New Tips & Techniques
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 23 Dec 2014 00:51:07 GMTTue, 23 Dec 2014 00:51:07 GMTThe latest Tips & Techniques applications added to the Application Centerhttp://www.mapleprimes.com/images/mapleapps.gifNew Tips & Techniques
http://www.maplesoft.com/applications/TipsAndTechniques
Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
http://www.maplesoft.com/applications/view.aspx?SID=153726&ref=Feed
The elementary functions include the six trigonometric and hyperbolic functions and their inverses. For all but five of these 24 functions, Maple's derivative (correct on the complex plane) agrees with the real-variable form found in the standard calculus text. For these five exceptions, this article explores two issues: (1) Does Maple's derivative, restricted to the real domain, agree with the real-variable form; and (2), to what extent do both forms agree on the complex plane.<img src="/view.aspx?si=153726/thumb.jpg" alt="Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions" align="left"/>The elementary functions include the six trigonometric and hyperbolic functions and their inverses. For all but five of these 24 functions, Maple's derivative (correct on the complex plane) agrees with the real-variable form found in the standard calculus text. For these five exceptions, this article explores two issues: (1) Does Maple's derivative, restricted to the real domain, agree with the real-variable form; and (2), to what extent do both forms agree on the complex plane.153726Wed, 10 Dec 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots
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Naive simplification of f(z) = sqrt(z - 1) sqrt(z + 1) to F(z) = sqrt(z<sup>2</sup> - 1) results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of f and F.<img src="/view.aspx?si=153697/thumb.jpg" alt="Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots" align="left"/>Naive simplification of f(z) = sqrt(z - 1) sqrt(z + 1) to F(z) = sqrt(z<sup>2</sup> - 1) results in a pair of functions that agree on only part of the complex plane. The enhanced ability of Maple 18 to find and display branch cuts of composite functions is used in this article to explore the branch cuts and regions of agreement/disagreement of f and F.153697Tue, 11 Nov 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezGroebner Bases: What are They and What are They Useful For?
http://www.maplesoft.com/applications/view.aspx?SID=153693&ref=Feed
Since they were first introduced in 1965, Groebner bases have proven to be an invaluable contribution to mathematics and computer science. All general purpose computer algebra systems like Maple have Groebner basis implementations. But what is a Groebner basis? And what applications do Groebner bases have? In this Tips and Techniques article, I’ll give some examples of the main application of Groebner bases, which is to solve systems of polynomial equations.<img src="/view.aspx?si=153693/thumb.jpg" alt="Groebner Bases: What are They and What are They Useful For?" align="left"/>Since they were first introduced in 1965, Groebner bases have proven to be an invaluable contribution to mathematics and computer science. All general purpose computer algebra systems like Maple have Groebner basis implementations. But what is a Groebner basis? And what applications do Groebner bases have? In this Tips and Techniques article, I’ll give some examples of the main application of Groebner bases, which is to solve systems of polynomial equations.153693Fri, 17 Oct 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganComputational Performance with evalhf and Compile: A Newton Fractal Case Study
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<p>This Tips and Techniques article focuses on the relative performance of Maple's various modes for floating-point computations. The example used here is the computation of a particular Newton fractal, which is easily parallelizable. We compute an image representation for this fractal under several computational modes, using both serial and multithreaded computation schemes.</p>
<p>This article is a follow up to a previous Tips and Techniques, <a href="http://www.maplesoft.com/applications/view.aspx?SID=153645">evalhf, Compile, hfloat and all that</a>, which discusses functionality differences amongst Maple's the different floating-point computation modes available in Maple.</p><img src="/view.aspx?si=153683/thumb.jpg" alt="Computational Performance with evalhf and Compile: A Newton Fractal Case Study" align="left"/><p>This Tips and Techniques article focuses on the relative performance of Maple's various modes for floating-point computations. The example used here is the computation of a particular Newton fractal, which is easily parallelizable. We compute an image representation for this fractal under several computational modes, using both serial and multithreaded computation schemes.</p>
