Mathematics Commons^™

Schaper Numbers, Palindrome Partitions, And Symmetric Functions, With Applications To Characters Of The Symmetric Group, Karlee J. Westrem

Dissertations, Master's Theses and Master's Reports

Finding the decomposition numbers for the symmetric group is a difficult problem and has led to many different research directions. For a given Specht module, the Schaper sum formula can be refined with knowledge of the Schaper number for the particular partition to give more precise information about the decomposition numbers. In Chapter 2, we give a combinatorial formula for the Schaper number when the partition has at most two columns. At the end, we provide a conjecture for the Schaper number for the partition of shaper (3^n).

Chapter 3 covers the joint work with my advisor, David Hemmer, and …

The Combinatorics Of Integer Partitions Enumerated By Some Exotic Weights, Hunter Waldron

Dissertations, Master's Theses and Master's Reports

Identities of integer partitions generally state that two dissimilar appearing families of partitions are in fact equinumerous when both are restricted to any fixed size. Euler's theorem is a classic example of such an identity, which equates the number of partitions with odd parts to the number of partitions with distinct parts. Lately, analogs of known partition identities involving weights other than size have begun to attract research interest. This dissertation is an investigation of two such weights. In Chapter 2, we study Schmidt weights, which count only parts with indices belonging to some given subset of the positive integers. …

The Inverse Scattering Transform For The Nonlinear Schrödinger Equation, Ivan Casas-Rocha

Honors Undergraduate Theses

The Nonlinear Schrödinger (NLS) Equation, iψ_t + 1/2 ψ_xx ± |ψ|²ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, …

Prime Factorization And Unit Calculations Of Quadratic Integer Rings, Gabriel F. Roca

Honors Undergraduate Theses

The failure of unique factorization in a ring leads to the investigation of the closest algebraic structure, which are prime ideals. Using generalizations that have helped solve questions such as Fermat's Last Theorem, there is interest to study the elements with a multiplicative inverse (units) via the geometry and arithmetic patterns that arise in quadratic integer rings, since they provide tools for other questions in mathematics, ranging from pure algebra to applications in cryptography, and more. Overall, the following thesis provides a small exposition on the theory of integral domains and some specific calculations.

Solvability Of Stochastic Linear-Quadratic Optimal Control Problems Under Partial Stabilizability Conditions, Al-Sadh Rahman Imadh

Honors Undergraduate Theses

Optimal Control Theory, a branch of Control Theory, is applicable in fields such as engineering, operations research, and economics. Stochastic Optimal Control deals with noisy systems and data using Ito’s formulation. Given a noisy system and a cost functional, the goal is to find a control that will minimize the cost. This thesis focuses on linear quadratic stochastic optimal control, and we explore state equations that are not stabilizable. We first address measurability concerns arising from the semigroup property of the state trajectory. The notions of partial stability and partial stabilizability are introduced, and we formulate their corresponding Lyapunov and …

Quantized Average Agreement Algorithms With Error Correction For Digraphs, Shuaib A. Mughal

Honors Undergraduate Theses

Multi-agent systems have become more and more prevalent as technology increasingly gets integrated into our daily lives. Some of these technological systems are large in size; for example, the smart grid where multiple devices are used to monitor and control different aspects of the energy grid. Another example is a team of autonomous systems deployed for a specific task. When these systems are spatially distributed, an important component of distributed algorithms is the ability for the agents to reach consensus on the global state of the system. Reaching agreement enables the spatially distributed agent make decisions or determine the next …

Abstractions Of The Game “Set”, Martin Grant

Master's Projects

Each card in the game of SET can be represented as a point in Z⁴₃, where Z₃ is the f ield of 3 elements; an in-game position without any SETs can be represented as a cap set. We find the largest cap sets in Zⁿ ₃ for n ≤ 4 and prove their uniqueness. Then, we provide more insight into Ellenberg and Gijswijt’s proof of upper bound for the maximum size of cap set in Fⁿ_q, where F_q is the field of q elements, which they find to be o(c^{n …}

