Mathematics Commons^™

A Survey Of Ai Methods For Geometry Preparation And Mesh Generation In Engineering Simulation, Steven Owen, Nathan Brown, Nikos Chrisochoides, Rao Garimella, Xianfeng Gu, Franck Ledoux, Na Lei, Roshan Quadros, Navamita Ray, Nick Winovich, Yongjie Jessica Zhang

Computer Science Faculty Publications

Artificial intelligence is beginning to reduce the manual effort in the CAD-to-mesh pipeline. Written for meshing and geometry practitioners with limited AI background, this survey organizes recent work by workflow step. We cover part classification and segmentation, mesh quality prediction, and defeaturing. We review AI guidance for unstructured meshing, block-structured meshing in 2D and 3D, and volumetric parameterization, including reconstruction from implicit or sampled geometry. We also discuss parallel mesh generation and scripting automation via reinforcement learning and large language models. Across these topics, AI complements established geometry and meshing algorithms rather than replacing them. We conclude with practical lessons …

When The Gorenstein Projective Modules Are Precovering: A Survey On The Existence Of The Gorenstein Projective Precovers, Alexis L. Mcburney

College of Graduate Studies: Theses & Dissertations

This thesis surveys known results on the existence of Gorenstein projective precovers and studies conditions under which the class of Gorenstein projective modules is precovering. In classical homological algebra, projective resolutions are used to study modules, while relative homological algebra replaces projective modules with a different class that must be precovering for resolutions to exist. Gorenstein homological algebra extends these ideas using Gorenstein projective modules. We review key developments by Jorgensen, Murfet and Salarian, Estrada, Iacob, and Yeomans, and Cortés and Saroch. We discuss the strongest known existence result, showing that Gorenstein projective precovers exist under suitable conditions; in particular, …

Fuchs' Problem For Quasi-Cyclic Groups, Dalen F. Elliott

Theses and Dissertations

Fuchs’ problem asks which groups can arise as the group of units of a ring. Although the finite cyclic case has been completely classified, much less is known in the infinite setting. This thesis contributes to this problem by investigating quasi-cyclic. (Pr¨ufer) groups and their finite direct products. We show that for every odd prime p, there is no commutative ring R such that R×∼= Cp∞. This obstruction arises from characteristic restrictions and the algebraic structure of finite fields. More generally, we prove that any group in which every element has order a power of an odd prime p and …

Symmetry In Latin Hypercubes, Levi Neiburger

Theses and Dissertations

Let [n] = {1, ..., n}. A hypercube H of order n and dimension d is a d-dimensional array whose nᵈ cells are indexed by [n]ᵈ. A hyperplane in H is obtained by fixing one coordinate, while allowing the remaining d–1 coordinates to vary. We wish to color each cell of H from a palette of n^d-1 colors such that each hyperplane is polychromatic.

Our main result is the following. Let n be sufficiently large. There exists a symmetric coloring of the d-dimensional hypercubes of order n whose all hyperplanes are polychromatic if and only if: …

Fostering A Growth Mindset In Mathematics: Faculty And Student Experiences, Yolanda G. Rush

Theses and Dissertations

According to the Center for Community College Student Engagement (2019), many students attending two-year institutions need productive persistence strategies, including the development of a growth mindset. Although some growth mindset interventions have been effective in improving academic achievement among students (Boaler, 2016; Canning et al., 2024) and persistence (Lewis, 2019) among students, especially those with developmental needs (Suh et al., 2019) and those in mathematics, little is known about the experiences of students and teachers (i.e., students’ perceptions of teachers’ intentions and implementation) as teachers work to foster a growth mindset culture (Murphy et al., 2021). In this dissertation, I …

Classifying Mathieu-Zhao Subspaces In Products Of Cyclic Rings, Sarah A. Huber

Theses and Dissertations

Mathieu-Zhao subspaces are a generalization of ideals in an algebra and were introduced by Wenhua Zhao in connection to the Jacobian conjecture and its variants. These subspaces have interesting properties, and often the problem of classification is hard. In this thesis, we investigate the structure of Mathieu-Zhao subspaces of the cartesian product of integers modulo powers of a prime p, Zpr × Zps . We will give a complete classification of the subgroups, maximal subgroups, Mathieu-Zhao subspaces, and maximal Mathieu-Zhao subspaces in these rings.

