Physical Sciences and Mathematics Commons^™

The Mathematical And Historical Significance Of The Four Color Theorem, Brock Bivens

Honors Theses

Researching how the Four Color Theorem was proved, its implications on the mathematical community, and interviews with working mathematicians to develop my own personal opinions on the significance of the Four Color Theorem.

Echolocation On Manifolds, Kerong Wang

Honors Theses

We consider the question asked by Wyman and Xi [WX23]: ``Can you hear your location on a manifold?” In other words, can you locate a unique point x on a manifold, up to symmetry, if you know the Laplacian eigenvalues and eigenfunctions of the manifold? In [WX23], Wyman and Xi showed that echolocation holds on one- and two-dimensional rectangles with Dirichlet boundary conditions using the pointwise Weyl counting function. They also showed echolocation holds on ellipsoids using Gaussian curvature.

In this thesis, we provide full details for Wyman and Xi's proof for one- and two-dimensional rectangles and we show that …

Centers Of N-Degree Poncelet Circles, Georgia Corbett

Honors Theses

Given a circle inscribed in a polygon inscribed in the unit circle, if one connects all the vertices with line segments we get a family of circles called a package of Poncelet circles, due to its connection to a theorem of Poncelet. We are interested in where the centers of the Poncelet circles can be. Specifically, we have shown that if one of the circles in the Poncelet package is centered at 0, then all of the circles must be centered at 0 as well. This was proven by Spitkovsky and Wegert in 2021 using elliptic integrals but we …

Modeling The Opioid Crisis In Virginia: A Differential Equations Model Assessing The Impact Of Medication-Assisted Treatment On The Addicted Population, Maniha Zehra Akram

Honors Theses

The opioid epidemic is prevalent in countless communities throughout the United States and has yet to be mitigated. Treatments for OUD (opioid use disorder) include Medication-Assisted Treatment (MAT) and treatment without medication (non-MAT), with the former being judged as more effective in terms of lower relapse rates, death rates, and criminal activity (U.S. Food & Drug Administration, 2023; SAMHSA, 2024). Motivated by the promising research on MAT, this paper models the relationship

between the treatment and addicted populations using a system of ordinary differential equations. In addition to producing closed-form equilibrium solutions, the model leads to the conclusion that expanding …

The Effect Of The Expanded Child And Dependent Care Tax Credit On Maternal Labor Supply, Abby Letocha

Honors Theses

Policies that subsidize childcare have many potential economic benefits such as mitigating the high cost of childcare, incentivizing families to have more children, increasing paid childcare participation, and increasing parental labor supply. In this paper, I focus on the effect of childcare subsidies on maternal labor supply through a tax policy expansion. The Child and Dependent Care Tax Credit (CDCTC) is the primary federal childcare subsidy in the United States, and it was temporarily expanded in 2021 under the American Rescue Plan Act. This expansion increased the generosity of the credit and made it fully refundable for the 2021 tax …

The Factors Affecting Fashion Trends Have Changing Over The Years By Different Age Groups & The Evolution Of Media., Omnahqiran Dazz

Honors Theses

Introduction:

Fashion trends undergo continuous evolution, influenced by factors such as age groups and the ever-changing landscape of media. This research delves into the intricate relationship between these elements. Initially driven by a passion for fashion, the project expanded to explore the profound impact of social media evolution over the past 15-20 years.

Objectives: Investigate changing fashion trends across age groups.

Examine the evolution of media.

Analyze the factors affecting current-day fashion trends.

Explore the influence of social media on fashion choices.

