New Tips & Techniques
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en-us2021 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 06 May 2021 00:56:12 GMTThu, 06 May 2021 00:56:12 GMTThe latest Tips & Techniques applications added to the Application Centerhttps://www.maplesoft.com/images/Application_center_hp.jpgNew Tips & Techniques
https://www.maplesoft.com/applications/TipsAndTechniques
Classroom Tips and Techniques: The Partial Fraction Decomposition
https://www.maplesoft.com/applications/view.aspx?SID=1753&ref=Feed
The algebraic technique of partial fraction decomposition typically appears first in the integral calculus course as part of the methodology of integrating rational functions; and second, in any course such as differential equations where Laplace transforms must be inverted. If the Laplace transform is used in an engineering course, partial fraction decompositions must generally be implemented over the complex field so that all factors are linear.
This column describes how to obtain the partial fraction decomposition in Maple, either with irreducible quadratic factors or with strictly linear factors. We also suggest some pedagogic devices for providing insight into the algebraic processes involved.<img src="https://www.maplesoft.com/view.aspx?si=1753/partialfrac.PNG" alt="Classroom Tips and Techniques: The Partial Fraction Decomposition" style="max-width: 25%;" align="left"/>The algebraic technique of partial fraction decomposition typically appears first in the integral calculus course as part of the methodology of integrating rational functions; and second, in any course such as differential equations where Laplace transforms must be inverted. If the Laplace transform is used in an engineering course, partial fraction decompositions must generally be implemented over the complex field so that all factors are linear.
This column describes how to obtain the partial fraction decomposition in Maple, either with irreducible quadratic factors or with strictly linear factors. We also suggest some pedagogic devices for providing insight into the algebraic processes involved.https://www.maplesoft.com/applications/view.aspx?SID=1753&ref=FeedFri, 25 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezEigenpairs: What are they and how they are found
https://www.maplesoft.com/applications/view.aspx?SID=154291&ref=Feed
Clearly, Maple can compute eigenpairs (eigenvalues and eigenvectors) for a matrix, but of what help is Maple in getting across the concept of an eigenpair, and relating that insight to the standard algorithms students are expected to use to find them? This application is the companion Maple document to the webinar “Eigenpairs in Maple”, presented by Dr. Robert Lopez. In both the webinar and this application, he demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms.
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<B>View the Recorded Webinar:</B><BR>
<A HREF="/webinars/recorded/featured.aspx?id=1181">Eigenpairs in Maple</A><img src="https://www.maplesoft.com/view.aspx?si=154291/eigenpair.jpg" alt="Eigenpairs: What are they and how they are found" style="max-width: 25%;" align="left"/>Clearly, Maple can compute eigenpairs (eigenvalues and eigenvectors) for a matrix, but of what help is Maple in getting across the concept of an eigenpair, and relating that insight to the standard algorithms students are expected to use to find them? This application is the companion Maple document to the webinar “Eigenpairs in Maple”, presented by Dr. Robert Lopez. In both the webinar and this application, he demonstrates how Maple can enhance the task of teaching the eigenpair concept, and shows how Maple bridges the gap between the concept and the algorithms.
