Dr. Frank Wang: New Applications
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en-us2021 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemMon, 12 Apr 2021 21:24:44 GMTMon, 12 Apr 2021 21:24:44 GMTNew applications published by Dr. Frank Wanghttps://www.maplesoft.com/images/Application_center_hp.jpgDr. Frank Wang: New Applications
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In and Out of a Schwarzschild Black Hole
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In classical general relativity, an astronaut can cross the event horizon of a black hole without noticing it, while an external observer would witness the astronaut approaching but never reaching the event horizon as time tends to infinity. We integrate the differential equations of motion to show the difference between proper and coordinate times for a particle free falling into a Schwarzschild black hole, and present the particle world line in Kruskal and Penrose diagrams.<img src="https://www.maplesoft.com/view.aspx?si=154673/penrosediagram.png" alt="In and Out of a Schwarzschild Black Hole" style="max-width: 25%;" align="left"/>In classical general relativity, an astronaut can cross the event horizon of a black hole without noticing it, while an external observer would witness the astronaut approaching but never reaching the event horizon as time tends to infinity. We integrate the differential equations of motion to show the difference between proper and coordinate times for a particle free falling into a Schwarzschild black hole, and present the particle world line in Kruskal and Penrose diagrams.https://www.maplesoft.com/applications/view.aspx?SID=154673&ref=FeedMon, 08 Feb 2021 05:00:00 ZDr. Frank WangDr. Frank WangIntegration by Pullback
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The calculus of differential forms is a powerful tool that unifies and simplifies many mathematical concepts. We use the Pullback command in Maple's DifferentialGeometry package to perform double, triple, line and surface integrals in James Stewart's Calculus to demonstrate the method differential forms and show the connection among these integrals.<img src="https://www.maplesoft.com/view.aspx?si=154643/flux.png" alt="Integration by Pullback" style="max-width: 25%;" align="left"/>The calculus of differential forms is a powerful tool that unifies and simplifies many mathematical concepts. We use the Pullback command in Maple's DifferentialGeometry package to perform double, triple, line and surface integrals in James Stewart's Calculus to demonstrate the method differential forms and show the connection among these integrals.https://www.maplesoft.com/applications/view.aspx?SID=154643&ref=FeedSun, 24 May 2020 04:00:00 ZDr. Frank WangDr. Frank WangApplications of Differential Forms in Thermodynamics
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We use Maple's DifferentialGeometry package to derive Maxwell and other mathematical identities in thermodynamics.<img src="https://www.maplesoft.com/view.aspx?si=154631/gradS.png" alt="Applications of Differential Forms in Thermodynamics" style="max-width: 25%;" align="left"/>We use Maple's DifferentialGeometry package to derive Maxwell and other mathematical identities in thermodynamics.https://www.maplesoft.com/applications/view.aspx?SID=154631&ref=FeedSat, 11 Apr 2020 04:00:00 ZDr. Frank WangDr. Frank WangGödel's Universe
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In 1949, Kurt Gödel proposed a solution to Einstein's field equations that exhibited a rotation of matter (with a nonvanishing cosmological constant). This solution allows the existence of closed timelike curves, which implies the possibility of time travel. In this worksheet, we demonstrate the use of DifferentialGeometry and Tensor packages to calculate the Ricci tensor, Ricci scalar, and Einstein tensor. We also apply the calculus of variations to calculate the null geodesics (trajectories of light rays) in Gödel's universe. The notation is based on the presentation in the monograph of Stephen Hawking and George Ellis.<img src="https://www.maplesoft.com/view.aspx?si=154430/godelnullgeodesics.png" alt="Gödel's Universe" style="max-width: 25%;" align="left"/>In 1949, Kurt Gödel proposed a solution to Einstein's field equations that exhibited a rotation of matter (with a nonvanishing cosmological constant). This solution allows the existence of closed timelike curves, which implies the possibility of time travel. In this worksheet, we demonstrate the use of DifferentialGeometry and Tensor packages to calculate the Ricci tensor, Ricci scalar, and Einstein tensor. We also apply the calculus of variations to calculate the null geodesics (trajectories of light rays) in Gödel's universe. The notation is based on the presentation in the monograph of Stephen Hawking and George Ellis.https://www.maplesoft.com/applications/view.aspx?SID=154430&ref=FeedMon, 02 Apr 2018 04:00:00 ZDr. Frank WangDr. Frank WangPropagation of Plane Gravitational Waves
