Mathematics Commons^™

Quasi-Gorenstein Modules, Alexander York

Electronic Theses and Dissertations

This thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An R-module M of grade g will be quasi-Gorenstein if ExtiR(M, R) = 0 for i 6= g and there is an isomorphism M ∼= ExtgR(M, R). Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their …

Weierstrass Vertices And Divisor Theory Of Graphs, Ajani Ruwandhika Chulangi De Vas Gunasekara

Electronic Theses and Dissertations

Chip-firing games and divisor theory on finite, connected, undirected and unweighted graphs have been studied as analogs of divisor theory on Riemann Surfaces. As part of this theory, a version of the one-dimensional Riemann-Roch theorem was introduced for graphs by Matt Baker in 2007. Properties of algebraic curves that have been studied can be applied to study graphs by means of the divisor theory of graphs. In this research, we investigate the property of a vertex of a graph having the Weierstrass property in analogy to the theory of Weierstrass points on algebraic curves. The weight of the Weierstrass vertices …

Residuated Maps, The Way-Below Relation, And Contractions On Probabilistic Metric Spaces., M. Ryan Luke

Electronic Theses and Dissertations

In this dissertation, we will examine residuated mappings on a function lattice and how they behave with respect to the way-below relation. In particular, which residuated $\phi$ has the property that $F$ is way-below $\phi(F)$ for $F$ in appropriate sets. We show the way-below relation describes the separation of two functions and how this corresponds to contraction mappings on probabilistic metric spaces. A new definition for contractions is considered using the way-below relation.

Vertex Weighted Spectral Clustering, Mohammad Masum

Electronic Theses and Dissertations

Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to …

Extending Difference Of Votes Rules On Three Voting Models., Sarah Schulz King

Electronic Theses and Dissertations

In a voting situation where there are only two competing alternatives, simple majority rule outputs the alternatives with the most votes or declares a tie if both alternatives receive the same number of votes. For any non-negative integer k, the difference of votes rule M_k outputs the alternative that beats the competing alternative by more than k votes. Llamazares (2006) gives a characterization of the difference of votes rules in terms of five axioms. In this thesis, we extend Llamazares' result by completely describing the class of voting rules that satisfy only two out of his five axioms. …

Application Of Symplectic Integration On A Dynamical System, William Frazier

Electronic Theses and Dissertations

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic …

Observations On Convexity, Chad A. Huckaby

Electronic Theses and Dissertations

This thesis will explore convexity as it pertains to sets of complex-valued functions. These include preliminary looks at established linear and polynomially convex hulls, along with the development of new types of convex hulls. These types will include, but are not limited to the hulls determined by inversions, shift inversions, and Mobius transformations. A convex hull must be preceded by the set of functions involved. These hulls are the smallest convex sets that contain the original set. Justifications and precise definitions are included within the body of the work.

Roman Domination In Complementary Prisms, Alawi I. Alhashim

Electronic Theses and Dissertations

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. …

Peptide Identification: Refining A Bayesian Stochastic Model, Theophilus Barnabas Kobina Acquah

Electronic Theses and Dissertations

Notwithstanding the challenges associated with different methods of peptide identification, other methods have been explored over the years. The complexity, size and computational challenges of peptide-based data sets calls for more intrusion into this sphere. By relying on the prior information about the average relative abundances of bond cleavages and the prior probability of any specific amino acid sequence, we refine an already developed Bayesian approach in identifying peptides. The likelihood function is improved by adding additional ions to the model and its size is driven by two overall goodness of fit measures. In the face of the complexities associated …

Differentiating Between A Protein And Its Decoy Using Nested Graph Models And Weighted Graph Theoretical Invariants, Hannah E. Green

Electronic Theses and Dissertations

To determine the function of a protein, we must know its 3-dimensional structure, which can be difficult to ascertain. Currently, predictive models are used to determine the structure of a protein from its sequence, but these models do not always predict the correct structure. To this end we use a nested graph model along with weighted invariants to minimize the errors and improve the accuracy of a predictive model to determine if we have the correct structure for a protein.

On T-Restricted Optimal Rubbling Of Graphs, Kyle Murphy

Electronic Theses and Dissertations

For a graph G = (V;E), a pebble distribution is defined as a mapping of the vertex set in to the integers, where each vertex begins with f(v) pebbles. A pebbling move takes two pebbles from some vertex adjacent to v and places one pebble on v. A rubbling move takes one pebble from each of two vertices that are adjacent to v and places one pebble on v. A vertex x is reachable under a pebbling distribution f if there exists some sequence of rubbling and pebbling moves that places a pebble on x. A pebbling distribution where every …

Z2-Orbifolds Of Affine Vertex Algebras And W-Algebras, Masoumah Abdullah Al-Ali

Electronic Theses and Dissertations

Vertex algebras arose in conformal field theory and were first defined axiomatically by Borcherds in his famous proof of the Moonshine Conjecture in 1986. The orbifold construction is a standard way to construct new vertex algebras from old ones. Starting with a vertex algebra V and a group G of automorphisms, one considers the invariant subalgebra V^G (called G-orbifold of V), and its extensions. For example, the Moonshine vertex algebra arises as an extension of the Z₂-orbifold of the lattice vertex algebra associated to the Leech lattice.

