Mathematics Commons^™

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.

Effects Of The Gravity Gradient On The Path Of 1i/‘Oumuamua, Hannah R. Richardson

Poster Presentations

In October 2017, the asteroid 1I/’Oumuamua first passed into viewing range [1]. The asteroid is notable for being the first interstellar object to enter the solar system. 1I/’Oumuamua was also unusual in its geometry; it is thought to have an aspect ratio of 6:1 and a length of approximately 400 m [2] [3]. The asteroid was observed to experience a non-Keplerian acceleration estimated to be on the order of 1×10−6 m s-2 Several theories have been proposed for the cause of this acceleration, all of which are non-gravitational in nature: volatile outgassing, photon pressure, and solar winds [1][4]. However, none …

Graph Codes, Jacquelyn Franqui

Theses, Dissertations and Culminating Projects

The concept of graph codes, introduced in recent work by Alon, applies coding theory to graphs by defining codewords as graphs and measuring distances between them through structural differences. In ordinary coding theory, the distance between two words is the number of positions in which the two words differ. In graph codes, each word is a graph and the distance between two graphs is measured by structural properties of their difference. For a fixed graph H, an H-code is a collection F of graphs on vertex set [n] = {1, 2, . . . , n} such that the symmetric …

Inversion Numbers Of Oriented Graphs, Sam Harris

Theses, Dissertations and Culminating Projects

This thesis investigates recent results and conjectures regarding inversion numbers of oriented graphs. We define the inversion number of an oriented graph G as the minimum size of a decycling family of subsets of vertices of G. We focus on three areas: the maximum inversion number of a graph on n vertices, the inversion numbers of dijoins of two oriented graphs, and computational algorithms for deciding the inversion number. The main contribution of this thesis is a proposed construction of a graph with higher inversion number than that of any other currently known construction. We also provide explicit, rather than …

Analysis And Mathematical Recomposition Of The Art Of The Fugue, Joshua Cellar

Theses, Dissertations and Culminating Projects

The Art of the Fugue is a collection of 18 fugues and cannons written by Johann Sebastian Bach in the 1700s. Each piece starts with a single theme which is transformed, developed and mixed in fascinating mathematical combinations resulting in a distinctive musical style. In this paper, we use the method introduced in [7] to analyze 14 pieces from The Art of the Fugue. We focus on the variation of motifs from one piece to another as well as the complexities of mathematical transformations present throughout the entire collection. We also demonstrate how using maps of transformations and the rules …

2-Segal Sets And Pseudomonoids In The Bicategory Of Spans, Sophia E. Marx, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

In this survey article, we give an introduction to the notion of a 2-Segal set and prove that 2-Segal sets are equivalent to pseudomonoids in the bicategory of spans. The proof utilizes graphical techniques for 2-Segal sets and spans that should be useful in more general settings.
There are procedures for obtaining an associative algebra from a 2-Segal set (satisfying finiteness conditions). We describe these procedures and give several examples of algebras arising from 2-Segal sets.
Wherever possible, we avoid higher category theory so as to make the paper accessible to a wide audience.

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 …

Computer Vision In Soccer: Yolov11 Analytics Engine For Quantifying Game Strategy, Connor S. Maurer

Data Science Undergraduate Honors Theses

Single-shot object detection capabilities significantly reduce computational overhead for real-time computer vision in sports analytics at 60 FPS. YOLO11’s lightweight CNN gives promising accuracy while meeting the low-latency demand of dynamic soccer matches. As data-driven approaches take over the sport of soccer, efficient player tracking systems become critical for informing coach’s strategies. I prototype the ETL (Extract, Transform, Load) process of data collected from a single- shot detection program and evaluate its viability for estimating player fatigue. YOLO11 detects players, the ball, and other characteristics, with the output transformed by homography to estimate the positions in the real world. These …

Two-Player Trick-Taking Games On Bipartite Tournaments., Allan Bagley

College of Arts & Sciences Senior Theses

We consider a two-person game played on a bipartite tournament with equal size parts. The game is modeled after trick-taking card games like bridge or euchre. The two players, South and North, each receive one of the parts as their hand, and the arcs from a vertex in one hand beat the corresponding vertices in the other hand. The game is played over rounds called tricks where the number of tricks is equal to the number of vertices in each player’s hand. At the beginning of the game, one player is designated as the leader, and they play the first …

A Study Of Knots And Quandles, Zhaoqi Wu

Math and Computer Science Honors Theses

We explore the mathematical theory of knots through the lens of algebraic structures known as kei and quandles. We begin by introducing classical knot invariants and then study the fundamental kei of a knot as a tool for distinguishing knot types. We generalize this approach using various kinds of quandles, including Alexander and dihedral quandles, and investigate their associated polynomial invariants. We also examine the connection between quandles and group theory, as well as their algebraic representations in quandle rings. Moreover, we analyze idempotent elements in quandle rings over finite fields, providing both general results and specific examples.

