Mathematics Commons^™

Automorphism Groups Of N-Graded Vertex Algebras Associated With Cyclic Leibniz Algebras With Small Dimensions, Alexander M. Keene

Theses and Dissertations

A fundamental problem in the study of vertex (operator) algebras V is the determination of the group of (grading-preserving) N-graded vertex algebras associated with cyclic Leibniz algebras of dimensions 2 and 3 that were classified by C. Barnes, E. Martin, J. Service, and G. Yamskulna in [1].

In each case examined, investigation of the automorphism group relies on the key fact that the action of an automorphism σ is determined solely by its value at a single basis element b. Furthermore, we employ a result in [19] by H. Li and G. Yamskulna which states that we can determine the …

Centralized Deep Reinforcement Learning For Homogeneous Multi-Component Maintenance Optimization, Joseph W. Wittrock

Theses and Dissertations

This thesis explores an application of reinforcement learning (RL) in maintenance optimization. Recent advances in hardware-accelerated computation and deep learning have made RL a powerful tool for solving optimization problems which are too complex for traditional methods. Maintenance optimization involves improving the efficiency and effectiveness of maintenance activities through data-driven approaches, ultimately reducing costs and increasing asset availability. Making informed maintenance decisions is crucial to long-term sustainability.

A desirable maintenance policy maximizes a utility signal while minimizing the cost of maintenance. Techniques in sequential decision making such as dynamic programming (DP) and RL have found success in optimizing these maintenance …

Oer Ancient Egyptian Numerals Activity, Cynthia Huffman Ph.D.

Open Educational Resources - Math

Ancient Egypt captures our imagination not just because of its pyramids, but because of its incredible staying power. This civilization thrived for over 3,000 years, influencing cultures across the globe—including the very roots of mathematics. As Egyptologist Bob Brier puts it, “No civilization lasted so long, contributed so much, or repeatedly amazed as did ancient Egypt.”

Ready to crack the code of Ancient Egypt and the pharoahs? In this activity, you will learn to decipher Egyptian numerals by examining real photos from temples and museums. Master this skill, and next time you visit a museum exhibit on ancient Egypt, you …

Hyper B-Ary Representations Of A Number, Mingway Wang

Masters Theses

In this work, we summarize combinatorial, analytical, and computational properties of Stern's diatomic sequence, and do the same for a generalization of a combinatorial property of this sequence to higher bases. We then construct special ``bit conversion'' functions $g_{b}$ that allows us to find a semi-explicit formula for the generalized sequences. Lastly, we analyze some of the properties of these $g_{b}$ functions.

Cost-Effective Strategies For Feral Swine Control: Exploring Trapping Equipment Cooperatives, Phil Kenkel, Riza Radmehr, Rodney Holcomb, Mckenzie Boyce

Human–Wildlife Interactions

Feral swine (Sus scrofa) in the United States have become a growing concern, with a population of ≥6 million and causing annual damages of ≥$1.5 billion USD. Because of their high reproductive capacity, effective control methods are crucial to reducing or slowing their population growth. While remotely monitored and triggered traps have proven to be an effective control strategy, the cost of sophisticated trapping equipment, which can cost ≥$7,000 USD, is often cost prohibitive for many landowners. Despite the pressing need to address the challenge of making effective feral swine control methods more affordable, there are few studies …

Machine Learning Methods For Intrusion Detection And Response In Network Security, Ayomide Oyemaja

College of Graduate Studies: Theses & Dissertations

Intrusion Detection Systems (IDS) play a crucial role in computer network security by identifying malicious activities and potential cyberattacks. This thesis combines machine learning and cybersecurity by applying Reinforcement Learning (RL) in intrusion detection and response using the NSL-KDD dataset.

