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Theses/Dissertations

University of Central Florida

Mathematics

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Lattice-Valued T-Filters And Induced Structures, Frederick Reid May 2019

#### Lattice-Valued T-Filters And Induced Structures, Frederick Reid

##### Electronic Theses and Dissertations

A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional. Subcategories of compact and of complete spaces are ...

Estimation And Clustering In Statistical Ill-Posed Linear Inverse Problems, Rasika Rajapakshage Jan 2019

#### Estimation And Clustering In Statistical Ill-Posed Linear Inverse Problems, Rasika Rajapakshage

##### Electronic Theses and Dissertations

The main focus of the dissertation is estimation and clustering in statistical ill-posed linear inverse problems. The dissertation deals with a problem of simultaneously estimating a collection of solutions of ill-posed linear inverse problems from their noisy images under an operator that does not have a bounded inverse, when the solutions are related in a certain way. The dissertation defense consists of three parts. In the first part, the collection consists of measurements of temporal functions at various spatial locations. In particular, we study the problem of estimating a three-dimensional function based on observations of its noisy Laplace convolution. In ...

Two Ramsey-Related Problems, Jingmei Zhang Jan 2019

#### Two Ramsey-Related Problems, Jingmei Zhang

##### Electronic Theses and Dissertations

Extremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades. It deals with the problem of determining or estimating the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. In this dissertation, we focus on studying the minimum number of edges of certain co-critical graphs. Given an integer r ≥ 1 and graphs G; H1; : : : ;Hr, we write → G (H1; : : : ;Hr) if every r-coloring of the edges of G contains a monochromatic copy of Hi in color i for some i ϵ {1; : : : ; r}. A ...

Jan 2019

#### Mathematical Investigation Of The Spatial Spread Of An Infectious Disease In A Heterogeneous Environment, Arielle Gaudiello

##### Electronic Theses and Dissertations

Outbreaks of infectious diseases can devastate a population. Researchers thus study the spread of an infection in a habitat to learn methods of control. In mathematical epidemiology, disease transmission is often assumed to adhere to the law of mass action, yet there are numerous other incidence terms existing in the literature. With recent global outbreaks and epidemics, spatial heterogeneity has been at the forefront of these epidemiological models. We formulate and analyze a model for humans in a homogeneous population with a nonlinear incidence function and demographics of birth and death. We allow for the combination of host immunity after ...

Frames And Phase Retrieval, Ted Juste Jan 2019

#### Frames And Phase Retrieval, Ted Juste

##### Electronic Theses and Dissertations

Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring a frame's redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be ...

Semi-Analytical Solutions Of Non-Linear Differential Equations Arising In Science And Engineering, Mangalagama Dewasurendra Jan 2019

#### Semi-Analytical Solutions Of Non-Linear Differential Equations Arising In Science And Engineering, Mangalagama Dewasurendra

##### Electronic Theses and Dissertations

Systems of coupled non-linear differential equations arise in science and engineering are inherently nonlinear and difficult to find exact solutions. However, in the late nineties, Liao introduced Optimal Homotopy Analysis Method (OHAM), and it allows us to construct accurate approximations to the systems of coupled nonlinear differential equations. The drawback of OHAM is, we must first choose the proper auxiliary linear operator and then solve the linear higher-order deformation equation by spending lots of CPU time. However, in the latest innovation of Liao's "Method of Directly Defining inverse Mapping (MDDiM)" which he introduced to solve a single nonlinear ordinary ...

Jan 2019

#### Solution Of Linear Ill-Posed Problems Using Overcomplete Dictionaries, Pawan Gupta

##### Electronic Theses and Dissertations

In this dissertation, we consider an application of overcomplete dictionaries to the solution of general ill-posed linear inverse problems. In the context of regression problems, there has been an enormous amount of effort to recover an unknown function using such dictionaries. While some research on the subject has been already carried out, there are still many gaps to address. In particular, one of the most popular methods, lasso, and its variants, is based on minimizing the empirical likelihood and unfortunately, requires stringent assumptions on the dictionary, the so-called, compatibility conditions. Though compatibility conditions are hard to satisfy, it is well ...

Jan 2019

#### Rigorous Analysis Of An Edge-Based Network Disease Model, Sabrina Mai

Edge-based network disease models, in comparison to classic compartmental epidemiological models, better capture social factors affecting disease spread such as contact duration and social heterogeneity. We reason that there should exist infinitely many equilibria rather than only an endemic equilibrium and a disease-free equilibrium for the edge-based network disease model commonly used in the literature, as there do not exist any changes in demographic in the model. We modify the commonly used network model by relaxing some assumed conditions and factor in a dependency on initial conditions. We find that this modification still accounts for realistic dynamics of disease spread ...

#### Spectral Properties Of The Finite Hilbert Transform On Two Adjacent Intervals Via The Method Of Riemann-Hilbert Problem, Elliot Blackstone

##### Electronic Theses and Dissertations

In this dissertation, we study a self-adjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a so-called "integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular Riemann-Hilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of ...

