Physical Sciences and Mathematics Commons^™

Uconn Baseball Batting Order Optimization, Gavin Rublewski, Gavin Rublewski

Honors Scholar Theses

Challenging conventional wisdom is at the very core of baseball analytics. Using data and statistical analysis, the sets of rules by which coaches make decisions can be justified, or possibly refuted. One of those sets of rules relates to the construction of a batting order. Through data collection, data adjustment, the construction of a baseball simulator, and the use of a Monte Carlo Simulation, I have assessed thousands of possible batting orders to determine the roster-specific strategies that lead to optimal run production for the 2023 UConn baseball team. This paper details a repeatable process in which basic player statistics …

Dimension Theory Of Conformal Iterated Function Systems, Sharon Sneha Spaulding

Honors Scholar Theses

This thesis is an expository investigation of the conformal iterated function system (CIFS) approach to fractals and their dimension theory. Conformal maps distort regions, subject to certain constraints, in a controlled way. Let $\mathcal{S} = (X, E, \{\phi_e\}_{e \in E})$ be an iterated function system where $X$ is a compact metric space, $E$ is a countable index set, and $\{\phi_e\}_{e \in E}$ is a family of injective and uniformly contracting maps. If the family of maps $\{\phi_e\}_{e \in E}$ is also conformal and satisfies the open set condition, then the distortion properties of conformal maps can be extended to the …

Minimal Inscribed Polyforms, Jack Hanke

Honors Scholar Theses

A polyomino of size n is constructed by joining n unit squares together by their edge to form a shape in the plane. This thesis will first examine the formal definition of a polyomino and the common equivalence classes polyominos are enumerated under. We then turn to polyomino families, and provide exact enumeration results for certain families, including the minimal inscribed polyominos. Next we will generalize polyominos to polyforms, and provide novel formulae for polyform analogues of minimal inscribed polyominos. Finally, we discuss some further questions concerning minimal inscribed polyforms.

The Generalized Riemann Hypothesis And Applications To Primality Testing, Peter Hall

University Scholar Projects

The Riemann Hypothesis, posed in 1859 by Bernhard Riemann, is about zeros
of the Riemann zeta-function in the complex plane. The zeta-function can be repre-
sented as a sum over positive integers n of terms 1/ns when s is a complex number
with real part greater than 1. It may also be represented in this region as a prod-
uct over the primes called an Euler product. These definitions of the zeta-function
allow us to find other representations that are valid in more of the complex plane,
including a product representation over its zeros. The Riemann Hypothesis says that
all …

Sobolev Inequalities And Riemannian Manifolds, John Reever

Honors Scholar Theses

Sobolev inequalities, named after Sergei Lvovich Sobolev, relate norms in Sobolev spaces and give insight to how Sobolev spaces are embedded within each other. This thesis begins with an overview of Lebesgue and Sobolev spaces, leading into an introduction to Sobolev inequalities. Soon thereafter, we consider the behavior of Sobolev inequalities on Riemannian manifolds. We discuss how Sobolev inequalities are used to construct isoperimetric inequalities and bound volume growth, and how Sobolev inequalities imply families of other Sobolev inequalities. We then delve into the usefulness of Sobolev inequalities in determining the geometry of a manifold, such as how they can …

Math And Voting: Voting Methods, Fair Representation, And The Electoral College, Sarah Nelson

Honors Scholar Theses

Voting is in integral part of any functioning democracy, but there exist more than just one way to count votes. Some voting methods use only a voter’s top-choice candidate, while others require a ranking of all candidates, most-preferred to least-preferred, from each voter. We examine some of these ranked-choice voting methods, including the anti-plurality method, Hare’s method, and Coomb’s method.

