Mathematics Commons^™

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.

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 …

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 …

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 …

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 …

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.

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.

“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 …

High Moment And Pathwise Error Estimates For Fully Discrete Mixed Finite Element Approximations Of The Stochastic Stokes Equations With Multiplicative Noise, Liet Vo

School of Mathematical & Statistical Sciences Faculty Publications

This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler–Maruyama scheme for time discretization and its fully discrete mixed finite element discretization. Optimal rates of convergence are established for all pth moment errors for p ≥ 2 using a novel doubling of moments technique. The almost optimal rates of convergence are then obtained using Kolmogorov’s theorem based on the high moment error estimates. Unlike for the velocity error estimate, the high moment and pathwise error estimates for the pressure approximation are proved in a time-averaged norm. In addition, the …

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 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.

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 …

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 …

Agent-Based Modeling: Introduction And Actuarial Applications, Rick Gorvett

Mathematics and Economics Faculty Working Papers

Agent-based modeling (ABM) has become an important and valued approach to modeling complex systems. In this paper, I advocate for actuaries to recognize the complex systems-nature of socioeconomic and risk processes and for ABM models to become a regular resource in our actuarial toolkits. These models allow for the observation of potential macro-behavior emerging from the underlying agent-level micro-activity and characteristics. Therefore, ABM models can provide significant insight into the quantification of risk and the identification of optimal strategies. This paper is an introduction and guide to ABM models, and it includes several case studies to illustrate their utility.

A Selective Discontinuous Galerkin Implicit Particle-In-Cell Method For Plasma Simulation With Improved Interpolation, Siyu Wu, Yang Li, Hongtao Liu, Xiaoming He, Yong Cao

Mathematics and Statistics Faculty Research & Creative Works

This article dynamically incorporates multiple ideas into the existing direct implicit particle-in-cell (DIPIC) method for plasma simulation, in order to dramatically improve the DIPIC method for its local mesh refinement needs based on Cartesian meshes as well as its interpolation needs based on the locally refined meshes. One key tool is to utilize the selective discontinuous Galerkin method, which is based on the interior penalty discontinuous Galerkin formulation and the regular local finite element basis functions, as the electric field solver in the DIPIC simulation. This hybrid type finite element method combines the advantages of both continuous and discontinuous finite …

A Short Proof Of The Lemma Of The Logarithmic Derivative In Several Complex Variables, Qi Han, Jingbo Liu

All Faculty Scholarship (Archived)

In this work, we provide a concise proof of the logarithmic derivative lemma in Cⁿ. We start by giving a complete proof of a critical result by Biancofiore and Stoll (Ann. of Math. Stud., No. 100, 29–45, 1981) via matrix theory, which is central to this area of study. Then, we follow Li (Trans. Amer. Math. Soc. 363, 6257–6267, 2011) and Ye (Math. Z. 222, 81–95, 1996) to present the shortest proof to date.

Math 115: College Algebra Instructor Guide, Seth Lehman

Open Educational Resources

OER instructor guide for Math 115, College Algebra, Queens College

Conditional Quantization For Some Discrete Distributions, Edgar A. Gonzalez, Mrinal Kanti Roychowdhury, David A. Salinas, Vishal Veeramachaneni

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a conditional quantization. In this paper, we have determined the conditional quantization, first for two different finite discrete distributions with a same conditional set, and for a finite discrete distribution with two different conditional sets. Next, we have determined the conditional and unconditional quantization for an infinite discrete distribution with support {12𝑛:𝑛∈ℕ}. We have …

All Graphs Of Order N With Distinguishing Number N−1 Or N − 2, Andi Pujo Rahadi, Edy Tri Baskoro, Suhadi Wido Saputro

Theory & Applications of Graphs

Let G be a simple connected graph. The distinguishing number of G, denoted by D(G), is the least integer d such that G has a vertex d-labeling preserved only by the trivial automorphism. In this paper, we characterize all graphs of order n with distinguishing number n − 1, or n − 2.

Similarity Metrics, Metrics, And Conditionally Negative Definite Functions, Daniel Alpay, Liora Mayats-Alpay

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give an example of a similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.

Global Attractivity Of Positive Solutions Of Higher Order Nonlinear Difference Equations And Its Applications, Abdulaziz Almaslokh

Theses and Dissertations

Difference equations play an effective role in many mathematical models that represent some phenomena in different fields such as physics, economics, engineering and biology. From this perspective, studying and analyzing such equations are interesting issues for the advancement of scientific research. In particular, studying the asymptotic behavior of positive solutions of some population models that are a system of difference equations has become a necessity. In this dissertation, we examine the global attractivity of positive solutions of certain higher order nonlinear difference equations and apply these results to some population models.

Entanglement Distribution Of Universal Quantum Cloners, James B. Daughtry

Student Scholarship

Szab`o et al. [4] show that universal quantum cloning machines may be used to manipulate entanglement between bipartite states. Entanglement is measured with concurrence, C, and it is shown that as the fidelity of the cloner improves, more entanglement transfers from the reference to the cloned state. This thesis characterizes the nature of this entanglement transfer by analyzing the entanglement distribution of the cloner. In this analysis we will show that entanglement is generally destroyed after cloning except in cases where there is no cloning, two identical qubits are created, or there is complete qubit transfer. Along with …

On The Special Cases Of Carmichael’S Totient Conjecture, Christopher Saito

UNLV Theses, Dissertations, Professional Papers, and Capstones

Euler’s totient function, φ(n), is the arithmetic function defined as the number of positive integers less than or equal to n that are relatively prime to n. In his 1922 paper [3], Professor R. D. Carmichael conjectured that for each positive integer n, there exists at least one positive integer m̸ = n such that φ(m) = φ(n).In this thesis, we consider some relevant literature and explore Carmichael’s totient conjecture for particular values of φ(n) = k. Our main consideration will be the set X_k = {n ∈ N : φ(n) = k}. In identifying X_k for k = 2^t, …

The Concept Of T-Basis And Vector-Valued Hardy Classes, Bilal Bilalov, Sabina Rahib Sadigova

Turkish Journal of Mathematics

This paper introduces the concept of a t-basis generated by some bilinear mapping t(·;·). It is considered the vector-valued class L_p(X) := L_p(J; X), 1 ≤ p < +∞, where J = [−π, π] and X is a Banach space with the UMD property, and it is proven that the classical system of exponents {e^int}_n∈ℤ forms a t-basis for L_p(X), 1 < p < +∞. Using this fact, the Hardy vector classes _nH_p^±(X), 1 < p < +∞, different from the classical ones, are defined, and an equivalent definition of these classes is given and some of their properties are studied. In addition, the concept of t-Riesz property of a system of exponentials is introduced in L_p(X), 1 < p < +∞, and it is proved that this system has the t-Riesz property. A new method is given for establishing the Plemelj–Sokhotski formulas for X-valued Cauchy type integrals when X has the UMD property. An abstract analogue of the “1/4-Kadets” theorem is obtained for …

Independent Vs. Signed Musicians: Does Thier Status Affect Their Success?, Zachary Vincent

Honors Projects in Mathematics and Economics

As technology has advanced and become more widely available, and as the public learns more about the recording industry and its workings, more musicians are choosing not to sign with record labels, becoming independent musicians. This now raises the question: does a musician’s status of independent or signed determine their level of success? It has been previously found that independent musicians have a more difficult time competing with signed musicians due to the resources provided by record labels and the way labels negotiate with large streaming platforms, giving signed labels more opportunities to be seen in the public eye, but …