Digital Commons Network^™

The Relationship Between Foliations Of The Plane, Kaplan Diagrams, And Non-Hausdorff 1-Manifolds, Monique Justine Howe

Master's Theses

In this paper we seek to explain the relationship between foliations of the plane, Kaplan diagrams, and simply connected non-Hausdorff 1-manifolds with countable basis that are orientable with an ordering on branch points. We will walk the reader through the definitions of all of these objects, and provide examples with a focus on the motivating example of the Reeb foliation. This paper will describe and define the known bijection from the set of foliations of the plane, F, to the set of Kaplan diagrams, K, its inverse, and the one-to-one map from F to the set of simply connected non-Hausdorff …

Rewriting The Calculus 3 Workshop At San José State University, Moorea Lippert

Master's Theses

Calculus 3 at San José State University is a very important course for any student pursuing a STEM degree. However, the course is very challenging, so the university offers a supplemental workshop course for Calculus 3 students. Unfortunately, this workshop has not aligned with modern research in math education. This thesis documents efforts to recreate the workshop to better align with this modern research.

Monoids With K-Strictly Local Geodesic Languages, William M. Hong

Master's Theses

A paper by Gilman, Hermiller, Holt and Rees (2011) prove that a group G finitely generated by X is virtually free if and only if the language of geodesics words in G over X is k-strictly local if and only if the word problem is context-free. If M is a monoid finitely generated by X, we define Γu(M,X) to be the underlying, undirected graph of Γ(M,X). We prove that if the language of geodesics in Γu(M,X) based at the identity is k-strictly local, then the monoid and semi-group word problems are context-free.

Fermat's Last Theorem Over Quadratic Number Fields, Richie Tay

Master's Theses

Fermat’s Last Theorem states that the equation xn + yn = zn has no nontrivial integer solutions for n ≥ 3. While the rational case is completely solved, the situation over quadratic number fields is not as straightforward. For some exponents nontrivial solutions exist, while for others they do not. This thesis studies Fermat-type equations over quadratic number fields, combining classical methods with more modern techniques from the theory of elliptic curves. The thesis first reviews the rational cases n = 2, n = 3, n = 4, then develops the necessary background on quadratic number fields, including rings of …

Math 141 Calculus I (Queens College), Anisha Clarke

Open Educational Resources

This syllabus provides the outline for a calculus one course using OpenStax Calculus Volume I and no graphing calculator.

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified by using numbers in the ring of integers of non-Archimedean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.

Probabilistic Metric Dimension, Kyle Worley

Master's Theses

The metric dimension of a graph G is the smallest number of unique vertices (often called “marked vertices” or “landmarks”) such that each vertex has a unique distance to the marked vertices. If a set of vertices S ⊆ V (G) when marked gives all vertices unique distances, it is called a resolving set. Hence if G is a graph with a resolving set S, then for all v, u ∈ V (G) there exists s ∈ S such that d(v, s) ̸= d(u, s). The metric dimension dim(G) = min(|S|). Two widely studied variants exist, one where edges are …

Rational Non-Euclidean Triangles And Elliptic Curves, Tyler Morales

Master's Theses

We say a triangle △ is rational if its side lengths are rational. For a geometry of constant curvature κ, we say △ is a rational triangle if the generalized tangents of its side lengths are rational. We are able to parameterize rational triangles having the same inradius and semiperimeter on an plane curve, write a bijection between the rational points on this curve and triples of side lengths, and show that this curve is an elliptic curve. Then, we write a general transformation for our plane curve to short Weierstrass form to compute ranks and add points easily using …

Fundamental Solutions To The Fractional Heat Operator, Jacob Flores

Math Theses

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution - Part B: Professional Development Framework And Institutional Implementation, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

This paper presents the professional-development and institutional-implementation component of a three-paper program on introducing data-driven dynamical systems into undergraduate ordinary differential equations instruction at a primarily undergraduate institution. The companion paper \cite{SelvitellaAnderson2026} develops the mathematical and computational content, including regression, regularization, Dynamic Mode Decomposition, Sparse Identification of Nonlinear Dynamics, and a Van der Pol oscillator activity. The present paper asks a complementary implementation question: what departmental structures, graduate teaching assistant roles, professional-development activities, and course-support pathways are needed for such materials to become teachable within local undergraduate settings?

We describe a three-phase professional-development model for graduate teaching assistants in the …

A Dynamics Viewpoint On The Approximation Capabilities Of Infinite-Depth Neural Networks, Andreus Brammer

Electronic Theses and Dissertations 2020 - Present

Neural Ordinary differential equations(Neural ODEs) provide a continuous-time framework for learning dynamical systems by representing the evolution of a state variable through a parameterized differential equation. While most existing methods rely on numerical time stepping methods during training such as RK4, this work investigates an alternate spectral framework where both the solution and the time-dependent network parameters are represented by global polynomial expressions.