A Bayesian Complex-Valued Latent Variable Model Applied To Functional Magnetic Resonance Imaging, Chase J. Sakitis, D. Andrew Brown, Daniel B. Rowe

Mathematical and Statistical Science Faculty Research and Publications

In linear regression, the coefficients are simple to estimate using the least squares method with a known design matrix for the observed measurements. However, real-world applications may encounter complications such as an unknown design matrix and complex-valued parameters. The design matrix can be estimated from prior information but can potentially cause an inverse problem when multiplying by the transpose as it is generally ill-conditioned. This can be combat by adding regularizers to the model but does not always mitigate the issues. Here, we propose our Bayesian approach to a complex-valued latent variable linear model with an application to functional magnetic …

Betti Numbers Of Generic Ideals, Jason R. Howell

Electronic Theses & Dissertations (2024 - present)

We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe …

Offline Guessing Games With Two Numbers, Justin Sciullo, Lisa Shen, Sarah Zaske

Mathematics Undergraduate Research

In an offline guessing game, there is a player called the Questioner and a player called the Responder. The Responder first picks two distinct numbers from the set {1, 2, 3, . . . , n}. The Questioner then creates a set of questions of the form “How many of your numbers are in the set q_i ⊆ {1, 2, 3, . . . , n}?” and sends them to the Responder who answers them. The Questioner wins if they can guess the Responder’s numbers no matter which numbers the Responder chose. The Responder wins otherwise. We …

Guessing Games To Determine 1 Of Several Secret Numbers, Kyle Mckee, Julia Osmun, Dori Schlutt

Mathematics Undergraduate Research

A guessing game is a two-player game between a Responder and a Questioner. In a guessing game, the Responder is thinking of x ≥ 1 secret numbers in a set S of size n. The Questioner is tasked with asking questions of the form: “How many numbers are in Q ⊆ S?” until the Questioner knows at least 1 of the Responder’s secret numbers. We find a game-winning strategy for the Questioner with minimal possible questions for games with x = 2. We also find lower-bounds for x = 2 and x = 3.

Towards A Baker-Campbell-Hausdorff Theorem In Positive Characteristic, Nikolas A. Koutroulakis

College of Graduate Studies: Theses & Dissertations

We investigate whether there is an analog of the Baker-Campbell-Hausdorff (BCH) theorem for Lie algebras over fields of positive characteristic. We begin by introducing the proof of the BCH formula in characteristic zero. We then introduce the Artin-Hasse exponential and show that it is $p$-integral. Our main result provides sufficient conditions under which a BCH-type formula exists for the Artin-Hasse exponential in positive characteristic. Additionally, we derive a formula for computing an inverse of the Artin-Hasse exponential.

Fairness-Aware And Culturally Adaptive Machine Learning For Predicting Adolescent Substance Use, Stephanie Nworgu

Senior Honors Theses and Projects

Early initiation of substance use during adolescence poses significant risks to long-term health, educational attainment, and social outcomes, making early identification a critical public health priority. Machine learning models have increasingly been used to predict substance use risk; however, many such models do not explicitly examine whether predictive performance differs across demographic groups. This senior project examines the fairness of logistic regression models used to predict first-time alcohol use among adolescents. Using nationally representative survey data from the Youth Risk Behavior Surveillance System (YRBSS), pooled across the 2017, 2019, 2021, and 2023 survey cycles, this study develops logistic regression–based predictive …

Sickle-Cell Genotyping Cost Analysis In Amua And R, Nicholas P. Haley

Senior Honors Theses and Projects

Sickle cell disease (SCD) is a prevalent genetic disorder in the United States, with a significant economic burden due to the high costs of care, especially from chronic red blood cell (RBC) transfusions. These transfusions carry risks such as alloimmunization, which can lead to complications like delayed hemolytic transfusion reactions (DHTRs), further increasing healthcare costs in addition to increasing suGering. This study aims to aid in the evaluation of cost-eGective strategies for SCD management using Markov models, currently within TreeAge Pro software, which is proprietary. A decision model was adapted from the Kacker et al. (2013) study, which analyzed the …