<BR><BR>
<B>View the Recorded Webinar:</B><BR>
<A HREF="/webinars/recorded/featured.aspx?id=1181">Eigenpairs in Maple</A>https://www.maplesoft.com/applications/view.aspx?SID=154291&ref=FeedFri, 25 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Eigenvalue Problems for ODEs
https://www.maplesoft.com/applications/view.aspx?SID=4971&ref=Feed
Some boundary value problems for partial differential equations are amenable to analytic techniques. For example, the constant-coefficient, second-order linear equations called the heat, wave, and potential equations are solved with some type of Fourier series representation obtained from the Sturm-Liouville eigenvalue problem that arises upon separating variables. The role of Maple in the solution of such boundary value problems is examined. Efficient techniques for separating variables, and a way to guide Maple through the solution of the resulting Sturm-Liouville eigenvalue problems are shown.<img src="https://www.maplesoft.com/view.aspx?si=4971/R-23EigenvalueProblemsforODEs.jpg" alt="Classroom Tips and Techniques: Eigenvalue Problems for ODEs" style="max-width: 25%;" align="left"/>Some boundary value problems for partial differential equations are amenable to analytic techniques. For example, the constant-coefficient, second-order linear equations called the heat, wave, and potential equations are solved with some type of Fourier series representation obtained from the Sturm-Liouville eigenvalue problem that arises upon separating variables. The role of Maple in the solution of such boundary value problems is examined. Efficient techniques for separating variables, and a way to guide Maple through the solution of the resulting Sturm-Liouville eigenvalue problems are shown.https://www.maplesoft.com/applications/view.aspx?SID=4971&ref=FeedMon, 14 Aug 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Green's Functions for Second-Order ODEs
https://www.maplesoft.com/applications/view.aspx?SID=4820&ref=Feed
<p>For second-order ODEs, we compute the Green's function for both initial and boundary value problems. For the boundary value problem, we consider mixed and unmixed boundary conditions, of both homogeneous and nonhomogeneous types. In every case, we compare our solutions to direct solutions using Maple's dsolve command.</p><img src="https://www.maplesoft.com/view.aspx?si=4820/image.php.gif" alt="Classroom Tips and Techniques: Green's Functions for Second-Order ODEs" style="max-width: 25%;" align="left"/><p>For second-order ODEs, we compute the Green's function for both initial and boundary value problems. For the boundary value problem, we consider mixed and unmixed boundary conditions, of both homogeneous and nonhomogeneous types. In every case, we compare our solutions to direct solutions using Maple's dsolve command.</p>https://www.maplesoft.com/applications/view.aspx?SID=4820&ref=FeedTue, 04 Jul 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezUsing the New Interactive Plot Builder
https://www.maplesoft.com/applications/view.aspx?SID=154272&ref=Feed
In Maple, Clickable Math covers a broad collection of features aimed at providing the user with easily discoverable, natural functionality for scientific and mathematical computing, without requiring an understanding of Maple syntax, commands, or the Maple programming language. In Maple 2017, a new Interactive Plot Builder joins this collection, providing an easy-to-use interface for creating and customizing a wide variety of 2-D and 3-D plots. In this Tips and Techniques, I will discuss some particular aspects of the new Plot Builder assistant, and how it fits into the Clickable Math framework.<img src="https://www.maplesoft.com/view.aspx?si=154272/plotbuilder.png" alt="Using the New Interactive Plot Builder" style="max-width: 25%;" align="left"/>In Maple, Clickable Math covers a broad collection of features aimed at providing the user with easily discoverable, natural functionality for scientific and mathematical computing, without requiring an understanding of Maple syntax, commands, or the Maple programming language. In Maple 2017, a new Interactive Plot Builder joins this collection, providing an easy-to-use interface for creating and customizing a wide variety of 2-D and 3-D plots. In this Tips and Techniques, I will discuss some particular aspects of the new Plot Builder assistant, and how it fits into the Clickable Math framework.https://www.maplesoft.com/applications/view.aspx?SID=154272&ref=FeedFri, 16 Jun 2017 04:00:00 ZDave LinderDave LinderPassword Protection in Maple
https://www.maplesoft.com/applications/view.aspx?SID=154270&ref=Feed
In Maple, worksheets can be password protected so the users of your Maple application can benefit from the specialized routines you've created while the details remain hidden. This Tips and Techniques shows you how to protect your Maple content from editing and viewing, while still allowing others to execute the code within and obtain results.<img src="https://www.maplesoft.com/view.aspx?si=154270/password.PNG" alt="Password Protection in Maple" style="max-width: 25%;" align="left"/>In Maple, worksheets can be password protected so the users of your Maple application can benefit from the specialized routines you've created while the details remain hidden. This Tips and Techniques shows you how to protect your Maple content from editing and viewing, while still allowing others to execute the code within and obtain results.https://www.maplesoft.com/applications/view.aspx?SID=154270&ref=FeedTue, 13 Jun 2017 04:00:00 ZGraham JacksonGraham JacksonClassroom Tips and Techniques: The Lagrange Multiplier Method
