Optimization: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=1600
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 23 Feb 2020 20:58:21 GMTSun, 23 Feb 2020 20:58:21 GMTNew applications in the Optimization categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgOptimization: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=1600
Avoiding Scurvy on a Budget
https://www.maplesoft.com/applications/view.aspx?SID=154368&ref=Feed
Are you a broke college or university student? Do you live your life on ramen, and nothing but? Want a healthier diet while spending as little money as possible (and avoiding scurvy!) ? This application optimizes your groceries by finding the least cost diet that maintains your nutritional requirements, via constrained linear programing.<img src="https://www.maplesoft.com/view.aspx?si=154368/scurvy.jpg" alt="Avoiding Scurvy on a Budget" style="max-width: 25%;" align="left"/>Are you a broke college or university student? Do you live your life on ramen, and nothing but? Want a healthier diet while spending as little money as possible (and avoiding scurvy!) ? This application optimizes your groceries by finding the least cost diet that maintains your nutritional requirements, via constrained linear programing.https://www.maplesoft.com/applications/view.aspx?SID=154368&ref=FeedMon, 27 Nov 2017 05:00:00 ZMaplesoftMaplesoftMathematics for Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=Feed
This interactive electronic textbook in the form of Maple worksheets comprises two parts.
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Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019<img src="https://www.maplesoft.com/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" style="max-width: 25%;" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=FeedTue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieSteepest Ascent Method
https://www.maplesoft.com/applications/view.aspx?SID=154132&ref=Feed
This worksheet applies the gradient search method for multi-variable maximization problems. This is an update of our earlier version in the application center. This worksheet solves nonlinear optimization problems by the method of steepest ascent.
Given a function <EM>f(x,y)</EM> and a current point <EM>(`x__0`, `y__0`)</EM>, the search direction is taken to be the gradient of <EM>f(x,y)</EM> at <EM>(`x__0`, `y__0`)</EM>. The step length is computed by line search, i.e. as the step length that maximizes <EM>f(x,y)</EM> along the gradient direction.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Steepest Ascent Method" style="max-width: 25%;" align="left"/>This worksheet applies the gradient search method for multi-variable maximization problems. This is an update of our earlier version in the application center. This worksheet solves nonlinear optimization problems by the method of steepest ascent.
Given a function <EM>f(x,y)</EM> and a current point <EM>(`x__0`, `y__0`)</EM>, the search direction is taken to be the gradient of <EM>f(x,y)</EM> at <EM>(`x__0`, `y__0`)</EM>. The step length is computed by line search, i.e. as the step length that maximizes <EM>f(x,y)</EM> along the gradient direction.https://www.maplesoft.com/applications/view.aspx?SID=154132&ref=FeedThu, 07 Jul 2016 04:00:00 ZDr. William Hank Richardson<BR>Dr. Henk KoppelaarDr. William Hank Richardson<BR>Dr. Henk KoppelaarOptimize the Flight Path of a Pan-US Delivery Drone
https://www.maplesoft.com/applications/view.aspx?SID=154005&ref=Feed
You run a pan-US drone delivery service for a popular online retailer. You're given a list of zip codes across the US at which you need to drop off parcels, and want to optimize its journey so it travels the shortest distance.
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This application extracts the latitude and longitude of those zip codes from an SQLite database (the database is included with the application, and cross-references US zip codes against their latitude, longitude, city and state). It then performs a traveling salesman optimization and plots the shortest path on a map of the US.<img src="https://www.maplesoft.com/view.aspx?si=154005/drone.png" alt="Optimize the Flight Path of a Pan-US Delivery Drone" style="max-width: 25%;" align="left"/>You run a pan-US drone delivery service for a popular online retailer. You're given a list of zip codes across the US at which you need to drop off parcels, and want to optimize its journey so it travels the shortest distance.
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This application extracts the latitude and longitude of those zip codes from an SQLite database (the database is included with the application, and cross-references US zip codes against their latitude, longitude, city and state). It then performs a traveling salesman optimization and plots the shortest path on a map of the US.https://www.maplesoft.com/applications/view.aspx?SID=154005&ref=FeedWed, 02 Mar 2016 05:00:00 ZSamir KhanSamir KhanPacking Discs into a Circle
https://www.maplesoft.com/applications/view.aspx?SID=154006&ref=Feed
This application finds the best packing of unequal non-overlapping disks in a larger circle, such that the radius of the container is minimized. This is a difficult global optimization problem that demands strong solvers; this application uses Maple's Global Optimization Toolbox. You must have the Global Optimization Toolbox installed to use this application.
