Engineering Mathematics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=138
en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 08 Jul 2015 02:16:56 GMTWed, 08 Jul 2015 02:16:56 GMTNew applications in the Engineering Mathematics categoryhttp://www.mapleprimes.com/images/mapleapps.gifEngineering Mathematics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=138
Blutdruckwerte aus Langzeitmessung
http://www.maplesoft.com/applications/view.aspx?SID=153813&ref=Feed
<p>During a period of 24 hours the blood pressure of a patient at the University Hospital Aachen has been measured. Thus, we have a lot of Systole-, Diastole-, and Pulse-Values important for medical doctors teating sick patients. To analyse these "data" the Maple Program 16 has been used.</p><img src="/view.aspx?si=153813/9eba1a814642633a6f07c19980f3a0e8.gif" alt="Blutdruckwerte aus Langzeitmessung" align="left"/><p>During a period of 24 hours the blood pressure of a patient at the University Hospital Aachen has been measured. Thus, we have a lot of Systole-, Diastole-, and Pulse-Values important for medical doctors teating sick patients. To analyse these "data" the Maple Program 16 has been used.</p>153813Thu, 04 Jun 2015 04:00:00 ZProf. Josef Professor BettenProf. Josef Professor BettenSplinefunktion als FOURIER-Reihen
http://www.maplesoft.com/applications/view.aspx?SID=153796&ref=Feed
<p>Zunächst werden experimentelle Daten durch eine kubische <em>Splinefunktion </em>interpoliert. Die stückweise stetige <em>Splinefunktion</em> wird als <em>FOURIER-Reihe </em>dargestellt Dabei wird festgestellt, dass sich die <em>FOURIER-Reihe </em>mit steigender Anzahl der Reihenglieder immer szärker an die <em>Splinefunktion </em>anschmiegt. Bei unendlicher Anzahl der Reihenglieder fällt die <em>FOURIER-Reihe </em>mit der <em>Splinefunktion zusammen.</em></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Splinefunktion als FOURIER-Reihen" align="left"/><p>Zunächst werden experimentelle Daten durch eine kubische <em>Splinefunktion </em>interpoliert. Die stückweise stetige <em>Splinefunktion</em> wird als <em>FOURIER-Reihe </em>dargestellt Dabei wird festgestellt, dass sich die <em>FOURIER-Reihe </em>mit steigender Anzahl der Reihenglieder immer szärker an die <em>Splinefunktion </em>anschmiegt. Bei unendlicher Anzahl der Reihenglieder fällt die <em>FOURIER-Reihe </em>mit der <em>Splinefunktion zusammen.</em></p>153796Mon, 18 May 2015 04:00:00 ZProf. Josef BettenProf. Josef BettenEconomic Pipe Sizer for Process Plants
http://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="/applications/images/app_image_blank_lg.jpg" alt="Economic Pipe Sizer for Process Plants" 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>153659Fri, 15 Aug 2014 04:00:00 ZSamir KhanSamir KhanWelded Beam Design Optimization
http://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="/view.aspx?si=153592/0621a9aba622112f66506495e21f68d9.gif" alt="Welded Beam Design Optimization" 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>153592Fri, 30 May 2014 04:00:00 ZSamir KhanSamir KhanVisualization Free and Forced Harmonic Oscillations
http://www.maplesoft.com/applications/view.aspx?SID=153558&ref=Feed
<p>This worksheet focuses on how the symbolic,numerical and graphical power of CAS, maple can be used to explore and visualize with animation free harmonic.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Visualization Free and Forced Harmonic Oscillations" align="left"/><p>This worksheet focuses on how the symbolic,numerical and graphical power of CAS, maple can be used to explore and visualize with animation free harmonic.</p>153558Thu, 01 May 2014 04:00:00 ZAli Abu OamAli Abu OamAutomatic Speech Segmentation
http://www.maplesoft.com/applications/view.aspx?SID=153553&ref=Feed
<p>This worksheet demonstrates the use of the Forward-Backward Divergence model (FBD) in Automatic Speech Segmentation, and how it detects discontinuities in the voice signal. It illustrates in the example below how it is possible to enlarge some segments of the speech (vowels enlargement for instance). To realize this result, it is possible to visually and acoustically perceive the stationary segments of the speech signal.</p><img src="/view.aspx?si=153553/speech.png" alt="Automatic Speech Segmentation" align="left"/><p>This worksheet demonstrates the use of the Forward-Backward Divergence model (FBD) in Automatic Speech Segmentation, and how it detects discontinuities in the voice signal. It illustrates in the example below how it is possible to enlarge some segments of the speech (vowels enlargement for instance). To realize this result, it is possible to visually and acoustically perceive the stationary segments of the speech signal.</p>153553Thu, 17 Apr 2014 04:00:00 ZJocelyn MagneJocelyn MagneThe House Warming Model
http://www.maplesoft.com/applications/view.aspx?SID=153491&ref=Feed
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 MonaganCollision detection between toolholder and workpiece on ball nut grinding
http://www.maplesoft.com/applications/view.aspx?SID=153477&ref=Feed
<p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p><img src="/view.aspx?si=153477/Collision_Detection_image1.jpg" alt="Collision detection between toolholder and workpiece on ball nut grinding" align="left"/><p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p>153477Mon, 23 Dec 2013 05:00:00 ZGyörgy HegedûsGyörgy HegedûsClassroom Tips and Techniques: Mathematical Thoughts on the Root Locus
http://www.maplesoft.com/applications/view.aspx?SID=153452&ref=Feed
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: Gems 31-35 from the Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=147092&ref=Feed
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 LopezClassroom Tips and Techniques: Least-Squares Fits
http://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="/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" 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>140942Wed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezHardening of Aluminium Alloy AA 7075 T 7351
