Physical Sciences and Mathematics Commons^™

Coloring Pythagorean Triples And A Problem Concerning Cyclotomic Polynomials, Daniel White

Theses and Dissertations

One may easily show that there exist $O( \log n)$-colorings of $\{1,2, \ldots, n\}$ such that no Pythagorean triple with elements $\le n$ is monochromatic. In Chapter~\ref{CH:triples}, we investigate two analogous ideas. First, we find an asymptotic bound for the number of colors required to color $\{1,2,\ldots ,n\}$ so that every Pythagorean triple with elements $\le n$ is $3$-colored. Afterwards, we examine the case where we allow a vanishing proportion of Pythagorean triples with elements $\le n$ to fail to have this property.

Unrelated, in 1908, Schur raised the question of the irreducibility over $\Q$ of polynomials of the form …

Shimura Images Of A Family Of Half-Integral Weight Modular Forms, Kenneth Allan Brown

Theses and Dissertations

In 1973, Shimura introduced a family of maps between modular forms of half-integral weight and modular forms of even integral weight. We will give explicit formulas for the images of two different classes of modular forms under these maps. In contrast to Shimura's difficult analytic construction, our formulas will fall out of relatively simple combinatorial derivations. Using the Shimura correspondence, we will prove congruences for the eigenvalues of a family of eigenforms introduced by Garvan. Using deep results of Eichler and Shimura, we state these congruences in terms of the number of points on associated elliptic curves, and we provide …

Generalizations Of Sperner's Theorem: Packing Posets, Families Forbidding Posets, And Supersaturation, Andrew Philip Dove

Theses and Dissertations

Sperner's Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an $n$-set, $[n]:=\{1,2,\dots,n\}$, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset $P$, we are interested in maximizing the size of a family $\mathcal{F}$ of subsets of $[n]$, where each maximally connected component of $\mathcal{F}$ is a copy of $P$, and finding the extreme configurations that achieve this value. For instance, Sperner showed …

Analysis And Processing Of Irregularly Distributed Point Clouds, Kamala Hunt Diefenthaler

Theses and Dissertations

We address critical issues arising in the practical implementation of processing real point cloud data that exhibits irregularities. We develop an adaptive algorithm based on Learning Theory for processing point clouds from a stationary sensor that standard algorithms have difficulty approximating. Moreover, we build the theory of distribution-dependent subdivision schemes targeted at representing curves and surfaces with gaps in the data. The algorithms analyze aggregate quantities of the point cloud over subdomains and predict these quantities at the finer level from the ones at the coarser level.

The Compact Implicit Integration Factor Scheme For The Solution Of Allen-Cahn Equations, Meshack K. Kiplagat

Theses and Dissertations

In this thesis we apply the compact implicit integration factor (cIIF) scheme towards solving the Allen-Cahn equations with zero-flux or periodic boundary conditions. The Allen-Cahn equation is a second-order nonlinear PDE which has been the focus of many applications spanning a wide range of fields, such as in material science where it was first introduced to model the phase separation of two metallic alloys, and in biology to study population dynamics, just to name a few. The compact implicit integration method works by first transforming the PDE into a system of ODEs by discretizing the spatial derivatives using the central …

Deducing Vertex Weights From Empirical Occupation Times, David Collins

Theses and Dissertations

We consider the following problem arising from the study of human problem solving: Let $G$ be a vertex-weighted digraph with marked "start" and "end" vertices. Suppose that a random walker begins at the start vertex, steps to neighbors of vertices with probability proportional to their weights, and stops upon reaching the end vertex. Could one deduce the weights from the paths that many such walkers take? We analyze an iterative numerical solution to this reconstruction problem, in particular, given the empirical mean occupation times of the walkers. We then consider the existence of a choice of weights for a given …

The Weierstrass Approximation Theorem, Larita Barnwell Hipp

Theses and Dissertations

In this thesis we will consider the work began by Weierstrass in 1855 and several generalization of his approximation theorem since. Weierstrass began by proving the density of algebraic polynomials in the space of continuous real-valued functions on a finite interval in the uniform norm. His theorem has been generalized to an arbitrary compact Hausdorff space and the approximation with elements from more general algebras of continuous real-valued functions. We will consider proofs that use brute force and proofs based on convolutions and approximate identities, trudge through probability and the use of the Bernstein polynomials, and become intimately close to …

Sharp Bounds Associated With An Irreducibility Theorem For Polynomials Having Non-Negative Coefficients, Morgan Cole

Theses and Dissertations

Consider a polynomial f(x) having non-negative integer coefficients with f(b) prime for some integer b greater than or equal to 2. We will investigate the size of the coefficients of the polynomial and establish a largest such bound on the coefficients that would imply that f(x) is irreducible. A result of Filaseta and Gross has established sharp bounds on the coefficients of such a polynomial in the case that b = 10. We will expand these results for b in {8, 9, ..., 20}.

Spectral Analysis Of Randomly Generated Networks With Prescribed Degree Sequences, Clifford Davis Gaddy

Theses and Dissertations

Network science attempts to capture real-world phenomenon through mathematical models. The underlying model of a network relies on a mathematical structure called a graph. Having seen its early beginnings in the 1950's, the field has seen a surge of interest over the last two decades, attracting interest from a range of scientists including computer scientists, sociologists, biologists, physicists, and mathematicians. The field requires a delicate interplay between real-world modeling and theory, as it must develop accurate probabilistic models and then study these models from a mathematical perspective. In my thesis, we undertake a project involving computer programming in which we …

Applications Of The Lopsided Lovász Local Lemma Regarding Hypergraphs, Austin Tyler Mohr

Theses and Dissertations

The Lovász local lemma is a powerful and well-studied probabilistic technique useful in establishing the possibility of simultaneously avoiding every event in some collection. A principle limitation of the lemma's application is that it requires most events to be independent of one another. The lopsided local lemma relaxes the requirement of independence to negative dependence, which is more general but also more difficult to identify. We will examine general classes of negative dependent events involving maximal matchings of uniform hypergraphs, partitions of sets, and spanning trees of complete graphs. The results on hypergraph matchings (together with the configuration model of …

Study On Covolume-Upwind Finite Volume Approximations For Linear Parabolic Partial Differential Equations, Rosalia Tatano

Theses and Dissertations

In this thesis we solve two-dimensional linear parabolic partial differential equations with pure Dirichelet boundary conditions, using the bilinear covolume-upwind finite volume method on rectangular grids to discretize the spatial variables and the Crank-Nicholson method for the time variable. These PDEs provide a model for problems from various fields of engineering and applied sciences, such as unsteady viscous flow problems, the simulation of oil extraction from underground reservoirs, transport of air and ground water pollutants and modeling of semiconductor devices. Finite volume method has the important advantage of allowing the conversion of integrations over the control volume to integrations over …

Selected Research In Covering Systems Of The Integers And The Factorization Of Polynomials, Joshua Harrington

Theses and Dissertations

In 1960, Sierpi\'{n}ski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. Such integers are known as Sierpi\'{n}ski numbers. Letting $f(x)=ax^r+bx+c\in\mathbb{Z}[x]$, Chapter 2 of this document explores the existence of integers $k$ such that $f(k)2^n+d$ is composite for all positive integers $n$. Chapter 3 then looks into a polynomial variation of a similar question. In particular, Chapter~\ref{CH:FH} addresses the question, for what integers $d$ does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ with $f(1)\neq -d$ such that $f(x)x^n+d$ is reducible for all positive integers $n$. The last two chapters of …