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Under the condition of weak fields, Einstein's field equation of general relativity can be linearized. The metric perturbation of the flat Minkowski spacetime satisfies the wave equation, and its solution is similar to the solution for electromagnetic waves. This worksheet demonstrates the similarity and difference between electromagnetic waves, which are vector fields, and gravitational waves, which are tensor fields.<img src="https://www.maplesoft.com/view.aspx?si=154275/mode1.gif" alt="Propagation of Plane Gravitational Waves" style="max-width: 25%;" align="left"/>Under the condition of weak fields, Einstein's field equation of general relativity can be linearized. The metric perturbation of the flat Minkowski spacetime satisfies the wave equation, and its solution is similar to the solution for electromagnetic waves. This worksheet demonstrates the similarity and difference between electromagnetic waves, which are vector fields, and gravitational waves, which are tensor fields.https://www.maplesoft.com/applications/view.aspx?SID=154275&ref=FeedSun, 09 Jul 2017 04:00:00 ZDr. Frank WangDr. Frank WangEl Niño Temperature Anomalies Modeled by a Delay Differential Equation
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Delay differential equations are differential equations in which the derivative of the unknown function at a certain time depends on past values of the function and/or its derivatives. Max J. Suarez and Paul S. Schopf used such an equation to model the El Niño phenomenon. This worksheet demonstrate how Maple's dsolve command can be used to solve a delay differential equation numerically.<img src="https://www.maplesoft.com/view.aspx?si=154142/waves.png" alt="El Niño Temperature Anomalies Modeled by a Delay Differential Equation" style="max-width: 25%;" align="left"/>Delay differential equations are differential equations in which the derivative of the unknown function at a certain time depends on past values of the function and/or its derivatives. Max J. Suarez and Paul S. Schopf used such an equation to model the El Niño phenomenon. This worksheet demonstrate how Maple's dsolve command can be used to solve a delay differential equation numerically.https://www.maplesoft.com/applications/view.aspx?SID=154142&ref=FeedMon, 29 Aug 2016 04:00:00 ZDr. Frank WangDr. Frank WangMatrix Representation of Quantum Entangled States: Understanding Bell's Inequality and Teleportation
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In 1935, Einstein, Podolsky and Rosen published a paper revealing a counter-intuitive situation in quantum mechanics which was later known as the EPR paradox. The phenomenon involved an entangled state., which Schrodinger called "not one but the characteristic trait of quantum mechanics." In textbooks, entanglement is often presented in abstract notations. In popular accounts of quantum mechanics, entanglement is sometimes portrayed as a mystery or even distorted in a nearly pseudoscientific fashion. In this worksheet, we use Maple's LinearAlgebra package to represent quantum states and measurements in matrix form. The famous Bell's inequality and teleportation can be understood using elementary matrix operations.<img src="https://www.maplesoft.com/view.aspx?si=154100/8df7b465a1583cedab7e3e6452644591.gif" alt="Matrix Representation of Quantum Entangled States: Understanding Bell's Inequality and Teleportation" style="max-width: 25%;" align="left"/>In 1935, Einstein, Podolsky and Rosen published a paper revealing a counter-intuitive situation in quantum mechanics which was later known as the EPR paradox. The phenomenon involved an entangled state., which Schrodinger called "not one but the characteristic trait of quantum mechanics." In textbooks, entanglement is often presented in abstract notations. In popular accounts of quantum mechanics, entanglement is sometimes portrayed as a mystery or even distorted in a nearly pseudoscientific fashion. In this worksheet, we use Maple's LinearAlgebra package to represent quantum states and measurements in matrix form. The famous Bell's inequality and teleportation can be understood using elementary matrix operations.https://www.maplesoft.com/applications/view.aspx?SID=154100&ref=FeedMon, 09 May 2016 04:00:00 ZDr. Frank WangDr. Frank WangCalculating Gaussian Curvature Using Differential Forms
