Gedichte Musik Bilder Quellcodebibliothek Diashow Normaldarstellung
Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/cddinterface/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 24.5.2025 mit Größe 20 kB image not shown  

Quelle  chap4_mj.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/cddinterface/doc/chap4_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 (CddInterface) - Chapter 4: Attributes and properties</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="chap4"  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="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="chap3_mj.html">[Previous Chapter]</a>    <a href="chapInd_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap4.html">[MathJax off]</a></p>
<p><a id="X7DC480E57D26429A" name="X7DC480E57D26429A"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X7DC480E57D26429A">4 <span class="Heading">Attributes and properties</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7A2BE11B87B4A521">4.1 <span class="Heading">Attributes and properties of polyhedron</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8385F69A87131EAD">4.1-1 Cdd_Canonicalize</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85E888C97D4E105B">4.1-2 Cdd_V_Rep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8414F54E80B312C5">4.1-3 Cdd_H_Rep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CF2AEA1810DCDB9">4.1-4 Cdd_AmbientSpaceDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81C260A1830AB7D4">4.1-5 Cdd_Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82FBBDE082F8D36E">4.1-6 Cdd_GeneratingVertices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E50781780D8F8EB">4.1-7 Cdd_GeneratingRays</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A31F87C7903D209">4.1-8 Cdd_Equalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86B3B0468420F573">4.1-9 Cdd_Inequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X869C857285945057">4.1-10 Cdd_InteriorPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8040DAA2872555F5">4.1-11 Cdd_Faces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X84B3FAAE7ECF5ACF">4.1-12 Cdd_FacesWithFixedDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X872EC7F186282090">4.1-13 Cdd_FacesWithInteriorPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DE87B0984B3F177">4.1-14 Cdd_FacesWithFixedDimensionAndInteriorPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B448A2578CB679F">4.1-15 Cdd_Facets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F1FB2B17B561A45">4.1-16 Cdd_Lines</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7FABEAFE790C7ABD">4.1-17 Cdd_Vertices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D4B6C2F7E71756C">4.1-18 Cdd_IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79FC1D7D7C266683">4.1-19 Cdd_IsCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80784F1B7FE8419D">4.1-20 Cdd_IsPointed</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Attributes and properties</span></h3>

<p><a id="X7A2BE11B87B4A521" name="X7A2BE11B87B4A521"></a></p>

<h4>4.1 <span class="Heading">Attributes and properties of polyhedron</span></h4>

<p><a id="X8385F69A87131EAD" name="X8385F69A87131EAD"></a></p>

<h5>4.1-1 Cdd_Canonicalize</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Canonicalize</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a CddPolyhedron</p>

<p>The function takes a polyhedron and reduces its defining inequalities (generators set) by deleting all redundant inequalities (generators).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Cdd_PolyhedronByInequalities( [ [ 0, 2, 6 ], [ 0, 1, 3 ], [1, 4, 10 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Cdd_Canonicalize( A );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( B );</span>
H-representation
begin
   2 X 3  rational

   0   1   3
   1   4  10
end
</pre></div>

<p><a id="X85E888C97D4E105B" name="X85E888C97D4E105B"></a></p>

<h5>4.1-2 Cdd_V_Rep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_V_Rep</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a CddPolyhedron</p>

<p>The function takes a polyhedron and returns its reduced <span class="SimpleMath">\(V\)</span>-representation.</p>

<p><a id="X8414F54E80B312C5" name="X8414F54E80B312C5"></a></p>

<h5>4.1-3 Cdd_H_Rep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_H_Rep</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a CddPolyhedron</p>

<p>The function takes a polyhedron and returns its reduced <span class="SimpleMath">\(H\)</span>-representation.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1 ], [ 0, 5, 5 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Cdd_V_Rep( A );</span>
<Polyhedron given by its V-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( B );</span>
V-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   0   1   0
   0  -1   1
end
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= Cdd_H_Rep( B );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span>
H-representation
begin
   1 X 3  rational