How And Why Preservice Mathematics Teachers’ Respond To Students’ Statistical Thinking, Radhika Pathak Kharel

Theses and Dissertations

Researchers have found that how preservice teachers (PSTs) respond to students' thinking is crucial to their students' learning and to PSTs' own teaching effectiveness. Although there are some research studies on how teachers respond, there is relatively scarce research on why PSTs respond to students’ thinking the way they do. Such understanding is crucial for designing mathematics methods courses and other professional development opportunities that enable PSTs to create more thoughtful and effective teaching plans. I examined how and why PSTs respond in-the-moment to students’ statistical thinking. The study involved 16 PSTs who enrolled in a middle school mathematics methods …

The Spacetime Finite Element Method For Investigations Into Physics Ghost Systems And Time Parallel Preconditioning, Jax Wysong

Electronic Theses and Dissertations

This work operates on two fronts, focusing on interesting physical phenomena before turning our attention to an interesting numerical math problem. First, using the spacetime finite element method (FEM), we investigate a PDE system consisting of two Klein Gordon equations, which are coupled nonlinearly through the potential energy. The system contains a ghost (negative kinetic energy term). Systems such as these are generally deemed physically unstable, resulting in infinite energy in finite time. However, recent work has shown that this is not always the case. We investigate multiple scenarios arising from different initial conditions to characterize if/when a ghost system …

A Novel Bernstein Operational Matrix Approach For Tempered Fractional Differential Equations: Convergence And Stability Analysis, Jalal Al Hallak, Mohammed Alshbool, Ishak Hashim, Eddie Shahril Ismail, Shaher Momani

All Works

Tempered fractional differential equations (TFDEs) incorporate exponential decay into fractional operators to account for truncated memory and semi-long-range dependence in a variety of applications, including anomalous diffusion, viscoelasticity, transport phenomena, geophysical processes, and financial dynamics. In this work, a tempered fractional Bernstein method (TFBM) was proposed for the numerical solution of TFDEs involving Caputo-type derivatives. The proposed formulation combined a Bernstein polynomial approximation with an analytic representation of the Caputo–tempered fractional derivative through operational matrices. On this basis, two collocation-based variants were developed, namely, a Chebyshev-type method (TFBM-C) and a Legendre-type method (TFBM-L). For the linear setting, a convergence analysis …

Transformer-Based Language Models For Bitcoin Market Prediction And Interpretation, Erich Gozebina

Masters Theses

"In the context of a Master's thesis in applied mathematics, this work investigates compact transformer-based language models as an instrument for Bitcoin-related predictions and decision support. The work connects three topics: the structure of the Bitcoin system and its data, the mathematical and algorithmic foundations of deep autoregressive transformers, and the design of practical training pipelines for financial applications. On this foundation, a reproducible ETL pipeline for Bitcoin data is developed and two forecasting experiments are conducted with the compact language model nanochat. The first experiment approaches the prediction of next-day price movements through autoregressive next-token generation based on structured …

Learning Weibull Loss Severity Models From Truncated And Censored Data, Majed Alkhasha

Graduate Studies Theses and Dissertations 2026

In modern actuarial science and risk management, due to various loss control mechanisms, observed severity losses are typically left-truncated at the deductible, right-censored at the policy limit, and scaled by a pre-specified co-insurance factor. This results in two types of actuarial payment random variables: payment-per-payment (PPP) and payment-per-loss (PPL). To learn ground-up Weibull loss severity models from PPP and PPL sample data, we implement two estimation techniques: Maximum Likelihood Estimation (MLE) and the dynamic Method of Trimmed Moments (MTM). MLE is employed to obtain efficient estimates of the Weibull shape and scale parameters. However, MLE may assign unnecessarily large point …

An Energy-Stable Mixed Cg-Dg Scheme For A Full Shliomis Phase Field Model Of Two-Phase Ferrofluid Flows, Mahdi Gharehbaygloo

Doctoral Dissertations

"Ferrofluids are magnetic nanoparticle suspensions whose motion couples surface tension, flow field, magnetostatics, and magnetization dynamics. This dissertation develops, analyzes, and validates an energy-stable finite element method for a two-phase ferrofluid model that couples the Cahn-Hilliard equations with the full Shliomis model of single-phase ferrofluids, retaining its damping torque term, magnetic torque term, and magnetic stress term.

