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<!DOCTYPE html> <html lang="en"> <head> <title>heart -- an 8-dimensional economics problem</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="examples of polynomial systems" href="index.html">ExampleSystems</a> :: <a title="an 8-dimensional economics problem" href="_heart.html">heart</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_ipp.html">next</a> | <a href="_geneig.html">previous</a> | <a href="_ipp.html">forward</a> | <a href="_geneig.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>heart -- an 8-dimensional economics problem</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">heart(kk)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">kk</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, the coefficient ring</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, of the polynomials in the system</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This system was solved in May 2020, using <a title="solve a system of polynomial equations" href="../../NumericalAlgebraicGeometry/html/_solve__System_lp__List_rp.html">solveSystem</a> in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.</p> <p>There were 2 solutions found in 3.356 seconds (with a Bezout bound of 1458).</p> <p>Reference: "Determination of magnitudes, directions, and locations of two independent dipoles in a circular conducting region from boundary potential measurements" by C.V. Nelson and B.C. Hodgkin (pages 817-823), "Coefficient-Parameter Polynomial Continuation" by A.P. Morgan and A.J. Sommese (pages 123-160), "Mathematical reduction of a heart dipole model" by A.P. Morgan and A. Sommese and L.T. Watson (pages 407-410).</p> <p>See also: http://homepages.math.uic.edu/~jan/Demo/heart.html.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : heart(RR_53) o1 = {a + b - .63254, c + d + 1.34534, a*t + b*u - c*v - d*w + .836535, c*t + ------------------------------------------------------------------------ d*u + a*v + b*w - 1.73453, | -2ctv-2duw+2at+2bu-2av-2bw-1.35235 |, | ------------------------------------------------------------------------ 2atv+2buw+2ct+2du-2cv-2dw+.843453 |, | ------------------------------------------------------------------------ -6atv-6ctv-6buw-6duw+3at+3bu+3cv+3dw+.956345 |, | ------------------------------------------------------------------------ 6atv-6ctv+6buw-6duw+3ct+3du-3av-3bw-1.23425 |} o1 : List</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">heart</span>:</h2> <ul> <li><kbd>heart(Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="an 8-dimensional economics problem" href="_heart.html">heart</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">ExampleSystems/heart.m2:48:0</span>.</p> </div> </div> </div> </body> </html>
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