We designed and implemented a Q-learning framework where an agent learns to classify network traffic over time by interacting with the environment and receiving rewards based on detection accuracy. We also look at the importance of feature selection and classification techniques and how effective they are in improving model performance, reducing the complexity of computation, and producing more desirable results. …

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 …

Bayesian Networks For Safety-Critical Systems, Joseph Mietkiewicz

Theses

This thesis addresses a operational challenge in modern industrial operations: the increasing complexity of systems and the consequent cognitive burden on operators. As industrial technologies advance, the human-computer interface has become the primary conduit for information flow, playing a pivotal role in operational decision-making. However, the proliferation of data often leads to information overload, potentially compromising rather than enhancing operator performance. This research explores an approach to this pressing issue through the application of Bayesian networks as decision support systems in safety- critical scenarios. Our study employs a multi-faceted approach, combining theoretical modeling with empirical testing. Through collaboration with industry …

A New G Family: Properties, Characterizations, Different Estimation Methods And Port-Var Analysis For U.K. Insurance Claims And U.S. House Prices Data Sets, Ahmad M. Aboalkhair, Gholamhossein Hamedani, Nazar Ali Ahmed, Mohamed Ibrahim, Mohammad A. Zayed, Haitham M. Yousof

Mathematical and Statistical Science Faculty Research and Publications

This paper introduces a new class of probability distributions, termed the generated log exponentiated polynomial (GLEP) family, designed to enhance flexibility in modeling complex real financial data. The proposed family is constructed through a novel cumulative distribution function that combines logarithmic and exponentiated polynomial structures, allowing for rich distributional shapes and tail behaviors. We present comprehensive mathematical properties, including useful series expansions for the density, cumulative, and quantile functions, which facilitate the derivation of moments, generating functions, and order statistics. Characterization results based on the reverse hazard function and conditional expectations are established. The model parameters are estimated using various …

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 …

Bayesian Merged Utilization Of Grappa And Sense (Bmugs) For In-Plane Accelerated Reconstruction Increases Fmri Detection Power, Chase J. Sakitis, Daniel B. Rowe

Mathematical and Statistical Science Faculty Research and Publications

In fMRI, capturing brain activity during a task is dependent on how quickly the k-space arrays for each volume image are obtained. Acquiring the full k-space arrays can take a considerable amount of time. Under-sampling k-space reduces the acquisition time, but results in aliased, or “folded,” images after applying the inverse Fourier transform (IFT). GeneRalized Autocalibrating Partial Parallel Acquisition (GRAPPA) and SENSitivity Encoding (SENSE) are parallel imaging techniques that yield reconstructed images from subsampled arrays of k-space. With GRAPPA operating in the spatial frequency domain and SENSE in image space, these techniques have been separate but can …

Isochronous And Period-Doubling Diagrams For Symplectic Maps Of The Plane, T. Zolkin, S. Nagaitsev, I. Morozov, S. Kladov, Y. -K. Kim

Physics Faculty Publications

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more …

Theoretical And Applied Considerations In Dynamical Sampling, Brendan Miller

Graduate Research Theses & Dissertations

Dynamical sampling is an area of mathematics which studies the theory and applications of signals evolving through time, as well as the stable recovery of said signals from spatiotemporal measurements. This manuscript is devoted to problems which arise as theoretical considerations in signal recovery and as applications of using space-time measurements as opposed to static measurements.

For the former, we consider the problem of recovering an evolving signal via space-time measurements at irregular time intervals. To do so, we generalize the classic Kadec $1/4$ theorem using techniques of Banach $L^1(\R)$-modules. For the latter, we show how dynamical samples can be …

Methods For Finding High Quality And Optimal Solutions Of Binary Quadratic Optimization Problems, James Haas

Graduate Research Theses & Dissertations

BiqAlps is an extension of the BiqCrunch project that leverages its bounding procedures within a computer cluster environment. The primary goal of this work was to investigate whether distributing the branch-and-bound tree search across multiple CPUs could improve solve times. While some problem instances demonstrated speedups of up to fivefold, others showed no measurable improvement, revealing that parallelization benefits are problem-dependent.

Beyond parallelization, BiqAlps introduces enhanced heuristics for generating high-quality initial solutions and refining existing ones. Additionally, the framework explores the impact of propagating triangle inequality cuts to child nodes during search, with the aim of accelerating bounding processes and …

Cubic Blaschke Products, Alexandra Hill

Graduate Research Theses & Dissertations

It is well known that Blaschke products can be classified as elliptic, hyperbolic, or parabolic. Based on the classification of the Blaschke product, we know that the Julia set is either the entire unit circle or a Cantor subset of the unit circle. For unicritical Blaschke products of degree d, it has been shown that the parabolic functions are parameterized by an epicycloid with d-1 cusps, with the interior of the epicycloid giving rise to the elliptic parameters, and the relative complement of the epicycloid within the unit disk providing the hyperbolic parameters. In this dissertation, we study in more …

