Mathematics Commons^™

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.

Tailoring Evaluations Of Chronic Rhinosinusitis: Understanding Sleep And Its Effect On Memory Through Actigraphy, Donyea Moore, Rachel Nolte, Yitong Pepper Huang, Shreya Maharana, Pavan Nataraj, Bichun Ouyang, Mahboobeh Mahdavinia

Mathematics Sciences: Faculty Publications

Background/Objectives: Chronic rhinosinusitis (CRS) is a persistent inflammatory condition of the sinonasal mucosa lasting for at least three months. For patients, CRS-related sleep disturbances can significantly disrupt circadian rhythms, leading to further health complications such as cognitive impairment. Despite the well-documented sleep disturbances associated with CRS, there is limited research on objective assessment methods. Additionally, the severity of these issues can vary among patients. This study aims to assess sleep quality and timing in CRS patients and investigate their impact on cognition, providing guidance for personalized and tailored assessment and management of CRS. Methods: Our case–control study compares sleep patterns …

A Novel Fractional Order Model For Analyzing Counterterrorism Operations And Mitigating Extremism, Mutaz Mohammad, Isa Abdullahi Baba, Evren Hincal, Fathalla A. Rihan

All Works

This study examines the profound impact of terrorism on individuals and society by developing a fractional-order mathematical model to analyze and enhance counterterrorism efforts. The model accounts for the persistent and complex nature of extremist behavior, particularly emphasizing the importance of preventing violent extremism before it escalates into terrorism. Real-world data on terrorist activities in Nigeria – specifically from the Boko Haram insurgency – was used to calibrate and validate the model, ensuring its relevance and accuracy. The model reveals that the basic reproduction number (R0) plays a decisive role in determining the long-term success of counterterrorism strategies. Numerical simulations …

Steiner Coset Partitions Of Groups, Fusun Akman, Papa Sissokho

Faculty Publications – Mathematics

A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture, which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups H₁,...,H_r, is a coset partition of G that contains exactly one coset of each …

Radiation-Induced Cardiotoxicity In Hypertensive Salt-Sensitive Rats: A Feasibility Study, Dayeong An, Alison Kriegel, Suresh N. Kumar, Heather A. Himburg, Brian Fish, S. Klawikowski, Daniel B. Rowe, Marek Lenarczyk, John Baker, El Sayed H. Ibrahim

Mathematical and Statistical Science Faculty Research and Publications

Radiation therapy (RT) plays a vital role in managing thoracic cancers, though it can lead to adverse effects, including significant cardiotoxicity. Understanding the risk factors like hypertension in RT is important for patient prognosis and management. A Dahl salt-sensitive (SS) female rat model was used to study hypertension effect on RT-induced cardiotoxicity. Rats were fed a high-salt diet to induce hypertension and then divided into RT and sham groups. The RT group received 24 Gy of whole-heart irradiation. Cardiac function was evaluated using MRI and blood pressure measurements at baseline, 8 weeks and 12 weeks post-RT. Histological examination was performed …

A Novel Closed Monoidal Structure On The Nucleus Of A Profunctor, Samantha K. Jarvis

Dissertations, Theses, and Capstone Projects

We describe a novel closed monoidal structure on the nucleus of a profunctor enriched over a posetal category, a distinguished subcategory of the category of presheaves given by the invariant part of an adjunction induced by the profunctor. Our structure is motivated by a connection to a notion of type given by orthogonality. For specific examples, we consider the two-element enriching category {0,1} and the reals.

Parametrization Of Formal Norm Compatible Sequences, Joseph Dicapua

Dissertations, Theses, and Capstone Projects

We give a classification of power series parametrizing Lubin-Tate trace compatible sequences. This proof answers a question posed in the literature by Berger and Fourquaux. Lubin-Tate trace compatible sequences are a generalization of norm compatible sequences, which arise in Iwasawa theory and local class field theory. The result we prove generalizes the interpolation theorem proved by Coleman in the classical norm compatible sequence case. We also, jointly with Victor Kolyvagin, give a method for finding such series explicitly in certain special cases.

