide the original number with the prime numbers until you get the quotient one. The original number 17 is not divisible by the least prime numbers other than the number itself.\n\n17/17 = 1\n\nTherefore, the prime factors of 17 are 17.\n\n## Solved Examples on Factors of 17\n\nSome solved examples on factors of 17 are mentioned below.\n\nExample 1: Find out the common factors of 17 and 15.\n\nSolution: Factors of 17 = 1 and 17\n\nFactors of 15 = 1, 3, 5, and 15\n\nTherefore, the common factors of 17 and 15 are 1.\n\nExample 2: Sam has 17 chocolate chips, and he gives all of them to his 17 friends. How many chocolate chips did each of them get?\n\nSolution: Number of chocolate chips = 17\n\nNumber of friends = 17\n\nTotal number of chocolate chips received by each friend = 17/17 = 1\n\nTherefore, the total number of chocolate chips received by each friend is 1.\n\nWe hope this article on factors of 17 was useful to you. For more about activities, worksheets and games, explore worksheets for kids, math for kidspuzzles for kids, kids learning sections at Osmo.\n\n## Frequently Asked Questions on Factors of 17\n\n### What are the factors of 17?\n\nThe factors of 17 are 1 and 17.\n\n### What are the pair factors of 17?\n\nThe pair factors of 17 are (1, 17), whereas the negative pair factors of 17 are (-1, -17).\n\n### What are the prime factors of 17?\n\nThe prime factors of 17 are 17." ]

