Mathematics Commons^™

Asymptotic And Numerical Analysis Of Time-Dependent Wave Propagation In Dispersive Dielectric Media That Exhibit Fractional Relaxation, Matthew Frank Causley

Dissertations

This dissertation addresses electromagnetic pulse propagation through anomalously dispersive dielectric media. The Havriliak-Negami (H-N) and Cole-Cole (C-C) models capture the non-exponential nature of such dielectric relaxation phenomena, which is manifest in a variety of dispersive dielectric media. In the C-C model, the dielectric polarization is coupled to the time-dependent Maxwell's equations by a fractional differential equation involving the electric field. In the H-N case, a more general pseudo-fractional differential operator describes the polarization.

The development and analysis of a robust method for implementing such models is presented, with an emphasis on algorithmic efficiency. Separate numerical schemes are presented for C-C …

Numerical And Asymptotic Modeling Of Evolving Nonlinear Ocean Surface Wave Fields, Matt Malej

Dissertations

The main focus of this dissertation is the asymptotic and numerical modeling of nonlinear ocean surface wave fields. In particular, a development of an accurate numerical model for the evolution of nonlinear ocean waves, including extreme waves known as Rogue/Freak waves. Due to their elusive and destructive nature, the media often portrays Rogue waves as unimaginatively huge and unpredictable monsters of the sea. To address these concerns, derivations of asymptotically reduced models, based on the small wave steepness assumption, are presented and their corresponding numerical simulations via a Fourier pseudo-spectral method are discussed. The simulations are initialized with a well-known …

The Weak Cayley Table And The Relative Weak Cayley Table, Melissa Anne Mitchell

Theses and Dissertations

In 1896, Frobenius began the study of character theory while factoring the group determinant. Later in 1963, Brauer pointed out that the relationship between characters and their groups was still not fully understood. He published a series of questions that he felt would be important to resolve. In response to these questions, Johnson, Mattarei, and Sehgal developed the idea of a weak Cayley table map between groups. The set of all weak Cayley table maps from one group to itself also has a group structure, which we will call the weak Cayley table group. We will examine the weak Cayley …

Trends Over 5 Decades In U.S. Occupation-Related Physical Activity And Their Associations With Obesity, Timothy S. Church, Diana M. Thomas, Catrine Tudor-Locke, Peter T. Katzmarzyk, Conrad P. Earnest, Ruben Q. Rodarte, Corby K. Martin, Steven N. Blair, Claude Bouchard

Department of Mathematics Faculty Scholarship and Creative Works

Background: The true causes of the obesity epidemic are not well understood and there are few longitudinal population-based data published examining this issue. The objective of this analysis was to examine trends in occupational physical activity during the past 5 decades and explore how these trends relate to concurrent changes in body weight in the U.S.

Methodology/Principal Findings: Analysis of energy expenditure for occupations in U.S. private industry since 1960 using data from the U.S. Bureau of Labor Statistics. Mean body weight was derived from the U.S. National Health and Nutrition Examination Surveys (NHANES). In the early 1960's almost half …

Boundary Quasi-Orthogonality And Sharp Inclusion Bounds For Large Dirichlet Eigenvalues, A. H. Barnett, A. Hassell

Dartmouth Scholarship

We study eigenfunctions $\phi_j$ and eigenvalues $E_j$ of the Dirichlet Laplacian on a bounded domain $\Omega\subset\mathbb{R}^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E>0$ and the spectrum $\{E_j\}$ in terms of the boundary $L^2$-norm of a normalized trial solution u of the Helmholtz equation $(\Delta+E)u=0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all E greater than a small constant, and improve upon the best-known bounds of Moler–Payne by a factor of the wavenumber $\sqrt{E}$. One …

Noncomputable Functions In The Blub-Shub-Smale Model, Wesley Calvert, Ken Kramer, Russell Miller

Publications and Research

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more …

Darboux Integrability - A Brief Historial Survey, Ian M. Anderson

Mathematics and Statistics Faculty Presentations

No abstract provided.

Geometric Invariants For A Class Of Semi-Fredholm Hilbert Modules., Shibananda Biswas Dr.

Doctoral Theses

One of the basic problem in the study of a Hilbert module H over the ring of polynomials C[z] := C[z1, . . . , zm] is to find unitary invariants (cf. [15,7]) for H. It is not always possible to find invariants that are complete and yet easy to compute. There are very few instances where a set of complete invariants have been identified. Examples are Hilbert modules over continuous functions (spectral theory of normal operator), contractive modules over the disc algebra (model theory for contractive operator) and Hilbert modules in the class Bn for a bounded domain C …

Linear Algebra, Lawrence Stout

Scholarship

This book is designed to deal with all of the concepts of linear algebra first in R2, a simple context where algorithmic concerns are minimized and geometric intuition can be brought to bear. Then those same concepts are dealt with in full generality. I find that this conceptual front loading helps in the understanding of the rest of the material. Students can see easily what aspects of the plane are focused on when we think of it as a vector space; linear transformations can be understood as deformations of the plane of a very special type. Nothing more complicated than …

Editorial Announcement, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

[No abstract available]

Viscosity Of Bacterial Suspensions: Hydrodynamic Interactions And Self-Induced Noise, Shawn D. Ryan, Brian M. Haines, Leonid Berlyand, Falko Ziebert, Igor S. Aranson

