<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Homology</code>( <var class="Arg">T</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Homology</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex, or cubical complex, or simplicial complex <span class="SimpleMath">T</span> and a non-negative integer <span class="SimpleMath">n</span>. It returns the n-th integral homology of <span class="SimpleMath">T</span> as a list of torsion integers. If no value of <span class="SimpleMath">n</span> is input then the list of all homologies of <span class="SimpleMath">T</span> in dimensions 0 to Dimension(T) is returned .</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RipsHomology</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RipsHomology</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">p</var> )</td><tdclass="tdright">( function )</td></tr></table></div>
<p>Inputs a graph <span class="SimpleMath">G</span>, a non-negative integer <span class="SimpleMath">n</span> (and optionally a prime number <span class="SimpleMath">p</span>). It returns the integral homology (or mod p homology) in degree <span class="SimpleMath">n</span> of the Rips complex of <span class="SimpleMath">G</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bettinumbers</code>( <var class="Arg">T</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bettinumbers</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex, or cubical complex, simplicial complex or chain complex <span class="SimpleMath">T</span> and a non-negative integer <span class="SimpleMath">n</span>. The rank of the n-th rational homology group <span class="SimpleMath">H_n(T, Q)</span> is returned. If no value for n is input then the list of Betti numbers in dimensions 0 to Dimension(T) is returned .</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChainComplex</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex, or cubical complex, or simplicial complex <span class="SimpleMath">T</span> and returns the (often very large) cellular chain complex of <span class="SimpleMath">T</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CechComplexOfPureCubicalComplex</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a d-dimensional pure cubical complex <span class="SimpleMath">T</span> and returns a simplicial complex <span class="SimpleMath">S</span>. The simplicial complex <span class="SimpleMath">S</span> has one vertex for each d-cube in <span class="SimpleMath">T</span>, and an n-simplex for each collection of n+1 d-cubes with non-trivial common intersection. The homotopy types of <span class="SimpleMath">T</span> and <span class="SimpleMath">S</span> are equal.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PureComplexToSimplicialComplex</code>( <var class="Arg">T</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a d-dimensional pure cubical complex <span class="SimpleMath">T</span> or a d-dimensional pure permutahedral complex <span class="SimpleMath">T</span> together with a non-negative integer <span class="SimpleMath">k</span>. It returns the first <span class="SimpleMath">k</span> dimensions of a simplicial complex <span class="SimpleMath">S</span>. The simplicial complex <span class="SimpleMath">S</span> has one vertex for each d-cell in <span class="SimpleMath">T</span>, and an n-simplex for each collection of n+1 d-cells with non-trivial common intersection. The homotopy types of <span class="SimpleMath">T</span> and <span class="SimpleMath">S</span> are equal.</p>
<p>For a pure cubical complex <span class="SimpleMath">T</span> this uses a slightly different algorithm to the function CechComplexOfPureCubicalComplex(T) but constructs the same simplicial complex.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RipsChainComplex</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph <span class="SimpleMath">G</span> and a non-negative integer <span class="SimpleMath">n</span>. It returns <span class="SimpleMath">n+1</span> terms of a chain complex whose homology is that of the nerve (or Rips complex) of the graph in degrees up to <span class="SimpleMath">n</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorsToSymmetricMatrix</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorsToSymmetricMatrix</code>( <var class="Arg">M</var>, <var class="Arg">distance</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a matrix <span class="SimpleMath">M</span> of rational numbers and returns a symmetric matrix <span class="SimpleMath">S</span> whose <span class="SimpleMath">(i,j)</span> entry is the distance between the <span class="SimpleMath">i</span>-th row and <span class="SimpleMath">j</span>-th rows of <span class="SimpleMath">M</span> where distance is given by the sum of the absolute values of the coordinate differences.</p>
