Mathematics Commons^™

Bounding The Average Kissing Number, Including New Bounds For Three-Dimensional Binary Sphere Packings, Mark William Bockhaus

Theses and Dissertations

The average kissing number is defined as the supremum over all sphere packings of the value: two times the number of tangencies divided by the number balls in the packing. In this paper, we present a survey of the literature about bounding the average kissing number, beginning with the first non-trivial results, through the most up-to-date bounds. We then turn our focus to binary sphere packings: those which contain spheres of two different radii. We improve upon known bounds for the average kissing number for binary sphere packings in three-dimensions and find exact bounds for many packings.

Bayesian Change-Point Detection In Stock And Cryptocurrency Markets Using Shrinkage Priors, Yosalin Sanchez

Theses and Dissertations

Financial markets often undergo abrupt structural changes driven by political, economic, and geopolitical events, leading to substantial volatility. Detecting such change-points is crucial for identifying structural breaks, improving risk management, and enhancing forecasting performance in financial time series. This study proposes a Bayesian change-point detection framework that incorporates both the t-shrinkage prior and the Horseshoe shrinkage prior. These priors enforce strong regularization on successive differences in mean parameters, enabling the identification of piecewise constant structures in time series data. Posterior inference is conducted using Markov Chain Monte Carlo (MCMC) methods, specifically a Gibbs sampling algorithm, which iteratively samples from the …

On The Structure Of The Homotopy Lie Algebra Of Local Rings, Dawson M. Strong

Theses and Dissertations

This thesis investigates the construction and homological properties of the homotopy Lie algebra π(R) of a commutative local ring (R,m,k). Drawing upon the theoretical framework of differential graded (DG) algebras, we first establish the theory of minimal free resolutions and other standard topics in homological algebra. The core of this work details the iterative construction of the acyclic closure R⟨Y⟩ of k over R, which is achieved by the systematic adjunction of exterior and divided power variables to eliminate cycles in homology. We demonstrate that this acyclic closure serves as a minimal free resolution and provides the means to define …

Rainbow Dominating Sets Of Graphs, Samuel L. Powell

All Graduate Theses and Dissertations, Fall 2023 to Present

The content of this thesis may be compared to a stack of plates with a common design that are broken, one at a time. If the plates are broken into large enough pieces you would be able to choose one fragment from each broken plate to discover the entire design that the plates share. For example, if you were still missing the center of the design after taking a piece from some plates, you could look for the piece of the next broken plate with the region in question.

We study the question of how many broken plates might guarantee …

Gradient Based Optimization Methods For Robust Learning And Biomedical Signal Modeling, Jarrod Mau

All Graduate Theses and Dissertations, Fall 2023 to Present

This dissertation explores how modern artificial intelligence techniques can be used to better understand complex biological data. Specifically, it develops new machine learning based methods and applies them to two important biomedical problems: analyzing brain signals and studying protein behavior.

The first part of the work introduces a new machine learning approach designed to improve how computers classify structured data. Traditional neural networks are powerful but can sometimes generalize poorly. This research proposes a method that combines the flexibility of neural networks with the reliability of ensemble techniques, leading to more robust and accurate predictions across different types of datasets. …

When A Sum Of Cubes Equals The Square Of The Sum, Jessica M. Aguilar

Electronic Theses, Projects, and Dissertations

This thesis investigates extensions and structural generalizations of the classical identity \[ \sum_{k=1}^{n} k^3 = \left( \sum_{k=1}^{n} k \right)^2, \] traditionally attributed to Nicomachus of Gerasa. Despite its simple look, this cube - square identity reveals connections between combinatorics, multiplicative number theory, and Diophantine equations.

We begin by presenting an expanded combinatorial proof of the identity based on Stein’s rectangle - counting argument, clarifying the geometric structure underlying the formula. We then establish a multiplicative analogue using Euler’s divisor-counting function \( \tau(n) \), proving that \[ \sum_{d \mid n} \tau(d)^3 = \left( \sum_{d \mid n} \tau(d) \right)^2, \] thereby extending …

Examining Student Practices In A Technology-Mediated Math Classroom, Bradley T. Dodd

Electronic Theses, Projects, and Dissertations

It is important for math educators to make sense of student thinking in the classroom. Without opportunities to practice this skill math educators struggle to improve, particularly when students are engaging in mathematics using technology. As such, there is a need for video artifacts of students engaging with mathematics using technology for use in professional development activities (Lovett 2020). In accordance with Lovett et al.'s (2020) design principles for examining student practices in a technology-mediated environment, I carried out a study to determine whether these artifacts of student work could be created working with college undergraduates as participants. Students engaged …

Constraint-Aware Metaheuristic Optimization For Experimental Design, Benjamin N. Fuller

