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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 …
Rigorous Analysis Of An Edge-Based Network Disease Model, Sabrina Mai
Rigorous Analysis Of An Edge-Based Network Disease Model, Sabrina Mai
Honors Undergraduate Theses
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 …
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 …