Mathematics Commons^™

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 …

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 …

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.

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 …

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 Impact Of U.S. News And World Report College Rankings On Prospective Students’ Decisions And Enrollment Outcomes, Camilla Schneider

Honors Theses

The following paper further expands on previous research using more recent data which is important given changes in the last decade. This paper will answer the question: How do U.S. News and World Report college rankings impact college application and enrollment decisions for prospective students within American public and private colleges and universities? Application decisions refer to the choice to apply to a school while enrollment decisions refer to the choice to attend a school after being accepted. This paper analyzes Integrated Postsecondary Education Data System (IPEDs) admissions/applicant, fall enrollment, and financial data alongside historical USNWR rankings lists to investigate …

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 …

The Square Roots Of Some Classical Operators, Javad Mashreghi, Marek Ptak, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.

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).

Overdose Prevention Sites Placement Informed By Simulation, Jing Dong

Honors Theses

In Philadelphia, people are experiencing the greatest opioid crisis in a century. Plac- ing the Overdose Prevention Site (OPS) can alleviate this crisis. However, the journey to the successful launch of the first OPS in the USA is rough. It was first accused of having a collision with federal drug laws. While Safehouse won the lawsuit and the OPS was judged to be legal in 2020, other pressure rose afterward such as the against from the public and the COVID19, which delayed the plan to open the OPS. Without solid research on the effectiveness of OPS, we thought it is …

An Agent-Based Model To Evaluate The Effect Of Socioeconomic Status And Demographic Factors On Covid-19 Prevalence And Mortality, Jonathan Huang, Berke Nouri

Arts & Sciences Student Symposium

The purpose of the project is to discover how the prevalence and mortality of a pandemic change depending on a population’s demographic factors as well as various intervention policies through a NetLogo agent-based model. This model will simulate how demographic factors affect the course of COVID-19 (infection rate, recovery rate, and death rate). Demographic factors of interest will include population density, income distribution, age distribution, and number of hospital beds per capita. Intervention policies include vaccination, social distancing, mask wearing, mass testing, and quarantining. Conclusions about the effect of demographic factors on the infection, recovery, and death rate will be …

Emerald Ash Borer And The Application Of Biological Control In Virginia, Shuheng Chen, Yihui Wu, Shengjie Liu

Arts & Sciences Student Symposium

The emerald ash borer (Agrilus planipennis; EAB) is an invasive wood-boring beetle whose larvae feed on ash phloem. After only 1-5 years of infestation, the larvae create extensive tunnels under the bark that disrupt the tree’s ability to transport water and nutrients, which eventually girdles and kills the tree. Since 2008, EAB has spread to all but the eastern-most counties in Virginia. Bological control is one strategy to limit EAB populations. In this project we study control by native agents (woodpeckers) and imported agents (parasitoid wasps).

Mathematical models of host-parasitoid interactions and simulations based on both models and field studies …

Getting Over It, Thomas Kade, Kevorc Ibrahimian, Max Simpson

Arts & Sciences Student Symposium

The research extended a 2D motion planning system to three dimensional environments. The updated system is now able to plan the motion for robots over 3D terrains modeled by polyhedrons.

The School Mathematics Study Group: Lessons In Mathematics Education, Madeline Polhill

Arts & Sciences Student Symposium

This work argues that the "new math" project called the School Mathematics Study Group offers a valuable case study for mathematics educators seeking to venture into the future better informed about both the successes and failures of previous projects. Understanding this project requires recognizing that the School Mathematics Study Group was wholly a product of the forces—personal, educational, mathematical, and political—that shaped it. Admittedly, some of the SMSG's shortcomings resulted from its members' lack of understanding of the changes needed in mathematics education. Still, the majority of the SMSG's public vilification resulted through no fault of its own, but rather …

The Surface Diffusion And The Willmore Flow For Uniformly Regular Hypersurfaces, Jeremy Lecrone, Yuanzhen Shao, Gieri Simonett

Department of Math & Statistics Faculty Publications

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are C^1+α–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are C^1+α–close to a sphere, and we prove …

On Quasilinear Parabolic Equations And Continuous Maximal Regularity, Jeremy Lecrone, Gieri Simonett

Department of Math & Statistics Faculty Publications

We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

A Template For Success: Celebrating The Work Of Judith Grabiner, Della Dumbaugh, Adrian Rice

Department of Math & Statistics Faculty Publications

Judith Grabiner is a mathematician who specializes in the history of mathematics. She is currently the Flora Sanborn Pitzer Professor Emerita of Mathematics at Pitzer College, one of the Claremont Colleges in Claremont, California. She has authored more than forty articles, as well as three books: The Origins of Cauchy’s Rigorous Calculus (1981), The Calculus as Algebra: J.-L. Lagrange, 1736–1813 (1990), and A Historian Looks Back: The Calculus as Algebra and Selected Writings (2010), which won the Beckenbach Prize from the Mathematical Association of America in 2014. She deliv- ered an invited address titled “The Centrality of Mathemat- ics in …

Almost Difference Sets In 2-Groups, Xin Yutong

Honors Theses

Difference sets have been studied for decades due to their applications in digital communication, cryptography, algebra, and number theory. More recently, mathematicians have expanded their focus to the field of almost difference sets. Almost difference sets have similar functionalities with difference sets, yet with more potential of finding new constructions. In this paper I will introduce the definitions, properties, and applications of difference sets and almost difference sets, and discuss our effort and results in the exploration of almost difference sets in cyclic and non-cyclic groups.

Estimating Value-At-Risk Of An Unconventional Portfolio, Elizabeth N. Mejía-Ricart

Honors Theses

Since the 2008 financial crisis, interest rates and bond yields have been low all through the recovery and expansion that followed, and they are still low. As a result, more investors have been attracted to US equities, a space of possibly higher returns. However, these returns come with a potential downside: risk of loss. One of the methods to assess this potential downside is value-at-risk (VaR), which gained momentum in the late 1990s. At the time, the market risk amendment to the 1988 Basle Capital Accord required commercial banks with significant trading activities to put aside capital to cover market …

Computer-Assisted Coloring-Graph Generation And Structural Analysis, Wesley Su

Honors Theses

Graphs are a well studied construction in discrete math, with one of the most common areas of study being graph coloring. The graph coloring problem asks for a color to be assigned to each vertex in a graph such that no two adjacent vertices share a color. An assignment of k colors that meets these criteria is called a k-coloring. The coloring graph Ck(G) is defined as the graph where every vertex represents a valid k-coloring of graph G and edges exist between colorings that di↵er by one vertex. We call graph G the base graph of the k-coloring graph …