<p>This article is a follow up to a previous Tips and Techniques, <a href="http://www.maplesoft.com/applications/view.aspx?SID=153645">evalhf, Compile, hfloat and all that</a>, which discusses functionality differences amongst Maple's the different floating-point computation modes available in Maple.</p>153683Fri, 26 Sep 2014 04:00:00 ZDave LinderDave LinderGenerating random numbers efficiently
http://www.maplesoft.com/applications/view.aspx?SID=153662&ref=Feed
Generating (pseudo-)random values is a frequent task in simulations and other programs. For some situations, you want to generate some combinatorial or algebraic values, such as a list or a polynomial; in other situations, you need random numbers, from a distribution that is uniform or more complicated. In this article I'll talk about all of these situations.<img src="/view.aspx?si=153662/thumb.jpg" alt="Generating random numbers efficiently" align="left"/>Generating (pseudo-)random values is a frequent task in simulations and other programs. For some situations, you want to generate some combinatorial or algebraic values, such as a list or a polynomial; in other situations, you need random numbers, from a distribution that is uniform or more complicated. In this article I'll talk about all of these situations.153662Mon, 18 Aug 2014 04:00:00 ZDr. Erik PostmaDr. Erik Postmaevalhf, Compile, hfloat and all that
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Users sometimes ask how to make their floating-point (numeric) computations perform faster in Maple. The answers often include references to special terms such as evalhf, the Compiler, and option hfloat. A difficulty for the non-expert lies in knowing which of these can be used, and when. This Tips and Techniques attempts to clear up some of the mystery of these terms, by discussion and functionality comparison.<img src="/applications/images/app_image_blank_lg.jpg" alt="evalhf, Compile, hfloat and all that" align="left"/>Users sometimes ask how to make their floating-point (numeric) computations perform faster in Maple. The answers often include references to special terms such as evalhf, the Compiler, and option hfloat. A difficulty for the non-expert lies in knowing which of these can be used, and when. This Tips and Techniques attempts to clear up some of the mystery of these terms, by discussion and functionality comparison.153645Tue, 22 Jul 2014 04:00:00 ZDave LinderDave LinderCustom Plot Sizing and Shading
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<p>If the number of Online Help queries per topic or the number of click-throughs on errors for a particular area of functionality is anything to go by, then plotting is hands down the most significant functionality in Maple. I saw some data on those recently, and what leapt out was just how much plotting dominated.</p>
<p>When functionality is introduced that affects most kinds of 2D or 3D plots, then it likely affects a great many Maple users in important ways. While there are help pages on the new 2D plot sizing and 3D plot shading options in Maple 18, I find myself using these new options so often I feel that its important to mention them as tips for visualization techniques.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Custom Plot Sizing and Shading" align="left"/><p>If the number of Online Help queries per topic or the number of click-throughs on errors for a particular area of functionality is anything to go by, then plotting is hands down the most significant functionality in Maple. I saw some data on those recently, and what leapt out was just how much plotting dominated.</p>
<p>When functionality is introduced that affects most kinds of 2D or 3D plots, then it likely affects a great many Maple users in important ways. While there are help pages on the new 2D plot sizing and 3D plot shading options in Maple 18, I find myself using these new options so often I feel that its important to mention them as tips for visualization techniques.</p>153606Mon, 16 Jun 2014 04:00:00 ZDave LinderDave LinderCustom Plot Sizing and Shading
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Many Maple users, no matter what they are working on, make use of Maple’s plotting abilities, and so this Tips and Techniques highlights some small but useful new plotting features introduced in Maple 18. Maple 18 give you the ability to set the size of your plots, giving you more control over your document’s use of space, as well as the ability to set the colour gradients used in 3-D plots. In this Tips and Techniques, you will find a variety of example that show you how to take advantage of these new plot options.<img src="/view.aspx?si=153607/thumb.jpg" alt="Custom Plot Sizing and Shading" align="left"/>Many Maple users, no matter what they are working on, make use of Maple’s plotting abilities, and so this Tips and Techniques highlights some small but useful new plotting features introduced in Maple 18. Maple 18 give you the ability to set the size of your plots, giving you more control over your document’s use of space, as well as the ability to set the colour gradients used in 3-D plots. In this Tips and Techniques, you will find a variety of example that show you how to take advantage of these new plot options.153607Mon, 16 Jun 2014 04:00:00 ZDave LinderDave LinderClassroom Tips and Techniques: The Explore Command in Maple 18
