Mathematics Commons^™

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 …

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.

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 …

Monomial Quadratic Identities Of Hecke Eigenforms, Trevor Vilardi

All Dissertations

Duke and Ghate independently studied the question of when it is possible for the product of two eigenforms to be an eigenform. In this dissertation, we take up a generalization of that question, namely is it possible for the product of two eigenforms to be equal to a different product of two eigenforms? Under this formulation, the question becomes closer to one about unique factorization, i.e., how closely do eigenforms work like irreducible elements? Our conjecture is that there are only finitely many cases where the product of two eigenforms is equal to a different product of two eigenforms, and …

Intersectionality And Belonging: Higher Education Mathematics, Ashley Natalie

Mathematics and Statistics

Students with intersecting marginalized identities (such as race, gender, socioeconomic status, disability, or first-generation background) often face unique challenges in advanced mathematics that affect confidence, participation, amounting to a sense of belonging. These experiences remain underrepresented in mathematics education research. This qualitative, narrative-based study examines how these students experience classroom dynamics and belonging in upper-level mathematics courses. Semi-structured interviews will be analyzed thematically to identify patterns related to identity, classroom culture, instructor behavior, and peer interactions. Findings aim to highlight barriers and supportive practices, informing more inclusive teaching strategies and equitable learning environments in advanced mathematics.

Intersectionality And Belonging: Higher Education Mathematics, Ashley Williams

Mathematics and Statistics

Students with intersecting marginalized identities (such as race, gender, socioeconomic status, disability, or first-generation background) often face unique challenges in advanced mathematics that affect confidence, participation, amounting to a sense of belonging. These experiences remain underrepresented in mathematics education research. This qualitative, narrative-based study examines how these students experience classroom dynamics and belonging in upper-level mathematics courses. Semi-structured interviews will be analyzed thematically to identify patterns related to identity, classroom culture, instructor behavior, and peer interactions. Findings aim to highlight barriers and supportive practices, informing more inclusive teaching strategies and equitable learning environments in advanced mathematics.

A New Approach To Generate Combinatorial Patterns In Logical Analysis Of Data And Its Application To Predict College Retention, Salihah Ahmed E. Jaafari

Theses and Dissertations

Student retention and degree completion remain central challenges for higher-education institutions, with significant implications for student success, institutional effectiveness, and public accountability. While advances in predictive analytics have enabled earlier identification of students at risk of withdrawal, many commonly used machine learning approaches suffer from limited interpretability, constraining their practical usefulness for advising, intervention, and policy decision making. This dissertation addresses the problem of predicting student persistence by developing and evaluating optimization based, interpretable classification models within the Logical Analysis of Data (LAD) framework. Building on existing LAD formulations, this research introduces two novel pattern generation models, the Best Term …

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 …

Numerical Methods For Modeling Darcy-Forchheimer Flow In Heterogeneous Porous Media, Nate Mcnair

Open Access Theses & Dissertations

Classic approaches to modeling fluid flow in porous media rely on Darcy's Law, which assumes a linear relationship between volumetric flow rate and the pressure gradient. However, in recent years, applications such as enhanced geothermal systems have highlighted the need to model nonlinear flow behavior, since experimental and observational data show that, once flow velocity exceeds a certain threshold, the relationship between velocity and macroscopic pressure gradient becomes nonlinear. This requires alternative models that more accurately capture the physical behavior of these systems.The Darcy-Forchheimer model provides one such alternative by introducing a nonlinear velocity-pressure relationship. However, this nonlinearity creates additional …

An Ordinal Categorical Data Analysis Using The Stereotype Model Within A Bayesian Framework With A Logistic-Normal Prior, Nathaniel A. Sakyi

Open Access Theses & Dissertations

Ordinal categorical data are pervasive in applied research, yet their analysis is often compromised by modeling strategies that implicitly assume equidistant category spacing or impose restrictive structural constraints. Metric regression models and latent-variable threshold approaches routinely misrepresent ordinal information, leading to biased inference, distorted uncertainty quantification, and loss of structural insight. This paper develops a Bayesian framework for ordinal regression that explicitly accommodates \textbf{unequal spacing among ordered response categories} while preserving ordinal structure and interpretability.

The proposed approach builds on the stereotype regression model, which embeds ordinal categories into a latent one-dimensional continuum through estimable score parameters. While the stereotype …

Singularity-Enriched Neural Networks For Elliptic Problems In Polygonal Domains, Harshini Reddy Kodiganti

Open Access Theses & Dissertations

This thesis develops and analyzes neural-network-based solvers for the Poisson equation on polygonal domains, where re-entrant corners induce reduced solution regularity and challenge standard numerical methods.

Three neural formulations are investigated: Physics-Informed Neural Networks (PINNs), Physics-Informed Extreme Learning Machines (PIELMs), and Rank-Inspired Neural Networks (RINNs). PINNs rely on gradient-based optimization with automatic differentiation, while PIELMs employ randomly initialized hidden features with least-squares training, achieving significantly lower computational cost. RINNs extend this framework through covariance-driven orthogonalization to improve numerical conditioning and stability.

On convex polygonal domains, all three methods are benchmarked against analytical solutions. Both PIELM and RINN consistently achieve higher …

Exploring The Evolution Of Preservice Elementary Teachers' Mathematics Identity And Possible Selves: A Multi-Case Study Approach, Christa R. Mawn

Theses, Dissertations and Culminating Projects

This study explores the nature of preservice elementary teachers’ mathematics identity and possible selves and identifies shifts in their mathematics identity or possible selves over the course of a place value unit during the semester during a course on mathematics content for elementary teachers. Drawing on narrative identity and possible selves theory, this qualitative multi-case study examined the mathematics identity and possible selves of preservice elementary teachers enrolled in a mathematics content course. Course assignments were used as data sources and included written narratives, future-oriented reflections, drawings, and course artifacts. Individual cases were analyzed, and were followed by a cross-cases …

Decision Making For Large-Scale Problems Under Uncertainty And Conflict, Benjamin J. Hamlin

All Dissertations

Large-scale decision-making problems appear in many areas including long-range forecasting such as energy generation forecasting. Many such problems are subject to conflicting objectives and uncertain data, and can be modeled as linear optimization problems. We study novel theoretical results and algorithms for large-scale linear decision problems under conflict and uncertainty. First, we propose a parametric Benders decomposition algorithm for solving large-scale linear optimization problems with multiple objectives or deterministically uncertain objectives. Second, we extend the parametric Benders decomposition to a multi-stage setting, developing a parametric stochastic dual dynamic programming algorithm, which enables decision-making when conflicts and uncertainty have planning impacts …

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 …

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 …

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 …