Axon Initial Segment Plasticity Caused By Auditory Deprivation Degrades Time Difference Sensitivity In A Model Of Neural Responses To Cochlear Implants, Anna Jing , '24, Sylvia Xi , '25, Ivan Fransazov , '24, Joshua H. Goldwyn

Mathematics & Statistics Faculty Works

Synaptic and neural properties can change during periods of auditory deprivation. These changes may disrupt the computations that neurons perform. In the brainstem of chickens, auditory deprivation can lead to changes in the size and biophysics of the axon initial segment (AIS) of neurons in the sound source localization circuit. This is the phenomenon of axon initial segment (AIS) plasticity. Individuals who use cochlear implants (CIs) experience periods of hearing loss, and so we ask whether AIS plasticity in neurons of the medial superior olive (MSO), a key stage of sound location processing, would impact time difference sensitivity in the …

Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses

Master's Theses

This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

The Degrees Of The Irreducible Representations Of The Symmetric Group With Respect To Primes, Vanessa F. Beeler

Master's Theses

In this thesis, we explore the relationship between the degrees of irreducible matrix representations of the symmetric group and prime numbers. We start by building up the required background necessary to construct our results. We dive into topics of discrete mathematics including matrix representations, characters of representations, integer partitions, conjugacy classes, class functions, and tableaux. Some other key ingredients in our work include using character tables to classify the irreducible rep- resentations of the symmetric group and the hook-length formula to easily compute the degrees of these representations. Once we have laid the groundwork for our in- vestigation, we begin …

A Realizability Approach To Constructing Higher Types Via Classifiers, Benjamin Carrick Logsdon

Dartmouth College Ph.D Dissertations

We construct an interpretation of higher types into Peano arithmetic, showing in particular that every model of PA is a model of higher types. This is a reversal of Gödel’s Dialectica construction. We also define the classifier degrees, a degree structure which subsumes the Turing degrees, the enumeration degrees, and the many-one degrees. The classifier degrees boast a rich structure and many well-behaved operations.

Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac

Dartmouth College Ph.D Dissertations

In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …

Computability Theoretic Aspects Of Profinite Groups And Models Of Presburger Arithmetic, Jason Block

Dissertations, Theses, and Capstone Projects

Profinite groups, which are exactly the Galois groups, are all either finite or uncountable. However, all second countable profinite groups can be presented as the set of paths through a countable tree. We use these tree presentations to find upper bounds on the complexity of the existential theories of profinite groups, as well as to prove sharpness for these bounds. These complexity results enable us to distinguish the class of profinite groups that are isomorphic to a direct product of finite groups, for which we find an upper bound on the complexity of the entire first order theory. Additionally, given …

Numerical Methods For Approximating Line Integrals Over Implicitly Defined Curves, Raghd Alsaadawi

Theses

In this thesis, we develop and investigate a predictor-corrector method for the numerical tracing of implicitly defined curves. The study begins with the introduction of modified numerical integration techniques — specifically, the modified trapezoidal and modified midpoint rules — for evaluating the line integral of a vector field along an implicitly defined curve. Furthermore, we explore higher-order methods aimed at improving the accuracy of such integrals. Theoretical and numerical results, including asymptotic error expansions, are presented to support the analysis. In addition, several numerical experiments are carried out to illustrate the effectiveness and robustness of the proposed approaches.

Wavelet-Based Multi-Step Methods For Systems Of Differential Equations, Rashad Assad Hijji

Theses

Wavelets have been widely used in many areas of engineering and mathematics, including the development of multistep algorithms to solve initial value problems (IVPs) in the context of the Galerkin method using Daubechies' wavelets. The main scope of our work is to build a comprehensive framework for solving Systems of Differential equations using the compactly supported wavelets proposed by I. Daubechies. Wavelets are mathematical functions that decompose data into distinct frequency components, and each element is analyzed with a resolution that matches its scale. Compact support of Daubechies wavelets is key in allowing them to be computationally efficient for high-dimensional …

Predictors Of Nursing Students' Stress, Anxiety, And Depression During The Covid-19 Pandemic In A Hispanic-Serving University In South Texas: A Cross-Sectional Study, Maria I. Diaz, Eleftherios Gkioulekas, Nancy Nadeau

School of Mathematical & Statistical Sciences Faculty Publications

Background: In nursing education, there have been several studies on the impact of the COVID-19 pandemic on the ability of nursing students to cope while in nursing school.

