Mathematics Commons^™

The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal

Honors Theses

In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.

Sumset Lower Bounds In Abelian Groups, Van T. Huynh

Honors Theses

This thesis investigates sumset lower bounds across discrete and continuous settings. We begin with general inequalities in torsion-free abelian groups and then specialize to the integers modulo prime p, where we present the Cauchy–Davenport Theorem, which establishes the bound ∣A+B∣≥min(p,∣A∣+∣B∣−1). The equality case is further examined via Vosper's Theorem, which characterizes subsets attaining this bound as arithmetic progressions under suitable conditions. The continuous analogue in Euclidean spaces is then considered, where cardinality is replaced by Lebesgue measure. In this setting, the Brunn–Minkowski Inequality provides a sharp lower bound for the Lebesgue measure of A+B and serves as a geometric counterpart …

Redefining Certainty: Non-Euclidean Geometry And Theology Transformation Throughout The Intellectual Unrest Of The Early 1800s, Luke Bensinger

Honors Theses

To bridge the gap between mathematics and theology, it is necessary to explore their intersection and challenge the notion that these fields are incompatible. This study focuses on the 19th century, a period when non-Euclidean geometries emerged and disrupted mathematical certainty, while Protestant theologians such as Barton W. Stone and Alexander Campbell grappled with Calvinism and shifting theological perspectives. By analyzing mathematicians studying geometry, such as Gauss, Lobachevsky, and Riemann, this research examines how both disciplines balance change with enduring truths.

Algorithm Performance In The Search For Hamiltonian Cycles, Chance Davis

Honors Theses

The Hamiltonian cycle problem is ubiquitous in both computer science and graph theory: Given a connected graph, a solution would either confirm the existence of a cycle which visits each vertex only once or its nonexistence. The importance of this problem, as well as its difficulty, is described in the Clay Mathematics Institute’s Millenium Prize Problems and Karp’s 21 NP-complete problems. Despite its “hardness,” solutions to the Hamiltonian cycle problem are desired in logistics, electronic circuit design, and network routing, among other fields. In this work, we benchmark a promising exhaustive enumeration algorithm on various graphs, including ones derived from …

Toward Completeness Theorem For Guarded Kleene Algebra With Tests, Hung Pham

Honors Theses

Code refactoring is a fundamental practice in software engineering, in which a program is restructured without changing the actions it performs and the results it produces. To carry out refactoring with confidence, one requires a formal method for verifying that two programs are equivalent. Guarded Kleene Algebra with Tests (GKAT) provides such a framework, an algebraic system designed to reason about a natural class of programs, namely those in which every branch and loop is governed by a Boolean condition, such as if–else and while statements. Central to GKAT is a finite set of algebraic axioms for deriving program equivalences. …

Pursuer Evader Surveillance Game Control Theory And Motion Planning, Zijie Mu

Honors Theses

This thesis studies the pursuer evader surveillance game with a triangular obstacle in the short-term. In the game, the pursuer aims to maintain surveillance of the evader as long as possible while the evader aims to break surveillance in a finite time. We classify the player strategies into ideal ones and best admissible ones. The outcome of the game is determined by line of sight. We reduce the 4D game to a 3D game with an upward motion for a small time interval to terminate the game. When the evader starts outside the threshold, we show that there exists an …

From Shock To Routine: The Evolving Impact Of Shutdown-Related Sentiment On Stock Markets, Yiran Shao

Honors Theses

To address gaps in existing research, this paper selects two U.S. government shutdown periods, 2018-2019 and 2025, as research samples to explore the effect of policy uncertainty on sentiment. This paper primarily analyzes the following two research questions.

First, what is the correlation between government shutdown-related sentiment during the shutdown period and daily market fluctuations? Specifically, can the sentiment index constructed from shutdown-related news effectively predict the next-day stock return during the event period?

Second, does the market have a learning effect? That is, between 2018-2019 and 2025, has the relationship between shutdown-related emotions and market outcomes weakened, shortened the …

Mechanisms Driving Disparities In Income Mobility Across The Income Distribution, Joe Larkins

Honors Theses

This study examines intergenerational income persistence across the income distribution, testing whether mechanisms driving inequality differ between families in the top and bottom halves of the income distribution. Using data from the National Education Longitudinal Study of 1988 (NELS:88), a nationally representative longitudinal survey of 8th grade students and their parents, this research estimates an interaction model comparing parental income effects for children in advantaged versus disadvantaged economic circumstances. The analysis reveals that a $1,000 increase in parental income yields eight times greater income gains for children in the bottom half of the distribution compared to those in the top …

