Mathematics Commons^™

Opinion Formation About Childhood Immunization And Disease Spread On Networks, Shan Shan Zhao

Theses and Dissertations

People are physically and socially connected with each other. Those connections between people represent two, probably overlapping, networks: biological networks, through which physical contacts occur, or social network, through which information diffuse. In my thesis research, I am trying to answer that question in the context of pediatric disease spread on the biological network between households as well as within them and its relationship with information sharing on the social network of households (parents in that case) via "Information Cascades." I mainly focus on the Erdos-Renyi network model. In particular, I use two different but overlapping Erdos-Renyi networks for the …

Lie Symmetry To Nonlinear Oscillator Systems And Applications, Xiaoyan Li

Theses and Dissertations

In this paper, we apply the theory of Lie symmetry to study a generalized second-order nonlinear differential equation, which includes several physical nonlinear oscillators such as force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden-type equation and its hierarchy etc, and investigate the dynamical properties of this rather general equation. We identify and classify several new integrable cases for arbitrary values of exponents, which determine the tangent vector as well as the infinitesimal generator. Using the Lie point symmetry, we find the useful infinitesimal generators and canonical coordinates, and obtain the first integrals of the second-order nonlinear …

On Hypergroups Of Order At Most 6, Jordy C. Lopez

Theses and Dissertations

This thesis surveys recent results on hypergroups as defined by Frédéric Marty in [3] and [4] and their relation to association schemes as presented in [5]. We show that every association scheme is a hypergroup. Then, we compile a few general results on hypergroups needed for our investigation of hypergroups with three, four and six elements. From [1] and [7], we give examples of hypergroups that do not come from finite schemes and from no scheme at all. Our main result occurs when considering hypergroups S with six elements that have a non-normal closed subset T of order 2 with …

Isomorphisms Of Landau-Ginzburg B-Models, Nathan James Cordner

Theses and Dissertations

Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted A and B) that are constructed from a nondegenerate quasihomogeneous polynomial W and a related group of symmetries G. In 2013, Tay proved that given two polynomials W1, W2 with the same quasihomogeneous weights and same group G, the corresponding A-models built with (W1, G) and (W2, G) are isomorphic. An analogous theorem for isomorphisms between orbifolded B-models remains to be found. This thesis investigates isomorphisms between B-models using polynomials in two variables in search of such a theorem. In particular, several examples are given showing the relationship between continuous …

On The Dynamics Of Boolean Gene Regulatory Networks With Stochasticity, Yuezhe Li

Theses and Dissertations

Genes are responsible for producing proteins that are essential to the construction of complex biological systems. The mechanisms by which this production is regulated have long been the center of wide spread research efforts. Deterministic Boolean gene regulatory models have been a particularly effective avenue of research in this field. However these models fall short of accounting for variations in the gene functionality due to the uncertain internal or external environmental conditions. One of the recent attempts to overcome this weakness is by (Murrugarra, 2012), in which a probabilistic component is introduced as the fixed activation/degradation propensities at the cellular …

Modeling Individual Health Care Utilization, Matthew Aaron Webb

Theses and Dissertations

Health care represents an increasing proportion of global consumption. We discuss ways to model health care utilization on an individual basis. We present a probabilistic, generative model of utilization. Leveraging previously observed utilization levels, we learn a latent structure that can be used to accurately understand risk and make predictions. We evaluate the effectiveness of the model using data from a large population.

The Principles Of Effective Teaching Student Teachershave The Opportunity To Learn In An Alternativestudent Teaching Structure, Danielle Rose Divis

Theses and Dissertations

Research has shown that the focus of mathematics student teaching programs is typically classroom management and non-mathematics specific teaching strategies. However, the redesigned BYU student teaching structure has proven to help facilitate a greater focus on mathematics-specific pedagogy and student mathematics during post-lesson reflection meeting conversations. This study analyzed what specific principles of NCTM’s standards of effective teaching were discussed in the reflection meetings of this redesigned structure. This study found that the student teachers extensively discussed seven of the eight principles NCTM considers to be necessary for effective mathematics teaching. Other pedagogical principles pertaining to student mathematical learning not …

Construction And Isomorphism Of Landau-Ginzburg B-Model Frobenius Algebras, Matthew Robert Brown

Theses and Dissertations

Landau-Ginzburg Mirror Symmetry provides for the construction of two algebraic objects, called the A- and B-models. Special cases of these models–constructed using invertible polynomials and abelian symmetry groups–are well understood. In this thesis, we consider generalizations of the B-model, and specifically address the associativity of the multiplication in these models. We also prove an explicit B-model isomorphism for a class of polynomials in three variables.

