Mathematics Commons^™

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.

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 …

The Control Of The False Discovery Rate Under Structured Hypotheses, Gavin Lynch

Dissertations

The hypotheses in many multiple testing problems often have some inherent structure based on prior information such as Gene Ontology in gene expression data. However, few false discovery rate (FDR) controlling procedures take advantage of this inherent structure. In this dissertation, we develop FDR controlling methods which exploit the structural information of the hypotheses.

First, we study the fixed sequence structure where the testing order of the hypotheses has been pre-specified. We are motivated to study this structure since it is the most basic of structures, yet, it has been largely ignored in the literature on large scale multiple testing. …

Electro-Deformation Of A Moving Boundary: A Drop Interface And A Lipid Bilayer Membrane, Herve Nganguia

Dissertations

This dissertation focuses on the deformation of a viscous drop and a vesicle immersed in a (leaky) dielectric fluid under an electric field. A number of mathematical tools, both analytical and numerical, are developed for these investigations. The dissertation is divided into three parts. First, a large-deformation model is developed to capture the equilibrium deformation of a viscous spheroidal drop covered with non-diffusing insoluble surfactant under a uniform direct current (DC) electric field. The large- deformation model predicts the dependence of equilibrium spheroidal drop shape on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Results from the …

Data Analysis And Simulation For Warranties And Golf Handicaps, Sonia Bandha

Dissertations

In this dissertation, we discuss the application of data analysis and numerical simulation in order to gain insight into problems related to warranty cost management and the effectiveness of the golf handicap system. Despite the commonalities of the approaches, we will discuss these problems in turn.

For many moderately high value items with a substantial sales volume (such as automobiles), a warranty is used as an important element of marketing products as a better warranty typically signals higher product quality to customers. Much recent research on modeling and optimization of servicing costs for Non-Renewing Free Replacement Warranties (NR-FRW) assumes that …

Fwer Controlling Procedures For Testing Multiple Hypotheses With Hierarchical Structure And Applications In Clinical Trials, Zhiying Qiu

Dissertations

In applications of clinical trials, the hypotheses to be tested often exhibit a hierarchical structure, and are usually hierarchically ordered based on their importance, clinical relevance, or dose concentration, etc. Thus, they are tested in a pre-defined fixed sequence. In some more complex cases, the hypotheses to be tested are hierarchically grouped into several families, and thus the families are tested in a sequential order. Although such problems of structured multiple testing have received much attention and several popular FWER controlling procedures, such as conventional fixed sequence procedure, fallback procedure and gatekeeping procedure have been introduced, not much progress has …

Inverse Methods For Sound Speed Estimation In The Ocean, Tao Lin

Dissertations

This dissertation presents theoretical and computational approaches for estimating sound-speed in the ocean under different conditions. The first part of the dissertation discusses a fast approach for solving the inverse problem of estimating sediment sound-speed based on the Deift-Trubowitz trace formula. Under certain assumptions, this algorithm can recover the sound speed profile in the seabed using pressure field measurements in the water column at low frequencies. The inversion algorithm requires solving a non-linear integral equation. In the past, Stickler and Zhou employed a first order Born approximation for solving the integral equation. This work introduces two new methods. The first …

Basis Function Approaches For Two Dimensional Cochlear Models, Lihua Li

Dissertations

The human cochlea possesses the amazing ability of analyzing audio signals. The structures and mechanisms behind its characteristic response to sound stimuli has been an active area of research for decades. It has been demonstrated that mathematical cochlear modeling poses a promising alternative to discover the elusive activities in an in vivo cochlea. However, despite the successful application of numerical methods such as the Wentzel-Kramers-Brillouin (WKB) method, finite difference method (FDM) and finite element method (FEM), the critical effects of the choice of basis functions have not been studied exclusively for the numerical solutions of cochlea models. This work presents …

Dynamics Of Phase Locking In Neuronal Networks In The Presence Of Synaptic Plasticity, Zeynep Akcay