https://www.maplesoft.com/applications/view.aspx?SID=4811&ref=Feed
Maple has a number of graphical and analytical tools for studying and implementing the method of Lagrange multipliers. In this article, we demonstrate a number of these tools, indicating how they might be used pedagogically.<img src="https://www.maplesoft.com/view.aspx?si=4811/lagrange.PNG" alt="Classroom Tips and Techniques: The Lagrange Multiplier Method" style="max-width: 25%;" align="left"/>Maple has a number of graphical and analytical tools for studying and implementing the method of Lagrange multipliers. In this article, we demonstrate a number of these tools, indicating how they might be used pedagogically.https://www.maplesoft.com/applications/view.aspx?SID=4811&ref=FeedTue, 23 May 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezAn Introduction to Regression Analysis in Maple
https://www.maplesoft.com/applications/view.aspx?SID=154246&ref=Feed
Linear regression is one of the fundamental approaches for determining relationships between dependent and exploratory variables.
This worksheet covers several introductory examples for regression analysis in Maple.
The primary focus of the worksheet is the use of the Statistics:-Fit command to fit a model to data as well as generating reports on the fitted model. Additional examples include fitting a curve using the Optimization package.<img src="https://www.maplesoft.com/view.aspx?si=154246/regression.PNG" alt="An Introduction to Regression Analysis in Maple" style="max-width: 25%;" align="left"/>Linear regression is one of the fundamental approaches for determining relationships between dependent and exploratory variables.
This worksheet covers several introductory examples for regression analysis in Maple.
The primary focus of the worksheet is the use of the Statistics:-Fit command to fit a model to data as well as generating reports on the fitted model. Additional examples include fitting a curve using the Optimization package.https://www.maplesoft.com/applications/view.aspx?SID=154246&ref=FeedTue, 09 May 2017 04:00:00 ZDaniel SkoogDaniel SkoogClassroom Tips and Techniques: Roles for the Laplace Transform's Shifting Laws
https://www.maplesoft.com/applications/view.aspx?SID=1723&ref=Feed
The shifting laws for the Laplace transform are examined, and the argument is made that the transform of f(t) Heaviside(t - a) should be done with the third shifting law, reserving the second shifting law strictly for inverting functions of the form e^(-a s) F(s). It is needlessly complicated to apply the second shifting law to functions of the form f(t) Heaviside(t - a)<img src="https://www.maplesoft.com/view.aspx?si=1723/laplace.PNG" alt="Classroom Tips and Techniques: Roles for the Laplace Transform's Shifting Laws" style="max-width: 25%;" align="left"/>The shifting laws for the Laplace transform are examined, and the argument is made that the transform of f(t) Heaviside(t - a) should be done with the third shifting law, reserving the second shifting law strictly for inverting functions of the form e^(-a s) F(s). It is needlessly complicated to apply the second shifting law to functions of the form f(t) Heaviside(t - a)https://www.maplesoft.com/applications/view.aspx?SID=1723&ref=FeedTue, 25 Apr 2017 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Integration by Parts
https://www.maplesoft.com/applications/view.aspx?SID=1742&ref=Feed
Maple implements integration by parts with two different commands. One was designed in a pedagogical setting, and the other, for a "production" setting. In this article, we compare the functionalities of these two commands.<img src="https://www.maplesoft.com/view.aspx?si=1742/tutor.png" alt="Classroom Tips and Techniques: Integration by Parts" style="max-width: 25%;" align="left"/>Maple implements integration by parts with two different commands. One was designed in a pedagogical setting, and the other, for a "production" setting. In this article, we compare the functionalities of these two commands.https://www.maplesoft.com/applications/view.aspx?SID=1742&ref=FeedThu, 02 Mar 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Norm of a Matrix
https://www.maplesoft.com/applications/view.aspx?SID=1430&ref=Feed
The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.<img src="https://www.maplesoft.com/view.aspx?si=1430/thumb.jpg" alt="Classroom Tips and Techniques: Norm of a Matrix" style="max-width: 25%;" align="left"/>The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.https://www.maplesoft.com/applications/view.aspx?SID=1430&ref=FeedMon, 13 Feb 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezVisualizing Multiple Datasets with BubblePlot
https://www.maplesoft.com/applications/view.aspx?SID=154178&ref=Feed
The BubblePlot command can convey information about three dimensions of a multi-dimensional dataset using the horizontal axis, the vertical axis, and point (bubble) size. Moreover, if a dataset is a time series, BubblePlot can generate an animation that shows the movement of data points over a common period of time.
