Physical Sciences and Mathematics Commons^™

Elegance Or Alchemy? An International Cross-Case Analysis Of Faculty And Graduate Student Perceptions Of Mathematical Proofs, Brooke Nicole Denney

Masters Theses

Artist Marcel Duchamp once said, ``The painter is a medium who doesn’t realize what he is doing. No translation can express the mystery of sensibility, a word, still unreliable, which is nonetheless the basis of painting or poetry, like a kind of alchemy" (Moffitt, 2012). Just as there is a puzzling aspect of creating art or writing poetry, the aesthetic quality of mathematical proofs is a mysterious and ill-defined concept. Like many other subjective terms, it can be difficult to reach a consensus on what elegance means in a mathematical context. In this thesis, I try to better understand faculty …

Classification Results Of Hadamard Matrices, Gregory Allen Schmidt

Masters Theses

In 1893 Hadamard proved that for any n x n matrix A over the complex numbers, with all of its entries of absolute value less than or equal to 1, it necessarily follows that

|det(A)| ≤ n^n/2 [n raised to the power n divided by two],

with equality if and only if the rows of A are mutually orthogonal and the absolute value of each entry is equal to 1 (See [2], [3]). Such matrices are now appropriately identified as Hadamard matrices, which provides an active area of research in both theoretical and applied fields …

Mathematics Education From A Mathematicians Point Of View, Nan Woodson Simpson

Masters Theses

This study has been written to illustrate the development from early mathematical learning (grades 3-8) to secondary education regarding the Fundamental Theorem of Arithmetic and the Fundamental Theorem of Algebra. It investigates the progression of the mathematics presented to the students by the current curriculum adopted by the Rhea County School System and the mathematics academic standards set forth by the State of Tennessee.

On The Existence And Uniqueness Of Static, Spherically Symmetric Stellar Models In General Relativity, Josh Michael Lipsmeyer

Masters Theses

The "Fluid Ball Conjecture" states that a static stellar model is spherically symmetric. This conjecture has been the motivation of much work since first mentioned by Kunzle and Savage in 1980. There have been many partial results( ul-Alam, Lindblom, Beig and Simon,etc) which rely heavily on arguments using the positive mass theorem and the equivalence of conformal flatness and spherical symmetry. The purpose of this paper is to outline the general problem, analyze and compare the key differences in several of the partial results, and give existence and uniqueness proofs for a particular class of equations of state which represents …

Injective Modules And Divisible Groups, Ryan Neil Campbell

Masters Theses

An R-module M is injective provided that for every R-monomorphism g from R-modules A to B, any R-homomorphism f from A to M can be extended to an R-homomorphism h from B to M such that hg = [equals] f. That is one of several equivalent statements of injective modules that we will be discussing, including concepts dealing with ideals of rings, homomorphism modules, short exact sequences, and splitting sequences. A divisible group G is defined when for every element x of G and every nonzero integer n, there exists y in …

Phase Dynamics Of Locset Control Methodology, Brendan Neschke

Masters Theses

Single-mode fiber amplifiers produce diffraction-limited beams very efficiently. Maximum beam intensity requires that an array of these amplifiers have their beams coherently combined at the target. Optical path differences and noise adversely affect beam quality. An existing closed loop phase control methodology, called the locking of optical coherence by single-detector electronic-frequency tagging (LOCSET), corrects phase errors in real time by electronically detecting path length differences and sending signals to lithium niobate phase adjusters. Broadening the line-width using “jitter” of the input signal can increase the output power of an individual amplifier by suppressing nonlinearity. The system dynamics of LOCSET are …

Statistical Mechanics And Schramm-Loewner Evolution With Applications To Crack Propagation Processes, Christopher Borut Mesic

Masters Theses

Schramm-Loewner Evolution (SLE) has both mathematical and physical roots that extend as far back as the early 20th century. We present the progression of these humble roots from the Ideal Gas Law, all the way to the renormalization group and conformal field theory, to better understand the impact SLE has had on modern statistical mechanics. We then explore the potential application of the percolation exploration process to crack propagation processes, illustrating the interplay between mathematics and physics.

