Physical Sciences and Mathematics Commons^™

A Combinatorial Miscellany: Antipodes, Parking Cars, And Descent Set Powers, Alexander Thomas Happ

Theses and Dissertations--Mathematics

In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the 𝑟th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results …

Polytopes Associated To Graph Laplacians, Marie Meyer

Theses and Dissertations--Mathematics

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, P_G, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, P_D.

Basic properties of both families of simplices, P_G and P …

Bounded Point Derivations On Certain Function Spaces, Stephen Deterding

Theses and Dissertations--Mathematics

Let 𝑋 be a compact subset of the complex plane and denote by 𝑅^𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿^𝑝(𝑋) norm. We show that if a point 𝑥₀ admits a bounded point derivation on 𝑅^𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥₀. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if …

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Theses and Dissertations--Mathematics

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H¹-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, …

Hilbert Bases, Descent Statistics, And Combinatorial Semigroup Algebras, Mccabe J. Olsen

Theses and Dissertations--Mathematics

The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials.

Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Gröbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory.

In Chapter …