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Explicit Inverse Of Symmetric, Tridiagonal Near Toeplitz Matrices With Strictly Diagonally Dominant Toeplitz Part, Bakytzhan Kurmanbek, Yogi Erlangga, Yerlan Amanbek

All Works

Let T n = tridiag ( - 1 , b , - 1 ) , an n x n symmetric, strictly diagonally dominant tridiagonal matrix ( divided by b divided by > 2 ). This article investigates tridiagonal near-Toeplitz matrices T n & colone; [ t i , j ] , obtained by perturbing the ( 1 , 1 ) and ( n , n ) entry of T n . Let t 1 , 1 = t n , n = b not equal b . We derive exact inverses of T n . Furthermore, we demonstrate that these results …

Frobenius And Commutative Pseudomonoids In The Bicategory Of Spans, Ivan Contreras, Rajan Amit Mehta, Walker H. Stern

Mathematics Sciences: Faculty Publications

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in Span correspond, respectively, to paracyclic sets and Γ-sets satisfying the 2-Segal conditions. These results connect closely with work of the third author on A∞ algebras in ∞-categories of spans, as well as the growing body of work on higher Segal objects. Because …

A Conversation On Fundamental Data Literacy Concepts For Undergraduate Education, Anna Bargagliotti, Wendy Pothier, David Homa, Arshia Mathus, Keith Mccormick, Paula Payton, Brian Wright

Mathematics, Statistics and Data Science Faculty Works

As the demand for data literacy grows, integrating foundational data literacy skills into undergraduate education becomes increasingly essential. This panel presents six core themes for cultivating data literacy among undergraduate students, addressing its interdisciplinary nature. Drawing upon the collective expertise of the group, literature research, case studies, and interviews, the panel explores the need for data literacy across various disciplines and the challenges of integrating it into higher education curricula. The proposed fundamental concepts include understanding data’s role in daily life, distinguishing between inference and prediction, recognizing the potential for misleading data, exploring ethical considerations, and honing communication and storytelling …

Extremal Trees For Random Walks, Ben Bridenbaugh

Mathematics, Statistics, and Computer Science Honors Projects

A random walk is a sequence of adjacent vertices that are chosen uniformly at random from the neighbors of the previous vertex. An access time is the average length of time that a random walk takes to reach a target probability distribution from a starting probability distribution, given an optimal stopping rule. This paper deals with characterizing the trees of diameter d and on n vertices that extremize three different types of access times.

The Inverse Scattering Transform For The Nonlinear Schrödinger Equation, Ivan Casas-Rocha

Honors Undergraduate Theses

The Nonlinear Schrödinger (NLS) Equation, iψ_t + 1/2 ψ_xx ± |ψ|²ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, …

Prime Factorization And Unit Calculations Of Quadratic Integer Rings, Gabriel F. Roca

Honors Undergraduate Theses

The failure of unique factorization in a ring leads to the investigation of the closest algebraic structure, which are prime ideals. Using generalizations that have helped solve questions such as Fermat's Last Theorem, there is interest to study the elements with a multiplicative inverse (units) via the geometry and arithmetic patterns that arise in quadratic integer rings, since they provide tools for other questions in mathematics, ranging from pure algebra to applications in cryptography, and more. Overall, the following thesis provides a small exposition on the theory of integral domains and some specific calculations.

Solvability Of Stochastic Linear-Quadratic Optimal Control Problems Under Partial Stabilizability Conditions, Al-Sadh Rahman Imadh

Honors Undergraduate Theses

Optimal Control Theory, a branch of Control Theory, is applicable in fields such as engineering, operations research, and economics. Stochastic Optimal Control deals with noisy systems and data using Ito’s formulation. Given a noisy system and a cost functional, the goal is to find a control that will minimize the cost. This thesis focuses on linear quadratic stochastic optimal control, and we explore state equations that are not stabilizable. We first address measurability concerns arising from the semigroup property of the state trajectory. The notions of partial stability and partial stabilizability are introduced, and we formulate their corresponding Lyapunov and …

Quantized Average Agreement Algorithms With Error Correction For Digraphs, Shuaib A. Mughal

Honors Undergraduate Theses

Multi-agent systems have become more and more prevalent as technology increasingly gets integrated into our daily lives. Some of these technological systems are large in size; for example, the smart grid where multiple devices are used to monitor and control different aspects of the energy grid. Another example is a team of autonomous systems deployed for a specific task. When these systems are spatially distributed, an important component of distributed algorithms is the ability for the agents to reach consensus on the global state of the system. Reaching agreement enables the spatially distributed agent make decisions or determine the next …