Two methods are employed: the first uses Taylor series representations, and the second approach uses Chebyshev polynomial expansions together with fast spectral transforms and efficient polynomial multiplication techniques. In each formulation, the Neural ODE is transformed into …

Cultural Traits May Replace Human Mobility Data In Forecasting Covid-19 Mortality: A Deep Learning Approach, Saif Abbas, Tamer Oraby, Michael G. Tyshenko, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

The COVID-19 pandemic highlighted the need for accurate epidemic forecasting to support public health decision-making. Most existing approaches depend heavily on human mobility data, while largely neglecting population behavior shaped by socio-cultural norms. In this study, we analyze daily COVID-19 mortality and Google mobility data from 72 countries during the first 130 d of the pandemic, a period characterized by high uncertainty and behavioral heterogeneity. In particular, we examine whether Hofstede’s country-level cultural dimensions can serve as latent behavioral forecasters of mortality in lieu of dynamic mobility indicators. Using 100 d for training and 30 d for forecasting, we employ …

The Edge-Distinguishing Game, Nathaniel Benjamin, Elisa Benthem, Cooper Burkel, Marissa E. Chesser, Mike Janssen

Communications on Number Theory and Combinatorial Theory

In this paper, we introduce a graph coloring game called the Edge-Distinguishing Game (EDGe). The edge-distinguishing chromatic number of a graph is used to determine the moves each player can make. We determine which player has a winning strategy for particular graphs and graph families. Additionally, utilizing principles from game theory as well as previous work on a computational solution for the Game of Cycles.

Differentiating And Accommodating Ocd And Anxiety In Mathematical Contexts, Madeline F. Gavares

Interdivisional Studies Summer Fellows

Students in mathematics classrooms may experience emotional and cognitive challenges that affect how they engage with coursework and assessments. While math anxiety has been widely documented, less attention has been given to how obsessive-compulsive disorder (OCD), particularly checking behaviors, may influence students’ experiences in math settings. This study explores how students who report math anxiety and those who report OCD symptoms describe their engagement with math tasks and test-taking. Drawing on survey and interview data from students and teachers, this study examines how these experiences are articulated and perceived in academic math contexts, rather than evaluating ability or outcomes. By …

Faith-Informed Instruction In Secondary Mathematics: A Hermeneutic Phenomenological Study Of Lutheran Church–Missouri Synod Teachers, Jonathan Balsman

Doctoral Dissertations and Projects

The purpose of this hermeneutic phenomenological study was to understand the phenomenon of faith-informed instruction as experienced by secondary mathematics teachers in Lutheran Church–Missouri Synod (LCMS) schools in the United States. The theory guiding this study was Vygotsky’s theory of social constructivism, which frames educators’ instructional and theological meaning-making as socially and contextually developed through collaborative, relational learning environments. The central research question guiding this study was: What are the lived experiences of LCMS secondary mathematics teachers as they practice faith-informed instruction? This study used a hermeneutic phenomenological design to interpret teacher meaning-making and reflective practice. Participants were selected using …

Double-Scoring: Reliable Extraction Of Strong Lottery Tickets, Bryce Christopherson, Jack Baretz, Darian Colgrove, Salah Dandan

Mathematics Faculty Publications

The lottery ticket hypothesis proposes that large random neural networks contain sparse subnetworks that can match the performance of dense models after compara- ble training. A stronger version asserts that sufficiently overparameterized random networks contain subnetworks that are already accurate before any weight training. Existing theory establishes that such strong lottery tickets exist, but reliable ex- traction remains difficult. We revisit edge-popup, a frozen-weight score-training method for extracting strong tickets, and identify layerwise sparsity selection as a central bottleneck. We introduce double-scoring, an augmented score-space parameterization that replaces a layerwise sparsity search with optimization over enlarged score tensors. We prove …

A Bayesian-Optimized Ensemble Deep Learning Framework For Automated Detection And Classification Of Retinal Diseases In Ghana Using Oct Images, Gifty Duah, Eric Nyarko, Gideon Nana Amo, Theophilus Dwamena Frimpong, Anani Lotsi

