Mathematics Commons^™

Group-Based Integer Factorization: Theory And Performance, Phuong Cao

Honors Theses

Integer factorization, the problem of finding a nontrivial factor of a composite integer N=pq for large primes p,q, particularly at the size of RSA moduli, is a notoriously difficult challenge that takes classical methods such as Trial Division and Fermat’s Algorithm trillions of years to solve. This thesis studies four probabilistic algorithms that exploit algebraic group structures to achieve significantly better, subexponential efficiency for certain classes of N: Pollard’s p-1, Williams’ p+1, Lenstra’s Elliptic Curve Method, and Pell’s Conic Method. In each case, the algorithm operates on a group over ℤ/Nℤ that decomposes, via the Chinese Remainder Theorem, into corresponding …

A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu

HMC Senior Theses

The goal of this senior thesis is to explore general nonstandard analysis and some possible applications to 𝐶^*-algebras in functional analysis. More specifically, we shall define an approximate identity of a 𝐶^*-algebra using nonstandard analysis and study nonstandard hulls of internal 𝐶^*-algebra in the context of different unitizations. We shall also prove a few results for ideals in 𝐶^*-algebra using nonstandard definitions of approximate identities. We shall also briefly discuss the history and developments of nonstandard analysis.

Digital Technologies And Calculus: Students’ Peaks And Pits, Ricela Feliciano-Semidei, Kevin A. Palencia Infante, Alcibiades Bustillo-Zarate

Faculty Articles, Papers, and Other Scholarship

Understanding calculus students’ perspectives can provide valuable insights into their learning needs and help develop strategies to enhance persistence in STEM programs. Numerous studies have shown that the use of digital technologies (DT) influences student engagement, motivation, and mathematics achievement. In this project, we explored students’ perspectives on using DT in calculus. We interviewed eight calculus students from a midwestern doctorate-granting institution in the United States and used thematic analysis informed by the didactical tetrahedron, accounting for internal and external factors that may influence the teaching and learning process. Students reflected on their use of technology, identifying various benefits, including …

Soft-Constrained Variants Of T-Distributed Stochastic Neighbor Embedding For Global Structure Preservation, Joseph A. Balderas

Mathematics Dissertations

Dimensionality reduction (DR) is a fundamental tool in data science and machine learning that transforms high-dimensional data into a low-dimensional representation while preserving important structural properties of the original data. Among modern DR methods, t-distributed stochastic neighbor embedding (t-SNE) has become one of the most widely used techniques for visualization due to its strong ability to preserve local neighborhood structure and produce visually separated clusters. However, despite its popularity, t-SNE is well known to struggle with preserving global structure of data, often producing embeddings in which distances between clusters and neighborhoods do not accurately reflect relationships in the high-dimensional space. …

Global Weak Solutions Of Optical Variational Wave System, Shahrazad Hamed Alnafie

Graduate Theses, Dissertations, and Problem Reports (ETD)

The coupling of a variational wave equation with Maxwell’s equations gives rise to the optical variational wave system, a hyperbolic PDE system that models the director field of the nematic liquid crystals. This system presents unique analytical challenges that have not been addressed in the existing literature. In this dissertation, we study the one-dimensional case of this system.

We establish the global existence of conservative weak solutions to the associated Cauchy problem. The hyperbolic system is derived using the energy variational method. Through a sequence of suitable coordinate transformations, we reformulate the system while preserving its hyperbolicity. Due to degeneracy …

I Think, Therefore I Can: Self-Efficacy, Self-Regulation, And Sense Of Belonging In Introductory Physics, Danielle Christine Maldonado

Graduate Theses, Dissertations, and Problem Reports (ETD)

Cognitive and affective beliefs in physics classrooms are important indicators of student success beyond traditional measurements of academic performance. This body of work explores major beliefs and processes, including self-efficacy, self-regulation, and sense of belonging, as they relate to academic achievement among introductory undergraduate physics students. First, this dissertation explores the development of physics self-efficacy and physics sense of belonging, two highly correlated variables. This work identifies unique sources of physics self-efficacy and sense of belonging development from current and prior academic experiences, personality factors, and demographic variables. Then, relationships between early- semester self-efficacy and sense of belonging, mid-semester …

Classifying Surfaces With Handle Decomposition, Elizabeth Sipes

Murray State Theses and Dissertations

Among topological spaces, manifolds draw a lot of interest. An
n-manifold is a space that is locally like R^n. Manifolds of dimension 2
are called surfaces. Using handle decomposition, we decompose surfaces
into k-handles, where 0< =k< =2. Techniques such as handle sliding and
handle cancellation allow us to get a more favorable representation of
our surface. We use these tools and calculation of the fundamental group
to classify all compact surfaces.

