Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/modules/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 22.11.2024 mit Größe 12 kB image not shown  

Quelle  chap12_mj.html   Sprache: HTML

 
 products/sources/formale Sprachen/GAP/pkg/modules/doc/chap12_mj.html


<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (Modules) - Chapter 12: Exterior Algebra and Koszul Complex</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap12"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapC_mj.html">C</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap11_mj.html">[Previous Chapter]</a>    <a href="chap13_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap12.html">[MathJax off]</a></p>
<p><a id="X7BD010F3847B274E" name="X7BD010F3847B274E"></a></p>
<div class="ChapSects"><a href="chap12_mj.html#X7BD010F3847B274E">12 <span class="Heading">Exterior Algebra and Koszul Complex</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7A005D4E870C281D">12.1 <span class="Heading">Exterior Algebra: Constructor</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X787BB7FF85F0AD68">12.1-1 ExteriorPower</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7E09B9C5844FC31E">12.2 <span class="Heading">Exterior Algebra: Properties and Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X79C5FE077B58DF82">12.2-1 IsExteriorPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X87CF59278702A550">12.2-2 ExteriorPowerExponent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X8282D0D7800F63CC">12.2-3 ExteriorPowerBaseModule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7A2AC54B87C85695">12.3 <span class="Heading">Exterior Algebra: Element Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7FC4A5DC7B592D04">12.3-1 IsExteriorPowerElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X80D7B36379182854">12.4 <span class="Heading">Exterior Algebra: Element Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7C71C3C77F2E225D">12.4-1 Wedge</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X8236B4167E79F186">12.4-2 ExteriorPowerElementDual</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X85EDBA2783A1E984">12.4-3 SingleValueOfExteriorPowerElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X8050EFB77A600595">12.5 <span class="Heading">Koszul complex and Cayley determinant</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7D84C7AC809B453F">12.5-1 KoszulCocomplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X794C601787143D2D">12.5-2 CayleyDeterminant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12_mj.html#X7C72190C8331FADD">12.5-3 Gcd_UsingCayleyDeterminant</a></span>
</div></div>
</div>

<h3>12 <span class="Heading">Exterior Algebra and Koszul Complex</span></h3>

<p>What follows are several operations related to the exterior algebra of a free module:</p>


<ul>
<li><p>A constructor for the graded parts of the exterior algebra (<q>exterior powers</q>)</p>

</li>
<li><p>Several Operations on elements of these exterior powers</p>

</li>
<li><p>A constructor for the <q>Koszul complex</q></p>

</li>
<li><p>An implementation of the <q>Cayley determinant</q> as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>, which allows calculating greatest common divisors from finite free resolutions.</p>

</li>
</ul>
<p><a id="X7A005D4E870C281D" name="X7A005D4E870C281D"></a></p>

<h4>12.1 <span class="Heading">Exterior Algebra: Constructor</span></h4>

<p><a id="X787BB7FF85F0AD68" name="X787BB7FF85F0AD68"></a></p>

<h5>12.1-1 ExteriorPower</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPower</code>( <var class="Arg">k</var>, <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> module</p>

<p>Construct the <var class="Arg">k</var>-th exterior power of module <var class="Arg">M</var>.</p>

<p><a id="X7E09B9C5844FC31E" name="X7E09B9C5844FC31E"></a></p>

<h4>12.2 <span class="Heading">Exterior Algebra: Properties and Attributes</span></h4>

<p><a id="X79C5FE077B58DF82" name="X79C5FE077B58DF82"></a></p>

<h5>12.2-1 IsExteriorPower</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsExteriorPower</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>Marks a module as an exterior power of another module.</p>

<p><a id="X87CF59278702A550" name="X87CF59278702A550"></a></p>

<h5>12.2-2 ExteriorPowerExponent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPowerExponent</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an integer</p>

<p>The exponent of the exterior power.</p>

<p><a id="X8282D0D7800F63CC" name="X8282D0D7800F63CC"></a></p>

<h5>12.2-3 ExteriorPowerBaseModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPowerBaseModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg module</p>

<p>The module that <var class="Arg">M</var> is an exterior power of.</p>

<p><a id="X7A2AC54B87C85695" name="X7A2AC54B87C85695"></a></p>

<h4>12.3 <span class="Heading">Exterior Algebra: Element Properties</span></h4>

<p><a id="X7FC4A5DC7B592D04" name="X7FC4A5DC7B592D04"></a></p>

<h5>12.3-1 IsExteriorPowerElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsExteriorPowerElement</code>( <var class="Arg">x</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>Checks if the element <var class="Arg">x</var> is from an exterior power.</p>

<p><a id="X80D7B36379182854" name="X80D7B36379182854"></a></p>

<h4>12.4 <span class="Heading">Exterior Algebra: Element Operations</span></h4>

<p><a id="X7C71C3C77F2E225D" name="X7C71C3C77F2E225D"></a></p>

<h5>12.4-1 Wedge</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Wedge</code>( <var class="Arg">x</var>, <var class="Arg">y</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an element of an exterior power</p>

<p>Calculate <span class="SimpleMath">\(\textit{x} \wedge \textit{y}\)</span>.</p>

<p><a id="X8236B4167E79F186" name="X8236B4167E79F186"></a></p>

<h5>12.4-2 ExteriorPowerElementDual</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExteriorPowerElementDual</code>( <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an element of an exterior power</p>

<p>For <var class="Arg">x</var> in a q-th exterior power of a free module of rank n, return <span class="SimpleMath">\(\textit{x}*\)</span> in the (n-q)-th exterior power, as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>.</p>

<p><a id="X85EDBA2783A1E984" name="X85EDBA2783A1E984"></a></p>

<h5>12.4-3 SingleValueOfExteriorPowerElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SingleValueOfExteriorPowerElement</code>( <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a ring element</p>

<p>For <var class="Arg">x</var> in a highest exterior power, returns its single coordinate in the canonical basis; i.e. <span class="SimpleMath">\([\textit{x}]\)</span> as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>.</p>

<p><a id="X8050EFB77A600595" name="X8050EFB77A600595"></a></p>

<h4>12.5 <span class="Heading">Koszul complex and Cayley determinant</span></h4>

<p><a id="X7D84C7AC809B453F" name="X7D84C7AC809B453F"></a></p>

<h5>12.5-1 KoszulCocomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KoszulCocomplex</code>( <var class="Arg">a</var>, <var class="Arg">E</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> cocomplex</p>

<p>Calculate the <var class="Arg">E</var>-valued Koszul complex of <var class="Arg">a</var>.</p>

<p><a id="X794C601787143D2D" name="X794C601787143D2D"></a></p>

<h5>12.5-2 CayleyDeterminant</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CayleyDeterminant</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a ring element</p>

<p>Calculate the Cayley determinant of the complex <var class="Arg">C</var>, as defined in <a href="chapBib_mj.html#biBCQ11">[CQ11]</a>.</p>

<p><a id="X7C72190C8331FADD" name="X7C72190C8331FADD"></a></p>

<h5>12.5-3 Gcd_UsingCayleyDeterminant</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gcd_UsingCayleyDeterminant</code>( <var class="Arg">x</var>, <var class="Arg">y</var>[, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a ring element</p>

<p>Returns the greatest common divisor of the given ring elements, calculated using the Cayley determinant.</p>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap11_mj.html">[Previous Chapter]</a>    <a href="chap13_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapC_mj.html">C</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

100%


¤ Dauer der Verarbeitung: 0.18 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.