School of Mathematical & Statistical Sciences Faculty Publications

Retinal diseases pose a significant global health challenge due to their potential to cause severe visual impairment and blindness. This study aimed to develop a robust deep learning ensemble framework for the automated detection and classification of retinal diseases from optical coherence tomography (OCT) images. This study used OCT images from WATBORG Eye Services in Ghana, including glaucoma, macular edema, posterior vitreous detachment (PVD), and healthy eyes. The data preprocessing steps included augmentation, resizing, and one-hot encoding. The dataset was divided into training (56%), validation (14%), and testing (30%) sets using stratified sampling. Six convolutional neural network (CNN) architectures, Visual …

Biharmonic Curves In Warped Product Manifolds I ×FMN(C), Şaban Güvenç, Ci̇han Özgür

Turkish Journal of Mathematics

We explore the geometric properties of biharmonic curves in warped product manifolds of the form I ×_f Mⁿ(c), where I is an open interval and Mⁿ(c) is a space of constant curvature. By establishing a main theorem, we analyze four distinct cases to reveal deeper curvature-related characteristics of these curves, including situations where they are slant. Finally, we construct three examples in I ×_f S²(1).

Generalized Submaximality And Paracompactness In Ideal Topological Spaces Via Δβ*-𝔍-Open Sets, Chawalit Boonpok, Areeyuth Sama-Ae

Turkish Journal of Mathematics

This paper explores the notions of δβ^*-𝔍-submaximality and δβ^*-𝔍-paracompactness within ideal topological spaces, viewing them as natural generalizations of the traditional concepts of submaximality and paracompactness. These concepts arise naturally in the context of ideal topology, where an ideal 𝔍 on a topological space provides additional structure for analyzing topological properties. It focuses on the study of submaximal spaces by employing δβ^*-𝔍-open sets, which serve as fundamental building blocks for understanding the topological behavior in ideal spaces. Furthermore, the work provides multiple characterizations of δβ^*-𝔍-paracompact spaces and examines how this property behaves …

Notes On The Geometry Of Electromagnetic Fields And Maxwell’S Equations Along A Nonnull Curves In Nonflat-3d Space Forms MQ3(C), Fatma Almaz, Cumali̇ Eki̇ci̇

Turkish Journal of Mathematics

In this paper, the directional derivatives in accordance with the orthonormal frame {T^→, N^→, B^→} are defined in M_q³(c), the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the bending elastic energy function for the same particle in M_q³(c) according to curve γ(s, ξ, η) and we tried to interpret it from a geometric perspective of the energy for unit vector fields. Also, Maxwell’s equations are expressed for the electric and magnetic field vectors …

Propagation Of Dirac Spherical Waves In The Expanding Universe, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

The explicit formulas for the spherical solutions of the Dirac equation in the expanding universe are given. The initial value of the solution can be, in particular, a wave function of the hydrogen-like atom or a spherical wave in the Minkowski space, that then propagates in the Friedmann-Lemaître-Robertson-Walker space-time, which is expanding with the de Sitter scale factor.

Observations On Recurrent Loss In The Neural Network Model Of A Partial Differential Equation: The Advection–Diffusion Equation, Jonah A. Reeger

Faculty Publications

A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic exploration of the stability of the solutions generated by these ML approaches. Here, a recurrent network is introduced that matches precisely the evaluation of a multi-step method paired with a collocation method for approximating spatial derivatives in the advection–diffusion equation. This allows for two things: (1) the use of traditional tools for analyzing the stability of a numerical method for solving PDEs and (2) bringing to bear …

Comparative Analysis Of Traditional And Deep Learning Time Series Architectures For Influenza A Infectious Disease Forecasting, Edmund Fosu Agyemang, Hansapani Rodrigo, Vincent Agbenyeavu

School of Mathematical & Statistical Sciences Faculty Publications

Influenza A remains a major cause of respiratory mortality worldwide, motivating accurate forecasting to support timely preparedness and resource allocation. This study presents a comparative evaluation of two traditional seasonal time series baselines, ARIMA and Holt–Winters exponential smoothing (ETS), and six deep learning (DL) architectures (Simple RNN, LSTM, GRU, BiLSTM, BiGRU, and a Transformer) for forecasting monthly Influenza A case counts in the United States. Data from January 2009 to December 2023 were analyzed, using January 2009 to December 2022 for training and January 2023 to December 2023 for out-of-sample testing. Models were tuned using a validation split and assessed …

Assessing The Role Of Model Complexity In Virtual Clinical Trial Outcomes, Jana L. Gevertz, Joanna R. Wares

