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One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that ...
In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes.It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs.
In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes.One of the assumptions is a non-triviality condition: If the building is an n‑dimensional abstract simplicial complex, and if k < n, then every k‑simplex of the building must be contained in at least three n‑simplices.
A simplicial complex is a topological space of a certain kind, constructed by "gluing together" 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 theory.
Let Δ be an abstract simplicial complex of dimension d − 1 with f i i-dimensional faces and f −1 = 1. These numbers are arranged into the f-vector of Δ, = (,, …,).An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.
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.
A simplicial 3-complex. In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration).
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.