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The Explore functionality, which provides an interactive experience with parameter-dependent plots and expressions, has been significantly enhanced in Maple 18. In this Tips and Techniques article, I will focus on some key usage points of using the Explore command with plots, including explorations based on simple Maple plots as well as user-defined plotting procedures.<img src="/view.aspx?si=153552/thumb.jpg" alt="Classroom Tips and Techniques: The Explore Command in Maple 18" align="left"/>The Explore functionality, which provides an interactive experience with parameter-dependent plots and expressions, has been significantly enhanced in Maple 18. In this Tips and Techniques article, I will focus on some key usage points of using the Explore command with plots, including explorations based on simple Maple plots as well as user-defined plotting procedures.153552Wed, 16 Apr 2014 04:00:00 ZDave LinderDave LinderInternet Page Ranking Algorithms
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In this guest article in the Tips and Techniques series, Dr. Michael Monagan explains how internet pages are ranked.<img src="/view.aspx?si=153532/thumb.jpg" alt="Internet Page Ranking Algorithms" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan explains how internet pages are ranked.153532Thu, 20 Mar 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganThe Mortgage Payment Problem: Approximating a Discrete Process with a Differential Equation
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In this guest article in the Tips and Techniques series, Dr. Michael Monagan uses mortgage interest to test how well a differential equation models what is essentially a discrete process.<img src="/view.aspx?si=153511/thumb.jpg" alt="The Mortgage Payment Problem: Approximating a Discrete Process with a Differential Equation" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan uses mortgage interest to test how well a differential equation models what is essentially a discrete process.153511Thu, 20 Feb 2014 05:00:00 ZProf. Michael MonaganProf. Michael MonaganThe House Warming Model
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In this guest article in the Tips and Techniques series, Dr. Michael Monagan discusses a model of heat-flow in a house, and shows how he uses this model in his class.<img src="/view.aspx?si=153491/thumb.jpg" alt="The House Warming Model" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan discusses a model of heat-flow in a house, and shows how he uses this model in his class.153491Wed, 22 Jan 2014 05:00:00 ZProf. Michael MonaganProf. Michael MonaganMeasuring Water Flow of Rivers
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In this guest article in the Tips & Techniques series, Dr. Michael Monagan discusses the art and science of measuring the amount of water flowing in a river, and relates his personal experiences with this task to its morph into a project for his calculus classes.<img src="/view.aspx?si=153480/thumb.jpg" alt="Measuring Water Flow of Rivers" align="left"/>In this guest article in the Tips & Techniques series, Dr. Michael Monagan discusses the art and science of measuring the amount of water flowing in a river, and relates his personal experiences with this task to its morph into a project for his calculus classes.153480Fri, 13 Dec 2013 05:00:00 ZProf. Michael MonaganProf. Michael MonaganClassroom Tips and Techniques: Locus of Eigenvalues
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If P(s) is a parameter-dependent square matrix, what is the locus of its eigenvalues as s varies from, say, 0 to 1? For a non-square P, the eigenvalues can become complex, so the loci could exist as curves in the real or complex planes. To avoid these difficulties, consider only real symmetric matrices for which the loci of eigenvalues are real curves, but curves that could intersect. What does it mean to trace an individual eigenvalue of P(0) to P(1) if the eigenvalue has algebraic multiplicity more than 1?<img src="/view.aspx?si=153463/thumb.jpg" alt="Classroom Tips and Techniques: Locus of Eigenvalues" align="left"/>If P(s) is a parameter-dependent square matrix, what is the locus of its eigenvalues as s varies from, say, 0 to 1? For a non-square P, the eigenvalues can become complex, so the loci could exist as curves in the real or complex planes. To avoid these difficulties, consider only real symmetric matrices for which the loci of eigenvalues are real curves, but curves that could intersect. What does it mean to trace an individual eigenvalue of P(0) to P(1) if the eigenvalue has algebraic multiplicity more than 1?153463Fri, 15 Nov 2013 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Mathematical Thoughts on the Root Locus
