Mathematics Commons^™

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 …

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.

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}.

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 …

Students' Development And Use Of Internal Representations When Solving Algebraic Tasks, Laban J. Cross

Theses and Dissertations

The difficulty in observing, recording, and examining internal representations has been well documented (Goldin & Shteingold, 2001). However, the important role that these internal representations play in the learning and understanding of mathematical concepts has been noted (Yackel, 2000). This study sought to develop a framework for examining the internalization patterns of school-aged children, gain a deeper understanding of how these internal representations are developed and used, and search for internalization patterns associated with successful generalizations.

Semi-structured interviews were conducted with six sixth-grade students and six tenth-grade students. During these interviews the students were asked to solve a series of …

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 …

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 …

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 …

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 …

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 …

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 …

Parametrizing Finite Frames And Optimal Frame Completions, Miriam J. Poteet

Theses and Dissertations

Frames are used in many signal processing applications. We consider the problem of constructing every frame whose frame operator has a given spectrum and whose vectors have prescribed lengths. For a given spectrum and set of lengths, we know when such a frame exists by the Schur-Horn Theorem; it exists if and only if its spectrum majorizes its squared lengths. We provide a more constructive proof of Horn's original result. This proof is based on a new method for constructing any and all frames whose frame operator has a prescribed spectrum and whose vectors have prescribed lengths. Constructing all such …

Methods Of Music Classification And Transcription, Jonathan Peter Baker

Theses and Dissertations

We begin with an overview of some signal processing terms and topics relevant to music analysis including facts about human sound perception. We then discuss common objectives of music analysis and existing methods for accomplishing them. We conclude with an introduction to a new method of automatically transcribing a piece of music from a digital audio signal.

A Maple Program For Computing Landau-Ginzburg A- And B-Models And An Exploration Of Mirror Symmetry, Evan D. Merrell

Theses and Dissertations

Mirror symmetry has been a significant area of research for geometry and physics for over two decades. Berglund and Hubsch proposed that for a certain family of singularities W, the so called "transposed" singularity WT should be the mirror partner of W. cite{BH} The techniques for constructing the orbifold LG models to test this conjecture were developed by FJR in cite{FJR} with a cohomological field theory generalized from the study of r-spin curves. The duality of LG A- and B-models became more elaborate when Krawitz cite{Krawitz} generalized the Intriligator-Vafa orbifold B-model to include contributions from more than one sector.This thesis …

Total Character Groups, Chelsea Lorraine Kennedy

Theses and Dissertations

The total character of a finite group G is the sum of the irreducible characters of G. When the total character of a finite group can be written as a monic polynomial with integer coefficients in an irreducible character of G, we say that G is a total character group. In this thesis we examine the total character of the dicyclic group of order 4n, the non-abelian groups of order p^3, and the symmetric group on n elements for all n ≥ 1. The dicyclic group of order 4n is a total character group precisely when n is congruent to …

Unknotting Tunnels Of Hyperbolic Tunnel Number N Manifolds, Stephan Daniel Burton

Theses and Dissertations

Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n−2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n−1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the …

K-S-Rings, Emma Louise Turner

Theses and Dissertations

For a finite group G we study certain rings called k-S-rings, one for each non-negative integer k, where the 1-S-ring is the centralizer ring of G. These rings have the property that the (k+1)-S-ring determines the k-S-ring. We show that the 4-S-ring determines G when G is any group with finite classes. We show that the 3-S-ring determines G for any finite group G, thus giving an answer to a question of Brauer. We show the 2-characters defined by Frobenius and the extended 2-characters of Ken Johnson are characters of representations of the 2-S-ring of G. We find the character …

Spread Option Pricing With Stochastic Interest Rate, Yi Luo

Theses and Dissertations

In this dissertation, we investigate the spread option pricing problem with stochastic interest rate. First, we will review the basic concept and theories of stochastic calculus, give an introduction of spread options and provide some examples of spread options in different markets. We will also review the market efficiency theory, arbitrage and assumptions that are commonly used in mathematical finance. In Chapter 3, we will review existing spread pricing models and term-structure models such as Vasicek Mode, and the Heath-Jarrow-Morton framework. In Chapter 4, we will use the martingale approach to derive a partial differential equation for the price of …

New Computational Techniques In Fjrw Theory With Applications To Landau Ginzburg Mirror Symmetry, Amanda Francis

Theses and Dissertations

Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A and B models, which are created based on a choice of an affine singularity with a group of symmetries. Both models are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras), and they are also Frobenius manifolds. We give a result relating stabilization of singularities in classical singularity to its counterpart in the Landau-Ginzburg setting. The A model comes from so-called FJRW theory and can be de fined up to a full …

Persistence And Foliation Theory And Their Application To Geometric Singular Perturbation, Ji Li

Theses and Dissertations

Persistence problem of compact invariant manifold under random perturbation is considered in this dissertation. Under uniformly small random perturbation and the condition of normal hyperbolicity, the original invariant manifold persists and becomes a random invariant manifold. The random counterpart has random local stable and unstable manifolds. They could be invariantly foliated thanks to the normal hyperbolicity. Those underlie an extension of the geometric singular perturbation theory to the random case which means the slow manifold persists and becomes a random manifold so that the local global structure near the slow manifold persists under singular perturbation. A normal form for a …

