Real Analysis: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 02 Apr 2015 03:12:46 GMTThu, 02 Apr 2015 03:12:46 GMTNew applications in the Real Analysis categoryhttp://www.mapleprimes.com/images/mapleapps.gifReal Analysis: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=160
Car Loan Calculator
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<p>This loan calculator facilitates the life of a borrower. By entering the <strong>purchase price</strong>, the <strong>down payment</strong>, the <strong>number of years</strong> it takes to repay the loan, the <strong>payment frequency</strong>, the <strong>annual interest rate</strong>, and clicking on the "<strong>calculate</strong>" buttom, the calculator will give you the <strong>amount of payment</strong> for each payment period.</p>
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<p>Want to make a loan? Try it out and see how things change with respect to each element.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Car Loan Calculator" align="left"/><p>This loan calculator facilitates the life of a borrower. By entering the <strong>purchase price</strong>, the <strong>down payment</strong>, the <strong>number of years</strong> it takes to repay the loan, the <strong>payment frequency</strong>, the <strong>annual interest rate</strong>, and clicking on the "<strong>calculate</strong>" buttom, the calculator will give you the <strong>amount of payment</strong> for each payment period.</p>
<p> </p>
<p>Want to make a loan? Try it out and see how things change with respect to each element.</p>145174Wed, 27 Mar 2013 04:00:00 ZZinan WangZinan WangClassroom Tips and Techniques: Best Taylor-Polynomial Approximations
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In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.<img src="/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" align="left"/>In the early 90s, Joe Ecker (Rensselaer Polytechnic Institute) provided a Maple solution to the problem of determining for a given function, which expansion point in a specified interval yielded the best quadratic Taylor polynomial approximation, where "best" was measured by the L<sub>2</sub>-norm. This article applies Ecker's approach to the function <em>f(x)</em> = sinh<em>(x)</em> – <em>x e<sub>-3x</sub>,</em> -1 ≤ <em>x</em> ≤ 3, then goes on to find other approximating quadratic polynomials.136471Tue, 14 Aug 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezPhénomène de Runge - subdivision de Chebychev
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<p>On observe d'abord la divergence du polynôme de Lagrange interpolant la fonction densité de probabilité de la loi de Cauchy lorsque la <strong>subdivision est équirépartie</strong> sur [-1;1]. C'est le <u>phénomène de Runge</u>.<br />
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On observe ensuite qu'en choisissant une <strong>subdivision de Chebychev</strong> le phénomène de divergence au voisinage des bornes disparait.<br />
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Cette activité a été réalisé dans le cadre de la préparation à l'agrégation interne de mathématiques de Rennes le 10 Mars 2010.<br />
Les nouveaux programmes du concours incitent à proposer des exercices utilisant les TICE. Il semble difficile de proposer une preuve convaincante du phénomène de Runge pour une épreuve orale. Ceci justifie de ne s'en tenir qu'à la seule observation.</p><img src="/view.aspx?si=35301/thumb.jpg" alt="Phénomène de Runge - subdivision de Chebychev" align="left"/><p>On observe d'abord la divergence du polynôme de Lagrange interpolant la fonction densité de probabilité de la loi de Cauchy lorsque la <strong>subdivision est équirépartie</strong> sur [-1;1]. C'est le <u>phénomène de Runge</u>.<br />
<br />
On observe ensuite qu'en choisissant une <strong>subdivision de Chebychev</strong> le phénomène de divergence au voisinage des bornes disparait.<br />
<br />
Cette activité a été réalisé dans le cadre de la préparation à l'agrégation interne de mathématiques de Rennes le 10 Mars 2010.<br />
Les nouveaux programmes du concours incitent à proposer des exercices utilisant les TICE. Il semble difficile de proposer une preuve convaincante du phénomène de Runge pour une épreuve orale. Ceci justifie de ne s'en tenir qu'à la seule observation.</p>35301Fri, 26 Mar 2010 04:00:00 ZKERNIVINEN SebastienKERNIVINEN SebastienFractal Dimension and Space-Filling Curves (with iterated function systems)
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By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.<img src="/view.aspx?si=4869/thumb.png" alt="Fractal Dimension and Space-Filling Curves (with iterated function systems)" align="left"/>By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.4869Fri, 16 Feb 2007 00:00:00 ZProf. Mark MeyersonProf. Mark MeyersonFinite Trigonometric Representation of Legendre Polynomials by Cosine Polynomials
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The main goal of present worksheet is to obtain first n terms of finite trigonometric Legendre polynomials by cosine polynomials with n- fold arguments, whereas Maple provides (orthopoly[P]) - Legendre and Jacobi polynomials in the power form.<img src="/view.aspx?si=1738/legendre.JPG" alt="Finite Trigonometric Representation of Legendre Polynomials by Cosine Polynomials" align="left"/>The main goal of present worksheet is to obtain first n terms of finite trigonometric Legendre polynomials by cosine polynomials with n- fold arguments, whereas Maple provides (orthopoly[P]) - Legendre and Jacobi polynomials in the power form.1738Mon, 29 May 2006 00:00:00 ZKonstantine NinidzeKonstantine NinidzeA Counter-Example on the Density of Integers