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<p>Riemannian geometry is customarily developed by tensor methods, which is not necessarily the most computationally efficient approach. Using the language of differential forms, Elie Cartan's formulation of the Riemannian geometry can be elegantly summarized in two structural equations. Essentially, the local curvature of the manifold is a measure of how the connection varies from point to point. This Maple worksheet uses the <strong>DifferentialGeometry</strong> package to solves three problems in Harley Flanders' book on differential forms to demonstrate the implementation of Cartan's method. </p><img src="https://www.maplesoft.com/view.aspx?si=153720/c119c404932805fdc4af274016b48a13.gif" alt="Calculating Gaussian Curvature Using Differential Forms" style="max-width: 25%;" align="left"/><p>Riemannian geometry is customarily developed by tensor methods, which is not necessarily the most computationally efficient approach. Using the language of differential forms, Elie Cartan's formulation of the Riemannian geometry can be elegantly summarized in two structural equations. Essentially, the local curvature of the manifold is a measure of how the connection varies from point to point. This Maple worksheet uses the <strong>DifferentialGeometry</strong> package to solves three problems in Harley Flanders' book on differential forms to demonstrate the implementation of Cartan's method. </p>https://www.maplesoft.com/applications/view.aspx?SID=153720&ref=FeedTue, 09 Dec 2014 05:00:00 ZDr. Frank WangDr. Frank WangAlexander Friedmann's Cosmic Scenarios
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<p>The Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p><img src="https://www.maplesoft.com/view.aspx?si=142459/friedmannscenario.jpg" alt="Alexander Friedmann's Cosmic Scenarios" style="max-width: 25%;" align="left"/><p>The Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p>https://www.maplesoft.com/applications/view.aspx?SID=142459&ref=FeedSun, 20 Jan 2013 05:00:00 ZDr. Frank WangDr. Frank WangNumerical Solution of a Mechanics Braintwister Problem
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<p>In 1995, Boris Korsunsky published a collection of what he called "braintwisters" physics problems. In 2011, Norman Paris and Michael L. Broide presented a comprehensive analysis of one of the mechanics problems involving the coupled motion of two blocks. This worksheet demonstrates how to use Maple to derive the equations of motion using the calculus of variations, and to solve the differential equations numerically. </p><img src="https://www.maplesoft.com/view.aspx?si=131117/131117_thumb.jpg" alt="Numerical Solution of a Mechanics Braintwister Problem" style="max-width: 25%;" align="left"/><p>In 1995, Boris Korsunsky published a collection of what he called "braintwisters" physics problems. In 2011, Norman Paris and Michael L. Broide presented a comprehensive analysis of one of the mechanics problems involving the coupled motion of two blocks. This worksheet demonstrates how to use Maple to derive the equations of motion using the calculus of variations, and to solve the differential equations numerically. </p>https://www.maplesoft.com/applications/view.aspx?SID=131117&ref=FeedThu, 23 Feb 2012 05:00:00 ZDr. Frank WangDr. Frank WangThe Hawk-Dove-Retaliator Game
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<p>In 1973, John Maynard Smith and George R. Price published a paper entitled "The Logic of Animal Conflict" in Nature, in which they formalized the concept of evolutionarily stable strategies (ESS) and launched the field of Evolutionary Game Theory. Subsequently, Maynard Smith published a book Evolution and the Theory of Games to present his ideas in a coherent form. This worksheet demonstrates how to use Maple to visualize the Hawk-Dove-Retaliator game---one of the most important examples of game theory. This worksheet can be modified for other two-player three-strategy games. <br /></p><img src="https://www.maplesoft.com/view.aspx?si=98755/maple_icon.jpg" alt="The Hawk-Dove-Retaliator Game" style="max-width: 25%;" align="left"/><p>In 1973, John Maynard Smith and George R. Price published a paper entitled "The Logic of Animal Conflict" in Nature, in which they formalized the concept of evolutionarily stable strategies (ESS) and launched the field of Evolutionary Game Theory. Subsequently, Maynard Smith published a book Evolution and the Theory of Games to present his ideas in a coherent form. This worksheet demonstrates how to use Maple to visualize the Hawk-Dove-Retaliator game---one of the most important examples of game theory. This worksheet can be modified for other two-player three-strategy games. <br /></p>https://www.maplesoft.com/applications/view.aspx?SID=98755&ref=FeedMon, 08 Nov 2010 05:00:00 ZDr. Frank WangDr. Frank WangRichard Dawkins' Battle of the Sexes Model