Hadwiger Numbers And Gallai-Ramsey Numbers Of Special Graphs, Christian Bosse Jan 2019

#### Hadwiger Numbers And Gallai-Ramsey Numbers Of Special Graphs, Christian Bosse

##### Electronic Theses and Dissertations

This dissertation explores two separate topics on graphs. We first study a far-reaching generalization of the Four Color Theorem. Given a graph G, we use chi(G) to denote the chromatic number; alpha(G) the independence number; and h(G) the Hadwiger number, which is the largest integer t such that the complete graph K_t can be obtained from a subgraph of G by contracting edges. Hadwiger's conjecture from 1943 states that for every graph G, h(G) is greater than or equal to chi(G). This is perhaps the most famous conjecture in Graph Theory and remains open ...

#### Variational Inclusions With General Over-Relaxed Proximal Point And Variational-Like Inequalities With Densely Pseudomonotonicity, George Nguyen

##### Electronic Theses and Dissertations

This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a study of a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal ...

Quasi-Gorenstein Modules, Alexander York Jan 2018

#### Quasi-Gorenstein Modules, Alexander York

##### Electronic Theses and Dissertations

This thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An R-module M of grade g will be quasi-Gorenstein if ExtiR(M, R) = 0 for i 6= g and there is an isomorphism M ∼= ExtgR(M, R). Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their ...

Jan 2018

#### In Quest Of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, And Summation Identities For Entire Functions, Rajitha Puwakgolle Gedara

##### Electronic Theses and Dissertations

The title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used throughout. Also a brief historical recount is given on the origin of the Bernstein inequality, which dated back ...

Weierstrass Vertices And Divisor Theory Of Graphs, Ajani Ruwandhika Chulangi De Vas Gunasekara Jan 2018

#### Weierstrass Vertices And Divisor Theory Of Graphs, Ajani Ruwandhika Chulangi De Vas Gunasekara

##### Electronic Theses and Dissertations

Chip-firing games and divisor theory on finite, connected, undirected and unweighted graphs have been studied as analogs of divisor theory on Riemann Surfaces. As part of this theory, a version of the one-dimensional Riemann-Roch theorem was introduced for graphs by Matt Baker in 2007. Properties of algebraic curves that have been studied can be applied to study graphs by means of the divisor theory of graphs. In this research, we investigate the property of a vertex of a graph having the Weierstrass property in analogy to the theory of Weierstrass points on algebraic curves. The weight of the Weierstrass vertices ...

Jan 2018

#### I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, Jason L. Greuling

Research has shown that a frame for an n-dimensional real Hilbert space oﬀers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and suﬃcient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will oﬀer phase retrieval. In this thesis, we will explore and provide what necessary and suﬃcient ...

On Saturation Numbers Of Ramsey-Minimal Graphs, Hunter M. Davenport Jan 2018

#### On Saturation Numbers Of Ramsey-Minimal Graphs, Hunter M. Davenport

Dating back to the 1930's, Ramsey theory still intrigues many who study combinatorics. Roughly put, it makes the profound assertion that complete disorder is impossible. One view of this problem is in edge-colorings of complete graphs. For forbidden graphs H1,...,Hk and a graph G, we write G "arrows" (H1,...,Hk) if every k-edge-coloring of G contains a monochromatic copy of Hi in color i for some i=1,2,...,k. If c is a (red, blue)-edge-coloring of G, we say c is a bad coloring if G contains no red K3or ...

Sampling And Reconstruction Of Spatial Signals, Cheng Cheng Jan 2017

#### Sampling And Reconstruction Of Spatial Signals, Cheng Cheng

##### Electronic Theses and Dissertations

Digital processing of signals f may start from sampling on a discrete set Γ, f →(f(ϒη))ϒηεΓ. The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and ...

Coloring Graphs With Forbidden Minors, Martin Rolek Jan 2017

#### Coloring Graphs With Forbidden Minors, Martin Rolek

##### Electronic Theses and Dissertations

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger's Conjecture from 1943 which states that every graph with no Kt-minor is (t − 1)-colorable. This conjecture has been proved true for t ≤ 6, but remains open for all t ≥ 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no Kt-minor is (2t − 6)- colorable for t = 7, 8, 9, in ...

Jan 2017

#### Scaling Of Spectra Of Cantor-Type Measures And Some Number Theoretic Considerations, Isabelle Kraus

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.

Gallai-Ramsey Numbers For C7 With Multiple Colors, Dylan Bruce Jan 2017

#### Gallai-Ramsey Numbers For C7 With Multiple Colors, Dylan Bruce

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H1, ..., Hk, we write G → (H1, ..., Hk), or G → (H)k when H1 = ··· = Hk = H, if every k-edge-coloring of G contains a monochromatic Hi in color i for some i ∈ {1,...,k}. The Ramsey number rk(H1, ..., Hk) is the ...

