Mathematics Commons^™

Krylov Subspace Spectral Methods For Pdes In Polar And Cylindrical Geometries, Megan Richardson

Dissertations

As a result of stiff systems of ODEs, difficulties arise when using time stepping methods for PDEs. Krylov subspace spectral (KSS) methods get around the difficulties caused by stiffness by computing each component of the solution independently. In this dissertation, we extend the KSS method to a circular domain using polar coordinates. In addition to using these coordinates, we will approximate the solution using Legendre polynomials instead of Fourier basis functions. We will also compare KSS methods on a time-independent PDE to other iterative methods. Then we will shift our focus to three families of orthogonal polynomials on the interval …

Mathematical Modeling Of Membrane Filtration, Pejman Sanaei

Dissertations

The purpose of this thesis is to formulate and investigate new mathematical models for membrane filtration. The work presented is divided into six chapters. In the first chapter the problem is introduced and motivated. In the second chapter, a new mathematical model for flow and fouling in a pleated membrane filter is presented. Pleated membrane filters are widely used in many applications, and offer significantly better surface area to volume ratios than equal area unpleated membrane filters. However, their filtration characteristics are markedly inferior to those of equivalent unpleated membrane filters in dead-end filtration. While several hypotheses have been advanced …

Qualitative Modeling Of Chaotic Logical Circuits And Walking Droplets: A Dynamical Systems Approach, Aminur Rahman

Dissertations

Logical circuits and wave-particle duality have been studied for most of the 20th century. During the current century scientists have been thinking differently about these well-studied systems. Specifically, there has been great interest in chaotic logical circuits and hydrodynamic quantum analogs.

Traditional logical circuits are designed with minimal uncertainty. While this is straightforward to achieve with electronic logic, other logic families such as fluidic, chemical, and biological, naturally exhibit uncertainties due to their inherent nonlinearity. In recent years, engineers have been designing electronic logical systems via chaotic circuits. While traditional boolean circuits have easily determined outputs, which renders dynamical models …

Pedagogical Moves As Characteristics Of One Instructor’S Instrumental Orchestrations With Tinkerplots And The Ti-73 Explorer: A Case Study, James L. Kratky

Dissertations

Those supporting contemporary reform efforts for mathematics education in the United States have called for increased use of technologies to support student-centered learning of mathematical concepts and skills. There is a need for more research and professional development to support teachers in transitioning their instruction to better meet the goals of such reform efforts.

Instrumental approaches to conceptualizing technology use in mathematics education, arising out of the theoretical and empirical work in France and other European nations, show promise for use to frame studies on school mathematics in the United States. Instrumental genesis is used to describe the bidirectional and …

Hamiltonian Bifurcations In Schrodinger Trimers, Casayndra H. Basarab

Dissertations

The phase space of the three-mode discrete NLS in the nonlinear regime with periodic boundary conditions is investigated by reducing the degree of freedom from three down to two. The families of standing waves are enumerated and normal forms are used to describe several families of relative periodic orbits whose topologies change due to Hamiltonian Hopf bifurcations and transcritical bifurcations. The Hamiltonian Hopf bifurcation occurs when eigenvalues on the imaginary axis collide and split and has two types: elliptic and hyperbolic. These two types arise in the DNLS problem, and the families of periodic orbits are discussed as a conserved …

Structural Exploration And Inference Of The Network, Ruihua Cheng

Dissertations

This dissertation consists of two parts. In the first part, a learning-based method for classification of online reviews that achieves better classification accuracy is extended. Automatic sentiment classification is becoming a popular and effective way to help online users or companies to process and make sense of customer reviews. The method combines two recent developments. First, valence shifters and individual opinion words are combined as bigrams to use in an ordinal margin classifier. Second, relational information between unigrams expressed in the form of a graph is used to constrain the parameters of the classifier. By combining these two components, it …

Numerical Simulations Of Dense Granular Systems With And Without Cohesive Effects, Lenka Kovalcinova

Dissertations

Granular materials are collections of objects ranging from sand grains that form sand piles or even sand castles to collections of large objects such as a group of meteors in outer space. The considered range of sizes of granular particles is such that the effect of thermal fluctuations is not relevant. However, the interaction between the particles may be very complex, involving inelasticity and friction, in addition to repulsive and possibly attractive interaction forces. These interactions that may be history dependent, make the systems that consist of a large number of particles complex to analyze and difficult to understand using …

Efficient High-Order Integral Equation Methods For The Heat Equation, Shaobo Wang

Dissertations

Efficient high-order integral equation methods have been developed for solving the boundary value problems of the heat equation with complex geometries in two and three dimensions. First of all, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials. Some advantages of the integral formulation as compared with standard finite difference and finite element methods include reduction of the dimension of the problem by one, high order accuracy, unconditional stability, insensitivity to different geometries, and elimination of truncating the computational domain and the need of artificial …

Hybrid Chebyshev Polynomial Scheme For The Numerical Solution Of Partial Differential Equations, Balaram Khatri Ghimire

Dissertations

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including …

Chromatic Connectivity Of Graphs, Elliot Laforge

Dissertations

No abstract provided.