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[ "", null, "RS Aggarwal Solutions for Class 9 Chapter 14: Areas of Triangles and Quadrilaterals\n\nThe students of Class 9 can use the RS Aggarwal Solutions as a valuable resource from exam point of view. The main objective of providing exercise wise solutions is to clear the doubts that arise in students while solving the problems. RS Aggarwal Solutions for Class 9 contains solutions which are prepared by expert faculties in a descriptive manner based on the CBSE syllabus.\n\nThe solutions are designed in such a way that they are easily understandable by the students to score well in the exams. Our experts have prepared the solutions according to the CBSE guidelines and evaluation process. RS Aggarwal Solutions for Class 9 Chapter 14 Areas of Triangles and Quadrilaterals are provided here.\n\nRS Aggarwal Solutions for Class 9 Chapter 14: Areas of Triangles and Quadrilaterals Download PDF", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "Access RS Aggarwal Solutions for Class 9 Chapter 14: Areas of Triangles and Quadrilaterals\n\nExercise 14 page: 533\n\n1. Find the area of the triangle whose base measures 24cm and the corresponding height measures 14.5cm.\n\nSolution:\n\nIt is given that b = 24 cm and h = 14.5 cm\n\nWe know that the\n\nArea of triangle = ½ × b × h\n\nBy substituting the values\n\nArea of triangle = ½ × 24 × 14.5\n\nSo we get\n\nArea of triangle = 174 cm2\n\nTherefore, the area of the triangle is 174 cm2.\n\n2. The base of a triangular field is three times its altitude. If the cost of sowing the field at ₹ 58 per hectare is ₹ 783, find its base and height.\n\nSolution:\n\nTake x and the height and 3x as the base of the triangular field\n\nWe know that\n\nArea of triangle = ½ × b × h\n\nBy substituting the values\n\nArea of triangle = ½ × x × 3x\n\nSo we get\n\nArea of triangle = 3/2 x2\n\n1 hectare = 1000 sq. metre\n\nIt is given that\n\nCost of sowing the field per hectare = ₹ 58\n\nTotal rate of sowing the field = ₹ 783\n\nSo we can find the total cost by\n\nTotal cost = Area of the field × ₹ 58\n\nBy substituting the values\n\n(3/2) x2 × (58/10000) = 783\n\nBy cross multiplication\n\nx2 = (783/58) × (2/3) × 10000\n\nOn further calculation\n\nx2 = 90000\n\nBy taking the square root\n\nx = √90000\n\nSo we get\n\nx = 300 m\n\nBase = 3 × 300 = 900 m\n\nTherefore, base = 900 m and height = 300 m.\n\n3. Find the area of the triangle whose sides are 42cm, 34cm and 20cm in length. Hence, find the height corresponding to the longest side.\n\nSolution:\n\nConsider a = 42 cm, b = 34 cm and c = 20 cm\n\nSo we get", null, "So we get\n\nArea = 336 cm2\n\nIt is given as\n\nb = Longest side = 42 cm\n\nConsider h as the height corresponding to the longest side\n\nWe know that\n\nArea of the triangle = ½ × b × h\n\nBy substituting the values\n\n½ × b × h = 336\n\nOn further calculation\n\n42 × h = 336 × 2\n\nSo we get\n\nh = (336 × 2)/42\n\nBy division\n\nh = 16 cm\n\nTherefore, Area = 336 cm2 and the height corresponding to the longest side is 16cm.\n\n4. Calculate the area of the triangle whose sides are 18cm, 24cm and 30cm in length. Also, find the length of the altitude corresponding to the smallest side.\n\nSolution:\n\nIt is given that a = 18cm, b = 24cm and c = 30cm\n\nSo we get", null, "So we get\n\nArea = 216 cm2\n\nIt is given as\n\nb = Smallest side = 18 cm\n\nConsider h as the height corresponding to the smallest side\n\nWe know that\n\nArea of the triangle = ½ × b × h\n\nBy substituting the values\n\n½ × b × h = 216\n\nOn further calculation\n\n18 × h = 216 × 2\n\nSo we get\n\nh = (216 × 2)/18\n\nBy division\n\nh = 24 cm\n\nTherefore, Area = 216 cm2 and the length of the altitude corresponding to the smallest side is 24 cm.\n\n5. Find the area of a triangular field whose sides are 91m, 98m and 105m in length. Find the height corresponding to the longest sides.\n\nSolution:\n\nConsider a = 91m, b = 98m and c = 105m\n\nSo we get", null, "So we get\n\nArea = 4116 m2\n\nIt is given as\n\nb = Longest side = 105 m\n\nConsider h as the height corresponding to the longest side\n\nWe know that\n\nArea of the triangle = ½ × b × h\n\nBy substituting the values\n\n½ × b × h = 4116\n\nOn further calculation\n\n105 × h = 4116 × 2\n\nSo we get\n\nh = (4116 × 2)/105\n\nBy division\n\nh = 78.4 m\n\nTherefore, Area = 4116 m2 and the height corresponding to the longest sides is 78.4 m.\n\n6. The sides of a triangle are in the ratio 5: 12: 13 and its perimeter is 150m. Find the area of the triangle.\n\nSolution:\n\nConsider the sides of the triangle as 5x, 12x and 13x\n\nIt is given that perimeter of the triangle = 150m\n\nWe can write it as\n\n5x + 12x + 13x = 150m\n\nOn further calculation\n\n30x = 150\n\nSo we get\n\nx = 15/30\n\nBy division\n\nx = 5m\n\nBy substituting the value of x we get\n\n5x = 5 (5) = 25m\n\n12x = 12 (5) = 60m\n\n13x = 13 (5) = 65m\n\nConsider a = 25m, b = 60m and c = 65m\n\nSo we get", null, "By multiplication\n\nArea = 750 sq. m\n\nTherefore, the area of the triangle is 750 sq. m.\n\n7. The perimeter of a triangular field is 540m and its sides are in the ratio 25: 17: 12. Find the area of the field. Also, find the cost of ploughing the field at ₹ 5 per m2.\n\nSolution:\n\nConsider a, b, c as the sides of a triangle in the ratio 25: 17: 12\n\nIt can be written as\n\na: b: c = 25: 17: 12\n\nSo we get\n\na = 25, b = 17 and c = 12\n\nIt is given that\n\nPerimeter = 540 m\n\nSo we get\n\n25x + 17x + 12x = 540\n\nOn further calculation\n\n54x = 540\n\nBy division\n\nx = 10\n\nBy substituting the value of x\n\na = 25x = 25(10) = 250m\n\nb = 17x = 17(10) = 170m\n\nc = 12x = 12(10) = 120m\n\nWe know that", null, "Area = 9000m2\n\nIt is given that the cost of ploughing the field is ₹ 5 per m2\n\nSo the cost of ploughing 9000m2 = 5 × 9000\n\nBy multiplication\n\nThe cost of ploughing 9000m2 = ₹ 45000\n\nTherefore, the area of the field is 9000m2 and the cost of ploughing the field is ₹ 45000.\n\n8. Two sides of a triangular field are 85m and 154m in length and its perimeter is 324m. Find\n\n(i) the area of the field and\n\n(ii) the length of the perpendicular from the opposite vertex on the side measuring 154m.\n\nSolution:\n\nIt is given that two sides of a triangular field are 85m and 154m\n\nConsider the third side as x m\n\nIt is given that the perimeter = 324m\n\nWe can write it as\n\n85 + 154 + x = 324\n\nOn further calculation we get\n\nx = 324 – 85 – 154\n\nBy subtraction\n\nx = 85m\n\nConsider a = 85m, b = 154m and c = 85m", null, "By multiplication\n\nArea = 2772 m2\n\nWe know that\n\nArea of the triangle = ½ × b × h\n\nBy substituting the values\n\n½ × 154 × h = 2772\n\nOn further calculation\n\n77 × h = 2772\n\nSo we get\n\nh = 2772/77\n\nBy division\n\nh = 36 m\n\nTherefore, area of the field is 2772 m2 and the length of the perpendicular from the opposite vertex on the side measuring 154m is 36m.\n\n9. Find the area of an isosceles triangle each of whose equal sides measures 13cm and whose base measures 20cm.\n\nSolution:\n\nConsider a = 13cm, b = 13cm and c = 20cm\n\nSo we get", null, "Therefore, area of an isosceles triangle is 83.06cm2.\n\n10. The base of an isosceles triangle measures 80cm and its area is 360cm2. Find the perimeter of the triangle.\n\nSolution:\n\nConsider △ ABC as an isosceles triangle with AL perpendicular to BC\n\nIt is given that BC = 80cm and area = 360 cm2\n\nWe know that area of a triangle = ½ × b × h\n\nBy substituting the values\n\n½ × BC × AL = 360 cm2\n\nSo we get\n\n½ × 80 × AL = 360 cm2\n\nOn further calculation\n\n40 × h = 360\n\nBy division\n\nh = 9cm", null, "We know that BL = ½ × BC\n\nBy substituting the values\n\nBL = ½ × 80\n\nBy division\n\nBL = 40cm and AL = 9cm\n\nUsing the Pythagoras theorem\n\nWe know that\n\na = √ (BL2 + AL2)\n\nBy substituting the values\n\na = √ (402 + 92)\n\nSo we get\n\na = √ (1600 + 81)\n\na = √ 1681\n\nSo we get\n\na = 41 cm\n\nSo the perimeter of the isosceles triangle = 41 + 41 + 8\n\nWe get\n\nPerimeter of the isosceles triangle = 162 cm\n\nTherefore, the perimeter of the triangle is 162 cm.\n\n11. The perimeter of an isosceles triangle is 32cm. The ratio of the equal side to its base is 3:2. Find the area of the triangle.\n\nSolution:\n\nWe know that the ratio of the equal side to its base is 3:2\n\nFor an isosceles triangle the ratio of sides can be written as 3: 3: 2\n\nSo we get\n\na: b: c = 3: 3: 2\n\nConsider a = 3x, b = 3x and c = 2x\n\nWe know that the perimeter = 32cm\n\nIt can be written as\n\n3x + 3x + 2x = 32\n\nOn further calculation\n\n8x = 32\n\nBy division\n\nx = 4\n\nBy substituting the value of x\n\na = 3x = 3(4) = 12cm\n\nb = 3x = 3(4) = 12cm\n\nc = 2x = 2(4) = 8cm\n\nSo we get", null, "Therefore, area of the triangle is 32√2 cm2.\n\n12. The perimeter of a triangle is 50cm. One side of the triangle is 4 cm longer than the smallest side and the third side is 6cm less than twice the smallest side. Find the area of the triangle.\n\nSolution:\n\nConsider the three sides of a triangle as a, b and c with c as the smallest side\n\nSo we get\n\na = c + 4\n\nIn the same way\n\nb = 2c – 6\n\nIt is given that perimeter of a triangle = 50cm\n\nWe know that\n\na + b + c = 50\n\nBy substituting the values\n\n(c + 4) + (2c – 6) + c = 50\n\nOn further calculation\n\n4c – 2 = 50\n\n4c = 50 + 2\n\n4c = 52\n\nBy division\n\nc = 13\n\nBy substituting the value of c\n\na = c + 4 = 13 + 4\n\na = 17cm\n\nb = 2c – 6 = 2(13) – 6\n\nb = 20cm\n\nSo we get", null, "Therefore, area of the triangle is 20√30 cm2.\n\n13. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 13m, 14m, 15m. The advertisements yield an earning of ₹ 2000 per m2 a year. A company hired one of its walls for 6 months. How much rent did it pay?\n\nSolution:\n\nConsider three sides of the wall as a = 13m, b = 14m and c = 15m\n\nSo we get", null, "It is given that the advertisements yield per year = ₹ 2000 per m2\n\nSo the rent for 6 months = ₹ 1000 per m2\n\nTotal rent paid for advertisements for 6 months = ₹ (1000 × 84) = ₹ 84000\n\nTherefore, the total rent paid in 6 months is ₹ 84000.\n\n14. The perimeter of an isosceles triangle is 42cm and its base is 1 ½ times each of the equal sides. Find\n\n(i) the length of each side of the triangle,\n\n(ii) the area of the triangle, and\n\n(iii) the height of the triangle. (Given, √7 = 2.64.)\n\nSolution:\n\nWe know that the lateral sides of an isosceles triangle are of equal length\n\nConsider the length of lateral side as x cm\n\nIt is given that base = 3/2 × x cm\n\n(i) It is given that the perimeter = 42cm\n\nWe can write it as\n\nx + x + 3/2 x = 42cm\n\nBy multiplying the entire equation by 2\n\n2x + 2x + 3x = 84cm\n\nOn further calculation\n\n7x = 84\n\nBy division\n\nx = 12 cm\n\nSo the length of lateral side = 12 cm\n\nBase = 3/2 x = 3/2 (12)\n\nSo we get the base = 18cm\n\nTherefore, the length of each side of the triangle is 12cm, 12 cm and 18cm.\n\n(ii) Consider a = 12cm, b = 12cm and c = 18cm", null, "Therefore, the area of the triangle is 71.28cm2.\n\n(iii) We know that\n\nArea of a triangle = ½ × b × h\n\nBy substituting the values\n\n71.28 = ½ × 18 × h\n\nOn further calculation\n\n71.28 = 9 × h\n\nBy division\n\nh = 7.92cm\n\nTherefore, the height of the triangle is 7.92cm.\n\n15. If the area of an equilateral triangle is 36√3 cm2, find its perimeter.\n\nSolution:\n\nConsider a as the length of a side of an equilateral triangle\n\nWe know that\n\nArea of an equilateral triangle = (√3 × a2)/4 sq units\n\nIt is given that the area = 36√3 cm2\n\nWe can write it as\n\n(√3 × a2)/4 = 36√3\n\nOn further calculation\n\na2 = (36√3 × 4)/ √3\n\nSo we get\n\na2 = 36 × 4\n\nBy multiplication\n\na2 = 144\n\nBy taking the square root\n\na = √144\n\nSo we get\n\na = 12 cm\n\nBy substituting the value of a\n\nPerimeter = 3a = 3 (12) = 36cm\n\nTherefore, the perimeter of the equilateral triangle is 36cm.\n\n16. If the area of an equilateral triangle is 81√3 cm2, find its height.\n\nSolution:\n\nConsider a as the length of a side of an equilateral triangle\n\nWe know that\n\nArea of an equilateral triangle = (√3 × a2)/4 sq units\n\nIt is given that the area = 81√3 cm2\n\nWe can write it as\n\n(√3 × a2)/4 = 81√3\n\nOn further calculation\n\na2 = (81√3 × 4)/ √3\n\nSo we get\n\na2 = 324\n\nBy taking the square root\n\na = √324\n\nSo we get\n\na = 18 cm\n\nWe know that the height of an equilateral triangle = √3/2 a\n\nBy substituting the value of a\n\nHeight of an equilateral triangle = √3/2 (18) = 9√3 cm\n\nTherefore, the height of an equilateral triangle is 9√3 cm.\n\n17. Each side of an equilateral triangle measures 8cm. Find\n\n(i) the area of the triangle, correct to 2 places of decimal and\n\n(ii) the height of the triangle, correct to 2 places of decimal. (Take √3 = 1.732)\n\nSolution:\n\n(i) Consider a as the side of equilateral triangle i.e. a = 8cm\n\nArea of an equilateral triangle = (√3 × a2)/4 sq units\n\nSo we get\n\nArea of the equilateral triangle = (√3 × 82)/4\n\nOn further calculation\n\nArea of the equilateral triangle = (√3 × 64)/4\n\nSo we get\n\nArea of the equilateral triangle = √3 × 16\n\nBy substituting the value of √3\n\nArea of the equilateral triangle = 1.732 × 16 = 27.712\n\nBy correcting to two places of decimal\n\nArea of the equilateral triangle = 27.71 cm2\n\n(ii) We know that\n\nHeight of an equilateral triangle = √3/2 a\n\nBy substituting the value\n\nHeight of an equilateral triangle = √3/2 × 8\n\nSo we get\n\nHeight of an equilateral triangle = √3 × 4\n\nBy substituting the value of √3\n\nHeight of an equilateral triangle = 1.732 × 4 = 6.928\n\nBy correcting to two places of decimal\n\nHeight of an equilateral triangle = 6.93 cm\n\n18. The height of an equilateral triangle measures 9cm. Find its area, correct to 2 places of decimal. (Take √3 = 1.732)\n\nSolution:\n\nConsider a as the side of an equilateral triangle\n\nWe know that\n\nHeight of an equilateral triangle = √3/2 a\n\nIt is given that height = 9cm\n\nSo we get\n\n√3/2 a = 9\n\nOn further calculation\n\na = (9 × 2)/ √3\n\nMultiplying both numerator and denominator by √3\n\nWe get\n\na = (9 × 2 × √3) / (√3 × √3)\n\nSo we get\n\nBase = a = 6√3cm\n\nWe know that the area of an equilateral triangle = ½ × b × h\n\nBy substituting the values\n\nArea of the equilateral triangle = ½ × 6√3 × 9\n\nOn further calculation\n\nArea of the equilateral triangle = 27√3 cm2\n\nBy substituting the value of √3\n\nArea of the equilateral triangle = 27 × 1.732 = 46.764\n\nBy correcting to 2 places of decimal\n\nArea of the equilateral triangle = 46.76 cm2\n\nTherefore, the area of an equilateral triangle = 46.76 cm2.\n\n19. The base of a right-angled triangle measures 48 cm and its hypotenuse measures 50 cm. Find the area of the triangle.\n\nSolution:\n\nIt is given that\n\nBase = BC = 48 cm\n\nHypotenuse = AC = 50 cm\n\nConsider AB = x cm", null, "Using the Pythagoras theorem\n\nAC2 = AB2 + BC2\n\nBy substituting the values\n\n502 = x2 + 482\n\nOn further calculation\n\nx2 = 502 – 482\n\nSo we get\n\nx2 = 2500 – 2304\n\nBy subtraction\n\nx2 = 196\n\nBy taking the square root\n\nx = √196\n\nSo we get\n\nx = 14cm\n\nWe know that the area of a right angled triangle = ½ × b × h\n\nBy substituting the values\n\nArea of a right angled triangle = ½ × 48 × 14\n\nOn further calculation\n\nArea of a right angled triangle = 24 × 14\n\nBy multiplication\n\nArea of the triangle = 336 cm2\n\nTherefore, the area of the triangle = 336 cm2.\n\n20. Find the area of the shaded region in the figure given below.", null, "Solution:\n\nConsider △ ABD\n\nUsing Pythagoras theorem\n\nAB2 = AD2 + BD2\n\nBy substituting the values\n\nAB2 = 122 + 162\n\nOn further calculation\n\nAB2 = 144 + 256\n\nAB2 = 400\n\nBy taking out the square root\n\nAB = √400\n\nSo we get\n\nAB = 20cm\n\nWe know that the area of △ ABD = ½ × b × h\n\nIt can be written as\n\nArea of △ ABD = ½ × AD × BD\n\nBy substituting the values\n\nArea of △ ABD = ½ × 12 × 16\n\nOn further calculation\n\nArea of △ ABD = 96 cm2\n\nConsider △ ABC", null, "So the area of the shaded region = Area of △ ABC – Area of △ ABD\n\nBy substituting the values\n\nArea of the shaded region = 480 – 96 = 384 cm2\n\nTherefore, the area of the shaded region is 384 cm2.\n\n21. The sides of a quadrilateral ABCD taken in order are 6cm, 8cm, 12 cm and 14cm respectively and the angle between the first two sides is a right angle. Find its area. (Given, √6 = 2.45.)\n\nSolution:\n\nConsider ABCD as a quadrilateral", null, "It is given that AB = 6cm, BC = 8cm, CD = 12cm and AD = 14cm\n\nConsider △ ABC\n\nUsing the Pythagoras theorem\n\nAC2 = AB2 + BC2\n\nBy substituting the values\n\nAC2 = 62 + 82\n\nOn further calculation\n\nAC2 = 36 + 64\n\nAC2 = 100\n\nBy taking out the square root\n\nAC = √100\n\nSo we get\n\nAC = 10cm\n\nWe know that the area of △ ABC = ½ × b × h\n\nIt can be written as\n\nArea of △ ABC = ½ × AB × BC\n\nBy substituting the values\n\nArea of △ ABC = ½ × 6 × 8\n\nOn further calculation\n\nArea of △ ABC = 24 cm2\n\nWe know that AC = 10cm, CD = 12cm and AD = 14cm\n\nIt can be written as a = 10cm, b = 12cm and c = 14cm\n\nSo we get", null, "So the area of quadrilateral ABCD = Area of △ ABC + Area of △ ACD\n\nBy substituting the values\n\nArea of quadrilateral ABCD = 24 + 58.8\n\nArea of quadrilateral ABCD = 82.8 cm2\n\nTherefore, the area of quadrilateral ABCD is 82.8 cm2.\n\n22. Find the perimeter and area of a quadrilateral ABCD in which BC = 12 cm, CD = 9 cm, BD = 15cm, DA = 17 cm and ∠ ABD = 90o.", null, "Solution:\n\nConsider △ ABD\n\nUsing the Pythagoras theorem\n\nAD2 = AB2 + BD2\n\nBy substituting the values\n\n172 =AB2 + 152\n\nOn further calculation\n\nAB2 = 289 – 225\n\nBy subtraction\n\nAB2 = 64\n\nBy taking out the square root\n\nAB = √64\n\nSo we get\n\nAB = 8cm\n\nWe know that\n\nPerimeter of quadrilateral ABCD = AB + BC + CD + AD\n\nBy substituting the values\n\nPerimeter = 8 + 12 + 9 + 17\n\nPerimeter = 46cm\n\nWe know that area of △ ABD = ½ × b × h\n\nIt can be written as\n\nArea of △ ABD = ½ × AB × BD\n\nBy substituting the values\n\nArea of △ ABD = ½ × 8 × 15\n\nOn further calculation\n\nArea of △ ABD = 60 cm2\n\nConsider △ BCD\n\nWe know that BC = 12cm, CD = 9cm and BD = 15cm\n\nIt can be written as a = 12cm, b = 9cm and c = 15cm\n\nSo we get", null, "So the area of quadrilateral ABCD = Area of △ ABD + Area of △ BCD\n\nBy substituting the values\n\nArea of quadrilateral ABCD = 60 + 54 = 114 cm2\n\nTherefore, the perimeter is 46cm and the area is 114 cm2.\n\n23. Find the perimeter and area of the quadrilateral ABCD in which AB = 21 cm, ∠ BAC = 90o, AC = 20cm, CD = 42 cm and AD = 34 cm.", null, "Solution:\n\nConsider △ BAC\n\nUsing the Pythagoras theorem\n\nBC2 = AC2 + AB2\n\nBy substituting the values\n\nBC2 =202 + 212\n\nOn further calculation\n\nBC2 = 400 + 441\n\nBC2 = 841\n\nBy taking out the square root\n\nBC = √841\n\nSo we get\n\nBC = 29cm\n\nWe know that\n\nPerimeter of quadrilateral ABCD = AB + BC + CD + AD\n\nBy substituting the values\n\nPerimeter = 21 + 29 + 42 + 34\n\nPerimeter = 126cm\n\nWe know that area of △ ABC = ½ × b × h\n\nIt can be written as\n\nArea of △ ABC = ½ × AB × AC\n\nBy substituting the values\n\nArea of △ ABC = ½ × 21 × 20\n\nOn further calculation\n\nArea of △ ABC = 210 cm2\n\nConsider △ ACD\n\nWe know that AC = 20cm, CD = 42cm and AD = 34cm\n\nIt can be written as a = 20cm, b = 42cm and c = 34cm\n\nSo we get", null, "So the area of quadrilateral ABCD = Area of △ ABC + Area of △ ACD\n\nBy substituting the values\n\nArea of quadrilateral ABCD = 210 + 336 = 546 cm2\n\nTherefore, the perimeter is 126cm and the area is 546 cm2.\n\n24. Find the area of the quadrilateral ABCD in which BCD is an equilateral triangle, each of whose sides is 26cm, AD = 24cm and ∠ BAD = 90o. Also, find the perimeter of the quadrilateral. (Given, √3 = 1.73.)", null, "Solution:\n\nIn △ ABD\n\nUsing the Pythagoras theorem\n\nBD2 = AB2 + AD2\n\nBy substituting the values\n\n262 =AB2 + 242\n\nOn further calculation\n\nAB2 = 676 – 576\n\nBy subtraction\n\nAB2 = 100\n\nBy taking out the square root\n\nAB = √100\n\nSo we get\n\nBase = AB = 10cm\n\nWe know that area of △ ABD = ½ × b × h\n\nBy substituting the values\n\nArea of △ ABD = ½ × 10 × 24\n\nOn further calculation\n\nArea of △ ABD = 120 cm2\n\nWe know that the area of △ BCD = √3/4 a2\n\nBy substituting the values\n\nArea of △ BCD = (1.73/4) (26)2\n\nSo we get\n\nArea of △ BCD = 292.37 cm2\n\nSo we get area of quadrilateral ABCD = Area of △ ABD + Area of △ BCD\n\nBy substituting the values\n\nArea of quadrilateral ABCD = 120 + 29237\n\nArea of quadrilateral ABCD = 412.37 cm2\n\nThe perimeter of quadrilateral ABCD = AB + BC + CD + DA\n\nBy substituting the values\n\nPerimeter = 10 + 26 + 26 + 24\n\nSo we get\n\nPerimeter = 86cm\n\nTherefore, the area is 412.37 cm2 and perimeter is 86cm.\n\n25. Find the area of a parallelogram ABCD in which AB = 28 cm, BC = 26 cm and diagonal AC = 30 cm.", null, "Solution:\n\nIn △ ABC\n\nTake a = 26cm, b = 30cm and c = 28cm\n\nSo we get", null, "We know that the diagonal divides the parallelogram into two equal area\n\nSo the area of parallelogram ABCD = Area of △ ABC + Area of △ ACD\n\nIt can be written as\n\nArea of parallelogram ABCD = 2(Area of △ ABC)\n\nBy substituting the values\n\nArea of parallelogram ABCD = 2 (336)\n\nBy multiplication\n\nArea of parallelogram ABCD = 672 cm2\n\nTherefore, the area of parallelogram ABCD is 672 cm2.\n\n26. Find the area of a parallelogram ABCD in which AB = 14cm, BC = 10cm and AC = 16cm. (Given, √3 = 1.73.)", null, "Solution:\n\nIn △ ABC\n\nTake a = 10cm, b = 16cm and c = 14cm\n\nSo we get", null, "We know that a diagonal in a parallelogram divides it into two equal area\n\nSo we get\n\nArea of parallelogram ABCD = Area of △ ABC + Area of △ ACD\n\nIt can be written as\n\nArea of parallelogram ABCD = 2(Area of △ ABC)\n\nBy substituting the values\n\nArea of parallelogram ABCD = 2 (40 √3)\n\nSo we get\n\nArea of parallelogram ABCD = 80√3\n\nBy substituting √3\n\nArea of parallelogram ABCD = 80 (1.73)\n\nSo we get\n\nArea of parallelogram ABCD = 138.4 cm2\n\nTherefore, the area of parallelogram ABCD is 138.4 cm2.\n\n27. In the given figure, ABCD is a quadrilateral in which diagonal BD = 64cm, AL ⊥ BD and CM ⊥ BD such that AL = 16.8 cm and CM = 13.2 cm. Calculate the area of the quadrilateral ABCD.", null, "Solution:\n\nWe know that\n\nArea of △ ABD = ½ × b × h\n\nIt can be written as\n\nArea of △ ABD = ½ × BD × AL\n\nBy substituting the values\n\nArea of △ ABD = ½ × 64 × 16.8\n\nOn further calculation\n\nArea of △ ABD = 537.6 cm2\n\nWe know that\n\nArea of △ BCD = ½ × b × h\n\nIt can be written as\n\nArea of △ BCD = ½ × BD × CM\n\nBy substituting the values\n\nArea of △ BCD = ½ × 64 × 13.2\n\nOn further calculation\n\nArea of △ BCD = 422.4 cm2\n\nSo we get area of quadrilateral ABCD = Area of △ ABD + Area of △ BCD\n\nBy substituting the values\n\nArea of quadrilateral ABCD = 537.6 + 422.4\n\nArea of quadrilateral ABCD = 960 cm2\n\nTherefore, the area of quadrilateral ABCD is 960 cm2.\n\n28. The area of a trapezium is 475 cm2 and its height is 19cm. Find the lengths of its two parallel sides if one side is 4cm greater than the other.\n\nSolution:\n\nConsider x cm as the smaller side of the trapezium\n\nSo the larger parallel can be written as (x + 4) cm\n\nWe know that the area of trapezium = ½ × sum of parallel sides × height\n\nBy substituting the values\n\n475 = ½ × (x + (x + 4)) × 19\n\nOn further calculation\n\n25 = ½ × (2x + 4)\n\nSo we get\n\n50 = 2x + 4\n\nBy subtraction\n\n2x = 50 – 4\n\n2x = 46\n\nBy division\n\nx = 23cm\n\nSo we get Larger parallel side (x + 4) = 23 + 4 = 27cm\n\nTherefore, the length of two parallel sides is 23cm and 27cm.\n\n29. In the given figure, a △ ABC has been given in which AB = 7.5 cm, AC = 6.5 cm and BC = 7cm. On base BC, a parallelogram DBCE of the same area as that of △ ABC is constructed. Find the height DL of the parallelogram.", null, "Solution:\n\nConsider △ ABC\n\nIt is given that AB = 7.5cm, BC = 7cm and AC = 6.5cm\n\nTake a = 7.5cm, b = 7cm and c = 6.5 cm\n\nSo we get", null, "We know that the area of parallelogram DBCE = Area of △ ABC\n\nIt can be written as\n\nBC × DL = 21\n\nBy substituting the values\n\n7 × DL = 21\n\nBy division\n\nDL = 3cm\n\nTherefore, the height DL of the parallelogram is 3cm.\n\n30. A field is in the shape of a trapezium having parallel sides 90 m and 30 m. These sides meet the third side at right angles. The length of the fourth side is 100 m. If it costs ₹ 5 to plough 1m2 of the field, find the total cost of ploughing the field.\n\nSolution:", null, "Construct BT parallel to CD\n\nConsider △ BTC\n\nUsing the Pythagoras theorem\n\nBC2 = BT2 + CT2\n\nBy substituting the values\n\n1002 =BT2 + 602\n\nOn further calculation\n\nBT2 = 10000 – 3600\n\nBy subtraction\n\nBT2 = 6400\n\nBy taking out the square root\n\nBT = √6400\n\nSo we get\n\nAD = BT = 80m\n\nWe know that\n\nArea of field = Area of trapezium ABCD\n\nSo we get\n\nArea of trapezium ABCD = ½ × Sum of parallel sides × height\n\nIt can be written as\n\nArea of trapezium ABCD = ½ × (AB + CD) × AD\n\nBy substituting the values\n\nArea of trapezium ABCD = ½ × (30 + 90) × 80\n\nOn further calculation\n\nArea of trapezium ABCD = 120 × 40\n\nBy multiplication\n\nArea of trapezium ABCD = 4800 m2\n\nIt is given that the cost of ploughing 1m2 is ₹ 5\n\nSo the cost of ploughing 4800 m2 = ₹ (5 × 4800)\n\nThe cost of ploughing 4800 m2 = ₹ 24000\n\nTherefore, the total cost of ploughing the field is ₹ 24000.\n\n31. A rectangular plot is given for constructing a house, having a measurement of 40m long and 15m in the front. According to the laws, a minimum of 3m wide space should be left in the front and back each and 2m wide space on each of the other sides. Find the largest area where house can be constructed.\n\nSolution:\n\nIt is given that the length of rectangular plot = 40m\n\nWidth = 15m\n\nBy keeping 3m wide space both in front and back\n\nThe length becomes = 40 – 3 – 3 = 34m\n\nBy keeping 2m wide space both the sides\n\nWidth = 15 – 2 – 2 = 11m\n\nSo the largest area where the house can be constructed = 34 × 11 = 374 m2\n\nTherefore, the largest area where the house can be constructed is 374 m2.\n\n32. A rhombus-shaped sheet with perimeter 40cm and one diagonal 12cm, is painted on both sides at the rate of ₹5 per cm2. Find the cost of painting.\n\nSolution:", null, "Consider ABCD as the rhombus shaped sheet\n\nIt is given that the perimeter = 40cm\n\nIt can be written as\n\n4 (side) = 40cm\n\nSo we get\n\nSide = 10cm\n\nWe get\n\nAB = BC = CD = AD = 1cm\n\nConsider the diagonal AC = 12cm\n\nWe know that the diagonals of a rhombus bisect each other at right angles\n\nSo we get\n\nAO = OC = 6cm\n\nConsider △ AOD\n\nUsing the Pythagoras theorem\n\nAD2 = OD2 + AO2\n\nBy substituting the values\n\n102 =OD2 + 62\n\nOn further calculation\n\nOD2 = 100 – 36\n\nBy subtraction\n\nOD2 = 64\n\nBy taking out the square root\n\nOD = √64\n\nSo we get\n\nOD = 8cm\n\nWe know that\n\nBD = 2OD\n\nSo we get\n\nBD = 2(8)\n\nBD = 16cm\n\nWe know that\n\nArea of rhombus ABCD = ½ (Product of diagonals)\n\nIt can be written as\n\nArea of rhombus ABCD = ½ × AC × BD\n\nBy substituting the values\n\nArea of rhombus ABCD = ½ × 12 × 16\n\nBy multiplication\n\nArea of rhombus ABCD = 96 cm2\n\nIt is given that the cost of painting is ₹5 per cm2\n\nSo the cost of paining both sides of rhombus = ₹5 × (96 + 96)\n\nWe get\n\nThe cost of painting both sides of rhombus = ₹5 × 192 = ₹960\n\nTherefore, the cost of painting is ₹960.\n\n33. The difference between the semiperimeter and the sides of a △ ABC are 8cm, 7cm and 5cm respectively. Find the area of the triangle.\n\nSolution:\n\nConsider a, b and c as the sides of a triangle and s as the semi perimeter\n\nSo we get\n\ns – a = 8cm\n\ns – b = 7cm\n\ns – c = 5cm\n\nWe know that\n\n(s – a) + (s – b) + (s – c) = 8 + 7 + 5\n\nOn further calculation\n\n3s – (a + b + c) = 20\n\nWe know that a + b + c = 2s\n\n3s – 2s = 2\n\nSo we get\n\ns = 20\n\nBy substituting the value of s\n\na = s – 8 = 20 – 8 = 12cm\n\nb = s – 7 = 20 – 7 = 13cm\n\nc = s – 5 = 20 – 5 = 15cm", null, "Therefore, the area of the triangle is 20√14 cm2.\n\n34. A floral design on a floor is made up of 16 tiles, each triangular in shape having sides 16cm, 12cm and 20cm. Find the cost of polishing the tiles at ₹1 per sq cm.", null, "Solution:\n\nConsider a = 16cm, b = 12cm and c = 20cm\n\nSo we get", null, "So the area of one tile = 96cm2\n\nThe area of 16 triangular tiles = 96 (16) = 1536 cm2\n\nIt is given that the cost of polishing the tiles per sq cm = ₹1\n\nSo the cost of polishing all the tiles = ₹ (1 × 1536) = ₹ 1536\n\nTherefore, the cost of polishing the tiles is ₹ 1536.\n\n35. An umbrella is made by stitching 12 triangular pieces of cloth, each measuring (50cm × 20cm × 50cm). Find the area of the cloth used in it.", null, "Solution:\n\nConsider a = 50cm, b = 20cm and c = 50cm\n\nSo we get", null, "So the area of one piece of cloth = 490 cm2\n\nWe know that the area of 12 pieces = 12 (490)\n\nBy multiplication\n\nThe area of 12 pieces = 5880 cm2\n\nTherefore, the area of cloth used in it is 5880 cm2.\n\n36. In the given figure, ABCD is a square with diagonal 44cm. How much paper of each shade is needed to make a kite given in the figure.", null, "Solution:\n\nWe know that\n\nArea of square sheet ABCD = ½ × diagonal2\n\nBy substituting the value\n\nArea of square sheet ABCD = ½ × 44 × 44 = 968 cm2\n\nFrom the figure we know that the\n\nArea of yellow sheet = Area of region I + Area of region II\n\nIt can be written as\n\nArea of yellow sheet = ½ × Area of square sheet ABCD\n\nBy substituting the value\n\nArea of yellow sheet = ½ × 968 = 484 cm2\n\nFrom the figure we know that the\n\nArea of red sheet = Area of region IV\n\nIt can be written as\n\nArea of red sheet = 1/4 × Area of square sheet ABCD\n\nBy substituting the value\n\nArea of red sheet = 1/4 × 968 = 242 cm2\n\nConsider △ AEF\n\nWe know that AE = 20cm, EF = 14cm and AF = 20cm\n\nConsider a = 20cm, b = 14cm and c = 20cm\n\nSo we get", null, "From the figure we know that\n\nArea of green sheet = Area of region III + Area of region V\n\nIt can be written as\n\nArea of green sheet = ¼ × Area of square sheet ABCD + 131.25\n\nBy substituting the values\n\nArea of green sheet = ¼ × 968 + 131.25\n\nOn further calculation\n\nArea of green sheet = 242 + 131.25\n\nArea of green sheet = 373.25 cm2\n\nTherefore, the area of yellow sheet is 484 cm2, area of red sheet is 242 cm2 and the area of green sheet is 373.25cm2.\n\n37. A rectangular lawn, 75m by 60m, has two roads, each road 4m wide, running through the middle of the lawn, one parallel to length and the other parallel to breadth, as shown in the figure. Find the cost of gravelling the roads at ₹ 50 per m2.", null, "Solution:\n\nWe know that for the road ABCD\n\nIt is given that length = 75m and breadth = 4m\n\nWe know that\n\nArea of road ABCD = l × b\n\nBy substituting the values\n\nArea of road ABCD = 75 × 4\n\nSo we get\n\nArea of road ABCD = 300 m2\n\nWe know that for the road PQRS\n\nIt is given that length = 60m and breadth = 4m\n\nWe know that\n\nArea of road PQRS = l × b\n\nBy substituting the values\n\nArea of road PQRS = 60 × 4\n\nSo we get\n\nArea of road PQRS = 240 m2\n\nWe know that for road EFGH\n\nIt is given that side = 4m\n\nArea of road EFGH = side2\n\nBy substituting the value\n\nArea of road EFGH = 42 = 16 m2\n\nThe total area of the road for gravelling = Area of road ABCD + Area of road PQRS – Area of road EFGH\n\nBy substituting the values\n\nTotal area of the road for gravelling = 300 + 340 – 16\n\nSo we get\n\nTotal area of the road for gravelling = 524 m2\n\nIt is given that the cost of gravelling the road = ₹ 50 per m2\n\nSo the cost of gravelling 524 m2 road = ₹ (50 × 524) = ₹ 26200\n\nTherefore, the cost of gravelling the road is ₹ 26200.\n\n38. The shape of the cross section of a canal is a trapezium. If the canal is 10m wide at the top, 6m wide at the bottom and the area of its cross section is 640m2, find the depth of the canal.\n\nSolution:\n\nIt is given that the area of cross section = 640 m2\n\nWe know that\n\nLength of top + Length of bottom = sum of parallel sides\n\nIt can be written as\n\nLength of top + Length of bottom = 10 + 6 = 16m\n\nWe know that\n\nArea of cross section = ½ × sum of parallel sides × height\n\nBy substituting the values\n\n640 = ½ × 16 × height\n\nSo we get\n\nHeight = (640 × 2)/16\n\nBy division\n\nHeight = 80m\n\nTherefore, the depth of the canal is 80m.\n\n39. Find the area of a trapezium whose parallel sides are 11m and 25m long, and the nonparallel sides are 15m and 13m long.\n\nSolution:", null, "From the point C construct CE || DA\n\nWe know that ADCE is a parallelogram having AE || DC and AD || EC with AD = 13m and D = 11m\n\nIt can be written as\n\nAE = DC = 11m and EC = AD = 13m\n\nSo we get\n\nBE = AB – AE\n\nBy substituting the values\n\nBE = 25 – 11 = 14m\n\nConsider △ BCE\n\nWe know that BC = 15m, CE = 13m and BE = 14m\n\nTake a = 15m, b = 13m and c = 14m\n\nSo we get", null, "We know that\n\nArea of △ BCE = ½ × BE × CL\n\nBy substituting the values\n\n84 = ½ × 14 × CL\n\nOn further calculation\n\n84 = 7 × CL\n\nBy division\n\nCL = 12m\n\nWe know that\n\nArea of trapezium ABCD = ½ × sum of parallel sides × height\n\nIt can be written as\n\nArea of trapezium ABCD = ½ × (AB + CD) × CL\n\nBy substituting the values\n\nArea of trapezium ABCD = ½ × (11 + 25) × 12\n\nOn further calculation\n\nArea of trapezium ABCD = 36 × 6\n\nBy multiplication\n\nArea of trapezium ABCD = 216 m2\n\nTherefore, the area of trapezium ABCD is 216 m2.\n\n40. The difference between the lengths of the parallel sides of a trapezium is 8cm, the perpendicular distance between these sides is 24cm and the area of the trapezium is 312cm2. Find the length of each of the parallel sides.\n\nSolution:\n\nConsider x cm as the smaller parallel side\n\nSo the longer parallel side can be written as (x + 8) cm\n\nIt is given that height = 24cm\n\nArea = 312 cm2\n\nWe know that\n\nArea of trapezium = ½ × sum of parallel sides × height\n\nBy substituting the values\n\n312 = ½ × (x + x + 8) × 24\n\nOn further calculation\n\n312 = 12 × (2x + 8)\n\nBy division\n\n2x + 8 = 26\n\nSo we get\n\n2x = 26 – 8\n\nBy subtraction\n\n2x = 18\n\nBy division\n\nx = 9cm\n\nSo we get\n\nx + 8 = 9 + 8 = 17cm\n\nTherefore, the length of each of the parallel sides is 9cm and 17cm.\n\n41. A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120m and 44m. If one of the sides of the parallelogram measures 66m, find its corresponding altitude.\n\nSolution:\n\nIt is given that\n\nArea of parallelogram = Area of rhombus\n\nWe can write it as\n\nBase × altitude = ½ × product of diagonals\n\nBy substituting the values\n\n66 × altitude = ½ × 120 × 44\n\nOn further calculation\n\n66 × altitude = 60 × 44\n\nSo we get\n\nAltitude = (60 × 44)/66\n\nAltitude = 40 m\n\nTherefore, the corresponding altitude is 40m.\n\n42. A parallelogram and a square have the same area. If the sides of the square measure 40m and altitude of the parallelogram measures 25m, find the length of the corresponding base of the parallelogram.\n\nSolution:\n\nIt is given that\n\nArea of parallelogram = Area of square\n\nWe know that\n\nBase × altitude = side2\n\nBy substituting the value\n\nBase × 25 = 402\n\nOn further calculation\n\nBase = (40 × 40)/25\n\nSo we get\n\nBase = 64m\n\nTherefore, the length of the corresponding base of the parallelogram is 64m.\n\n43. Find the area of the rhombus one side of which measures 20cm and one of whose diagonals is 24cm.\n\nSolution:\n\nConsider ABCD as a rhombus with AC and BD as diagonals which intersect each other at O", null, "The diagonals of a rhombus bisect at right angles\n\nConsider △ AOD\n\nUsing the Pythagoras theorem\n\nAD2 = OD2 + AO2\n\nBy substituting the values\n\n202 =OD2 + 122\n\nOn further calculation\n\nOD2 = 400 – 144\n\nBy subtraction\n\nOD2 = 256\n\nBy taking out the square root\n\nOD = √256\n\nSo we get\n\nOD = 16cm\n\nWe know that BD = 2OD\n\nSo we get\n\nBC = 2 (16) = 32cm\n\nWe know that\n\nArea of rhombus ABCD = ½ × AC × BD\n\nBy substituting the values\n\nArea of rhombus ABCD = ½ × 24 × 32\n\nOn further calculation\n\nArea of rhombus ABCD = 384 cm2\n\nTherefore, the area of rhombus ABCD is 384 cm2.\n\n44. The area of a rhombus is 480cm2, and one of its diagonals measures 48cm. Find\n\n(i) the length of the other diagonal,\n\n(ii) the length of each of its sides, and\n(iii its perimeter.\n\nSolution:\n\n(i) It is given that\n\nArea of rhombus = 480 cm2\n\nIt can be written as\n\n½ × diagonal 1 × diagonal 2 = 480\n\nSo we get\n\n½ × 48 × diagonal 2 = 480\n\nOn further calculation\n\nDiagonal 2 = 20cm\n\n(ii) Consider AC = 48cm and BD = 20cm", null, "The diagonals of a rhombus bisect at right angles\n\nConsider △ AOD\n\nUsing the Pythagoras theorem\n\nAD2 = OD2 + AO2\n\nBy substituting the values\n\nAD2 =242 + 102\n\nOn further calculation\n\nAD2 = 576 + 100\n\nBy taking out the square root\n\nSo we get\n\nWe know that AD = BC = CD = AD = 26cm\n\nTherefore, the length of each side of rhombus is 26cm.\n\n(iii) We know that\n\nPerimeter of a rhombus = 4 (side)\n\nBy substituting the value\n\nPerimeter of a rhombus = 4 (26)\n\nSo we get\n\nPerimeter of a rhombus = 104 cm\n\nRS Aggarwal Solutions for Class 9 Maths Chapter 14: Areas of Triangles and Quadrilaterals\n\nChapter 14, Areas of Triangles and Quadrilaterals, has 1 exercise with answers in PDF format which can be downloaded by the students to improve their exam preparation. Some of the topics which are covered in RS Aggarwal Solutions Chapter 14 are:\n\n• Introduction\n• Area of Triangles\n• Areas of Quadrilaterals\n• Area of a quadrilateral whose one diagonal and lengths of perpendiculars from opposite vertices to the diagonal are given\n\nRS Aggarwal Solutions Class 9 Maths Chapter 14 – Exercise list\n\nExercise 14 Solutions 44 Questions\n\nRS Aggarwal Solutions Class 9 Maths Chapter 14 – Areas of Triangles and Quadrilaterals\n\nThe 14th chapter mainly talks about finding the areas of triangles and quadrilaterals using different formulas. The students mainly require a proper study material in order to score well in the exams. So the RS Aggarwal Solutions for Class 9 Maths Chapter 14 Areas of Triangles and Quadrilaterals will help you understand the concepts which are covered in it.\n\nMathematics is a subject which requires lots of practice to understand the methods used to solve the problems. The students can download the solutions which are prepared based on the current syllabus. The area of triangles and quadrilaterals are used in architecture for designing and construction of buildings." ]