Mathematics and Statistics Faculty Publications

The viscosity of a suspension of swimming bacteria is investigated analytically and numerically. We propose a simple model that allows for efficient computation for a large number of bacteria. Our calculations show that long-range hydrodynamic interactions, intrinsic to self-locomoting objects in a viscous fluid, result in a dramatic reduction of the effective viscosity. In agreement with experiments on suspensions of Bacillus subtilis, we show that the viscosity reduction is related to the onset of large-scale collective motion due to interactions between the swimmers. The simulations reveal that the viscosity reduction occurs only for relatively low concentrations of swimmers: Further …

The Falc-Loop Web Server For Protein Loop Modeling, Evangelos A. Coutsias, Juns Ko, Dongseon Lee, Hahnbeom Park, Julian Lee, Chaok Seok

Branch Mathematics and Statistics Faculty and Staff Publications

The FALC-Loop web server provides an online interface for protein loop modeling by employing an ab initio loop modeling method called FALC (fragment assembly and analytical loop closure). The server may be used to construct loop regions in homology modeling, to refine unreliable loop regions in experimental structures or to model segments of designed sequences. The FALC method is computationally less expensive than typical ab initio methods because the conformational search space is effectively reduced by the use of fragments derived from a structure database. The analytical loop closure algorithm allows efficient search for loop conformations that fit into the …

The Quantum Dialectic, Logan Kelley

Pitzer Senior Theses

A philosophic account of quantum physics. The thesis is divided into two parts. Part I is dedicated to laying the groundwork of quantum physics, and explaining some of the primary difficulties. Subjects of interest will include the principle of locality, the quantum uncertainty principle, and Einstein's criterion for reality. Quantum dilemmas discussed include the double-slit experiment, observations of spin and polarization, EPR, and Bell's theorem. The first part will argue that mathematical-physical descriptions of the world fall short of explaining the experimental observations of quantum phenomenon. The problem, as will be argued, is framework of the physical descriptive schema. Part …

Pruitt's Estimates In Banach Space, Philip S. Griffin

Mathematics - All Scholarship

Pruitt's estimates on the expectation and the distribution of the time taken by a random walk to exit a ball of radius r are extended to the infinite dimensional setting. It is shown that they separate into two pairs of estimates depending on whether the space is type 2 or cotype 2. It is further shown that these estimates characterize type 2 and cotype 2 spaces.

Implementation Of An Inexpensive 3d Scanner, Robert Michael Sivley

Mahurin Honors College Capstone Experience/Thesis Projects

The purpose of this project was to make 3D scanning technology accessible to the general public at an affordable price. To accomplish this, a scanner was designed that utilized structured-light projection. Coupled with triangulation, this provided an inexpensive, image-based modeling technique. Software was developed to analyze the required images and generate a 3D model. Users can edit analysis parameters during runtime to better optimize results for specific images. The 3D models generated are stored according to the .obj standard and can be opened in any commercial 3D modeling software. Results were positive, but several issues exist with the chosen scanning …

Universal Hypergraphs., Michael Deren

Electronic Theses and Dissertations

In this thesis, we study universal hypergraphs. What are these? Let us start with defining a universal graph as a graph on n vertices that contains each of the many possible graphs of a smaller size k < n as an induced subgraph. A hypergraph is a discrete structure on n vertices in which edges can be of any size, unlike graphs, where the edge size is always two. If all edges are of size three, then the hypergraph is said to be 3-uniform. If a 3-uniform hypergraph can have edges colored one of a colors, then it is called a …

A Predictive Model Which Uses Descriptors Of Rna Secondary Structures Derived From Graph Theory., Alissa Ann Rockney

Electronic Theses and Dissertations

The secondary structures of ribonucleic acid (RNA) have been successfully modeled with graph-theoretic structures. Often, simple graphs are used to represent secondary RNA structures; however, in this research, a multigraph representation of RNA is used, in which vertices represent stems and edges represent the internal motifs. Any type of RNA secondary structure may be represented by a graph in this manner. We define novel graphical invariants to quantify the multigraphs and obtain characteristic descriptors of the secondary structures. These descriptors are used to train an artificial neural network (ANN) to recognize the characteristics of secondary RNA structure. Using the ANN, …

Set: The Probabilities And Possibilities, Tabitha K. Bollinger

Undergraduate Theses and Capstone Projects

The card game SET involves finding groups o f three cards called SETs. Choices are based upon the individual card characteristics, including shape, pattern, number, and color. Previously, the maximum number o f cards that can be played without creating a SET has been determined as 20 cards by extensive computer work. This report further explored the probabilities and possibilities o f the game. Using discrete mathematics and probability, we explored how many SETs are possible and what strategies led to the most points. Additionally, this project exercised undergraduate logic and reasoning to generalize the results in order to be …

Algebraic Solutions To Overdefined Systems With Applications To Cryptanalysis, Eric Crockett

Mathematical Sciences Technical Reports (MSTR)

Cryptographic algorithms are based on a wide variety of difficult problems in mathematics. One of these problems is finding a solution to a system of multivariate quadratic equations (MQ). A generalization of this problem is to find a solution to a system of higher order non-linear equations. Both of these problems are NP-hard over any field. Many cryptosystems such as AES, Serpent, Toyocrypt, and others can be reduced to some form of the MQ problem. In this paper we analyze the relinearization and XL algorithms for solving overdetermined systems of non-linear equations, as well as two variations of the XL …

A Range And Existence Theorem For Pseudomonotone Perturbations Of Maximal Monotone Operators, Vy Khoi Le

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we prove a range and existence theorem for multivalued pseudomonotone perturbations of maximal monotone operators. We assume a general coercivity condition on the sum of a maximal monotone and a pseudomonotone operator instead of a condition on the pseudomonotone operator only. An illustrative example of a variational inequality in a Sobolev space with variable exponent is given.