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Illustrate the solution of a problem of finding a counter-example involving density of sets of integers; emphasizing analytic, numerical and graphical methods of illustrating the solution.<img src="/view.aspx?si=1708/density.jpg" alt="A Counter-Example on the Density of Integers" align="left"/>Illustrate the solution of a problem of finding a counter-example involving density of sets of integers; emphasizing analytic, numerical and graphical methods of illustrating the solution.1708Mon, 23 Jan 2006 00:00:00 ZProf. Steven DunbarProf. Steven DunbarConvex Geometry
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A Maple package for convex geometry. It can deal with polytopes and, more generally, with all kinds of polyhedra of arbitrary dimension.<img src="/view.aspx?si=4507//applications/images/app_image_blank_lg.jpg" alt="Convex Geometry" align="left"/>A Maple package for convex geometry. It can deal with polytopes and, more generally, with all kinds of polyhedra of arbitrary dimension.4507Mon, 17 May 2004 14:56:00 ZMatthias FranzMatthias FranzAdvanced Math Packages: IntegerRelations, QDifferenceEquations, Generating Functions, and more
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Maple 9 furthers Maplesoft's commitment to supply the best mathematical algorithms known within the research community. There are new math packages in Maple 9 for integer relations, generating functions, q-difference equations, scientific error analysis and FFT. Users will also enjoy improved efficiency with the inclusion of the GNU Multiprecision (GMP) Libraries, new routines from the Numerical Algorithms Group (NAG), dynamic hashtables, and more.<img src="/view.aspx?si=4385//applications/images/app_image_blank_lg.jpg" alt="Advanced Math Packages: IntegerRelations, QDifferenceEquations, Generating Functions, and more" align="left"/>Maple 9 furthers Maplesoft's commitment to supply the best mathematical algorithms known within the research community. There are new math packages in Maple 9 for integer relations, generating functions, q-difference equations, scientific error analysis and FFT. Users will also enjoy improved efficiency with the inclusion of the GNU Multiprecision (GMP) Libraries, new routines from the Numerical Algorithms Group (NAG), dynamic hashtables, and more.4385Thu, 15 May 2003 14:05:59 ZMaplesoftMaplesoftInterval Arithmetic: Complete Package
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<p>The Interval Arithmetic PowerTool provides basic data types and operations for interval arithmetic as well as features for interval computation.</p><p>With Interval Arithmetic, you can define intervals and easily perform type checking, convert Maple expressions to interval arithmetic and more. You can enclose the ranges of two or three-dimensional functions using a combination of the following methods:</p><ul><li>interval evaluation</li><li">mean value form</li><li>monotony properties</li><li>Taylor forms</li></ul><p>This PowerTool offers the Interval Newton Method for the computation/enclosure of all zeros of a continuously differentiable real function. Types and procedures for both real intervals and complex interval arithmetic are also available.</p><p>IntpakX v 1.0 is an upgraded version of the Maple package intpakX for interval arithmetic. Intpak was initially written by R. Corless and A. Connell of the University of Western Ontario. IntpakX was written by I. Geulig and <a href="mailto:%20markus.grimmer@math.uni-wuppertal.de%20">W. Kraemer</a> and was updated by <a href="mailto:%20markus.grimmer@math.uni-wuppertal.de%20">M. Grimmer</a> to IntpakX v1.0.</p><img src="/view.aspx?si=1389//applications/images/app_image_blank_lg.jpg" alt="Interval Arithmetic: Complete Package" align="left"/><p>The Interval Arithmetic PowerTool provides basic data types and operations for interval arithmetic as well as features for interval computation.</p><p>With Interval Arithmetic, you can define intervals and easily perform type checking, convert Maple expressions to interval arithmetic and more. You can enclose the ranges of two or three-dimensional functions using a combination of the following methods:</p><ul><li>interval evaluation</li><li">mean value form</li><li>monotony properties</li><li>Taylor forms</li></ul><p>This PowerTool offers the Interval Newton Method for the computation/enclosure of all zeros of a continuously differentiable real function. Types and procedures for both real intervals and complex interval arithmetic are also available.</p><p>IntpakX v 1.0 is an upgraded version of the Maple package intpakX for interval arithmetic. Intpak was initially written by R. Corless and A. Connell of the University of Western Ontario. IntpakX was written by I. Geulig and <a href="mailto:%20markus.grimmer@math.uni-wuppertal.de%20">W. Kraemer</a> and was updated by <a href="mailto:%20markus.grimmer@math.uni-wuppertal.de%20">M. Grimmer</a> to IntpakX v1.0.</p>1389Fri, 22 Nov 2002 09:27:45 ZMarkus GrimmerMarkus GrimmerThe VariationalCalculus Package
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The new VariationalCalculus package in Maple 8 provides routines for solving problems in the calculus of variations, which studies nature's most "efficient" curves and surfaces. Examples include: find the shortest path between two points on a 3-D surface, shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time, and find the shape of a soap film having minimum surface area spanning a given wire frame.
Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation.<img src="/view.aspx?si=4269//applications/images/app_image_blank_lg.jpg" alt="The VariationalCalculus Package" align="left"/>The new VariationalCalculus package in Maple 8 provides routines for solving problems in the calculus of variations, which studies nature's most "efficient" curves and surfaces. Examples include: find the shortest path between two points on a 3-D surface, shape a ramp between two heights such that a ball rolling down it reaches the bottom in minimum time, and find the shape of a soap film having minimum surface area spanning a given wire frame.
Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation.4269Thu, 25 Apr 2002 14:28:54 ZMaplesoftMaplesoftNew Advanced Mathematics Packages in Maple 8 (Updated 4/25/02)
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We showcase some of the new Maple 8 packages for advanced mathematics: Calculus of variations, Symbolic-numeric algorithms for polynomials (SNAP), matrix polynomial algebra, modular linear algebra, and conversion routines for special functions<img src="/view.aspx?si=4255//applications/images/app_image_blank_lg.jpg" alt="New Advanced Mathematics Packages in Maple 8 (Updated 4/25/02)" align="left"/>We showcase some of the new Maple 8 packages for advanced mathematics: Calculus of variations, Symbolic-numeric algorithms for polynomials (SNAP), matrix polynomial algebra, modular linear algebra, and conversion routines for special functions4255Mon, 15 Apr 2002 15:01:19 ZMaplesoftMaplesoftLambert W function
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Still more fun results on the Lambert W function
This worksheet explores some recent results related to the function W(x), which satisfies W(x)*exp(W(x)) = x<img src="/view.aspx?si=4204//applications/images/app_image_blank_lg.jpg" alt="Lambert W function" align="left"/>Still more fun results on the Lambert W function
This worksheet explores some recent results related to the function W(x), which satisfies W(x)*exp(W(x)) = x4204Fri, 18 Jan 2002 10:20:55 ZRobert CorlessRobert CorlessDaubechies Wavelets
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This worksheet demonstrates the use of Maple for exploring the properties of the Daubechies scaling function, along with the family of wavelets associated with this function. The compact support, averaging, orthogonality, and regularity properties of are explored in this worksheet, along with graphs of the mother, daughter, and son wavelets.<img src="/view.aspx?si=4191/965.jpg" alt="Daubechies Wavelets" align="left"/>This worksheet demonstrates the use of Maple for exploring the properties of the Daubechies scaling function, along with the family of wavelets associated with this function. The compact support, averaging, orthogonality, and regularity properties of are explored in this worksheet, along with graphs of the mother, daughter, and son wavelets.4191Tue, 18 Dec 2001 14:57:47 ZProf. Edward AboufadelProf. Edward AboufadelINTERVALs Package
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This package is meant to give a front end package to Maple's already existing interval arithmetic tools, including the interval object and the command . It over-rides the operations +, -, *, /, ^, and the functions and and the various trigonometric, hyprbolic and their inverse functions and the exponential and logrithmic functions. It solves two problems: the need to call evalr on the result of operations, and it solves the problem of cancelation of equal objects.
<img src="/view.aspx?si=4143//applications/images/app_image_blank_lg.jpg" alt="INTERVALs Package" align="left"/>This package is meant to give a front end package to Maple's already existing interval arithmetic tools, including the interval object and the command . It over-rides the operations +, -, *, /, ^, and the functions and and the various trigonometric, hyprbolic and their inverse functions and the exponential and logrithmic functions. It solves two problems: the need to call evalr on the result of operations, and it solves the problem of cancelation of equal objects.