Abstractions Of The Game “Set”, Martin Grant

Master's Projects

Each card in the game of SET can be represented as a point in Z⁴₃, where Z₃ is the f ield of 3 elements; an in-game position without any SETs can be represented as a cap set. We find the largest cap sets in Zⁿ ₃ for n ≤ 4 and prove their uniqueness. Then, we provide more insight into Ellenberg and Gijswijt’s proof of upper bound for the maximum size of cap set in Fⁿ_q, where F_q is the field of q elements, which they find to be o(c^{n …}

Generalizations Of Some Q-Product Identities Of Ramanujan And Others, Timothy J. Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

By considering certain limiting cases of a WP-Bailey chain discovered by Andrews, and also limiting cases of certain classical summation formulae for basic hypergeometric series, we derive new expressions for certain Lambert series in terms of basic hypergeometric series. In some cases, the resulting series involve an arbitrary Bailey pair. This allows for the derivation of new basic hypergeometric expansions for some q-products and series that Ramanujan expressed in terms of Lambert series. Some of Ramanujan’s identities are extended to more general relations containing one or more free parameters.

Polar Decompositions And Spectral Properties Of Linear Operator Pencils, Slaviša Djordjević, Jaewoong Kim, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair T=(T1,T2) . Using this decomposition, we define the generalized spherical Aluthge transforms for the linear pencil of an ordered pair T=(T1,T2) and give spectral properties between the linear pencil and the generalized spherical Aluthge transform of it. In detail, we study whether the spectral properties of the linear pencil of an ordered pair T=(T1,T2) are invariant under the generalized spherical Aluthge transform.

The Taylor Resolution Over A Skew Polynomial Ring, Luigi Ferraro, Desiree Martin, W. Frank Moore

School of Mathematical & Statistical Sciences Faculty Publications

Let k be a field and let I be a monomial ideal in the polynomial ring Q = k[x1, . . . , xn]. In her thesis, Taylor introduced a complex which provides a finite free resolution for Q/I as a Q-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring R. Under the hypothesis that the skew commuting parameters defining R are roots of unity, we prove as an application that …

Editorial Insights: Reflections On The Volume And Charge To The Field, Heather Howell, Meghan Shaughnessy, Barbara Stengal, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Gregory Benoit, Yvonne Lai

School of Mathematical & Statistical Sciences Faculty Publications

In this, the final chapter of Section 3, the authors document an interview that took place in June of 2024, as the editorial team wrapped up reviews of the chapters submitted to this volume. The interview was led by two of the advisory board members, Meghan Shaughnessy and Barbara Stengal, and included the editorial group, Heather Howell, Carrie Wilkerson Lee, Liza Bondurant, and Bima Sapkota, and other members of the advisory board, Greg Benoit and Yvonne Lai. The motivation to create this interview-based chapter was to capture some of the authors' own learning and reflection as they brought the volume …

Oer Ancient Egyptian Numerals Activity, Cynthia Huffman Ph.D.

Open Educational Resources - Math

Ancient Egypt captures our imagination not just because of its pyramids, but because of its incredible staying power. This civilization thrived for over 3,000 years, influencing cultures across the globe—including the very roots of mathematics. As Egyptologist Bob Brier puts it, “No civilization lasted so long, contributed so much, or repeatedly amazed as did ancient Egypt.”

Ready to crack the code of Ancient Egypt and the pharoahs? In this activity, you will learn to decipher Egyptian numerals by examining real photos from temples and museums. Master this skill, and next time you visit a museum exhibit on ancient Egypt, you …

Fairness-Aware And Culturally Adaptive Machine Learning For Predicting Adolescent Substance Use, Stephanie Nworgu

Senior Honors Theses and Projects

Early initiation of substance use during adolescence poses significant risks to long-term health, educational attainment, and social outcomes, making early identification a critical public health priority. Machine learning models have increasingly been used to predict substance use risk; however, many such models do not explicitly examine whether predictive performance differs across demographic groups. This senior project examines the fairness of logistic regression models used to predict first-time alcohol use among adolescents. Using nationally representative survey data from the Youth Risk Behavior Surveillance System (YRBSS), pooled across the 2017, 2019, 2021, and 2023 survey cycles, this study develops logistic regression–based predictive …

Sickle-Cell Genotyping Cost Analysis In Amua And R, Nicholas P. Haley

Senior Honors Theses and Projects

Sickle cell disease (SCD) is a prevalent genetic disorder in the United States, with a significant economic burden due to the high costs of care, especially from chronic red blood cell (RBC) transfusions. These transfusions carry risks such as alloimmunization, which can lead to complications like delayed hemolytic transfusion reactions (DHTRs), further increasing healthcare costs in addition to increasing suGering. This study aims to aid in the evaluation of cost-eGective strategies for SCD management using Markov models, currently within TreeAge Pro software, which is proprietary. A decision model was adapted from the Kacker et al. (2013) study, which analyzed the …