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Packing optimization is industrially important, with applications in pallet loading, the arrangement of fiber optic cables in a tube, or the placing of blocks on a circuit board.<img src="https://www.maplesoft.com/view.aspx?si=154006/discs_circle.png" alt="Packing Discs into a Circle" style="max-width: 25%;" align="left"/>This application finds the best packing of unequal non-overlapping disks in a larger circle, such that the radius of the container is minimized. This is a difficult global optimization problem that demands strong solvers; this application uses Maple's Global Optimization Toolbox. You must have the Global Optimization Toolbox installed to use this application.
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Packing optimization is industrially important, with applications in pallet loading, the arrangement of fiber optic cables in a tube, or the placing of blocks on a circuit board.https://www.maplesoft.com/applications/view.aspx?SID=154006&ref=FeedWed, 02 Mar 2016 05:00:00 ZSamir KhanSamir KhanDiet Optimization
https://www.maplesoft.com/applications/view.aspx?SID=154004&ref=Feed
This application finds the least-cost diet that fulfills a specific set of nutritional requirements It has a default basket of foods (with an associated set of nutritional data), but foods can be added or removed, with changes remembered from prior saved sessions.
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Stigler studied this problem heuristically in the 1940s, but only the development of modern optimization algorithms gave relatively quick accurate solutions. The linear programming techniques implemented in this application are now widely used to create practical diet plans from accepted nutritional guidelines.<img src="https://www.maplesoft.com/view.aspx?si=154004/Diet_Optimization.png" alt="Diet Optimization" style="max-width: 25%;" align="left"/>This application finds the least-cost diet that fulfills a specific set of nutritional requirements It has a default basket of foods (with an associated set of nutritional data), but foods can be added or removed, with changes remembered from prior saved sessions.
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Stigler studied this problem heuristically in the 1940s, but only the development of modern optimization algorithms gave relatively quick accurate solutions. The linear programming techniques implemented in this application are now widely used to create practical diet plans from accepted nutritional guidelines.https://www.maplesoft.com/applications/view.aspx?SID=154004&ref=FeedWed, 02 Mar 2016 05:00:00 ZSamir KhanSamir KhanEconomic Pipe Sizer for Process Plants
https://www.maplesoft.com/applications/view.aspx?SID=153659&ref=Feed
<p>Pipework is a large part of the cost of a process plant. Plant designers need to minimize the total cost of this pipework across the lifetime of the plant. The total overall cost is a combination of individual costs related to the:</p>
<ul>
<li>pipe material,</li>
<li>installation, </li>
<li>maintenance, </li>
<li>depreciation, </li>
<li>energy costs for pumping, </li>
<li>liquid parameters, </li>
<li>required flowrate,</li>
<li>pumping efficiencies,</li>
<li>taxes,</li>
<li>and more.</li>
</ul>
<p>The total cost is not a simple linear sum of the individual costs; a more complex relationship is needed.</p>
<p>This application uses the approach described in [1] to find the pipe diameter that minimizes the total lifetime cost. The method involves the iterative solution of an empirical equation using <a href="/support/help/Maple/view.aspx?path=fsolve">Maple’s fsolve function</a> (the code for the application is in the Startup code region).</p>
<p>Users can choose the pipe material (carbon steel, stainless steel, aluminum or brass), and specify the desired fluid flowrate, fluid viscosity and density. The application then solves the empirical equation (using Maple’s fsolve() function) and returns the economically optimal pipe diameter.</p>
<p>Bear in mind that the empirical parameters used in the application vary as economic conditions change. Those used in this application are correct for 1998 and 2008.</p>
<p><em>[1]: "Updating the Rules for Pipe Sizing", Durand et al., Chemical Engineering, January 2010</em></p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Economic Pipe Sizer for Process Plants" style="max-width: 25%;" align="left"/><p>Pipework is a large part of the cost of a process plant. Plant designers need to minimize the total cost of this pipework across the lifetime of the plant. The total overall cost is a combination of individual costs related to the:</p>
<ul>
<li>pipe material,</li>
<li>installation, </li>
<li>maintenance, </li>
<li>depreciation, </li>
<li>energy costs for pumping, </li>
<li>liquid parameters, </li>
<li>required flowrate,</li>
<li>pumping efficiencies,</li>