All Graduate Theses and Dissertations, Fall 2023 to Present

Designing experiments becomes much more challenging when many variables and strict constraints are involved, as is common in modern science and engineering. This thesis introduces a new computational and mathematical framework that efficiently searches for optimal experiments in complex, high-dimensional spaces where traditional methods fail. By combining geometric techniques with flexible optimization algorithms like particle swarm optimization, our methods handle difficult constraints while scaling to real-world problems. Built in the high-performance Julia programming language and released as open-source software, this work bridges advanced theory with practical tools, offering researchers a powerful and accessible way to design better experiments under realistic …

Developmental Mathematics As A Potential Barrier To Degree Completion In Community Colleges, Cynthia Bernice Fletcher

Graduate Theses and Dissertations

Developmental mathematics is often seen as a barrier to student progression in community colleges, especially for students pursuing Associate of Science degrees requiring math coursework. This quantitative, ex post facto, non-experimental study examined how developmental mathematics factors predicted Associate of Science degree completion at a two-year community college in the West South-Central United States. Specifically, it assessed how academic performance in developmental mathematics, placement method, math pathway, and number of developmental math courses related to Associate of Science degree completion, as well as differences across student subgroups. Archival institutional data were used for first-time-in-college students across four cohorts: 2018, 2019, …

On The Algebraicity Of The Tic-Tac-Toe Matroid And Homogeneous Mixed Bowtie Systems, Benjamin R. Allen

Electronic Theses and Dissertations

This thesis is presented in two parts. First, we explore whether the class of algebraic matroids is closed under duality, a decades-old open question. We consider the Tic-Tac-Toe matroid as a potential candidate to answer the open question. The Tic-Tac-Toe matroid is known to satisfy many of the necessary conditions for a matroid to be algebraic and has a non-algebraic dual.  Second, we focus on decompositions of the complete mixed graph into mixed bowties. A complete mixed graph has between every pair of vertices an undirected edge and antiparallel arcs. A mixed bowtie is a graph consisting of two 3-cycles …

Analysis Of Collective Behavior In Living And Nonliving Systems, Kaitlyn Cohan

Theses, Dissertations and Culminating Projects

This thesis aims at understanding the phenomenon of of self-organization in complex dissipative systems, living and nonliving. Dissipative systems are characterized by their search for energy, interactions with their surroundings and the production of entropy, all of which result in the creation of stable structures or patterns, which persist as long as the initial environmental conditions are maintained. The two specific models that we chose to study here are (a) Futbol (or Soccer) and (b) a chemical system involving free-floating menthol crystals floating on a fluid surface to represent nonliving systems. Using experiments and mathematical models, we will try to …

The Tones And Frequencies Of A Glockenspiel, Edward Czudak

Theses, Dissertations and Culminating Projects

In this thesis, I will analyze the tones and frequencies of a glockenspiel based on the results of Warburton’s 1954 article, “The Vibration Of Rectangular Plates,” [12]. I will be measuring and modeling the frequencies of a glockenspiel and connect them with the analytical and expected numerical solutions. The solutions will be based on the plate equation along with the boundary conditions of a rectangle free on all edges. I will look at the high modes of vibration for the bars and their corresponding overtones in the acoustic spectrum.

Volume 17, Christian O’Neill, Kyara Greene, Savva Sidorov, Laura Bisaillon, Luke Clemmer, Hannah Gordon, Kitt Benson, Taylor Blount, Rachel Danzitz, Nicholas Duellman, Chase Gionis, Hima Fernando, Seth Franzyshen, Onyx Gonzalez, Bryan Lin, Samantha Start, Ysabel Wells, Maggie Duncan

Incite: The Journal of Undergraduate Scholarship

Introduction Dr. Amorette Barber, Director, Office of Student Research

From the Editor Dr. Hannah Dudley-Shotwell

Cover Artist’s Statement Maggie Duncan

On Mentoring Dr. Yulia Uryadova

Ukrainian Resistance in the Face of Russification: Nestor Makhno and Anarchism

by Christian O’Neill

Life Vest by Kyara Greene

Isolation and 16S rRNA Identification of Bacteria from Fire Department Connection Pipe by Savva Sidorov

The Effectiveness of Planned Exercise in Reducing ADHD Symptoms in Children by Laura Bisaillon & Luke Clemmer

Linguistic Analysis on Confidence and Communication Strategies with Disparities Between Sign Fluency and Hearing Impairment by Hannah Gordon

Freedmen in Indian Territory by Kitt …

Analyzing The Evolution Of Science: Topological Cycles And Community Detection In Knowledge Networks, Frances C. Mcconnell

Mathematics, Statistics, and Computer Science Honors Projects

How scientific knowledge grows and organizes itself is a central question in the study of science. This thesis uses tools from topology and network science to detect and characterize knowledge gaps—places in a field’s literature where related concepts do not co-occur. We develop a metric to quantify the degree of interdisciplinarity of each gap, using the community structure of the underlying network as a proxy for subfields. Across a wide range of fields, gaps reliably span multiple subfields and evolve in recognizable temporal patterns, highlighting new insights into how scientific fields are structured and their stage of development.

