Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/fining/doc/   (GAP Algebra Version 4.15.1©)  Datei vom 27.6.2023 mit Größe 45 kB image not shown  

Quelle  chap14.html   Sprache: HTML

 

 products/Sources/formale Sprachen/GAP/pkg/fining/doc/chap14.html


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

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (FinInG) - Chapter 14: Subgeometries of projective spaces</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<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="chap14"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chapA.html">A

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap13.html">[Previous Chapter]</a>    <a href="chapA.html"of at A/>fromA<> ajava.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 56

<p id="mathjaxlink" class="pcenter"><a href="chap14_mj.html">[MathJax on]</a></p>
<p><a id="X79A3CC6E85E72EC1" name="X79A3CC6E85E72EC1"></a></p>
<div class="ChapSects"><a href="chap14.html#X79A3CC6E85E72EC1">14 <span class="Heading">Subgeometries of projective(ee <Func="GeneralizedReedSolomonCode""" 
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7D237EAD8797C140">14.1 <span class="Heading">Particular Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> A>r/A  be a<java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 43
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7D908542820D2FBE">14.1-2 <span class="Heading">Categories for elements and java.lang.StringIndexOutOfBoundsException: Index 172 out of bounds for length 10
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X79A3CC6E85E72EC1">14.2 <span class="Heading">Subgeometries of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8204DC137E1B54B6">14.2-1 CanonicalSubgeometryOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X81EB7C0B838B53B2">14.2-2 RandomFrameOfProjectiveSpace/Description>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X824D4BA67AE1923B">14.2-3 IsFrameOfProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7A88800C80667E22">14.2-4 java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 13
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X82EB5BE77F9F686A">14.3 <span class="Heading">Basic operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8058FF3479158445">14.3-1 <span class="Heading">Underlying vector space and ambient projective space</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7DC6B3E57C74A153">14Example>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X79CC2F0483575105">14.3-3 <span class="Heading">Projective dimension and rank</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X85437D577DE97AEF">14.3-4 <span java.lang.StringIndexOutOfBoundsException: Range [0, 124) out of bounds for length 15
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8550F38F7FB123DB">14.3-5 CollineationFixingSubgeometry</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7836EC02824B9425">14.4 <span class="Heading">Constructing elements of a subgeometry</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"gap> a:=rimitiveRootF; a^; b+b3+; 
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X78219A547B858E2E">14.4-2 ExtendElementOfSubgeometry</a></span>
<span class="ContSS"><br /><span class="nocss">  </0*(java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7B1757048405DD29">14.4-4 <span class="Heading">Flags</span></a>
</span>
</div></div>
<div class=ist([0.]i>^i);
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X78F858C8863C7721">14.5-1 <span class="Heading">Groups of collineations</span></a>
</span>
</div></div>
</div>

<h3>14 <span class="Heading">Subgeometries of projective spaces</span></h3>

<p>Let <span class="SimpleMath">S=(P,L,I)</span> be a point-line incidence geometry. In case <span class="SimpleMath">S</span> is a projective space over a finite field, it is clear that every line (and every subspace as well) can be identified with the set of points incident with it. Furthermore, the incidence relation <span class="SimpleMath">I</span> is then symmetrised containment. To define a subgeometry mathematically, we follow <a href="chapBib.html#biBDembowski68">[Dem97]</a>. Let <span class="SimpleMath">P' ⊂ P</span> and let <span class="SimpleMath">L'</span> be a set of subsets of <span class="SimpleMath">P'</span>, such that every <span class="SimpleMath">l' ∈ L'</span> is a subset of exactly one line <span class="SimpleMath">l ∈ L</span>. If <span class="SimpleMath">S'=(P',L',I)</span> is a projective space again, then we call <span class="SimpleMath">S'</span> a <em>subgeometry</em> of <span class="SimpleMath">S</span>. Note java.lang.StringIndexOutOfBoundsException: Index 954 out of bounds for length 71

<p>A typical example of a subgeometry is a Baer subplane of a projective plane. In this example, with <span class="SimpleMath">S'</span> the Baer subplane of the projective plane <span class="SimpleMath">S</span>, one could say that a pointgap>R2=(,,];