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[ "Schematics, Example Code and Q&A\n\nThe Rev1 and Rev0 versions of the ESP32 has a couple of issues with the ADC. There is a topic on noise pickup https://forum.ezsbc.com/viewtopic.php?f=12&t=8816 that describes that issue. In addition to that issue the ADC is not as linear as it should be but the non-linearity is repeatable and can be removed with some numerical manipulation. The code below corrects the raw readings from the ADC and returns a corrected, linearized reading. The code is not particularly fast because it uses floating point functions. If you care about speed you can put the function into a spreadsheet and generate a lookup table.\n\nCode: Select all\n`#define ADC_Ch 36void setup() {  Serial.begin(115200);}void loop() {  Serial.println( ReadVoltage( ADC_Ch ), 3 ) ;  Serial.println( analogRead( ADC_Ch ) ) ;  delay( 1000 ) ;}double ReadVoltage(byte  pin ){  double reading = analogRead( pin ) ; // Reference voltage is 3v3 so maximum reading is 3v3 = 4095 in range 0 to 4095  if( reading < 1 || reading > 4095) {    return 0 ;   }  // return -0.000000000009824 * pow( reading,3 )  +  0.000000016557283 * pow( reading, 2 ) + 0.000854596860691 * reading + 0.065440348345433 ;  return -0.000000000000016 * pow( reading, 4 ) + 0.000000000118171 * pow( reading, 3 )- 0.000000301211691 * pow( reading,2 )+ 0.001109019271794 * reading + 0.034143524634089 ;} // More complex, improved polynomial or simpler polynomial, use either/* ADC readings v voltage// Polynomial curve fit, based on this raw data: *   464     0.5 *  1088     1.0 *  1707     1.5 *  2331     2.0 *  2951     2.5  *  3775     3.0 */`\nDaniel\n\nPosts: 119\nJoined: Tue Nov 13, 2012 5:10 pm", null, "" ]

