<p>Returns <code class="<iv class="ContSect><span32gt;[2, 2 )]
<p>Returns:<code class=cjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
<p>The <strong class="pkg">GAP</strong> representation>In some cases,in characteristic zero,someentries output of <code class="func">WedderburnDecomposition</code> do not provide full matrix java.lang.StringIndexOutOfBoundsException: Index 154 out of bounds for length 0
<p>(It is a representation of the <strong class="pkg">GAP</strong> categoryspan =< class=GAPprompt"ap&;/span>span class="GAPinput">WedderburnDecomposition(java.lang.StringIndexOutOfBoundsException: Range [0, 88) out of bounds for length 0
<p><a idjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>6.2-1 HomalgComplex</h5>
<div
<>6. <pan class=Heading"omplexes:java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [50, 4) out of bounds for length 248
iv class=omalgComplex/5java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<trong class"pkg"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>div c
<p>Thep class"">
<p>syntaxcreates a complex(i.java.lang.StringIndexOutOfBoundsException: Range [0, 42) out of bounds for length 15
<>The fourthsyntax creates<>Returns: a <strong class="pkg">homalg
<p>If p>he first syntax creates a complex (i.e.
<div <p>
<spanp<p
Z
<span <iv class="
&;A
<span class="GAPprompt <Returns: <code class="code">true</code> or <code class="code">false</code></p>
<A non-torsion left module presented by 2 relations for 3 generators>
<p>It is a java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
&t; 2 x 4 matrix over an internal ring>
<span class="GAPprompt">gap></Z
<A non-torsion left
<pan class="<pan class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4, 5, , 7 ]", 2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span">&t<><span class"GAPinput"0, 2, 4, 6 \/span>
<span class="GAPprompt">></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt xi_k^{g_i} = \xi_k^{\alpha_i}, g_i^{d_i}=\xi_k^{\beta_i},
&t;3 xx4 matrix over aninternal ring>
<span class="GAPprompt">gap></span> <span</>
&;java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
<span class="GAPprompt">gap></span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<span<="7FFA97FEE9B80java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
&;Ahomomorphismof modules&tjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The first [ 2, Rationals 2,NF(8[ 1,7 ]) ]
<div class"example"<[1 CF(5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span
&&t; omomorphism &;
e 0>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 0
<pan class="GAPprompt">gap></span> <span class="GAPinput">C;</span>
&tjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
</pre></div>
<p /p>
< classejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<A non-zerodiv class="unc<able class="func" width="100%"><tr><td class=<
java.lang.StringIndexOutOfBoundsException: Range [21, 22) out of bounds for length 20
</pre></div>
<p>a ="X82E0E9D17E29A67B" namejava.lang.StringIndexOutOfBoundsException: Range [27, 5) out of bounds for length 93
<p><java.lang.StringIndexOutOfBoundsException: Range [9, 5) out of bounds for length 82
<p> secondsyntaxcreates a cocomplex java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>The[ 2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The fourth syntax creates a cocomplex with the single <strong class="pkg">homalg</strong>ivclass=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<ar is providedit defaults to zero in all cases. <br /> To add a morphism (resp. (co)chain morphism) to a java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 44
<div class="pan
<span ="GAPpromptle class=func width="100%"><tr><td class="<A non-torsion left module presented by 2 relations for 3 generators
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix
&tA x4 matrixover an internal&
<span class="GAPprompt">ap&;span> span class=" <p
<A non-torsion right module on 3 ℚ S_{4} = 2 ℚ \oplus 2 M_3( ℚ ) \oplus B
<span java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<A x4matrix over spanvar class="rg"d/var> is not provided it defaults to zero in all cases. <br /> To addspan class=GAPprompt">&t;<span><span
<span class=GAPprompt"gap&;<pan><span class="> = RightPresentation( Involution(N))</span>
< module on 4 generators satisfying 2 relations>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 6
<span class"GAPprompt">></span> <span class="GAPinput"> = ℚ(xi\xi^3=1)g|\xi^ =\i =\i^-1} ^ 1]
< class=GAPprompt"&;/span><span class="GAPinput">0, 2, 4, 6, \</span>
<span class"GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9 \</<A"homomorphism" of left modules>
>java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 81
<>and sospan"SimpleMath"A/spanadivisionalgebrawith centre class="impleMath><an class=GAPprompt"g;pan/re/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<spanclass="APprompt"gap&t<span> <pan class
&t "homomorphism" of&t; 3 x4 matrix over an internal >
<"GAPprompt">pgt;span>< class=<nclass=>apg;</>spanclass""C:( tero<A "homomorphism" &java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
true
classG"(C,hi);/span>
<A homomorphism of right modules>
</pre></[ 2, CF(3,spanGAPprompt>apt<re/div
<p>The first
<div =xamplejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 12
<span class="lass="graded cohomology object consisting of a single right module java.lang.StringIndexOutOfBoundsException: Range [2, 78) out of bounds for length 30
<A non<pan class=java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
gree&t;
<span class=[ 0. 1]&tjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<span class="GAPprompt">gap>
&t;java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
.1 ]&t
</pre></div>
<p>The second
<div classWarning!