<p>The same considerations apply for classical polar spaces. These consideration have implications for the behaviour of certain operations in <strong class="pkg">FinInG</strong>, e.g. when computing the span and meet of different elements.</p>

<p>Using geometry morphisms, and more particular a function like <code class="file">java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 70

<p>A subgeometry in a projective space is completely determined by a frame of the projective space and a subfield of the base field of the projective space. The <em>standard frame</em> in an <span class="SimpleMath">n</span>-dimensional projective space <span class="SimpleMath">PG(n,q)</span> is the set of <span class="SimpleMath">n+2</span> points represented by <span class="SimpleMath">(1,0,...,0),(0,1,...,0),...,(0,0,...,1),(1,1,...,1)</span>. The subgeometry determined by the standard frame will be called <em>canonical</em>. Note that different frames may determine the same subgeometry (over a fixed subfield).</p>

<p>For a given subfield <span class="SimpleMath">GF(q') ⊂ GF(q)</span>, the canonical subgeometry determined by the standard frame in <span class="SimpleMath">PG(n,q)</span> is mathematically spoken the image of the <strong class="pkg">FinInG</strong> geometry morphism <code java.lang.StringIndexOutOfBoundsException: Index 281 out of bounds for length 56

<p>For an arbitrary frame of <span class="SimpleMath">PG(n,q)</span> and a subfield <span class="SimpleMath">GF(q')</span>, there exists a natural collineation of <span class="SimpleMath">PG(n,q)</span> which fixes the subgeometry pointwise. This collineation is the conjugation of the Frobenius automorphism by the unique collineation mapping the defining frame of the subgeometry to the standard frame of <span class="SimpleMath">PG(n,q)</span>, i.e. the frame defining the canonical subgeometry over <span class="SimpleMath">GF(q')</span>. Upon construction of a subgeometry, both collineations will be computed, and are of use when dealing with the full collineation group of a subgeometry. As for any incidence geometry in <strong class="pkg">FinInG</strong>, operations to compute this collineation group as well as particular action functions for subgeometries are provided.</p>

<p>Subgeometries of projective spacesgap> =eneralizedReedSolomonListDecoder(,r2 time;Uncommentthe to ";"tosee 

<p><a id="X7D237EAD8797C140" name="X7D237EAD8797C140"></a></p>

<h4>14.1 <span class="Heading">Particular Categories</span></h4>

<p><a id="X84BB6A0D7DABB9EB"[[  a^9a^ ^13a^ ^ ^  a12a^ ^ ^8a^7 ^ ^ , 

<h5>14.1-1 IsSubgeometryOfProjectiveSpace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSubgeometryOfProjectiveSpace</code></td><td class="tdright">( category )</td></tr></table></div>
<p>This[00  0  0    0  00 0   ]

<p><a id="X7D908542820D2FBE" name="X7D908542820D2FBE"></a></p>

<h5>14.1-2 <span class="Heading">Categories for elements and collections of elements</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSubspaceOfSubgeometryOfProjectiveSpace</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSubspacesOfSubgeometryOfProjectiveSpace</code></td><td class="tdright">( category )</td></tr></table></div>
<p>A subspace of a subgeometry belongs to the category <code class="code">IsSubspaceOfSubgeometryOfProjectiveSpace</code>.</p>

<p><a id="X79A3CC6E85E72EC1" name="X79A3CC6E85E72EC1"></a></p>

<h4>14.2 <span class="Heading">Subgeometries of projective spaces</span></h4>

<p><a id="X8204DC137E1B54B6" name="X8204DC137E1B54B6"></a></p>

<h5>14.2-1 CanonicalSubgeometryOfProjectiveSpace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft">rue
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CanonicalSubgeometryOfProjectiveSpace</code>( <var class="Arg">pg</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )>=2]c2;
<p>Returns: a subgeometry of <var class="Arg">pg</var></p>

<p>This operation returns the subgeometry of <var java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 33