http://www.maplesoft.com/applications/view.aspx?SID=140361&ref=Feed
<p><span id="ctl00_mainContent__documentViewer"><span><span class="body summary">This worksheet is concerned with the hardening of aluminium alloy, the behaviour of which can be expressed by a simple power law with two hardening parameters. Based upon experimental data these parameters have been determined by both a linear regrssion and the nonlinear <em>MARQUARDT-LEVENBERG algorithm.</em></span></span></span></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Hardening of Aluminium Alloy AA 7075 T 7351" align="left"/><p><span id="ctl00_mainContent__documentViewer"><span><span class="body summary">This worksheet is concerned with the hardening of aluminium alloy, the behaviour of which can be expressed by a simple power law with two hardening parameters. Based upon experimental data these parameters have been determined by both a linear regrssion and the nonlinear <em>MARQUARDT-LEVENBERG algorithm.</em></span></span></span></p>140361Wed, 14 Nov 2012 05:00:00 ZJosef BettenJosef BettenLAGRANGE Interpolation
http://www.maplesoft.com/applications/view.aspx?SID=139076&ref=Feed
<p>Zur Aufstellung von Interpolationspolynomen n - ten Grades bei n + 1 gegebenen Messpunkten wird die LAGRANGEsche Interpolationsmethode ausführlich diskutiert und mit anderen Verfahren verglichen. Als numerisches Beispiel wird ein Polynom fünften Grades mut sechs Stützpunkten konstruiert.</p>
<p><em>Keywords: </em>LAGRANGEsche Grundfunktionen, Determinantenmethode, lineare, kubische und Splinefunktionen fünften und höheren Grades, Glättung von Interpolationspolynomen hohen Grades, Formfunktionen vom LAGRANGEschen Typ für finite Elemente</p><img src="/view.aspx?si=139076/b14d4036a98db9e3dc88c1915625b142.gif" alt="LAGRANGE Interpolation" align="left"/><p>Zur Aufstellung von Interpolationspolynomen n - ten Grades bei n + 1 gegebenen Messpunkten wird die LAGRANGEsche Interpolationsmethode ausführlich diskutiert und mit anderen Verfahren verglichen. Als numerisches Beispiel wird ein Polynom fünften Grades mut sechs Stützpunkten konstruiert.</p>
<p><em>Keywords: </em>LAGRANGEsche Grundfunktionen, Determinantenmethode, lineare, kubische und Splinefunktionen fünften und höheren Grades, Glättung von Interpolationspolynomen hohen Grades, Formfunktionen vom LAGRANGEschen Typ für finite Elemente</p>139076Thu, 01 Nov 2012 04:00:00 ZJosef BettenJosef BettenClassroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package
http://www.maplesoft.com/applications/view.aspx?SID=134198&ref=Feed
The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.<img src="/view.aspx?si=134198/thumb.jpg" alt="Classroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package" align="left"/>The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.134198Mon, 14 May 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezParameterizing Motion along a Curve
http://www.maplesoft.com/applications/view.aspx?SID=130465&ref=Feed
<p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" align="left"/><p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>130465Wed, 08 Feb 2012 05:00:00 ZShawn HedmanShawn HedmanClassroom Tips and Techniques: An Undamped Coupled Oscillator
http://www.maplesoft.com/applications/view.aspx?SID=129521&ref=Feed
<p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p><img src="/view.aspx?si=129521/thumb.jpg" alt="Classroom Tips and Techniques: An Undamped Coupled Oscillator" align="left"/><p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p>129521Tue, 10 Jan 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezDamage Effective Stress Concepts
http://www.maplesoft.com/applications/view.aspx?SID=129456&ref=Feed
<p>During the last two or three decades many scientists have devoted much effort to the stress analysis in a damaged material, and the notation <em>damage effective stress </em>has been introduced. </p>
<p>In the following some various <em>damage effective stress concepts </em>should be reviewed.</p><img src="/view.aspx?si=129456/428320\12aff582b7c2a4ef667abfccba992187.gif" alt="Damage Effective Stress Concepts" align="left"/><p>During the last two or three decades many scientists have devoted much effort to the stress analysis in a damaged material, and the notation <em>damage effective stress </em>has been introduced. </p>
<p>In the following some various <em>damage effective stress concepts </em>should be reviewed.</p>129456Sun, 08 Jan 2012 05:00:00 ZProf. Josef BettenProf. Josef BettenSimple Harmonic Motion
http://www.maplesoft.com/applications/view.aspx?SID=87640&ref=Feed
<p>The aim of this topic is to visualize the motion of a simple pendulum which consist of a small mass m
suspended by a light inextensible cord of length L from a fixed support.
</p><img src="/view.aspx?si=87640/harmonic_sm.png" alt="Simple Harmonic Motion" align="left"/><p>The aim of this topic is to visualize the motion of a simple pendulum which consist of a small mass m
suspended by a light inextensible cord of length L from a fixed support.
</p>87640Fri, 09 Dec 2011 05:00:00 ZAli Abu OamAli Abu OamClassroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem
http://www.maplesoft.com/applications/view.aspx?SID=128444&ref=Feed
<p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A - lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>-1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p><img src="/view.aspx?si=128444/thumb.jpg" alt="Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem" align="left"/><p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A - lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>-1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p>128444Tue, 06 Dec 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezThe Origin of Complex Numbers
http://www.maplesoft.com/applications/view.aspx?SID=126618&ref=Feed
The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.<img src="/applications/images/app_image_blank_lg.jpg" alt="The Origin of Complex Numbers" align="left"/>The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.126618Fri, 14 Oct 2011 04:00:00 ZDr. John MathewsDr. John Mathews