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<div class="ChapSects"><a href="chap4.html#X83BBAA668672A76D">4 <span class="Heading">Projective Spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7862BC887D20B37A">4.1 <span class="Heading">Projective Spaces and basic operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79B440FF7EFBA661">4.1-1 IsProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7962DA507C64FCBA">4.1-2 ProjectiveSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84FDF25D797B874B">4.1-3 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BCBA564829D9E89">4.1-4 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D544D7985A4572D">4.1-5 UnderlyingVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8606750A8586DF8D">4.1-6 AmbientSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8016E6857D53F2ED">4.2 <span class="Heading">Subspaces of projective spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82E9593B8074AECB">4.2-1 VectorSpaceToElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8461BCEF862B9A7B">4.2-2 EmptySubspace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84FDF25D797B874B">4.2-3 ProjectiveDimension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87657AEF7E2C50F9">4.2-4 ElementsOfIncidenceStructure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87E64DA67C3D6661">4.2-5 <span class="<
</span>
<span class="ContSS"><br /><span class="nocss"> &<p(K1)S<>i</> S<></sub> <class=S></span><>/>={1, < class"java.lang.StringIndexOutOfBoundsException: Range [127, 126) out of bounds for length 145
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X870D9D9A7F11806F"<p>(K2 S<ub<sub>sup>/> <span =S"><S<sub>/ub> 1,for <span =">,<./>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84E3985A8700B302">4.2-8 Coordinates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CE4FD76820B503A">4.
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86628227863989E5">4.2-10 HyperplaneByDualCoordinates</a></span>
<=C"< /=>n;nbsp;</span><a href="chap4.html#X801A95907B13F447">4.2-11 EquationOfHyperplane</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8606750A8586DF8D">4.2-12 AmbientSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BCBA564829D9E89">4.2-13 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79730D657AB219DB"<p>Remarkably, group s>2< quadrangle. java.lang.StringIndexOutOfBoundsException: Range [146, 145) out of bounds for length 330
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D3C5D3B7AA4DE28">4.2-15 RandomSubspace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X875BE2957FAF6209">4.2-16 Span</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8469B54180FE1E4C">4.2-17 Meet</a></span>
<span class="ContSS"><br
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7AEFC2C57F10C3A7">4.2-19 IsEmptyFlag</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A453E0E861F2C94">4.2-20 IsChamberOfIncidenceStructure</a></span>
</div></div<()1 java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 82
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BD8312C85784503">4.3 <span class="Headingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7FFA08DA85C5251C">4.3-1 java.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 0
<span <p>(iii () the ()<pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81A8365A7FE68447"
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 34
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a<>(a java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 124
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EF8910F82B45EC7">4.4-2 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EBA57FC7CCF8449">4.4-3 List</a></span>
</><
</div>
<h3>4 <span class="Heading">Projective Spaces</span></h3>
<p>In this chapter we describe how to use <strong java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><a id="X7862BC887D20B37A" name="X7862BC887D20B37A"></a></p>
<h4>4.1 <span class="Heading">Projective Spaces and basic operations</span></h4>
<p>A <em>p> java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 226
<p><a id="X79B440FF7EFBA661" name="X79B440FF7EFBA661"></a></p>
<h5>4.1-1 IsProjectiveSpace</h5>
<div class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This category is a subcategory of <code class="code">IsLieGeometry</code>, and contains all finite Desarguesian p><strong class="pkg">FinInG</strong> provides functions to construct elation generalised quadrangles directly from a Kantor family. The constructed generalised quadrangles are generalised polygons in the sense of <strong class="pkg">FinInG</strong>, i.e. all generic operations described in Sections <a href="chap12.html#X864C966D8184A9C0"><span class="RefLink">12.3</span></a> and <a
<p>e a "hapBib.bjava.lang.StringIndexOutOfBoundsException: Range [67, 65) out of bounds for length 159
<p><a id="X7962DA507C64FCBA" name="X7962DA507C64FCBA"></a></p>
<h5>4.1-2 ProjectiveSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveSpace</code>( <var class="Arg">d</var>, <var class="Arg">F</var> )</td><<div class="unc"<class"unc"width=100%>><class=tdleft"><code class="func">‣ IsEGQByKantorFamily</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; ProjectiveSpace</code>( <var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation <p><code class="ode"</>java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 447