This study provides invaluable insights for fashion designers, brands, and retailers, aiding in the development of effective market …

Aging An Ancient Maya Population From Actuncan, Belize Using Dental X-Rays, Kaitlyn Nicole Cash

Honors Theses

My goal is to determine an accurate age at death estimation of an ancient Maya population from the archaeological site of Actuncan, Belize. This was done by measuring the lengths of the coronal pulp cavities in the individuals’ teeth. I used X-Ray images to measure the coronal pulp cavities of the teeth and estimated age using multiple regression formulae for premolars, molars, and incisors to make age estimations. The formulae came from two studies, Ikeda et al. (1985) and Drusini (2008), that form the basis of my research. Ikeda et al. (1985) was the first to use this aging method, …

An Application Of The Pagerank Algorithm To Ncaa Football Team Rankings, Morgan Majors

Honors Theses

We investigate the use of Google’s PageRank algorithm to rank sports teams. The PageRank algorithm is used in web searches to return a list of the websites that are of most interest to the user. The structure of the NCAA FBS football schedule is used to construct a network with a similar structure to the world wide web. Parallels are drawn between pages that are linked in the world wide web with the results of a contest between two sports teams. The teams under consideration here are the members of the 2021 Football Bowl Subdivision. We achieve a total ordering …

The Hilbert Sequence And Its Associated Jacobi Matrix, Caleb Beckler

Honors Theses

In this project, we investigate positive definite sequences and their associated Jacobi matrices in Hilbert space. We set out to determine the Jacobi matrix associated to the Hilbert sequence by methods described in Akhiezer’s book The Classical Moment Problem. Using methods in Teschl’s book Jacobi Operators and Completely Integrable Nonlinear Lattice, we determine the essential spectrum of the corresponding Jacobi matrix.

Exploring The Structure Of Partial Difference Sets With Denniston Parameters, Nicolas Ferree

Honors Theses

In this work, we investigate the structure of particular partial difference sets (PDS) of size 70 with Denniston parameters in an elementary abelian group and in a nonelementary abelian group. We will make extensive use of character theory in our investigation and ultimately seek to understand the nature of difference sets with these parameters. To begin, we will cover some basic definitions and examples of difference sets and partial difference sets. We will then move on to some basic theorems about partial difference sets before introducing a group ring formalism and using it to explore several important constructions of partial …

The 2015 Ncaa Cost-Of-Attendance Stipend And Its Effects On Institutional Financial Aid Packages, Sara Greene

Honors Theses

In 2015, the National Collegiate Athletic Association (NCAA) allowed “Cost of Attendance” (COA) stipends to be offered to athletic recruits for Division I schools. These stipends are intended to allow schools to grant aid to student-athletes beyond a full-ride scholarship to cover additional costs imposed on student-athletes. These stipends created an opportunity for the “Autonomy” Power 5 programs to utilize a competitive tactic to try to win over the top recruits. There is evidence that these COA stipends have caused an increase in the estimated cost of attendance reported by the university. This paper examines if the COA stipends have …

The Parental Labor Gap: The Impact Of Daycare Access On The Parental Labor Force During The Covid-19 Pandemic, Acacia Wyckoff

Honors Theses

In the two years since the COVID-19 pandemic began, the landscape for work has shifted dramatically. Many companies and employers switched to telework when the pandemic hit, and many still do not require workers to come into the office. Research suggests these COVID-induced changes have led to a closing of the gap in childcare duties between men and women in households. Comparing parents in positions with telework eligibility versus in-person positions, Heggeness and Suri (2022) found that while telework improved the labor participation rate of mothers slightly, there was still a major gap in labor force participation between mothers and …

From Big Farm To Big Pharma: A Differential Equations Model Of Antibiotic-Resistant Salmonella In Industrial Poultry Populations, Rilyn Mckallip

Honors Theses

Antibiotics are used in poultry production as prophylaxis, curative treatment, and growth promotion. The first use is as prophylaxis, or prevention of common bacterial diseases. The crowded conditions in concentrated animal feeding operations necessitate management of infectious disease to ensure overall animal health and the profitability of such operations. In these farms, between 20,000 and 125,000 birds are raised in shed-like enclosures [3], with an average of less than one square foot of space per chicken [34]. Antibiotics are currently used in chicken farms to manage and prevent common bacterial diseases such as respiratory and digestive tract infections, as well …