   0  1  1
end
<span class="GAPprompt">gap></span> <span class="GAPinput">D:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1, 34, 22, 43 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 11, 2, 2, 54, 53, 221 ], [33, 23, 45, 2, 40, 11 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_V_Rep( D );</span>
<Polyhedron given by its V-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( last );</span>
V-representation
linearity 2, [ 5, 6 ]
begin
   6 X 6  rational

   1  -743/14   369/14    11/14        0        0
   0    -1213      619       22        0        0
   0       -1        1        0        0        0
   0      764     -390      -11        0        0
   0   -13526     6772       99      154        0
   0  -116608    59496     1485        0      154
end
</pre></div>

<p><a id="X7CF2AEA1810DCDB9" name="X7CF2AEA1810DCDB9"></a></p>

<h5>4.1-4 Cdd_AmbientSpaceDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_AmbientSpaceDimension</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The dimension of the ambient space of the polyhedron(i.e., the space that contains <span class="SimpleMath">\(P\)</span>).</p>

<p><a id="X81C260A1830AB7D4" name="X81C260A1830AB7D4"></a></p>

<h5>4.1-5 Cdd_Dimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Dimension</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The dimension of the polyhedron, where the dimension, <span class="SimpleMath">\(\mathrm{dim}(P)\)</span>, of a polyhedron <span class="SimpleMath">\(P\)</span> is the maximum number of affinely independent points in <span class="SimpleMath">\(P\)</span> minus 1.</p>

<p><a id="X82FBBDE082F8D36E" name="X82FBBDE082F8D36E"></a></p>

<h5>4.1-6 Cdd_GeneratingVertices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_GeneratingVertices</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The reduced generating vertices of the polyhedron</p>

<p><a id="X7E50781780D8F8EB" name="X7E50781780D8F8EB"></a></p>

<h5>4.1-7 Cdd_GeneratingRays</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_GeneratingRays</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: list</p>

<p>The output is the reduced generating rays of the polyhedron</p>

<p><a id="X7A31F87C7903D209" name="X7A31F87C7903D209"></a></p>

<h5>4.1-8 Cdd_Equalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Equalities</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>The output is the reduced equalities of the polyhedron.</p>

<p><a id="X86B3B0468420F573" name="X86B3B0468420F573"></a></p>

<h5>4.1-9 Cdd_Inequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Inequalities</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The output is the reduced inequalities of the polyhedron.</p>

<p><a id="X869C857285945057" name="X869C857285945057"></a></p>

<h5>4.1-10 Cdd_InteriorPoint</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_InteriorPoint</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>The output is an interior point in the polyhedron</p>

<p><a id="X8040DAA2872555F5" name="X8040DAA2872555F5"></a></p>

<h5>4.1-11 Cdd_Faces</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Faces</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this list is a again a list, contains the dimension and linearity of the face defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X84B3FAAE7ECF5ACF" name="X84B3FAAE7ECF5ACF"></a></p>

<h5>4.1-12 Cdd_FacesWithFixedDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_FacesWithFixedDimension</code>( <var class="Arg">P</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and a positive integer <em>d</em>. The output is a list. Every entry in this list is the linearity of an <em>d</em>- dimensional face of <em>P</em> defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X872EC7F186282090" name="X872EC7F186282090"></a></p>

<h5>4.1-13 Cdd_FacesWithInteriorPoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_FacesWithInteriorPoints</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this list is a again a list, contains the dimension, linearity of the face defined as a polyhedron over the same system of inequalities and an interior point in the face.</p>

<p><a id="X7DE87B0984B3F177" name="X7DE87B0984B3F177"></a></p>

<h5>4.1-14 Cdd_FacesWithFixedDimensionAndInteriorPoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_FacesWithFixedDimensionAndInteriorPoints</code>( <var class="Arg">P</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and a positive integer <em>d</em>. The output is a list. Every entry in this list is a again a list, contains the linearity of the face defined as a polyhedron over the same system of inequalities and an interior point in this face.</p>