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PG</code>( <var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a projective space</p>
<p><var class="Arg">d</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<h512.63 =java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 60
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveSpace(3,GF(3));</span>
ProjectiveSpace(3, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsKantorFamily</code>( <var class="Arg">g</var>, <var class="Arg">f</var>, <var class="Arg">fstar</var> )</td><td class="tdright">( operation )</td></tr></table></div>
ProjectiveSpace(3, 3)
</pre></div>
<p><a id="X84FDF25D797B874B" name="X84FDF25D797B874B"></a></pp> is ajava.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 407
<h5>4.1-3 ProjectiveDimension</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">ps</var> )</td><td class="tdright">( attribute )</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">ps</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleftjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: the projective dimension of the <div class="func"><table class="func" width="100<tr>td class"dleft">code class=func>#8227;EGQByKantorFamily<java.lang.StringIndexOutOfBoundsException: Range [125, 124) out of bounds for length 281
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(<p>Returns: a generalised quadrangle</p>
ProjectiveSpace(5, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(ps);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension(ps);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank(ps);</span>
5
<pre><>
<p><a id="X7BCBA564829D9E89" name="X7BCBA564829D9E89"></a></p>
<h5>4.1-4 BaseField</h5>
<div class="func"><table class="func" width="100%"><tr
<p>Returns: returns the base field for the projective space <var class="Arg">ps</var></p>
<div class="<div class="example><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseField(ProjectiveSpace(3,81));</span>
GF(3^4)
</pre></div>
<p><a id="X7D544D7985A4572D" name="X7D544D7985A4572D"></a></p>
<h5>4.1-5 UnderlyingVectorSpace;pc 27 3 &;
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code < =>gt<><java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 139
<p>Returns: a vector space</p>
<p>If <var class="Arg">ps</var> is a projective space of dimension <span <span class="GAPprompt">gap></span> <span class="GAPinput">flist2 := [ Group g.^*.] Group(g. .^*.3 ]) <span>
<div<pan ">gt</><"> java.lang.StringIndexOutOfBoundsException: Range [77, 76) out of bounds for length 138
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(4,7);</span>
<spanclass="GAPprompt">gap></span> <span class="GAPinput">IsKantorFamily( g, flist1, flist2 );</span>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 34
( GF(7)^5 )
</pre><# java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 32
<p><a id="X8606750A8586DF8D" name="java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 4
<h5>4.1-6 AmbientSpace</h5>
<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">ps</var> )</td><td class="tdright">( attribute )</td></tr>&tEGQ of order [3,3 ]andbasepoint0>
<p>Returns: a projective space</p>
<p>The ambient space of a projective space <var class="Arg">ps</var> is the java.lang.StringIndexOutOfBoundsException: Range [0, 86) out of bounds for length 73
<p><a id="X8016E6857D53F2ED" name="IsGeneralisedQuadrangle", "IsElationGQ", "IsElationGQByKantorFamily" ]
<h4>4.2 <span class="Heading">Subspaces of projective spaces</span></h4>
<p>The elements of a projective space PG(n,q) are the subspaces of a suitable dimension. The empty subspace, also called the trivial subspace, has dimension <span class< of2of&EGQof[ 3 &;
<p><a id="X82E9593B8074AECB" name="X82E9593B8074AECB"></a></p>
<h5>4.2-1 VectorSpaceToElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 67
<p>ReturnsIjava.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 62
<p><var class="Arg">geo</var> is a projective space, and <var class="Arg"
<div class="pre></
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(6,7);</span>
ProjectiveSpace(6, 7)
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [3,5,6,0,
[ Z(7), Z(7)^5, Z(7)^3, 0*Z(7h5>1265<=> java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 94
<span class="GAPprompt">gap></span> <
<a point in ProjectiveSpace(6, 7)>
<span class="nc"< "java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 279
[142.131]
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(3,4);</span>
ProjectiveSpace(3, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [1,1,0,1]*Z(4)^d "> java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 253
[ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,v);</span>
<a point in ProjectiveSpace(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0,1],[0,1,1,0]]*Z(4)^0;</span>
2)0, 0*() 0*(2, Z2^ ,[0*() Z()0Z2)0 0*()]]
<span class="GAPprompt">gap></span> <span class="GAPinput">line := VectorSpaceToElement(ps,mat);</span>
<a line in ProjectiveSpace(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">e := VectorSpaceToElement(ps,[]);</span>
