Physical Sciences and Mathematics Commons^™

A Markov Model For Baseball With Applications, Daniel Joseph Ursin

Theses and Dissertations

In this work we confirm a Markov chain model of baseball for 2013 Major League Baseball batting data. We describe the transition matrices for individual player data and their use in generating single and nine-inning run distributions for a given lineup. The run distribution is used to calculate the expected number of runs produced by a lineup over nine innings. We discuss batting order optimization heuristics to avoid computation of distributions for the 9! = 362, 880 distinct lineups for 9 players. Finally, we describe an implementation of the algorithms and review their performance against actual game data.

Analyzing State Attempts At Implementing The Common Core State Standards For High School Geometry: Case Studies Of Utah And New York, Edward Steltenpohl

Theses and Dissertations

This study analyzes two state attempts at aligning curricula to the Common Core State Standards (CCSS) in secondary school geometry. The education departments of Utah and New York have approved curricula aimed at aligning to the Common Core State Standards: the Mathematics Vision Project (MVP) and EngageNY (ENY) respectively. This study measures the extent to which those curricula align with the content demands of the relevant Common Core Standards. The results indicate that, while the two curricula vary in structure and assumptions about learners, each one aligns well with the Common Core State Standards in secondary school geometry. We conclude …

Contractible N-Manifolds And The Double N-Space Property, Pete Sparks

Theses and Dissertations

We are interested in contractible n-manifolds M which decompose or split as M = A union B where A,B, and A intersect B are all homeomorphic to Euclidean n-space or A,B, and A intersect B are all homeomorphic to the n-dimensional unit ball. We introduce a 4-manifold M containing a spine which splits as A union B with A,B, and A intersect B all collapsible which in turn implies M splits as the union of two 4-balls whose intersection is also a 4-ball. From M we obtain a countably infinite collection of distinct 4-manifolds all of which split this way. …

The Existence Of A Discontinuous Homomorphism Requires A Strong Axiom Of Choice, Michael Steven Andersen

Theses and Dissertations

Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We …

Pro-Covering Fibrations Of The Hawaiian Earring, Nickolas Brenten Callor

Theses and Dissertations

Let H be the Hawaiian Earring, and let H denote its fundamental group. Assume (Bi) is an inverse system of bouquets of circles whose inverse limit is H. We give an explicit bijection between finite normal covering spaces of H and finite normal covering spaces of Bi. This bijection induces a correspondence between a certain family of inverse sequences of these covering spaces. The correspondence preserves the inverse limit of these sequences, thus offering two methods of constructing the same limit. Finally, we characterize all spaces that can be obtained in this fashion as a particular type of fibrations of …

A Covering System With Minimum Modulus 42, Tyler Owens

Theses and Dissertations

We construct a covering system whose minimum modulus is 42. This improves the previous record of 40 by P. Nielsen.

Fake Real Quadratic Orders, Richard Michael Oh

Theses and Dissertations

The study of fake real quadratic orders is fascinating as their class group structure is similiar to real quadratic fields. Statistical data strongly agree with the heuristics of Cohen and Lenstra of real quadratics with class number one. We will investigate why this holds true as well as explore other analogues to open conjecture on real quadratic fields, such as the Ankeny-Artin-Chowla Conjecture, and present various results that mark the similiarities between real quadratic fields and fake real quadratic orders. Fake real quadratics are defined by inverting an ideal above any prime p which is split in OK where K …

The Non-Existence Of A Covering System With All Moduli Distinct, Large And Square-Free, Melissa Kate Bechard

Theses and Dissertations

The work in this thesis is based on a paper written by Bob Hough in 2013. This thesis addresses the conjecture posed by Erdos that the least modulus for a covering system can be arbitrarily large. Hough proves the least modulus cannot be arbitrarily large. In this thesis, we present Hough’s proof for the case of square-free moduli.

Explorations In Elementary And Analytic Number Theory, Scott Michael Dunn

Theses and Dissertations

In this dissertation, we investigate two distinct questions in number theory. Each question is dedicated its own chapter.