Infimum Dimension Nash Embeddings For 2d Projective Shape Analysis, Robert L. Paige, Vic Patrangenaru

Mathematics and Statistics Faculty Research & Creative Works

Vector embeddings make complicated data extracted from networks, words and images, more amendable to data science applications. At the present time, the Veronese-Whitney (VW) matrix embedding of the real projective space is the state of the art for making inference about digital images from an uncalibrated camera, such as a cell phone or security camera. In this work we consider vector embeddings for the projective shape data and in particular determine the minimum dimension isometric (distance-preserving or Nash) vector embedding for a projective space. We determine such an embedding for the projective plane in closed-form. From this embedding we determine …

A Fully Discrete Semi-Implicit Numerical Scheme And Its Optimal Error Estimates For Cahn-Hilliard-Mhd Model With Variable Density, Dongmei Duan, Fuzheng Gao, Xiaoming He, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes and analyzes a fully discrete semi-implicit unconditionally energy stable numerical scheme to solve the Cahn-Hilliard Magnetohydrodynamics (Cahn-Hilliard-MHD) model with variable density. The unconditional energy stability and optimal L² error estimates are established for the fully discrete scheme. Major challenges in error estimation arise from the variable density, the strong nonlinearities, and the multi-physics coupling of the model. Under the mathematical induction framework, the Ritz quasi-projection and the Stokes quasi-projection, proposed in [SIAM J. Numer. Anal., 61(3):1218-1245, 2023], are utilized to avoid the gradient terms of the projection errors. The H⁻¹ superconvergence error estimates of Ritz …

Gliders On The Sca Model, Alexa Renner

Mathematical Sciences Technical Reports (MSTR)

The Stranded Cellular Automata (SCA) model consists of a grid of cells which can each contain between zero and two strands apiece and two turning rules that control when strands turn and when they cross. While patterns on this model have been studied previously, such research has not needed an algebraic description of the model. We provide a formal algebraic definition of patterns on the model, define gliders on the model in a way which is semi-compatible with definitions of gliders in other cellular automata models, and classify all 1- and 2-stranded gliders on this model. In addition, we prove …

Integer-Valued Time Series Model Via Copula-Based Bivariate Skellam Distribution, Mohammed Alqawba, Norou Diawara, Mame Mor Sene

Mathematics & Statistics Faculty Publications

Time series analysis is crucial for modeling and forecasting diverse real-world phenomena. Traditional models typically assume continuous-valued data; however, many applications involve integer-valued series, often including negative integers. This paper introduces an approach that combines copula theory with the bivariate Skellam distribution to handle such integer-valued data effectively. Copulas are widely recognized for capturing complex dependencies among variables. By integrating copulas, our proposed method respects integer constraints while modeling positive, negative, and temporal dependencies accurately. Through simulation and an empirical study on a real-life example, we demonstrate that our class of models performs well. This approach has broad applicability in …

Temporal Variational Graph Autoencoder For Influenza Evolution, Soumya Chauhan

2026 Research Poster Competition

Frequent mutations in influenza virus surface proteins can increase infectivity while evading human and vaccine immunity, causing seasonal epidemics. The CDC annually evaluates thousands of virus strains to predict mutated sequences likely to be dominant in the next season, which creates a need for methods that better capture how viral mutations evolve over time. In this study, we represent influenza protein sequences as a network-like graph, creating connections if sequences are collected a week apart and only differ by one mutation. This method explicitly considers time information as part of the evolution, while other existing methods analyze mutated sequences without …

Lie-Galois Theory, Giovanni Reed

Honors Undergraduate Theses

Differential equations are a much-studied topic in the field of mathematics, as well as other sciences, such as engineering, economics, and biology. While much is known concerning these, there is still a large gap in our knowledge about such equations. It is important therefore, both to mathematics and other sciences, that we gain a more complete knowledge of differential equations, in particular the nature of their solutions. In this research, we investigate the solution space of linear ordinary differential equations (ODEs) from the standpoint of differential algebra. Differential algebra allows an ODE to be treated similarly to a polynomial, allowing …

Dominating Hadwiger's Conjecture For 2k2-Free Graphs, Thomas Tibbetts

Honors Undergraduate Theses

A dominating K_t minor in a graph �� is a sequence (��₁,…,��_t) of pairwise disjoint non-empty connected subgraphs of ��, such that for 1≤��< ��≤��, every vertex in ��_j has a neighbor in ��_i. Replacing “every vertex in ��_j” by “some vertex in ��_j” retrieves the standard definition of a ��_t minor. The strengthened notion was introduced by Illingworth and Wood in 2024, who asked whether every graph with chromatic number �� contains a dominating ��_t minor. This is a substantial strengthening of the celebrated Hadwiger’s Conjecture, which asserts that every …