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[ "# Basics: Kinematics\n\n**pre reqs:** *none*\n\nOften I will do some type of analysis that I think is quite cool. But there is a problem. I keep having to make a choice. Either go into all the little details, or skip over them. My goal for this blog is to make each post such that someone could learn some physics, but I also don’t want it to go too long. So, instead of continually describing different aspects of basic physics – I will just do it once. Then, when there is a future post using those ideas, I can just refer to this post. Get it?\n\nFine. On with the first idea – kinematics. Kinematics typically means a description of motion (not what causes that motion). In particular, kinematics looks at position, velocity, and acceleration. In this post, I will try to stay in one dimension. This will make things look simpler without really losing too much. Later, when I talk about vectors, I will make it all better.\n\n**Position**\n\nIn one dimension, position is a location on the axis (x or y or whatever you want to call it). Note that this really doesn’t tell you much because the origin of this axis (I will call it x) is completely arbitrary. Where is x = 0 meters in the room you are in now? Anything you say could be correct. The coordinate system is just make believe, you know, like Peter Pan.\nSo, position isn’t that useful. **Change** in position IS important. Also, change in position is independent of your coordinate system.\n\n**Velocity**\n\nHow fast the position changes is a measure of velocity. In the x-direction only, this can be written as:\n\nA couple of points:\n– Since this is the velocity in the x-direction, it can be positive or negative.\n– Don’t confuse this with the velocity vector, which can only have a positive magnitude. Technically, this v-x only has a positive magnitude only.\n– What about speed vs. average velocity? Usually, the definition is that speed is how far you have gone divided by the time. Average velocity is your change in position over time. Here is an example of how these two would be different.\n\nSuppose in this diagram, something moved from one red dot to the other following the purple path. The average velocity would simply be:\n\nWhile the average speed would be:\n\nOk. Enough about that. There are much more things to talk about.\n\n**Average Velocity and Position**\n\nIt may seem obvious, but I will say it anyway. Using average velocity, I can write an expression that determines the final position given the initial position, the change in time and the average velocity:\n\n-Note that this is true. Most people will say that this only works if the velocity is constant, but since I put AVERAGE velocity, it always works. I will come back to this shortly.\n-What about instantaneous velocity? This is the velocity when the time interval gets really small.\n\n**Acceleration**\n\nJust as velocity is the change in position, acceleration is the change in velocity. This would make a great ACT analogy question (for any ACT writers that read this).\nAverage acceleration is:\n\nI could also talk about average and instantaneous accelerations, but truthfully many useful situations can be approximated as constant acceleration. In such a case, the average and instantaneous accelerations are the same.\n\n**Kinematic Equations**\n\nYou will often see much stuff in textbooks about kinematic equations. The kinematic equations are some equations that relate position, velocity, acceleration and time. Most people will claim that these are derived from calculus (which they can be), but that isn’t necessary. Let me start with average velocity:\n\nIf the velocity is changing, then I can write:\n\nand substitute this in for average velocity:\n\nNow, I want to get rid of the v2. I can use the definition of acceleration:\n\nand substituting this in:\n\nAnd that is the common kinematics equation everyone thinks of (not using calculus). The next step would be to algebraically manipulate the above equations to get one that is independent of time – I will not go through that exercise, but I will list it:\n\nSo that is it. The “kinematic equations”. Use them wisely.\n\n**Update** I just want to make one thing clear. The above kinematics equations are only valid when the assumption that the average velocity is v1 plus v2 divided by two. This is obviously true when there is a constant acceleration. This is also mostly true when the time interval is very short. Ok, I feel better now.\n\n## One thought on “Basics: Kinematics”\n\n1.", null, "Peter Teuben says:\n\nYou can continue this game. After acceleration comes jerk, followed by snap, crackle and pop. Once you use numerical integrations these ‘taylor coefficients’ come in handy sometimes.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]

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[ "# identifying algebraic expressions worksheet\n\nAlgebra 1 Worksheets | Basics for Algebra 1 Worksheets. 8 Pics about Algebra 1 Worksheets | Basics for Algebra 1 Worksheets : Combining Like Terms Worksheets | Math-Aids.Com, Multi Step Inequalities worksheets and also √ 20 Variables Worksheets 5th Grade | Simple Template Design.\n\n## Algebra 1 Worksheets | Basics For Algebra 1 Worksheets", null, "www.math-aids.com\n\nmath aids algebra worksheets numbers sets number basics identifying worksheet printable maths pdf integers adding\n\n## Simplifying Variable Expressions Worksheet", null, "worksheets.ambrasta.com\n\nsimplifying algebraic nidecmege\n\n## Combining Like Terms Worksheets | Math-Aids.Com", null, "www.pinterest.co.uk\n\naids distributive subtracting simplifying sequences\n\n## Triangles Worksheets", null, "www.mathworksheets4kids.com\n\ntriangles algebra sides triangle perimeter missing worksheets value length sheet mathworksheets4kids\n\n## Worksheet - Combine Like Terms Or Simplify Expressions By Shannon Balthazor", null, "www.teacherspayteachers.com\n\n## √ 20 Variables Worksheets 5th Grade | Simple Template Design", null, "restaurantecop3.com\n\n5th variables\n\n## Two Step Inequalities Worksheets", null, "www.mathworksheets4kids.com\n\nworksheet inequalities step templates periodic table translating phrases translate verbal sample template mathworksheets4kids writing worksheets algebraic into expression phrase business\n\n## Multi Step Inequalities Worksheets", null, "www.mathworksheets4kids.com\n\nstep multi inequalities solving worksheet worksheets answer key inequality sheet solution mathworksheets4kids circle solutions each values identifying\n\n√ 20 variables worksheets 5th grade. Aids distributive subtracting simplifying sequences. Worksheet terms expressions combine simplify radiohead pinkpop 1996 steven festival" ]

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[ "# What to conclude when most results are statistically significant to fail to reject null hypothesis but not all?\n\nI have sampled 8 bags of a certain brand of candy to compare the color distributions of the candies. I have 4 bags for each size of bag, 8 oz and 1.9 lb. The bags were paired randomly. Here are my hypotheses:\n\n$\\ \\ \\ \\ H_0: The \\ distribution \\ of \\ each \\ color \\ of \\ candies \\ is \\ equal \\ in \\ all \\ sizes \\ of \\ bags.\\\\ \\ \\ \\ \\ H_A: The \\ distribution \\ of \\ each \\ color \\ of \\ candies \\ is \\ not \\ equal \\ in \\ all \\ sizes \\ of \\ bags.$\n\nI then ran 4 chi-square tests for each pair of bags, generating 4 p-values. With an alpha level of .05, 3 of the pairs suggest I fail to reject my null hypothesis but one suggests I reject it. What is the best way to draw a conclusion from this? Should I overall fail to reject the null hypothesis because the majority shows this?\n\n• You state you have four bags of each size. On what basis, then, have you paired them?\n– whuber\nJun 5, 2017 at 17:40\n• There are 8 total bags, 4 8 oz ones and 4 1.9 lb ones. None of them are equal in size. Jun 7, 2017 at 0:55\n• How, then, do you manage to pair them? There are 9 ways to construct a set of pairings of the 8-oz bags and the 1.9-lb bags: how did you select one of them?\n– whuber\nJun 7, 2017 at 13:15\n• It was randomized. Jun 9, 2017 at 1:39\n• That is crucial information for answering your question. Please include it in your post. But why randomly pair the bags? That adds no information and precludes the use of more powerful techniques. What exactly is your null hypothesis?\n– whuber\nJun 9, 2017 at 14:46\n\nIf all of your null hypothesis are, in reality, true, then your probability of rejecting in at least one of your experiments is\n\n$$1 - 0.95^4 \\approx 0.19$$\n\nSo there about a 20% chance you would find at least one rejection in your experiment, even if all of the bags had an equal distribution of colors. Not too unlikely; how you decide to act now depends on the costs of being wrong.\n\nI suggest you eat 20% of the candy.\n\nWouldn't it be (1-.95)^4?\n\nI think I got it right:\n\n• Probability of one experiment falsely rejecting: $0.05$\n• Probability of one experiment not falsely rejecting: $0.95$\n• Probability of all experiments not falsely rejecting: $0.95^4$\n• Probability of at least one experiment falsely rejecting: $1 - 0.95^4$\n• Wouldn't it be (1-.95)^4? Jun 4, 2017 at 23:08\n• @ZodiacZubeda Edited my answer. Jun 4, 2017 at 23:11\n• @ZodiacZubeda, for completeness: (1-95)^4 would be the probability that all bags falsely reject the null hypothesis (which is quite small). Jun 5, 2017 at 9:16\n\nIf you are trying to test if distribution depends on the bag -or, equivalently, if all bags are random samples from the same population- performing tests on pairs of bags is not going to work, because it can yield contradictory results -as you found- and because probability of type I errors is going to build up due to the multiple comparisons problem -as Mathew Durry's answer explains and as the XKCD comic demonstrates in a different context.\n\nYou can avoid this problem by performing a single test using all bags: a chi-square test for homogeneity, which will tell you whether there are significant differences between bags.\n\nPlease notice that most online examples of this test use just a pair of samples, but it works equally fine for more samples. Furthermore, the test is the same as the chi-square test for independence (just interpretation is a bit different), so you can find information under both names.\n\nIf homogeneity test shows that there are significant differences between bags, you might be interested on knowing between which bags there are significant differences. Then, paired tests can be useful, but to prevent the multiple comparisons problem to happen again, you need to make corrections. I would suggest Bonferroni correction because of its simplicity.\n\nAnyway, if your bags are just random bags taken from a shop shelf, knowing which one is significantly different is uninteresting and the homogeneity test should be enough for your purposes.\n\nAfter you explain the results in the Results chapter, you can state in the discussion that one result was found significant. You can provide your interpretation of the results based on literature and suggest number of plausbile explanations to the reader." ]

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[ "# Integrals #6 and #7\n\n#### Random Variable\n\nChallenge Problems:\n\n$$\\displaystyle \\int^{b}_{a} \\frac{e^{x /a} - e^{b /x}}{x} \\ dx$$\n\n$$\\displaystyle \\int^{1}_{0} \\sqrt{2x^{3}-3x^2-x+1} \\ dx$$\n\nEDIT: I don't like the second integral. It doesn't even make any sense unless you state that $$\\displaystyle \\sqrt{2x^{3}-3x^2-x+1}$$ is a real-valued function. Sorry. It's a bad problem.\n\nModerator edit: Approved Challenge question.\n\nLast edited by a moderator:\n\n#### chiph588@\n\nMHF Hall of Honor\nChallenge Problems:\n\n$$\\displaystyle \\int^{b}_{a} \\frac{e^{x /a} - e^{b /x}}{x} \\ dx$$\n$$\\displaystyle \\int^{b}_{a} \\frac{e^{x /a} - e^{b /x}}{x} \\ dx =0$$\n\nA simple Taylor/Laurent expansion of $$\\displaystyle e^{x /a}$$ and $$\\displaystyle e^{b /x}$$ should prove my assertion.\n\n#### Random Variable\n\n$$\\displaystyle \\int^{b}_{a} \\frac{e^{x /a} - e^{b /x}}{x} \\ dx =0$$\n\nA simple Taylor/Laurent expansion of $$\\displaystyle e^{x /a}$$ and $$\\displaystyle e^{b /x}$$ should prove my assertion.\nCould you elaborate?\n\nThe solution I had in mind is even simpler than that.\n\n#### simplependulum\n\nMHF Hall of Honor\nIf my calculations are correct , the integrals are both zero .\n\nFor the first one ,\n\nSub. $$\\displaystyle x = \\frac{ab}{t}$$ in the second integral .\n\nand we will find that it is equal to\n\n$$\\displaystyle \\int_a^b \\frac{e^{x/a}}{x}~dx - \\int_a^b \\frac{e^{t/a}}{t}~dt$$\n\nAnd the second integral , i find that it is identical to\n\n$$\\displaystyle \\sqrt{(2x-1)[(x-\\frac{1}{2})^2 - \\frac{5}{4}]}$$\n\nSub. $$\\displaystyle x = 1-t$$ and we have\n\n$$\\displaystyle I = -I$$\n\n$$\\displaystyle I = 0$$\n\n•", null, "chiph588@\n\n#### Random Variable\n\nIf my calculations are correct , the integrals are both zero .\n\nFor the first one ,\n\nSub. $$\\displaystyle x = \\frac{ab}{t}$$ in the second integral .\n\nand we will find that it is equal to\n\n$$\\displaystyle \\int_a^b \\frac{e^{x/a}}{x}~dx - \\int_a^b \\frac{e^{t/a}}{t}~dt$$\n\nAnd the second integral , i find that it is identical to\n\n$$\\displaystyle \\sqrt{(2x-1)[(x-\\frac{1}{2})^2 - \\frac{5}{4}]}$$\n\nSub. $$\\displaystyle x = 1-t$$ and we have\n\n$$\\displaystyle I = -I$$\n\n$$\\displaystyle I = 0$$\n\nNice.\n\nIf for the first integral you made the same substitution but didn't break it up into two integrals, you would get I = -I.\n\nAnd for the second integral, Maple claims the answer is a complex number and Mathematica doesn't know what to do with it. The issue is the cube root of negative real numbers.\n\n#### chiph588@\n\nMHF Hall of Honor\n$$\\displaystyle \\int^{1}_{0} \\sqrt{2x^{3}-3x^2-x+1} \\ dx$$\nLet $$\\displaystyle x=t+\\tfrac12$$\n\nWe get $$\\displaystyle \\int^{1}_{0} \\sqrt{2x^{3}-3x^2-x+1} \\ dx = \\int^{\\tfrac12}_{-\\tfrac12} \\sqrt{2(t+\\tfrac12)^{3}-3(t+\\tfrac12)^2-(t+\\tfrac12)+1} \\ dx =$$ $$\\displaystyle \\int^{\\tfrac12}_{-\\tfrac12} \\sqrt{2t^3-\\tfrac52t} \\ dx = 0$$ since $$\\displaystyle \\sqrt{2t^3-\\tfrac52t}$$ is odd.\n\nLast edited:" ]