<li>taxes,</li>
<li>and more.</li>
</ul>
<p>The total cost is not a simple linear sum of the individual costs; a more complex relationship is needed.</p>
<p>This application uses the approach described in [1] to find the pipe diameter that minimizes the total lifetime cost. The method involves the iterative solution of an empirical equation using <a href="/support/help/Maple/view.aspx?path=fsolve">Maple’s fsolve function</a> (the code for the application is in the Startup code region).</p>
<p>Users can choose the pipe material (carbon steel, stainless steel, aluminum or brass), and specify the desired fluid flowrate, fluid viscosity and density. The application then solves the empirical equation (using Maple’s fsolve() function) and returns the economically optimal pipe diameter.</p>
<p>Bear in mind that the empirical parameters used in the application vary as economic conditions change. Those used in this application are correct for 1998 and 2008.</p>
<p><em>[1]: "Updating the Rules for Pipe Sizing", Durand et al., Chemical Engineering, January 2010</em></p>https://www.maplesoft.com/applications/view.aspx?SID=153659&ref=FeedFri, 15 Aug 2014 04:00:00 ZSamir KhanSamir KhanOptimizing the Design of a Coil Spring
https://www.maplesoft.com/applications/view.aspx?SID=153608&ref=Feed
<p>The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.</p>
<p>Maple lets you increase the number of digits used in calculations; hence numerically difficult problems, like this, can be solved.</p>
<p>This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency and the maximum stress, and a loading condition.</p>
<ul>
<li>the minimum deflection, </li>
<li>the minimum surge wave frequency, </li>
<li>the maximum stress, </li>
<li>and a loading condition.</li>
</ul>
<p>The design variables are the</p>
<ul>
<li>diameter of the wire, </li>
<li>the outside diameter of the spring,</li>
<li>and the number of coils</li>
</ul>
<p> Reference: "Introduction to Optimum Design", Jasbir S. Arora, 3<sup>rd</sup> Edition 2012.</p><img src="https://www.maplesoft.com/view.aspx?si=153608/695d991fff8fb4975d1e1dcd90bb771d.gif" alt="Optimizing the Design of a Coil Spring" style="max-width: 25%;" align="left"/><p>The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.</p>
<p>Maple lets you increase the number of digits used in calculations; hence numerically difficult problems, like this, can be solved.</p>
<p>This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency and the maximum stress, and a loading condition.</p>
<ul>
<li>the minimum deflection, </li>
<li>the minimum surge wave frequency, </li>
<li>the maximum stress, </li>
<li>and a loading condition.</li>
</ul>
<p>The design variables are the</p>
<ul>
<li>diameter of the wire, </li>
<li>the outside diameter of the spring,</li>
<li>and the number of coils</li>
</ul>
<p> Reference: "Introduction to Optimum Design", Jasbir S. Arora, 3<sup>rd</sup> Edition 2012.</p>https://www.maplesoft.com/applications/view.aspx?SID=153608&ref=FeedTue, 17 Jun 2014 04:00:00 ZSamir KhanSamir KhanPacking Circles into a Triangle
https://www.maplesoft.com/applications/view.aspx?SID=153596&ref=Feed
<p>This application finds the best packing and largest radius of equal-sized circles, such that they fit in a pre-defined triangle. One solution, as visualized by this application, is given below.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="http://www.maplesoft.com/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 310 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations. The vertices of the triangle can also be modified</p>
<p>Applications like this are used to stress-test global optimizers.</p><img src="https://www.maplesoft.com/view.aspx?si=153596/2ac6ca1378717b3d939f3d8107616b35.gif" alt="Packing Circles into a Triangle" style="max-width: 25%;" align="left"/><p>This application finds the best packing and largest radius of equal-sized circles, such that they fit in a pre-defined triangle. One solution, as visualized by this application, is given below.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="http://www.maplesoft.com/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 20 circles generates 310 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations. The vertices of the triangle can also be modified</p>
<p>Applications like this are used to stress-test global optimizers.</p>https://www.maplesoft.com/applications/view.aspx?SID=153596&ref=FeedWed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanCircle Packing in an Ellipse
https://www.maplesoft.com/applications/view.aspx?SID=153598&ref=Feed
<p>This application optimizes the packing of circles in an ellipse, such that the area of the ellipse is minimized. A typical solution is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 35 circles generates 666 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>