Error, <v> does not represent any elementdiv =f ""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 225
<function "unknown">( <arguments> )
called from read-eval loop at line 17 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class=">For technical reasons of java.lang.StringIndexOutOfBoundsException: Range [79, 78) out of bounds for length 912
</pre></div>
<p><a id="X8461BCEF862B9A7B" name="X8461BCEF862B9A7B"></a></p>
<h5>4.2-2 EmptySubspace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EmptySubspace</code>( <var java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 0
<p>Returns: the trivial subspace in the projective <var class="Arg">ps</var></p>
<p>The object returned by this operation is contained in every projective subspace of the projective space <var class="Arg">ps</var>, but is not an&;c size27 3g;
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">e := EmptySubspace(PG(5,9));</span>
&;&java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(PG(5,9, (.12g2 .3]];<java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
;a (,9&java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 40
<span class="GAPprompt">gap></span> <span class="GAPinput">e*p;</span&; [3 0g;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
st `* from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
called from read-eval loop at f1*2
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<pan =G">brk>/span> <pan ="APinput>quit<span
<span class="GAPprompt">gap></span> <span class="GAPinput">e in p;</span>
true
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
</pre></div>
<< id""F">a>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<h5>4.2-3 ProjectiveDimension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveDimension java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 101
<p>Returns; class1ofl; ,3]java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 74
<"<java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG
ProjectiveSpace(2, 5)
<span class="GAPprompt">gap></span> <span class="GAPprompt">gap></span> <span class="GAPinput">S := flist2[2];</span>
[ [ Z(5)^0, Z(5)^0&t;c group of size 9 2 generatorsjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
<span class="GAPprompt">gap></span> <span class="GAPinput">line := span class="GAPprompt">gap></span> <span class="GAPinput">m := ObjectToElement(egqjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
<a line in ProjectiveSpace(2, 5)&&taline of 2 of &t java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 74
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(line)< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 94
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension(line);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,[1,2,3]*Z(5)^0);</span>
<a point in ProjectiveSpace(2, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 4
0
< class="GAPprompt">gap></span> <span class="GAPinput">Dimension(p);</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(EmptySubspace(ps[ lta line 1 &E [,3 ]and basepoint 0>>,
-1
</pre></div>
<p><a id="X87657AEF7E2C50F9" name="X87657AEF7E2C50F9"></a></p>
<h5>4.2-4 ElementsOfIncidenceStructure</h5>
<div class="func"><table class="func" width="100%"><tr><td &t;a lineof class 1 of <EGQoforder[3 3 and basepoint0&t;gt;]
<p><pan =">ap&t;<> < class="GAPinput">pts1 := List(Points(m));</span>
<p>For the projective space <var class="Arg">ps</var> of <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint
<div <a po&t; point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(6,7 la 2oflt; java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 79
ProjectiveSpace(6, 7)
< class"GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,1);</span>
<points of ProjectiveSpace(6, 7)>
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 104
<lines of ProjectiveSpace(6, 7)>
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,3);</span>
&planes (,7)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,4);</span>
< of (6,7&;
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,5);</span>
&;proj 4- ( )>
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,6);</span>
&proj - 6,)&;
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,7);</span>
Error, <ps> has no span class="GAPprompt">gap>> <pan class=Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 106
<function "unknown">( <arguments> )
called from read-[f2^2, f1*2^2, f1^2*2^ (roup( [ ,f2^* )^2)]
you can 'quit;' to quit to outer loop, or
you can 'eturn'to
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
</pre></div>
<p><a id="X87E64DA67C3D6661" name="X87E64DA67C3D6661"></a></p>
<h5>4.2-5 <span class="Heading">Short names for ElementsOfIncidenceStructure</span></h5>
<div class="func"><table class="func" width="100%"><tr><p><a id="X7DCD7EAB839BD97F" name="X7DCD7EAB839BD97F"></a>
<div class="func"><table class="func"
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 89
<div class="func"><table class="func" width="100%"><tr><
<=""<= =100"<r><d class="tdleft"<code class=func"&8227 Hyperplanes/code(var=Arg><var>)/d< class"dright"(n;nbsp)/d>/>/<div>