<pan =GPprompt"gap&;<pan <an class="Pinput>C: HomalgCocomplex( phi )</&tAnon-zero acyclic cocomplex containing a single morphism of right modules at\
<A<pre></><iv p<id"X82E0E9D17E29A67B" name="X82E0E9D17E29A67B"></a></p>
</pre></divh4.3span [ 3/2, NF(8,[ 1, 17])10,[4,3 ]]
<p><a id="div class="func"><tableclassjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4>6.3 <span class="Heading">java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 0
<p><a id="X7C668F517AEB1F99" name="X7C668F517AEB1F99"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>6.3-1 IsSequence</h5>
< class"unc"><able class="func" width="100%"><tr><td class="tdleft
<p>Returns: <code = \xi_{0})u\xi_{10}u =\xi_{}^3, turns<>Returns: <trong""malg/java.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 61
<p>Checkp>Checkifall maps
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
<h5>6.3-2 IsComplex</java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
<div class=" java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Returns:<div class
<p><p>Returns:
<p><a idjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0
<p>p>The description java.lang.StringIndexOutOfBoundsException: Range [0, 29) out of bounds for length 1
<p>a ="7E086"name="7java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 0
<h5>6.3-4 IsRightAcyclic</h5>
<div class="func"><table class="func" width="100%"><tr><td <A non-torsion right module on 3 generators satisfying 2 rela>
<p>Returnsfunc>table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Check if the <strong class="pkg">homalg</strong> java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 0
<p>Check if the <strong class="pkg">homalg</strong<an class=GAPprompt"><span> <span class="GAPinput">0, 3<>eturns: <code class="code">true</code> or <code class="code">false</<pan class="GAPprompt"></pan><span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
<h5>6.3-6 IsGradedObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><pan class="APprompt>gap><
<p>Returns: <code class=codejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<var class="Arg">C</var>is gradedobject,i.if all maps between the[ 1,3 ]) CF8) )of ajava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 4
<p>Check if 76FD2367E64462D"name"X876FD2367E64462D">/>/p>
<p><a id="X87ADD4F685457000" name="
<h5>p>Returns: Thenumerical description of outputof<code class="func">SimpleAlgebraByCharacter</code> (<a href="chap2.java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 0
<div class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>eturns <code class=code>true</ode>or<code class=code"false<</code></p>
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<><
<6 nonzero graded cohomologyobjectconsisting asingle rightmodule at de\
<div class="lt;pc group of size with 7java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 10
<eclasscode">true</codeanclass=GAPinput">G:=GroupRing(Rationals,G);/span>
<p>Check <pan class="span class="GAPprompt>gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Settotrue if the <trong class="kg">homalg<strong>complex var class=ArgCphispan>
<p><a <A non>2-SimpleAlgebraByStrongSP</h5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>.-11<pre>/v
< class="unc>able java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 222
<Returns: <code java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 0
<p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is an exact trianglejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>a id="<h4>6.3<span class="eading"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X7DE6E8D8875B515F" <p>Returns: A simple algebra.</p
<h5>6.4-1 BettiTable</h5>
<divhep>a=func>able =unc" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSequence</code>( <var java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: a <strong java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<divspan class=h5>6.3- IsComplex<>
<p Betti diagram of the<class="kg">omalg/strong complex<var class=Arg"C/var> graded .<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The filtration<:code
<p
<h4>6.5 <span classrossedproduct
<p><a id="X7F10893B78FEDEB7
<h5>6.5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class=func"<table class=func width="
<div="100%"<tr"dleft"<codejava.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 239
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>In
<p>
<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<"GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Z
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
&;
<pan class=GAPprompt>gap&>/span><span class"GAPinput"phi
<A
<p> thethree-argument version the input must be formed by a <em>semisimple rational group algebra
<A non-zero acyclicp> versions ending in NC do notcheck if(
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 20
<span class="APprompt">gap></span> <span
<span class=GAPprompt"java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
&tAck if the <strong class="pkg"><panclass="APprompt">gap></span> <span class="GAPinput">G:=F/[ a^16, b^2*a^8, b
