Mathematics Commons^™

Prism: Priming Relationships In Syntax And Mathematics, Michelle J. Zhu

Honors Scholar Theses

The implementation of sheltered education programs has become increasingly prevalent in schools as new research in second language acquisition emerges. Yet despite this growing attention to ESL instructional practice, far less is known about the cognitive processes that underlie how bilingual students engage with academic content. This study investigates cross-domain structural priming between mathematical and linguistic processing in monolingual and bilingual individuals, focusing on how bilingual experience influences syntactic attachment preferences. Prior research suggests that mathematical and linguistic structures share underlying cognitive representations, and that exposure to structures in one domain can influence processing in another. Additionally, bilingualism has been …

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 …

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.

Course Insight Portfolio: Math 1080, A Gentle Introduction To Infinite Series, Igor Luis Aureliano, Ross George

Publications

This lesson aims to introduce the concept of an infinite series through geometric representations and partial sums to help students develop an intuitive understanding of convergence. A common misconception students may have when first encountering infinite series is the belief that “a sum of infinitely many things must be infinite”. Through the study of a convergent geometric series, students will investigate how the limit of partial sums can produce a finite value, thereby addressing this misconception and developing a more accurate understanding of infinite series.

Course Insight Portfolio: Math 1020 Business Calculus I, Ibrahim Khalilullah, M.G.M. Al Faruque, Jessica Ratovondranto, Leslie Spahr

Publications

This portfolio presents an overview of the student educational context, key course concepts, and some approaches to clarify student misconceptions associated with MATH 1020 (Business Calculus I). The portfolio includes a concept map illustrating the major topics of the course and an overview of common student misconceptions, and lesson plans designed to address those misconceptions through research informed instructional practices.

The concept map highlights the interconnected nature of topics such as functions, limits, continuity, differentiability, derivatives, optimization, graph interpretation, and applications of calculus. Particular attention is given to areas where students commonly experience difficulty, including distinguishing domain, codomain, and range; …

Course Insight Portfolio Math 1040: Precalculus And Introductory Differential Calculus, Jessica Ayers, Morgan Hayes, Meghan Kerrick, Uthman Rasaq, Drishty Singh

Publications

This portfolio includes four primary elements. First is a document titled “Where did they come from? Where are they going?” that provides a description of the types of students who are likely to take the course, followed by the “Course Concept Map” that visually depicts the flow of topics and possible misconceptions in MATH 1040. Additionally, the “Misconceptions Overlay” document lists articles that offer interesting discussion and further insight into the ideas students may bring to MATH 1040 or develop throughout the course. Two lesson plans are included that cover “Continuity and Introduction to Classifying Discontinuities” as well as an …

Course Insight Portfolio: Math 1060 Calculus Of One Variable I, Minh Ha, Dinaniaina Florence Rafidimanantsoa, Preston Sessoms, Rishabh Shukla

Publications

The following portfolio is a representation of the course MATH 1060, Calculus I, taught at Clemson University. The contents can be broken down into 3 major components.

• The first being an analysis of the students who may typically be in the course, the prerequisites required to take the course, and what paths students have after completing the course. This analysis is based upon the course curriculum at Clemson, and as a result will likely not align with that of other universities

. • The second major component is a map of the concepts discussed in MATH 1060 as well …

The Fundamental Group: A Geometric Perspective, Kayla M. Bittenbinder

All NMU Master's Theses

This thesis explores the deep mathematical connection between the flexible, continuous world of topology and the rigid, distance-preserving world of metric geometry. We begin by constructing the fundamental group of a topological space, using the concept of loops and loop homotopy to identify a global topological invariant such as a hole or puncture. We then transition to the geometric realm of metric spaces and isometries, demonstrating how the isometry group of a space algebraically encodes its rigid symmetries. To bridge these two distinct mathematical frameworks, we utilize the construction of the universal covering space. We pull back the metric from …

Combinatorial Statistics Witnessing An Infinite Family Of Congruences For A Sum Of Partition Functions, Jena M. Gregory

Theses and Dissertations

In 2007, Kronholm established The Interval Theorem, infinite families of congruences in arithmetic progression, modulo any prime ��, for ��(��, ��), the function enumerating the partitions of �� into parts whose sizes come from the set {1, 2, … , ��}. In 2022, Eichhorn, Kronholm, and Larsen proved there are combinatorial statistics described in terms of the multiplicities of the part sizes that witness Kronholm’s Interval Theorem. Here, “witness" means given a congruence of the form ��(��, ��) ≡ 0 (mod ��), we can use these statistics to classify the set of partitions of �� into �� equally sized subsets …

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 …