The Automorphism Group Of The Halved Cube, Benjamin B. Mackinnon

Theses and Dissertations

An n-dimensional halved cube is a graph whose vertices are the binary strings of length n, where two vertices are adjacent if and only if they differ in exactly two positions. It can be regarded as the graph whose vertex set is one partite set of the n-dimensional hypercube, with an edge joining vertices at hamming distance two. In this thesis we compute the automorphism groups of the halved cubes by embedding them in R n and realizing the automorphism group as a subgroup of GLn(R). As an application we show that a halved cube is a circulant graph if …

Classification Of Compact 2-Manifolds, George H. Winslow

Theses and Dissertations

It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we …

Chebyshev Inversion Of The Radon Transform, Jared Cameron Szi

Theses and Dissertations

In its two-dimensional form, the Radon transform of an image (function) is a collection of projections of the image which are parameterized by a set of angles (from the positive x-axis) and distances from the origin. Computational methods of the Radon transform are important in many image processing and computer vision problems, such as pattern recognition and the reconstruction of medical images. However, computability requires the construction of a discrete analog to the Radon transform, along with discrete alternatives for its inversion. In this paper, we present discrete analogs using classical methods of Chebyshev polynomial reconstruction, along with a new …

On A Constant Associated With The Prouhet-Tarry-Escott Problem, Maria E. Markovich

Theses and Dissertations

For n a positive integer, the Prouhet-Tarry-Escott Problem asks for two different sets of n positive integers for which the sum of the kth powers of the elements of one set is equal to the sum of the kth powers of the elements of the second set for each positive integer k < n. For n > 12, it is not known whether such sets exist. I will give some background on this problem and then show how Newton polygons can be used to determine information on the size of the 2-adic value of a certain constant associated with the problem.

An Analysis Of The Impact Of The Csu Peer Leader Program On Student Performance In 2014-2015, Harrison E. Sharitt

Theses and Dissertations

In efforts to retain students and increase student performance, Columbus State implemented a Peer Leader program. With access to archival data on the CSU Peer Leader Program, the goal of this project was to determine if the program at CSU had a significant impact on students’ course performance. The impact of the program was assessed by performing multiple regression analyses and ANOVA. Preliminary results showed that Columbus State’s Peer Leader program had a small, but statistically significant, positive impact on students’ final course grades.

Automated Conjecturing Approach For Benzenoids, David Muncy

Theses and Dissertations

Benzenoids are graphs representing the carbon structure of molecules, defined by a closed path in the hexagonal lattice. These compounds are of interest to chemists studying existing and potential carbon structures. The goal of this study is to conjecture and prove relations between graph theoretic properties among benzenoids. First, we generate conjectures on upper bounds for the domination number in benzenoids using invariant-defined functions. This work is an extension of the ideas to be presented in a forthcoming paper. Next, we generate conjectures using property-defined functions. As the title indicates, the conjectures we prove are not thought of on our …

Automated Conjecturing Approach To The Discrete Riemann Hypothesis, Alexander Bradford

Theses and Dissertations

This paper is a study on some upper bounds of the Mertens function, which is often considered somewhat of a ``mysterious" function in mathematics and is closely related to the Riemann Hypothesis. We discuss some known bounds of the Mertens function, and also seek new bounds with the help of an automated conjecture-making program named CONJECTURING, which was created by C. Larson and N. Van Cleemput, and inspired by Fajtowicz's Dalmatian Heuristic. By utilizing this powerful program, we were able to form, validate, and disprove hypotheses regarding the Mertens function and how it is bounded.