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[ "leancheck-0.6.1: Cholesterol-free property-based testing\n\nSafe Haskell Safe Haskell2010\n\nTest.LeanCheck.Tiers\n\nDescription\n\nLeanCheck is a simple enumerative property-based testing library.\n\nThis module provides advanced functions for manipulating tiers. Most definitions given here are exported by Test.LeanCheck, except: listCons, choices, setChoices and bagChoices.\n\nSynopsis\n\n# Additional tiers constructors\n\nlistCons :: Listable a => ([a] -> b) -> [[b]] Source #\n\nGiven a constructor that takes a list, return tiers of applications of this constructor.\n\nThis is basically a type-restricted version of cons1. You should use cons1 instead: this serves more as an illustration of how setCons and bagCons work (see source).\n\nsetCons :: Listable a => ([a] -> b) -> [[b]] Source #\n\nGiven a constructor that takes a set of elements (as a list), lists tiers of applications of this constructor.\n\nA naive Listable instance for the Set (of Data.Set) would read:\n\ninstance Listable a => Listable (Set a) where\ntiers = cons0 empty \\/ cons2 insert\n\nThe above instance has a problem: it generates repeated sets. A more efficient implementation that does not repeat sets is given by:\n\n tiers = setCons fromList\n\nAlternatively, you can use setsOf direclty.\n\nbagCons :: Listable a => ([a] -> b) -> [[b]] Source #\n\nGiven a constructor that takes a bag of elements (as a list), lists tiers of applications of this constructor.\n\nFor example, a Bag represented as a list.\n\nbagCons Bag\n\nnoDupListCons :: Listable a => ([a] -> b) -> [[b]] Source #\n\nGiven a constructor that takes a list with no duplicate elements, return tiers of applications of this constructor.\n\nmaybeCons0 :: Maybe b -> [[b]] Source #\n\nmaybeCons1 :: Listable a => (a -> Maybe b) -> [[b]] Source #\n\nmaybeCons2 :: (Listable a, Listable b) => (a -> b -> Maybe c) -> [[c]] Source #\n\n# Products of tiers\n\nproduct3 :: [[a]] -> [[b]] -> [[c]] -> [[(a, b, c)]] Source #\n\nLike ><, but over 3 lists of tiers.\n\nproduct3With :: (a -> b -> c -> d) -> [[a]] -> [[b]] -> [[c]] -> [[d]] Source #\n\nLike productWith, but over 3 lists of tiers.\n\nproductMaybeWith :: (a -> b -> Maybe c) -> [[a]] -> [[b]] -> [[c]] Source #\n\nTake the product of lists of tiers by a function returning a Maybe value discarding Nothing values.\n\n# Tiers of lists\n\nlistsOf :: [[a]] -> [[[a]]] Source #\n\nTakes as argument tiers of element values; returns tiers of lists of elements.\n\nlistsOf [[]] == [[[]]]\nlistsOf [[x]] == [ [[]]\n, [[x]]\n, [[x,x]]\n, [[x,x,x]]\n, ...\n]\nlistsOf [[x],[y]] == [ [[]]\n, [[x]]\n, [[x,x],[y]]\n, [[x,x,x],[x,y],[y,x]]\n, ...\n]\n\nbagsOf :: [[a]] -> [[[a]]] Source #\n\nTakes as argument tiers of element values; returns tiers of size-ordered lists of elements possibly with repetition.\n\nbagsOf [,,,...] =\n[ [[]]\n, []\n, [[0,0],]\n, [[0,0,0],[0,1],]\n, [[0,0,0,0],[0,0,1],[0,2],[1,1],]\n, [[0,0,0,0,0],[0,0,0,1],[0,0,2],[0,1,1],[0,3],[1,2],]\n, ...\n]\n\nsetsOf :: [[a]] -> [[[a]]] Source #\n\nTakes as argument tiers of element values; returns tiers of size-ordered lists of elements without repetition.\n\nsetsOf [,,,...] =\n[ [[]]\n, []\n, []\n, [[0,1],]\n, [[0,2],]\n, [[0,3],[1,2],]\n, [[0,1,2],[0,4],[1,3],]\n, ...\n]\n\nCan be used in the constructor of specialized Listable instances. For Set (from Data.Set), we would have:\n\ninstance Listable a => Listable (Set a) where\ntiers = mapT fromList \\$ setsOf tiers\n\nnoDupListsOf :: [[a]] -> [[[a]]] Source #\n\nTakes as argument tiers of element values; returns tiers of lists with no repeated elements.\n\nnoDupListsOf [,,,...] ==\n[ [[]]\n, []\n, []\n, [[0,1],[1,0],]\n, [[0,2],[2,0],]\n, ...\n]\n\nproducts :: [[[a]]] -> [[[a]]] Source #\n\nTakes the product of N lists of tiers, producing lists of length N.\n\nAlternatively, takes as argument a list of lists of tiers of elements; returns lists combining elements of each list of tiers.\n\nproducts [xss] = mapT (:[]) xss\nproducts [xss,yss] = mapT (\\(x,y) -> [x,y]) (xss >< yss)\nproducts [xss,yss,zss] = product3With (\\x y z -> [x,y,z]) xss yss zss\n\nlistsOfLength :: Int -> [[a]] -> [[[a]]] Source #\n\nTakes as argument an integer length and tiers of element values; returns tiers of lists of element values of the given length.\n\nlistsOfLength 3 [,,,,...] =\n[ [[0,0,0]]\n, [[0,0,1],[0,1,0],[1,0,0]]\n, [[0,0,2],[0,1,1],[0,2,0],[1,0,1],[1,1,0],[2,0,0]]\n, ...\n]\n\n# Tiers of pairs\n\ndistinctPairs :: [[a]] -> [[(a, a)]] Source #\n\nTakes as argument tiers of element values; returns tiers of pairs with distinct element values.\n\nWhen argument tiers have no repeated elements:\n\ndistinctPairs xss = xss >< xss suchThat uncurry (/=)\n\ndistinctPairsWith :: (a -> a -> b) -> [[a]] -> [[b]] Source #\n\ndistinctPairs by a given function:\n\ndistinctPairsWith f = mapT (uncurry f) . distinctPairs\n\nunorderedPairs :: [[a]] -> [[(a, a)]] Source #\n\nTakes as argument tiers of element values; returns tiers of unordered pairs where, in enumeration order, the first element is less than or equal to the second.\n\nThe name of this function is perhaps a misnomer. But in mathematics, an unordered pair is a pair where you don't care about element order, e.g.: (1,2) = (2,1). This function will enumerate canonical versions of such pairs where the first element is less than the second.\n\nThe returned element pairs can be seen as bags with two elements.\n\nWhen argument tiers are listed in Ord:\n\ndistinctPairs xss = xss >< xss suchThat uncurry (<=)\n\nunorderedPairsWith :: (a -> a -> b) -> [[a]] -> [[b]] Source #\n\nunorderedPairs by a given function:\n\nunorderedPairsWith f = mapT (uncurry f) . unorderedPairs\n\nunorderedDistinctPairs :: [[a]] -> [[(a, a)]] Source #\n\nTakes as argument tiers of element values; returns tiers of unordered pairs where, in enumeration order, the first element is strictly less than the second.\n\nThe returned element pairs can be seen as sets with two elements.\n\nWhen argument tiers are listed in Ord:\n\ndistinctPairs xss = xss >< xss suchThat uncurry (<)\n\nunorderedDistinctPairsWith :: (a -> a -> b) -> [[a]] -> [[b]] Source #\n\nunorderedPairs by a given function:\n\nunorderedDistinctPairsWith f = mapT (uncurry f) . unorderedDistinctPairs\n\ndeleteT :: Eq a => a -> [[a]] -> [[a]] Source #\n\nDelete the first occurence of an element in a tier.\n\nFor tiers without repetitions, the following holds:\n\ndeleteT x = normalizeT . (suchThat (/= x))\n\nnormalizeT :: [[a]] -> [[a]] Source #\n\nNormalizes tiers by removing an empty tier from the end of a list of tiers.\n\nnormalizeT [xs0,xs1,...,xsN,[]] = [xs0,xs1,...,xsN]\n\nNote this will only remove a single empty tier:\n\nnormalizeT [xs0,xs1,...,xsN,[],[]] = [xs0,xs1,...,xsN,[]]\n\ncatMaybesT :: [[Maybe a]] -> [[a]] Source #\n\nConcatenate tiers of maybes\n\nmapMaybeT :: (a -> Maybe b) -> [[a]] -> [[b]] Source #\n\n# Tiers of choices\n\nchoices :: [[a]] -> [[(a, [[a]])]] Source #\n\nLists tiers of choices. Choices are pairs of values and tiers excluding that value.\n\nchoices [[False,True]] == [[(False,[[True]]),(True,[[False]])]]\nchoices [,,]\n== [ [(1,[[],,])]\n, [(2,[,[],])]\n, [(3,[,,[]])] ]\n\nEach choice is sized by the extracted element.\n\nsetChoices :: [[a]] -> [[(a, [[a]])]] Source #\n\nLike choices but lists tiers of strictly ascending choices. Used to construct setsOf values.\n\nsetChoices [[False,True]] == [[(False,[[True]]),(True,[[]])]]\nsetChoices [,,]\n== [ [(1,[[],,])]\n, [(2,[[],[],])]\n, [(3,[[],[],[]])]\n]\n\nbagChoices :: [[a]] -> [[(a, [[a]])]] Source #\n\nLike choices but lists tiers of non-decreasing (ascending) choices. Used to construct bagsOf values.\n\nbagChoices [[False,True]] =\n[ [(False,[[False,True]]), (True,[[True]])]\n]\nbagChoices [,,,...] =\n[ [(1,[,,,...])]\n, [(2,[[ ],,,...])]\n, [(3,[[ ],[ ],,...])]\n, ...\n]" ]

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[ "# KeapNotes blog\n\nStay hungry, Stay Foolish.\n\n0%\n\nPandas数据类型简单的分有Series和DataFrame两类，我粗略的理解相当于R语言的向量和数据框\n\nSeries可以接受字典dict、ndarry以及标量值等数据来源，比如：\n\n``pd.Series({'b': 1, 'a': 0, 'c': 2})``\n\n``````s = pd.Series(np.random.randn(5), index=['a', 'b', 'c', 'd', 'e'])\ns.index #索引\ns.dtype #数据类型``````\n\n``pd.Series(5., index=['a', 'b', 'c', 'd', 'e'])``\n\n``````s = pd.Series(np.random.randn(5), index=['a', 'b', 'c', 'd', 'e'])\ns\ns[:3]\ns[s > s.median()]\ns[[4, 3, 1]]``````\n\n``````s['a']\ns['e'] = 12.\n'e' in s``````\n\n``````s.get('f') == None\n>>True``````\n\n``````np.exp(s)\ns + s\ns[1:] + s[:-1]``````\n\nDataFrame跟Series差不多，也支持多个数据结构的来源，如：\n\n• 1D ndarray，list，dicts或Series的Dict\n• two-dimensional ndarray\n• ndarray\n• Series\n• DataFrame\n\n``````df = pd.DataFrame({'foo1': np.random.randn(5),\n'foo2': np.random.randn(5)})\ndf.columns``````\n\n``````data = np.array([[1, 2],\n[3, 4]])\ndf = pd.DataFrame(data = data,\nindex = [\"Row1\",\"Row2\"],\ncolumns = [\"Col1\",\"Col2\"])\nprint(df)\nprint(df.shape)\nprint(len(df.index))\nprint(list(df.columns.values))``````\n\n``pd.DataFrame(np.nan, index=[0,1,2,3], columns=[\"A\",\"B\"])``\n\n`DataFrame.from_dict`函数，其中加不加index有略微区别（前者A/B是column，后者加了`orient`参数的话，A/B则变成了index，此时需要再设定column）\n\n``````df = pd.DataFrame.from_dict(dict([('A', [1, 2, 3]), ('B', [4, 5, 6])]))\ndf = pd.DataFrame.from_dict(dict([('A', [1, 2, 3]), ('B', [4, 5, 6])]),\norient='index', columns=['one', 'two', 'three'])``````\n\n`DataFrame.from_records`函数则是用于ndarray类型的数据：\n\n``````data = np.array([(1, 2., 'Hello'), (2, 3., 'World')])\npd.DataFrame.from_records(data, index = (\"a\",\"b\"))``````\n\n``````df = pd.DataFrame(np.array([[1,2,3], [4,5,6], [7,8,9]]), columns = [\"A\",\"B\",\"C\"])\n\n# 切片行\ndf[:1]\n# 指定列\ndf[\"A\"]``````\n\n``````# 1行1列的值\ndf.iloc # 等价于 df.iloc[0,0]\ndf.loc['A'] # 等价于 df.loc[:,'A']\n# 切片行/列\ndf.iloc # 等价于 df.iloc[:] 或者 df.loc\ndf.loc[:,\"A\"] # 等价于 df.loc[:][\"A\"]``````\n\n``````df.loc = [10,11,12] # 添加行\ndf[\"D\"] = df.index # 添加列\ndf.loc[:, \"D\"] = df.index # 同上 ``````\n\n``````df.at[0,'A']\ndf.iat[0,0]``````\n\nDataFrame可以通过`set_index`方法，可以设置单索引和复合索引（类似于交叉表的形式。。。个人觉得），而`reset_index`则是逆操作\n\n``df.set_index('C')``\n\n``````df.drop(\"D\", axis=1, inplace=True)\ndf.drop(df.columns, axis=1) # 同上\ndf.drop(3, axis=0, inplace = True)\ndf.drop(df.index, axis=0) # 同上``````\n\n``del df['A']``\n\n``df.drop_duplicates([\"A\"], keep='last')``\n\n``````df.rename(index = {0:1, 1:2, 2:3})\ndf.rename(columns = {\"A\":\"col1\", \"B\":\"col2\", \"C\":\"col3\", \"D\":\"col4\"}, inplace = True)``````\n\n``df.isnull()``\n\n``````df.dropna()\ndf.dropna(axis=1, thresh=2)``````\n\n``````df.fillna(value=100)\ndf.fillna(value={\"A\":100}) # 指定列中的缺失值``````\n\n``````df.replace([1,2,3,4], [11,12,13,14])\ndf.replace({1:2, 2:3})\ndf.replace('[A-Z]', 0, regex = True)``````\n\n``````df[\"A\"].apply(doubler) # 对列\ndf.loc.apply(doubler) # 也可以对行``````\n\n• `pivot()`\n• `stack()``unstack()`\n• `melt()`\n\n``````for index, row in df.iterrows() :\nprint(row[\"A\"], row[\"B\"])``````" ]

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[ "# llvm-prefer-isa-or-dyn-cast-in-conditionals¶\n\nLooks at conditionals and finds and replaces cases of cast<>, which will assert rather than return a null pointer, and dyn_cast<> where the return value is not captured. Additionally, finds and replaces cases that match the pattern var && isa<X>(var), where var is evaluated twice.\n\n// Finds these:\nif (auto x = cast<X>(y)) {}\n// is replaced by:\nif (auto x = dyn_cast<X>(y)) {}\n\nif (cast<X>(y)) {}\n// is replaced by:\nif (isa<X>(y)) {}\n\nif (dyn_cast<X>(y)) {}\n// is replaced by:\nif (isa<X>(y)) {}\n\nif (var && isa<T>(var)) {}\n// is replaced by:\nif (isa_and_nonnull<T>(var.foo())) {}\n\n// Other cases are ignored, e.g.:\nif (auto f = cast<Z>(y)->foo()) {}\nif (cast<Z>(y)->foo()) {}\nif (X.cast(y)) {}" ]