Purpose statement: The goal of this study is to assess undergraduate nursing students' support mechanisms as predictors of stress, anxiety, and depression during the COVID-19 pandemic within a Hispanic-serving institution in South Texas.

Methods: Across-sectional design was used in this study. An online survey using self-reported questionnaires was used to gather data from an undergraduate nursing student cohort during the Fall 2021 semester. Linear regression was used to identify the predictors of …

“What Makes It Eigen-Esque-Ish?”: A Form-Function Analysis Of The Development Of Eigentheory Concepts In A Quantum Mechanics Course, Megan Wawro, Kaitlyn Stephens Serbin

School of Mathematical & Statistical Sciences Faculty Publications

Eigentheory concepts are central in mathematics and physics; they serve multiple functions, such as symbolizing physical phenomena and facilitating mathematical computations. Words and meanings associated with eigentheory develop and vary over time, as do their associated symbols. In this study, we investigate how “eigen” develops over time in one quantum mechanics course by analyzing form-function relations (Saxe, 1999) for eigentheory concepts over 22 class sessions. We share results concerning our microgenetic and ontogenetic analyses of the creation of form-function relations and their shifts over time by characterizing the continuity and discontinuity of the various functions and forms associated with concepts …

Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie

Master's Theses

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …

On Diffeomorphism Groups Of Surfaces, Madeleine Goertz

Master's Theses

Let $M$ be a closed, connected, smooth manifold. What are the symmetries of $M$? From a geometric viewpoint, the symmetries of $M$ are precisely its isometries, the self maps which preserve lengths and angles. In the smooth category, the symmetries of $M$ are its diffeomorphisms, the self maps which are smooth and have a smooth inverse. Does expanding our notion of symmetries to include diffeomorphisms result in ``more'' symmetries in a meaningful sense? As a formal conjecture, the claim is that the isometry group of $M$ is a deformation retract of the diffeomorphism group of $M$. If $M$ is the …

Topiarism: The Kernel Embedding Of Distributions Applied To Modern Portfolio Theory, Stephen G. Cook

Master's Theses

The method of kernel embedding of distributions on a set $\Omega$ into a reproducing kernel Hilbert space is a method of studying the space of measures on a set $\Omega$ using Hilbert space geometry. Because positively weighted portfolios can be interpreted as nonnegative probability measures on the space of assets, we are able to apply this technique to portfolio theory. In this thesis, we discuss the theory of "topiarism", the study of positively weighted probability measures on compact sets under the kernel embedding of distributions. Given a specific payoff function $\psi$ on the set of assets $\Omega$, we optimize a …

On The Design Of A Framework For Large-Scale Exploratory Graph Analytics, Oliver Andres Alvarado Rodriguez

Dissertations

Large-scale exploratory graph analytics merges data science with high-performance computing to extract critical insights from network-representable data. Data scientists routinely analyze data from the natural, social, and computing sciences by representing it as networks, or graphs, where objects become vertices and their relationships become edges. This representation allows data scientists to add graph analytics to their toolbox. However, designing tools for large-scale exploratory graph analytics is challenging due to the complexities of graph algorithms, such as high communication in distributed systems and large memory demands. These challenges can lead to overly complex software, which limits usability and development to a …

From Neural Networks To Large Language Models: Innovations In Financial Ai, Mathematical Reasoning, And Structured Data Representation, Junyi Ye

Dissertations

This dissertation explores the evolution and application of artificial intelligence techniques across three critical domains: financial modeling, mathematical reasoning, and structured data analysis. The dissertation presents seven research projects that chart a progression from specialized neural architectures to sophisticated large language models (LLMs), contributing novel methodologies and frameworks at each stage.

In the financial domain, the research first introduces TS-Mixer, a MLP-based architecture for time-series forecasting that captures both feature relationships and temporal dependencies through a simple yet effective design, outperforming more complex models in S&P500 index prediction. The dissertation then presents DySTAGE, a dynamic graph representation learning framework that …

A Novel Framework For Dynamic Graph Representation Learning With Mamba, Ashish Pandey

Theses

Dynamic graph embedding is a key technique for modeling temporal dependencies in evolving networks. While transformer-based models perform well, their quadratic complexity limits scalability on long graph sequences. This thesis compares transformer approaches with the Mamba architecture-a linear-complexity state-space model—for temporal graph embedding.