First, we consider arithmetic progressions in the polygonal numbers with a fixed number of sides. We will show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Additionally we show that there are infinitely many three-term arithmetic progressions starting with an arbitrary polygonal number.

Second, we will show certain irreducibility criteria for polynomials. Let f(x) be a polynomial with non-negative integer coefficients such that f(b) is prime for some integer 2 ≤ b ≤ …

On The Group Of Transvections Of Ade-Diagrams, Marvin Jones

Theses and Dissertations

In this thesis we examine symplectic spaces with forms generated by the ADEdiagrams. Specifically, we determine the generators of the group of transvections for each space under the standard basis, S, of Kⁿ (where K is a field with characteristic 0) and the hyperbolic basis, H, we get from the classification theorem of symplectic spaces. Further, we examine how the generators of these groups are related via g : G_f,S ! SL(Z)_n where g(X) = P⁻¹XP where P is the change of basis matrix for S to H.

Option Pricing For A General Stock Model In Discrete Time, Cindy Lynn Nichols

Theses and Dissertations

{As there are no arbitrage opportunities in an efficient market, the seller of an option must find a risk neutral price. This thesis examines different characterizations of this option price. In the first characterization, the seller forms a hedging portfolio of shares of the stock and units of the bond at the time of the option's sale so as to reduce his risk of losing money. Before the option matures, the present value stock price fluctuates in discrete time and, based on those changes, the seller alters the content of the portfolio at the end of each time period. The …

Newton's Method Backpropagation For Complex-Valued Holomorphic Neural Networks: Algebraic And Analytic Properties, Diana Thomson La Corte

Theses and Dissertations

The study of Newton's method in complex-valued neural networks (CVNNs) faces many difficulties. In this dissertation, we derive Newton's method backpropagation algorithms for complex-valued holomorphic multilayer perceptrons (MLPs), and we investigate the convergence of the one-step Newton steplength algorithm for the minimization of real-valued complex functions via Newton's method. The problem of singular Hessian matrices provides an obstacle to the use of Newton's method backpropagation to train CVNNs. We approach this problem by developing an adaptive underrelaxation factor algorithm that avoids singularity of the Hessian matrices for the minimization of real-valued complex polynomial functions.

To provide experimental support for the …

Statistical Hyperbolicity Of Relatively Hyperbolic Groups, Jeremy Osborne

Theses and Dissertations

In this work, we begin by defining what it means for a group to be statistically hyperbolic. We then give several examples of groups, including non-elementary hyperbolic groups, which either are statistically hyperbolic or are not. Following that, we define what it means for a group to be relatively hyperbolic. Finally, in the main portion of this work, we show that groups which are relatively hyperbolic, with a few additional conditions in place, must also be statistically hyperbolic.

P_4-Decomposability In Regular Graphs And Multigraphs, David Joshua Mendell

Theses and Dissertations

The main objective of this thesis is to review and expand the study of graph decomposability. An H-decomposition of a graph G=(V,E) is a partitioning of the edge set, $E$, into edge-disjoint isomorphic copies of a subgraph H. In particular we focus on the decompositions of graphs into paths. We prove that a 2,4 mutligraph with maximum multiplicity 2 admits a C_2,C_3-free Euler tour (and thus, a decomposition into paths of length 3 if it has size a multiple of 3) if and only if it avoids a set of 15 forbidden structures. We also prove that …

Connecting Galois Representations With Cohomology, Joseph Allen Adams

Theses and Dissertations

In this paper, we examine the conjecture of Avner Ash, Darrin Doud, David Pollack, and Warren Sinnott relating Galois representations to the mod p cohomology of congruence subgroups of the general linear group of n dimensions over the integers. We present computational evidence for this conjecture (the ADPS Conjecture) for the case n = 3 by finding Galois representations which appear to correspond to cohomology eigenclasses predicted by the ADPS Conjecture for the prime p, level N, and quadratic nebentype. The examples include representations which appear to be attached to cohomology eigenclasses which arise from D8, S3, A5, and S5 …

Hecke Eigenvalues And Arithmetic Cohomology, William Leonard Cocke

Theses and Dissertations

We provide algorithms and documention to compute the cohomology of congruence subgroups of the special linear group over the integers when n=3 using the well-rounded retract and the Voronoi decomposition. We define the Sharbly complex and how one acts on a k-sharbly by the Hecke operators. Since the norm of a sharbly is not preserved by the Hecke operators we also examine the reduction techniques described by Gunnells and present our implementation of said techniques for n=3.

