The notion of simplicial complexes is usually introduced at the graduate level, as part of the study of algebraic topology. So the main objective of this paper is to organize and present materials found in the literature so that it is accessible to students at the undergraduate level. We first generalize linear algebra to affine linear algebra. Then using these results, we define, and subsequen...

A fast algorithm for calculating the simplicial depth of a single parameter value of a polynomial regression model and for calculating the maximum simplicial depth within an affine subspace of the parameter space or a polyhedron is presented. Since the maximum simplicial depth estimator is not unique, l1 and l2 methods are used to make the estimator unique. This estimator is compared with other...

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices o...

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

We show that the quantum family of all maps from a finite space to a finite dimensional compact quantum semigroup has a canonical quantum semigroup structure. Let X be a finite space and Y be a compact space. Then the space F (X,Y ) of all maps from X to Y , is a compact space with compact-open topology. The notion of quantum family of all maps, defined by Woronowicz [8] and Soltan [5], is a no...

Homology theories for algebraic varieties are often constructed using simplicial sets of algebraic cycles. For example, Bloch’s higher Chow groups and motivic cohomology, as defined by Suslin and Voevodsky [17], are given in this fashion. In this paper, we construct homology groups Hi(X,G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG×X, an...