Green's Functions And Fixed Points Of Quasiregular Maps, Mark Broderius

Graduate Research Theses & Dissertations

This dissertation is on the dynamics of planar quasiregular maps of degree 2. These maps have three parameters. One of these parameters determines the distortion, one of them determines the orientation, and the last one is a shift parameter. Our first objective is to construct a new dynamical version of Green's functions for these maps. These functions provide information about the escaping sets of our quasiregular maps and are used to prove topological results about them. We show that the boundary of the escaping set is either connected or consists of infinitely many components. Additionally, we prove that if one …

Facial Reduction Of Semidefinite Relaxations Of Binary Quadratic Problems, Ian Sugrue

Graduate Research Theses & Dissertations

A standard technique for bounding a combinatorial optimization problem is to form a semidefinite relaxation of the problem by "lifting" the problem into a higher dimensional space of symmetric matrices. For some combinatorial problems such as maximum k-cluster, the resulting semidefinite relaxation is not strictly feasible. In the context of the maximum k-cluster problem, we present an equivalent, strictly feasible semidefinite relaxation by way of the dimensionality reduction technique known as facial reduction. We provide a recipe to form this semidefinite relaxation directly, bypassing the lifting and facial reduction steps entirely. We then present our work on generalizing this result, …

Frobenius And Commutative Pseudomonoids In The Bicategory Of Spans, Ivan Contreras, Rajan Amit Mehta, Walker H. Stern

Mathematics Sciences: Faculty Publications

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in Span correspond, respectively, to paracyclic sets and Γ-sets satisfying the 2-Segal conditions. These results connect closely with work of the third author on A∞ algebras in ∞-categories of spans, as well as the growing body of work on higher Segal objects. Because …

Quantification Of Changes With Combined Shape Mode Analysis And Swimming Simulations, Kelli E. Gutierrez, Becca Thomases, Paulo E. Arratia, Robert D. Guy

Mathematics Sciences: Faculty Publications

Many different microswimmers propel themselves using flagella that beat periodically. The shape of the flagellar beat and swimming speed have been observed to change with fluid rheology. We quantify changes in the flagellar waveforms of Chlamydomonas reinhardtii in response to changes in fluid viscosity using (i) shape mode analysis and (ii) a full swimmer simulation to analyse how shape changes affect the swimming speed and to explore the dimensionality of the shape space. By decomposing the gait into the time‑independent mean shape and the time‑varying stroke, we find that the flagellar mean shape substantially changes in response to viscosity, while …

Characterizations Of Stability Via Morse Limit Sets, Jacob D. Garcia

Mathematics Sciences: Faculty Publications

Subgroup stability is a strong notion of quasiconvexity that generalizes convex cocompactness in a variety of settings. In this paper, we characterize stability of a subgroup by properties of its limit set on the Morse boundary. Given H < G, both finitely generated, H is stable exactly when all the limit points of H are conical, or equivalently when all the limit points of H are horospherical, as long as the limit set of H is a compact subset of the Morse boundary for G We also demonstrate an application of these results in the settings of the mapping class …

Topological Symmetry Groups Of The Generalized Petersen Graphs, Angelynn Álvarez, Erica Flapan, Mark Hunnell, John Hutchens, Emille Lawrence, Paul Lewis, Candice Price, Ruth Vanderpool

Mathematics Sciences: Faculty Publications

The topological symmetry group TSG(Γ) of an embedded graph Γ in S³ is the subgroup of the automorphism group of the graph which is induced by homeomorphisms of (S³,Γ). If we restrict to orientation-preserving homeomorphisms then we obtain the orientation-preserving topological symmetry group TSG₊(Γ). In this paper, we determine all groups that can beTSG(Γ) or TSG₊(Γ) for some embedding Γ of a generalized Petersen graph other than the exceptional graphs P(12,5) and P(24,5).

Circadian Variation In Mgmt Promoter Methylation And Expression Predicts Sensitivity To Temozolomide In Glioblastoma, Maria F. Gonzalez-Aponte, Yitong Huang, William A. Leidig, Tatiana Simon, Omar H. Butt, Marc D. Ruben, Albert H. Kim, Joshua B. Rubin, Erik D. Herzog, Olivia J. Walch

Mathematics Sciences: Faculty Publications

Purpose Recent studies show that glioblastoma (GBM) is more sensitive to temozolomide (TMZ) in the morning. In cells, inhibiting O6-Methylguanine-DNA-Methyltransferase (MGMT) abolished time-dependent TMZ efficacy, suggesting that circadian regulation of this DNA repair enzyme underlies daily TMZ sensitivity. Here, we tested the hypotheses that MGMT promoter methylation and protein abundance vary with time-of-day in GBM, resulting in daily rhythms in TMZ efficacy.