#### Computational Study Of Traveling Wave Solutions And Global Stability Of Predator-Prey Models, Yi Zhu

##### Electronic Theses and Dissertations

In this thesis, we study two types of reaction-diffusion systems which have direct applications in understanding wide range of phenomena in chemical reaction, biological pattern formation and theoretical ecology. The first part of this thesis is on propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order $m$ without decay. The second is chemical reaction of order $m$ with a decay of order ...

Jan 2016

#### Weighted Low-Rank Approximation Of Matrices:Some Analytical And Numerical Aspects, Aritra Dutta

##### Electronic Theses and Dissertations

This dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing state-of-the-art unweighted and weighted low-rank approximation algorithms. Classical principal component analysis (PCA) is constrained to have equal weighting on the elements of the matrix, which might lead to a degraded design in some problems. To address this fundamental flaw in ...

#### Building Lax Integrable Variable-Coefficient Generalizations To Integrable Pdes And Exact Solutions To Nonlinear Pdes, Matthew Russo

##### Electronic Theses and Dissertations

This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one ...

Jan 2016

#### A Mathematical Model For Feral Cat Ecology With Application To Disease., Jeff Sharpe

##### Electronic Theses and Dissertations

We formulate and analyze a mathematical model for feral cats living in an isolated colony. The model contains compartments for kittens, adult females and adult males. Kittens are born at a rate proportional to the population of adult females and mature at equal rates into adult females and adult males. Adults compete with each other in a manner analogous to Lotka-Volterra competition. This competition comes in four forms, classified by gender. Native house cats, and their effects are also considered, including additional competition and abandonment into the feral population. Control measures are also modeled in the form of per-capita removal ...

Modeling Rogue Waves In Deep Water, Maria Strawn Jan 2016

#### Modeling Rogue Waves In Deep Water, Maria Strawn

##### Electronic Theses and Dissertations

The evolution of surface waves in deep water is governed by the nonlinear Schrodinger (NLS) equation. Spatially periodic breathers (SPBs) and rational solutions of the NLS equation are used as typical models for rogue waves since they exhibit many features of rogue waves. A major component of the dissertation is the stability of solutions of the NLS equation. We address the stability of the rational solutions of the NLS equation used to model rogue waves using squared eigenfunctions of the associated Lax Pair. This allows us to contrast to the existing results for SPBs. The stability of the constant amplitude ...

#### Structure-Preserving Finite Difference Methods For Linearly Damped Differential Equations, Ashish Bhatt

##### Electronic Theses and Dissertations

Differential equations (DEs) model a variety of physical phenomena in science and engineering. Many physical phenomena involve conservative or dissipative forces, which manifest themselves as qualitative properties of DEs that govern these phenomena. Since only a few and simplistic models are known to have exact solutions, approximate solution techniques, such as numerical integration, are used to reveal important insights about solution behavior and properties of these models. Numerical integrators generally result in undesirable quantitative and qualitative errors . Standard numerical integrators aim to reduce quantitative errors, whereas geometric (numerical) integrators aim to reduce or eliminate qualitative errors, as well, in order ...

Interval Edge-Colorings Of Graphs, Austin Foster Jan 2016

#### Interval Edge-Colorings Of Graphs, Austin Foster

##### Electronic Theses and Dissertations

A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher conferences so that every person's conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and ...

#### Analysis Of Employment And Earnings Using Varying Coefficient Models To Assess Success Of Minorities And Women, Amanda Goedeker

##### Electronic Theses and Dissertations

The objective of this thesis is to examine the success of minorities (black, and Hispanic/Latino employees) and women in the United States workforce, defining success by employment percentage and earnings. The goal of this thesis is to study the impact gender, race, passage of time, and national economic status reflected in gross domestic product have on the success of minorities and women. In particular, this thesis considers the impact of these factors in Science, Technology, Engineering and Math (STEM) industries. Varying coefficient models are utilized in the analysis of data sets for national employment percentages and earnings.

Modeling Network Worm Outbreaks, Evan Foley Jan 2015

#### Modeling Network Worm Outbreaks, Evan Foley

##### Electronic Theses and Dissertations

Due to their convenience, computers have become a standard in society and therefore, need the utmost care. It is convenient and useful to model the behavior of digital virus outbreaks that occur, globally or locally. Compartmental models will be used to analyze the mannerisms and behaviors of computer malware. This paper will focus on a computer worm, a type of malware, spread within a business network. A mathematical model is proposed consisting of four compartments labeled as Susceptible, Infectious, Treatment, and Antidotal. We shall show that allocating resources into treating infectious computers leads to a reduced peak of infections across ...

On The Theory Of Zeta-Functions And L-Functions, Almuatazbellah Awan Jan 2015

#### On The Theory Of Zeta-Functions And L-Functions, Almuatazbellah Awan

##### Electronic Theses and Dissertations

In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's L-function. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Number Theorem and the Riemann Hypothesis. These two topics constitute the main theme of this thesis. For the Prime Number Theorem, we provide a thorough discussion that compares and ...