On Crown-Free Set Families, Diffusion State Difference, And Non-Uniform Hypergraphs, Edward Lawrence Boehnlein

Theses and Dissertations

We present results in three different arenas of discrete mathematics. Let La(n, H) denote the cardinality of the largest family on the Boolean lattice that does not contain H as a subposet. Denote π(H) := limn→∞ La(n,H) (bn/ n2c) . A crown O2k for k ≥ 2 is a poset on 2 levels whose Hasse diagram is a cycle. Griggs and Lu (2009) showed π(O4k) = 1 for k ≥ 2. Lu (2014) proved π(O2k) = 1 for odd k ≥ 7. We prove that the maximum size of a O6 -free family, when restricted to the middle two levels …

Binary Quartic Forms Over Fp, Daniel Thomas Kamenetsky

Theses and Dissertations

Let Vp denote the five dimensional vector space of binary quartic forms over the finite field Fp, with p a prime greater than 3. There is a natural action of the group GL1(Fp)×GL2(Fp) on Vp. This action partitions Vp into orbits, the number of which increases with p. In this thesis, we determine explicitly, for a given p, the number of orbits under the action of GL1(Fp) × GL2(Fp) on Vp. Moreover, we determine the size of each orbit and the general structure of the forms each orbit contains. We also introduce an application of understanding these orbits to the …

Some Extremal And Structural Problems In Graph Theory, Taylor Mitchell Short

Theses and Dissertations

This work considers three main topics. In Chapter 2, we deal with König-Egerváry graphs. We will give two new characterizations of König-Egerváry graphs as well as prove a related lower bound for the independence number of a graph. In Chapter 3, we study joint degree vectors (JDV). A problem arising from statistics is to determine the maximum number of non-zero elements of a JDV. We provide reasonable lower and upper bounds for this maximum number. Lastly, in Chapter 4 we study a problem in chemical graph theory. In particular, we characterize extremal cases for the number of maximal matchings in …

Structure Of The Stable Marriage And Stable Roommate Problems And Applications, Joe Hidakatsu

Theses and Dissertations

The well-known Gale-Shapley algorithm is a solution to the stable marriage problem, but always results in the same stable marriage, regardless of how the algorithm is executed. Robert Irving and Paul Leather constructed the rotation poset, whose downward closed sets are in one-to-one correspondence with the set of stable marriage assignments. We discuss how to use the rotation poset to find the k-optimal matching, and prove that a k-optimal matching is the same as a minimum regret matching for high enough k. Finally, Dan Gusfield defines the rotation poset for the stable roommate problem, and uses it to efficiently enumerate …

Modeling Of Structural Relaxation By A Variable-Order Fractional Differential Equation, Su Yang

Theses and Dissertations

In physical point of view, relaxation usually describes the return from a perturbed system into equilibrium and each process has its own characteristic relaxation time. In 1946, Tool first formulated the notion of fictive temperature to characterize the structure of a glass-forming melt. Since then, people used to simulate structural relaxation by first order model. Since fractional-based models have not widely applied in modeling the fictive temperature, I want to explore the the possibility of modeling structural relaxation by fractional differential equation.

In this thesis, I will first introduce the definitions of two different kinds of fractional derivatives: Riemann-Liouville fractional …

Modeling, Simulation, And Applications Of Fractional Partial Differential Equations, Wilson Cheung

Theses and Dissertations

The Black-Scholes model is commonly used to track the price of European options with respect to maturity in many financial markets. This model degenerates into a partial differential equation that relates the European-style option price to the underlying price and time of expiry. Black-Scholes assumes that underlying prices satisfy a geometric Brownian motion.