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[ "# Appending an extra column in\n\nCould you please tell me how to surface plot with those data files using Matplotlib?..i.e both reading the data files and then surface plotting in matplotlib?\n\nSayan\n\n···\n\nOn 11 April 2013 02:25, Maximilian Trescher <faucon@…3667…> wrote:\n\nHi,\n\nnp.savetxt(fname, np.array([pp_za,pv_za]).T, ‘%f’)\n\ndoes maybe\n\nnp.savetxt(fname, np.array([pp_za,pv_za, np.ones(1000)*t]).T, ‘%f’)\n\ndoes what you want? (replace 1000 with approppriate length)\n\n`````` if pp_za[i] < 0:\n``````\n`````` pp_za[i] = 2 - abs(pp_za[i])\n``````\n``````if pp_za[i] > 2:\n``````\n`````` pp_za[i] = pp_za[i] % 2\n``````\n\nin general\n\npp_za[pp_za < 0] = 2 - abs(pp_za[pp_za < 0])\n\npp_za[pp_za > 2] = pp_za[pp_za > 2] % 2\n\nis shorter and much faster.\n\nMax\n\nSayan Chatterjee\nDept. of Physics and Meteorology\n\nIIT Kharagpur\nLal Bahadur Shastry Hall of Residence\nRoom AB 205\nMob: +91 9874513565\nblog: www.blissprofound.blogspot.com\n\nHi,\n\nI'm not sure what kind of plot you want to create from this data, but\nhttp://matplotlib.org/mpl_toolkits/mplot3d/tutorial.html\n\nMax\n\nPS: Sorry for not sending my first mail to the list, but you can see it\nbelow\n\n···\n\nAm 11/04/2013 04:54, schrieb Sayan Chatterjee:\n\nCould you please tell me how to surface plot with those data files using\nMatplotlib?...i.e both reading the data files and then surface plotting\nin matplotlib?\n\nSayan\n\nOn 11 April 2013 02:25, Maximilian Trescher <faucon@…3667… > <mailto:faucon@…3667…>> wrote:\n\nHi,\n>\n> *np.savetxt(fname, np.array([pp_za,pv_za]).T, '%f')*\n\ndoes maybe\n\nnp.savetxt(fname, np.array([pp_za,pv_za, np.ones(1000)*t]).T, '%f')\ndoes what you want? (replace 1000 with approppriate length)\n\n> if pp_za[i] < 0:\n> pp_za[i] = 2 - abs(pp_za[i])\n> if pp_za[i] > 2:\n> pp_za[i] = pp_za[i] % 2\n\nin general\n\npp_za[pp_za < 0] = 2 - abs(pp_za[pp_za < 0])\npp_za[pp_za > 2] = pp_za[pp_za > 2] % 2\n\nis shorter and much faster.\n\nMax\n\n--\n\n--------------------------------------------------------------------------\n*Sayan Chatterjee*\nDept. of Physics and Meteorology\nIIT Kharagpur\nLal Bahadur Shastry Hall of Residence\nRoom AB 205\nMob: +91 9874513565\nblog: www.blissprofound.blogspot.com <http://www.blissprofound.blogspot.com>" ]

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[ "calculator.org\n\n# property>frequency\n\n## What Is Frequency?", null, "Frequency is understood as the number of repeating events in a given unit of time; a measure of how frequently something occurs. For every frequency, there is a period, which is the duration of time between one event of the same type and the next. This period is the reciprocal of the frequency. This is written as:\n\nT = 1/f\n\nwhere T is the period and f is the frequency. The SI unit for frequency is the hertz (Hz), which is equal to 1/s, or once per second. Frequency is encountered many times in physics. A few highly important instances are listed here.\n\nAll waves in physics are said to have frequency. This frequency is inversely proportional to wavelength by the phase speed, or speed that a wave travels in a given medium. This medium depends on the wave in consideration. Sound, for example, travels through the air, solids, and liquids. We write the frequency of a physical wave as:\n\nf = v/λ\n\nwhere v is the phase speed and the wavelength is given by λ. Let’s use sound as an example. The frequency we are measuring is the time between one compressed zone of air molecules and the next. Sound waves are alternating changes in air pressure that deviate from the usual air pressure. The same goes for sounds travelling through a liquid, which has local changes in compression that propagate the sound wave. A sound travelling through a solid, however, takes on a different character. It is considered to be a transverse wave that represents local changes in shear stress.\n\nThe phase speed of sound in air is 343 meters per second. Sound waves can have many different frequencies, which for us, are considered to be their pitch. The human audible range of sound frequencies begins at 20 Hz and ends at 20,000 Hz. Other animals are able to pick up on sounds outside our audible frequency range. Dogs, for example, are able to hear higher pitched sounds than we are.\n\nLight is another physical wave that has a frequency. For all electromagnetic waves (including light), we assume that the phase speed v is equal to the speed of light, c (3*108 m/s), so that:\n\nf = c/λ\n\nAgain, there are many different frequencies of light in the electromagnetic spectrum. The light visible to the human eye lies between 790 to 400 terahertz, and we are most sensitive to the greenish light at 540 terahertz. It must be noted that visible light is only a small portion of the entire electromagnetic spectrum. There are UV rays, Infrared rays, microwaves, radio waves, gamma rays, and X rays. The entire electromagnetic spectrum has frequencies that run between 10 Hz to 1024 Hz.\n\nBookmark this page in your browser using Ctrl and d or using one of these services: (opens in new window)" ]

[ null, "https://www.calculator.org/images/properties/frequency.jpg", null ]

[ "Gamasutra is part of the Informa Tech Division of Informa PLC\n\nThis site is operated by a business or businesses owned by Informa PLC and all copyright resides with them. Informa PLC's registered office is 5 Howick Place, London SW1P 1WG. Registered in England and Wales. Number 8860726.", null, "", null, "", null, "Contents\nA Non-Integer Power Function on the Pixel Shader", null, "Latest Jobs\nNovember 19, 2019", null, "Latest Blogs", null, "Press Releases\nNovember 19, 2019\nGames Press", null, "About", null, "Gama Network\nIf you enjoy reading this site, you might also want to check out these UBM Tech sites:", null, "Features\n\n# A Non-Integer Power Function on the Pixel Shader\n\nJuly 31, 2002", null, "Page 1 of 5", null, "Raising a variable to a non-integer power is an operation often encountered in computer graphics. It is required, for example, to compute the specular term in various shading schemes. It can also be used to produce a smooth conditional function useful in many pixel shaders. In most cases, the input variable falls between 0 and 1 while the exponent is greater than 1 and often quite large.\n\nLike many other techniques, a quality gain can be achieved by computing the function per pixel rather than per vertex. This gain is very noticeable when using large exponents since the function varies a lot and sampling it at each vertex is bound to miss visually important details (see Figure 1).\n\nTherefore, we are particularly interested in finding a way to compute such a function on the pixel shader. Like any pixel shader trick, it is important to minimize the number of textures and blending stages since these are very limited resources. This text presents a simple shader trick that performs a good per pixel approximation of a non-integer power function. The technique works for input values between 0 and 1 and supports large exponents. The presented shader does not require any texture look-up and is scalable, making it possible to spend more instructions in order to decrease the error or to reach greater exponents.\n\nWe first consider and analyze two typical techniques used to compute a power function on the pixel shader. We then expose some mathematical background used throughout the text. Finally, we show how the algorithm can be used to perform smooth conditional functions and complex bump-mapped Phong shading. The actual implementation of the approximation as a pixel shader program is discussed in detail.", null, "", null, "", null, "", null, "", null, "", null, "Figure 1. Gouraud shading (left) and Phong shading (right)\n\nWhen confronted with the problem of computing a power function on the pixel shader, two simple techniques come to mind. First, it seems possible to proceed through a 1D texture look-up, and second, applying successive multiplications looks promising.\n\nTexture Look-Up\nLinearly interpolated textures can be thought of as piecewise linear functions. In particular, 1D textures with a linear filter are really a function taking a value between 0 and 1 and mapping it onto another value in the same range . This looks promising for our problem since an input between 0 and 1 raised to any power greater than 0 yields a result between 0 and 1.\n\nListing 1 shows a piece of code that builds a 16-bit monochrome 1D texture of resolution Resi to compute xn:\n\nvoid ComputePowerTexture( int Resi, double n, unsigned short* Texture )\n{\nint i;\nfor( i=0; i\nTexture[i] = (unsigned short)( pow( (double)i/Resi, n ) * USHRT_MAX );\n}\n\nListing 1. C Function to create a 1D power texture\n\nOnce this texture has been constructed, a simple texture look-up pixel shader can be used to perform the computation, provided that the value to raise to power n is placed in an interpolated texture coordinate. The pixel shader at Listing 2 shows how to apply the power function on the result of a dot product, like it is often required. Note that this code only works for pixel shader versions 1.2 and 1.3. The code for version 1.4 is presented in Listing 3.\n\nps.1.2\n\ntex t0 ; Texture #0 look-up a vector\ntexdp3tex t1, t0 ; Use texture coordinates #1 as a 3D vector,\n; performs dot product between this 3D vector and t0,\n; using the result, look-up power function in 1D texture #1\n\nmov r0, t1 ; emit the result\n\nListing 2. Code computing a power function using a 1D texture\n(pixel shader versions 1.2 and 1.3)\n\nps.1.4\n\ntexld r0, t0 ; Texture #0 look-up a vector\ntexcrd r1.rgb, t1 ; Load texture coordinates #1 as a 3D vector\n\ndp3 r0, r0, r1 ; Performs dot product between this 3D vector and r0\n\nphase\n\ntexld r1, r0.x ; using the result, look-up power function in 1D texture #1\n\nmov r0, r1 ; emit the result\n\nListing 3. Code computing a power function using a 1D texture\n\nVarious problems exist with that particular technique:\n\n• it uses up one texture stage, which may make it unfit to algorithms requiring many textures,\n• changing the value of the exponent n requires regenerating the texture, unless a 2D texture is used in which case a limited number of predefined exponents can be used,\n• for pixel shaders versions less than 1.4, a 1D texture look-up cannot be applied to intermediate computation results unless multiple passes are used.\n\nThis last limitation is often a major drawback since, in usual cases, the power function must be preceded by a vector renormalization (such as done with a cube map) and a dot product. With large exponents, the vector renormalization is especially important. This is due to the fact that the maximum value of a dot product is the product of the length of the two vectors. If one of these is not normalized, the dot product can never reach 1. When raised to a large exponent, a value smaller than 1 will rapidly move toward 0. This translates to visual details being washed out. Figure 2 shows a vector interpolated with and without normalization, followed by a dot product, and then raised to a high power. It is obvious that the detail (for example a specular spot) is washed out in the second version.", null, "", null, "", null, "", null, "", null, "Figure 2. Result of not normalizing vectors before applying a power function.\n\nSuccessive Multiplications\nSince raising a value to an integer power simply requires multiplying a value with itself a number of times, it seems possible to approximate a non-integer power function through successive multiplication steps.\n\nFor example, the pixel shader at Listing 4 shows how to raise t0 to power 16. Analyzing this scheme indicates that log2 n multiplications are required to raise a variable to the power n, when n is a power of 2:\n\n ps.1.0 tex t0 mul r0, t0, t0 mul r0, r0, r0 mul r0, r0, r0 mul r0, r0, r0 ; r0 = t0 *t0 = t0^2 ; r0 = t0^2*t0^2 = t0^4 ; r0 = t0^4*t0^4 = t0^8 ; r0 = t0^8*t0^8 = t0^16\n\nListing 4. Power function using successive multiplications\n\nListing 5 shows a pixel shader that raises t0 to power 31. Analyzing this shows that, in general, for n in the range [2a,2a+1), the algorithm can require 2a multiplications and a temporary variables2.\n\n ps.1.1 tex t0 mul t1, t0, t0 mul t2, t1, t1 mul t3, t2, t2 mul r0, t3, t3 mul r0, r0, t3 mul r0, r0, t2 mul r0, r0, t1 mul r0, r0, t0 ; t1 = t0^2 ; t2 = t0^4 ; t3 = t0^8 ; r0 = t0^16 ; r0 = t0^16 * t0^8 = t0^24 ; r0 = t0^24 * t0^4 = t0^28 ; r0 = t0^28 * t0^2 = t0^30 ; r0 = t0^30 * t0 = t0^31\n\nListing 5. Power function with a non-power-of-2 exponent\nusing successive multiplications\n\nThe limitations of this technique are the following:\n\n• only supports discrete changes in the exponent, making it impossible to change the value of n in a continuous fashion,\n• requires a lot of instructions for a large exponent,\n• requires a lot of instructions and temporary variables for non power of 2 exponents.\n\nThese last two problems often limit the usefulness of successive multiplications, since practical exponents have a tendency to be large and are usually not powers of 2.\n\nThe Need for a New Trick\nAlthough the 1D texture look-up and the successive multiplications techniques have no limitations in common, both are too restrictive to be really useful in the general case. In particular, none of them is suited to large exponents. In the case of 1D textures, the impossibility to perform a per pixel renormalization before the look-up makes it unsuitable, and for successive multiplications the number of instructions required for large exponents is prohibitive.\n\nThe power function approximation technique presented in the rest of the text addresses all of the preceding issues. We therefore think it can often be used as an efficient alternative in pixel shaders that require a power function.", null, "Page 1 of 5", null, "### Top Stories", null, "" ]