Group-Based Integer Factorization: Theory And Performance, Phuong Cao

Honors Theses

Integer factorization, the problem of finding a nontrivial factor of a composite integer N=pq for large primes p,q, particularly at the size of RSA moduli, is a notoriously difficult challenge that takes classical methods such as Trial Division and Fermat’s Algorithm trillions of years to solve. This thesis studies four probabilistic algorithms that exploit algebraic group structures to achieve significantly better, subexponential efficiency for certain classes of N: Pollard’s p-1, Williams’ p+1, Lenstra’s Elliptic Curve Method, and Pell’s Conic Method. In each case, the algorithm operates on a group over ℤ/Nℤ that decomposes, via the Chinese Remainder Theorem, into corresponding …

Pursuer Evader Surveillance Game, Zijie Mu

Honors Theses

This thesis studies the pursuer evader surveillance game with a triangular obstacle in the short-term. In the game, the pursuer aims to maintain surveillance of the evader as long as possible while the evader aims to break surveillance in a finite time. We classify the player strategies into  ideal ones and best admissible ones. The outcome of the game is determined by line of sight. We reduce the 4D game to a 3D game with a boundary separating two different local regimes. When the evader starts inside the threshold, we show that there exists an admissible evader that maintains an …

Education And The Maternal Childcare Gap: Evidence From The Uk Covid-19 Pandemic, Lara Bakhaya

Honors Theses

This paper examines whether college education shapes working mothers’ access to remote work, flexible working, and childcare hours in the United Kingdom, and whether COVID-19 school closures amplified these inequalities. Using data from the UK Time Use Survey (2016–2021), a repeated cross-sectional diary dataset spanning the pre-pandemic period and five COVID-19 waves, this paper estimates weighted logistic and ordinary least squares regressions on a sample of married or cohabiting, employed mothers. School closures serve as a natural experiment, providing an exogenous shock to caregiving demands that affected all mothers simultaneously regardless of education level. College education significantly predicted working from …

Spillover Effects Of Medicare Advantage On Fee-For-Service Post-Acute Care Spending, Nyel Bangash

Honors Theses

Does the growth of Medicare Advantage reduce fee-for-service post-acute care spending through practice-pattern spillovers, or do observed spending differences primarily reflect favorable selection? Using a county-level panel of roughly 2,700 counties (2014–2023) and a two-way fixed effects specification, I find that a one percentage-point increase in MA penetration is associated with $9.54 less per-capita standardized FFS spending. Spending per episode falls while participation rates remain stable, consistent with practice-pattern spillovers rather than compositional changes from selection. Welfare indicators from County Health Rankings, CDC PLACES, and CMS Care Compare show no evidence that spending reductions harm health or care quality. The …

Asymptotics Of Discrete Convolution Powers And Applications To Difference Schemes, Pedro Henrique Alves Silva Dos Santos

Honors Theses

In this thesis we provide Gaussian Estimates and Local Limit Theorems describing the asymptotic behavior of convolution powers of a class of complex-valued functions on $\mathbb{Z}^d$. Convolution powers arise naturally in the study of partial differential equations, as well as in random walks in probability theory. In particular, they are connected to the stability theory of difference schemes used to approximate solutions to partial differential equations. We take inspiration from the work of Vidar Thomée on stability theory to restrict our attention to convolution powers of functions whose Fourier Transforms satisfy certain local expansions. We then combine the Cauchy Integral …

Enumerating Matrices Of A Given Order Over Finite Fields, Thor Richard Gabrielsen

Honors Theses

This thesis derives a generating function that describes the matrices of a given multiplicative order over finite fields of a given order assuming that the order of a field is not a divisor of the desired order of the matrix. This is done by using the power series known as the cycle index derived from the rational canonical form. This index can be factored, and using the fact that the minimal polynomial divides some polynomial of the form x^k − 1 we can show that the minimal polynomial is squarefree. This sufficiently restricts the form of the rational canonical form …

Exploration Into Homotopy Equivalence And Vietoris-Rips Complexes, Alexander J. Mobley

Honors Theses

An important question in discrete topology and geometry is how to recover the structure and characteristics of a manifold when only given a finite set of points sampled from that manifold. Thus, if mathematicians have a point cloud of data which induces a discrete metric space, they look for some structure to these points. Understanding the structure of these points often gives needed insight to solve this problem. One such way to determine structure to these points is to use these points to construct a Vietoris-Rips complex. In abstract, Latschev shows that this allows us to recover the homotopy type …

Ritt’S Theorem And The Chebyshev Polynomials, Eric Neuhaus

Honors Theses

We explore polynomial composition through Ritt’s Theorem and mondromy groups. J.F. Ritt asked the question of when a polynomial can be written as the composition of two other polynomials of smaller, nontrivial degree. Ritt’s Theorem provides a method for determining whether or not a polynomial is composite by observing the actions of its associated monodromy group, a group of permutations acting upon the roots of a given polynomial. In this work, we study the monodromy groups of the Chebyshev polynomials of the first and second kinds to uncover their compositional properties.