From The Hopf Fibration To Instantons: Geometry In Gauge Theory, Emily D. Wessman

Undergraduate Honors Capstone Projects

This paper explores the relationship between topology, differential geometry, and gauge theory through the study of Yang-Mills theory and its solutions, known as instantons. Beginning with the Hopf fibration, we show how principal fiber bundles encode topological information and appear in physical contexts such as electromagnetism. In particular, we consider how the fibration of S³ over CP¹ ≅ S² represents the Dirac magnetic monopole, and how the Chern number associated with the bundle is exactly the winding number for the monopole.

We then develop the framework of gauge theory, focusing on connections on principal SU(2) bundles …

Fractsynth: Exploring Glitch Timbres With Chaos Theory, Aidan Roach

The Transdisciplinary STEAM+ Journal

In this paper, I explore how chaos theory can be used to design a new kind of synthesizer with the primary focus of producing glitchy, unpredictable sounds. Glitch music embraces abstract sound design, malfunctioning electronics, and randomness as the main compositional elements. However, most synthesizers rely on stable, repetitive oscillators that often sound too controlled. To challenge this, I developed FractSynth, a real-time synthesizer that uses chaotic attractors–including the Logistic Map, Henon Map, and Lorenz System–as modulation sources for frequency, amplitude, and tone. The software also features real-time Lyapunov Exponent Tracking, which gives users a direct visual of how …

Interactions Of Dance And Mathematics: A Closer Look At Mathematical Sequences As Scores For Choreography From A Choreographic Lens, Caroline Wolfe

The Transdisciplinary STEAM+ Journal

This paper investigates the embodiment of infinite mathematical sequences through concert contemporary dance, examining how choreography can serve both as an artistic and analytical tool for exploring numerical patterns. Grounded in interdisciplinary literature on mathematical visualization in choreography, the study centers on two original choreographic works presented and performed at Missouri State University: Golden Ratio Sequence (Spring 2025), which maps the Fibonacci sequence and the natural applications of the Golden Ratio onto a spiral floor pattern, and Tetrahedral Numbers (Fall 2024), which employs an accumulation score to embody three-dimensional number growth through layered movement motifs and body created tetrahedra. The …

Getting Your Master's, Is It Worth It?, Vy Le, Madyson Schreifels

Celebrating Scholarship and Creativity Day (2018-)

This project uses a differential equation model to evaluate whether earning a master’s degree is financially worthwhile. Using UC Berkeley’s Master of Computer Science program as a case study, we modeled student loan repayment and compared long-term earnings between bachelor’s and master’s degree holders. The model predicts monthly loan payments and estimates the break-even point where the higher salary from the master’s degree outweighs the total investment cost. Results suggest that the degree becomes financially beneficial after approximately 15 years.

Bibliography Of Religious Faith And The Mathematical Sciences, Calvin Jongsma

Faculty Work Comprehensive List

This Bibliography is offered as a helpful resource for anyone who wishes to thoughtfully explore the connections between religious faith and the mathematical sciences. Searching the database for a topic of interest will bring up items with that focus. An entry’s attached PDF (when publicly available) can be opened and read while in the database. Alternatively, each item contains a URL/web link to a location where one can either read the item or retrieve information about how to obtain a copy of it.

An Analysis Of A Family Of Root Finding Methods, Arabella Fuzak

Celebrating Scholarship and Creativity Day (2018-)

This thesis analyzes the behavior of a family of iterative root-finding methods, the Hansen-Patrick Family, which has a parameter, alpha, to create methods such as Newton’s methods, Halley’s method, and Euler’s method. By varying the parameter alpha, with both real and complex values, this project examines how the parameter can change convergence, divergence, and stability for different functions. This is shown by basin maps and Mandelbrot-like sets to visualize this behavior and classify points based on whether they converge to a root, diverge to infinity, or remain bounded without converging.

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.