A Conceptual Framework For Cognitive Engagement And Human Agency In Ai-Mediated Learning, Sam Rhodes, Zuhal Yilmaz, Terrie Galanti, Younggon Bae, Victoria Delaney, Zareen Gul Aga, Stephen Hutt

School of Mathematical & Statistical Sciences Faculty Publications

The purpose of this chapter is to propose a conceptual framework that unpacks AI use in instructional practice across three interrelated dimensions: human agency, cognitive demand, and AI orchestration. We ground this work in historical perspectives on technology's mediating role in education. We then propose the conceptual framework and three characterizations of the use of AI in school instruction: cognitive load reducer, cognitive capacity builder, and cognitive companion. We then illustrate a worked example of each, including a curricular example and sample student prompts, and conclude with questions that educators might ask themselves as they consider if and how to …

Fully Discrete Finite Element Methods For The Stochastic Kuramoto–Sivashinsky Equation With Multiplicative Noise, Hung D. Nguyen, Liet Vo

School of Mathematical & Statistical Sciences Faculty Publications

We investigate a fully discrete finite element approximation for the stochastic Kuramoto–Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler–Maruyama scheme in time. Rigorous error estimates are established for two distinct noise regimes. In the case of bounded multiplicative noise, we prove optimal strong convergence rates in full expectation. The analysis relies crucially on a stochastic Gronwall inequality and an exponential stability estimate for the PDE solution, which together control the interplay between the nonlinear drift and the multiplicative stochastic forcing. For general multiplicative noise, where boundedness no longer holds, we derive sub-optimal convergence …

Circle Actions On Oriented 4-Manifolds, Donghoon Jang, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

In this present paper, we consider an action of the circle group on a compact oriented 4-manifold. We derive the Atiyah–Hirzebruch formula for the manifold, and associate a graph in terms of data on the fixed point set. We show in the case of isolated fixed points that if an abstract graph satisfies the Atiyah–Hirzebruch formula, then there exists a corresponding 4-dimensional oriented S1-manifold.

Dataless Neural Networks For Boolean Satisfiability And Network Optimization, Andrew Evan Gautier

Graduate Theses, Dissertations, and Problem Reports (ETD)

Combinatorial optimization problems (COPs) require searching over a finite solution space subject to constraints, with the goal of satisfying an objective function. They arise in operations research, scheduling, resource allocation, circuit design, and many other fields. Many problems in combinatorial optimization (including satisfiability and network design) are NP-hard. Traditionally, researchers have built approximate solvers that return near- optimal solutions efficiently by developing increasingly sophisticated heuristics and meta- heuristics. Deep learning has provided new opportunities for improving combinatorial solvers by leveraging neural guidance to prune the search space. Traditional neural networks have distinct drawbacks in this context: separate training and …

Studies On The Depth Formula And On Reducing Dimensions, Brian Mccourt Laverty

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation presents the author’s recent research, conducted under the supervision of Professor Olgur Celikbas, and based on two articles—one published and one in progress. These works develop two closely related research directions in commutative algebra. Together, they contribute to the subject by addressing aspects of existing conjectures, establishing new results, and introducing methods for studying homological invariants.

The first research direction concerns the depth formula, namely the equality \[ \depth_R(M)+\depth_R(N)=\depth(R)+\depth_R(M\otimes_RN) \] where $M$ and $N$ are finitely generated $R$-modules. A classical result of Auslander \cite{Aus} shows that the depth formula holds provided that either $M$ or $N$ has finite …

A New Parsimony-Based Method For Reconstructing Sequences Of Developmental Events And Estimating Rates Of Intraspecific Variation, With Applications To Trilobites, Alexander Benjamin Bradley

Graduate Theses, Dissertations, and Problem Reports (ETD)

Ontogenetic Sequence Analysis (OSA) is a method for reconstructing multiple sequences of abrupt phenotypic transformations (developmental events) for sampled populations of any kind of organism. It makes use of cladistic parsimony software to produce a network, or graph, whose vertices represent unique phenotypes at different stages of maturity (“semaphoronts”) and whose edges describe the developmental events occurring between semaphoronts. It is a powerful tool for constraining the range of developmental variation possible for a given species and generating sample statistics capable of identifying both the most common and the outlier ontogenetic sequences. However, OSA has seen little use since its …