<p>Returns: The elements of <var class="Arg">ps</var> of respective type 1, 2, 3, 4, anddiv "><java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 248
<p>An error message is produced when the projective space <var class="Arg">ps</var> has no elements of a required type.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(6,13);</span>
ProjectiveSpace(6, 13)
<span class="GAPprompt">gap></span> <span class="GAPinput">Points(ps);</span>
<points of ProjectiveSpace(6, 13)>
<span class="GAPprompt">gap><p> attribute=">/java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 710
<lines of ProjectiveSpace(6, 13)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Planes(ps);</span>
<planes of ProjectiveSpace(6, 13)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Solids(ps);</span>
<solids of ProjectiveSpace(6, 13)>
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 26
<proj. 5-subspaces of ProjectiveSpace(6, 13)>
<java.lang.StringIndexOutOfBoundsException: Range [37, 5) out of bounds for length 83
ProjectiveSpace(2, 2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Hyperplanes(ps);</span>
<lines <pc group 27gt;
</pre></div>
<p>a id="7904128479BDFCC9" name="X7904128479BDFCC9"></a></p>
<h5>4.2-6 <span class="Heading">Incidence and containment</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsIncident</code>( <var class="Arg">el1</var>, <var class="Arg">el2</var> )</td
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; \*</code>( <var class="Arg">el1</var>, <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 159 out of bounds for length 45
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&<;pan =java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 96
<p>Returns: true or false</p>
<p>Recall that for projective spaces, incidence is symmetrized containment, where the empty subspace and the whole projective space are excluded java.lang.StringIndexOutOfBoundsException: Range [0, 147) out of bounds for length 112
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(5,9);</span>
ProjectiveSpace(5, 9)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,[1,1,1,1,0,0]*Z(9)^0);</span>
<a point in ProjectiveSpace(5, 9)>
<span < "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 112
<a line in ProjectiveSpace(5, 9)>
<span class="GAPprompt">gap></span> <span ( ]
<a plane in ProjectiveSpace(5, 9)>
<span class<pan "gapgt>=GAPinput">java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 94
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l lt; 1ofl;[, and 0gt&;java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
true
<span class="GAPprompt&; java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 79
true
<span class="GAPprompt">gap></span> <span class="GAPinput">p in l;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l in p;</span>
false
<<apoint of class 1 &t; oforder[3,3]and java.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 79
false
<span class="GAPprompt">gap></span> <span class="GAPinput">l * plane;<&t; java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 79
false
<span class="GAPprompt">gap></span> <span class="GAPinput">l in plane;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">e := EmptySubspace(ps);</span>
< empty subspace >
<span class="GAPprompt">gap></span> <span class="GAPinput">e * l;</span>
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `*' on 2 arguments&t; point l;EGQ of java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 79
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
called from read-eval loop at line 21 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 77
<span class="GAPprompt">gap></span> <span class="GAPinput">e in l;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l in ps;</span>
true
</pre></div>
<870D9D9A7F11806F>/><>
<h5>4.2-7 StandardFrame</h5>
<=>java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 222
<p>Returns: the standard frame of the projective space <var class="Arg">ps</var></p>
<div class="example"><pre>
<span class="GAPprompt">gap>&; of lEGQ ,3 java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 79
[ <a point in ProjectiveSpace(5, 4)>, <a point in <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>
<a point in &ta 1 java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 79
<a point in ProjectiveSpace(5, 4)>, <a point in ProjectiveSpace(5, 4)>,
<a point in ProjectiveSpace(5, 4)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span a java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 79
[ NewRowVector(IsCVecRep,GF(2,2),[Z(2)^0,0*Z(2),0*Z(2),0*Z(2&tajava.lang.StringIndexOutOfBoundsException: Range [16, 13) out of bounds for length 79
, NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),Z(2)^0,0*Z(2) ;class &java.lang.StringIndexOutOfBoundsException: Range [44, 31) out of bounds for length 79
0*Z(2),]), NewRowVector(IsCVecRep,GF(2, &t; java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 79
0*Z(2),0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),0*Z(2),0*Z(2),
Z(2)^0,0*Z(2),0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),0*Z(2),
0*Z(2),0*Z(2),Z(2)^0,0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),
0*Z(2),0*Z(2),0*Z(2),0*Z(2),Z(2)^0,]), NewRowVector(IsCVecRep,GF(2,2),[
Z(2)^0,Z(2)^0,Z(2)^0,Z(2)^0,Z(2)^0,Z(2)^0,]) ]