Lebesgue’S Theory Of Integration: An Introduction To Modern Integration Theory, Nathanael C. Krug

Senior Honors Theses

The depth and richness of integration theory far surpasses the introductory material presented in an elementary calculus classroom. The work of Bernhard Riemann and Henri Lebesgue demonstrates just a small sample of the richness of the field of analysis. In formulating and contrasting the Riemann and Lebesgue integrals, students can gain an enriched and well-rounded introduction to integration theory. This not only deepens understanding and love for previously learned material, but also enables further study within the fields of analysis and measure theory. An introductory primer to integration theory equips students with the tools needed to continue their exploration of …

Detection, Mapping, And Spraying Of Carolina Redroots In Cranberry Bogs Using Ai And Autonomous Drones, Duwon Ham, Bishal Neupane, Thien Ba Nguyen, Thanh Nguyen, Hieu D. Nguyen, Thierry Besancon

STEM Student Research Symposium Posters

Use artificial intelligent and autonomous drones to automatically detect Carolina Redroots in cranberry bogs, create density maps of the weed, and perform spot spraying.

Finiteness Of Torsion Loci And Normal Functions Arising From Cycles On Hadamard Products, Devin Akman

Arts & Sciences Graduate Student Theses and Dissertations

The unifying theme of this dissertation is normal functions. In the first chapter, we study an invariant of knot exteriors and similar manifolds called the A-polynomial through the lens of vanishing loci of normal functions. Using a special case of the Zilber-Pink conjecture, proven in the second chapter, we show that only finitely many irreducible Laurent polynomials of bounded overgenus appear as factors of A-polynomials. In the third chapter, we construct normal functions from hypergeometric variations of Hodge structure and compute their regulators. Finally, we determine under which conditions incomplete motivic cohomology cycles on families complete to cycles on their …

Complexity Of The Zero Set Of A Matrix Schubert Ideal, Cesar Julian Meza

Arts & Sciences Graduate Student Theses and Dissertations

T-varieties are normal varieties equipped with an action of an algebraic torus T. When the action is effective, the complexity of a T-variety X is dim(X)−dim(T). Matrix Schubert varieties, introduced by Fulton in 1992, are T-varieties consisting of n×n matrices satisfying certain constraints on the ranks of their submatrices. In this dissertation, we focus on the complexity of certain torus-fixed affine subvarieties of matrix Schubert varieties. Concretely, given a matrix Schubert variety X_w where w∈S_n, we study the complexity of Y_w obtained by the decomposition X_w = Y_w ×C^k with k as large as possible. Building on results by Escobar–Mészáros …

Rosenzweig's Elements And Universal History, Martin Zwick

Complex Systems Faculty Publications and Presentations

This paper uses Rosenzweig’s conception of the three elements of God, World, and Human from The Star of Redemption in a model of universal history. The model also has some relation to Rosenzweig’s geopolitical essay, Globus, which is very different from the meta-historical Star. Based on a systems-theoretic schema of events and processes, the model views human history in terms of three linked processes: a primary process labeled “World” – the origin and development of human society embedded in nature, a secondary process labeled “God” – the origin and development of the Axial religions and philosophies, and a …

Boolean Rank Via Monomial Ideals And Neural Ideals, Juliann Marie Geraci

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

This dissertation develops new connections between Boolean matrix factorization, combinatorial neural codes, and commutative algebra. The central goal is to understand how structural and algebraic invariants can be used to measure and bound complexity in discrete data. We begin by studying the Boolean matrix factorization (BMF) problem, which seeks to express a binary matrix as a product over the Boolean semiring with minimal inner dimension, known as the Boolean rank. By interpreting binary matrices as bipartite graphs, we relate Boolean rank to biclique covers and introduce algebraic techniques to study this quantity. We associate to a matrix its edge ideal …

Halo: High Autonomous Low-Swap Operations, Sloan Hatter, Blake Gisclair

Mathematics and System Engineering Student Publications

Orbital object detection is a vital aspect of space operations, particularly for identifying satellite components. Convolutional Neural Networks (CNNs) are typically used for such operations by running onboard models directly on satellite systems. However, a new neural network architecture, known as Vision Transformers (ViTs), have shown greater effectiveness due to their ability to capture global context. One main issue of deploying systems with such capabilities is resource allocation. One solution is to run models on a Low-SWaP system; however, this results in inefficient performance. To enable efficient ViT operations on Low-SWaP systems, the model must be scaled down through quantization, …

Stochastic Network Resilience Under Random Failures, Blake Gisclair

Mathematics and System Engineering Student Publications

This project investigates threshold-crossing probabilities in stochastic networks. Using probabilistic modeling and transform-based analytical methods, the work derives expressions that characterize when cumulative losses exceed prescribed limits with the goal of providing insight into the relationship between local random behavior and global network risk.

Determining Material Transport Method For Moon Colony Using Differential Equations, Scott Meeson, Anthonie Page, Matas Vaitkevicius

Mathematics and System Engineering Student Publications

NASA Artemis Missions: NASA’s Artemis program marks a fundamental shift from short-term lunar exploration to sustained, permanent colonization.