Digital Commons Network^™

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 …

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 …

Machine Learning-Based Intrusion Detection System For Iot Networks Using The Rt-Iot 2022 Dataset, Bukunmi Ebenezer Afolabi

Theses, Dissertations and Capstones

The rapid expansion of the Internet of Things (IoT) has transformed modern computing by enabling seamless connectivity among heterogeneous devices across diverse application domains. However, this increased interconnectivity has significantly enlarged the attack surface of IoT networks, exposing them to a wide range of sophisticated cyber threats. Conventional security mechanisms often lack the capability to detect emerging attacks in real time, thereby necessitating the development of intelligent Intrusion Detection Systems (IDS) capable of accurately identifying malicious network activities. This study developed and evaluated a machine learning-based intrusion detection framework for multiclass IoT attack detection using the RT-IoT2022 dataset. The dataset …

On Values Taken By Characters Of Finite Groups, Christopher William Herbig

Graduate Research Theses & Dissertations

The characters of finite groups are a fundamental tool for analyzing the structure of finite groups and are interesting in their own right. In particular, we are concerned with how certain assumptions on the values taken by characters impact the structure of finite groups and vice versa. After establishing notation and stating some fundamental results in Chapter 2, this work begins with a presentation of results in relation to a conjecture of N. N. Hung and P. H. Tiep on fields generated by character values. In particular, we have found a large family of counterexamples to the conjecture by taking …

The Effects Of Problem-Based Learning On Mathematical Creativity And Self-Efficacy Of High School Students, Margaret Remus

Graduate Research Theses & Dissertations

Mathematical creativity is recognized as an important component of mathematics education; however, there is limited understanding of how instructional approaches can effectively support this skill among high school students, particularly through problem-based learning (PBL). This study examined the effects of problem-based learning on mathematical creativity among high school students.

This study employed a mixed-methods approach using an alternating treatment design with two groups of participants. Group 1 received the PBL intervention followed by a control condition, while Group 2 received the control condition followed by the PBL intervention. Quantitative data were collected through a mathematical creativity test measuring flexibility, fluency, …

Quantum Mechanics As A Framework For Data Assimilation And Its Application To Atmospheric Parameterization, David Freeman

Dartmouth College Ph.D Dissertations

Quantum mechanics, as a mathematical system, can be understood as a generalization of classical probability theory. Quantum Mechanical Data Assimilation (QMDA) is a method in which classical dynamical systems are embedded into a quantum mechanical setting, with an associated data assimilation scheme leveraging the operator algebraic setting. In this dissertation, the algebraic structure underlying the operator theoretic formulation of QMDA is discussed. A procedure for closure of dynamical systems based on QMDA, known as Quantum Mechanical Closure (QMCl), is then constructed, and the procedures for constructing the quantum embeddings and implementing QMCl in practice are laid out and implemented for …

Analyzing Sleep Architecture And Brain State Transitions Via Hidden Markov Models On Fmri Data, Brileigh Jay Cates

Masters Theses

Sleep is associated with systematic changes in brain activity and functional connectivity observable in functional magnetic resonance imagining (fMRI) signals. Because subjects often fall asleep during resting-state experiments, the absence of vigilance monitoring can confound the interpretation of resting-state dynamics. Although electroencephalography (EEG) is the gold standard for sleep staging, simultaneous EEG-fMRI acquisition is not always feasible.

This study investigates whether sleep stages can be inferred directly from fMRI using a probabilistic latent-state framework. Hidden Markov Models (HMMs) are applied to blood-oxygen-level-dependent (BOLD) time series to identify latent brain states and their temporal transitions. Inferred states are aligned with EEG-derived …

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 …

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, …

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 …

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 …

Traveling Waves For A Diffusive Sir Epidemic Model With Delay In The Diffusion Term, William Barker

Faculty Scholarship

This paper investigates the existence of traveling waves in a diffusive SIR model with delay incorporated in the diffusion terms and a nonlinear incidence rate with delay. By employing a cross-iteration scheme and partial monotonicity conditions, we establish that the existence of quasi-upper and lower solutions, along with suitable super and sub-solutions, provides sufficient conditions for the existence of a traveling wavefront. This existence result is obtained via Schauder’s fixed-point theorem. Furthermore, given an appropriate basic reproduction number, the traveling wavefront transitions from the disease-free steady state to the endemic steady state. To illustrate our approach, we explicitly construct super- …

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 …

Computational Insights Into Orthotropic Fracture: Crack-Tip Fields In Strain-Limiting Materials Under Non-Uniform Loads, Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

School of Mathematical & Statistical Sciences Faculty Publications

A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasi-linear elliptic BVP. This BVP is solved using a robust numerical scheme combining …

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 …

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 …

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.

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 …

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 …

Categories, Homology And Sheaves For Hypergraphs, Robert Green

Electronic Theses & Dissertations (2024 - present)

Hypergraphs are a prominent tool for representing networks with connections among three or more entities. There is an inherent flexibility that allows hypergraphs to more naturally represent certain types of networks than graphs or simplicial complexes can on their own. This flexibility, however, comes at a cost, as there is a zoo of various categories and homology theories that are applicable to hypergraphs. The first chapter of this dissertation explores various categorical perspectives on hypergraphs, focusing on what the natural notion of morphism between hypergraphs should be. It also contains an exploration of the functoriality of vertex-edge duality in these …

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 …

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 …

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, …

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 …

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 …

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 …

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.

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 …