An Analysis Of The Patterns Of Crime And Socioeconomic Status Visualized Through Self-Organized Maps, Jason Carlin Kaufman

Masters Theses

This work is research to explore the association of spatial patterns between crime and socioeconomic status (SES) through the use of self-organized maps (SOM). It had been found that the spatial patterns of crime could be associated with those of socioeconomic, and this work sought to further these analyses in order to better understand how crime patterns and SES were related. To explore this association, patterns of crime and SES were examined in three cities: Nashville, TN; Portland, OR; and Tucson, AZ. Three SOMs were used in each city: one to analyze the patterns of crime, a second to analyze …

The Complementing Condition In Elasticity, Lavanya Ramanan

Masters Theses

We consider a boundary value problem of nonlinear elasticity on a domain [omega] in R³ [3-dimensional space] and compute the Complementing Condition for the linearized equations at a point X₀ [x zero] on boundary of omega. We assume a stored energy function depending on the first and third invariants of the deformation F and that the strong-ellipticity condition holds in [omega] . A surface traction boundary condition is imposed at X_0.
The Complementing Condition is calculated from a system of 3 second-order ordinary differential equations (0 less than and equal to t less than infinity) with boundary …

Finite Topological Spaces, Jimmy Edward Miller

Masters Theses

A thesis on some fundamental aspects of Algebraic Topology on Finite Topological Spaces, specifically 4 point spaces.

Validation Of Weak Form Thermal Analysis Algorithms Supporting Thermal Signature Generation, Elton Lewis Freeman

Masters Theses

Extremization of a weak form for the continuum energy conservation principle differential equation naturally implements fluid convection and radiation as flux Robin boundary conditions associated with unsteady heat transfer. Combining a spatial semi-discretization via finite element trial space basis functions with time-accurate integration generates a totally node-based algebraic statement for computing. Closure for gray body radiation is a newly derived node-based radiosity formulation generating piecewise discontinuous solutions, while that for natural-forced-mixed convection heat transfer is extracted from the literature. Algorithm performance, mathematically predicted by asymptotic convergence theory, is subsequently validated with data obtained in 24 hour diurnal field experiments for …

On The Spherical Symmetry Of Perfect-Fluid Stellar Models In General Relativity, Joshua M Brewer

Masters Theses

It is well known in Newtonian theory that static self-gravitating perfect fluids in a vacuum are necessarily spherically symmetric. The necessity of spherical symmetry of perfect-fluid static spacetimes with constant density in general relativity is shown.

On Decision Making: Bayesian And Stochastic Optimization Approaches, Yang Shen

Masters Theses

Decision analysis provides a framework for searching an optimal solution under uncertainties and potential risks. This thesis focuses on two problems arising in transportation engineering and computer sciences, respectively.

First, it is considered a centralized controller which imposes actions on a number of interacting subsystems. Employing an appropriate Markov Decision Process framework, we establish that the Pareto optimal solution of each subsystem will be optimal for the entire system. Synthetic data have been taken into account for verifying this claim.

Next, we focus on a supercomputing problem utilizing a hierarchical Bayesian model. We estimate an optimal solution in order to …

Contractible Theta Complexes Of Graphs, Chelsea Marian Mcamis

Masters Theses

We examine properties of graphs that result in the graph having a contractible theta complex. We classify such properties for tree graphs and graphs with one loop and we introduce examples of graphs with such properties for tree graphs and graphs with one or two loops. For more general graphs, we show that having a contractible theta complex is not an elusive property, and that any skeleton of a graph with at least three loops can be made to have a contractible theta complex by strategically adding vertices to its skeleton.