Length Bias Estimation Of Small Businesses Lifetime, Simeng Li

Honors Theses

Small businesses, particularly restaurants, play a crucial role in the economy by generating employment opportunities, boosting tourism, and contributing to the local economy. However, accurately estimating their lifetimes can be challenging due to the presence of length bias, which occurs when the likelihood of sampling any particular restaurant's closure is influenced by its duration in operation. To address the issue, this study conducts goodness-of-fit tests on exponential/gamma family distributions and employs the Kaplan-Meier method to more accurately estimate the average lifetime of restaurants in Carytown. By providing insights into the challenges of estimating the lifetimes of small businesses, this study …

Graphs, Adjacency Matrices, And Corresponding Functions, Yang Hong

Honors Theses

Stable polynomials, in the context of this research, are two-variable polynomials like $p(z_1,z_2) = 2 - z_1 - z_2$ that are guaranteed to be non-zero if both input variables have an absolute value less than one in the complex plane. Stable polynomials are used in a variety of mathematical fields, thus finding ways to construct stable polynomials is valuable. An important property of these polynomials is whether they have boundary zeros, which are points in the complex plane where the polynomial equals zero and both variables have an absolute value of 1. Overall, it is challenging to find stable polynomials …

Finding The Common Denominator: Understanding The Shared Experiences Of Female Math Majors, Abigail R. Rosenbaum

Honors Theses

Despite efforts to increase gender diversity in STEM fields, women remain underrepresented in mathematics, especially in advanced academic and research positions. This study aimed to explore the experiences of female math majors as they attempt to navigate this male-dominated space. Through qualitative interviews with seven female math majors, two female math professors, and a focus group with education majors at Woodbridge College, small liberal arts college in the United States, several common themes were identified that define the experiences of female math majors. The findings suggest that math is held at an elevated status in society and that there is …

Elliptic Curves Over Finite Fields, Christopher S. Calger

Honors Theses

The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of F_q-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …

A View Into Secondary Education Mathematics, Thomas Krieger Jr.

Honors Theses

Teaching methods, and the effects they can have on students, are important to consider for a classroom because when teaching you should allow for every student to have an opportunity. Every student should feel encouraged in the classroom, however not every method may allow for that. An important task for a teacher is to find out how to reach their students in their classroom; be it adapting methods or choosing when to implement one item over another. This task differs with every student that enters the classroom as no student is the same. Every students’ differences stem from their academic …

A Fractal Geometry For Hydrodynamics, Jonah Mears

Honors Theses

Experiments have shown that objects with uneven surfaces, such as golf balls, can have less drag than those with smooth surfaces. Since fractal surfaces appear naturally in other areas, it must be asked if they can produce less drag than a traditional surface and save energy. Little or no research has been conducted so far on this question. The purpose of this project is to see if fractal geometry can improve boat hull design by producing a hull with low friction.

Relative Energy Comparison For Various Water Clusters Using Mp2, Df-Mp2, And Ccsd(T):Mp2 Methods, Qihang Wang

Honors Theses

The study of water clusters is an important area of research in many disciplines, such as biology, physical chemistry, and environmental studies. However, due to the difficulty in studying larger water clusters, such as clathrate hydrates, it is beneficial to obtain accurate descriptions of smaller water clusters to use as models for larger systems via computational methods. By starting with small water clusters, such as (H2O)₆, and moving into larger systems it is possible to build up data on various water structures that can determine the energetics of the various geometries within a certain number of water molecules. …

Classifying Pretzel Links Obtained By Strong Fusion, Jonathan Homan

Honors Theses

A link is a collection of circles embedded into 3-dimensional space. Pretzel links are an important family of links which comprises those links that fit a general form that includes many of the most common links. The strong fusion of a link joins two components of the link via a band and adds an unknotted circle about the band [4]; this naturally arises in the study of concordance and has been used to model biological phenomena such as site specific recombination in DNA [2]. Here we present a complete and original classification of those pretzel links which can be obtained …

Gl(1|1) Graph Connections, Andrea Bourque

Honors Theses

No abstract provided.

Opers On The Projective Line, Wronskian Relations, And The Bethe Ansatz, Ty J. Brinson

Honors Theses

No abstract provided.

