Digital Commons Network^™

The Global Orbit ∞-Category And Applications To Assembly Maps, Zoë Pope

Electronic Theses & Dissertations (2024 - present)

We reformulate the foundations of assembly maps in the context of the global orbit ∞-category of all discrete groups. We first show that the ∞-categorical slice of the global orbit ∞-category over any fixed group G is equivalent to the orbit 1-category of G, and we also frame the subgroup 1-category of G in this global context. We then use the afore-mentioned equivalence to redefine assembly maps as counits of the adjunction between left Kan extension and restriction, and give purely ∞-categorical and conceptual proofs of known results, such as the Transitivity Principle. Additionally, we give equivalent formulations for what …

Entropic Dynamics Approach To The Classical Limit Of Quantum Mechanics: Decoupling Of The Center Of Mass Motion For A Mesoscopic Particle, Fatimah Judayba

Electronic Theses & Dissertations (2024 - present)

In the Entropic Dynamics (ED) approach, quantum mechanics is derived from the principles of entropic inference and information geometry. The ED approach differs from other interpretations by making a clear commitment to distinguishing which variables are ontic (real) and which are epistemic. The classical limit for the center of mass (CM) coordinate is achieved for a large number of particles, M →∞, while Planck’s constant ℏ remains finite. Typically, the emergence of the classical limit requires decoherence through interactions with the external environment. In this work, we investigate whether the classical behavior of the CM coordinate in a mesoscopic system …

The Ellipsoidal Principal Semi-Axis Geometry Of The Solution To An Ivp For A Matrix Diffusion Pde, Brandon M. Fox

Electronic Theses & Dissertations (2024 - present)

We analyze the underlying geometry of the solution to an IVP for a matrix diffusion PDE. We first derive the fundamental solution to the PDE. We then determine the unique solution to the IVP. From there, we begin analyzing its underlying geometry. We first observe that the geometry exhibits an ellipsoidal nature. Furthermore, we observe that it is described by the principal semi-axis geometry of the ellipsoids associated with the solution. This conclusion follows from applying the Principal Axis Theorem to the associated ellipsoids to establish their principal semi-axis geometry, as governed by the eigenstructure of the matrix. This thesis …

Likelihood-Based Inference For Random Networks With Changepoints, Daniel Cirkovic, Tiandong Wang, Xianyang Zhang

Mathematical and Statistical Science Faculty Research and Publications

Generative, temporal network models play an important role in analyzing the dependence structure and evolution patterns of complex networks. Due to the complicated nature of real network data, it is often naive to assume that the underlying data-generative mechanism itself is invariant with time. Such observation leads to the study of changepoints or sudden shifts in the distributional structure of the evolving network. In this paper, we propose a likelihood-based methodology to detect changepoints in undirected, affine preferential attachment networks where, upon introduction, a new node selects one old to attach to with probability proportional to its degree. In particular, …

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: …

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 …

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.

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 …

A Mathematical Frameworks For Singular, Nonlinear Phenomena: Applications To Nematocyst Firing And Inhomogeneous Nls With Coulomb Potential, Abdulrahman Alharbi

Theses and Dissertations

Nematocysts are specialized cellular organelles found in all cnidarians, including corals and jellyfish, as well as in some single-celled protists such as dinoflagellates. These organelles display remarkable diversity in morphology and function, playing roles in prey capture and defense. The firing of a nematocyst is one of the fastest accelerations in nature, yet the underlying physical mechanisms remain not fully understood. In this work, we address key questions: how sufficient force is generated to overcome the fluid boundary layer, whether fluid–structure interaction models can reproduce observed dynamics, and what mechanisms trigger discharge.

