Mathematics Commons^™

Algebraic And Combinatorial Properties Of Schur Rings Over Cyclic Groups, Andrew F. Misseldine

Theses and Dissertations

In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of …

Identification Of Transcriptionally Quiescent Regions In The Neurospora Crassa Genome, Katie Marie Groskreutz

Theses and Dissertations

Sexual reproduction and genetic exchange via meiosis are important and highly conserved processes in many living organisms. Occasionally, complications occur during meiosis that can result in chromosome abnormalities. In humans, improper chromosome development can cause life altering disorders such as Down Syndrome, Edwards Syndrome, and Patau Syndrome. Unfortunately, despite its importance, gaps remain in our knowledge of how this process works. For instance, little is known about how homolog identification occurs and what proteins identify matching chromosomes during pairing. This fundamental process occurs early during meiosis and ensures proper development of gametes.

Understanding the proteins involved during homolog pairing may …

A Volume Bound For Montesinos Links, Kathleen Arvella Finlinson

Theses and Dissertations

The hyperbolic volume of a knot complement is a topological knot invariant. Futer, Kalfagianni, and Purcell have estimated the volumes of Montesinos link complements for Montesinos links with at least three positive tangles. Here we extend their results to all hyperbolic Montesinos links.

The Minimum Rank Of Schemes On Graphs, William Nelson Sexton

Theses and Dissertations

Let G be an undirected graph on n vertices and let S(G) be the class of all real-valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let V = {1, 2, . . . , n} be the vertex set of G. A scheme on G is a function f : V → {0, 1}. Given a scheme f on G, there is an associated class of matrices Sf (G) = {A ∈ S(G)|aii = 0 if and only if f(i) = 0}. A scheme f is said …

Representations Associated To The Group Matrix, Joseph Aaron Keller

Theses and Dissertations

For a finite group G = {g_0 = 1, g_1,. . ., g_{n-1}} , we can associate independent variables x_0, x_1, . . ., x_{n-1} where x_i = x_{g_i}. There is a natural action of Aut(G) on C[x_0, . . . ,x_{n-})]. Let C_1, . . . , C_r be the conjugacy classes of G. If C = {g_{i_1}, g_{i_2}, . . . , g_{i_u }} is a conjugacy class, then let x(C) = x_{i_1} + x_{i_2} + . . . + x_{i_u}. Let ρG be the representation of Aut(G) on C[x_0, . . . , x_(n-1)]/〈x(C_1), . . . …

Turán Problems On Non-Uniform Hypergraphs, Jeremy Travis Johnston

Theses and Dissertations

A non-uniform hypergraph H = (V, E) consists of a vertex set V and an edge set E ⊆ 2 V; the edges in E are not required to all have the same cardinality. The set of all cardinalities of edges in H is denoted by R(H), the set of edge types. For a fixed hypergraph H, the Turán density π(H) is defined to be the maximum Lubell value of a graph G (in the limit) which is H-free and such that R(G) ⊆ R(H). The Lubell function, is the expected number of edges in G hit by a random …

Independence Polynomials, Gregory Matthew Ferrin

Theses and Dissertations

In this thesis, we investigate the independence polynomial of a simple graph G. In addition to giving several tools for computing these polynomials and giving closed-form representations of these polynomials for common classes of graphs, we prove two results concerning the roots of independence polynomials. The first result gives us the unique root of smallest modulus of the independence polynomial of a graph. The second result tells us that all the roots of the independence polynomial of a claw-free graph fall on the real line.

