Mathematics Commons^™

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, …

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.

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, …

An Association Test For Ordinal Outcomes In Clustered Data With Informative Cluster Size, Hasika K. Wickrama Senevirathne, Sandipan Dutta

Mathematics & Statistics Faculty Publications

In cluster-correlated data, the number of observations in a cluster can be associated with the outcome from that cluster. This phenomenon is known as informative cluster size which can occur in cluster-randomized clinical trial data. Several studies have found that ignoring the issue of informative cluster size can produce biased results in the analysis of clustered data. Most of the existing methods for addressing informative cluster size are suited to continuous outcomes. However, ordinal outcomes and covariates are often encountered in clustered data obtained from large clinical studies. The existing methods for ordinal association testing in clustered data can produce …

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 …

Mg-Spair: Multi-Grade Sparse-Guided Implicit Representation For Training-Data-Free Image Restoration, Jianmin Liao, Lei Huang, Ronglong Fang, Ashley Prater-Bennette, Lixin Shen, Yuesheng Xu

Mathematics & Statistics Faculty Publications

MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade residual hierarchy that progressively refines the reconstruction from low to high spatial frequencies across grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g., ℓ0 type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a …

Juncture Models: Trivariate Nonlinear Regression Models And Application To In-Hospital Quality Of Care, Manasi Sheth, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

Mortality, readmission, and length of stay (LOS) are commonly used measures for quality of hospital care. These measures are correlated with each other, so we propose a new way to combine these three variables to evaluate hospital care quality of care. In this paper, we investigate the joint probability distribution of these three variables using what we named as the Juncture Distribution Models and characteristics of the Juncture models.

Odd Colorings Of Graphs With Maximum Degree And Surface Conditions, Matthew Kyle

Graduate Studies Theses and Dissertations 2026

A proper vertex coloring of a graph is said to be odd if, for each vertex, there is a color that is present an odd number of times in its neighborhood. Introduced as a softening of other coloring problems, this problem has recently seen much research interest. It has been conjectured that all planar graphs are odd-5-colorable. Further research has suggested bounds for odd coloring numbers of graphs with different maximum degree conditions, different girths, and on different surfaces; we progress results in all of these areas. We improve an upper bound for odd coloring numbers of graphs of maximum …

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 …

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 …

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 …