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(2,25);</span>
ProjectiveSpace(2, 25)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := CanonicalSubgeometryOfProjectiveSpace(pg,GF(5));</span>
Subgeometry PG(2, 5) of ProjectiveSpace(2, 25)
<span class="GAPprompt">gap></span> <span class="GAPinput">CategoriesOfObject(sub);</span>
[ "IsIncidenceStructure", "IsIncidenceGeometry", "IsLieGeometry", 
  "IsProjectiveSpace", "IsSubgeometryOfProjectiveSpace" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(3,3^6);</span>
ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := CanonicalSubgeometryOfProjectiveSpace(pg,3^2);</span>
Subgeometry PG(3, 9) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 1
Subgeometry PG(3, 27) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := CanonicalSubgeometryOfProjectiveSpace(pg,3^6);</span>
rojectiveSpace(3, 729)
 
</pre></div>

<p><a id="X81EB7C0B838B53B2" name="X81EB7C0B838B53B2"></a></p>

<h5>14.2-2 RandomFrameOfProjectiveSpace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomFrameOfProjectiveSpace</code>( <var class="Arg">pg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a set of points of <var class="Arg">pg</var>, being a frame. Note that the returned object is also a set in the GAP sense, i.e. an ordered list without duplicates.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(1,5^5);</span>
ProjectiveSpace(1, 3125)
<span class="GAPprompt">gap></span> <span class="GAPinput">frame := RandomFrameOfProjectiveSpace(pg);</span>
[ <a point in ProjectiveSpace(1, 3125)>, <a point in ProjectiveSpace(1, 3125)>
    , <a point in ProjectiveSpace(1, 3125)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(frame);</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(6,2^2);</span>
ProjectiveSpace(6, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">frame := RandomFrameOfProjectiveSpace(pg);</span>
[<point 6,4>, &;point in ProjectiveSpace(6, 4)>, 
  <a point in java.lang.StringIndexOutOfBoundsException: Range [74, 32) out of bounds for length 86
  <a point in ProjectiveSpace(6, 4)>, <a point in ProjectiveSpace(6, 4)>, 
  <a point in ProjectiveSpace(6, 4)>, <a point in ProjectiveSpace(6, 4)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(frame);</span>
8
 
</pre></div>

<p><a id="X824D4BA67AE1923B" name="X824D4BA67AE1923B"></a></p>

<h514.- </

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsFrameOfProjectiveSpace</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: true or false</p>

<p>When <var class="Arg">list</var> is a list of points of a projective space, this operation returns true if and only if <var class="Arg">list</var> constitutes a frame of the projective space. It is checked as well whether all points in <var class="Arg">list</var> belong to the same projective space.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(1,7^3);</span>
ProjectiveSpace(1, 343)
<span class="GAPprompt">gap></span> <span class="GAPinput">p1 := java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 42
<a point in ProjectiveSpace(1, 343)>
<span class="GAPprompt">gap></span> <span class="GAPinput">p2 := VectorSpaceToElement(pg,[1,2]*Z(7)^0);</span>
&t;apoint in (,343)&;
<span class="GAPprompt">gap></span> <span class="GAPinput">p3 := VectorSpaceToElement(pg,[1,3]*Z(7)^0);</span>
<a point in ProjectiveSpace(1, 343)>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFrameOfProjectiveSpace([p1,p2,p3]);</span>
true
 
<pre>/>

<p><a id="X7A88800C80667E22" name="X7A88800C80667E22"></a></p>

<h5>14.2-4 SubgeometryOfProjectiveSpaceByFrame</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SubgeometryOfProjectiveSpaceByFrame</code>( <var class="Arg">pg</var>, <var class="Arg">list</var>, <var class="Arg">field</var> )</td><td class="tdright"C:GeneralizedReedSolomonCode(,,;MinimumDistance()
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><z:=Zero(F);
<p>Returns: a subgeometry of <var class="Arg">pg</var></p>

<p>The argument <var class="Arg">pg</var> is a projective space which is not a subgeometry itself, the argument <var class="Arg">list</var> is a list of points of <var class="Arg">pg</var> defining a frame of <var class="Arg">pg</var>, and finally the argument <var class="Arg">field</var> is a subfield of the base field of <var class="Arg">pg</var>. Alternatively, the argument <var class="Arg">q</var> is the order of a subfield of the base field of <var class="Arg">pg</var>. This method returns the subgeometry defined by the frame in <var class="Arg">list</var> and the subfield <var class="Arg">field</var> of the subfield <span class="SimpleMath">GF(<var class="Arg">q</var>)</span>. This method checks whether the subfield <var class="Arg">field</var> or the field <span class


