Physical Sciences and Mathematics Commons^™

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.

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.

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 ≤ …

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 …

Ranking World Class Chess Players Using Only Results From Head-To-Head Games, Sterling Swygert

Senior Theses

This honors thesis explores a method of ranking the world’s top ten chess grand- masters using only the outcomes of games containing only players in that very set. This method allows for players in a single era to be quickly ranked via algorithmic and numerical means, including very specific information, from a statistical stand- point. Furthermore, unlike the rating systems that are commonly used, the Elo and the Glicko systems, this method is Classicist in its statistical approach, rather than Bayesian. Finally, this ranking method also differs from others as it limits the infor- mation to games between the individuals …

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.