After the U.S. stock market crash of 1987, this assumption becomes inaccurate as it fails to represent the behavior of S&P 500 European vanilla option prices. Specifically, under the measure of moneyness, the volatility smirk does not flatten out and the resulting conditional return distribution does not …

Fast Methods For Variable-Coefficient Peridynamic And Non-Local Diffusion Models, Che Wang

Theses and Dissertations

In previous studies, scientists developed the classical solid mechanic theory. The model has been widely used in scientific research and practical production. The main assumption of the classical theory of solid mechanics is that all internal forces act through zero distance. Because of this assumption, the mathematical model always leads to partial differential equations, which meet with problems when describing the spontaneous formation of discontinuities and other singularities. A peridynamic model was proposed as a reformation of solid mechanics [40, 41, 43, 44, 45], which leads to a non-local framework that does not explicitly involve the notion of deformation gradients, …

A Survey Of The Kinetic Monte Carlo Algorithm As Applied To A Multicellular System, Michael Richard Laughlin

Theses and Dissertations

We explore the origins and implementation of the Kinetic Monte Carlo method on a system of cells suspended in a liquid media. The situation presented herein has applications in the emerging field of biofabrication, which may have lasting impacts in medical science. The theory behind the method is explained in detail, starting with its emergence in the 1960s, and two major improvements to the scaling of the method are presented, along with a restriction to a special case. Finally, we give the results of several simulations.

The Solenoid And Warsawanoid Are Sharkovskii Spaces, Tyler Willes Hills

Theses and Dissertations

We extend Sharkovskii's theorem concerning orbit lengths of endomorphisms of the real line to endomorphisms of a path component of the solenoid and certain subspaces of the Warsawanoid. In particular, Sharkovskii showed that if there exists an orbit of length 3 then there exist orbits of all lengths. The solenoid is the inverse limit of double covers over the circle, and the Warsawanoid is the inverse limit of double covers over the Warsaw circle. We show Sharkovskii's result is true for path components of the solenoid and certain subspaces of the Warsawanoid.

Compression Bodies And Their Boundary Hyperbolic Structures, Vinh Xuan Dang

Theses and Dissertations

We study hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. We consider individual hyperbolic structures as well as special regions in the space of all such hyperbolic structures. We use some properties of the boundary hyperbolic structures on C to establish an interesting property of cusp shapes of tunnel number one manifolds. This extends a result of Nimershiem in [26] to the class of tunnel number one manifolds. We also establish convergence results on the geometry of compression bodies. This extends the work of Ito in [13] from the punctured-torus case …

The Bourbaki-Jacobson Correspondence, Jose R. Vera

Theses and Dissertations

A general ring theoretic correspondence between subrings of the endomorphism ring of the additive group of a commutative field will be established. This correspondence (called Bourbaki-Jacobson Correspondence) provides the ordinary Galois correspondence when applied to specific group rings. Throughout this thesis, we will work with finite dimensional field extensions.

Alternating Direction Implicit Method With Adaptive Grids For Modeling Chemotaxis In Dictyostelium Discoideum, Christopher F. Loomis

Theses and Dissertations

Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons …

Superconvergence Of Finite Element Approximations For Second-Order Elliptic Problems By L^2-Projection Methods, Anna Harris

Theses and Dissertations

The superconvergence of finite element approximations is an important and useful phenomenon and has been an active research area in numerical analysis. A brief definition of the superconvergence of the finite element methods (FEMs) is methods to improve the convergence rate by post-processing an exiting finite element solution so that the new approximation is closer to the exact solution than the existing finite element solution. The purpose of this dissertation is to investigate the superconvergence of different finite element methods for second-order elliptic problems. The finite element methods include the conforming finite element method (CFEM), the non-conforming finite element method …

A Nonabelian Landau-Ginzburg B-Model Construction, Ryan Thor Sandberg

Theses and Dissertations

The Landau-Ginzburg (LG) B-Model is a significant feature of singularity theory and mirror symmetry. Krawitz in 2010, guided by work of Kaufmann, provided an explicit construction for the LG B-model when using diagonal symmetries of a quasihomogeneous, nondegenerate polynomial. In this thesis we discuss aspects of how to generalize the LG B-model construction to allow for nondiagonal symmetries of a polynomial, and hence nonabelian symmetry groups. The construction is generalized to the level of graded vector space and the multiplication developed up to an unknown factor. We present complete examples of nonabelian LG B-models for the polynomials x^2y + y^3, …

Nonlocally Maximal Hyperbolic Sets For Flows, Taylor Michael Petty

Theses and Dissertations

In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.