In this thesis we consider two problems. First, …

On Zero-Sum Rado Numbers For The Equation Ax_1 + X_2 = X_3, Nicholas Brown

Electronic Theses and Dissertations

For every positive integer $a$, let $n = R_{ZS}(a)$ be the least integer, provided it exists, such that for every coloring \[ \Delta : \{1, 2, ..., n\} \rightarrow \{0, 1, 2\},\] there exist three integers $x_1, x_2, x_3$ (not necessarily distinct) such that \[ \Delta(x_1) + \Delta(x_2) + \Delta(x_3) \equiv 0\ (mod\ 3) \] and \[ ax_1 +x_2 = x_3.\] If such an integer does not exist, then $R_{ZS}(a) = \infty.$ The main results of this paper are \[R_{ZS}(2) = 12\] and a lower bound is found for $R_{ZS}(a)$ where $a \geq 2$.

The Element Spectrum Of A Graph, Milisha Hart-Simmons

Electronic Theses and Dissertations

Characterizations of graphs and matroids that have cycles or circuits of specified cardinality have been given by authors including Edmonds, Junior, Lemos, Murty, Reid, Young, and Wu. In particular, a matroid with circuits of a single cardinality is called a Matroid Design. We consider a generalization of this problem by assigning a weight function to the edges of a graph. We characterize when it is possible to assign a positive integer value weight function to a simple 3-connected graph G such that the graph G contains an edge that is only in cycles of two different weights. For example, as …

Orthosymmetric Maps And Polynomial Valuations, Stephan Christopher Roberts

Electronic Theses and Dissertations

We present a characterization of orthogonally additive polynomials on vector lattices as orthosymmetric multilinear maps. Our proof avoids partitionaly orthosymmetric maps and results that represent orthogonally additive polynomials as linear maps on a power. We also prove band characterizations for order bounded polynomial valuations and for order continuous polynomials of order bounded variation. Finally, we use polynomial valuations to prove that a certain restriction of the Arens extension of a bounded orthosymmetric multilinear map is orthosymmetric.

Asymptotic Properties Of Polynomials Orthogonal Over Multiply Connected Domains, James A. Henegan

Electronic Theses and Dissertations

We investigate the asymptotic behavior of polynomials orthogonal over certain multiply connected domains. Each domain that we consider has an analytic boundary and is, in a strong sense, conformally equivalent to a canonical type of multiply connected domain called a circular domain. The two most general results involve the construction of a series expansion and an integral representation for these polynomials. We show that the integral representation can be utilized to derive more specific results when the domain of orthogonality is circular. In this case, we shed light on the manner in which the holes in the domain of orthogonality …

Sampling And Reconstruction Of Spatial Signals, Cheng Cheng

Electronic Theses and Dissertations

Digital processing of signals f may start from sampling on a discrete set Γ, f →(f(ϒη))ϒηεΓ. The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and …

Orthogonal Polynomials On An Arc Of The Unit Circle With Respect To A Generalized Jacobi Weight: A Riemann-Hilbert Method Approach, Lynsey Cargile Naugle

Electronic Theses and Dissertations

We investigate the asymptotic behavior of polynomials orthogonal over a symmetric arc of the unit circle with respect to a generalized Jacobi-type weight. Full asymptotic expansions for the orthogonal polynomials are obtained at every point of the complex plane. Our method of proof is based on a characterization of the orthogonal polynomials as solutions of a 2X2 matrix Riemann-Hilbert problem, which extends to the unit circle the original Riemann-Hilbert characterization for orthogonal polynomials on the real line, first discovered by Fokas, Its, and Kitaev. In order to extricate the behavior of the polynomials from its Riemann-Hilbert matrix representation, we follow …

Coloring Graphs With Forbidden Minors, Martin Rolek

Electronic Theses and Dissertations

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger's Conjecture from 1943 which states that every graph with no Kt-minor is (t − 1)-colorable. This conjecture has been proved true for t ≤ 6, but remains open for all t ≥ 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no Kt-minor is (2t − 6)- colorable for t = …

Filtering Problems In Stochastic Tomography, Tyler Gomez

Electronic Theses and Dissertations

Distinguishing signal from noise has always been a major goal in probabilistic analysis of data. Such is no less the case in the field of medical imaging, where both the processes of photon emission and their rate of absorption by the body behave as random variables. We explore methods by which to extricate solid conclusions from noisy data involving an X-ray transform, long the mathematical mainstay of such tools as computed axial tomography (CAT scans). Working on the assumption of having some prior probabilities assigned to various states a body can be found in, we introduce and make rigorous an …

Quantum Metrics On Approximately Finite-Dimensional Algebras, Konrad Aguilar

Electronic Theses and Dissertations

Our dissertation focuses on bringing approximately finite-dimensional (AF) algebras into the realm of noncommutative metric geometry. We construct quantum metric structures on unital AF algebras equipped with a faithful tracial state, and prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite dimensional C*-algebras for the quantum Gromov-Hausdorff propinquity. We then study the geometry, for the quantum propinquity, of three natural classes of AF algebras equipped with our quantum metrics: the UHF algebras, the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional …

Banach Spaces From Barriers In High Dimensional Ellentuck Spaces, Gabriel Girón-Garnica

Electronic Theses and Dissertations

We construct new Banach spaces using barriers in high dimensional Ellentuck spaces following the classical framework under which a Tsirelson type norm is defined from a barrier in Ellentuck space. It is shown that these spaces contain arbitrary large copies of lⁿ_infinity and specific block subspaces isomorphic to l_p. We also prove that they are l_p-saturated and not isomorphic to each other. Finally, a study of alternative norms for our spaces is presented.

An Algorithm For The Machine Calculation Of Minimal Paths, Robert Whitinger

Electronic Theses and Dissertations

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R³, but also to the general case of finding minimal functionals on hypersurfaces in Rⁿ associated with an arbitrary metric.

Newsvendor Models With Monte Carlo Sampling, Ijeoma W. Ekwegh

Electronic Theses and Dissertations

Newsvendor Models with Monte Carlo Sampling by Ijeoma Winifred Ekwegh The newsvendor model is used in solving inventory problems in which demand is random. In this thesis, we will focus on a method of using Monte Carlo sampling to estimate the order quantity that will either maximizes revenue or minimizes cost given that demand is uncertain. Given data, the Monte Carlo approach will be used in sampling data over scenarios and also estimating the probability density function. A bootstrapping process yields an empirical distribution for the order quantity that will maximize the expected profit. Finally, this method will be used …

Multilevel Models For Longitudinal Data, Aastha Khatiwada

Electronic Theses and Dissertations

Longitudinal data arise when individuals are measured several times during an ob- servation period and thus the data for each individual are not independent. There are several ways of analyzing longitudinal data when different treatments are com- pared. Multilevel models are used to analyze data that are clustered in some way. In this work, multilevel models are used to analyze longitudinal data from a case study. Results from other more commonly used methods are compared to multilevel models. Also, comparison in output between two software, SAS and R, is done. Finally a method consisting of fitting individual models for each …

Global Supply Sets In Graphs, Christian G. Moore

Electronic Theses and Dissertations

For a graph G=(V,E), a set S⊆V is a global supply set if every vertex v∈V\S has at least one neighbor, say u, in S such that u has at least as many neighbors in S as v has in V \S. The global supply number is the minimum cardinality of a global supply set, denoted γgs (G). We introduce global supply sets and determine the global supply number for selected families of graphs. Also, we give bounds on the global supply number for general graphs, trees, and grid graphs.

Takens Theorem With Singular Spectrum Analysis Applied To Noisy Time Series, Thomas K. Torku

Electronic Theses and Dissertations

The evolution of big data has led to financial time series becoming increasingly complex, noisy, non-stationary and nonlinear. Takens theorem can be used to analyze and forecast nonlinear time series, but even small amounts of noise can hopelessly corrupt a Takens approach. In contrast, Singular Spectrum Analysis is an excellent tool for both forecasting and noise reduction. Fortunately, it is possible to combine the Takens approach with Singular Spectrum analysis (SSA), and in fact, estimation of key parameters in Takens theorem is performed with Singular Spectrum Analysis. In this thesis, we combine the denoising abilities of SSA with the Takens …

Neighborhood-Restricted Achromatic Colorings Of Graphs, James D. Chandler Sr.

Electronic Theses and Dissertations

A (closed) neighborhood-restricted 2-achromatic-coloring of a graph G is an assignment of colors to the vertices of G such that no more than two colors are assigned in any closed neighborhood. In other words, for every vertex v in G, the vertex v and its neighbors are in at most two different color classes. The 2-achromatic number is defined as the maximum number of colors in any 2-achromatic-coloring of G. We study the 2-achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds.

The Complete Structure Of Linear And Nonlinear Deformations Of Frames On A Hilbert Space, Devanshu Agrawal

Electronic Theses and Dissertations

A frame is a possibly linearly dependent set of vectors in a Hilbert space that facilitates the decomposition and reconstruction of vectors. A Parseval frame is a frame that acts as its own dual frame. A Gabor frame comprises all translations and phase modulations of an appropriate window function. We show that the space of all frames on a Hilbert space indexed by a common measure space can be fibrated into orbits under the action of invertible linear deformations and that any maximal set of unitarily inequivalent Parseval frames is a complete set of representatives of the orbits. We show …

Follower And Extender Sets In Symbolic Dynamics, Thomas Kelly French

Electronic Theses and Dissertations

Given a word w in the language of a one-dimensional shift space X, the follower set of w, denoted F_X(w), is the set of all right-infinite sequences which follow w in some point of X. Extender sets are a generalization of follower sets and are defined similarly. To a given shift space X, then, we may associate a follower set sequence {|F_X(n)|} which records the number of distinct follower sets in X corresponding to words of length n. Similarly, we may define an extender set sequence {|E …