Search results
Results from the WOW.Com Content Network
The standard simplex or probability simplex [2] is the (k − 1)-dimensional simplex whose vertices are the k standard unit vectors in , or in other words {: + + =, =, …,}. In topology and combinatorics , it is common to "glue together" simplices to form a simplicial complex .
Let K be a geometric simplicial complex (GSC). A subdivision of K is a GSC L such that: [1]: 15 [2]: 3 |K| = |L|, that is, the union of simplices in K equals the union of simplices in L (they cover the same region in space). each simplex of L is contained in some simplex of K.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc.
For each k ≤ n, this has a subcomplex , the k-th horn inside , corresponding to the boundary of the n-simplex, with the k-th face removed. This may be formally defined in various ways, as for instance the union of the images of the n maps Δ n − 1 → Δ n {\displaystyle \Delta ^{n-1}\rightarrow \Delta ^{n}} corresponding to all the other ...
(Simplex Aircraft Co (founders: E J & F W Allen), Defiance, Ohio, United States) Simplex K-2-C Red Arrow; Simplex K-3-C Red Arrow; Simplex K-2-S Red Arrow; Simplex W-2-S Red Arrow; Simplex R-2-D Red Arrow Dual Plane a.k.a. Simplex Racer; Simplex S-2 Kite [4] Simplex Special [4] Simplex W-5-C [4]
Let K and L be two geometric simplicial complexes (GSC). A simplicial map of K into L is a function : such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex , ((())).
There is a 0-simplex of N(C) for each object of C. There is a 1-simplex for each morphism f : x → y in C. Now suppose that f: x → y and g : y → z are morphisms in C. Then we also have their composition gf : x → z. A 2-simplex. The diagram suggests our course of action: add a 2-simplex for this commutative triangle.