Two frameworks are proposed: DG-Mamba and GDG-Mamba. DG-Mamba uses standard GCN-based spatial encoding, while GDG-Mamba incorporates domain-aware edge features using Graph Isomorphism Network with Edge Convolution (GraphGINE). Experiments on UCI, Reality Mining, Slashdot, Bitcoin-OTC, and SBM datasets show that Mamba-based models match or exceed transformer performance, especially on graphs with high temporal variability.

The thesis also applies …

The Jacod-Yor Theorem For Sigma Martingales And The Second Fundamental, Moritz Sohns

Journal of Stochastic Analysis

In this paper, we prove the Jacod-Yor Theorem for sigma martingales, a class of processes that generalize local martingales and play a pivotal role in financial mathematics. While the Jacod-Yor Theorem has been extensively studied for L2-martingales, martingales, and local martingales, no prior version exists for sigma martingales. Our result establishes the connection between sigma martingales and their martingale representation properties, addressing a critical gap in the literature. As an application, we prove the Second Fundamental Theorem of Asset Pricing for markets where price processes are modeled as sigma martingales.

The Witten Deformation And Proper Cocompact Lie Group Actions, Hao Zhuang

Arts & Sciences Graduate Student Theses and Dissertations

We study the interactions between the Witten deformation of the de Rham exterior differentiation and topological invariants in two scenarios of proper Lie group actions. In the first scenario, we work on a closed oriented manifold admitting an action by a compact connected Lie group. Using a special Morse-Bott function invariant under the group action, we deform the de Rham exterior derivative and get the associated Witten Laplacian. Applying asymptotic analysis, we localize the kernel of the Witten Laplacian around the critical components of the invariant Morse-Bott function. Finally, we build the chain isomorphism between the invariant Thom-Smale complex and …

The Mckay-Navarro Conjecture For The Prime 2, L. Ruhstorfer, A. A. Schaeffer Fry

Mathematics: Faculty Scholarship

We complete the proof of the McKay-Navarro conjecture (also known as the Galois-McKay conjecture) for the prime 2, by completing the proof of the inductive McKay-Navarro conditions introduced by Navarro-Späth-Vallejo for this prime.

Lower Bounds For The Total Distance $K$-Domination Number Of A Graph, Randy R. Davila

Theory & Applications of Graphs

For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total distance $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than itself. The \emph{total distance $k$-domination number} of $G$ is the minimum cardinality of a total $k$-dominating set in $G$ and is denoted by $\gamma_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $\gamma_t(G)$; that is, $\gamma_t(G) = \gamma_{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize …

The Integer-Antimagic Spectra Of A Weak Join Of Hamiltonian Graphs, Ugur Odabasi, Dan Roberts, Richard M. Low

Theory & Applications of Graphs

A simple graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is \emph{$\mathbb{Z}_{k}$-antimagic} if there exists a function $f: E(G) \to \mathbb{Z}_{k} \backslash \{0\}$ such that the induced function $f^+(v)=\sum_{uv\in E(G)} f(uv)$ is injective. The \textit{integer-antimagic spectrum} of a graph $G$ is the set IAM$(G) = \{k: G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic and } k \geq 2\}$. A \emph{weak join} of vertex-disjoint graphs is the collection of the graphs with additional simple edges (possibly none) between the original graphs. In this paper, we characterize IAM$(H)$ where $H$ is a weak join of Hamiltonian graphs.

Prime Labelings On A 3xn Grid Graph, Stephen J. Curran, Matt A. Ollis

Theory & Applications of Graphs

It is conjectured that the mxn grid graph has a prime labeling for all positive integers m and n. It is known that for any prime p and any integer n such that 1≤n≤p², there exists a prime labeling on the pxn grid graph P_m x P_n. Also, it is known that the ladder P₂ x P_n has a prime labeling for all positive integers n. We assume that Goldbach's Even Conjecture and a strengthened variant of Lemoine's Conjecture are true in order to show that the 3xn grid graph P …