The spatial discretization is a mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) formulation. It uses continuous ��2 elements for the phase field, chemical potential, velocity, and magnetostatic potential, discontinuous ��2 elements for the magnetization, and discontinuous ��1 elements for the pressure. The …

Approximation Of Solution Of Certain Nonlinear Problems With Applications, Faisal Aijaz

Master of Science in Mathematics

In the thesis, we established the strong convergence of inertial iterative scheme designed to approximate the common solutions of the equilibrium and the fixed-point problem for non-expansive mappings in Hilbert spaces. Through extensive numerical analysis, we analyze the efficiency of the algorithm and demonstrate that the incorporation of the inertial technique yields a significantly faster convergence rate than existing comparable schemes. Furthermore, we examine the effect of initial value on the convergence dynamics, offering visual and comparative assessments against classical methods. To highlight the practical utility and versatility of our framework, we apply the algorithm to signal restoration problems, and …

Metric And Dominant Metric Dimension Of Weakly Zero-Divisor Graphs, Adnan Hussain

Master of Science in Mathematics

This thesis investigates the metric dimension and dominant metric dimension of weakly zero-divisor graphs associated with commutative rings. The study focuses on determining resolving sets and dominating resolving sets for several classes of Γ(Z𝑛) graphs. These results establish connections between algebraic properties of rings and combinatorial characteristics of their associated graphs.

Machine Learning For Predictive Energy And Emissions Modeling Of Vehicles And Power Grids In The United States, S M Tanvir Faysal Alam Chowdhoury

Dissertations

The environmental benefits of electric vehicle (EV) adoption depend on more than replacing internal combustion engine vehicles with electric powertrains. EV adoption reshapes electricity demand, interacts with regional generation mixes, and influences travel behavior and congestion, creating a coupled transportation-energy system in which vehicle and power-plant emissions must be evaluated together. This dissertation develops machine-learning frameworks for predicting energy consumption and emissions from vehicles and power grids under rising EV adoption. The first component forecasts grid emissions from EV charging. Using simulation data from NREL's Cambium database, a Prophet-based time-series framework predicts carbon dioxide, nitrous oxide, and methane emission rates …

Multivariate Quantile Autoregression-Mixed Data Sampling (Mvqar-Midas) Modeling Of Cost Of Living And Supply Chain Dynamics In Canada., Patrick Gbolonyo

Theses and Dissertations (Comprehensive)

In recent years, the rising cost of living as a result of persistent inflationary pressures, disruptions in the global supply chains, and changes in the macroeconomic landscape has become a critical topic of discussion. To address this, we move beyond a mean-based framework and employ a quantile regression approach. This allows the persistence of each series and the transmis- sion of shocks between the Consumer Price Index (CPI) (the total CPI which is a percentage change over the past 12 months), the Interest Rate (IR)(the target for the overnight rate), the New Housing Price Index (NHPI), and high-frequency supply chain …

Geovig And Purevig: Geometry-Aware Architectures For Efficient Computer Vision, Omar Ismail

Theses and Dissertations (Comprehensive)

Deploying deep learning models for medical image analysis on mobile devices requires a balance between inference latency, memory footprint, and delineating anatomical boundaries with high accuracy. While Convolutional Neural Networks (CNNs) and mobile Vision Transformers (ViTs) offer efficiency, they often struggle to model the irregular, non-local geometric structures inherent in biological tissues without incurring prohibitive computational costs. In this thesis, we introduce GeoViG (Geometric Vision Graph), an architecture that bridges the gap between efficient grid-based processing and explicit Geometric Deep Learning. GeoViG introduces a novel transition from high-resolution pixel grids to low-resolution dynamic graphs via a SpreadEdgePool operator, a geometry-aware …

The Classical Limit In Geometric Quantization By Group Extension, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Physical systems as a rule are associated with a symmetry group. The group approach to geometric quantization makes use of this to introduce a quantization by means of group extension. This procedure is discussed and applied to a physical system whose group law has its origin with the Galilean group. The main intention is to investigate the classical limit and its relationship under this approach to geometric quantization. The classical quantization conditions are obtained based on a deeper foundation.