<p><a id="X7B448A2578CB679F" name="X7B448A2578CB679F"></a></p>

<h5>4.1-15 Cdd_Facets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Facets</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this is the linearity of a facet defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X7F1FB2B17B561A45" name="X7F1FB2B17B561A45"></a></p>

<h5>4.1-16 Cdd_Lines</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Lines</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this is the linearity of a ray (<span class="SimpleMath">\(1\)</span>-dimensional face) defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X7FABEAFE790C7ABD" name="X7FABEAFE790C7ABD"></a></p>

<h5>4.1-17 Cdd_Vertices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Vertices</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this list is the linearity of a vertex defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X7D4B6C2F7E71756C" name="X7D4B6C2F7E71756C"></a></p>

<h5>4.1-18 Cdd_IsEmpty</h5>

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

<p>The output is <code class="code">true</code> if the polyhedron is empty and <code class="code">false</code> otherwise</p>

<p><a id="X79FC1D7D7C266683" name="X79FC1D7D7C266683"></a></p>

<h5>4.1-19 Cdd_IsCone</h5>

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

<p>The output is <code class="code">true</code> if the polyhedron is cone and <code class="code">false</code> otherwise</p>

<p><a id="X80784F1B7FE8419D" name="X80784F1B7FE8419D"></a></p>

<h5>4.1-20 Cdd_IsPointed</h5>

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

<p>The output is <code class="code">true</code> if the polyhedron is pointed and <code class="code">false</code> otherwise</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">poly:= Cdd_PolyhedronByInequalities( [ [ 1, 3, 4, 5, 7 ], [ 1, 3, 5, 12, 34 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 9, 3, 0, 2, 13 ]  ], [ 1 ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_InteriorPoint( poly );</span>
[ -194/75, 46/25, -3/25, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_FacesWithInteriorPoints( poly );</span>
[ [ 3, [ 1 ], [ -194/75, 46/25, -3/25, 0 ] ], [ 2, [ 1, 2 ],
[ -62/25, 49/25, -7/25, 0 ] ], [ 1, [ 1, 2, 3 ],
[ -209/75, 56/25, -8/25, 0 ] ], [ 2, [ 1, 3 ], [ -217/75, 53/25, -4/25, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Dimension( poly );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_IsPointed( poly );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_IsEmpty( poly );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Faces( poly );</span>
[ [ 3, [ 1 ] ], [ 2, [ 1, 2 ] ], [ 1, [ 1, 2, 3 ] ], [ 2, [ 1, 3  ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly1 := Cdd_ExtendLinearity( poly, [ 1, 2, 3 ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( poly1 );</span>
H-representation 
linearity 3, [ 1, 2, 3 ]
begin
   3 X 5  rational

   1   3   4   5   7 
   1   3   5  12  34 
   9   3   0   2  13 
end
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Dimension( poly1 );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Facets( poly );</span>
[ [ 1, 2 ], [ 1, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_GeneratingVertices( poly );</span>
[ [ -209/75, 56/25, -8/25, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_GeneratingRays( poly );</span>
[ [ -97, 369, -342, 75 ], [ -8, -9, 12, 0 ],
[ 23, -21, 3, 0 ], [ 97, -369, 342, -75 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Inequalities( poly );</span>
[ [ 1, 3, 5, 12, 34 ], [ 9, 3, 0, 2, 13 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Equalities( poly );</span>
[ [ 1, 3, 4, 5, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">P := Cdd_FourierProjection( poly, 2);</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( P );</span>
H-representation 
linearity 1, [ 3 ]
begin 
   3 X 5  rational

    9    3    0    2   13 
   -1   -3    0   23  101 
    0    0    1    0    0 
end
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap3_mj.html">[Previous Chapter]</a>    <a href="chapInd_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="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>

98%


¤ Dauer der Verarbeitung: 0.11 Sekunden  ¤

*© 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.