Alexander And Conway Polynomials Of Torus Knots, Katherine Ellen Louise Agle

Masters Theses

We disprove the conjecture that if K is amphicheiral and K is concordant to K', then C_K'(z)C_K'(iz)C_K\(z²) is a perfect square inside the ring of power series with integer coefficients. The Alexander polynomial of (p,q)-torus knots are found to be of the form A_T(p,q)(t)= (f(t^q))/(f(t)) where f(t)=1+t+t²+...+t^p-1. Also, for (pⁿ,q)-torus knots, the Alexander polynomial factors into the form A_T(pⁿ ,q)=f(t)f(t^p)f(t^p2 )...f(t^pn-2 )f(t^pn-1 ). A new conversion from the Alexander polynomial to the …

On A Quantum Form Of The Binomial Coefficient, Eric J. Jacob

Masters Theses

A unique form of the quantum binomial coefficient (n choose k) for k = 2 and 3 is presented in this thesis. An interesting double summation formula with floor function bounds is used for k = 3. The equations both show the discrete nature of the quantum form as the binomial coefficient is partitioned into specific quantum integers. The proof of these equations has been shown as well. The equations show that a general form of the quantum binomial coefficient with k summations appears to be feasible. This will be investigated in future work.

On Cyclotomic Primality Tests, Thomas Francis Boucher

Masters Theses

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomicprimality test, and shortly after, in 1981, a simplified and more efficient versionwas presented by H.W. Lenstra for the Bourbaki Seminar. Later, in 2008, ReneSchoof presented an updated version of Lenstra's primality test. This thesis presents adetailed description of the cyclotomic primality test as described by Schoof, along withsuggestions for implementation. The cornerstone of the test is a prime congruencerelation similar to Fermat's \little theorem" that involves Gauss or Jacobi sumscalculated over cyclotomic fields. The algorithm runs in very nearly polynomial time.This primality test is currently one of …

On The Behavior Of The Asymptotics Of Robertson-Walker Cosmologies As A Function Of The Cosmological Constant, Noah Thomas Schaefferkoetter

Masters Theses

An analysis of the Einstein Field Equations within a Robertson-Walker Cosmology. More specifically, what values of the cosmological constant will result in a Big Bang.

The Origins Of Mathematical Societies And Journals, Eric S. Savage

Masters Theses

We investigate the origins of mathematical societies and journals. We argue that the origins of today’s professional societies and journals have their roots in the informal gatherings of mathematicians in 17th century Italy, France, and England. The small gatherings in these nations began as academies and after gaining government recognition and support, they became the ancestors of the professional societies that exist today. We provide a brief background on the influences of the Renaissance and Reformation before discussing the formation of mathematical academies in each country.

Some Congruence Modulo 2 Statements Of Primitive Conway Vassiliev Invariants., James M. Dawson

Masters Theses

Polynomial knot invariants can often be used to define Vassiliev invariants on singu- lar knots. Here Vassiliev invariants form the Conway, Jones, HOMFLY, and Kauffman polynomials are explored. Also, some explanation is given about how symbols of the Jones and Conway polynomial can evaluated on suitable chord diagrams. These in- variants are further used to find expressions that are congruent modulo 2 to some low degree invariants derived from the Primitive Conway polynomial.

Biochemical Reactions Instigating Vision – Parameter Sensitivity Analysis, Sophonie Ketcha Tchoua

Masters Theses

The purpose of this work is to investigate the sensitivity of parameters involved in a cascade of biochemical reactions occurring in photoreceptor cells in the retina of the eye. This cascade constitutes the first stage of the elaborate process of vision, by which light captured in a photoreceptor generates an electrical signal. It is this signal that travels to the brain enabling vision.

Sensitivity on parameters was performed on an ODE model of the biochemical cascade using two methods. One method used SimLab, a statistical sensitivity analysis program. We found that there are at most five important parameters out of …

A Note On Hamel Bases, Jeremy S. Higdon

Masters Theses

The purpose of this paper is to discuss certain properties of Hamel bases. In particular, we reprove and generalize a theorem of R. Mabry (Aequations Mathematicae 71 (2006) p. 294-299) on the non-existence of nontrivial Hamel bases closed under multiplication.