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(3,3^6);</span>
ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">frameWeightCodewordc1-;
[ <a point in ProjectiveSpace(3, 729)>, <a point in ProjectiveSpace(3, 729)>, 
  <a point in ProjectiveSpace(3, 729)>, <a point in ProjectiveSpace(3, 729)>, 
  <a point in ProjectiveSpace(3, 729)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sub1 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,GF(3));</span>
Subgeometry PG(3, 3) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub2 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,3^2);</span>
Subgeometry PG(3, 9) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub3 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,3^3);</span>
Subgeometry PG(3, 27) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub4 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,3^6);</span>
ProjectiveSpace(3, 729)

</pre></div>

<p><a id="X82EB5BE77F9F686A" name="X82EB5BE77F9F686A"></a></p>

<h4>14.3 <span class="Heading">Basic operations</span></h4>

<p><a id="X8058FF3479158445" name="X8058FF3479158445"></a></p>

<h5>14.3-1 <span class="Heading">Underlying vector space and ambient projective space</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UnderlyingVectorSpace</code>( <var class="Arg">sub</var> )</td><td class="tdright">(<ManSection>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AmbientSpace</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">P</span> be a projective space over the field <span class="SimpleMath">F</span>. Let <var class="Arg">sub</var> be a subgeometry of <span class="SimpleMath">P</span> over the subfield <span class="SimpleMath">F'</span><unc Name=BitFlipDecoder" Arg=" C r "/>


<div class="example"><pre>
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub1 := CanonicalSubgeometryOfProjectiveSpace(pg,5);</span>
Subgeometry PG(2, 5) of ProjectiveSpace(2,
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingVectorSpace(pg);</span>
( GF(5^6)^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingVectorSpace(sub1);</span>
( GF(5^6)^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">AmbientSpace(sub1The iterative   
ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub2 := CanonicalSubgeometryOfProjectiveSpace(pg,5^3);</span>
Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap></span> <span class="GAPinput">AmbientSpace(sub2);</span>
ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingVectorSpace(sub2);</span>
( GF(5^6)^3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">frame := RandomFrameOfProjectiveSpace(pg);</span>
[ <a point in ProjectiveSpace(2, 15625)>, 
  <a point in ProjectiveSpace(2, 15625)>, 
  <a point in ProjectiveSpace(2, 15625)>, 
  <a point in ProjectiveSpace(2, 15625)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sub3 := SubgeometryOfProjectiveSpaceByFrame(pg,frame,5^2);</span>
Subgeometry PG(2, 25) of ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap></span> <span class="GAPinput">AmbientSpace(sub3);</span>
ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingVectorSpace(sub3);</span>
( GF(5^6)^3 )

</pre></div>

<p><a id="X7DC6B3E57C74A153" name="X7DC6B3E57C74A153"></a></p>

<h5>14.3-2 DefiningFrameOfSubgeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DefiningFrameOfSubgeometry</code>( <var class="Arg">sub</var> )</td><td class="tdright(Warning:  cangive java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 56
<p>Returns: a set of projective points</p>

<p>This attribute returns a frame of the ambient space of <var class="Arg">sub</var> defining it. Note that different frames might define the same subgeometry, but the frame used to constructed <var class="Arg">sub</var> is stored at construction, and 


