Mathematics Commons^™

Parameterized Polynomial Systems: Monodromy, Sparse Polynomials, And Solutions, Julianne Barnhart

All Dissertations

The lift of a loop in the base space of a branched cover to the cover induces a permutation of points in a fibre. The monodromy group of the branched cover is the permutation group generated by all such permutations. When loops are restricted to a particular subset of the base space, the corresponding permutation group induced by these loops is the restricted monodromy group. Monodromy groups encode structure and symmetries of many enumerative problems. We describe the relationship between the restricted monodromy group and the monodromy group of the original branched cover. Our main result is a local-to-global property: …

Galois Action And Arithmetic In Algebraic Number Fields, Jared Kettinger

All Dissertations

This dissertation explores the arithmetic of numerous algebraic objects living within an algebraic number field from submonoids of the integers up to localizations of the ring of integers. We begin with a study of factorization in proper orders using an element-theoretic approach. In Chapter 2, by defining a natural generalization of the Davenport constant, we are able to determine the elasticity of certain orders whose integral closure is a unique factorization domain. In Chapter 3, using ideal-theoretic analogues, we are able to significantly broaden the scope of our results and the literature on factorization in orders. In particular, we give …

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.

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 …

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 …

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, …

Fixed Perimeter Analogues Of Some Partition Results, Gabriel Gray, Emily Payne, Holly Swisher, Ren Watson

School of Mathematical & Statistical Sciences Faculty Publications

Euler's partition identity states that the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook length (perimeter) n. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.

Efficient Solvers For Phase Field Models, Raymond Obeng

Open Access Theses & Dissertations

Phase field models provide a versatile framework for describing phase transitions and pattern formation in materials, enabling the study of complex phenomena such as crystallization, grain growth, and defect dynamics. Among these models, the Phase Field Crystal (PFC) equation has gained significant attention due to its ability to capture atomic-scale structures while evolving on diffusive time scales. However, the numerical solution of the PFC equation presents substantial challenges arising from its high-order nature and the large-scale, coupled linear systems generated by discretization.

In particular, standard formulations of the discretized PFC system lead to non-symmetric and often ill-conditioned linear systems, which …

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.

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.

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.

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.

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 …

Cooperation And Conflict: Oswald Veblen And Abraham Flexner, Juan Carvajalino, Della Dumbaugh

Department of Math & Statistics Faculty Publications

In 1929, Caroline Bamberger Fuld and her brother, Louis Bamberger, sold their department stores, Bamberger & Co., at a particularly timely moment. They invested their sizeable fortune in a new academic initiative in America, an institute devoted to research that would go by the name of the Institute for Advanced Study. This paper introduces two colleagues, Abraham Flexner and Oswald Veblen, and explores the cooperation and conflict between them as they worked alongside one another to shape the Institute for Advanced Study and the School of Mathematics within it.

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.

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 …

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 …

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.

The Invisible Shield: Why Our Stomach Doesn’T Digest Itself, Kevon Findley

Mathematics and System Engineering Student Publications

The stomach contains highly acidic gastric fluid with a very low pH.  A thin mucus lining protects the stomach from self-digestion and keeps nearby epithelial cells near a pH 7.  A weakened mucus layer is associated with conditions such as Gastritis and Peptic Ulcer Disease.

How can the protective mucus barrier be maintained and be of stable thickness?

Area Approximation Of Jordan Curves Via The Isoperimetric Inequality And Cauchy-Crofton Formula, Nikita Veselkin, Rashad Kaiyal

Mathematics Colloquium Series

We combine the Isoperimetric Inequality (Dido's Problem) and the Cauchy-Crofton formula to approximate the area enclosed by Jordan curves in the 2D plane. The Cauchy-Crofton formula provides a consistent estimate of a curve's length, which the Isoperimetric Inequality then uses to approximate the enclosed area. We additionally present software that automates the Cauchy-Crofton length computation, making the method practical for real-world use. Empirical testing validates the accuracy of this combined approach, with potential applications in tumor segmentation from MRI scans.

Theory And Simulations Of Delayed Stochastic And Deterministic Models Of Prion Diseases, Gangadhara Boregowda, Omar Sharif, Daniel Gutierrez Iii, Allegra Simmons, Laurent Pujo-Menjouet, Tamer Oraby, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Neurodegenerative diseases (NDs), such as Alzheimer’s, Parkinson’s, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein (PrPC), produced by neurons, misfolds into a toxic form, known as scrapie prion protein (PrPSc). PrPSc induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial …

Massively Parallel Kalman Filtering: Scaling State Estimation Via Cuda Kernels, Joseph Campione, Peter Chim, Hannah Depuydt, William Johnston, Jackson Phillips, Brian White

Mathematics & Computer Science Student Scholarship

This project addresses the challenge of state estimation in real-world conditions by fusing data from multiple sensors using the Kalman Filter. To ensure numerical stability, we use the Joseph Form covariance update, which guarantees valid results but introduces significant computational overhead due to its complexity. To overcome this limitation, we implement a parallelized solution using custom CUDA kernels on a GPU, distributing matrix operations across thousands of threads rather than relying on sequential CPU execution. Through systematic benchmarking across matrix sizes ranging from 4×4 to 4096×4096, we identify a crossover region where GPU performance surpasses CPU efficiency. This work shows …

Creating A Random Number Generator Harnessing: Lava Lamps As A Source Of Randomness, Rachel Barter, Nicolas Guerra, Sarah O'Connor

Mathematics & Computer Science Student Scholarship

No abstract provided.

The Euler Characteristic, Cara Admiraal

SPARK Symposium Presentations

The Euler characteristic is an example of a topological invariant most famously Leonard Euler proved that for any convex polyhedron with $v$ vertices, $f$ faces, and $e$ edges, $v-e+f=2$. In this presentation, we will extend his ideas to define the Euler characteristic for surfaces.