<p>The constraints and ellipse parameterization are taken from "Packing circles within ellipses", Birgin et al., International Transactions in Operational Research , Volume 20, Issue 3, pages 365–389, May 2013.</p><img src="https://www.maplesoft.com/view.aspx?si=153598/5f52383daddaeb53aec548d14ebd6ce0.gif" alt="Circle Packing in an Ellipse" style="max-width: 25%;" align="left"/><p>This application optimizes the packing of circles in an ellipse, such that the area of the ellipse is minimized. A typical solution is visualized here.</p>
<p>This is a difficult global optimization problem and demands strong solvers. This application uses Maple's <a href="/products/toolboxes/globaloptimization/">Global Optimization Toolbox</a>.</p>
<p>Circle packing (and packing optimization in general) is characterized by a large optimization space and many constraints; for this application, 35 circles generates 666 constraint equations.</p>
<p>The number of circles can be increased to create an increasingly complex problem; Maple automatically generates the symbolic constraint equations.</p>
<p>Applications like this are used to stress-test global optimizers.</p>
<p>The constraints and ellipse parameterization are taken from "Packing circles within ellipses", Birgin et al., International Transactions in Operational Research , Volume 20, Issue 3, pages 365–389, May 2013.</p>https://www.maplesoft.com/applications/view.aspx?SID=153598&ref=FeedWed, 04 Jun 2014 04:00:00 ZSamir KhanSamir KhanWelded Beam Design Optimization
https://www.maplesoft.com/applications/view.aspx?SID=153592&ref=Feed
<p>A rigid member is welded onto a beam, with a load applied to the end of the member. The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.</p>
<p>The design of the beam is optimized to minimize the production costs by varying the weld and member dimensions.</p>
<p>The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints.</p>
<p>The application uses Maple’s non-linear optimizers</p><img src="https://www.maplesoft.com/view.aspx?si=153592/0621a9aba622112f66506495e21f68d9.gif" alt="Welded Beam Design Optimization" style="max-width: 25%;" align="left"/><p>A rigid member is welded onto a beam, with a load applied to the end of the member. The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.</p>
<p>The design of the beam is optimized to minimize the production costs by varying the weld and member dimensions.</p>
<p>The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints.</p>
<p>The application uses Maple’s non-linear optimizers</p>https://www.maplesoft.com/applications/view.aspx?SID=153592&ref=FeedFri, 30 May 2014 04:00:00 ZSamir KhanSamir KhanOptimizing the Design of a Fuel Pod with NX and Maple
https://www.maplesoft.com/applications/view.aspx?SID=153573&ref=Feed
<p>A manufacturer has designed a fuel pod in NX. The fuel pod has a hemispherical and conical end, and a cylindrical mid-section. To minimize material costs, the manufacturer wants to minimize the surface area of the fuel pod while maintaining the existing volume.</p>
<p>This application:</p>
<ul>
<li>pulls the current dimensions of the fuel pod (radius of the hemispherical end, length of the cylindrical midsection, and height of the conical end) from the NX CAD model, </li>
<li>calculates the current volume of the fuel pod,</li>
<li>optimizes the dimensions to minimize the surface area while maintaining the existing volume,</li>
<li>and pushes the optimized dimensions back into the NX CAD model.</li>
</ul>
<p>NOTE: To use this application, you must</p>
<ul>
<li>have a supported version of NX installed, </li>
<li>load canisterOptimization.prt in NX (this is the CAD model of the fuel pod),</li>
<li>ensure the NX-Maple link works correctly.</li>
</ul><img src="https://www.maplesoft.com/view.aspx?si=153573/fuelpod.jpg" alt="Optimizing the Design of a Fuel Pod with NX and Maple" style="max-width: 25%;" align="left"/><p>A manufacturer has designed a fuel pod in NX. The fuel pod has a hemispherical and conical end, and a cylindrical mid-section. To minimize material costs, the manufacturer wants to minimize the surface area of the fuel pod while maintaining the existing volume.</p>
<p>This application:</p>
<ul>
<li>pulls the current dimensions of the fuel pod (radius of the hemispherical end, length of the cylindrical midsection, and height of the conical end) from the NX CAD model, </li>
<li>calculates the current volume of the fuel pod,</li>
<li>optimizes the dimensions to minimize the surface area while maintaining the existing volume,</li>
<li>and pushes the optimized dimensions back into the NX CAD model.</li>
</ul>
<p>NOTE: To use this application, you must</p>
<ul>
<li>have a supported version of NX installed, </li>
<li>load canisterOptimization.prt in NX (this is the CAD model of the fuel pod),</li>
<li>ensure the NX-Maple link works correctly.</li>
</ul>https://www.maplesoft.com/applications/view.aspx?SID=153573&ref=FeedWed, 07 May 2014 04:00:00 ZSamir KhanSamir KhanClassroom Tips and Techniques: Bivariate Limits - Then and Now
