Consider a function on a

simplicial complex, f : K [right arrow] M where we define the filtration as the sublevei set [f.sup.-1](--[infinity], [alpha]].

Given a graph G, an elimination ordering is an ordering [[upsilon].sub.0], ..., [[upsilon].sub.l] of the vertices [V.sub.G] so that [[upsilon].sub.i] is a

simplicial vertex of the induced sub-graph on [[upsilon].sub.0], ..., [[upsilon].sub.i] for every i = 1, ..., l.

A

simplicial complex K consists of a set of objects, V(K), called vertices and a set, S(K), of finite nonempty subsets of V(K), called simplices such that (i) any nonempty subset of a simplex is also a simplex, (ii) every one element set {v}, where v [member of] V(K), is a simplex, and (iii) the intersection of any two simplices is also a simplex.

Let (X, [tau]; Y) be a triple where X and Y are finite n-dimensional complexes, [tau] is a free

simplicial involution on X for any map f : X [right arrow] Y with Coin(f,f [??] [tau]) = {[x.sub.1],[tau]([x.sub.1]),***, [x.sub.m],[tau]([x.sub.m])} we define the Borsuk-Ulam coincidence set for the pair (f, [tau]), as the set of pairs:

However it is still of interest to investigate if a self-consistent and self-contained discrete electromagnetic theory can be developed on

simplicial mesh using DEC, which has been accomplished based on Yee grid [3,4].

Jurkiewicz, "Four-dimensional

simplicial quantum gravity," Physics Letters.

Plassmann, "Local optimization-based

simplicial mesh untangling and improvement," International Journal for Numerical Methods in Engineering, vol.

We denote the intermediate Delaunay complex, D(t), as the Delaunay complex of B(t), and it is not a

simplicial complex.

Note that plugging in any of the representations of Theorem 6 into (42) yields the polyhedral star-shaped polar coordinate, star-spherical coordinate, and facet-content,

simplicial and dual norm representations of the generalized uniform distribution, respectively.

This obstacle-free motion surface can be modelled, for instance, according to Euclidean buildings, other abstract

simplicial complexes or CAT(0) cube complexes.

One then says a few words about the

simplicial complexes attached to the quadrangulation, which can be thought of as the dual of Baryshnikov's Stokes polytopes.