<p>Optionally, a function distance(v,w) can be entered as a second argument. This function has to return a rational number for each pair of rational vectors <span class="SimpleMath">v,w</span> of length Length(M[1]).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EulerCharacteristic</code>( <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex, or cubical complex, or simplicial complex <span class="SimpleMath">T</span> and returns its Euler characteristic.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalSimplicesToSimplicialComplex</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list L whose entries are lists of vertices representing the maximal simplices of a simplicial complex. The simplicial complex is returned. Here a "vertex" is a GAP object such as an integer or a subgroup.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SkeletonOfSimplicialComplex</code>( <var class="Arg">S</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a simplicial complex <span class="SimpleMath">S</span> and a positive integer <span class="SimpleMath">k</span> less than or equal to the dimension of <span class="SimpleMath">S</span>. It returns the truncated <span class="SimpleMath">k</span>-dimensional simplicial complex <spanclass="SimpleMath">S^k</span> (and leaves <span class="SimpleMath">S</span> unchanged).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PathComponentsOfSimplicialComplex</code>( <var class="Arg">S</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a simplicial complex <span class="SimpleMath">S</span> and a nonnegative integer <span class="SimpleMath">n</span>. If <span class="SimpleMath">n=0</span> the number of path components of <span class="SimpleMath">S</span> is returned. Otherwise the n-th path component is returned (as a simplicial complex).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuillenComplex</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">G</span> and returns, as a simplicial complex, the order complex of the poset of non-trivial elementary abelian subgroups of <span class="SimpleMath">G</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatrixToIncidenceMatrix</code>( <var class="Arg">S</var>, <var class="Arg">t</var> )</td><tdclass="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricMatrixToIncidenceMatrix</code>( <var class="Arg">S</var>, <var class="Arg">t</var>, <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a symmetric integer matrix S and an integer t. It returns the matrix <span class="SimpleMath">M</span> with <span class="SimpleMath">M_ij=1</span> if <span class="SimpleMath">I_ij</span> is less than <span class="SimpleMath">t</span> and <span class="SimpleMath">I_ij=1</span> otherwise.</p>
<p>An optional integer <span class="SimpleMath">d</span> can be given as a third argument. In this case the incidence matrix should have roughly at most <span class="SimpleMath">d</span> entries in each row (corresponding to the <span class="SimpleMath">d</span> smallest entries in each row of <span class="SimpleMath">S</span>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IncidenceMatrixToGraph</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a symmetric 0/1 matrix M. It returns the graph with one vertex for each row of <span class="SimpleMath">M</span> and an edges between vertices <span class="SimpleMath">i</span> and <span class="SimpleMath">j</span> if the <span class="SimpleMath">(i,j)</span> entry in <span class="SimpleMath">M</span> equals 1.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CayleyGraphOfGroup</code>( <var class="Arg">G</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> and a set <span class="SimpleMath">A</span> of generators. It returns the Cayley graph.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PathComponentsOfGraph</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph <span class="SimpleMath">G</span> and a nonnegative integer <span class="SimpleMath">n</span>. If <span class="SimpleMath">n=0</span> the number of path components is returned. Otherwise the n-th path component is returned (as a graph).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContractGraph</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph <span class="SimpleMath">G</span> and tries to remove vertices and edges to produce a smaller graph <span class="SimpleMath">G' such that the indlusion G' → G</span> induces a homotopy equivalence <span class="SimpleMath">RG → RG' of Rips complexes. If the graph G is modified the function returns true, and otherwise returns false.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimplicialMap</code>( <var class="Arg">K</var>, <var class="Arg">L</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimplicialMapNC</code>( <var class="Arg">K</var>, <var class="Arg">L</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs simplicial complexes <span class="SimpleMath">K</span> , <span class="SimpleMath">L</span> and a function <span class="SimpleMath">f: K!.vertices → L!.vertices</span> representing a simplicial map. It returns a simplicial map <span class="SimpleMath">K → L</span>. If <span class="SimpleMath">f</span> does not happen to represent a simplicial map then SimplicialMap(K,L,f) will return fail; SimplicialMapNC(K,L,f) will not do any check and always return something of the data type "simplicial map".</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimplicialNerveOfGraph</code>( <var class="Arg">G</var>, <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph <span class="SimpleMath">G</span> and returns a <span class="SimpleMath">d</span>-dimensional simplicial complex <span class="SimpleMath">K</span> whose 1-skeleton is equal to <span class="SimpleMath">G</span>. There is a simplicial inclusion <span class="SimpleMath">K → RG</span> where: (i) the inclusion induces isomorphisms on homotopy groups in dimensions less than <span class="SimpleMath">d</span>; (ii) the complex <span class="SimpleMath">RG</span> is the Rips complex (with one <span class="SimpleMath">n</span>-simplex for each complete subgraph of <span class="SimpleMath">G</span> on <span class="SimpleMath">n+1</span> vertices).</p>
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