<h3>2 <span class="Heading">Bits and Pieces</span></h3>
<p>This chapter contains a few very basic functions which are needed for space group calculations and were missing in standard <strong class="pkg">GAP</strong>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignRat</code<>: vector same with in ="SimpleMath">(01\/><p>
<p>Returns: sign of the rational number <ar="Arg">x</var(Standard <strong="pkg">GAP</strong currently hascode class="code">SignInt</>)<p
ass class width<>< ="tdleft>codeclass27; VectorModOnev)
(nbspmethodnbsp;/>
<p>Returns: Rational vector of the same length with enties in <span class="SimpleMath">\<>/>
<p>For a rational vector <var class="Arg">v</var>, this returns the vector with all entries taken "mod 1".</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SignRat((-4)/(-2));</span>
1
<span class="GAPprompt">gap></span> <<p>Returns: <code class"true/odeif v class"rg</var is square and < class"eyw>false
-1
<span classjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
[/1,/9 /6 15]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSquareMat</code>( <var class="Arg">matrix</var> )</td><td class="tdright">( method )</td></tr></pReturnsNumber in matrix class>atrixvar if it squareand< classkeyw</> otherwise<>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">matrix</var> is a square matrix and <code class
<p><a id="X78C932A48515EF10" name, Matrix not called
<h5>2.1-4 DimensionSquareMatjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" widthp>Affine of/> span=SimpleMath(\<spanis alinear and span=SimpleMath\\)/> is avector< classpkgGAP/strong affine byspan="SimpleMath">\+1)/span timesspanclass"\n1\< class=SimpleMath>(M\
<>eturns inthe <var=Arg>atrixvar>ifitsquarecodeclasskeyw>ail/code> otherwise</p>
<h5>2.2-1 LinearPartOfAffineMatOnRightp><aid"" name"X86FBE5B77C2F9442">/a<p> affine with dimensions asas class="Arg>at/p>
<div classp> basis changevar="Arg">transform/var an< class="">\(n\)</span> dimensional induces atransformation affine on space.Ifvar class"Arg>at></var is everything except the row columnpjava.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
<div class
<span="GAPprompt">gap;<spanspanclass>:[01,10]</>
[
<span="GAPprompt">>/> < class"m=[/,,0,0230],101]
[ [ pGiven <var="Arg>java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 175
< ="">&;</> <spanclass="java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0
[span=GAPprompt&;<span=":()
java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 12
<p><a id=/>
<span[, , ,0 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
<div class="func"><table class ="GAPprompt/div>
<pReturns [,, 0,0,[, 0 ], ,, java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
<div="example"><pre
<="GAPprompt">/pan ="APinput">m:TranslationOnRightFromVector,,]);<java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111
<n
[ 1p>: signthe <var="Arg">x/> (Standard< class=pkg</strong only <code="code>/code
[ 0,
[ 0, 0, 1, 0 ],
[ 1, 2, 3, 1] ]
<span class class"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1,0,p>Returns Rationalvector lengthwithin< class"">\([,1))><pjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
<spanjava.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 0
[ ,, 0 >turns definite/p>
</pre></div>
<p> id="8A0B0637F269E37" name"X84A0B0637F269E37"></a>/pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4span>223-2 < class"Heading">/pan<java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
<p> <<a==" =X7BB083A57C474F45p
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: Symmetric
<p>ForpReturns:vector <codeclass"Lengthv1+/java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
"< ="func7(Arg<var class>/>[ ==">tdclass="">(;methodnbsp)//r
<p>Returns: vector of length <code class="code">Length(v1)+1</code></p>
<p>Calculates NC does test =GAPpromptgt< class"/>
<p><<p><a
<h54WhichSideOfHyperplane3
<<div Matrix from
<div java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<p>Returns:pAffinemappings -,8
<-
<<aid
divp<aid"nameX838946957FC75C17"<a<pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<span class="GAPprompt">gap&p>: linear the matrix
[,
< class
[ 5, -10, class< class" width=10%
< =Arg">transform<><iv=>table="" width"<< =tdleft"< class&27 </> <var"">point, class>/ /td="">;methodtd>/><>
=gaps>span>{23*12]/java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
-5
<span
1
<span ="/ var"group: matrix
<
<Letvarjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
<idname< classGAPprompt&pan ="Dsplay(;
h4span=> </span></h4>
<p><a id 1 , 0/>
<h5>2.4-1 PointGroupRepresentatives</h5>
<div class [1,00,0], , , ,0] , ,1 0] 4, 4,1 ]
< class""><table="func" width"100"><tr><td class"func#27 PointGroupRepresentatives/odevar="Arg">group/> )</><td class="tdright">( )/></></table/div>
<p>Returns: list of matrices</p>
<p>Given an <code>2.- GramianOfAverageScalarProductFromFiniteMatrixGroup</h5
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