Our research investigates models based on osmotic pressure …

Near Real-Time Adaptive Isotropic And Anisotropic Image-To-Mesh Conversion For Cerebral Aneurysm Simulations, Kevin Garner, Chander Sadasivan, Nikos Chrisochoides

Computer Science Faculty Publications

This paper presents two performance optimization techniques for a mesh adaptation method that is designed to help streamline the discretization of complex vascular geometries within the numerical modeling process. This method is integrated into a pipeline with an image-to-mesh conversion tool to generate adaptive anisotropic meshes from segmented medical images. The pipeline is shown to satisfy quality, fidelity, smoothness, and robustness requirements while providing near real-time performance for medical image-to-mesh conversion. Tested with two brain aneurysm cases and utilizing up to 96 CPU cores within a single, multicore node on Purdue University’s Anvil supercomputer, the parallel adaptive anisotropic meshing method …

Exponentially Many Correspondence Colourings Of Planar And Locally Planar Graphs, Luke Postle, Evelyne Smith-Roberge

Faculty Publications – Mathematics

We show that there exists a constant c>0 such that if G is a planar graph with a 5-correspondence assignment (L,M), then G has at least 2c⋅v(G) distinct (L,M)-colourings. This confirms a conjecture of Langhede and Thomassen. More broadly, we introduce a general method showing how hyperbolicity theorems for certain families of critical graphs can be used to derive lower bounds on the number of colourings of the associated class of planar graphs. Hence our main result follows from this method plus a technical theorem (that we proved in a previous paper) involving the hyperbolicity …

Shrinking Attachment Spaces, Anastasia M. Clements

West Chester University Graduate Theses, Dissertations, and Final Projects

Gluing constructions such as pushouts and other colimits are often used to attach spaces to- gether in algebraic topology. The weak topology is a natural choice of topology for attachment spaces in the context of CW-complexes and simplicial complexes because of its universal property but is insufficient for gluing together infinitely many spaces and preserving topological properties like compactness or metrizability. In this thesis, we introduce a modification of the weak topology called the shrinking attachment topology, which is defined on a space Y constructed by attaching an infinite sequence of spaces B₁, B₁, B_{3 …}

Adaptive Multi-Grade Deep Learning For Highly Oscillatory Fredholm Integral Equations Of The Second Kind, Jie Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.

A Composite Narxnn Approach To Photovoltaic Power Forecasting With Integrated Weather Inputs And Uncertainty Quantification, Denisse Urenda Castañeda, Sharmin Abdullah, Jackson Morgan, Honglun Xu, Michael Pokojovy, Tzu-Liang Tseng

Mathematics & Statistics Faculty Publications

Solar photovoltaics (PV) are a major source of sustainable energy. Yet, their power output is highly sensitive to environmental variability, particularly solar irradiance, cloud cover, wind, and temperature. Accurate forecasting of PV power is essential for efficient grid integration and energy planning, especially in applications requiring reliable longer-term forecasting rather than one-step-ahead predictions. This study presents a PV power forecasting approach using Nonlinear Autoregressive models with Exogenous Inputs (NARX), integrating large-scale numerical weather historical data as exogenous variables. Although NARX models effectively capture temporal dependencies, they can become overly dependent on historical power values, reducing responsiveness to real-time weather changes. …

A New Parallel-In-Time Direct Inverse Method For Nonlinear Differential Equations, Nail K. Yamaleev, Subhash Paudel

Mathematics & Statistics Faculty Publications

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at …

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, …

Arrangements Of N Planes Resulting In One Bounded Tetrahedral Chamber, Ava Knight

Williams Honors College, Honors Research Projects

This paper investigates the combinatorial geometry of plane arrangements in three-dimensional space, focusing on configurations that produce exactly one bounded tetrahedral chamber. We define T(n) as the number of face-combinatorial equivalence classes of arrangements of n planes in ℝ³ containing exactly one bounded tetrahedral chamber. Known values — T(3) = 0, T(4) = 1, and T(5) = 2 — are established through direct construction, while T(6) remains an open problem. This paper contributes experimental evidence toward resolving T(6) by systematically extending the two valid 5-plane arrangements and verifying, through a plane removal argument, that each yields a valid plane configuration …

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 …

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 …

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.

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 …

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 …

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.

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 …

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 …