In the following example, datasets containing information on Gross Domestic Product at Power Purchasing Parity, Life Expectancy, and Population are retrieved for selected countries and visualized.<img src="https://www.maplesoft.com/view.aspx?si=154178/BubblePlot.png" alt="Visualizing Multiple Datasets with BubblePlot" style="max-width: 25%;" align="left"/>The BubblePlot command can convey information about three dimensions of a multi-dimensional dataset using the horizontal axis, the vertical axis, and point (bubble) size. Moreover, if a dataset is a time series, BubblePlot can generate an animation that shows the movement of data points over a common period of time.
In the following example, datasets containing information on Gross Domestic Product at Power Purchasing Parity, Life Expectancy, and Population are retrieved for selected countries and visualized.https://www.maplesoft.com/applications/view.aspx?SID=154178&ref=FeedMon, 17 Oct 2016 04:00:00 ZDaniel SkoogDaniel SkoogWorking with Thermophysical Data: Dew-Point and Wet-Bulb Temperature of Air
https://www.maplesoft.com/applications/view.aspx?SID=154054&ref=Feed
Maple can perform calculations and generate visualizations involving thermophysical properties of pure fluids, humid air, and mixtures. Using the dew-point and web-bulb temperature of air as an example, this Tips and Techniques application demonstrates how to access thermophysical properties data, perform calculations that include units, and visualize the results on a psychrometric chart.
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Atmospheric air contains varying levels of water vapor. Weather reports often quantify the water content of air with its relative humidity; this is the amount of water in air, divided by the maximum amount of water air can hold at the same temperature.
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Given the temperature and the relative humidity of air, you can calculate:
<UL>
<LI>the temperature below which water condenses out of air - this is known as the dew-point
<LI>the coldest temperature you can achieve through evaporative cooling - this is known as the wet-bulb temperature
</UL><img src="https://www.maplesoft.com/view.aspx?si=154054/webbulb.PNG" alt="Working with Thermophysical Data: Dew-Point and Wet-Bulb Temperature of Air" style="max-width: 25%;" align="left"/>Maple can perform calculations and generate visualizations involving thermophysical properties of pure fluids, humid air, and mixtures. Using the dew-point and web-bulb temperature of air as an example, this Tips and Techniques application demonstrates how to access thermophysical properties data, perform calculations that include units, and visualize the results on a psychrometric chart.
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Atmospheric air contains varying levels of water vapor. Weather reports often quantify the water content of air with its relative humidity; this is the amount of water in air, divided by the maximum amount of water air can hold at the same temperature.