</pre></div>
<p><a lt;a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
<h5> <apoint of class 2 of &t; of ,3 ]and basepoint 0>gt;java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
<div class="func"><table class="func" &t;a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
<p>Returns: the homogeneous <a point of class 2 of <EGQ[ 3 0&;gt;
<div class="example"><pre>
span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(PG(5,16)));</span>
<a point in ProjectiveSpace(5, 16)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Coordinates(p);</span>
[ Zl; java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 79
</pre></div>
<>aid="7E4FD76820B503A" name"7E4FD76820B503A"</>/>
<h5>4.2-9 DualCoordinatesOfHyperplane</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 79
<p>Returns: a list</p><a E java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 79
<p>The argument <var class="Arg">hyp</var> is a hyperplane of a projective space. This operation returns the dual< point of class2 of < order[3,3 ]andjava.lang.StringIndexOutOfBoundsException: Range [70, 67) out of bounds for length 79
<p><a id="X86628227863989E5" name="X86628227863989E5"></a></p>
<h5>4.2-10 HyperplaneByDualCoordinates</h5>
<div class="func"><table class="func< =""&<span> ""gpts,;/java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
<p>Returns: a hyperplane of[[&;a point 1 of &t;GQ of [ 3,3]and &&t;
<p>The argument <var class="Arg">pg</var> is a projective space, and <var class="Arg">list</var> is the coordinate vector of a point of <var class="Arg">pg</var>. This operation returns the hyperplane that has <var class="Arg">list</var> java.lang.StringIndexOutOfBoundsException: Index 240 out of bounds for length 83
<p><a id="X801A95907B13F447" name="X801A95907B13F447"></a<a java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 83
<h5>4.2-11 EquationOfHyperplane</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EquationOfHyperplane</code>( <var class="&;1 java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 83
<> equation thejava.lang.StringIndexOutOfBoundsException: Range [44, 42) out of bounds for length 93
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput lta of 1 lEGQ [3 &t;g;,
<a plane in ProjectiveSpace(3, 2)>
<">EquationOfHyperplane()</>
x_1+x_2
</pre></div>
<p><a id="X8606750A8586DF8D" name="X8606750A8586DF8D"></a></p>
<h5>4.2-12 AmbientSpace</h5>
<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">el</var> )</td><td&t; 1of&;ofjava.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 83
: space of var=><java.lang.StringIndexOutOfBoundsException: Range [77, 76) out of bounds for length 103
<p>This operation is also applicable on the <a point of class 1 of <EGQ of order java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 83
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(3,27);</span>
ProjectiveSpace(3, 27)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 83
<a point in ProjectiveSpace(3, 27)>
<span class="GAPprompt">gap>< &t; point of 1 &t; of order ,3 ] 0g;>java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
ProjectiveSpace(3, 27)
</pre></div>
<p><a id="X7BCBA564829D9E89" name="X7BCBA564829D9E89 <a point of class 1 of <EGQ order [ 3, 3 ] and basepoint 0>&t,
<h5>4.2-13 BaseField</h5>
<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">el</var> )</td><td class="tdright">( operation
<p>Returns: returns the base field of an element <var class="Arg">el</var> of a projective space</p>
<>This is java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 85
<div class="example"><pre>
<span class="GAPprompt">gap&;/pan <pan"": java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 83
ProjectiveSpace(5, 8)
<span class="APprompt"gapgt;/> <"> := ps,,,100,]*Z(2));</span>
<a point in ProjectiveSpace(5, 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseField(p);</span>
GF(2^java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
</pre></div>
<p><a id="X79730D657AB219DB" name="X79730D657AB219DB"></a></p>
<h5>4.2-14 Random</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Random</code>( <var class="Arg">elements</var> )</td><td lt; ;GQ ,3]java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 81
<p>Returns: a random element from the collection <var class="Arg">elements</var></p>
<p>The collection <var class="Arg">elements</var> is an object in the category <code class="code">IsElementsOfIncidenceStructure</code>, i.e. an java.lang.StringIndexOutOfBoundsException: Index 148 out of bounds for length 81
<div class="example"><pre>
<span class="GAPprompt">gap></span> &; point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
ProjectiveSpace(9, 49)
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(Lines(ps));</span>
<< 2java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 83
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(Points(ps));</span>
<a <&tapoint of 2 &EGQorder ,3 basepoint&& ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(Solids(ps));</span>
<a solid in ProjectiveSpace(9, 49)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(Hyperplanes(ps));</span>
<a proj. 8-space in ProjectiveSpace(9, 49)>
<span class="GAPprompt">gap></span> <span class=[&a of 1&E ofjava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 78