Diagonal Entry Restrictions In Minimum Rank Matrices, And The Inverse Inertia And Eigenvalue Problems For Graphs, Curtis G. Nelson

Theses and Dissertations

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in …

Weak Cayley Table Isomorphisms, Long Pham Bao Nguyen

Theses and Dissertations

We investigate weak Cayley table isomorphisms, a generalization of group isomorphisms. Suppose G and H are groups. A bijective map phi : G to H is a weak Cayley table isomorphism if it satisfies two conditions:(1) If x is conjugate to y, then phi(x) is conjugate to phi(y); (2) For all x, y in G, phi(xy) is conjugate to phi(x)phi(y).If there exists a weak Cayley table isomorphism between two groups, then we say that the two groups have the same weak Cayley table.This dissertation has two main goals. First, we wish to find sufficient conditions under which two groups have …

Using Predictive Analytics To Detect Major Problems In Department Of Defense Acquisition Programs, Austin W. Dowling

Theses and Dissertations

This research provides program analysts and Department of Defense (DoD) leadership with an approach to identify problems in real-time for acquisition contracts. Specifically, we develop optimization algorithms to detect unusual changes in acquisition programs’ Earned Value data streams. The research is focused on three questions. First, can we predict the contractor provided estimate at complete (EAC)? Second, can we use those predictions to develop an algorithm to determine if a problem will occur in an acquisition program or subprogram? Lastly, can we provide the probability of a problem occurring within a given timeframe? We find three of our models establish …

A Women-Only Comparision Of The U.S. Air Force Fitness Test And The Marine Combat Fitness Test, Tarah D. Mitchell

Theses and Dissertations

In 2009, Captain Thomas Worden determined the Air Force Physical Fitness Test (AFPFT) poorly predicted combat capability for his 86 study participants. With only 5 of these 86 volunteers being women, this limited Worden's findings to primarily men. This follow-on research investigated whether these results carried over to women. We recruited 61 female volunteers and compared their performance on the AFPFT to the Marine Combat Fitness Test, the proxy for combat capability. Like Worden's research, we discovered little association between the two (R² of 0.161). However, this association significantly increased (adj R² of 0.572) when utilizing the raw …

Quadratic Spline Approximation Of The Newsvendor Problem Optimal Cost Function, Christina Marie Burton

Theses and Dissertations

We consider a single-product dynamic inventory problem where the demand distributions in each period are known and independent but with density. We assume the lead time and the fixed cost for ordering are zero and that there are no capacity constraints. There is a holding cost and a backorder cost for unfulfilled demand, which is backlogged until it is filled by another order. The problem may be nonstationary, and in fact our approximation of the optimal cost function using splines is most advantageous when demand falls suddenly. In this case the myopic policy, which is most often used in practice …

Metacalibration In Geometric Optimization, Donald C. Sampson

Theses and Dissertations

A introduction to metacalibration methods and their application. This includes a new proof of the double bubble conjecture, new results in the area of equitent problems (isoperimetric problems with boundary), and comments on a mapping conjecture.

Subdivision Rules, 3-Manifolds, And Circle Packings, Brian Craig Rushton

Theses and Dissertations

We study the relationship between subdivision rules, 3-dimensional manifolds, and circle packings. We find explicit subdivision rules for closed right-angled hyperbolic manifolds, a large family of hyperbolic manifolds with boundary, and all 3-manifolds of the E^3,H^2 x R, S^2 x R, SL_2(R), and S^3 geometries (up to finite covers). We define subdivision rules in all dimensions and find explicit subdivision rules for the n-dimensional torus as an example in each dimension. We define a graph and space at infinity for all subdivision rules, and use that to show that all subdivision rules for non-hyperbolic manifolds have mesh not going to …

Dynamic Appointment Scheduling In Healthcare, Mckay N. Heasley

Theses and Dissertations

In recent years, healthcare management has become fertile ground for the scheduling theory community. In addition to an extensive academic literature on this subject, there has also been a proliferation of healthcare scheduling software companies in the marketplace. Typical scheduling systems use rule-based analytics that give schedulers advisory information from programmable heuristics such as the Bailey-Welch rule cite{B,BW}, which recommends overbooking early in the day to fill-in potential no-shows later on. We propose a dynamic programming problem formulation to the scheduling problem that maximizes revenue. We formulate the problem and discuss the effectiveness of 3 different algorithms that solve the …

Topological Properties Of Invariant Sets For Anosov Maps With Holes, Skyler C. Simmons

Theses and Dissertations

We begin by studying various topological properties of invariant sets of hyperbolic toral automorphisms in the linear case. Results related to cardinality, local maximality, entropy, and dimension are presented. Where possible, we extend the results to the case of hyperbolic toral automorphisms in higher dimensions, and further to general Anosov maps.