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(2,2^4);</span>
ProjectiveSpace216)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := CanonicalSubgeometryOfProjectiveSpace(pg,2);</span>
Subgeometry PG(2, 2) of ProjectiveSpace(2, 16)
<span class="GAPprompt">gap></span> <span class="GAPinput">frame := DefiningFrameOfSubgeometry(sub);</span>
[ <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)>, 
  <a point in java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 13
<span class="GAPprompt">gap></span> <span class="GAPinput">List(frame,x->Unpack(UnderlyingObject(x)));</span>
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">frame := RandomFrameOfProjectiveSpace(pg);</span>
[ <a point in ProjectiveSpace(2, <Example
  <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := SubgeometryOfProjectiveSpaceByFrame(pg,frame,2^2);</span>
Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)
<span class="GAPprompt">gap></span> <span class="GAPinput">def := DefiningFrameOfSubgeometry(sub);</span>
[ <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)>, 
  <a point in ProjectiveSpace(2, 16)>, <a point in ProjectiveSpace(2, 16)> ]
< class">gt;<span> span class="GAPinput">List(def,x->Unpack(UnderlyingObject(x)));</span>
[ [ Z(2)^0, 0*Z(2), Z(2^4)^9 ], [ Z(2)^0, Z(2^4)^4, Z(2^4)^11 ], 
  [ Z(2)^0, Z(2^4)^2, Z(2^2)^2 ], [ Z(2)^0, Z(2^2)^2, Z(2^4)^14 ] ]
<span class="GAPprompt">gap></span> <span
[ <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)>, 
  <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)>, 
  <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)>, 
  <a point in Subgeometry PG(2, 4) of ProjectiveSpace(2, 16)> ]

</pre></div>

<p><a id="X79CC2F0483575105" name="X79CC2F0483575105"></a></p>

<h5>14.3-3 <span class="Heading">Projective dimension and rank</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveDimension</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Dimension</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class""<able =func"width="100"<< class=tdleft"<class"func>#8227; <code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an integer</p>

<p>If <var class="Arg">sub</var> is a subgeometry of a projective space, then it is a projective space itself. Therefore, these three operations return the projective dimension of <var class="Arg">sub</var>, see also <a href="chap4.html#X84FDF25D797B874B"><span class="RefLink">4.1-3</span></a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(7,8^2);</span>
ProjectiveSpace(7, 64)
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := CanonicalSubgeometryOfProjectiveSpace(pg,8);</span>
Subgeometry PG(7, 8) of ProjectiveSpace(7, 64)
<span class="GAPprompt">gap></span> <span class=" C(vjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
7
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension(sub);</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank(sub);</span>
7

</pre></div>

<p><a id="X85437D577DE97AEF" name="X85437D577DE97AEF"></a></p>

<h5>14.3-4 <span class="Heading">Underlying algebraic structures</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UnderlyingVectorSpace</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td>      1  
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BaseField</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div<!--
<p>Returns: the first operation returns a vector space, the second and third operations return a finite field</p>

<p>The operations <code class="file">UnderlyingVectorSpace</code> and <code class="file">BaseField</code> are defined for projective spaces, see <a href=


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">pg := PG(3,3^6);</span>
ProjectiveSpace(3, 729)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">sub1 := CanonicalSubgeometryOfProjectiveSpace(pg,3^3);</span>
Subgeometry PG(3, 27) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">BaseField(sub1);</span>
GF(3^6)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">UnderlyingVectorSpace(sub1);</span>
( GF(3^6)^4 )
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
GF(3^3)
<span class="GAPpromptDescription>>
Subgeometry PG(3, 9) of ProjectiveSpace(3, 729)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">BaseField(sub2);</span>
GF(3^6)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">UnderlyingVectorSpace(sub2);</span>
( GF(3^6)^4 )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SubfieldOfSubgeometry(sub2);</span>
GF(3^2)

</pre></div>

<p><a id="X8550F38F7FB123DB" name="X8550F38F7FB123DB"></a></p>

<h5>14.3-5 CollineationFixingSubgeometry</h5>

< class"unc"<class"unc" width=100%>tr>java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 239
<p>Returns: a collineation of the ambient space of <var class="Arg">sub</var></p>

<p>Let <span class="SimpleMath">GF(q)</span> be the field over which <var class="Arg">sub</and M>  fx_1)..(_n))/.