Improving Research Software Engineering In Mathematics, Abram Miller

Honors Theses

Research Software Engineering is critical to modern mathematical research, enabling the creation, maintenance, and dissemination of computational tools that bridge theory and practice. However, the field faces systemic challenges, including insufficient funding, lack of institutional recognition, and gaps in training and infrastructure. This thesis investigates these challenges through two approaches: (1) a comparative survey study focused on mathematicians and (2) hands-on contributions to an open-source research software project.

The Improving Research Software Engineering in Mathematics survey, conducted from September 2024 to January 2025, adapts the survey framework developed by Carver et al. in A survey of the state of the …

Mathematical Melodies: The Exploration Of Music Using Fourier Signal Analysis, Courtney Francois

Honors Theses

Upon initial inspection, mathematics and music appear to be distinct disciplines lacking any connections to one another. However, many complex intersections lie beneath the surface. This paper seeks to explore these connections through the analysis of the Fourier series and the Fourier transformation of musical sound signals. Orthogonal functions of various frequencies are used to approximate and recover sound signals. Following the examination of the basis of Fourier signal analysis, MATLAB is utilized to visualize the connectedness of mathematics and music through sound signals from varied musical instruments and by performing Fourier signal analysis to the sound signals. Then, the …

A Multi-Scale Compartmental Model For Glucose Regulation And Diabetes Treatment, Andrew M. Watts

Honors Theses

Glucose is a fundamental energy source for cellular function, and its regulation is critical for maintaining metabolic stability in the human body. Glucose homeostasis is governed by a network of biochemical processes involving multiple organ systems that coordinate glucose production, storage, and uptake. Disruptions in this regulation contribute to metabolic disorders such as diabetes mellitus, underscoring the need for mathematical models that provide a mechanistic understanding of systemic glucose dynamics. Herein, we report a multi-time scale discrete-time dynamical systems model using compartmental difference equations to describe glucose concentrations across key physiological compartments. By quantitatively modeling glucose transport and metabolism, this …

The Impact Of Corporate Profits On Gdi Estimates: Forecasting And Data Revisions, Peterson Haas

Honors Theses

The National Income and Product Accounts (NIPAs) produced by the U.S. Bureau of Economic Analysis (BEA) provide key measures of U.S. economic activity, including gross domestic product (GDP) and gross domestic income (GDI). Although conceptually equivalent, GDP and GDI often diverge due to differences in source data and revision timing. This study focuses on corporate profits—a small but volatile component of GDI that is frequently revised, especially during the BEA’s annual (A1) and benchmark (C1) revisions. I examine how revisions to corporate profits influence GDI revisions and whether they can improve real-time estimates of GDI. First, I document the size …

Knighted Pawn's Tour, Morgan Wilson

Honors Theses

The Knighted Pawn’s Tour is a variant of the traditional Knight’s Tour problem, whichitself is an instance of the Hamiltonian cycle problem. The deviation from the Knight’s Tour problem is that a pawn begins on any field on the second rank (row), and advances with legal moves that can include captures to the opposite end of the board to become a knight. These fields, used by the pawn on the path to knighthood, are subsequently forbidden for the knight’s return to the starting field. The knight must traverse the rest of the chessboard, visiting each remaining field exactly once …

Affine Groups: A Functorial Perspective, Vladimir Khabaev

Honors Theses

A familiar construction associated to any commutative ringRwith1is its group of units, traditionally denoted by R^x= {u in R | uv = 1 for some v in R}. This is but one out of many ways to get a group from a ring. To see at least one other way, we need a mild change in perspective: units may instead be characterized as elements for which the linear transformation f(r) = u ⋅ r is an isomorphism of R as a module over itself. That is to say, R^x = GL(1, R …

The Anatomy Of A Reconstruction: From Fourier Space To Image Recovery In Computed Tomography, Charlotte P. Maurer