Theoretical Foundations And Applied Performance Of Periodicity-Aware Imputation: Variable Bandpass Block Bootstrap Methods For Incomplete Time Series, Asmaa Ahmad

Electronic Theses & Dissertations (2024 - present)

Time series data are prevalent across a wide range of disciplines, including health surveillance, public policy, and environmental monitoring. In the presence of underlying cyclical patterns, the integrity of time series analysis depends critically on the ability to detect, model, and impute structured missing data without compromising the temporal structure. This dissertation introduces and validates a novel imputation framework that integrates the Variable Bandpass Periodic Block Bootstrap (VBPBB) into multiple imputation procedures, improving the accuracy, robustness, and interpretability of time series models under high rates of missingness and noise. The overarching goal of this dissertation was to develop and evaluate …

Integrating Sentiment Analysis In Predictive Models: A Comparative Study On Game Popularity On Steam, Khaleefa Alhemeiri

CMC Senior Theses

Over the past decades, the gaming industry has managed to evolve into a multi-billion-dollar enterprise. Gaming platforms such as Steam foster unprecedented amounts of engagement among players worldwide daily. In this thesis, we investigate the effect of incorporating sentiment-driven metrics, specifically YouTube view counts and positive reviews, into predictive models for game popularity. In addition, by comparing our linear regression sentiment-based approach to the Bayesian hierarchical folded normal model used by De Luisa et al. (2021), we can understand the many differences, strengths, and limitations of each methodology. In our thesis, we focus on three games. Each is of varying …

Analyzing Political Sentiment On Micro-Blogging Data: A Lexicon And Machine Learning Approach To The 2024 U.S. Presidential Election, Ava Grey

CMC Senior Theses

This paper explores the trends in sentiment towards U.S. presidential candidates Kamala Harris and Donald Trump through micro-blogging social media text during the five months leading up to the election. Two datasets of varying sizes and origins were used to contextualize and validate analysis findings. The analyses include both a lexicon-based approach and a machine learning predictive method. Common sentiment analysis techniques like term frequency, term frequency inverse, various lexicons, and n-grams were utilized during the lexicon approach. During the modeling, a random forest was utilized in addition to the methods used during the lexicon approach. Results showed that overall …

Gorenstein Flat Preenvelopes Over Coherent Rings, Alec Moore

College of Graduate Studies: Theses & Dissertations

The existence of precovers and preenvelopes of Gorenstein flat modules is of great interest in the field of Gorenstein homological algebra. We give a sufficient condition in order for the class of Gorenstein flat modules to be preenveloping. More precisely, we prove that if the ring R is coherent such that every injective module has finite flat dimension, then every R-module has a Gorenstein flat preenvelope.

Golden Spirals Everywhere? Solutions For Fermi Questions, January 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It addresses the misconception that golden rectangles and spirals can be found in various natural and man-made structures, emphasizing the importance of understanding the properties of logarithmic spirals. The article provides mathematical explanations and solutions for questions related to pitch angles and self-similarity in logarithmic spirals, using examples like the nautilus shell and an ammonite-like stone. It concludes by referencing additional sources for further exploration of the golden ratio and golden spiral myths.

Enhancing The Accuracy Of Image Classification For Degenerative Brain Diseases With Cnn Ensemble Models Using Mel-Spectrograms, Sang-Ha Sung, Michael Pokojovy, Do-Young Kang, Woo-Yong Bae, Yeon-Jae Hong, Sangjin Kim

Mathematics & Statistics Faculty Publications

Alzheimer’s disease (AD) and Parkinson’s disease (PD) are prevalent neurodegenerative disorders among the elderly, leading to cognitive decline and motor impairments. As the population ages, the prevalence of these neurodegenerative disorders is increasing, providing motivation for active research in this area. However, most studies are conducted using brain imaging, with relatively few studies utilizing voice data. Using voice data offers advantages in accessibility compared to brain imaging analysis. This study introduces a novel ensemble-based classification model that utilizes Mel spectrograms and Convolutional Neural Networks (CNNs) to distinguish between healthy individuals (NM), AD, and PD patients. A total of 700 voice …