Physical Sciences and Mathematics Commons^™

Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen

Doctoral Dissertations

Let E₁ x E₂ over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E₁ x E₂, denoted as a_p(E₁) + a_p(E₂), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical …

Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, Lian Duan

Doctoral Dissertations

In this thesis, we prove that, a selfdual 3-dimensional Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to 3-dimensional Galois representations with ground field not equal to Q. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

Doctoral Dissertations

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main …

Asymptotic And Numerical Analysis Of Coherent Structures In Nonlinear Schrodinger-Type Equations, Cory Ward

Doctoral Dissertations

This dissertation concerns itself with coherent structures found in nonlinear Schrödinger-type equations and can be roughly split into three parts. In the first part we study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa-Holm NLS (CH-NLS) equation in both one and two space dimensions. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently used to obtain approximate solitary wave solutions for both the 1D and 2D CH-NLS equations. We then use direct numerical simulations to investigate the validity of these approximate solutions, their evolution, and their head-on …

From Optimization To Equilibration: Understanding An Emerging Paradigm In Artificial Intelligence And Machine Learning, Ian Gemp

Doctoral Dissertations

Many existing machine learning (ML) algorithms cannot be viewed as gradient descent on some single objective. The solution trajectories taken by these algorithms naturally exhibit rotation, sometimes forming cycles, a behavior that is not expected with (full-batch) gradient descent. However, these algorithms can be viewed more generally as solving for the equilibrium of a game with possibly multiple competing objectives. Moreover, some recent ML models, specifically generative adversarial networks (GANs) and its variants, are now explicitly formulated as equilibrium problems. Equilibrium problems present challenges beyond those encountered in optimization such as limit-cycles and chaotic attractors and are able to abstract …

Modeling Of Hiv, Sir And Sis Epidemics On Time Scales And Oscillation Theory, Gülşah Yeni

Doctoral Dissertations

"We study higher dimensional systems of first order dynamic equations on time scales together with their applications. In particular, we focus on epidemic models such as HIV (Human Immunodeficiency Virus), SIS (Susceptible-Infected-Susceptible) and SIR (Susceptible-Infected-Recovered).

First, we generalize the early studied continuous three dimensional linear model of drug therapy for HIV-1 decline on time scales in order to derive new discrete models that predict the total concentration of plasma virus as a function of time. We compare these models to explore the impact of the theory of time scales. After fitting the models to the data collected at a clinical …

Pressure Versus Impulse Graph For Blast-Induced Traumatic Brain Injury And Correlation To Observable Blast Injuries, Barbara Rutter

Doctoral Dissertations

"With the increased use of explosive devices in combat, blast induced traumatic brain injury (bTBI) has become one of the signature wounds in current conflicts. Animal studies have been conducted to understand the mechanisms in the brain and a pressure versus time graph has been produced. However, the role of impulse in bTBIs has not been thoroughly investigated for animals or human beings.

This research proposes a new method of presenting bTBI data by using a pressure versus impulse (P-I) graph. P-I graphs have been found useful in presenting lung lethality regions and building damage thresholds. To present the animal …

Dendrites Or Lambda-Dendroids As Generalized Inverse Limits, Faruq Abdullah Mena

Doctoral Dissertations

"This research centers on the study of generalized inverse limits. We show that all members of an infinite family of inverse limit spaces are homeomorphics to one particularly complicated inverse limit space known as "The Monster". Further, properties of factor spaces and graphs of bonding functions which are preserved in generalized inverse limit spaces with upper semi-continuous bonding functions with appropriate restrictions are investigated. Some of the properties are locally connectedness, hereditary decomposability, hereditary indecomposability, hereditary unicoherence, arc-likeness, and tree-likeness. The theorems are illustrated by several examples"--Abstract, page iv.

Decoupling Methods For The Time-Dependent Navier-Stokes-Darcy Interface Model, Changxin Qiu

Doctoral Dissertations

"In this research, several decoupling methods are developed and analyzed for approximating the solution of time-dependent Navier-Stokes-Darcy (NS-Darcy) interface problems. This research on decoupling methods is motivated to efficiently solve the complex Stokes-Darcy or NS-Darcy type models, which arise from many interesting real world problems involved with or even dominated by the coupled porous media flow and free flow. We first discuss a semi-implicit, multi-step non-iterative domain decomposition (NIDDM) to solve a coupled unsteady NS-Darcy system with Beavers-Joseph-Saffman-Jones (BJSJ) interface condition and obtain optimal error estimates. Second, a parallel NIDDM is developed to solve unsteady NS-Darcy model with Beavers-Joseph (BJ) …

New Reproducing Kernel Hilbert Spaces On Plane Regions, Their Properties, And Applications To Partial Differential Equations, Jabar S. Hassan

Doctoral Dissertations

"We introduce new reproducing kernel Hilbert spaces W₂^(m,n) (D) on unbounded plane regions D. We study linear non-homogeneous hyperbolic partial differential equation problems on D with solutions in various reproducing kernel Hilbert spaces. We establish existence and uniqueness results for such solutions under appropriate hypotheses on the driver. Stability of solutions with respect to the driver is analyzed and local uniform approximation results are obtained which depend on the density of nodes. The local uniform approximation results required a careful determination of the reproducing kernel Hilbert spaces on which the elementary …