Cost-Effective Strategies For Feral Swine Control: Exploring Trapping Equipment Cooperatives, Phil Kenkel, Riza Radmehr, Rodney Holcomb, Mckenzie Boyce

Human–Wildlife Interactions

Feral swine (Sus scrofa) in the United States have become a growing concern, with a population of ≥6 million and causing annual damages of ≥$1.5 billion USD. Because of their high reproductive capacity, effective control methods are crucial to reducing or slowing their population growth. While remotely monitored and triggered traps have proven to be an effective control strategy, the cost of sophisticated trapping equipment, which can cost ≥$7,000 USD, is often cost prohibitive for many landowners. Despite the pressing need to address the challenge of making effective feral swine control methods more affordable, there are few studies …

Studies On Convexity Of Dnf Formulae, Josue A. Ruiz

Electronic Theses & Dissertations (2024 - present)

In this dissertation, we investigate the problem of determining whether a Boolean formula given in disjunctive normal form (DNF) is convex. Although Boolean formulas have various applications, our research focuses on the practical application for rule-based access control policies, where policies are often expressed as a set of Boolean rules. Understanding the structural properties of such formulas is crucial for determining whether a policy can be efficiently represented within a specific access control model.

The main contribution of this research is the conception and analysis of convexity derived from the “gap problem.” In this context, convexity is characterized by the …

Betti Numbers Of Generic Ideals, Jason R. Howell

Electronic Theses & Dissertations (2024 - present)

We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe …

Comparison Of Methane Emission Quantification Methods At Landfills In New York State, Alexandra M. Catena

Electronic Theses & Dissertations (2024 - present)

Greenhouse gas (GHG) emission increases are a consequence of global population rise and economic growth. It is well known that carbon dioxide is the dominant GHG by mass, but methane has received increasing attention recently due to its high warming potential and relatively short lifetime of about a decade. Methane has several different types of both natural and anthropogenic sources including wetlands, waste management, energy, agriculture and farming, and wildfires, which makes tracking and reducing methane emissions a complex problem. The emission rates of each of these sources are highly uncertain due to differing emission estimation methods, which can be …

Geometric Properties Of Positive Definite Matrices: Means, Order, And Metrics, Blaine Dubois

UNF Graduate Theses and Dissertations

In this thesis, we study matrix means from a geometric point of view. In particular, we consider divergences of the form Tr[A + B − 2G(A, B)] for certain Geometric-Type matrix means G(A, B). We derive alternative formulations of this distance function through the application of one-sided inverses of G(A, B). When G(A, B) = A#B, we give a curve parametrization of the straight-line path between two points with respect to this semi-metric and present conditions under which this holds for other Geometric-Type means. For read- ability and self-containment, we introduce most of the preliminary concepts to build up to …

Theoretical Foundations And Applied Performance Of Periodicity-Aware Imputation: Variable Bandpass Block Bootstrap Methods For Incomplete Time Series, Asmaa Ahmad

Electronic Theses & Dissertations (2024 - present)

Time series data are prevalent across a wide range of disciplines, including health surveillance, public policy, and environmental monitoring. In the presence of underlying cyclical patterns, the integrity of time series analysis depends critically on the ability to detect, model, and impute structured missing data without compromising the temporal structure. This dissertation introduces and validates a novel imputation framework that integrates the Variable Bandpass Periodic Block Bootstrap (VBPBB) into multiple imputation procedures, improving the accuracy, robustness, and interpretability of time series models under high rates of missingness and noise. The overarching goal of this dissertation was to develop and evaluate …

Knörr Lattices For Elementary Abelian Groups Over Unramified Extensions Of Z_P, Marie Swartz

Graduate Research Theses & Dissertations

In this dissertation, we examine the existence of positive height Knörr RD-lattices for D an elementary abelian p-group and R an unramified extension of the p-adics Zp. Knörr lattices form an important subclass of the absolutely indecomposable lattices and have close connections to modular representation theory’s “local to global” behavior. We generalize many prior results, shown only when p = 2, to odd p when |D| = p3. Further, when p = 3, we show that a positive height Knörr RD lattice must have a projective summand when restricted to any maximal subgroup of D. The proof of this requires …

Geometric Function Theory On Uniformly Quasiconformally Homogeneous Domains, Allyson Mackenzie Hahn

Graduate Research Theses & Dissertations

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Uniformly quasiconformally homogeneous domains in $\mathbb{R}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\geq 1.$ In this dissertation we investigate three questions in this setting. The first is to show that quasiconformality and quasisymmetry with respect to the quasihyperbolic metric are equivalent. The second is to study normal quasiregular maps from such a domain into $S^n$ or $\mathbb{R}^n$ and determine they hold geometric properties such as a uniform Hölder condition. The third is to study preimages of a power-type mapping and show they are well-distributed in a precise sense.