4143Thu, 18 Oct 2001 13:48:11 ZDouglas HarderDouglas HarderSlowly diverging series: advanced theory and examples
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This worksheet demonstrates the use of Maple for calculating and illustrating the rates of divergence of slowly diverging series from advanced mathematical analysis. It illustrates how to use numerical calculations and plotting of the sequences of partial sums as well as Euler-Maclaurin and asymptotic expansions. It also illustrates the use of Maple in series comparison tests.<img src="/view.aspx?si=3691//applications/images/app_image_blank_lg.jpg" alt="Slowly diverging series: advanced theory and examples" align="left"/>This worksheet demonstrates the use of Maple for calculating and illustrating the rates of divergence of slowly diverging series from advanced mathematical analysis. It illustrates how to use numerical calculations and plotting of the sequences of partial sums as well as Euler-Maclaurin and asymptotic expansions. It also illustrates the use of Maple in series comparison tests.3691Tue, 19 Jun 2001 00:00:00 ZSteven DunbarSteven DunbarNumerical approximation of functions
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The ``numapprox'' package contains various procedures for developing numerical approximations of functions. Examples of its use are presented here using the Maple worksheet facility. Note, some of the calculations done in this worksheet, in particular, the plots, require some time to compute<img src="/view.aspx?si=3650//applications/images/app_image_blank_lg.jpg" alt="Numerical approximation of functions" align="left"/>The ``numapprox'' package contains various procedures for developing numerical approximations of functions. Examples of its use are presented here using the Maple worksheet facility. Note, some of the calculations done in this worksheet, in particular, the plots, require some time to compute3650Tue, 19 Jun 2001 00:00:00 ZK. GeddesK. GeddesFourier Integrals
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This section illustrates Section 10.8 in Kreyszig 's book (8th ed.)<img src="/view.aspx?si=3598/FourierIntegrals_16.gif" alt="Fourier Integrals" align="left"/>This section illustrates Section 10.8 in Kreyszig 's book (8th ed.)3598Mon, 18 Jun 2001 00:00:00 ZAlain GorielyAlain GorielyTaylor, Legendre, and Bernstein polynomials
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This worksheet illustrates the convergence of the Taylor (power) series for a function bounded and defined on (-1,1). You can modify the number of terms that will be used, and then view the animation as the terms are gradually added from lowest to highest power. To get started, just press enter after all of the statements. Then try your own f(x) and play with N.
<img src="/view.aspx?si=3486//applications/images/app_image_blank_lg.jpg" alt="Taylor, Legendre, and Bernstein polynomials" align="left"/>This worksheet illustrates the convergence of the Taylor (power) series for a function bounded and defined on (-1,1). You can modify the number of terms that will be used, and then view the animation as the terms are gradually added from lowest to highest power. To get started, just press enter after all of the statements. Then try your own f(x) and play with N.
3486Mon, 18 Jun 2001 00:00:00 ZJim HerodJim HerodFourier Series: Square Wave
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This section illustrates Section 10.2 in Kreyszig 's book (8th ed.) <img src="/view.aspx?si=3594//applications/images/app_image_blank_lg.jpg" alt="Fourier Series: Square Wave" align="left"/>This section illustrates Section 10.2 in Kreyszig 's book (8th ed.) 3594Mon, 18 Jun 2001 00:00:00 ZAlain GorielyAlain GorielyMathematics in the movie "Good Will Hunting"
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When Professor Lambeau first announces the prize problem on the blackboard in the corridor, which is to be solved by Will the next evening, the blackboard in the classroom contains Parseval's Theorem from Fourier Analysis, as well as a part of its proof. Lambeau refers to the prize problem as an "Advanced Fourier System", but it turns out to be a second year problem in algebraic graph theory, to be solved in four stages. The problem is easily solved by Maple.<img src="/view.aspx?si=3512/goodwill.gif" alt="Mathematics in the movie "Good Will Hunting"" align="left"/>When Professor Lambeau first announces the prize problem on the blackboard in the corridor, which is to be solved by Will the next evening, the blackboard in the classroom contains Parseval's Theorem from Fourier Analysis, as well as a part of its proof. Lambeau refers to the prize problem as an "Advanced Fourier System", but it turns out to be a second year problem in algebraic graph theory, to be solved in four stages. The problem is easily solved by Maple.3512Mon, 18 Jun 2001 00:00:00 ZBert JagersBert Jagers