<proj. 5-subspaces of ProjectiveSpace(9, 49)>
<span class=GAPprompt">ap>/span>java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 82
<a proj. 5-space in ProjectiveSpace(9, 49)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last)&t;line1& java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 78
[[Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^14,Z(7^2)^ &t;a java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 78
Z(7)^5,]
2^,
Z(7^2)^27,]
[0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),Z(7^2) l; class java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 78
Z(7&t; 1 of& , 3]java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 78
[0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),Z(7),Z(7^2nd0g&;
14,java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
[0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),Z(7^2)^10, <a line of class 1 of <EGQ of order [ 3, 3 ] an 1&EGQorder[3,3]andbasepoint&t&,
Z(^)^29,java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
[0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,Z < java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 78
Z(7^2)^25,]
]
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomSubspace(ps,3);</span>
& l;a 1of lEGQ java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 78
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span>
[[Z(7)^0,0*Z(7),0*Z(7),0*Z(7),Z(7^2)^17,Z(7^2)^33,Z(7^2)^4,0*Z(7),Z(7^2),
Z(^^33,]
[0*Z(7),Z(7)^0,0*Z(7),0*Z(7),Z(7^2)^30,Z(7)^2,Z(7)^3,Z(7)^0,Z(7^2)^20,Z(7^2)^
42,]
[0*Z(7),0*Z(7),Z(7)^0,0*Z(7),Z(7^2)^20,Z(7^2)^30,Z(7^2)^11,Z(7^2)^39,Z( < of1&EGQ ,3]java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 78
Z(7),]
[0*Z(7),0*Z(7),0*Z(7),Z(7)^0,Z(7^2)^21,Z(7)^0 l; java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 78
Z(7^2)^9,]
]
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomSubspace(ps,7);</span>
<a &t; 1&EGQjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 78
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span>
[[Z(7)^0,0*Z(7),0*Z(7)<a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
35,]
[0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^43,Z(7^2),]
[Z()0Z(7),Z()^,0*(7,0*()0*Z()0*()0*(,(^)44,()^,
[0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^41,Z(7^2)^
10,]
[0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z,&t;line 1 l;java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 78
12,]
0()0*7,*()0*7)0*7)(77)^0,0*Z(,*(7)11,7^^
39,]
[0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),Z(7^2)^22,Z(7^2)^
10,]
[0*Z(7),0*Z(7),0 &t;line 1&EGQjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 78
22,]
]
<span class="GAPprompt">gap&&t; java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 78
<a plane in ProjectiveSpace(9, 49)>
<span class="GAPprompt">gap></span> <span class="GAPinput"><a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>
<a proj. 6-space in ProjectiveSpace(9, 49) &t; line of 1ofl;EGQ ,3 java.lang.StringIndexOutOfBoundsException: Range [67, 66) out of bounds for length 78
</><divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
" =java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<h5>4.2-15 RandomSubspace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomSubspace</code>( <var class="Arg">ps</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></ &a java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 78
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomSubspace</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation < line class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
<p>Returns: the first variant returns a random element of type <var class="Arg">i</var> of the projective space <var class="Arg">ps</var>. The second variant returns a random elementspan =gt< <=>java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 104
<div class="example"><pre>
<= PG(8,16)/java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 84
ProjectiveSpace(8, 16)
<span class="GAPprompt">gap&a 1 lEGQ [ 3, 3 ]and java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 82
<a point in ProjectiveSpace(8, 16)>
< class=">()<>
<a proj. 5-space in ProjectiveSpace(8, 16)>
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomSubspace(ps);</span>
<a proj. 7-space in ProjectiveSpace(8, 16)>
<span class="GAPprompt">gap></span> <span class=&t; java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 82
<a proj. 4-space in ProjectiveSpace(8, 16)>
< =>gt<java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 88
<a plane in ProjectiveSpace(8, 16)>
<span class="GAPprompt">gap></span> <span < line of class 1 of &;EGQ of order[3,3 ]and basepoint0gt;gt; ,
<a plane in ProjectiveSpace(8, 16)>
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomSubspace(ps);</span>
<a plane in ProjectiveSpace(8, 16)>
<pre>/
<p><a id="X875BE2957FAF6209" name="X875BE2957FAF6209"></a></p>
<h5>4.2-16 Span</h5>
<div class="unc"< class"unc=100"<<class"dleft"><code class="func">‣ Span</code>( <var class="Arg">u</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table &a java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 82