Resolving Classes And Resolvable Spaces In Rational Homotopy Theory, Timothy L. Clark

Dissertations

A class of topological spaces is called a resolving class if it is closed under weak equivalences and homotopy limits. Letting R(A) denote the smallest resolving class containing a space A, we say X is A-resolvable if X is in R(A), which induces a partial order on spaces. These concepts are dual to the well-studied notions of closed class and cellular space, where the induced partial order is known as the Dror Farjoun Cellular Lattice. Progress has been made toward illuminating the structure of the Cellular Lattice. For example: Chachólski, Parent, and Stanley have shown that it …

Methods For The Direct Simulation Of Nanoscale Film Breakup And Contact Angles, Kyle Mahady

Dissertations

This thesis investigates direct simulation of fluids with free surfaces and contact lines, with a focus on capturing nanoscale physics in a continuum based computational framework. Free surfaces and contact lines have long presented some of the most challenging problems in computational fluid dynamics. Extensive progress has been made in recent years, and a wide variety of different methods are currently employed for direct simulation in these contexts. The complexity of the full governing equations for such flows poses significant challenges in terms of analytical techniques, and leads to lengthy computational times for direct simulations. For these reasons, reduced models …

Investigation Of Infinite-Dimensional Dynamical System Models Applicable To Granular Flows, Hao Wu

Dissertations

Recently Blackmore, Samulyak and Rosato developed a class of infinite-dimensional dynamical systems in the form of integro-partial differential equations, which have been called the BSR models. The BSR models were originally derived to model granular flows, but they actually have many additional applications in a variety of fields. BSR models have already been proven to be completely integrable infinite-dimensional Hamiltonian dynamical systems for perfectly elastic interactions in the case of one space dimension, but the well-posedness question of these systems is at least partially answered for the first time here. In particular, dynamical systems of the BSR type are proven …

Solution Of Nonlinear Time-Dependent Pde Through Componentwise Approximation Of Matrix Functions, Alexandru Cibotarica

Dissertations

Exponential propagation iterative (EPI) methods provide an efficient approach to the solution of large stiff systems of ODE, compared to standard integrators. However, the bulk of the computational effort in these methods is due to products of matrix functions and vectors, which can become very costly at high resolution due to an increase in the number of Krylov projection steps needed to maintain accuracy. In this dissertation, it is proposed to modify EPI methods by using Krylov subspace spectral (KSS) methods, instead of standard Krylov projection methods, to compute products of matrix functions and vectors. This improvement allowed the benefits …

Edge Colorings Of Graphs And Their Applications, Daniel Johnston

Dissertations

Edge colorings have appeared in a variety of contexts in graph theory. In this work, we study problems occurring in three separate settings of edge colorings.

For more than a quarter century, edge colorings have been studied that induce vertex colorings in some manner. One research topic we investigate concerns edge colorings belonging to this class of problems. By a twin edge coloring of a graph G is meant a proper edge coloring of G whose colors come from the integers modulo k that induce a proper vertex coloring in which the color of a vertex is the sum of …

Methods For Two-Sample Comparisons From Censored Time-To-Event Data, Nubyra Ahmed

Dissertations

In the analysis of censored survival data, it is frequently of interest to determine the efficacy of a treatment or new method over a control or existing method. For this purpose, one may report estimates of the two survival functions or, more specifically, their difference, accompanied by simultaneous confidence bands (SCBs). Alternatively, or in addition, one may conduct hypothesis testing for the difference of the two survival functions.

The first project exploits two bootstrap methods to develop new Wald-type SCBs for the difference of survival functions. The censored data bootstrap is employed to obtain nonparametric SCBs for the difference of …

Multiple Testing Procedures For Complex Structured Hypotheses And Directional Decisions, Anjana Grandhi

Dissertations

Several multiple testing procedures are developed based on the inherent structure of the tested hypotheses and specific needs of data analysis. Incorporating the inherent structure of the hypotheses results in development of more powerful and situation-specific multiple testing procedures than existing ones. The focus of this dissertation is on developing multiple testing procedures that utilize the information on this structure of the hypotheses and aims at answering research questions while controlling appropriate error rates.

In the first part of the thesis, a mixed directional false discovery rate (mdFDR) controlling procedure is developed in the context of uterine fibroid gene expression …

An Efficient Boundary Integral Method For Stiff Fluid Interface Problems, Oleksiy Varfolomiyev

Dissertations

The purpose of this thesis is to formulate and investigate a boundary integral method for the solution of the internal waves/Rayleigh-Taylor problem. This problem describes the evolution of the interface between two immiscible, inviscid, incompressible, irrotational fluids of different density in three dimensions. A mathematical model of this interfacial flow problem in 3D is derived. The motion of the interface and fluids is driven by the action of a gravity force, surface tension at the interface, elastic bending and/or a prescribed far-field pressure gradient. The presented models include derived equations for the evolution of the interface and dipole density on …

Time Integration Methods Of Fundamental Solutions And Approximate Fundamental Solutions For Nonlinear Elliptic Partial Differential Equations, Corey Leon Jones

Dissertations

A time-dependent method is coupled with the Method of Approximate Particular Solutions (MAPS) of Delta-shaped basis functions, the Method of Fundamental Solutions (MFS), and the Method of Approximate Fundamental Solutions (MAFS) to solve a second order nonlinear elliptic partial differential equation (PDE) on regular and irregular shaped domains. The nonlinear PDE boundary value problem is first transformed into a time-dependent quasilinear problem by introducing a fictitious time. Forward Euler integration is then used to ultimately convert the problem into a sequence of time-dependent linear nonhomogeneous modified Helmholtz boundary value problems on which the superposition principle is applied to split the …

Spectrally Equivalent Matrix Polynomials: Non-Standard Representations And Preservation Of Structure, Vasilije Perovic

Dissertations

Abstract attached as separate file.