Department of Math & Statistics Faculty Publications

Virtual clinical trials (VCTs) hold significant promise for improving the drug development process, yet their predictive reliability depends critically on design decisions that remain poorly understood. This study examines how model complexity influences VCT outcomes, as well as how the choice of prior parameter distributions and virtual patient inclusion criteria affects those outcomes. Using oncolytic virotherapy treatment of murine tumors as a case study, we compared a relative hierarchy of three mathematical models of increasing complexity under different parameter priors (uniform and normal distributions) and two inclusion methods (accept-or-reject and accept-or-perturb). Our results demonstrate that the simplest model produces a …

Operator Theoretic Methods For Coupled Pde In General Geometries, Yuhao Mu

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

This thesis develops and applies operator theoretic methods applied to coupled PDE analysis in general geometries, in which the coupling involves interchange across a boundary. The general geometries refer to domains with merely Lipschitz continuous boundaries (e.g. any non-convex polygon), as opposed to those with any further smoothness conditions on the boundary. The key results include a new pressure elimination method for coupled fluid–structure interaction (FSI) PDEs with boundary interchange, which allows an explicit semigroup representation of the PDE in terms of just the fluid and structure variables. This novel technique, valid over arbitrary bounded Lipschitz domains, leads to a …

On Universal C∗ Algebras Associated To Operator Spaces And Generalized Crossed Products, Sayan Kansa Banik

Doctoral Theses

In my thesis, we study generalized crossed product constructions of the group \( C^* \)-algebra \( C^*(G) \) with respect to certain completely positive maps, where \( G \) is assumed to be a discrete amenable group. We also investigate the universal \( C^* \)-algebra \( \mathcal{E}_\alpha \) introduced by Hirshberg, which is constructed from \( C^*(G) \) and a pure injective homomorphism \( \alpha \colon G \to G \). In particular, we analyze its relationship with Exel's construction of generalized crossed products associated with the endomorphism of \( C^*(G) \) induced by \( \alpha \), together with an appropriate …

Pseudodifferential Absorbing Boundary Conditions For Waves, Lauren Taylor

Mechanical Engineering Theses

Absorbing boundary conditions (ABCs) are required to truncate the computational domain when performing Finite Element Analyses of exterior acoustic problems where physical domain is unbounded. ABC is applied on a fictitious boundary containing the scatterer and ideally allows outgoing waves to leave without non-physical reflections. Preventing artificial reflections is essential to benefit from the accuracy of the numerical method used. Otherwise, ABC acts as a reflective surface, and artificial reflections distort the solution in the entire domain which is not recoverable by any type of refinement. Pseudodifferential ABCs were used to describe the Dirichlet-to-Neumann map which maps known boundary values …

Analytic Solutions To Oceanographic Kinematic Property Equations, James Turbett

Theses and Dissertations

With the recent advent of large submesoscale drifter deployments, interpreting the timeseries of kinematic properties (KP) - divergence, vorticity, shear, and normal strain rates observed therein is a pressing issue. The evolution equations for the four KP are derived from the two-dimensional momentum conservation equations. The resulting equations, a nonlinear coupled system of ordinary differential equations, capture the submesoscale motion of drifters along fixed ocean surfaces, with time and space scales on the order of days and kilometers. This thesis builds theory around KP timeseries behavior by solving for the analytic solutions in the particular cases that one KP is …

Open Neighbourhood Edge Degree And Metric Descriptors Qspr And Multi Criteria Analysis Of Chemicals And Antiviral Drugs, Gayathri A Ms

Theses and Dissertations

Chemical Graph theory offers powerful computational methods for evaluating chemical structures, significantly impacting cheminformatics and drug analysis. It provides a robust mathematical framework to represent, model, and analyse molecular systems through topological indices and graph invariants. These indices serve as effective numerical descriptors, encapsulating significant structural information that enhances the prediction of physicochemical and biological properties. This thesis utilizes mathematical modeling, algorithmic computation, and decision making process to derive extensive insights to drug discovery and molecular analysis.

The thesis introduces a new class of topological indices ONE1, ONE2, ONE3, ONE4, ONE5,ONE6 and ONE7 based on open-neighbourhood-edge-degree and it evaluates the …

Coupled-Pendulum Modeling In An Ode Class: An Assignment On Fourier-Initialized Gradient Descent In Machine Learning, Huy Truong, Andrew Bennett

CODEE Journal

As data-driven methods are increasingly used in science and engineering, students benefit from learning to integrate machine learning techniques with traditional mathematical modeling. We present a hands-on extra-credit assignment for an undergraduate ordinary differential equations (ODE) course that enables students to compare classical analytical methods with data-driven approaches on the same physical system. Using a coupled-pendulum system---two pendulums connected by a spring---with real experimental data acquired via video tracking of a real physical setup, students work through three models in a guided Jupyter notebook with all code provided. First, they fit a neural network with Fourier features as a purely …