Mathematics Commons^™

A Bridge Too Low, John Adam

Mathematics & Statistics Faculty Publications

The article "A bridge too low" in the Physics Teacher journal discusses a low bridge near the River Greta in Keswick, England, with an arch shaped like a semiellipse. It presents questions about the maximum height a person of a certain height can walk under the bridge without hitting their head, the cross-sectional area of the arch, its eccentricity, and perimeter. The article also mentions the approximation by Indian mathematician Srinivasa Ramanujan for the perimeter of an ellipse and invites readers to find the answers online.

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 …

Weakly Reversible Deficiency Zero Realizations Of Polynomial Dynamical Systems, Neal Ryan Buxton

Graduate Theses, Dissertations, and Problem Reports (ETD)

Weakly reversible, deficiency zero (WR0) systems form a large class of polynomial ODEs modeling chemical reaction networks. Their behavior is exceptionally stable: they have unique positive steady states, which are locally asymptotically stable (they are also conjectured to be globally asymptotically stable). This powerful result (the Deficiency Zero Theorem) applies to a large class of high dimensional, nonlinear polynomial dynamics, and is independent of the choice of parameters in the model. In this dissertation we present two algorithms that expands the scope of the Deficiency Zero Theorem to:

1. Networks with WR0 realizations, i.e. networks that are not necessarily WR0 …

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 …

Theoretical And Applied Considerations In Dynamical Sampling, Brendan Miller

Graduate Research Theses & Dissertations

Dynamical sampling is an area of mathematics which studies the theory and applications of signals evolving through time, as well as the stable recovery of said signals from spatiotemporal measurements. This manuscript is devoted to problems which arise as theoretical considerations in signal recovery and as applications of using space-time measurements as opposed to static measurements.

For the former, we consider the problem of recovering an evolving signal via space-time measurements at irregular time intervals. To do so, we generalize the classic Kadec $1/4$ theorem using techniques of Banach $L^1(\R)$-modules. For the latter, we show how dynamical samples can be …

Methods For Finding High Quality And Optimal Solutions Of Binary Quadratic Optimization Problems, James Haas

Graduate Research Theses & Dissertations

BiqAlps is an extension of the BiqCrunch project that leverages its bounding procedures within a computer cluster environment. The primary goal of this work was to investigate whether distributing the branch-and-bound tree search across multiple CPUs could improve solve times. While some problem instances demonstrated speedups of up to fivefold, others showed no measurable improvement, revealing that parallelization benefits are problem-dependent.

Beyond parallelization, BiqAlps introduces enhanced heuristics for generating high-quality initial solutions and refining existing ones. Additionally, the framework explores the impact of propagating triangle inequality cuts to child nodes during search, with the aim of accelerating bounding processes and …

Cubic Blaschke Products, Alexandra Hill

Graduate Research Theses & Dissertations

It is well known that Blaschke products can be classified as elliptic, hyperbolic, or parabolic. Based on the classification of the Blaschke product, we know that the Julia set is either the entire unit circle or a Cantor subset of the unit circle. For unicritical Blaschke products of degree d, it has been shown that the parabolic functions are parameterized by an epicycloid with d-1 cusps, with the interior of the epicycloid giving rise to the elliptic parameters, and the relative complement of the epicycloid within the unit disk providing the hyperbolic parameters. In this dissertation, we study in more …

Green's Functions And Fixed Points Of Quasiregular Maps, Mark Broderius

Graduate Research Theses & Dissertations

This dissertation is on the dynamics of planar quasiregular maps of degree 2. These maps have three parameters. One of these parameters determines the distortion, one of them determines the orientation, and the last one is a shift parameter. Our first objective is to construct a new dynamical version of Green's functions for these maps. These functions provide information about the escaping sets of our quasiregular maps and are used to prove topological results about them. We show that the boundary of the escaping set is either connected or consists of infinitely many components. Additionally, we prove that if one …

Facial Reduction Of Semidefinite Relaxations Of Binary Quadratic Problems, Ian Sugrue

Graduate Research Theses & Dissertations

A standard technique for bounding a combinatorial optimization problem is to form a semidefinite relaxation of the problem by "lifting" the problem into a higher dimensional space of symmetric matrices. For some combinatorial problems such as maximum k-cluster, the resulting semidefinite relaxation is not strictly feasible. In the context of the maximum k-cluster problem, we present an equivalent, strictly feasible semidefinite relaxation by way of the dimensionality reduction technique known as facial reduction. We provide a recipe to form this semidefinite relaxation directly, bypassing the lifting and facial reduction steps entirely. We then present our work on generalizing this result, …

Spectral Theory And The Gelfand Transform, Brenden Schlader

All Graduate Theses, Dissertations, and Other Capstone Projects

The overall goal of this thesis is to study spectral and Gelfand theory as it relates to unital Banach and C∗-algebras. In the first part, we develop the necessary algebraic, analytic, and topological background relevant to the content of this work. We also discuss concrete examples of algebras frequently used in the subsequent sections. In the second part of this thesis, we develop spectral theory by first defining the spectrum of an algebra through the characterization of the invertible and noninvertible elements. In particular, we establish properties of the commutative unital Banach algebra ℓ1(Z). We also establish fundamental results such …

Introducing Triangular Groups And Further Results On Magic Groups, Nicholas Charles Fleece

Doctoral Dissertations

In this research, we discuss two new topics in group theory. First, we define an n-magic square in a group to be a nxn array of group elements whose rows, columns, and diagonals have the same product. This definition is akin to the idea of magic squares in the integers. Groups that have an n-magic square are said to be n-magic. We begin with some preliminary results and focus much of our attention on 3-magic groups, though we also give some results for higher n. Through a series of propositions, we ultimately prove a characterization theorem for 3-magic finitely generated …

A Geometric Characterization Of N-Graded Manifolds And The Frobenius Theorem, Henrique Bursztyn, Miquel Cueca, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

This paper studies graded manifolds with local coordinates concentrated in non-negative degrees. We provide a canonical description of these objects in terms of classical geometric data and, building on this geometric viewpoint, we prove the Frobenius theorem for distributions in this graded setting.

Runner Locating With Multiple Probes, Kola Akinrele, Axel Brandt, Elizabeth Breen, Catherine Erbes, Matthew Lee, Patrick O’Doherty, Weston Rainer

2025 Faculty Bibliography

In the runner locating variation of cops and robbers on a graph, a chaser attempts to locate an invisible runner by probing a single vertex v each turn, from which the chaser learns the runner’s distance. The runner is then permitted to stay at his current vertex or move to an adjacent vertex other than v. A graph is locatable if the chaser is able to locate the runner in a finite number of turns, and the location number of a graph is the minimum number of turns necessary to determine the runner’s location regardless of the runner’s evasion strategy. …

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 …

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 …

Rings With Q-Torsionfree Canonical Modules, Naoki William Endo, Laura P. Ghezzi, Shiro S. Goto, Jooyoun Hong Phd, Shin-Ichiro Iai, Toshinori Kobayashi, Naoyuki Matsuoka, Ryo Takahashi

Publications and Research

Let A be a Noetherian local ring with canonical module K A . We characterize A when K A is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3 . If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K A is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein …

Vanishing Of Local Cohomology With Applications To Hodge Theory, Scott Hiatt

Mathematics Faculty Publications

Let H = ((H, F •), L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U . By M. Saito’s theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M ∈ HM (U ). Let X be a compactification of U , and j : U ↪→ X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M ∈ DbM HM (X). In particular, we study the graded pieces of the de Rham complex GrF p DR(j+M) …