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[ "# WASP AI ANNOUNCES 18 PHD POSITIONS IN AI-MATH\n\nApplication deadlines passed for all positions.\n\nWallenberg AI, Autonomous Systems and Software Program (WASP) is Sweden’s largest individual research program ever. WASP provides a platform for academic research and education, fostering interaction with Sweden’s leading technology companies. Part of the initiative in AI within WASP deals with increasing our understanding of fundamental mathematical principles behind AI.\n\nWe now offer up to 18 university PhD positions at seven university sites, with focus on mathematics behind AI.\n\nFor further descriptions of positions and to apply, follow the links to the coordinating universities. The positions included in this call are at Chalmers, KTH, Linköping university, Lund university, Stockholm university, Umeå university and Uppsala university. Please note different final dates for applications.\n\nChalmers\n\nOptimization algorithms forachine learning:\nOptimization methods have played a major role in modern machine learning: in particular, many of the recent successes rest on the development of highly scalable stochastic optimization methods such as stochastic gradient descent (SGD) and accelerated versions of it. The goal of this project is to explore further ideas around these concepts and apply them to fundamental ML problems such as matrix completion and optimal transport.\n\nFour positions in mathematics for AI within the following areas/projects:\nLearning with noisy labels, Deep learning and statistical model choice, Deepest learning using stochastic partial differential equations, Quantum deep learning and renormalization: A group-theoretic approach to hierarchical feature representations.\n\nUnderstanding deep learning: From theory to algorithms.\nThe purpose of this project is to increase our theoretical understanding of deep neural networks. This will be done by relying on tools of information theory and focusing on specific tasks that are relevant to computer vision.\n\nKTH Royal Institute of Technology\n\nAlgebraic Topology and Mathematical Statistics:\nAI is often identified with the ability to simplify the data while retaining its information content relevant for the decision making.  Homological invariants are examples of simplifying tools particularly suitable for encoding shape. The challenge is to adopt homological information for statistical analysis.\n\nInformation theory for AI and machine learning:\nThe project studies the interaction between information theory and mathematical methods for learning and statistical inference, with applications to artificial intelligence and machine learning, especially deep learning. New theoretical principles and guidelines for algorithm development and mathematical performance analysis will be developed, with information theory as the scientific basis.\n\nCombinatronics:\nProject concerns studies of combinatorial structures, such as directed graphs and convex polytopes, and studies on combinatorial notions of causality. The candidate should have knowledge of and an interest in combinatorics and will be part of the combinatorics group.\n\nGoing Beyond State of the Art in Integer Linear Programming by Using Conflict Driven Search:\nIn this project we want to develop mathematical methods for efficient 0-1 integer linear programming (a.k.a. pseudo-Boolean SAT solving) based on the methods for conflict-driven clause learning algorithms used for SAT solving, and then implement these methods of reasoning in new solvers. Our ultimate goal is to build a proof-of-concept engine for pseudo-Boolean optimization and integer linear programming, based on new, original ideas, that would have the potential to be competitive in the relevant domains with packages such as SCIP or even CPLEX and Gurobi.\n\nApplied and computational mathematics:\nGenerative models, such as variational auto-encoders and generative adversarial networks, are being developed with the aim to produce samples of complex data structures such as images, natural language, voice and sound, video, times series, financial data, etc. This project will study mathematical aspects of generative models, related to large deviations theory and stochastic computational methods.\n\nUp to two positions in computer science:\nConstraint Satisfaction Problems (CSP) are a well-known class of computational problems and many problems encountered in AI (but also in other fields of computer science, mathematics, and elsewhere) are instances of the CSP. The computational complexity of CSP problems has been intensively studied and several breakthrough results have recently been presented. These results are typically based on utilising methods from universal algebra and mathematical logic. This project aims at developing new mathematical methods for analysing the computational complexity of CSPs.\n\nLund university\n\nOptimization methods for matrix factorization, dimensionality reduction and deep networks:\nIn recent years there has been a dramatic increase in performance of recognition, classification and segmentation systems due to deep learning approaches, in particular convolutional nets. Furthermore, training with these models requires solving very large scale non-convex optimization problems and it is unclear under what conditions these can be reliably solved. In this project we will study mathematical optimization methods for learning and dimensionality reduction approaches related to matrix factorization. We will develop effective and reliable algorithms that scale well beyond today’s standard and apply these to computer vision applications.\nNote that only the project ”Optimization methods for matrix factorization, dimensionality reduction and deep networks” is a WASP-position in this ad.\n\nModelling and analysis of multi-scale neuronal networks\nThe project is within the area of mathematics at the intersection of probability theory, analysis and dynamical systems. The goal is the development and study of the dynamical systems which model processes compatible with brain functions, as learning,  memory, cognition, decision making. The project will focus on modelling and analysis of multi-scale complex systems which combine molecular/particle level with macro-dynamics.\n\nStockholm university\n\nCombining quantitative and qualitative analysis of multi-player games:\nThe project aims to explore the mathematical aspects of a general framework for modelling of intelligent multi-agent systems with both quantitative and qualitative goals and constraints. The main objectives of the project are to develop and apply mathematical methods for analysing dynamic multi-stage games modelled by concurrent game models with payoffs and guards.\n\nUmeå University\n\nDevelopment of mathematical methods and tools for investigation and future development of deep neural networks:\nThe project will in particular investigate a specific type of deep neural network which is among the most successful and can be related to evolution equations. The research project involves mathematical and algorithmic development as well as practical application and testing using state of the art software.\n\nUppsala university\n\nNew mathematical knowledge about the intrinsic behaviour of neural networks when subjected to noise:\nOne of the core aims is to develop the mathematical foundation for understanding how the distribution of information carrying capacity for a network’s internal weights can be utilized in order to optimize its performance.\n\nRobust learning of geometric equivariances:\nDeep convolutional neural networks (CNNs) are showing impressive results in a variety of tasks in image data analysis and understanding. This project explores ways to further improve their performance by representing and exploiting their properties, such as known equivariances, as well as learning equivariances from data. The project builds on, and extends recent very promising works on Geometric deep learning and combines this with Manifold learning, to produce truly learned equivariances without the need for engineered solutions.\n\nView all calls" ]

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[ "## Electrical Engineering Basics Questions\n\nToday we are Going to Provide You Electrical engineering interview questions from Electrical Engineering Basics Chapter.\n\nAll matter is composed of - Atoms\n\nTo be a conductor, the substance must contain some mobile electrons\n\nConductivity depends on the number of electrons. - (Yes/No)\n\nThe energy level of an electron decreases as its distance from the nucleus increases.  - (Yes/No)\n\nAtoms with fewer than four valence electrons are good conductors.\n\nAtoms with more than four valence electrons are poor conductors.\n\nAtoms with four valence electrons are - semi-conductors\n\nMass of an electron is approximately equal to 1/1000 mass of the proton - (Yes/No)\n\nThe diameter of an electron = 10-15 Meter (Yes/No)\n\nWhen an electron is removed from a neutral atom, this atom becomes positively charged and is called a - Positive Ion\n\nThe controlled movement of electrons through a substance is called - Current\n\nElectromotive force is the force that causes a current of electricity to flow.\n\nAlso Read - Basic of Electrical Engineering and Basic Electrical Formulas\n\nThe is a unit of potential difference and electromotive force - Volt\n\nElectron-volt is a unit in terms of which the energies of atomic particles are expressed. (Yes/No)\n\nThe opposition to the flow of electrons is called electrical resistance\n\nThe practical unit of electric resistance is mho. (Yes/No)\n\nResistivity of a material may be defined as the resistance between the opposite faces of a meter cube of that material.\n\nThe reciprocal of specific resistance of a material is called its conductivity\n\nThe unit of conductivity is ohm/ meter. (Yes/No)\n\nThe resistance of metal conductors increases with rising of temperature. (Yes/No)\n\nThe resistance of semiconductors and all electrolytes decreases as the temperature rises.\n\nA linear resistor is one which Obey's Ohm's law.\n\nAt very low temperature, some metals acquire zero electrical resistance and zero magnetic induction, the property known as super-conductivity\n\nA fixed resistor is the simplest type of resistor and its value is constant and unchangeable.\n\nAlso ReadImportant Topic on A.C. Fundamentals\n\nA variable resistor is commonly called a control. (Yes/No)\n\nThe most common type of special resistor is the fusible type. (Yes/No)\n\nIn a series combination of resistors, the current flowing through each resistor is different. (Yes/No\n\nThe resistivity of pure semiconductors is of the order of 1 ohm-meters. (Yes/No)\n\nThe symbol marked R8 represents a fixed resistor.\n\nThe part represented by the symbol marked R1 is a Variable resistor.\n\nThis is Part 1 of Electrical Engineering Basics Questions with Answers, these Questions and Answers are Very Important For Your Exam and Interview. Keep Visiting ElectricalNotes4u for Latest Electrical Engineering MCQ, Notes, Interview Questions with Answers.\n\nShare:\n\n1.", null, "1.", null, "" ]

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[ "# If $(p)$ is a proper subset of a proper ideal $I$, then is $I$ prime?\n\nLet $$R$$ be the ring of algebraic integers of a quadratic imaginary number field $$\\mathbb Q[\\sqrt{d}]$$ for a negative square-free integer $$d$$. For a prime integer $$p$$, $$(p)$$ is a prime ideal or is the product $$P \\overline P$$ of some prime ideal $$P$$ and $$\\overline P$$, the ideal consisting of the complex conjugates of elements of $$P$$. Why does this mean if $$(p)$$ is a proper subset of a proper ideal $$I$$ of $$R$$, then $$I$$ is prime?\n\n• If $$(p)$$ is a prime ideal, then $$(p)$$ is a maximal ideal so $$(p)=I$$.\n\n• I don't know how to say $$(p)=P \\overline P \\subset I \\subset R$$ implies $$I$$ is a prime ideal.\n\n• Our definition of a prime ideal $$P$$ is that $$P$$ is nonzero and if the product $$CD$$ of two ideals $$C$$ and $$D$$ is a subset of $$P$$, then $$C$$ or $$D$$ is a subset of $$P$$.\n\n• I'm afraid what you claim is not true. $I=(p)$ clearly contains $(p)$, but is not always prime. For an explicit example, consider $I=(2)$ in $\\mathbb Z[i]$. Then $(1+i)^2\\in I$ but $1+i\\not\\in I$, so $I$ is not prime. – Wojowu Jan 20 at 9:29\n• @Wojowu I changed to $(p)=P \\overline P \\subset I \\subset R$ – Ekhin Taylor R. Wilson Jan 20 at 10:02\n• I see, now it makes more sense. – Wojowu Jan 20 at 10:04\n\nHere is a straightforward proof. Since we are in a quadratic field, it's not hard to see that $$R/(p)$$ has $$p^2$$ elements (since, as a group, $$R$$ is free abelian on two generators). If $$I$$ is a proper ideal properly containing $$(p)$$, then the quotient $$R/I$$ is isomorphic to a quotient of $$R/(p)$$ by the image of $$I$$ modulo $$(p)$$. From there it's clear $$R/I$$ has $$p$$ elements, so is a field, implying $$I$$ is maximal, hence prime.\nSuppose that $$I$$ is a proper ideal such that $$P\\overline P\\subsetneq I$$ (notice that the inclusion should be strict, otherwise $$P\\overline P=I$$ is a counterexample). If $$M$$ is a maximal ideal containing $$I$$ then $$P\\overline P\\subsetneq M$$ implies either $$P\\subsetneq M$$ or $$\\overline{P}\\subsetneq M$$. WLOG we have $$P\\subsetneq M$$ and hence $$P$$ is a prime ideal which is neither $$(0)$$ or maximal. This contradicts the fact that the ring of integers have dimension $$1$$ (i.e., every nonzero prime ideal is maximal).\nA more elucidative proof would be using the properties of the ideal factorization in number fields. If $$P\\overline{P}\\subsetneq I$$ then we have that $$I$$ properly divides $$P\\overline{P}$$ and hence either $$I=P$$ or $$I=\\overline{P}$$." ]

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[ "To be able to describe the motion of a body in mathematical terms, we must first understand the type of motion it is undergoing. The equation(s) of motion to be used will vary depending on the situation under consideration.\n\n## 1. Linear motion\n\nThere are three general cases of linear motion characterised by the type of acceleration it is undergoing.\n\n### No acceleration or deceleration – Constant speed\n\nIf the object’s speed is fixed at all times, then it does not experience any acceleration or deceleration at all. Its speed will remain constant as time changes. Remember that there may or may not be forces acting on it but there is no net force causing it to accelerate. The equation of motion used in this case is simple, in the form of:\n\n$$v=s/t$$\n\nwhere,\n\n$$v$$: speed or velocity (in m/s or ft/s)\n\n$$s$$: distance travelled or displacement (in m or ft)\n\n$$t$$: time taken to travel that distance (in s)\n\nExample 1\nA person takes 10 minutes to walk from point A to point B. Calculate the distance separating the two points, assuming that he walks at a constant speed of 1.5 m/s the whole time.\n\nAnswer: First, the time taken needs to be converted to seconds to remain consistent with units. By plugging in the values into the equation, we then get a distance of 900 m.\n\n### Constant acceleration or deceleration – Varying Speed\n\nIn this case, the body accelerates at a constant rate throughout its motion. The speed it travels will therefore change as time advances.\n\nTypical graphs for this type of motion are as shown below.\n\nThe following formula of motion are then used:\n\n$$v=u+at$$\n\n$$s=ut+\\frac{1}{2}at^2$$\n\n$$v^2=u^2+2as$$\n\nwhere,\n\n$$v$$: final velocity (m/s)\n\n$$u$$: initial velocity (m/s)\n\n$$a$$: rate of acceleration. If body is in deceleration, then use a minus sign ($$m/s^2$$)\n\n$$t$$: time taken (s)\n\n$$s$$: distance travelled (m)\n\nExample 2\nAn object is held at rest 12 m above the ground. The object is released at time $$t=0 s$$ and falls freely under gravity. Given that the acceleration due to gravity is 9.81 $$m/s^2$$, find its speed midway between its starting point and the ground.\n\nAnswer: Using the third formula of motion ($$v^2=u^2+2as$$) and the quantities $$a=9.81 m/s^2$$, $$s=6m$$, $$u=0 m/s$$; we then get an answer of 10.85 m/s.\n\n### Varying Acceleration\n\nEach object undergoing this type of motion will have a specific equation describing its displacement, velocity and acceleration at specific points in time. By knowing one of these equations, it is possible to determine the others by integrating or differentiating with respect to time. For this type of motion, it is important to keep in mind the following:\n\n• Speed ($$v$$) is the rate of change of distance ($$s$$) with time ($$t$$) or in calculus terms: $$\\frac{ds}{dt}$$\n• Acceleration ($$a$$) is the rate of change of speed ($$v$$) with time ($$t$$) or in calculus terms: $$\\frac{{d^2}s}{d{t^2}}$$\n\nExample 3\nA particle moves in space with its speed given by the equation $$v= 4t^3 + 6$$. Find its speed, distance travelled and acceleration at time t = 5 s.\n\nAnswer: The speed can be found easily by plugging t = 5 s in the above equation, giving $$v=506 m/s$$. The acceleration can also be found by differentiating $$v$$ and using t = 5 s. To find the distance travelled, we need to integrate $$v$$ using limits of t = 0s and t = 5 s.\n\n.\n\n## 2. Circular Motion\n\nConsider a body that is moving in circular motion about point O as shown below.", null, "If the body takes a time $$t$$ to make an angle $$\\theta$$ about point O, then we can find its angular speed using:\n\n$$\\omega = \\theta/t$$\n\nThe period $$T$$ for the object to complete a full revolution is given by:\n\n$$T = \\frac{2\\pi}{\\omega}$$\n\nThe distance travelled $$s$$ is calculated using:\n\n$$s = r\\theta$$\n\nThe linear speed can also be found by dividing the last equation by time taken $$t$$:\n\n$$\\frac{s}{t} = r\\frac{\\theta}{t}$$\n\n$$v = r\\omega$$\n\nFor the body to remain in circular motion, there needs to be a force pulling it towards the centre. Otherwise, the object will move out of orbit in a straight line according to Newton’s First law of motion. This force is also known as the centripetal force and it induces an acceleration on the body. The direction of the acceleration is towards the centre which is the same as that of the centripetal force. The centripetal acceleration can be calculated as:\n\n$$a = \\frac{v^2}{r}$$\n\nor\n\n$$a = r\\omega^2$$\n\n## 3. Projectile Motion\n\nThe motion of a projectile thrown at an angle $$x$$ degrees relative to the ground is shown below.", null, "The velocity vector $$u$$ can be decomposed to its 2 components $$u_x$$ and $$u_y$$.\n\n$$u_x = u\\cos{x}$$\n\n$$u_y = u\\sin{x}$$\n\nAs soon as a projectile is released, its trajectory will depend on two things: its initial velocity and the acceleration due to gravity." ]