Methods We assessed daily rhythms in MGMT promoter methylation in GBM in vitro and retrospectively analyzed MGMT methylation status in human GBM biopsies collected at different times of day. Next, we measured MGMT and BMAL1 protein …

A Geometric Characterization Of N-Graded Manifolds And The Frobenius Theorem, Henrique Bursztyn, Miquel Cueca, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

This paper studies graded manifolds with local coordinates concentrated in non-negative degrees. We provide a canonical description of these objects in terms of classical geometric data and, building on this geometric viewpoint, we prove the Frobenius theorem for distributions in this graded setting.

Numerical Analysis And The Deal.Ii Implementation Of An Efficient And Robust Parameterized Ensemble Algorithm For Navier Stokes Flow In The Convection-Dominated Regimes, Brandiece Berry

All ETDs from UAB

Ensemble methods are widely used to approximate solutions for the Navier-Stokes Equations to account for inherent uncertainties within the system. This work goes about the numerical analysis and efficiency assessment of a fully discretized Ensemble Eddy Viscosity (EEV) model for the incompressible Stochastic Navier-Stokes Equations. The model is efficient due to the use of ensemble methods. Through a single coefficient matrix for $J$ realizations, we reduce computational cost and thereby ensure computational efficiency. It is found that the model is robust with the inclusion of the eddy viscosity term, as an additional uncertainty quantification. Finally, we verify theoretical convergence rates …

Early Entering Research And Writing (Eerw) In College-Level Statistics, Rebecca Fang

2025 Faculty Bibliography

With the high demand for data-driven knowledge, students today need to comprehend skills in problem-solving, critical thinking, communication, and collaboration. However, in traditional statistics lectures, students learn descriptive algorithms, hypothesis, and linear regression at their early stage of college. Unfortunately, they often analyze numbers without considering the context, purpose, audience, and the meaning of the numbers. Therefore, providing students the early opportunity to research and write with data in college will allow them to deepen their understanding of the subjects, to think critically about the data and the results, to communicate effectively with diverse audience, and to prepare for their …

Securing Secrets: Exploring The Aes Encryption And Key Security Capabilities Of Chatgpt, Kayla Taylor

Student Works

The development and increasing accessibility of generative artificial intelligence (AI) tools and large language models (LLMs) have allowed cryptographers to explore a variety of cryptanalysis problems in dynamic and interactive ways. Prompt engineering, the process by which input text is tested and refined to elicit a desired response from LLMs, is a nascent area of research that remains largely unexplored in many contexts, including cryptography. This study will explore the potential applications and limitations of prompt engineering in the context of Advanced Encryption Standard (AES) encryption and key security with OpenAI’s ChatGPT (GPT-4o) through two main objectives: First, given a …

Comparison Of Methane Emission Quantification Methods At Landfills In New York State, Alexandra M. Catena

Electronic Theses & Dissertations (2024 - present)

Greenhouse gas (GHG) emission increases are a consequence of global population rise and economic growth. It is well known that carbon dioxide is the dominant GHG by mass, but methane has received increasing attention recently due to its high warming potential and relatively short lifetime of about a decade. Methane has several different types of both natural and anthropogenic sources including wetlands, waste management, energy, agriculture and farming, and wildfires, which makes tracking and reducing methane emissions a complex problem. The emission rates of each of these sources are highly uncertain due to differing emission estimation methods, which can be …

Studies On Convexity Of Dnf Formulae, Josue A. Ruiz

Electronic Theses & Dissertations (2024 - present)

In this dissertation, we investigate the problem of determining whether a Boolean formula given in disjunctive normal form (DNF) is convex. Although Boolean formulas have various applications, our research focuses on the practical application for rule-based access control policies, where policies are often expressed as a set of Boolean rules. Understanding the structural properties of such formulas is crucial for determining whether a policy can be efficiently represented within a specific access control model.

The main contribution of this research is the conception and analysis of convexity derived from the “gap problem.” In this context, convexity is characterized by the …