Hölder Extensions For Non-Standard Fractal Koch Curves, Joshua Taylor Fetbrandt

Theses and Dissertations

Let K be a non-standard fractal Koch curve with contraction factor α. Assume α is of the form α = 2+1/m for some m ∈ N and that K is embedded in a larger domain Ω. Further suppose that u is any Hölder continuous function on K. Then for each such m ∈ N and iteration n ≥ 0, we construct a bounded linear operator Πn which extends u from the prefractal Koch curve Kn into the whole of Ω. Unfortunately, our sequence of extension functions Πnu are not bounded in norm in the limit because the upper bound is …

Convolutions And Convex Combinations Of Harmonic Mappings Of The Disk, Zachary M. Boyd

Theses and Dissertations

Let f_1, f_2 be univalent harmonic mappings of some planar domain D into the complex plane C. This thesis contains results concerning conditions under which the convolution f_1 ∗ f_2 or the convex combination tf_1 + (1 − t)f_2 is univalent. This is a long-standing problem, and I provide several partial solutions. I also include applications to minimal surfaces.

A New Public-Key Cryptosystem, Christopher James Hettinger

Theses and Dissertations

Public key cryptosystems offer important advantages over symmetric methods, but the most important such systems rely on the difficulty of integer factorization (or the related discrete logarithm problem). Advances in quantum computing threaten to render such systems useless. In addition, public-key systems tend to be slower than symmetric systems because of their use of number-theoretic algorithms. I propose a new public key system which may be secure against both classical and quantum attacks, while remaining simple and very fast. The system's action is best described in terms of linear algebra, while its security is more naturally explained in the context …

An Analysis And Comparison Of The Common Core State Standards For Mathematics And The Singapore Mathematics Curriculum Framework, Heidi Ann Ertl

Theses and Dissertations

In this analysis and comparison we look at the Common Core State Standards for Mathematics and the Singapore Mathematics Curriculum Framework, standards documents that guide primary and secondary mathematics education in the United States and Singapore respectively. The official Common Core State Standards for Mathematics website claims that the standards have been developed to be "internationally benchmarked, so that all students are prepared for the 21st century". Singapore has recently been recognized as a world leader in mathematics education. We investigate the claim that the Common Core State Standards for Mathematics are internationally benchmarked by comparing the Common Core State …

Teachers' Beliefs And Practices Regarding Homework: An Examination Of The Cognitive Domain Embedded In Third Grade Mathematics Homework, Pandora Dell Bedford

Theses and Dissertations

The purpose of this phenomenological study was to gain a better understanding of third grade math teachers''beliefs and practices regarding homework, to explain how teachers''beliefs and practices regarding homework aligned to the framework of the Revised Bloom's'Taxonomy Cognitive Domain, and to determine the administrative influences on homework practices. The data were collected during October and November 2013. Six third grade math teachers (primary unit of analysis) and four principals (secondary unit of analysis) were interviewed from Dell School District. Each participant (teacher and principal) was interviewed for approximately one hour. A second meeting was set at a later time with …

On The Generalized Ince Equation, Ridha Moussa

Theses and Dissertations

We investigate a Hill differential equation with trigonometric polynomial coefficients. we are interested in solutions which are even or odd and have period π or semi-period π. With one of the mentioned boundary conditions, our equation constitute a regular Sturm-Liouville eigenvalue problem. Using Fourier series representation each one of the four Sturm-Liouville operators is represented by an infinite banded matrix. In the particular cases of Ince and Lam equations, the four infinite banded matrices become tridiagonal. We then investigate the problem of coexistence of periodic solutions and that of existence of polynomial solutions.