Phantom Maps, Decomposability, And Spaces Meeting Particular Finiteness Conditions, James Schwass

Dissertations

The purpose of this dissertation is to extend principles for detecting the existence of essential phantom maps into spaces meeting particular finiteness conditions. Zabrodsky shows that a space Y having the homotopy type of a finite CW complex is the target of essential phantom maps if and only if Y has a nontrivial rational homology group. We show this observation holds on the collection of finite generalized CW complexes. Similarly, Iriye shows a finite-type, simply connected suspension space is the target of essential phantom maps if and only if it has a nontrivial rational homology group. We show this observation …

Wavelet Analysis Of Non-Stationary Signals With Applications, Maria Dorothea Van Der Walt

Dissertations

The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang et al in 1998, is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF's), after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu and H.-T. Wu in 2011, is first applied to …

Hybrid Meshless Method For Numerical Solution Of Partial Differential Equations, Jeanette Marie Monroe

Dissertations

A meshless method for solving partial differential equations (PDEs) which combines the method of fundamental solutions (MFS) and the method of particular solutions (MPS) is formulated and tested. The hybrid method finds a numerical approximation by solving only one system of equations as opposed to the two-stage method of fundamental solutions and method of particular solutions. This new approach, denoted MFS-MPS, one-stage MFS-MPS, or hybrid method, can be applied to a wide variety of PDEs including PDEs with variable coefficients. The MFS-MPS can simplify Helmholtz-type differential operators to Laplacian-type differential operators providing flexibility and simplification to calculating particular solutions and …

Lnference On Differences In K Means For Data With Excess Zeros And Detection Limits, Haolai Jiang

Dissertations

Many data have excess zeros or unobservable values falling below detection limit. For example, data on hospitalization costs incurred by members of a health insurance plan will have zeros for the percentage who did not get sick. Benzene exposure measurements on petroleum re nery workers have some exposures fall below the limit of detection. Traditional methods of inference like one-way ANOVA are not appropriate to analyze such data since the point mass at zero violates typical distribution assumptions.

For testing for equality of means of k distributions, we will propose a likelihood ratio test that accounts for excess zeros or …

Modular Monochromatic Colorings, Spectra And Frames In Graphs, Chira Lumduanhom

Dissertations

Abstract attached as separate document.

Confidence Bands For Survival Curves Using Model Assisted Cox Regression, Shoubhik Mondal

Dissertations

The goal of this dissertation is to develop informative subject-specific simultaneous confidence bands (SCBs) for survival functions from right censored data. The approach is based on an extension of semiparametric random censorship models (SRCMs) to Cox regression, which produces reliable and more informative SCBs. SRCMs derive their rationale from their ability to utilize parametric ideas within the random censorship environment. Incorporating SRCMs into the existing framework produces more powerful procedures to analyze right censored data. The first part of the project focuses on proposing new estimators of Cox regression parameters and the cumulative baseline hazard function, and deriving their large …

Efficient Domain Decomposition Algorithms For The Solution Of The Helmholtz Equation, Dawid Midura

Dissertations

The purpose of this thesis is to formulate and investigate new iterative methods for the solution of scattering problems based on the domain decomposition approach. This work is divided into three parts. In the first part, a new domain decomposition method for the perfectly matched layer system of equations is presented. Analysis of a simple model problem shows that the convergence of the new algorithm is guaranteed provided that a non-local, square-root transmission operator is used. For efficiency, in practical simulations such operators need to be localized. Current, state of the art domain decomposition algorithms use the localization technique based …

Mathematical Methods Of Analysis For Control And Dynamic Optimization Problems On Manifolds, Robert J. Kipka

Dissertations

Mathematical Methods Of Analysis For Control And Dynamic Optimization Problems On Manifolds Driven by applications in fields such as robotics and satellite attitude control, as well as by a need for the theoretical development of appropriate tools for the analysis of geometric systems, problems of control of dynamical systems on manifolds have been studied intensively during the past three decades. In this dissertation we suggest new mathematical techniques for the study of control and dynamic optimization problems on manifolds. This work has several components including: an extension of the classical Chronological Calculus to control and dynamical systems which are merely …

On Eulerian Irregularity And Decompositions In Graphs, Eric Andrews

Dissertations

Abstract attached as separate file.

Lie Loops Associated With Gl(ℋ), ℋ A Separable Infinite Dimensional Hilbert Space, Alper Bulut

Dissertations

Abstract attached as separate file.