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<p>"Battle of the sexes" is the title of Chapter 9 in Richard Dawkins' Selfish Gene, in which he advocated the gene-centered view of evolution. The central thesis is that an organism is expected to evolve to maximize the number of copies of its genes passed on globally, and as a result, populations will tend towards an evolutionarily stable strategy (ESS). The concept of ESS was introduced by John Maynard Smith, in collaboration with George R. Price. They used the branch of mathematics called "game theory" to understanding the logic of animal conflict. Dawkins took their method of analyzing aggresive contests among animals of the same species and applied it to sex; he found a stable proportion of males and females playing some hypothetical strategies. In the second edition of Selfish Gene, Dawkins acknowledged a misstatement, based on Peter Schuster and Karl Sigmund's differential equations corresponding to Dawkins' game. The ratio Dawkins found was not asymptotically stable, but oscillatory. This worksheet utilizes Maple's <strong>LinearAlgebra</strong>, <strong>VectorCalculus</strong>, <strong>DEtools</strong> packages, in addition to other basic commands, to investigate the type and stability of the fixed point of the equations by Schuster and Sigmund. This example serves to supplement the topic "nonlinear differential equations and stabilities" covered in standard undergraduate textbooks such as that by W. E. Boyce and R. C. DiPrima. </p><img src="https://www.maplesoft.com/view.aspx?si=95974/95974.png" alt="Richard Dawkins' Battle of the Sexes Model" style="max-width: 25%;" align="left"/><p>"Battle of the sexes" is the title of Chapter 9 in Richard Dawkins' Selfish Gene, in which he advocated the gene-centered view of evolution. The central thesis is that an organism is expected to evolve to maximize the number of copies of its genes passed on globally, and as a result, populations will tend towards an evolutionarily stable strategy (ESS). The concept of ESS was introduced by John Maynard Smith, in collaboration with George R. Price. They used the branch of mathematics called "game theory" to understanding the logic of animal conflict. Dawkins took their method of analyzing aggresive contests among animals of the same species and applied it to sex; he found a stable proportion of males and females playing some hypothetical strategies. In the second edition of Selfish Gene, Dawkins acknowledged a misstatement, based on Peter Schuster and Karl Sigmund's differential equations corresponding to Dawkins' game. The ratio Dawkins found was not asymptotically stable, but oscillatory. This worksheet utilizes Maple's <strong>LinearAlgebra</strong>, <strong>VectorCalculus</strong>, <strong>DEtools</strong> packages, in addition to other basic commands, to investigate the type and stability of the fixed point of the equations by Schuster and Sigmund. This example serves to supplement the topic "nonlinear differential equations and stabilities" covered in standard undergraduate textbooks such as that by W. E. Boyce and R. C. DiPrima. </p>https://www.maplesoft.com/applications/view.aspx?SID=95974&ref=FeedThu, 12 Aug 2010 04:00:00 ZDr. Frank WangDr. Frank WangVisualizing the Laplace-Runge-Lenz Vector
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The vector treatment of Kepler's first law presented in most calculus textbooks is based on the existence of a constant vector that is associated with the exact inverse square force law. Such a treatment is not a general substitute for the methods based on the differential equations of motion. This worksheet demonstrates how to use Maple to solve the differential equations governing planetary motion, and how to visualize the Laplace-Runge-Lenz vector which is peculiar to the force law of the form k/r^2.<img src="https://www.maplesoft.com/view.aspx?si=19187/thumb.gif" alt="Visualizing the Laplace-Runge-Lenz Vector" style="max-width: 25%;" align="left"/>The vector treatment of Kepler's first law presented in most calculus textbooks is based on the existence of a constant vector that is associated with the exact inverse square force law. Such a treatment is not a general substitute for the methods based on the differential equations of motion. This worksheet demonstrates how to use Maple to solve the differential equations governing planetary motion, and how to visualize the Laplace-Runge-Lenz vector which is peculiar to the force law of the form k/r^2.https://www.maplesoft.com/applications/view.aspx?SID=19187&ref=FeedMon, 02 Mar 2009 00:00:00 ZDr. Frank WangDr. Frank WangApplication of the Lambert W Function to the SIR Epidemic Model