Efficient Algorithms For Nearest Correlation Matrix Computation With Missing Data, Ibrahim Eniola Oyeyinka

Graduate Research Theses & Dissertations

This thesis investigates efficient algorithms for computing the Nearest Correlation Matrix (NCM) under incomplete financial data. Correlation matrices are vital in portfolio optimization and risk management, yet empirical estimates often violate symmetry, positive semidefiniteness, and unit diagonal conditions due to missing observations. Two projection-based methods are analyzed: the Modified Alternating Projections (MAP) and Anderson Acceleration (AA). Theoretical analysis using convex optimization and normal cone characterization supports numerical evaluation on synthetic and real-world stock-return matrices (550×550, 2020–2025). Missing data are modeled through Missing Completely at Random (MCAR) and Not Missing at Random (NMAR) mechanisms. The results show that AA converges faster …

Bytes, Banter, And The Bible: An Interdisciplinary Account Of Objective Meaning, Cameron Bonin

Senior Honors Theses

The claim that the Bible has objective meaning is contested in a postmodern world. This claim can be more persuasively defended when it is addressed by insights from multiple disciplines. In particular, the field of computer science is apt to illuminate the concept of meaning through its reflection on the nature of languages and its concern with the accurate transmission of information. By synthesizing insights from the field of computer science, such as that of Claude Shannon, with Nicholas Wolterstorff’s use of speech-act theory, the concept of meaning can be understood more clearly. Consequently, this synthesis assists in answering questions …

Estimating The Gender Wage Gap: A Comparative Analysis Of Different Estimators, Xinran Zhang

Mathematics, Statistics, and Computer Science Honors Projects

The gender wage gap between males and females has been well studied by labor economists. We take a multi-prong approach to evaluate three estimators —a regression-imputation estimator, a weighting estimator, and a doubly robust estimator—in estimating the gender wage gap. Using the Panel Study of Income Dynamics, we conduct an empirical study of the estimators’ performances. In a simulation study, we evaluate the properties of estimators and study whether bootstrapping is an appropriate measure of the uncertainty of each estimator. The findings show that while the estimators provide different results, the doubly robust estimator provides reliable and consistent results under …

Action This Day: The Mathematics And Machinations That Bested The German Enigma, Jonah Weinbaum

Dartmouth College Master’s Theses

This thesis presents a comprehensive and chronological overview of cryptographic techniques designed to break Enigma, beginning in 1932 and culminating in the creation of the Turing-Welchman Bombe. We discuss the mathematical theory and electromechanical implements used to decode one of history's greatest ciphers.

Reexamining the Bombe through the lens of modern group theory, we critique Alan Turing's estimation of the number of "stops" that the Bombe produces for various plaintext-ciphertext pairing structures. To address its limitations, we introduce a new framework for estimating the number of stops by extending John Dixon's theorem concerning the probability that uniformly distributed elements of …

Numerical Analysis And The Deal.Ii Implementation Of An Efficient And Robust Parameterized Ensemble Algorithm For Navier Stokes Flow In The Convection-Dominated Regimes, Brandiece Berry

All ETDs from UAB

Ensemble methods are widely used to approximate solutions for the Navier-Stokes Equations to account for inherent uncertainties within the system. This work goes about the numerical analysis and efficiency assessment of a fully discretized Ensemble Eddy Viscosity (EEV) model for the incompressible Stochastic Navier-Stokes Equations. The model is efficient due to the use of ensemble methods. Through a single coefficient matrix for $J$ realizations, we reduce computational cost and thereby ensure computational efficiency. It is found that the model is robust with the inclusion of the eddy viscosity term, as an additional uncertainty quantification. Finally, we verify theoretical convergence rates …

Bayesian Networks For Safety-Critical Systems, Joseph Mietkiewicz

Theses

This thesis addresses a operational challenge in modern industrial operations: the increasing complexity of systems and the consequent cognitive burden on operators. As industrial technologies advance, the human-computer interface has become the primary conduit for information flow, playing a pivotal role in operational decision-making. However, the proliferation of data often leads to information overload, potentially compromising rather than enhancing operator performance. This research explores an approach to this pressing issue through the application of Bayesian networks as decision support systems in safety- critical scenarios. Our study employs a multi-faceted approach, combining theoretical modeling with empirical testing. Through collaboration with industry …