Tree-Like Spaces With The Fixed-Point Property, Realized As Inverse Limits Of Special Finite Trees, Jennifer Katherine Aust

Masters Theses

In this paper, we explore some properties of inverse limit sequences on sub-spaces of Euclidean n-space. We address some well-known examples, in particular the example by David Bellamy of the "tree-like" continuum that does not have the fixed-point property. We highlight some spaces with the fixed-point property that are between snake-like continua and Bellamy's example in their level of complexity. Specifically, we prove the fixed-point property for inverse limits of limit sequences on the unit interval and on the n-ad (in two configurations), and for inverse limits that can be mapped via a continuous function with small point pre-images to …

Transformations Of Differential Equations And Applications, Despina Andrea Stavri

Masters Theses

The Sturm-Liouville problems introduced by Jacques Charles Francois Sturm and Joseph Liouville in the 19^thcentury are very important in applied mathematics. Singular Sturm-Liouville problems and two physical problems are discussed. The two physical problems are solved using two methods: Exact and asymptotic solutions. Also, the Sturm-Liouville problem's are classified using the Weyl-Kodaira Theorem. A general transformation of a third order differential equation is introduced. Oscillation and non-oscillation theorems on an infinite interval for a third order differential equation are stated. New versions of these theorems are introduced for the special cases and examples are given to illustrate them. …

Application Of The Green’S Function For Solutions Of Third Order Nonlinear Boundary Value Problems, Shannon Mathis Morrison

Masters Theses

Green’s functions are used to prove a collection of existence and uniqueness theorems for third order nonlinear boundary value problems. Several examples of Green’s functions for both second and third order boundary value problems are given. Various applications of the existence theorems are presented in detail.

A Study Of The Homology Of Subset Spaces And Their Connection To The K-Sat Problem In Computer Science, Oliver J. Thistlethwaite

Masters Theses

It is the purpose of this thesis to introduce an idea for studying questions of computer science via topology. We begin by describing the homology of a certain space constructed from a given subset of the power set of any finite set. We then discuss how this relates to the k-SAT problem in computer science.

We shall use computers as a tool to calculate the homology groups as well as the Euler characteristic of some of these spaces. Due to the sheer number of calcu- lations needed, doing the necessary computations by hand is both impractical and impossible.

In addition, …

Variational And Partial Differential Equation Models For Color Image Denoising And Their Numerical Approximations Using Finite Element Methods, Miun Yoon

Masters Theses

Image processing has been a traditional engineering field, which has a broad range of applications in science, engineering and industry. Not long ago, statistical and ad hoc methods had been main tools for studying and analyzing image processing problems. In the past decade, a new approach based on variational and partial differential equation (PDE) methods has emerged as a more powerful approach. Compared with old approaches, variational and PDE methods have remarkable advantages in both theory and computation. It allows to directly handle and process visually important geometric features such as gradients, tangents and curvatures, and to model visually meaningful …

Nikolski's Approach To The Theorems Of Beurling And Nyman Regarding Zeros Of The Riemann Ζ-Function, Jared R. Bunn

Masters Theses

In this thesis we present the proof of a theorem by Nikolai Nikolski. This theorem leads to a more general theorem by Nikolski regarding zero free regions of the Riemann ζ-function. This theorem is an improvement on the theorems that Nyman and Beurling proved in the nineteen fifties. Nikolski’s approach uses, in addition to step function approximations introduced by Nyman, distance functions to give more flexibility, including possible numerical experiments. The introduction discusses the Riemann Hypothesis, which always surrounds any study of the Riemann ζ-function.

The background material discussed in this thesis gives all the necessary prerequi- sites …

Coarse Structures And Higson Compactification, Christian Stuart Hoffland

Masters Theses

FOLLOWING John Roe in his Lectures on Coarse Geometry, we begin by describing the large-scale structure of metric spaces by means of coarse maps between them, those being maps which preserve distances at large scales. Using these techniques, we demonstrate that the real numbers and the integers have the same large scale structure--or are coarsely equivalent--but that the real line is coarsely equivalent to neither the Euclidean plane nor the set of positive real numbers. Following a generalization of these concepts for general topological spaces with the introduction of an abstract coarse structure on the space, we show, among …

The Classification Of Links Of Up To And Including Thirteen Crossings, Dylan Joshua Faullin

Masters Theses

The prime links of up to 13 crossings are classified up to unoriented equivalence.