  
  [1X33 [33X[0;0YFinite metric spaces and their filtered complexes[133X[101X
  
  
  [1X33.1 [33X[0;0Y [133X[101X
  
  [1X33.1-1 CayleyMetric[101X
  
  [33X[1;0Y[29X[2XCayleyMetric[102X( [3Xg[103X, [3Xh[103X, [3XN[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCayleyMetric[102X( [3Xg[103X, [3Xh[103X ) [32X function[133X
  
  [33X[0;0YInputs two permutations [22Xg,h[122X and optionally the degree [22XN[122X of a symmetric group
  containing  them.  It returns the minimum number of transpositions needed to
  express [22Xg*h^-1[122X as a product of transpositions.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutMetrics.html[107X) [133X
  
  [1X33.1-2 HammingMetric[101X
  
  [33X[1;0Y[29X[2XHammingMetric[102X( [3Xg[103X, [3Xh[103X, [3XN[103X ) [32X function[133X
  [33X[1;0Y[29X[2XHammingMetric[102X( [3Xg[103X, [3Xh[103X ) [32X function[133X
  
  [33X[0;0YInputs two permutations [22Xg,h[122X and optionally the degree [22XN[122X of a symmetric group
  containing  them. It returns the number of integers moved by the permutation
  [22Xg*h^-1[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X33.1-3 KendallMetric[101X
  
  [33X[1;0Y[29X[2XKendallMetric[102X( [3Xg[103X, [3Xh[103X, [3XN[103X ) [32X function[133X
  [33X[1;0Y[29X[2XKendallMetric[102X( [3Xg[103X, [3Xh[103X ) [32X function[133X
  
  [33X[0;0YInputs two permutations [22Xg,h[122X and optionally the degree [22XN[122X of a symmetric group
  containing  them.  It  returns the minimum number of adjacent transpositions
  needed  to  express  [22Xg*h^-1[122X  as  a  product  of  adjacent transpositions. An
  adjacent transposition has the for [22X(i,i+1)[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X33.1-4 EuclideanSquaredMetric[101X
  
  [33X[1;0Y[29X[2XEuclideanSquaredMetric[102X( [3Xv[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs two vectors [22Xv,w[122X of equal length and returns the sum of the squares of
  the  components  of  [22Xv-w[122X.  In  other  words,  it  returns  the square of the
  Euclidean distance between [22Xv[122X and [22Xw[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X33.1-5 EuclideanApproximatedMetric[101X
  
  [33X[1;0Y[29X[2XEuclideanApproximatedMetric[102X( [3Xv[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs  two vectors [22Xv,w[122X of equal length and returns a rational approximation
  to  the  square  root of the sum of the squares of the components of [22Xv-w[122X. In
  other words, it returns an approximation to the Euclidean distance between [22Xv[122X
  and [22Xw[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../tutorial/chap5.html[107X) , 2 ([7X../tutorial/chap10.html[107X) [133X
  
  [1X33.1-6 ManhattanMetric[101X
  
  [33X[1;0Y[29X[2XManhattanMetric[102X( [3Xv[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs  two  vectors [22Xv,w[122X of equal length and returns the sum of the absolute
  values  of  the components of [22Xv-w[122X. This is often referred to as the taxi-cab
  distance between [22Xv[122X and [22Xw[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutMetrics.html[107X) [133X
  
  [1X33.1-7 VectorsToSymmetricMatrix[101X
  
  [33X[1;0Y[29X[2XVectorsToSymmetricMatrix[102X( [3XL[103X ) [32X function[133X
  [33X[1;0Y[29X[2XVectorsToSymmetricMatrix[102X( [3XL[103X, [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  list  [22XL[122X  of  vectors  and  optionally  a metric [22XD[122X. The default is
  [22XD=ManhattanMetric[122X.  It  returns  the  symmetric  matrix  whose  i-j-entry is
  [22XS[i][j]=D(L[i],L[j])[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap10.html[107X) ,            2
  ([7X../www/SideLinks/About/aboutMetrics.html[107X) [133X
  
  [1X33.1-8 SymmetricMatDisplay[101X
  
  [33X[1;0Y[29X[2XSymmetricMatDisplay[102X( [3XS[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSymmetricMatDisplay[102X( [3XL[103X, [3XV[103X ) [32X function[133X
  
  [33X[0;0YInputs an [22Xn × n[122X symmetric matrix [22XS[122X of non-negative integers and an integer [22Xt[122X
  in  [22X[0  ..  100][122X. Optionally it inputs a list [22XV=[V_1, ... , V_k][122X of disjoint
  subsets of [22X[1 .. n][122X. It displays the graph with vertex set [22X[1 .. n][122X and with
  an edge between [22Xi[122X and [22Xj[122X if [22XS[i][j] < t[122X. If the optional list [22XV[122X is input then
  the  vertices  in  [22XV_i[122X  will  be  given  a common colour distinct from other
  vertices.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutMetrics.html[107X) [133X
  
  [1X33.1-9 SymmetricMatrixToFilteredGraph[101X
  
  [33X[1;0Y[29X[2XSymmetricMatrixToFilteredGraph[102X( [3XS[103X, [3Xt[103X, [3Xm[103X ) [32X function[133X
  
  [33X[0;0YInputs  an  integer  symmetric matrix [22XS[122X, a positive integer [22Xt[122X and a positive
  integer [22Xm[122X. The function returns a filtered graph of filtration length [22Xt[122X. The
  [22Xk[122X-th  term  of  the filtration is a graph with one vertex for each row of [22XS[122X.
  There  is  an  edge  in this graph between the [22Xi[122X-th and [22Xj[122X-th vertices if the
  entry [22XS[i][j][122X is less than or equal to [22Xk*m/t[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X   1  ([7X../tutorial/chap5.html[107X) ,  2  ([7X../tutorial/chap10.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutPersistent.html[107X) [133X
  
  [1X33.1-10 PermGroupToFilteredGraph[101X
  
  [33X[1;0Y[29X[2XPermGroupToFilteredGraph[102X( [3XS[103X, [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  permutation  group  [22XG[122X and a metric [22XD[122X defined on permutations. The
  function  returns  a  filtered  graph.  The [22Xk[122X-th term of the filtration is a
  graph  with  one vertex for each element of the group [22XG[122X. There is an edge in
  this  graph  between  vertices  [22Xg[122X  and [22Xh[122X if [22XD(g,h)[122X is less than some integer
  threshold  [22Xt_k[122X.  The  thresholds [22Xt_1 < t_2 < ... < t_N[122X are chosen to form as
  long  a  sequence as possible subject to each term of the filtration being a
  distinct graph.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