Prospect Theory Preferences In Noncooperative Game Theory, Philip Leclerc

Theses and Dissertations

The present work seeks to incorporate a popular descriptive, empirically grounded model of human preference under risk, prospect theory, into the equilibrium theory of noncooperative games. Three primary, candidate definitions are systematically identified on the basis of classical characterizations of Nash Equilibrium; in addition, three equilibrium subtypes are defined for each primary definition, in order to enable modeling of players' reference points as exogenous and fixed, slowly and myopically adaptive, highly flexible and non-myopically adaptive. Each primary equilibrium concept was analyzed both theoretically and empirically; for the theoretical analyses, prospect theory, game theory, and computational complexity theory were all summoned …

Ramp Loss Svm With L1-Norm Regularizaion, Eric Hess

Theses and Dissertations

The Support Vector Machine (SVM) classification method has recently gained popularity due to the ease of implementing non-linear separating surfaces. SVM is an optimization problem with the two competing goals, minimizing misclassification on training data and maximizing a margin defined by the normal vector of a learned separating surface. We develop and implement new SVM models based on previously conceived SVM with L_1-Norm regularization with ramp loss error terms. The goal being a new SVM model that is both robust to outliers due to ramp loss, while also easy to implement in open source and off the shelf mathematical programming …

Counting Threshold Graphs And Finding Inertia Sets, Christopher Abraham Guzman

Theses and Dissertations

This thesis is separated into two parts: threshold graphs and inertia sets. First we present an algorithmic approach to finding the minimum rank of threshold graphs and then progress to counting the number of threshold graphs with a specific minimum rank. Second, we find an algorithmic and more automated way of determining the inertia set of graphs with seven or fewer vertices using theorems and lemmata found in previous papers. Inertia sets are a relaxation of the inverse eigenvalue problem. Instead of determining all the possible eigenvalues that can be obtained by matrices with a specific zero/nonzero pattern we restrict …

Record Linkage, Stasha Ann Bown Larsen

Theses and Dissertations

This document explains the use of different metrics involved with record linkage. There are two forms of record linkage: deterministic and probabilistic. We will focus on probabilistic record linkage used in merging and updating two databases. Record pairs will be compared using character-based and phonetic-based similarity metrics to determine at what level they match. Performance measures are then calculated and Receiver Operating Characteristic (ROC) curves are formed. Finally, an economic model is applied that returns the optimal tolerance level two databases should use to determine a record pair match in order to maximize profit.

Minimum Rank Problems For Cographs, Nicole Andrea Malloy

Theses and Dissertations

Let G be a simple graph on n vertices, and let S(G) be the class of all real-valued symmetric nxn matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The smallest rank achieved by a matrix in S(G) is called the minimum rank of G, denoted mr(G). The maximum nullity achieved by a matrix in S(G) is denoted M(G). For each graph G, there is an associated minimum rank class, MR(G) consisting of all matrices A in S(G) with rank A = mr(G). Although no restrictions are applied to the diagonal entries of …

A Topics Analysis Model For Health Insurance Claims, Jared Anthony Webb

Theses and Dissertations

Mathematical probability has a rich theory and powerful applications. Of particular note is the Markov chain Monte Carlo (MCMC) method for sampling from high dimensional distributions that may not admit a naive analysis. We develop the theory of the MCMC method from first principles and prove its relevance. We also define a Bayesian hierarchical model for generating data. By understanding how data are generated we may infer hidden structure about these models. We use a specific MCMC method called a Gibbs' sampler to discover topic distributions in a hierarchical Bayesian model called Topics Over Time. We propose an innovative use …

Examining Middle School Students' Statistical Thinking While Working In A Technological Environment, Melissa Arnold Scranton

Theses and Dissertations

Examining Middle School Students' Statistical Thinking

While Working in a Technological Environment

Melissa Arnold Scranton

The purpose of this study was to gain a better understanding of how students think in a technological environment. This was accomplished by exploring the differences in the thinking of students while they worked in a technological environment and comparing this to their work in a paper and pencil environment. The software program TinkerPlots: Dynamic Data Exploration (Konold & Miller, 2005), a construction tool that middle school students use for data analysis was the technological environment. In both environments, types of critical, creative, and statistical …

A Class Of Univalent Convolutions Of Harmonic Mappings, Matthew Daniel Romney

Theses and Dissertations

A planar harmonic mapping is a complex-valued function ƒ : D → C of the form ƒ(x+iy) = u(x,y) + iv(x,y), where u and v are both real harmonic. Such a function can be written as ƒ = h+g where h and g are both analytic; the function w = g'/h' is called the dilatation of ƒ. This thesis considers the convolution or Hadamard product of planar harmonic mappings that are the vertical shears of the canonical half-plane mapping p;(z) = z/(1-z) with respective dilatations e^iθz and e^ipz, θ, p ∈ R. We prove that any such convolution is univalent. …