<BR><BR>
Given the temperature and the relative humidity of air, you can calculate:
<UL>
<LI>the temperature below which water condenses out of air - this is known as the dew-point
<LI>the coldest temperature you can achieve through evaporative cooling - this is known as the wet-bulb temperature
</UL>https://www.maplesoft.com/applications/view.aspx?SID=154054&ref=FeedWed, 09 Mar 2016 05:00:00 ZSamir KhanSamir KhanTips and Techniques: Working with Finitely Presented Groups in Maple
https://www.maplesoft.com/applications/view.aspx?SID=153852&ref=Feed
This Tips and Techniques article introduces Maple's facilities for working with finitely presented groups. A finitely presented group is a group defined by means of a finite number of generators, and a finite number of defining relations. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute with many infinite groups.<img src="https://www.maplesoft.com/view.aspx?si=153852/thumb.jpg" alt="Tips and Techniques: Working with Finitely Presented Groups in Maple" style="max-width: 25%;" align="left"/>This Tips and Techniques article introduces Maple's facilities for working with finitely presented groups. A finitely presented group is a group defined by means of a finite number of generators, and a finite number of defining relations. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute with many infinite groups.https://www.maplesoft.com/applications/view.aspx?SID=153852&ref=FeedTue, 25 Aug 2015 04:00:00 ZMaplesoftMaplesoftTime Series Analysis: Forecasting Average Global Temperatures
https://www.maplesoft.com/applications/view.aspx?SID=153791&ref=Feed
Maple includes powerful tools for accessing, analyzing, and visualizing time series data. This application works with global temperature data to demonstrate techniques for analyzing time series data sets using the TimeSeriesAnalysis package, including visualizing trends and modeling future global temperatures.<img src="https://www.maplesoft.com/view.aspx?si=153791/thumb.jpg" alt="Time Series Analysis: Forecasting Average Global Temperatures" style="max-width: 25%;" align="left"/>Maple includes powerful tools for accessing, analyzing, and visualizing time series data. This application works with global temperature data to demonstrate techniques for analyzing time series data sets using the TimeSeriesAnalysis package, including visualizing trends and modeling future global temperatures.https://www.maplesoft.com/applications/view.aspx?SID=153791&ref=FeedTue, 21 Apr 2015 04:00:00 ZDaniel SkoogDaniel SkoogTips and Techniques: 3-D Model Import/Export and Printing
https://www.maplesoft.com/applications/view.aspx?SID=153770&ref=Feed
Maple can import from and export to several popular graphics formats. In this tips and techniques, you’ll learn about importing and exporting 3-D graphics files. Examples include printing Maple graphics on 3-D printers.<img src="https://www.maplesoft.com/view.aspx?si=153770/thumb.jpg" alt="Tips and Techniques: 3-D Model Import/Export and Printing" style="max-width: 25%;" align="left"/>Maple can import from and export to several popular graphics formats. In this tips and techniques, you’ll learn about importing and exporting 3-D graphics files. Examples include printing Maple graphics on 3-D printers.https://www.maplesoft.com/applications/view.aspx?SID=153770&ref=FeedFri, 13 Mar 2015 04:00:00 ZStephen ForrestStephen ForrestClassroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions
https://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="https://www.maplesoft.com/view.aspx?si=153726/thumb.jpg" alt="Classroom Tips and Techniques: Real and Complex Derivatives of Some Elementary Functions" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=153726&ref=FeedWed, 10 Dec 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots
https://www.maplesoft.com/applications/view.aspx?SID=153697&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=153697/thumb.jpg" alt="Classroom Tips and Techniques: Branch Cuts for a Product of Two Square-Roots" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=153697&ref=FeedTue, 11 Nov 2014 05:00:00 ZDr. Robert LopezDr. Robert LopezGroebner Bases: What are They and What are They Useful For?
https://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="https://www.maplesoft.com/view.aspx?si=153693/thumb.jpg" alt="Groebner Bases: What are They and What are They Useful For?" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=153693&ref=FeedFri, 17 Oct 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganComputational Performance with evalhf and Compile: A Newton Fractal Case Study
https://www.maplesoft.com/applications/view.aspx?SID=153683&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=153683/thumb.jpg" alt="Computational Performance with evalhf and Compile: A Newton Fractal Case Study" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=153683&ref=FeedFri, 26 Sep 2014 04:00:00 ZDave LinderDave Linder