We introduce CASC: a new, modern, and header-only C++ library which provides a data structure to represent arbitrary dimension abstract simplicial complexes (ASC) with user-defined classes stored directly on the simplices at each dimension. This is accomplished by using the latest C++ language features including variadic template parameters introduced in C++11 and automatic function return type deduction from C++14. Effectively CASC decouples the representation of the topology from the interactions of user data. We present the innovations and design principles of the data structure and related algorithms. This includes a metadata aware decimation algorithm which is general for collapsing simplices of any dimension. We also present an example application of this library to represent an orientable surface mesh.

Recently, covering arrays have been the subject of considerable research attention as they hold both theoretical interest and practical importance due to their applications to testing. In this thesis, we perform the first comprehensive study of a generalization of covering arrays called variable strength covering arrays, where we dictate the interactions to be covered in the array by modeling them as facets of an abstract simplicial complex. We outline the necessary background in the theory of hypergraphs, combinatorial testing, and design theory that is relevant to the study of variable strength covering arrays. We then approach questions that arise in variable strength covering arrays in a number of ways. We demonstrate their connections to hypergraph homomorphisms, and explore the properties of a particular family of abstract simplicial complexes, the qualitative independence hypergraphs. These hypergraphs are tightly linked to variable strength covering arrays, and we determine and identify several of their important properties and subhypergraphs. We give a detailed study of constructions for variable strength covering arrays, and provide several operations and divide-and-conquer techniques that can be used in building them. In addition, we give a construction using linear feedback shift registers from primitive polynomials of degree 3 over arbitrary finite fields to find variable strength covering arrays, which we extend to strength-3 covering arrays whose sizes are smaller than many of the best known sizes of covering arrays. We then give an algorithm for creating variable strength covering arrays over arbitrary abstract simplicial complexes, which builds the arrays one row at a time, using a density concept to guarantee that the size of the resultant array is asymptotic in the logarithm of the number of facets in the abstract simplicial complex. This algorithm is of immediate practical importance, as it can be used to create test suites for combinatorial testing. Finally, we use the Lovasz Local Lemma to nonconstructively determine upper bounds on the sizes of arrays for a number of different families of hypergraphs. We lay out a framework that can be used for many hypergraphs, and then discuss possible strategies that can be taken in asymmetric problems.

Let B be a finite pseudodisk collection in the plane. By the principle of inclusion-exclusion, the area or any other measure of the union is $$\mu \left( { \cup B} \right) = \sum\limits_{\sigma \in 2^B - \left\{ {\not 0} \right\}} {( - 1)^{card \sigma - 1} \mu \left( { \cap \sigma } \right)} .$$ . We show the existence of a two-dimensional abstract simplicial complex, ź ⊆ 2B, so the above relation holds even if ź is substituted for 2B. In addition, ź can be embedded in ź2 so its underlying space is homotopy equivalent to int ź B, and the frontier of ź is isomorphic to the nerve of the set of boundary contributions.

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on h h -vectors of simplicial polytopes, we show that h h -vectors of pure rank- d d simplicial complexes that have this property satisfy h 0 ≤ h 1 ≤ ⋯ ≤ h [ d / 2 ] h_{0} \leq h_{1} \leq \cdots \leq h_{[d/2]} and h i ≤ h d − i h_{i} \leq h_{d-i} for 0 ≤ i ≤ [ d / 2 ] 0 \leq i \leq [d/2] . We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the h h -vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [Face numbers inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices, Pacific Journal of Math. 154 (1992), 253-264] that the h h -vector of a matroid complex satisfies the above two sets of inequalities.