Strong Q-Analogues For Values Of The Dirichlet Beta Function, Ankush Goswami, Tim Huber

School of Mathematical & Statistical Sciences Faculty Publications

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan’s Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong q-analogues by virtue of their reduction to the classical beta evaluations as 𝑞→1− and explicit evaluations at CM points for |𝑞|< 1. We also determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

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 …

Optimal Quantization On Spherical Surfaces: Continuous And Discrete Models—A Beginner-Friendly Expository Study, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets ofthe sphere, emphasizing both continuous and discrete settings. We first present a detailedgeometric and analytical foundation for intrinsic quantization on the unit sphere, includingdefinitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation,the Fréchet mean, spherical Voronoi regions, centroid conditions, and quantizationdimensions. Building upon this framework, we develop explicit continuous and discretequantization models on spherical curves, namely great circles, small circles, and greatcircular arcs—supported by rigorous derivations and pedagogical exposition. For uniformcontinuous distributions, we compute optimal …

Gain-Delay Decoupling In Human Stimulus Frequency Otoacoustic Emissions Reveals Constraints On Cochlear Nonlinear Amplification, Yoshita Sharma, Arturo Moleti, Renata Sisto, Teresa Botti, Hansapani Rodrigo, Sri Mishra

School of Mathematical & Statistical Sciences Faculty Publications

Purpose: The nonlinear cochlear amplifier, driven by outer hair cells, underlies the remarkable sensitivity and frequency selectivity of the mammalian auditory system. Stimulus frequency otoacoustic emissions (SFOAEs) provide a noninvasive window into these active cochlear processes, yet the relationship between emission gain and delay across stimulus levels remains incompletely understood. This study examined the level dependence of SFOAEs in normal-hearing human listeners to characterize cochlear nonlinear response properties. We tested how emission gain and delay vary with stimulus level and estimated the frequency of the apical–basal transition associated with the breakdown of the approximate local scaling symmetry.

Methods: SFOAEs were …

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.

All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

Exploring Caring Pedagogy Through Self-Study: A Mathematics Educator’S Journey, Kien H. Lim

Departmental Papers (Math)

This self-study examines my efforts to cultivate a positive learning environment in a capstone mathematics course for preservice middle school teachers. Motivated by earlier negative course evaluations and guided by heart-centered pedagogy, I explored how my beliefs, emotions, and instructional practices evolved as I sought to integrate kindness, foster mathematical thinking, and support students’ well-being. Drawing on dialogue transcripts, reflective journals, and reconstructed tutoring notes, I analyzed tensions between content coverage and relational pedagogy, as well as between deficit- and strength-based perspectives of students. Through critical dialogue and reflective inquiry, I gained deeper insight into my identity as a caring …

Creating A Positive Learning Environment In A Math Course For Preservice Teachers, Kien H. Lim, Hilda Y. Sotelo

Departmental Papers (Math)

This study examines how a mathematics instructor cultivated a positive learning environment in a synchronous online course for 23 preservice teachers. Guided by self-determination theory, growth mindset, and mathematical caring, the course integrated high-cognitive-demand tasks, collaborative problem solving, flexible policies, and motivational supports. Data sources included reflection logs (Likert-scale items and open-ended responses) and a focus group; the instructor’s reflexive self-study informed result interpretation. Wilcoxon signed-rank tests indicated significant gains in students’ sense of inclusion and trust, though some instructor- related ratings declined late in the semester. Qualitative findings emphasized students’ appreciation for flexibility, encouragement, and growth-oriented feedback. Despite time …