<p>Returns: an element or the empty subspace or the whole space</p>
<p>When <var class="Arg">u</var> and <var class="Arg">v</var> are elements of a projective space. This function returns the span of the two elements. When <var class="Arg">list</var> is a list of java.lang.StringIndexOutOfBoundsException: Index 203 out of bounds for length 84
<div <aline ofclass1 &t; order [3,3 basepoint 0>>,
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(3,3);</span>
ProjectiveSpace(3, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Planes(ps));</span>
&;a(,)&;
<span class="GAPprompt">gap></span> <span class="GAPinput">q := Random(Planes(ps));</span>
<a plane in ProjectiveSpace(3, 3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := Span( lt; Eof , java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 82
ProjectiveSpace(3, 3)
<span"APprompt"gap&t<span <panclass"APinput"> = Span([,];/>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">t := Span( <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0> 1 lt ,3] java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 82
&java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 84
<span class="GAPprompt">gap></span> <span class="GAPinput">t = p;</span>
java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 4
<span class="GAPprompt">gap></span> <span class="GAPinput">Span(ps,p);</span>
ProjectiveSpace(3, 3)
</pre></div>
<p><a id="X8469B54180FE1E4C" name="X8469B54180FE1E4C"></a></p>
<h5>4.2-17 Meet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Meet</code>( <var class="Arg">u</var>, <var class="Arg">v</var> &n style='color:red'>a java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 82
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Meet</code>( lt; 1java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 82
<Returns theempty orjava.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 67
<p>When <var class="Arg">u</var> and <var class="Arg">v</var> are elementsl; java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 84
<div class="example"><pre>
ProjectiveSpace(7, 8)
<[ <a linejava.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 82
<a solid in ProjectiveSpace(7, 8)>
ss"GAPprompt"gap&t</pan <pan "APinput">q := Random(Solids(ps));</span>
<a solid in ProjectiveSpace(7, 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := Meet(p,q);</span>
< empty subspace >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(s);</span>
< empty subspace >
<span class="<pre></>
<a proj. 6-space in ProjectiveSpace(7, 8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Meet(p,r);</span>
<a plane in ProjectiveSpace(7, 8)>
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0
<a planeh5>267 <span=H>,- java.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 106
<span class="GAPprompt">gap></span> <span class="GAPinput">Meet([java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<>et = {<>/:=1. } "><java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 345
</pre></div>
<p><a id="X7E204A78815C46DD" name="X7E204A78815C46DD"></a></p>
<h5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FlagOfIncidenceStructure</code>( <var class="Arg">ps</var>, <var class="Arg">els</var> )</td><td class="tdright">( operation )</td></tr></table></div>
< java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 242
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ProjectiveSpace(12,11);</span>
ProjectiveSpace(12, 11)
<span class="GAPprompt">gap></span> <span class="GAPinput">s1 := RandomSubspace(ps,8);</span>
<a proj. 8-space in ProjectiveSpace(12, 11)>
<span class="GAPprompt">gap></span> <span class="GAPinput">s2 := RandomSubspace(s1,6);</span>
<a proj. 6-space in ProjectiveSpace(12, 11)>
<pan class="APprompt">ap&t;/span> <span class=GAPinput> : RandomSubspace(s2,4);</span>
<a proj. 4-space in ProjectiveSpace(12, 11)>
<span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Random(Solids(s3));</span>
&;solid ProjectiveSpace(12,11&t;
<span class="GAPprompt">gap></span> <span class="GAPinput">s5 := Random(Points(s4));</span>
<a point in ProjectiveSpace
<span class="GAPprompt">gap></span> <span class="GAPinput">flag := FlagOfIncidenceStructure(ps,[s1,s3,s2,p>Define J = {A(i): i= 1 ... q} ∪ {A(∞)} and J<sup>*</sup> = {A<sup>*</sup>(i): i= 1 ... q} ∪ {A<sup>*</sup>(
<a pA resultsof java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 427
<""g;>=Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 83
ProjectiveSpace(4, 5)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(ps));</span>
<a point in ProjectiveSpace(4, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := Random(Lines(ps));</span>
<a line in ProjectiveSpace(4, 5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">v := Random(Solids(ps));</span>
<a solid in ProjectiveSpace(4, 5)>
<t">ap;<span><class=G">:=java.lang.StringIndexOutOfBoundsException: Range [97, 94) out of bounds for length 114
Error, <els> does not determine a flag> called from
<function "unknown">( <
called from read-eval loop at line 19 of *stdin*
ou qloop,java.lang.StringIndexOutOfBoundsException: Range [41, 42) out of bounds for length 41
you can 'return;' to continue
<classGjava.lang.StringIndexOutOfBoundsException: Range [27, 25) out of bounds for length 77