The Minimum Rank Problem For Outerplanar Graphs, John Henry Sinkovic

Theses and Dissertations

Given a simple graph G with vertex set V(G)={1,2,...,n} define S(G) to be the set of all real symmetric matrices A such that for all i not equal to j, the ijth entry of A is nonzero if and only if ij is in E(G). The range of the ranks of matrices in S(G) is of interest and can be determined by finding the minimum rank. The minimum rank of a graph, denoted mr(G), is the minimum rank achieved by a matrix in S(G). The maximum nullity of a graph, denoted M(G), is the maximum nullity achieved by a matrix …

Bounding The Norm Of Matrix Powers, Daniel Ammon Dowler

Theses and Dissertations

In this paper I investigate properties of square complex matrices of the form A^k, where A is also a complex matrix, and k is a nonnegative integer. I look at several ways of representing A^k. In particular, I present an identity expressing the k^th power of the Schur form T of A in terms of the elements of T, which can be used together with the Schur decomposition to provide an expression of A^k. I also explain bounds on the norm of A^k, including some based on the element-based expression of …

Integral Traces Of Weak Maass Forms Of Genus Zero Odd Prime Level, Nathan Eric Green

Theses and Dissertations

Duke and Jenkins defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.

Spectral Stability Of Weak Detonations In The Majda Model, Jeffrey James Hendricks

Theses and Dissertations

Using analytical and numerical Evans-function techniques, we examine the spectral stability of weak-detonation-wave solutions of Majda's scalar model for a reacting gas mixture. We provide a proof of monotonicity of solutions. Using monotonicity we obtain a bound on possible unstable eigenvalues for weak-detonation-wave solutions that improves on the more general bound given by Humpherys, Lyng, and Zumbrun. We use a numerical approximation of the Evans function to search for possible unstable eigenvalues in the bounded region obtained by the energy estimate. For the parameter values tested, our results combined with the result of Lyng, Raoofi, Texier, and Zumbrun demonstrate that …

The Frobenius Manifold Structure Of The Landau-Ginzburg A-Model For Sums Of An And Dn Singularities, Rachel Megan Webb

Theses and Dissertations

In this thesis we compute the Frobenius manifold of the Landau-Ginzburg A-model (FJRW theory) for certain polynomials. Specifically, our computations apply to polynomials that are sums of An and Dn singularities, paired with the corresponding maximal symmetry group. In particular this computation applies to several K3 surfaces. We compute the necessary correlators using reconstruction, the concavity axiom, and new techniques. We also compute the Frobenius manifold of the D3 singularity.

Market Dynamics With Non-Homogeneous Poisson Processes, Preston T. Redd

Theses and Dissertations

The Bertrand Duopoly model for demand in economics is a well-used model. Although this model has important insights towards pricing strategy, it does not accurately depict true market behaviors. In this paper, we will examine the advantages and disadvantages of the current model and its assumptions.We then take a whole new approach towards modeling this phenomena, using Poisson processes to model the demand of goods. We will discuss why this is a better approach and explain how we can extend this to better understand pricing strategies and market dynamics. We then apply our findings to the newsvendor problem, a commonly …

Modelling Infertility With Markov Chains, Rebecca Dorff

Theses and Dissertations

Infertility affects approximately 15% of couples. Testing and interventions are costly, in time, money, and emotional energy. This paper will discuss using Markov decision and multi-armed bandit processes to identify a systematic approach of interventions that will lead to the desired baby while minimizing costs.

Investment-Consumption With A Randomly Terminating Income, James Benjamin Taylor Jr.