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[ "# PITI Calculator\n\nBy Anna Szczepanek, PhD\nLast updated: Mar 02, 2020\n\nOur PITI calculator helps you evaluate the total monthly cost of your mortgage, which is crucial when you want to determine how much money you can afford to borrow to buy a new house. Here you can not only learn how to calculate PITI but also understand what PITI is and why it matters.\n\n## What is PITI?\n\nPITI is an acronym of:\n\n1. Principal - the amount of your loan (not including interest). Most likely, it is the cost of your house minus your down payment.\n2. Interest - the rate at which the lender charges you for borrowing money. To learn more, check out our interest rate calculator.\n3. Tax - property tax charged at the municipal level. You can find the exact amount on the website of the county where the house is located, or get it from your real estate agent.\n4. Insurance - protection for your property in case of fire, lightning, break-in, flooding, random acts of God, etc. Most often, it is not required by law, yet demanded by the lender.\n\nTaking all of these into account, the PITI mortgage calculator gives you an accurate idea of your loan's monthly repayments.\n\n## Why does PITI matter?\n\nPITI allows you to calculate your Debt-to-Income (DTI) ratio, which helps determine what amount of money you can safely borrow. The specific maximum value of DTI that will be deemed acceptable by a lender depends on your region, yet most lenders use the DTI 28% rule as a first estimate when they decide whether or not to loan you money.\n\n## How to calculate PITI?\n\nBased on the principal loan amount, interest rate, the annual tax amount and insurance cost, our calculator determines your PITI using the following formula:\n\n`PITI = monthly tax + monthly insurance + monthly mortgage payment`\n\nwhere:\n\n1. Monthly tax is your annual tax amount divided by 12.\n2. Monthly insurance is your annual insurance cost divided by 12.\n3. Monthly mortgage payment is calculated based on your principal loan amount and annual interest rate. How? We have a mortgage payment calculator for that :)\n\nFor instance, let's assume that each year \\$3,600 is needed to cover the property tax, and the insurance cost is \\$1,200. Your expected monthly tax and insurance costs are therefore equal to `\\$3600 / 12 = \\$300` and `\\$1200 / 12 = \\$100`, respectively. If you borrow \\$200,000 for 30 years at 7%, your monthly mortgage payment will be %1,330.60. All that's left to do is to add together these three terms to get your PITI estimation: `PITI = \\$300 + \\$100 + \\$1330.60 = \\$1730.60`\n\n## How does our PITI mortgage calculator work?\n\nThe exact formula implemented in this PITI calculator is:\n\n`PITI = t / 12 + i / 12 + P * r / 12 * (1 + r / 12)ⁿ / [(1 + r / 12)ⁿ - 1]`\n\nwhere:\n\n• t is the annual tax amount, so t/12 is the monthly tax amount.\n• i is the annual insurance cost, so i/12 is the monthly insurance cost.\n• P is the principal loan amount.\n• r is the annual interest rate. Hence, r/12 is the monthly interest rate.\n• n is the number of monthly repayments.\n\nNote that the term `P * r / 12 * (1 + r / 12)ⁿ / [(1 + r / 12)ⁿ - 1]` is the exact same formula as the loan payment formula, where an amount, P, is borrowed for n months at an annual rate, r.\n\nAnna Szczepanek, PhD\nPrincipal\n\\$\nTerm\nyrs\nmos\nInterest rate\n%\nTax\n\\$\nInsurance\n\\$\nPITI\n\\$/month\nPeople also viewed…\n\n### Coffee kick\n\nA long night of studying? Or maybe you're on a deadline? The coffee kick calculator will tell you when and how much caffeine you need to stay alert after not sleeping enough 😀☕ Check out the graph below!\n\n### Interest rate\n\nThe Interest Rate Calculator is a device that computes various types of interest rate, from the point of view of either a loan or a deposit account.", null, "" ]

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[ "# Difference between revisions of \"Surface types\"\n\n## Elementary surfaces\n\nNotes:\n\n• Elementary surfaces refer here to surfaces that can be represented by a single equation.\n\n### Planes\n\nNotes:\n\n• Parametric form of the general plane is assumed if four values are provided in the surface card. With six values the plane is assumed to be defined by three points.\n• The positive side for a plane defined by three points is determined by the order in which the points are entered (see the right-hand rule).\nSurface name Parameters Surface equation Description\npx x0", null, "$S(x) = x - x_0$ Plane perpendicular to x-axis at x = x0\npy y0", null, "$S(y) = y - y_0$ Plane perpendicular to y-axis at y = y0\npz z0", null, "$S(z) = z - z_0$ Plane perpendicular to z-axis at z = z0\nplane A, B, C, D", null, "$S(x,y,z) = Ax+ By + Cz - D$ General plane in parametric form\nplane x1, y1, z1, x2, y2, z2, x3, y3, z3 General plane defined by three points\n\nNotes:\n\n• cyl is the same surface as cylz\n• Infinite cylinders use the same names as the truncated cylinders.\n• With three values provided in the surface card for cylx, cyly, cylz or cyl the surface is an infinite cylinder. With five values the surface is an truncated cylinder.\nSurface name Parameters Surface equation Description\ncylx y0, z0, r", null, "$S(y,z) = (y - y_0)^2 + (z - z_0)^2 - r^2$ Infinite cylinder parallel to x-axis, centred at (y0,z0), radius r\ncyly x0, z0, r", null, "$S(x,z) = (x - x_0)^2 + (z - z_0)^2 - r^2$ Infinite cylinder parallel to y-axis, centred at (x0,z0), radius r\ncylz, cyl x0, y0, r", null, "$S(x,y) = (x - x_0)^2 + (y - y_0)^2 - r^2$ Infinite cylinder parallel to z-axis, centred at (x0,y0), radius r\ncylv x0, y0, z0, u0, v0, w0, r", null, "$S(x,y,z) = (1-u_0^2)(x - x_0)^2 + (1-v_0^2)(y - y_0)^2 + (1-w_0^2)(z - z_0)-r^2$ Infinite cylinder, parallel to (u0,v0,w0), centred at (x0,y0,z0), radius r\nsph x0, y0, z0, r", null, "$S(x,y,z) = (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 - r^2$ Sphere, centred at (x0,y0,z0), radius r\ncone x0, y0, z0, r, h", null, "$S(x,y,z) = (x - x_0)^2 + (y - y_0)^2 - \\left(1 - (z - z_0)/h\\right)r^2$ Half cone parallel to z-axis, base at (x0,y0,z0), base radius r, height h (distance from base to vertex)\nquadratic A, B, C, D, E, F, G, H, I, J", null, "$S(x,y,z) = Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fzx + Gx + Hy + Iz + J$ General quadratic surface in parametric form\n\nNotes:\n\n• Serpent can handle circular and elliptical torii. Radii r1 and r2 must be set equal (denoted in the surface equations as r) to describe a circular torus surface.\nSurface name Parameters Surface equation Description Notes\ninf -", null, "$S(y,x,z) = -\\infty$ All space Can not be used in root universe\ntorx x0, y0, z0, R, r, r", null, "$S(x,y,z) = \\left(R - \\sqrt{(y - y_0)^2 + (z - z_0)^2}\\right)^2 + (x - x_0)^2 - r^2$ Circular torus with major radius R perpendicular to x-axis (revolving radius), centred at (x0, y0, z0), minor radius r (inner radius).\nx0, y0, z0, R, r1, r2", null, "$S(x,y,z) = \\dfrac{\\left(R - \\sqrt{(y - y_0)^2 + (z - z_0)^2}\\right)^2}{r_2^2} + \\dfrac{(x - x_0)^2}{r_1^2} - 1$ Elliptic torus with major radius R perpendicular to x-axis (revolving radius) centred at (x0, y0, z0), vertical (x-)semi-axis r1 and horizontal (y-/z-)semi-axis r2.\ntory x0, y0, z0, R, r, r", null, "$S(x,y,z) = \\left(R - \\sqrt{(x - x_0)^2 + (z - z_0)^2}\\right)^2 + (y - y_0)^2 - r^2$ Circular torus with major radius R perpendicular to y-axis (revolving radius), centred at (x0, y0, z0), minor radius r (inner radius).\nx0, y0, z0, R, r1, r2", null, "$S(x,y,z) = \\dfrac{\\left(R - \\sqrt{(x - x_0)^2 + (z - z_0)^2}\\right)^2}{r_2^2} + \\dfrac{(y - y_0)^2}{r_1^2} - 1$ Elliptic torus with major radius R perpendicular to y-axis (revolving radius) centred at (x0, y0, z0), vertical (y-)semi-axis r1 and horizontal (x-/z-)semi-axis r2.\ntorz x0, y0, z0, R, r, r", null, "$S(x,y,z) = \\left(R - \\sqrt{(x - x_0)^2 + (y - y_0)^2}\\right)^2 + (z - z_0)^2 - r^2$ Circular torus with major radius R perpendicular to z-axis (revolving radius), centred at (x0, y0, z0), minor radius r (inner radius).\nx0, y0, z0, R, r1, r2", null, "$S(x,y,z) = \\dfrac{\\left(R - \\sqrt{(x - x_0)^2 + (y - y_0)^2}\\right)^2}{r_2^2} + \\dfrac{(z - z_0)^2}{r_1^2} - 1$ Elliptic torus with major radius R perpendicular to z-axis (revolving radius) centred at (x0, y0, z0), vertical (z-)semi-axis r1 and horizontal (x-/y-)semi-axis r2.\n\n## Derived surface types\n\nNotes:\n\n• Derived surfaces refer here to surfaces composed of two or more elementary types.\n\n### Truncated cylinders\n\nNotes:\n\n• Truncated cylinders use the same names as the infinite cylinders.\n• With five values provided in the surface card for cylx, cyly, cylz or cyl the surface is an truncated cylinder. With three values the surface is an infinite cylinder.\nSurface name Parameters Composed of Description\ncylx y0, z0, r, x0, x1 Infinite cylinder + two planes Infinite cylinder parallel to x-axis, centred at (y0,z0), radius r, truncated between [x0, x1]\ncyly x0, z0, r, y0, y1 Infinite cylinder + two planes Infinite cylinder parallel to y-axis, centred at (x0,z0), radius r, truncated between [y0, y1]\ncylz, cyl x0, y0, r, z0, z1 Infinite cylinder + two planes Infinite cylinder parallel to z-axis, centred at (x0,y0), radius r, truncated between [z0, z1]\n\n### Regular prisms\n\nNotes:\n\n• All prisms are parallel to z-axis, and they can be rotated using surface transformations.\n• Infinite and truncated triangular prisms use the same name, and are composed by three or five planes, respectively. Infinite prism is assumed if three/four values are provided in the surface card tric. With six values the surface is assumed to be a truncated (equilateral) triangular prism.\n• Triangular prisms orientation, ori, corresponds to the cardinal direction of the non-aligned vertex of the triangle. Default orientation is North. It follows the scheme: W-S-E-N (W=1, S=2, E=3, N=4).\nSurface name Parameters Composed of Description\ntric x0, y0, d, ori three planes Infinite (equilateral) triangular prism parallel to z-axis, centred at (x0,y0), half-width d, orientation ori (optional)\nx0, y0, d, ori, z0, z1 five planes Truncated (equilateral) triangular prism parallel to z-axis, centred at (x0,y0), half-width d, orientation ori, truncated between [z0, z1]\nsqc x0, y0, d four planes Infinite square prism parallel to z-axis, centred at (x0,y0), half-width d\nrect x0, x1, y0, y1 four planes Infinite rectangular prism parallel to z-axis, between [x0, x1] and [y0, y1]\nhexxc x0, y0, d six planes Infinite hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to x-axis, half-width d\nhexyc x0, y0, d six planes Infinite hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to y-axis, half-width d\nhexxprism x0, y0, d, z0, z1 eight planes Truncated hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to x-axis, half-width d, truncated between [z0, z1]\nhexyprism x0, y0, d, z0, z1 eight planes Truncated hexagonal prism parallel to z-axis, centred at (x0,y0), flat surface perpendicular to y-axis, half-width d, truncated between [z0, z1]\nocta x0, y0, d1, d2 eight planes Infinite octagonal prism parallel to z-axis, centred at (x0,y0), half-widths d1 and d2. If the last value is omitted: d = d1 = d2\ndode x0, y0, d1, d2 twelve planes Infinite dodecagonal prism parallel to z-axis, centred at (x0,y0), half-widths d1 and d2. If the last value is omitted: d = d1 = d2\n\n### 3D polyhedra\n\nNotes:\n\n• The description of parallelepiped may be wrong.\nSurface name Parameters Composed of Description\ncube x0, y0, z0, d six planes Cube, centred at (x0,y0,z0), half-width d\ncuboid x0, x1, y0, y1, z0, z1 six planes Cuboid, between [x0, x1], [y0, y1] and [z0, z1]\nppd x0, y0, z0, Lx, Ly, Lz, αx, αy, αz six planes Parallelepiped, with corner at (x0, y0, z0) and edges of length Lx, Ly and Lz at angles αx, αy and αz (in degrees) with respect to the coordinate axes\n\n### Other derived surface types\n\nSurface name Parameters Description\npad x0, y0, r1, r2, α1, α2 Sector from α1 to α2 (in degrees) of a cylinder parallel to z-axis, centred at (x0,y0), between radii r1 and r2\ncross x0, y0, l, d Cruciform prism parallel to z-axis, centered at (x0,y0), half-width l, half-thickness d\ngcross x0, y0, d1, d2, ... Prism parallel to z-axis, centred at (x0,y0), formed by planes at distances dn from the center (\"generalized cruciform prism\", see figure below)\nhexxap x0, y0, wd, dw, a Surface for simplified modeling of hexagonal fuel assembly angle pieces. Consists of two infinite hexagonal prisms parallel to z-axis, centered at (x0, y0), flat surfaces perpendicular to x-axis, with the outer hexagon having outer half-width of wd, and the surface perpendicular thickness is dw (the inner hexagon half width is wd - dw), and each half-section of each angle piece with width of a, measured from the tip of the angle piece angle to the flat surface of the angle piece.\nhexyap x0, y0, wd, dw, a Surface for simplified modeling of hexagonal fuel assembly angle pieces. Consists of two infinite hexagonal prisms parallel to z-axis, centered at (x0, y0), flat surfaces perpendicular to y-axis, with the outer hexagon having outer half-width of wd, and the surface perpendicular thickness is dw (the inner hexagon half width is wd - dw), and each half-section of each angle piece with width of a, measured from the tip of the angle piece angle to the flat surface of the angle piece.\ninvolute x0, y0, r0, θ1, θ2, r1, r2 Involute parallel to z-axis, centred at (x0,y0), involute radius r0, and involute starting angle θ0 (defined by θ1 and θ2, first and second involute angles, respectively, in radians), and, limited by an inner and an outer cylinder with radii r1 and r2, respectively.\nvessel x0, y0, r, z1, z2, h1, h2 Vessel-like surface based on a infinite cylinder parallel to z-axis centred at (x0, y0), radius r, truncated between [z1,z2], and two semi-ellipsoids: [bottom] centred at (x0,y0,z1), with semi-axes r, r, h1, truncated at z1; [top] centred at (x0,y0,z2), with semi-axes r, r, h2, truncated at z2. If the last value is omitted: h = h1 = h2\n\n### Rounded corners\n\nInfinite prisms:\n\n• sqc\n• hexxc\n• hexyc\n• cross\n\nAllow defining rounded corners. The radius is then provided as the last surface parameter (s in figure below):\n\n## MCNP-equivalent surfaces\n\nNotes:\n\n• Additional surfaces included to simplify input conversion between Serpent and MCNP.\n• For description, see Chapter 5, Section 5.3 of the MCNP6.3 User Manual.\nSurface name Equivalent surface in MCNP\nbox BOX\nckx K/X\ncky K/Y\nckz K/Z\nmplane P (form defined by three points)\nrcc RCC\nx X\ny Y\nz Z\n\n## User-defined surfaces\n\nNotes:\n\n• Remember to make a backup of your subroutine before installing new updates.\n• If you have a working surface routine that might be useful for other users as well, contact the Serpent team and we'll include it in the next update as a built-in type.\nSurface name Parameters Description\nusr p1, p2, ... User-defined surface, implemented in source file \"usersurf.c\". The subroutine receives the number and list of surface parameters as input. Instructions are included as comments in the source file." ]

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[ "Cody\n\n# Problem 43. Subset Sum\n\nSolution 114142\n\nSubmitted on 17 Jul 2012\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   Pass\n%% v = [2, 3, 5]; n = 8; correct = [2, 3]; actual = subset_sum(v, n); assert(isequal(actual, correct))\n\n2   Fail\n%% v = [5, 3, 2]; n = 2; correct = 3; actual = subset_sum(v, n); assert(isequal(actual, correct))\n\nError: Assertion failed.\n\n3   Pass\n%% v = [2, 3, 5]; n = 4; correct = []; actual = subset_sum(v, n); assert(isequal(actual, correct))\n\n4   Fail\n%% v = [1, 1, 1, 1, 1]; n = 5; correct = [1, 2, 3, 4, 5]; actual = subset_sum(v, n); assert(isequal(actual, correct))\n\nError: Assertion failed.\n\n5   Fail\n%% v = [1, 2, 3, 4, 100]; n = 100; correct = 5; actual = subset_sum(v, n); assert(isequal(actual, correct))\n\nError: Assertion failed.\n\n6   Pass\n%% v = [-7, -3, -2, 8, 5]; n = 0; correct = [2, 3, 5]; actual = subset_sum(v, n); assert(isequal(actual, correct))\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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