<span class="GAPprompt">gap></span> <span class="GAPinput">flag := FlagOfIncidenceStructure(ps,[]);</span>
<a flag of ProjectiveSpace(4, 5)>
</pre></div>
<p><a id="X7AEFC2C57F10C3A7" name="X7AEFC2C57F10C3A7"></a></p>
<h5>4.2-19 IsEmptyFlag</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 63
<p>Returns: return true if <var class
<p><a id="X7A453E0E861F2C94" name="<div class="func"><table class="func" width="100%"><tr ="> ""&<span style='color: green'>8227 /ode>(< java.lang.StringIndexOutOfBoundsException: Range [133, 131) out of bounds for length 219
<h5>4.2-20 IsChamberOfIncidenceStructure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsChamberOfIncidenceStructure</code>( <var class="Arg">flag</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: true if <var class="Arg">flag</var> is a chamber</p>
<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(3,13);</span>
(,13)
<span class="GAPprompt">gap></span> <span class="GAPinput">plane := Random(Planes(ps));</span>
<a plane in ProjectiveSpace(3, 13)>
<span class="GAPprompt">gap></span> <span class="GAPinput">line := Random(Lines(plane));</span>
<a line in ProjectiveSpace(3, 13)>
<span class="GAPprompt">gap></span> <span class="p>Some famous q-clans are built in. We refer to ... for more on these.</>
<a point in ProjectiveSpace(3, 13)>
<span class="GAPprompt">gap></span> <span class="GAPinput">flag := FlagOfIncidenceStructure(ps,[point,line,plane]);</span>
<a flag of ProjectiveSpace(3, 13)>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsChamberOfIncidenceStructure(flag);</span>
true
</h5>610 /
<p><a id=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4>4.3 <span class="Heading">Shadows of Projective Subspaces</span></h4>
<p><a id="X7FFA08DA85C5251C" name="X7FFA08DA85C5251C
<h5>.- /5
<div class="func"><table class="
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classp><id="name">/>
<p>Returns: the elements of type <var class="Arg">i</var> incident with <var class="Arg">el</var>. The
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PG(4,3);</span>
ProjectiveSpace(4, 3)
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 97
<a plane in ProjectiveSpace(4, 3)>
<span class="GAPpromptp> java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 76
<shadow points in ProjectiveSpace(4, 3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(shadowpoints);</span>
[lt; (,3&,&a java.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 86
<a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>,
<a point in ProjectiveSpace(4, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<a point in ProjectiveSpace(4, 3)>, <<div class=e><>
<a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)&java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 80
<a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>,
<a point in ProjectiveSpace(4, 3)> ]
<s,plane,2);</span
<shadow lines in ProjectiveSpace(4, 3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Listspan "">gap&;/><=Gjava.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 104
[ <a line in ProjectiveSpace(4, < java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 91
<a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>,
<a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>,
<a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>,
<a line in ProjectiveSpace(4, #I Computed Kantorfamily NowcomputingEGQ..
<a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>,
<a line in ProjectiveSpace(4, 3)> ]
</pre></div>
<p>name=X7"<a><p>
<h5>4.3-2 ShadowOfFlag</h5>
<div class==java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 106
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ShadowOfFlag</code>( <var class="Arg">ps</var>, <var class="Arg">flag</var><java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 43
<p>Returns: the type <var class="Arg">i</var> shadow elements of the flag <var class="Arg">flag</var>, i.e. the elements of type <var class="Arg">i26127360
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps26127360
( 7java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,[1,0,0,0,0,0]*Z(7)^0);</span>
<a point in ProjectiveSpace(5, 7)>
<span<
<a line in ProjectiveSpace(5, 7)>
<span class="APprompt"gap&t<span> <="">clan : FisherThasWalkerKantorBettenqClan(1);/pan
<a plane in ProjectiveSpace(5, 7)>
>ap<java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 114
<a flag of ProjectiveSpace(5, 7)>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := ShadowOfFlag(ps,flag,4);</span>
<shadow solids in ProjectiveSpace(5, 7)>
<span
<shadow solids in ProjectiveSpace(5, 7)>
</pre></div>
<p><a id="X81A8365A7FE68447" name="X81A8365A7FE68447"></a></p>
<h5>4.3-3 ElementsIncidentWithElementOfIncidenceStructure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ElementsIncidentWithElementOfIncidenceStructure</code>( <var class="Arg">el</var>, <var class="Arg">i<p>A <em>flock</m is partitionof the of a quadratic in , . Each conic is determined by a plane, and each plane is determined ajava.lang.StringIndexOutOfBoundsException: Range [197, 196) out of bounds for length 657
<p>Returns: the elements of type <var class="Arg">i</var> incident with <var class="Arg">el</var>, in other words, the type < | | |