Dissertations

The behavior generated by neuronal networks depends on the phase relationships of its individual neurons. Observed phases result from the combined effects of individual cells and synaptic connections whose properties change dynamically. The properties of individual cells and synapses can often be characterized by driving the cell or synapse with inputs that arrive at different phases or frequencies, thus producing a feed-forward description of these properties. In this study, a recurrent network of two oscillatory neurons that are coupled with reciprocal synapses is considered. Feed-forward descriptions of the phase response curves of the neurons and the short-term synaptic plasticity properties …

A Modeling Study Of The History-Dependence Of Conduction Delay In Unmyelinated Axons, Yang Zhang

Dissertations

Conduction delay in an axon is the time required for an action potential to propagate between two positions. It is a function of the axon’s passive membrane properties, voltage-gated ion channels and the Na+/K+ pump, and can be substantially affected by neuromodulators. The conduction delay of action potential, generated by the pyloric dilator (PD) neuron unmyel i nated motor axon in the stomatogastric nervous system, shows significant variability with ongoing bursting or Poisson stimulation. When the axon is stimulated, the mean value (Dmean) and coefficient variation of conduction delay (CV-D) slowly increase with time (slow timescale effect), and the relationship …

Particle Filtering For Frequency Estimation From Acoustic Time-Series In Dispersive Media, Nattapol Aunsri

Dissertations

Acoustic signals propagating in the ocean carry information about geometry and environmental parameters within the propagation medium. Accurately retrieving this information leads us to effectively estimate parameters that are of utmost importance in environmental studies, climate monitoring, and defense. This dissertation focuses on the development of sequential Bayesian filtering methods to obtain accurate estimates of instantaneous frequencies using Short Term Fourier Transforms within the acoustic field measured at an array of hydrophones, which can be used in a subsequent step for the estimation of propagation related parameters. We develop a particle filter to estimate these frequencies along with modal amplitudes, …

Using Prior Knowledge And Learning From Experience In Estimation Of Distribution Algorithms, Mark Walter Hauschild

Dissertations

Estimation of distribution algorithms (EDAs) are stochastic optimization techniques that explore the space of potential solutions by building and sampling explicit probabilistic models of promising candidate solutions. One of the primary advantages of EDAs over many other stochastic optimization techniques is that after each run they leave behind a sequence of probabilistic models describing useful decompositions of the problem. This sequence of models can be seen as a roadmap of how the EDA solves the problem. While this roadmap holds a great deal of information about the problem, until recently this information has largely been ignored. My thesis is that …

The Effects Of Standards-Based Grading On Student Performance In Algebra 2, Rachel Beth Rosales

Dissertations

The use of standards-based grading in American public schools is increasing, offering students, parents, and teachers a new way of measuring and communicating about student achievement and performance. Parents indicate an appreciation for this method of grading, and students at the elementary grades (K-6) have improved standardized test scores in reading and math as a result of its implementation. This study seeks to determine whether standards-based grading has the same effect on students at the high school level (grades 9-12) by comparing end-of-course test scores and posttest scores of Algebra 2 students enrolled in a standards-based graded classroom with to …

A Pattern In The Lusternik-Schnirelmann Category Of Rational Spaces, Julie Dare Houck

Dissertations

The Lusternik-Schnirelmann (or LS) category of a space is one less than the number of contractible open sets with which we can cover the space. If we look at the LS categories of the skeleta of a CW complex, we find a sequence of dimensions where the LS category changes. I discuss whether certain of these "category sequences" (defined in the paper, "Categorical Sequences", by Nendorf, Scoville, and Strom) could be realized as the categorical sequences of rational spaces. I first reduce from looking at all rational spaces to only Postnikov sections of finite wedges of spheres. Using the Leray-Serre …

Iterative Solvers For Large, Dense Matrices, Eowyn Wilhelmina Cenek

Dissertations

Stochastic Interpolation (SI) uses a continuous, centrally symmetric probability distribution function to interpolate a given set of data points, and splits the interpolation operator into a discrete deconvolution followed by a discrete convolution of the data. The method is particularly effective for large data sets, as it does not suffer from the problem of oversampling, where too many data points cause the interpolating function to oscillate wildly. Rather, the interpolation improves every time more data points are added. The method relies on the inversion of relatively large, dense matrices to solve Annx = b for x. Based on the probability …

The Application Of Bayesian Adaptive Design And Markov Model In Clinical Trials, Xiaoyu Lu

Dissertations

In this research, two new designs in clinical trials are proposed. The first problem is a new Bayesian adaptive dose-finding design and its application in an oncology clinical trial. This design is used for phase IB studies with the biomarker as the endpoint and with the fewer patients. The second problem is another new Bayesian adaptive dose-finding design with longitudinal analysis and its application in phase II depression clinical trial. This design is best fit for phase II dosing-finding clinical trials with clinical endpoints. MTD information has been obtained before the trials.