Experiments On Temporal Variable Step Bdf2 Algorithms, Anja Katrin Denner

Theses and Dissertations

Efficient algorithms for solving stiff PDEs are of great interest. For developing such an algorithm step sizes should vary in both space and time. We have to understand each separately first before putting it together, and this thesis is dedicated to developing a sharper notion of the performance of a variable step size BDF2 scheme for some examples. We find suitable parameters for the variable step size algorithm proposed by Jannelli and Fazio in their respective paper concerning adaptive stiff solvers at low accuracy and complexity. Finally, we make a short excursion on the stability of BDF2 for the Allen-Cahn …

Optimal Reinsurance Strategy With Bivariate Pareto Risks, Evelyn Susanne Gaus

Theses and Dissertations

In an insurance, one is often concerned with risks and extreme events which can cause large losses. The Pareto distribution is often used in actuarial sciences for modeling large losses. This thesis extends the study of Cai and Wei (2011) by considering a two-line business model with positive dependence through stochastic ordering (PDS) risks, where the risks are bivariate Pareto distributed. Cai and Wei (2011) showed that in individual reinsurance treaties the excess-of-loss treaty is the optimal reinsurance form for an insurer with PDS risks. We derive explicit expressions for the optimal retention levels in the excess-of-loss treaty by considering …

A Limit Theorem For The Squared Norm Of Empirical Distribution Functions, Alexander Nerlich

Theses and Dissertations

There are many limit theorems which involve empirical distribution functions.

This thesis is dedicated to prove a limit theorem for the squared L2-norm of two empirical distribution functions.

Incorporating Krylov Subspace Methods In The Etdrk4 Scheme, Jeffrey H. Allen

Theses and Dissertations

A modification of the (2,2)-Pade algorithm developed by Wade et al. for implementing the exponential time differencing fourth order Runge-Kutta (ETDRK4) method is introduced. The main computational difficulty in implementing the ETDRK4 method is the required approximation to the matrix exponential. Wade et al. use the fourth order (2,2)-Pade approximant in their algorithm and in this thesis we incorporate Krylov subspace methods in an attempt to improve efficiency. A background of Krylov subspace methods is provided and we describe how they are used in approximating the matrix exponential and how to implement them into the ETDRK4 method. The (2,2)-Pade and …

The Pekeris Method For Lithium: Possibilities And Obstructions, Marcel Kreuter

Theses and Dissertations

It is widely believed that the properties of atoms and molecules are accurately described by the Schrödinger equation, at least in so far as relativistic effects may be neglected. Extracting these properties from the equation in practice, however, can be a highly challenging task. In 1958, Chaim L. Pekeris developed a method for computing the ground state energy of the Helium atom. This thesis surveys the possibilities and obstructions that occur when one tries to compute the ground state energy of Lithium using Pekeris's method.

Mathematical Modeling Of Competition For Light And Nutrients Between Phytoplankton Species In A Poorly Mixed Water Column, Thomas George Stojsavljevic

Theses and Dissertations

Phytoplankton live in a complex environment with two essential resources forming various gradients. Light supplied from above is never homogeneously distributed in a body of water due to refraction and absorption from biomass present in the ecosystem and from other sources. Nutrients in turn are typically supplied from below. In poorly mixed water columns phytoplankton can be heterogeneously distributed forming various layering patterns. The relationship between the location and the thickness of the layers is an open problem of interest. Here we present three models which study how competition for light and resources can form common layering patterns seen in …

A Mathematical Model Of Moisture Movement And Bacterial Growth In Two-Dimensional Porous Medium, Rachel Elizabeth Tewinkel

Theses and Dissertations

Bacterial growth in sand is of concern in regard to the health of beaches. A mathematical model is presented that represents the movement of moisture and the growth of bacteria through a beach. Simulations were run by numerically solving Richards Equation using a Finite Volume Method in order to track moisture movement. A model of moisture-dependent bacterial growth was then implemented. These simulations show that elevated bacteria counts following rain events do not necessarily result from bacteria in the body of water, but can also be sourced from the sand. Additionally, four different moisture-dependent bacterial growth models are compared to …

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 …