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">pg := PG(2,7^3);</span>
ProjectiveSpace(2, 343)ap F=16;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">sub := CanonicalSubgeometryOfProjectiveSpace(pg,GF(7));</span>
Subgeometry PG(2, 7) of ProjectiveSpace(2, 343)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">coll := CollineationFixingSubgeometry(sub);</span>
&lt; a collineation: &lt;cmat 3x3 over GF(7,3)&gt;, F^7&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order(coll);</span>
3

</pre></div>

<p><a id="X7836EC02824B9425" name="X7836EC02824B9425"></a></p>

<h4>14.4 <span class="Heading">Constructing elements of a subgeometry</span></h4>

<p><a id="X82E9593B8074AECB" name="X82E9593B8074AECB"></a></p>

<h5>14.4-1 VectorSpaceToElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; VectorSpaceToElement</code>( <var class="Arg">Fvars);
<p>Returns: a subspace of a subgeometry</p>

<p><var class="Arg">sub</var> is a subgeometry of a projective space, and <var class="Arg">v</var> is java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 16


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">pg := PG(2,5^6);</span>
ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">vecs := [ [ Z(5)^0, Z(5^6)^13972  [0 0 0000   ,*() ,
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[ Z(5)^0, Z(5^6)^9384, Z(5^6)^1372 ],</span>
<span class="GAPprompt">&gt;</=rimitiveRoot();b:a^;b^+^1;
<Pts:List(0.14,->^ii)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">frame := List(vecs,x-&
an>gt;VectorSpaceToElement(pg,x));</span>
[ &lt;a point in ProjectiveSpace(2, 15625)&gt;, 
  &lt;a point in ProjectiveSpace(2, 15625)&gt;, 
  &lt;a point in ProjectiveSpace(2, 15625)&gt;, 
  &lt;a point in ProjectiveSpace(2, 15625)&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">sub := SubgeometryOfProjectiveSpaceByFrame(pg,frame,5^3);</span>
Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">VectorSpaceToElement(sub,[0,0,java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 61
&lt; empty subspace &gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinputr:=(r;
[ Z(5)^0, Z(5^6)^8584, Z(5^6)^13650 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">VectorSpaceToElement(sub,vec);</span>
&lt;a point in Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">vec := [ [ Z(5)^0, 0*Z(5), Z(5^6)^5740 ], [ 0*Z(5), Z(5)^0, Z(5^6)^15250 ] ];</span>
[ ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">VectorSpaceToElement(sub,vec);</span>
Error, &lt;obj&gt; does not determine an element in &A>>/>   is the distance <>v/>search in A><A.
VectorSpaceToElementForSubgeometries( geom, v 
 ) at ./pkg/fining/lib/subgeometries.gi:400 called from
&lt;function "unknown"&gt;( &lt;arguments&gt; )
 called from read-eval loop at line 19 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class="GAPbrkprompt">brk&gt;</span> <span class="GAPinput">quit;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">vec := [ [ Z(5)^0, 0*Z(5), Z(5^6)^8268 ], [ 0*Z(5), Z(5)^0, Z(5^6)^1472 ] ];</span>
[ [ Z(5)^0, 0*Z(5), Z(5^6)^8268 ], [ 0*Z(5), Z(5)^0, Z(5^6)^1472 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">VectorSpaceToElement(sub,vec);</span>
&lt;a line in Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">VectorSpaceToElement(sub,vecs);</span>
Subgeometry PG(2, 125) of ProjectiveSpace(2, 15625)

</pre></div>

<p><a id="X78219A547B858E2E" name="X78219A547B858E2E"></a></p>

<h5>14.4-2 ExtendElementOfSubgeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; ExtendElementOfSubgeometry</code>( <var class="Arg">el</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<p>Returns: a subspace of a projective space</p>

<p>The argument <var class="Arg">el</var> is an element of a subgeometry <span class="SimpleMath">P'</span> with ambient projective space <span class="SimpleMath">P</span>. The projective space is defined over a field <span class="SimpleMath">F</span>,/java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10


