Physical Sciences and Mathematics Commons^™

Finite Groups In Which Pronomality And 𝔉-Pronormality Coincide, Adolfo Ballester-Bolinches, James C. Beidleman, Arnold D. Feldman, Matthew F. Ragland

Mathematics Faculty Publications

For a formation 𝔉, a subgroup U of a finite group G is said to be 𝔉-pronormal in G if for each g ∈ G, there exists x ∈ ⟨U, U^g⟩ ^𝔉 such that U^x = U^g. If 𝔉 contains 𝔑, the formation of nilpotent groups, then every 𝔉-pronormal subgroup is pronormal and, in fact, 𝔑-pronormality is just classical pronormality. The main aim of this paper is to study classes of finite soluble groups in which pronormality and 𝔉-pronormality coincide.

Molecular Network Control Through Boolean Canalization, David Murrugarra, Elena S. Dimitrova

Mathematics Faculty Publications

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges …

Convergence Rates And Hölder Estimates In Almost-Periodic Homogenization Of Elliptic Systems, Zhongwei Shen

Mathematics Faculty Publications

For a family of second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded C^1,α domain.

Deletion-Induced Triangulations, Clifford T. Taylor

Theses and Dissertations--Mathematics

Let d > 0 be a fixed integer and let A ⊆ ℝ^d be a collection of n ≥ d + 2 points which we lift into ℝ^d+1. Further let k be an integer satisfying 0 ≤ k ≤ n-(d+2) and assign to each k-subset of the points of A a (regular) triangulation obtained by deleting the specified k-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector defined by Gelfand, Kapranov, and Zelevinsky by assigning to each …

Homogenization Of Stokes Systems And Uniform Regularity Estimates, Shu Gu, Zhongwei Shen

Mathematics Faculty Publications

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L^∞ estimates for the pressure as well as a Liouville property for solutions in ℝ^d. We also obtain the boundary W^1,p estimates in a bounded C¹ domain for any 1 < p < ∞.

Combinatorial Potpourri: Permutations, Products, Posets, And Pfaffians, Norman B. Fox

Theses and Dissertations--Mathematics

In this dissertation we first examine the descent set polynomial, which is defined in terms of the descent set statistics of the symmetric group. Algebraic and topological tools are used to explain why large classes of cyclotomic polynomials are factors of the descent set polynomial. Next the diamond product of two Eulerian posets is studied, particularly by examining the effect this product has on their cd-indices. A combinatorial interpretation involving weighted lattice paths is introduced to describe the outcome of applying the diamond product operator to two cd-monomials. Then the cd-index is defined for infinite posets, with …

Analysis And Constructions Of Subspace Codes, Carolyn E. Troha

Theses and Dissertations--Mathematics

Random network coding is the most effcient way to send data across a network, but it is very susceptible to errors and erasures. In 2008, Kotter and Kschischang introduced subspace codes as an algebraic approach to error correcting in random network coding. Since this paper, there has been much work in constructing large subspace codes, as well as exploring the properties of such codes. This dissertation explores properties of one particular construction and introduces a new construction for subspace codes. We begin by exploring properties of irreducible cyclic orbit codes, which were introduced in 2011 by Rosenthal et al. As …

Free Resolutions Associated To Representable Matroids, Nicholas D. Armenoff

Theses and Dissertations--Mathematics

As a matroid is naturally a simplicial complex, one can study its combinatorial properties via the associated Stanley-Reisner ideal and its corresponding free resolution. Using results by Johnsen and Verdure, we prove that a matroid is the dual to a perfect matroid design if and only if its corresponding Stanley-Reisner ideal has a pure free resolution, and, motivated by applications to their generalized Hamming weights, characterize free resolutions corresponding to the vector matroids of the parity check matrices of Reed-Solomon codes and certain BCH codes. Furthermore, using an inductive mapping cone argument, we construct a cellular resolution for the matroid …

Unimodality Questions In Ehrhart Theory, Robert Davis

Theses and Dissertations--Mathematics

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h^*-vector.Although various sufficient conditions have been found, necessary conditions remain a challenge. Highly-structured polytopes, such as the polytope of real doubly-stochastic matrices, have been proven to possess unimodal h^*-vectors, but the same is unknown even for small variations of it.

In this dissertation, we mainly consider two particular classes of polytopes: reflexive simplices and the polytope of symmetric real doubly-stochastic matrices. For the first class, we discuss an operation that preserves reflexivity, integral closure, and unimodality of the h^{* …}

Determinantal Ideals From Symmetrized Skew Tableaux, Bill Robinson

Theses and Dissertations--Mathematics

We study a class of determinantal ideals called skew tableau ideals, which are generated by t x t minors in a subset of a symmetric matrix of indeterminates. The initial ideals have been studied in the 2 x 2 case by Corso, Nagel, Petrovic and Yuen. Using liaison techniques, we have extended their results to include the original determinantal ideals in the 2 x 2 case, as well as special cases of the ideals in the t x t case. In particular, for any skew tableau ideal of this form, we have defined an elementary biliaison between it and one …

Polyhedral Problems In Combinatorial Convex Geometry, Liam Solus

Theses and Dissertations--Mathematics

In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem.

In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a …