Honors Theses

This thesis develops the mathematical foundations of computed tomography (CT) reconstruction through the lens of harmonic analysis. Beginning with the Schwartz class, we introduce the Fourier transform and its role in expressing the Radon transform and its inversion via a fractional Laplacian. After constructing the Radon transform in general dimension R^d, we specialize to the cases d = 2 and d = 3, demonstrating explicit inversion formulas and the associated instability in lower dimensions. For its computational advantages, we study filtered back-projection using classical low-pass filters (Ram-Lak, Shepp–Logan, Cosine, Gaussian) and formulate a discrete reconstruction algorithm grounded in …

On Linear Invariants Of Hypergraphs, Clara Chaplin

Honors Theses

We introduce linear invariants of hypergraphs as a way to study hypergraphs by their tensor representations. Our primary research goal is to determine what information linear invariants capture about the hypergraphs they arise from. We first investigate the centroid, which is shown to determine the connected components of a hypergraph. Next, we study the derivations of a hypergraph, and use this linear invariant to define a quotient operator $Q_\mathrm{Der}$ on the collection of all hypergraphs. This operator is shown to be a closure operator in that $Q_\mathrm{Der}(Q_\mathrm{Der}(\mathcal{H}))=Q_\mathrm{Der}(\mathcal{H})$ for any hypergraph $\mathcal{H}$. We apply the operator $Q_\mathrm{Der}$ to synthetically generated hypergraphs, …

Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan

Honors Theses

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …

Analysis Of The Snake Cube Puzzle And Adjacency Criteria, Trey Matus

Honors Theses

The snake cube is a puzzle that consists of straight and turn pieces attached by a string that folds into a n x n x n cube. Finding solutions for large cubes is difficult, so finding necessary conditions for solutions is crucial. Using computational algorithms and mathematical proofs, I find improved bounds for the maximum number of straight pieces in a given puzzle size. In particular, I introduce and apply the adjacency criterion to identify groups of puzzles that are unsolvable. Additionally, I find a connection between the number of puzzles that adhere to the adjacency criterion and generalized Fibonacci …

The Uniform Triadic Transformation And Claude Debussy’S Music, Davielle Smith

Honors Theses

Beginning with Claude Debussy’s Berceuse Héroïque, I analyzed each triad using chord symbols along with their Roman numeral representation (where applicable).1 Next, I used integer notation and modular arithmetic, where each pitch name is given a number such that C is 0 and B is 11, creating a set, the integers (mod 12).

This leads to the development of pitch-class sets where the “mode” of a triad or arbitrary number of pitches is expressed in terms of interval spacings. In many cases, such as that of 12-tone or other serialized music, pitch class sets explain many of the composer’s artistic …

A Comparison Of Assessment Experiences Between Standards-Based Practices And Traditional Practices Within Secondary Mathematics Classrooms, Emily Mayes

Honors Theses

The purpose of this research was to compare assessment experiences and find ways to improve those experiences for students in two mathematics classrooms: one classroom that employs Standards-Based Grading and one classroom that uses traditional grading practices. The research examines students’ perceptions regarding their level of preparation, their anxiety levels surrounding assessment, the validity of assessments, and using assessments and grading practices to give accurate indications of student progress in their learning, given the students’ perceptions. Students in both settings voluntarily and anonymously participated in completing pre- and post-assessment free-response surveys which asked questions about students’ assessment experiences. This research …

How To Explain Allen-Manandhar’S Method To Beginner Mathematicians : A Convergence Analysis Of A Hybrid Method For Variable-Coefficient Boundary Value Problems, Rebecca Scariano

Honors Theses

In this project, analogies are employed to make complex math concepts approachable to beginners who may only have a basic understanding of calculus and linear algebra. Serving as the focal point of this project, Allen-Manandhar’s method solves an equation, known as an ordinary differential equation (ODE). The mentioned equation with its coefficients is comparable to a pie recipe with ingredients. With the outcome to a recipe seen as its solution, the solution to our pie recipe is a perfectly baked pie, as in without error. The chosen method for baking a pie then classifies as its baking approach that when …

A Tale Of Two Toroidal Graphs, Akshat Gulgulia

Honors Theses

A graph is toroidal if it can be embedded on a torus which is a doughnut-shaped surface. Two well-known examples of toroidal graphs are the complete graph K5 and the complete bipartite graph K3,3. In this thesis we elucidate the association of the subject matter with two renowned enigmas in graph theory, namely the Five Princes Problem and the Three Utilities Problems. Additionally, we look at their association with several renowned theorems in topological graph theory. We explore the link between these two graphs and a contemporary labeling concept.