<div class="example"><pre>
<span class="GAPpromptR2:=PolynomialRingF,[,y);
ProjectiveSpace(3, 3125)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">frame := RandomFrameOfProjectiveSpace(pg);</span>
[ &lt;a point in ProjectiveSpace(3, 3125)&gt;, &lt;a point in ProjectiveSpace(3, 3125)&gt;
    , &lt;a point in ProjectiveSpace(3, 3125)&gt;, 
  &lt;a point in ProjectiveSpace(3, 3125)&gt;, 
  &lt;a point in ProjectiveSpace(3, 3125)&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">sub := SubgeometryOfProjectiveSpaceByFrame(pg,frame,5);</span>
Subgeometry PG(3, 5) of ProjectiveSpace(3, 3125)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">p := Random(Points(sub));</span>
&lt;a point in Subgeometry PG(3, 5) of ProjectiveSpace(3, 3125)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l := Random(Lines(p));</span>
&lt;a line in Subgeometry PG(3, 5) of ProjectiveSpace(3, 3125)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">p * l;</span>
Des>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">q := ExtendElementOfSubgeometry(p);</span>
&lt;a point in ProjectiveSpace(3, 3125)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">qspace<C<>.If <vA   java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 63
Error, &lt;x&gt; and &lt;y&gt; do not belong to the same geometry called from
x in y at ./pkg/fining/lib/projectivespace.gi:1658 called from
IsIncident( b, a ) at ./pkg/fining/lib/geometry.gi:439 called from
&lt;function "unknown"&gt;( &lt;arguments&gt; )
 called from read-eval loop at line 15 of *stdin*
you can 'quit;C>Syndrome</C> then returns an error.
you can 'return;' to continue
<span class="GAPbrkprompt">brk&gt;</span> <span class="GAPinput">quit;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">m := ExtendElementOfSubgeometry(l);</span>
&lt;a line in ProjectiveSpace(3, 3125)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">q * m;</span>
true
<span class="GAPprompt">gap&gt;</span> <span a java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 49
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">UnderlyingObject(l) = UnderlyingObject(m);</span>
true

</pre></div>

<p><a id="X799DB77886B8ABDB" name="X799DB77886B8ABDB"></a></p>

<h5>14.4-3 AmbientGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; AmbientGeometry</code>(
<p>Returns: an incidence geometry</p>

<p>For <var class="Arg">el</var> an element of a subgeometry <span class="SimpleMath">P</span>, which is also a projective space, this operation returns <span class="SimpleMath">P</span>.</p>

<p><a id="X7B1757048405DD29" name="X7B1757048405DD29"></a></p>

<h5>14.4-4 <span class="Heading">Flags</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; FlagOfIncidenceStructure</code>( <var class="Arg">sub</var>, <var class="Arg">els</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td><unc =SyndromeTable " 
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; IsEmptyFlag</code>( <var class="Arg">flag</var> )</td><td class="tdright">(&nbsp;vAcan  inthesyndrome 
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; IsChamberOfIncidenceStructure</code>( <var class="Arg">flag</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>Returns: true or false</p>

<p>These operations are defined for projective spaces and so they are also applicable to subgeometries.</p>

<p><a id="X80503DDC8270EE69" name="X80503DDC8270EE69"></a></p>

<h4>14.5 <span class="Heading">Groups and actions</span></h4>

<p>Let <span class="SimpleMath">P'</span> be a subgeometry of <span class="SimpleMath">P</span>. Although one could argue that any semilinear map inducing a collineation preserving <span class="SimpleMath">P'</span> java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<p><a id="X78F858C8863C7721" name="X78F858C8863C7721"></a></p>

<h5>14.-1 <span class"Heading"Groups of collineations</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; CollineationGroup</code>( <var class="Arg">sub</var> )</td><td class="tdright">(&>nbsp;operation&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; ProjectivityGroup</code>( <var class="Arg">sub</var> )</td><td class="tdright">(&>nbsp;operation&nbsp;)</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&pan>#8227; SpecialProjectivityGroup</code>( <var class="Arg">sub</var> )</td><td class="tdright">(&nbsp;operation&nbsp;)</td></tr></table></div>
<p>Returns: a group of collineations</p>


<div class="chlinkprevnextbot">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap13.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a hrefthepermutationautomorphism <><M  >/>

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

Messung
CHZG
8510010095

¤ Dauer der Verarbeitung: 0.37 Sekunden  ¤

*© Formatika GbR, Deutschland






Wurzel

Suchen

PVS Prover

Isabelle Prover


Cephes Mathematical Library

Vienna Development Method

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 und die Messung sind noch experimentell.