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Yield Prediction Of Pv Solar Energy Systems And Its Application In The Energy Grid For Operational Efficiency, Pablo Bustamante

Open Access Theses & Dissertations

The generation of power from photovoltaic (PV) solar panels is influenced by a multitude of factors. These include the tilt and orientation of the solar panels, the latitude of their location, and the prevailing climate and weather conditions. Additionally, shading at specific locations, particularly if the panels are not part of a solar facility, can significantly impact their efficiency. The quality and efficiency of the panels themselves, along with the preventive maintenance of both the solar panels and associated components such as inverters and trackers, are also critical. Finally, the overall system design and installation play a vital role in …

Multiscale Integration Of Receptor-Ligand Dynamics Into Discrete And Continuous Tumor Growth Models With Application To Tyrosine Kinase Inhibitor Treatment, Romasa Qasim

Open Access Theses & Dissertations

The epidermal growth factor (EGF) receptor cascade plays a crucial role in the survival and proliferation of tumor cells. Tyrosine kinase inhibitors (TKIs) are a class of drugs that inhibit epidermal growth factor receptors (EGFRs), thereby preventing the downstream signal transduction. Despite their importance, models that link spatial receptor dynamics to tumor growth remain scarce. Further, TKIs act through selective mechanisms, inhibiting active, inactive, or all receptor states, which poses a challenge to traditional modeling approaches.

We propose to numerically study two mathematical models incorporating receptor-dynamics into cancer models to describe the impact of EGFR overexpression and TKIs. The first …

Maximal Independent Set Algorithms Within Procedural Planar Maps: A Large-Scale Evaluation, Chaucer Ihrig

Honors College Theses

Analysis of a childhood game has led us to the problem of maximum independent sets in planar graphs. We wrote a graph creation utility using R to generate a random planar map and its dual graph. This utility then finds a graph’s maximal independent set using a variety of six algorithms. We investigate statistical connections between graph structure, colorability, and the maximal independent sets found using these algorithms over an incredibly large and procedurally generated dataset. We find one can always win the coloring game if the resultant graph is two-colorable. The algorithms perform statistically and practically significantly better on …

An Algebraic-Combinatorial Proof Of A Bézout-Type Inequality For Mixed Volumes Of Three-Dimensional Zonoids, Gennadiy Averkov, Ivan Soprunov

Mathematics and Statistics Faculty Publications

We present a new algebraic-combinatorial approach to proving a Bézout-type inequality for zonoids in dimension three, which has recently been established by Fradelizi, Madiman, Meyer, and Zvavitch. Our approach hints at connections between inequalities for mixed volumes of zonoids and real algebra and matroid theory.

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 …

Bibliography Of Christianity And Mathematics Update, Calvin Jongsma

Faculty Work Comprehensive List

This was a banquet talk given at the Twenty-fourth Biennial Conference of the Association of Christians in the Mathematical Sciences held at Dordt University on May 28-June 1, 2024.

It provides an update on the progress being made in the author’s project of producing a second edition of Bibliography of Christianity and Mathematics. This work is now being developed as a Zotero database with a slightly broader focus on Religious Faith and the Mathematical Sciences and with a much expanded list of books, articles, and blog posts on the topic. The author hopes to make the database public sometime …

Setting Up Students For Success: Analysis Of Effectiveness For Mathematics Placement, Jeff Carvell, Jason N.E. Ho, Dave Klanderman, Sarah Klanderman

Faculty Work Comprehensive List

How do we effectively and equitably place students into math classes in a way that provides them the best chance of success? As many higher education institutions veer away from placement based on standardized testing, many departments are seeking placement alternatives that will properly support students. Additionally, math placement determines not only a student’s mathematics courses but also influences their progress in related fields, including physics, chemistry, engineering, and more. This paper will describe three different existing placement systems at each of our liberal arts institutions as well as the affordances and constraints of each approach. Further, we analyze data …

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 …