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Lattice-Valued T-Filters And Induced Structures, Frederick Reid
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 …
Spectral Properties Of The Finite Hilbert Transform On Two Adjacent Intervals Via The Method Of Riemann-Hilbert Problem, Elliot Blackstone
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 $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the …
Semi-Analytical Solutions Of Non-Linear Differential Equations Arising In Science And Engineering, Mangalagama Dewasurendra
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 differential …
Variational Inclusions With General Over-Relaxed Proximal Point And Variational-Like Inequalities With Densely Pseudomonotonicity, George Nguyen
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 …
Hadwiger Numbers And Gallai-Ramsey Numbers Of Special Graphs, Christian Bosse
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 even for graphs G with alpha(G) …
Two Ramsey-Related Problems, Jingmei Zhang
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 …
Estimation And Clustering In Statistical Ill-Posed Linear Inverse Problems, Rasika Rajapakshage
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 …
Spatial Models With Specific Error Structures, Nathaniel Adu
Spatial Models With Specific Error Structures, Nathaniel Adu
Electronic Theses and Dissertations
The purpose of this dissertation is to study the first order autoregressive model in the spatial context with specific error structures. We begin by supposing that the error structure has a long memory in both the i and the j components. Whenever the model parameters alpha and beta equal one, the limiting distribution of the sequence of normalized Fourier coefficients of the spatial process is shown to be a function of a two parameter fractional Brownian sheet. This result is used to find the limiting distribution of the periodogram ordinate of the spatial process under the null hypothesis that alpha …
Frames And Phase Retrieval, Ted Juste
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 of …
Mathematical Investigation Of The Spatial Spread Of An Infectious Disease In A Heterogeneous Environment, Arielle Gaudiello
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 …
Solution Of Linear Ill-Posed Problems Using Overcomplete Dictionaries, Pawan Gupta
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 …
Analytical And Numerical Investigations Of The Kudryashov Generalized Kdv Equation, William Hilton
Analytical And Numerical Investigations Of The Kudryashov Generalized Kdv Equation, William Hilton
Electronic Theses and Dissertations
This thesis concerns an analytical and numerical study of the Kudryashov Generalized Korteweg-de Vries (KG KdV) equation. Using a refined perturbation expansion of the Fermi-Pasta-Ulam (FPU) equations of motion, the KG KdV equation, which arises at sixth order, and general higher order KdV equations are derived. Special solutions of the KG KdV equation are derived using the tanh method. A pseudospectral integrator, which can handle stiff equations, is developed for the higher order KdV equations. The numerical experiments indicate that although the higher order equations exhibit complex dynamics, they fail to reach energy equipartition on the time scale considered.
Quasi-Gorenstein Modules, Alexander York
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 …
Weierstrass Vertices And Divisor Theory Of Graphs, Ajani Ruwandhika Chulangi De Vas Gunasekara
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 …
In Quest Of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, And Summation Identities For Entire Functions, Rajitha Puwakgolle Gedara
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 …
Coloring Graphs With Forbidden Minors, Martin Rolek
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 = …
Filtering Problems In Stochastic Tomography, Tyler Gomez
Filtering Problems In Stochastic Tomography, Tyler Gomez
Electronic Theses and Dissertations
Distinguishing signal from noise has always been a major goal in probabilistic analysis of data. Such is no less the case in the field of medical imaging, where both the processes of photon emission and their rate of absorption by the body behave as random variables. We explore methods by which to extricate solid conclusions from noisy data involving an X-ray transform, long the mathematical mainstay of such tools as computed axial tomography (CAT scans). Working on the assumption of having some prior probabilities assigned to various states a body can be found in, we introduce and make rigorous an …
Sampling And Reconstruction Of Spatial Signals, Cheng Cheng
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 sparse signals, and …
A Mathematical Model For Feral Cat Ecology With Application To Disease., Jeff Sharpe
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 …
On Randic Energy Of Graphs, Brittany Burns
On Randic Energy Of Graphs, Brittany Burns
Electronic Theses and Dissertations
In this research, we explore the subject of graph energy. We first discuss the connections between linear algebra and graph theory and review some important definitions and facts of these two fields. We introduce graph energy and provide some historical perspectives on the subject. Known results of graph energy are also mentioned and some relevant results are proven. We discuss some applications of graph energy in the physical sciences. Then, Randic energy is defined and results are given and proved for specific families of graphs. We focus on simple, connected graphs that are commonly studied in graph theory. Also, the …
Analysis Of Employment And Earnings Using Varying Coefficient Models To Assess Success Of Minorities And Women, Amanda Goedeker
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.
Interval Edge-Colorings Of Graphs, Austin Foster
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 teachers, and …
Structure-Preserving Finite Difference Methods For Linearly Damped Differential Equations, Ashish Bhatt
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 …
Comparing The Variational Approximation And Exact Solutions Of The Straight Unstaggered And Twisted Staggered Discrete Solitons, Daniel Marulanda
Comparing The Variational Approximation And Exact Solutions Of The Straight Unstaggered And Twisted Staggered Discrete Solitons, Daniel Marulanda
Electronic Theses and Dissertations
Discrete nonlinear Schrödinger equations (DNSL) have been used to provide models of a variety of physical settings. An application of DNSL equations is provided by Bose-Einstein condensates which are trapped in deep optical-lattice potentials. These potentials effectively splits the condensate into a set of droplets held in local potential wells, which are linearly coupled across the potential barriers between them [3]. In previous works, DNLS systems have also been used for symmetric on-site-centered solitons [11]. A few works have constructed different discrete solitons via the variational approximation (VA) and have explored their regions for their solutions [11, 12]. Exact solutions …
Weighted Low-Rank Approximation Of Matrices:Some Analytical And Numerical Aspects, Aritra Dutta
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
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 …
Modeling Rogue Waves In Deep Water, Maria Strawn
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 …
Computational Study Of Traveling Wave Solutions And Global Stability Of Predator-Prey Models, Yi Zhu
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 …
Propagation Failure In Discrete Inhomogeneous Medium Using A Caricature Of The Cubic, Elizabeth Lydon
Propagation Failure In Discrete Inhomogeneous Medium Using A Caricature Of The Cubic, Elizabeth Lydon
Electronic Theses and Dissertations
Spatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle- rosis. We construct steady-state, single front solutions by employing a piecewise linear reaction term. Using a combination of Jacobi-Operator theory and the Sherman-Morrison formula we de- rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain conditions outlined in their construction. The range of parameter values that satisfy these conditions constitutes the interval of propagation failure, determining under what circumstances a front becomes pinned in the media. Our exact solutions represent …
Modeling Network Worm Outbreaks, Evan Foley
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 …