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The basic framework of mathematical epidemiology is the SIR model. The letters in the acronym represent the three primary states that any member of a population can occupy with respect to a disease: susceptible, infectious, and removed. This worksheet demonstrates the use of the Lambert W function (LambertW), which was included in Maple in its early years, for the phase transition in the SIR model.<img src="https://www.maplesoft.com/view.aspx?si=7088/1.jpg" alt="Application of the Lambert W Function to the SIR Epidemic Model" style="max-width: 25%;" align="left"/>The basic framework of mathematical epidemiology is the SIR model. The letters in the acronym represent the three primary states that any member of a population can occupy with respect to a disease: susceptible, infectious, and removed. This worksheet demonstrates the use of the Lambert W function (LambertW), which was included in Maple in its early years, for the phase transition in the SIR model.https://www.maplesoft.com/applications/view.aspx?SID=7088&ref=FeedTue, 06 Jan 2009 00:00:00 ZDr. Frank WangDr. Frank WangRosalind Franklin's X-ray diffraction Photograph of DNA
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In the April 25, 1953 issue of Nature, James D. Watson and Francis H. C. Crick announced one of the most significant scientific discoveries ever: the double helix model for the structure of DNA. They could not have proposed their celebrated structure for DNA without access to experimental results obtained by Rosalind Franklin, particularly this crucial X-ray diffraction photograph taken by her. This worksheet use Maple to show why Franklin's photograph suggests a helix structure.<img src="https://www.maplesoft.com/view.aspx?si=4902/appviewer.aspx.jpg" alt="Rosalind Franklin's X-ray diffraction Photograph of DNA" style="max-width: 25%;" align="left"/>In the April 25, 1953 issue of Nature, James D. Watson and Francis H. C. Crick announced one of the most significant scientific discoveries ever: the double helix model for the structure of DNA. They could not have proposed their celebrated structure for DNA without access to experimental results obtained by Rosalind Franklin, particularly this crucial X-ray diffraction photograph taken by her. This worksheet use Maple to show why Franklin's photograph suggests a helix structure.https://www.maplesoft.com/applications/view.aspx?SID=4902&ref=FeedTue, 01 May 2007 00:00:00 ZDr. Frank WangDr. Frank WangDemonstrating Soliton Interactions using 'pdsolve'
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The term "soliton" was introduced in a paper by Zabusky and Kruskal published in Physical Review Letters.[1] By solving the Korteweg-de Vries equation (KdV equation) numerically, solitary-wave pulses propagating in nonlinear dispersive media are observed. This worksheet demonstrates soliton interactions using pdsolve.<img src="https://www.maplesoft.com/view.aspx?si=1733/solitonimage.jpg" alt="Demonstrating Soliton Interactions using 'pdsolve'" style="max-width: 25%;" align="left"/>The term "soliton" was introduced in a paper by Zabusky and Kruskal published in Physical Review Letters.[1] By solving the Korteweg-de Vries equation (KdV equation) numerically, solitary-wave pulses propagating in nonlinear dispersive media are observed. This worksheet demonstrates soliton interactions using pdsolve.https://www.maplesoft.com/applications/view.aspx?SID=1733&ref=FeedMon, 01 May 2006 00:00:00 ZDr. Frank WangDr. Frank WangStandard Map on a Torus
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Maple worksheet for producing a standard map on a torus is developed. A brief introduction of Hamiltonian mechanics using a simple pendulum system as an example is provided, followed by a discussion of a kicked rotor system which consists of a simple pendulum with the potential energy turned on in delta-function pulses. The integration of the kicked rotor problem yields the standard map, which has characteristics of a large class of systems. Plotting the standard map on a torus facilitates a three-dimensional visualization the Kolmogorov-Arnold-Moser (KAM) theory of chaos in a Hamiltonian system. An animation demonstrating chaos created by homoclinic tangle near a hyperbolic point is contained in this worksheet.<img src="https://www.maplesoft.com/view.aspx?si=1703/SMTorus.jpg" alt="Standard Map on a Torus" style="max-width: 25%;" align="left"/>Maple worksheet for producing a standard map on a torus is developed. A brief introduction of Hamiltonian mechanics using a simple pendulum system as an example is provided, followed by a discussion of a kicked rotor system which consists of a simple pendulum with the potential energy turned on in delta-function pulses. The integration of the kicked rotor problem yields the standard map, which has characteristics of a large class of systems. Plotting the standard map on a torus facilitates a three-dimensional visualization the Kolmogorov-Arnold-Moser (KAM) theory of chaos in a Hamiltonian system. An animation demonstrating chaos created by homoclinic tangle near a hyperbolic point is contained in this worksheet.https://www.maplesoft.com/applications/view.aspx?SID=1703&ref=FeedTue, 10 Jan 2006 00:00:00 ZDr. Frank WangDr. Frank Wang