In adaptive dose-finding clinical trials, the strategy is …

Mathematical Models For Bistable Nematic Liquid Crystal Displays, Chenjing Cai

Dissertations

Bistable Liquid Crystal Displays (LCDs) offer the potential for considerable power savings, compared with conventional (monostable) LCDs. The existence of two (or more) stable field-free states that are optically-distinct means that contrast can be maintained in a display without an externally-applied electric field. An applied field is required only to switch the device from one state to the other, as needed. This dissertation focuses on theoretical models of a possible bistable nematic device, whose operating principle relies on controlling surface anchoring conditions. Switching between the two stable steady states is achieved by application of a transient electric field. A 1D …

A Numerical Method For Electro-Kinetic Flow With Deformable Interfaces, Manman Ma

Dissertations

We consider two-phase flow of ionic fluids whose motion is driven by an imposed electric field. At a fluid-fluid interface, a screening cloud of ions develops and forms an electro-chemical double layer or ‘Debye layer’. The applied electric field acts on the ionic cloud it induces, resulting in a strong slip flow near the interface. This is known as ‘induced-charge electro-kinetic flow’, and is an important phenomenon in microfluidic applications and in the manipulation of biological cells. The models with two different cases including the fast or slow charging time scales are studied both analytically and numerically. We address a …

Gp Representation Space Reduction Using A Tiered Search Scheme, John Joseph Aleshunas

Dissertations

The size and complexity of a GP representation space is defined by the set of functions and terminals used, the arity of those functions, and the maximal depth of candidate solution trees in the space. Practice has shown that some means to reduce the size or bias the search must be provided. Adaptable Constrained Genetic Programming (ACGP) can discover beneficial substructures and probabilistically bias the search to promote the use of these substructures. ACGP has two operating modes: a more efficient low granularity mode (1st order heuristics) and a less efficient higher granularity mode (2nd order heuristics). Both of these …

Weighted Shifts Of Finite Multiplicity, Daniel S. Sievewright

Dissertations

We will discuss the structure of weighted shift operators on a separable, infinite dimensional, complex Hilbert space. A weighted shift is said to have multiplicity n when all the weights are n x n matrices. To study these weighted shifts, we will investigate which operators can belong to the Deddens algebras and spectral radius algebras, which can be quite large. This will lead to the necessary and sufficient conditions for these algebras to have a nontrivial invariant subspace.

Modeling Cell Proliferation In A Perfusion Tissue Engineering Bioreactor, Jeffrey Vincent Pohlmeyer

Dissertations

In this dissertation we develop a comprehensive model to simulate a tissue engineering experiment. The experiment takes place in a bioreactor in which a cell seeded porous scaffold is placed, and the scaffold experiences a perfused flow of a nutrient-rich culture medium. The goal of the model is to assist experimentalists in evaluation of different parameter scenarios as the time needed to simulate an experiment is significantly less than the time needed for the experiment itself. We provide the full two-dimensional model development, as well as investigation into possible variations of specific model choices, and we demonstrate the robustness and …

The Goodness-Of-Fit Tests For Geometric Models, Feiyan Chen

Dissertations

We propose two types of goodness-of-fit tests for geometric distribution and for a bivariate geometric distribution called BGD(B&D), based on their probability generating function (PGF). The first type is a special-case application of the general testing procedure for discrete distributions proposed by Kocherlakota and Kocherlakota (1986). The second type utilizes the supremum of the absolute value of the standardized difference between the PGF’s maximum likelihood estimator (MLE) and its empirical counterpart as the test statistic. We verify the asymptotic properties of the test statistics for the first type of test and explore the asymptotic behaviors of the test statistics for …

Boolean And Profinite Loops, Benjamin Andrew Phillips

Dissertations

A Quasigroup G consists of a set G together with a binary operation ∗ : G × G → G such that, for any elements a, b ∈ G there are unique solutions to the equations a ∗ x = b and y ∗ a = b within G. A loop is a quasigroup which also contains a 2-sided identity element. More heuristically, loops are essentially non-associative groups. However, without an associative binary operation, some of the familiar properties found in groups, such as 2-sided inverse, need not be present in loops.