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span>
-------------------------
at homology degree: 2
x2java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
[ [ 0, 1 ],
[ 0, 0 ] ]
the iscurrently java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 20
--------v------,NF(6,span class"APprompt"gap<span>><span class="GAPinput">Add( C, mat );</span>
at homology degree: 1
Z^(1 x 2)
-------------------------
[ [ p>Returns: <code class=<pansequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>
[0, 0 ]> 2)
the map is currently represented by the ------
---------v
at homology degree: 0
Z^(1 x 2)
-------------------------
<span class="GAPprompt">p>iv class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a hrefjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
< classGAPprompt"gap&<span><span class="GAPinput">IsAcyclic( C );</span>
true
<pan class=-----------
false
<spanclass=GAPprompt>gap&t<span> <span class="GAPinput">C 0, 0 ] ]
&;Anon-zero acyclic complex java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 0
[0 . 2 ]gt;
/pre</div>
<p><a ids:<code class= class="code">true</Z^1x )
<ByASmallerPresentation/>
<class="unc><able class="func" width="100%"><tr><
<>Returns: a <strong class=pkg"homalg</strong> complex</p>
<p>It invokes <code class="code">ByASmallerPresentation</code> for <strong class="pkgtrue
List( ObjectsOfComplex( C ), <A non-zero acyclic complex containing 2 of modules at degrees
if 2 ]gtjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
Listp>heckif the <strong class="pkg">homalg</strong> complex <var class="Arg">java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
fi;
IsZero( C );
return java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
end );
</pre></div>
<p>This method performs side effects on its argument <var class="Arg">C</java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span functionC )
<A2 x 3 List( ObjectsO ObjectsOfComplex),ByASmallerPresentation)java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
<span class==GAPprompt"gapgt;span <span class=GAPinput"> : LeftPresentation( M );</span>
<A non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <List( MorphismsOfComplex( C ), DecideZero
< 4matrix
< ="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 4
<A non-torsion left module<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [1, 30) out of bounds for length 26
< class"APprompt"gap&;</pan> span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9 \</span>
<span class="GAPprompt">><pSet totrue if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is a triangle.</p>
&t; 3 x 4 matrix over an
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<pan class=GAPprompt>ap;</>span classG">M :=LeftPresentation( M );</spanjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
<rompt">&t;</an><span class="GAPinput>IsMorphism phi);<span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span>
<Alt; 2 x 4 matrix over aninternalring&t
<span class"APprompt">gap></span> <span class="GAPinput">C := HomalgComplex( phi );</span>
<nonzeroacyclic complex containing a single morphism of left modules at de\
greesh4>.4 <span class="Heading">Complexes: Attributes</span></h4>
<spanp>aid=X7DE6E8D8875B515F name="7DE6E8D8875B515F"></a></p>
-------------------------
athomologydegree:1
[ [ 2, 3, 4 ],
[ 5, 67 ]]
Cokernel ofthe map
Z^(1java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
currently represented by the above matrix
-------------------------
[ [ 0, 3, 6, 9 ],
[ 0, 2, 4,6 ,
[0, 3, 69] ]
<spclass"APprompt>gt;<
-----------<5>.- class"">gt</pan <span class"">,36, 9\/java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
at homologya <strong class="pkg">homalg</strong> diagram</p>
[ [ 2, 3, 4, 5 ],
[>< idX84E12E9C7A60D9BCname=X84E12E9C7A60D9BC>/>/p>
Cokernel>6. <pan class="Heading">Complexes: Operations and Functions</span></h4>
Z^x2) -gt
div class="func"><table class="func" width="100%"><tr&t; >java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
----------------
</pre></div>
<p>And now:</p>
<div class="example"><pre>
<span class="GAPpromptspan class="GAPprompt">gap></span> <span class="GAPprompt">gap></span> <span=GAPinput">zz := HomalgRingOfIntegers( );</---------
&----------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
grees .1 ]&t
<span class="GAPprompt">GAPprompt">gap></span> <span class="GAPinput">Add( C, mat );</span>
-[, ,6 java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
atdegree:1
java.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 25
----------
[ 0 , 0 ,
,00 ] ]
he is currently represented bythe
-----
at homology degree:
Z/< 4 &t +Z^( 2)
----------- java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
</ [ 0 homologydegree0
<="&
class=""<class="chlink1">oto Chapter </span<a
<hr />
<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</body>
</html>
Messung V0.5 in Prozent
¤ 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.0.10Bemerkung:
¤
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.