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Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.<img src="/view.aspx?si=153452/thumb.jpg" alt="Classroom Tips and Techniques: Mathematical Thoughts on the Root Locus" align="left"/>Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.153452Tue, 29 Oct 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Slider-Control of Parameters in an ODE
http://www.maplesoft.com/applications/view.aspx?SID=152112&ref=Feed
Several ways to provide slider-control of parameters in a differential equation are considered. In particular, the cases of one and two parameters are illustrated, and for the case of two parameters, a 2-dimensional slider is constructed.<img src="/view.aspx?si=152112/thumb.jpg" alt="Classroom Tips and Techniques: Slider-Control of Parameters in an ODE" align="left"/>Several ways to provide slider-control of parameters in a differential equation are considered. In particular, the cases of one and two parameters are illustrated, and for the case of two parameters, a 2-dimensional slider is constructed.152112Mon, 23 Sep 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface
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Four different techniques are given for obtaining a graph showing a surface with a normal and tangent plane attached. The work is a response to <a href="http://www.mapleprimes.com/questions/147681-A-Problem-About-Plot-The-Part-Of-The-Surface">a MaplePrimes question asked on May 25, 2013</a>.<img src="/view.aspx?si=150722/thumb.jpg" alt="Classroom Tips and Techniques: Drawing a Normal and Tangent Plane on a Surface" align="left"/>Four different techniques are given for obtaining a graph showing a surface with a normal and tangent plane attached. The work is a response to <a href="http://www.mapleprimes.com/questions/147681-A-Problem-About-Plot-The-Part-Of-The-Surface">a MaplePrimes question asked on May 25, 2013</a>.150722Tue, 20 Aug 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method
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The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.<img src="/view.aspx?si=149514/thumb.jpg" alt="Classroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method" align="left"/>The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.149514Tue, 16 Jul 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: The Sliding Ladder
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A January 10, 2013 post to <a href="http://www.mapleprimes.com/questions/142194-Sliding-Ladder-Animation" class="plainlink">MaplePrimes</a> asked for an animation of the trajectory traced by the center of a "sliding ladder." This month's article generalizes the solutions suggested by Adri van der Meer and Doug Meade, and shows the trajectory of an arbitrary point on the ladder as its top slides down a vertical wall and its bottom moves away from that wall along an orthogonal "floor." The location of the arbitrary point on the ladder is controlled by a slider, the animation being generated with the updated Explore command.<img src="/view.aspx?si=148714/thumb.jpg" alt="Classroom Tips and Techniques: The Sliding Ladder" align="left"/>A January 10, 2013 post to <a href="http://www.mapleprimes.com/questions/142194-Sliding-Ladder-Animation" class="plainlink">MaplePrimes</a> asked for an animation of the trajectory traced by the center of a "sliding ladder." This month's article generalizes the solutions suggested by Adri van der Meer and Doug Meade, and shows the trajectory of an arbitrary point on the ladder as its top slides down a vertical wall and its bottom moves away from that wall along an orthogonal "floor." The location of the arbitrary point on the ladder is controlled by a slider, the animation being generated with the updated Explore command.148714Fri, 21 Jun 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems 31-35 from the Red Book of Maple Magic
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Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initial-value problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.<img src="/view.aspx?si=147092/thumb.jpg" alt="Classroom Tips and Techniques: Gems 31-35 from the Red Book of Maple Magic" align="left"/>Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initial-value problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.147092Fri, 10 May 2013 04:00:00 ZDr. Robert LopezDr. Robert Lopez