The study of quasigroups and loops emerges …

Fast Algorithms For Brownian Dynamics Simulation With Hydrodynamic Interactions, Zhi Liang

Dissertations

In the Brownian dynamics simulation with hydrodynamic interactions, one needs to generate the total displacement vectors of Brownian particles consisting of two parts: a deterministic part which is proportional to the product of the Rotne-Prager-Yamakawa (RPY) tensor D [1, 2] and the given external forces F; and a hydrodynamically correlated random part whose covariance is proportional to the RPY tensor. To be more precise, one needs to calculate Du for a given vector u and compute √Dv for a normally distributed random vector v. For an arbitrary N-particle configuration, D is a 3N x 3N matrix and u, v are …

Analytical And Computational Methods For The Study Of Rare Event Probabilities In Dispersive And Dissipative Waves, Daniel S. Cargill

Dissertations

The main focus of this dissertation is the application of importance sampling (IS) to calculate the probabilities associated with rare events in nonlinear, large-dimensional lightwave systems that are driven by noise, including models for fiber-based optical communication system and mode-locked lasers. Throughout the last decade, IS has emerged as a valuable tool for improving the efficiency of simulating rare events in such systems. In particular, it has shown great success in simulating various sources of transmission impairments found in optical communication systems, with examples ranging from large polarization fluctuations resulting from randomly varying fiber birefringence to large pulse-width fluctuations resulting …

Characterizing And Supporting Change In Algebra Students' Representational Fluency In A Cas/Paper-And-Pencil Environment, Nicole L. Fonger

Dissertations

Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students’ RF in learning environments that utilize both computer algebra systems (CAS) and paper-and-pencil. The purpose of this research was to: (a) characterize change in ninth-grade algebra students’ RF in solving problems involving linear equations, and (b) determine conditions of a CAS and paper-and-pencil learning environment in which those students changed their RF.

Change in RF was measured by comparing results from initial to …

Variations On A Graph Coloring Theme, Bryan Phinezy

Dissertations

Historically, the subject of graph colorings has been the most popular research area in graph theory. There are many problems in mathematics and in real life that can be represented by a graph and whose solution involves finding a specific coloring of this graph. Our research consists of two parts: (1) combinatorial problems and vertex colorings and (2) distance-defined colorings. In this research, we show that certain combinatorial puzzles and problems can be placed in a graph coloring setting and graph colorings can be defined in terms of distance in graphs that are useful in applications.

Two vertices u and …

Rank-Based Estimation And Prediction For Mixed Effects Models In Nested Designs, Yusuf K. Bilgic

Dissertations

Hierarchical designs frequently occur in many research areas. The experimental design of interest is expressed in terms of fixed effects but, for these designs, nested factors are a natural part of the experiment. These nested effects are generally considered random and must be taken into account in the statistical analysis. Traditional analyses are quite sensitive to outliers and lose considerable power to detect the fixed effects of interest.

This work proposes three rank-based fitting methods for handling random, fixed and scale effects in k-level nested designs for estimation and inference. An algorithm, which iteratively obtains robust prediction for both scale …

Energy Methods For Reaction-Diffusion Problems, Xing Zhong

Dissertations

Nonlinear reaction-diffusion equations arise in many areas of applied sciences such as combustion modeling, population dynamics, chemical kinetics, etc. A fundamental problem in the studies of these equations is to understand the long time behavior of solutions of the associated Cauchy problem. These kinds of questions were originally studied in the context of combustion modeling.

For suitable nonlinearity and a monotone increasing one-parameter family of initial data starting with zero data, small values of the parameter lead to extinction, whereas large values of the parameter may lead to spreading, i.e., the solution converging locally uniformly to a positive spatially independent …