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            <td style="vertical
href="aboutPersistent.html"><small style="color: rgb(0, 0, 102);">Previous</small></a000064000064
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            <td
style="java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
style="font-weight: bold;">About HAP: Metric Spaces<br>
            </span></big></td>
            <td style="text-align: right; vertical-align: top;"><a
href="
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      <big><span style="font-weight: bold;"></span></big><br>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><big><span
style="font-weight: bold;">1. Metrics on Vectors</span></big><br>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The

      <a href="data.txt">data file</a> contains a list L of
400 vectorsi in <span style="font-weight: bold;">R</span><sup>3</sup>.
The followingcommandsproduce  400times400  whose
(i,j)-entry is the Manhattan distance between L[i] and L[j].
(Alternative choices java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 72
Hamming metric.)<br>
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style="vertical-align: top; background-color: java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
ReadPackage("hap","www/SideLinks/About/data.txt"); 000064000064000064000064000064000064000064000064000064000064000064000064
#This
reads
in
the
list
L<br>
gap> S:=VectorsToSymmetricMatrix(L,ManhattanMetricF
      </td>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
followingusesjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 71
400
vertices with edge between i and j precisely when <br>
      <br>
      <div style="text-java.lang.StringIndexOutOfBoundsException: Range [72, 66) out of bounds for length 72
      </div>
      <br>
where M is the maximum value of the entries in S. Thus the java.lang.StringIndexOutOfBoundsException: Range [72, 70) out of bounds for length 72
should be chosen in the range from 0 to 100. We choose FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
choose to give the first 200 vertices a common colour distinct from the
remainingvertices. Theshowsthatjava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 69
in one path-component of G(S,8), and the remaining 200 vertices lie in
a second path-component. Each path-component "has the shape" of a
cylinder
or annulus. <br>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
SymmetricMatDisplay(S,8,
[
[1..200],
[201..400]
]
);<br>
      <br>
      <div style="text-align: center;"><img
style="width: 600px; height: 284
      </div>
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      <td
style="vertical-align: top; background-color: rgb(F
following commands construct the graph G=G(S,10) and then display it. <br>
      </td>
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      <td
style="vertical-align: top; background-color: rgb(255java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
M; &;n; & &bsp;nbsp;&;&;nbsp;n n&bsp;n&&&&bsp;nbsp&;n;n&bsp&bsp&bsp&bsp;  n&bsp&bsp;nbsp;nbsp;n&bsp&bsp; nbsp;&; &bsp;bsp;;

      <br>
gap> G:=SymmetricMatrixToGraph(S,10*M/100); <br>
Graph on 400 vertices.<br>
      <br>
gap>
GraphDisplay(G);<br>
      <div style="text-align: center;"><img
style="width: 384px; height: 347px;" alt="" src="400graph.gif"><br>
      </div>
      </td>
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    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
java.lang.StringIndexOutOfBoundsException: Range [24, 3) out of bounds for length 76
of G to mean the simplicial complex
NG which has the same vertices and edges as G; a collection of vertices
is a
is connected by an edge in G. The following commands determine a
subgraph in Gsuch that  nerves java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 70
equivalent. The commands replace G by H and then display the subgraph. </td>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
ContractGraph(G);;<br>
gap> G;<br>
Graph on 248 vertices.<br>
      <br>
gap> GraphDisplay(G);<br>
      <div style="text-align: center;"><img
style="width: 384px; height: 178FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000064000064000064000064000064000064000064
src="400contractedgraph.gif"><br>
      </div>
      </td>
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    <000064FF
      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
methods for calculating the low-dimensional homology of NG. The second
method is more efficient in degrees 0 and 1 but has000064000064000064000064
implemented in higher degrees.<br>
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style="vertical-align: top; background-color: rgbF
#Method
One<br>
gap> NG:=SimplicialNerveOfGraph(G,3);;FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000064000064000064000064FFFFFFFFFFFF
gap> NG:=SimplicialComplexToRegularCWComplex(NG);<br>
Regular CW-complex of dimension 3<br>
gap> Homology(NG,0);<br>
[ 0, 0 ]<br>
gap> Homology(NG,1);<br>
[ 0, 0 ]<br>
      <br>
gap> #Method Two<br>
gap> C:=RipsChainComplex(G,1);<br>
Sparse chain
      <br>
gap> Bettinumbers(C,0);<br>
2<br>
gap> Bettinumbers(C,1);<br>
2<br>
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      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><big><span
style="font-weight: bold;">2.
      </td>
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      <td
style="vertical-align: top; background-colorF00064000064
are
a
number of standard metrics d(x,y) on permutations x, y <sub> </java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
as
the064000064000064000064000064000064000064000064000064000064000064000064
transpositions (i,i+1) needed to express x*y^-1), the Cayley metric (=
number
Hamming metric (= #{ i :  x*y^-1(i) differs from i  } ). The
following 2-subgroup  <sub>0<sub> with
respect to the Cayley metric.<br>
      </td>
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      <td
style=vertical-:top; color rgb(55  204;>java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
G:=SymmetricGroup(10);;     java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72

      <br>
gap> P:=FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
gap> P:=Elements(P);;<br>
gap> S:=NullMat(Size(P),Size(
gap> for i in [1..Size(P)] do<br>
> for FFFFFFFFFF
> S[i][j]:=CayleyMetric(P[i],P[j],10);<br>
FFFFFFFFFFFFFFF
gap> SymmetricMatDisplay(S,50);<br>
      <div style="text-align: center;"
      <img style="width: 355px; height: 384px;" alt=""
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 24
      <div style="text-align: left;"><br>
gap> SymmetricMatDisplay(S,15);<br>
      <br>
      <div style="text-align: FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
style="width: 355px; height: 384px;" alt="" src="sylowS1015.gif"><br>
      </divFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
      </div>
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            <td style="
href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br>
            </td>
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href="aboutTDA.html"><span style=000064
page</span><br>
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