Because of the variety of voting methods, we introduce criteria, which allow for an evaluation of the advantages and disadvantages of each method. The criteria give various definitions for what “good” or “bad” voting methods look like, depending on context. …

The Geometry Of Spacetime And Its Singular Nature, Filip Dul

Honors Scholar Theses

One hundred years ago, Albert Einstein revolutionized our understanding of gravity, and thus the large-scale structure of spacetime, by implementing differential geometry as the pri- mary medium of its description, thereby condensing the relationship between mass, energy and curvature of spacetime manifolds with the Einstein field equations (EFE), the primary compo- nent of his theory of General Relativity. In this paper, we use the language of Semi-Riemannian Geometry to examine the Schwarzschild and the Friedmann-Lemaˆıtre-Robertson-Walker met- rics, which represent some of the most well-known solutions to the EFE. Our investigation of these metrics will lead us to the problem of …

Efficient Coupling For Random Walk With Redistribution, Elizabeth Tripp

Honors Scholar Theses

What can be said on the convergence to stationarity of a finite state Markov chain that behaves 'locally' like a nearest-neighbor random walk on the set of integers? In this work, we looked to obtain sharp bounds for the rate of convergence to stationarity for a particular non-symmetric Markov chain. Our Markov chain is a variant of the simple symmetric random walk on the state space {0, ..., N} obtained by allowing transitions from 0 to J₀ and from N to J_N. We first looked at the case where J₀ and J_N are fixed, deterministic …

Schwarzschild Spacetime And Friedmann-Lemaitre-Robertson-Walker Cosmology, Zachary Cohen

Honors Scholar Theses

The advent of General Relativity via Einstein's field equations revolutionized our understanding of gravity in our solar system and universe. The idea of General Relativity posits that gravity is entirely due to the geometry of the universe -- that is, the mass distribution throughout the universe results in the "curving" of spacetime, which gives us the physics we see on a large scale. In the framework of General Relativity, we find that the universe behaves differently than was predicted in the model of gravitation developed by Newton. We will derive the general relativistic model for a simple system near a …

High Frequency Data: Modeling Durations Via The Acd And Log Acd Models, Lilian Cheung

Honors Scholar Theses

This thesis proposes a method of finding initial parameter estimates in the Log ACD₁ model for use in recursive estimation. The recursive estimating equations method is applied to the Log ACD₁ model to find recursive estimates for the unknown parameters in the model. A literature review is provided on the ACD and Log ACD models, and on the theory of estimating equations. Monte Carlo simulations indicate that the proposed method of finding initial parameter estimates is viable. The parameter estimation process is demonstrated by fitting an ACD model and a Log ACD model to a set of IBM …

Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna

Honors Scholar Theses

Finite fields, and the polynomial rings over them, have many neat algebraic properties and identities that are very convenient to work with. In this paper we will start by exploring said properties with the goal in mind of being able to use said properties to efficiently irreducibly factorize polynomials over these fields, an important action in the fields of discrete mathematics and computer science. Necessarily, we must also introduce the concept of an algorithm’s speed as well as particularly speeds of basic modular and integral arithmetic opera- tions. Outlining these concepts will have laid the groundwork for us to introduce …

Dirichlet's Theorem And Applications, Nicholas Stanford

Honors Scholar Theses

Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.

The proof of Dirichlet's theorem suggests a method for defining a notion of density of a set of primes, called its Dirichlet density, …

Weierstrass Points On Families Of Graphs, William D. Lindsay Jr.

Master's Theses

No abstract provided.

Computable Linear Orders And Turing Reductions, Whitney P. Turner

Master's Theses

This thesis explores computable linear orders through Turing Reductions and codes zero jump and zero double jump into linear orders using discrete, dense, and block linear relations.

Explicit And Implicit Methods In Solving Differential Equations, Timothy Bui

Honors Scholar Theses

Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.

The Devil’S Calculus: Mathematical Models Of Civil War, Ajay Shenoy

Honors Scholar Theses

In spite of the movement to turn political science into a real science, various mathematical methods that are now the staples of physics, biology, and even economics are thoroughly uncommon in political science, especially the study of civil war. This study seeks to apply such methods - specifically, ordinary differential equations (ODEs) - to model civil war based on what one might dub the capabilities school of thought, which roughly states that civil wars end only when one side’s ability to make war falls far enough to make peace truly attractive. I construct several different ODE-based models and then test …

The Hasse-Minkowski Theorem, Adam Gamzon

Honors Scholar Theses

The Hasse-Minkowski theorem concerns the classification of quadratic forms over global fields (i.e., finite extensions of Q and rational function fields with a finite constant field). Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921. He extended the research of his thesis to quadratic forms over all number fields in 1924. Historically, the Hasse-Minkowski theorem was the first notable application of p-adic fields that caught the attention of a wide mathematical audience. The goal of this thesis is to discuss the Hasse-Minkowski theorem over the rational numbers and over the rational function fields with a …