# Simplex

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SIMPLEX. Simple or single; as, charta simplex, is a deed-poll, of single deed. Jacob's L. Dict. h.t.

A Law Dictionary, Adapted to the Constitution and Laws of the United States. By John Bouvier. Published 1856.
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Its vertexes are the barycenters of the simplexes of K.
We will construct the topological specification using abstract simplexes and complexes.
The vertexes on the common boundary of the two simplexes are local process states that cannot distinguish between the two global states.
Since the sets of input vectors we consider are prefix-closed, we can collect input and output vectors into abstract chromatic complexes (i.e., sets of simplexes closed under containment).
The input complex corresponding to I, denoted I, is the collection of input simplexes S(1) corresponding to the input vectors of I.
The link, written lk(S, C), is the complex consisting of all simplexes in st(S, C) that contain no vertex of S.
(1) for every pair of n-simplexes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in A, there exists a sequence of simplexes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an (n - 1)-simplex, and
The figure's top half shows how the task specification [Delta] takes input simplexes to allowed output simplexes.
Let [I.sub.ij] and [O.sub.ij] denote the input and output simplexes where P has value i and Q has value j.
These three executions correspond to the three simplexes in the subdivision of [I.sub.01], which are carried to [O.sub.00], [O.sub.10], and [O.sub.11].
This complex is not 1-connected--note the "hole" created by the missing simplexes colored with all three values.
To highlight this distinction, we use symbols such as p and q for dimensions of arbitrary simplexes, while n, as usual, is one less than the number of processes, and m typically ranges between 0 and n.

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