A Contraction Based Approach To Tensor Isomorphism, Anh Kieu

Honors Theses

Tensor isomorphism is a hard problem in computational complexity theory. Tensor isomorphism arises not just in mathematics, but also in other applied fields like Machine Learning, Cryptography, and Quantum Information Theory (QIT). In this thesis, we develop a new approach to testing (non)-isomorphism of tensors that uses local information from "contractions" of a tensor to detect differences in global structures. Specifically, we use projective geometry and tensor contractions to create a labelling data structure for a given tensor, which can be used to compare and distinguish tensors. This contraction labelling isomorphism test is quite general, and its practical potential remains …

Decoding Cyclic Codes Via Gröbner Bases, Eduardo Sosa

Honors Theses

In this paper, we analyze the decoding of cyclic codes. First, we introduce linear and cyclic codes, standard decoding processes, and some standard theorems in coding theory. Then, we will introduce Gr¨obner Bases, and describe their connection to the decoding of cyclic codes. Finally, we go in-depth into how we decode cyclic codes using the key equation, and how a breakthrough by A. Brinton Cooper on decoding BCH codes using Gr¨obner Bases gave rise to the search for a polynomial-time algorithm that could someday decode any cyclic code. We discuss the different approaches taken toward developing such an algorithm and …

Representation Theory And Its Applications In Physics, Jakub Bystrický

Honors Theses

Representation theory is a branch of mathematics that allows us to represent elements of a group as elements of a general linear group of a chosen vector space by means of a homomorphism. The group elements are mapped to linear operators and we can study the group using linear algebra. This ability is especially useful in physics where much of the theories are captured by linear algebra structures. This thesis reviews key concepts in representation theory of both finite and infinite groups. In the case of finite groups we discuss equivalence, orthogonality, characters, and group algebras. We discuss the importance …

Decomposing Manifolds In Low-Dimensions: From Heegaard Splittings To Trisections, Suixin "Cindy" Zhang

Honors Theses

The decomposition of a topological space into smaller and simpler pieces is useful for understanding the space. In 1898, Poul Heegaard introduced the concept of a Heegaard splitting, which is a bisection of a 3-manifold. Heegaard diagrams, which describe Heegaard splittings combinatorially, have been recognized as a powerful tool for classifying 3-manifolds and producing important invariants of 3-manifolds. Handle decomposition, invented by Stephen Smale in 1962, describes how an n-manifold can be constructed by successively adding handles. In 2012, Gay and Kirby introduced trisections of 4-manifold, which are a four-dimensional analogues of Heegaard splittings in dimension three. Trisection diagrams give …

A Weighted Version Of Erdős-Kac Theorem, Unique Subedi

Honors Theses

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. A celebrated result of Erd{\H o}s and Kac states that $\omega(n)$ as a Gaussian distribution. In this thesis, we establish a weighted version of Erd{\H o}s-Kac Theorem. Specifically, we show that the Gaussian limiting distribution is preserved, but shifted, when $\omega(n)$ is weighted by the $k-$fold divisor function $\tau_k(n)$. We establish this result by computing all positive integral moments of $\omega(n)$ weighted by $\tau_k(n)$.

We also provide a proof of the classical identity of $\zeta(2n)$ for $n \in \mathbb{N}$ using Dirichlet's kernel.

2-Adic Valuations Of Square Spiral Sequences, Minh Nguyen

Honors Theses

The study of p-adic valuations is connected to the problem of factorization of integers, an essential question in number theory and computer science. Given a nonzero integer n and prime number p, the p-adic valuation of n, which is commonly denoted as νp(n), is the greatest non-negative integer ν such that p ν | n. In this paper, we analyze the properties of the 2-adic valuations of some integer sequences constructed from Ulam square spirals. Most sequences considered were diagonal sequences of the form 4n 2 + bn + c from the Ulam spiral with center value of 1. Other …

An Introduction To Obstacle Problems, Calvin Reedy

Honors Theses

The obstacle problem can be used to predict the shape of an elastic membrane lying over an obstacle in a domain Ω. In this paper we introduce and motivate a mathematical formulation for this problem, and give an example to demonstrate the need to search for solutions in non-classical settings. We then introduce Sobolev spaces as the proper setting for solutions, and prove that unique solutions exist in W1,2(Ω).