Theses and Dissertations

We develop a stochastic control model for an investor's optimal investment and consumption over an uncertain planning horizon when the investor is endowed with a defaultable income stream. The distributions of the random time of default and the random terminal time are prescribed by deterministic hazard rates, and the investor makes investments in a standard financial market with a bond and a stock, modeled by geometric Brownian motion. In addition, the investor purchases insurance against both default and the terminal date, the default insurance serving as a proxy for the investor's disutility for default. We approximate the original continuous-time problem …

Poincaré Polynomial Of Fjrw Rings And The Group-Weights Conjecture, Julian Boon Kai Tay

Theses and Dissertations

FJRW-theory is a recent advancement in singularity theory arising from physics. The FJRW-theory is a graded vector space constructed from a quasihomogeneous weighted polynomial and symmetry group, but it has been conjectured that the theory only depends on the weights of the polynomial and the group. In this thesis, I prove this conjecture using Poincaré polynomials and Koszul complexes. By constructing the Koszul complex of the state space, we have found an expression for the Poincaré polynomial of the state space for a given polynomial and associated group. This Poincaré polynomial is defined over the representation ring of a group …

American Spread Option Models And Valuation, Yu Hu

Theses and Dissertations

Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we …

Investigation Into The Ratio Of System Operations And Support Costs To Life-Cycle Costs For Department Of Defense Weapon Systems, Gary L. Jones

Theses and Dissertations

As the arsenal of weapons managed by the Department of Defense (DoD) ages, the country's leadership is forced to make decisions regarding what to do with current and projected funding to maintain military defensive and offensive capabilities. The acquisition community at many levels has stated that O&S costs are about 70 percent of the total life-cycle costs of the average system, the other 30 percent being spent in the acquisition phase. This golden ratio does not appear to come from empirical evidence. There has been little research into the actual ratio of O&S-to-acquisition costs. This research found a significant departure …

Determining Optimal Machine Replacement Events With Periodic Inspection Intervals, Theodore C. Shiveley

Theses and Dissertations

This research will examine the optimal maintenance and replacement policies for a generic machine with periodic inspection intervals. The considered reliability models consist of a single machine that can fail during operation or else may be found to be inoperative during regularly-scheduled maintenance inspections. A distinction will be made between spontaneously-occurring failures during operation and those that are discovered during inspections. Since the elapsed time between inspections is constant, the resulting stochastic reliability process becomes non-Markovian, and thus a Semi-Markov Decision Process (SMDP) framework must be employed in order to determine the cost-optimal stationary policy consisting of repair and replace …

Character Tables Of Metacyclic Groups, Dane Christian Skabelund

Theses and Dissertations

We show that any two split metacyclic groups with the same character tables are isomorphic. We then use this to show that among metacyclic groups that are either 2-groups or are of odd order divisible by at most two primes, that the dihedral and generalized quaternion groups of order 2^n, n = 3, are the only pairs that have the same character tables.

A Classification Tool For Predictive Data Analysis In Healthcare, Mason Lemoyne Victors

Theses and Dissertations

Hidden Markov Models (HMMs) have seen widespread use in a variety of applications ranging from speech recognition to gene prediction. While developed over forty years ago, they remain a standard tool for sequential data analysis. More recently, Latent Dirichlet Allocation (LDA) was developed and soon gained widespread popularity as a powerful topic analysis tool for text corpora. We thoroughly develop LDA and a generalization of HMMs and demonstrate the conjunctive use of both methods in predictive data analysis for health care problems. While these two tools (LDA and HMM) have been used in conjunction previously, we use LDA in a …

The Weierstrass Approximation Theorem, Larita Barnwell Hipp

Theses and Dissertations

In this thesis we will consider the work began by Weierstrass in 1855 and several generalization of his approximation theorem since. Weierstrass began by proving the density of algebraic polynomials in the space of continuous real-valued functions on a finite interval in the uniform norm. His theorem has been generalized to an arbitrary compact Hausdorff space and the approximation with elements from more general algebras of continuous real-valued functions. We will consider proofs that use brute force and proofs based on convolutions and approximate identities, trudge through probability and the use of the Bernstein polynomials, and become intimately close to …