Search results
Results from the WOW.Com Content Network
Geometric realization of a 3-dimensional abstract simplicial complex. In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family.
An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC.
A simplicial 3-complex. In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). ). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy th
A simplicial complex that has a sequence of collapses leading to a point is called collapsible. Every collapsible complex is contractible , but the converse is not true. This definition can be extended to CW-complexes and is the basis for the concept of simple-homotopy equivalence .
The order complex of a poset P is an ASC whose vertices are the elements of P and whose simplices are the chains of P. The first barycentric subdivision of an ASC K is the order complex of its face poset. [3]: 18-19 The order complex of the above poset is the closure of the following simplices: { {A} , {A,B} , {A,B,C} } { {A} , {A,C} , {A,B,C} }
Every flag complex is a clique complex: given a flag complex, define a graph G on the set of all vertices, where two vertices u,v are adjacent in G iff {u,v} is in the complex (this graph is called the 1-skeleton of the complex). By definition of a flag complex, every set of vertices that are pairwise-connected, is in the complex.
The search engine that helps you find exactly what you're looking for. Find the most relevant information, video, images, and answers from all across the Web.
Constructing the Čech complex of a set of points sampled from a circle. In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distribution it is drawn from.