https://www.maplesoft.com/applications/view.aspx?SID=145979&ref=Feed
An introductory overview of the functionalities in Maple's GraphTheory package.<img src="https://www.maplesoft.com/view.aspx?si=145979/thumb.jpg" alt="Classroom Tips and Techniques: Bivariate Limits - Then and Now" style="max-width: 25%;" align="left"/>An introductory overview of the functionalities in Maple's GraphTheory package.https://www.maplesoft.com/applications/view.aspx?SID=145979&ref=FeedWed, 17 Apr 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Introduction to Maple's GraphTheory Package
https://www.maplesoft.com/applications/view.aspx?SID=142357&ref=Feed
An introductory overview of the functionality in Maple's GraphTheory package.<img src="https://www.maplesoft.com/view.aspx?si=142357/thumb.jpg" alt="Classroom Tips and Techniques: Introduction to Maple's GraphTheory Package" style="max-width: 25%;" align="left"/>An introductory overview of the functionality in Maple's GraphTheory package.https://www.maplesoft.com/applications/view.aspx?SID=142357&ref=FeedThu, 17 Jan 2013 05:00:00 ZProf. Michael MonaganProf. Michael MonaganClassroom Tips and Techniques: Least-Squares Fits
https://www.maplesoft.com/applications/view.aspx?SID=140942&ref=Feed
<p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p><img src="https://www.maplesoft.com/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" style="max-width: 25%;" align="left"/><p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p>https://www.maplesoft.com/applications/view.aspx?SID=140942&ref=FeedWed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Best Taylor-Polynomial Approximations
https://www.maplesoft.com/applications/view.aspx?SID=136471&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=136471/image.jpg" alt="Classroom Tips and Techniques: Best Taylor-Polynomial Approximations" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=136471&ref=FeedTue, 14 Aug 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: An Inequality-Constrained Optimization Problem
https://www.maplesoft.com/applications/view.aspx?SID=135904&ref=Feed
<p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p><img src="https://www.maplesoft.com/view.aspx?si=135904/thumb.jpg" alt="Classroom Tips and Techniques: An Inequality-Constrained Optimization Problem" style="max-width: 25%;" align="left"/><p>This article shows how to work both analytically and numerically to find the global maximum of</p>
<p><em>w</em> = ƒ(<em>x, y, z</em>) ≡ <em>x</em><sup>2</sup>(1 + <em>x</em>) + <em>y</em><sup>2</sup>(1 + <em>y</em>) + z<sup>2</sup>(1 + <em>z</em>)</p>
<p>in that part of the first octant on, or below, the plane <em>x</em> + <em>y</em> + <em>z</em> = 6.</p>https://www.maplesoft.com/applications/view.aspx?SID=135904&ref=FeedMon, 16 Jul 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezStreet-fighting Math
https://www.maplesoft.com/applications/view.aspx?SID=129226&ref=Feed
This interactive Maple document contains a simple street-fighting game and performs a mathematical analysis of it, involving probability and game theory. The document is suitable for presentation in an undergraduate course on operations research, probability or linear programming. No knowledge of Maple is required.<img src="https://www.maplesoft.com/view.aspx?si=129226/fighter_sm.jpg" alt="Street-fighting Math" style="max-width: 25%;" align="left"/>This interactive Maple document contains a simple street-fighting game and performs a mathematical analysis of it, involving probability and game theory. The document is suitable for presentation in an undergraduate course on operations research, probability or linear programming. No knowledge of Maple is required.https://www.maplesoft.com/applications/view.aspx?SID=129226&ref=FeedThu, 29 Dec 2011 05:00:00 ZDr. Robert IsraelDr. Robert IsraelGreat Expectations
https://www.maplesoft.com/applications/view.aspx?SID=127116&ref=Feed
<p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p><img src="https://www.maplesoft.com/view.aspx?si=127116/expectation_thum.png" alt="Great Expectations" style="max-width: 25%;" align="left"/><p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p>https://www.maplesoft.com/applications/view.aspx?SID=127116&ref=FeedThu, 27 Oct 2011 04:00:00 ZClassroom Tips and Techniques: Steepest-Ascent Curves
https://www.maplesoft.com/applications/view.aspx?SID=123985&ref=Feed
Steepest-ascent curves are obtained for surfaces defined analytically and digitally.<img src="https://www.maplesoft.com/view.aspx?si=123985/thumb.jpg" alt="Classroom Tips and Techniques: Steepest-Ascent Curves" style="max-width: 25%;" align="left"/>Steepest-ascent curves are obtained for surfaces defined analytically and digitally.https://www.maplesoft.com/applications/view.aspx?SID=123985&ref=FeedTue, 19 Jul 2011 04:00:00 ZDr. Robert LopezDr. Robert Lopez