A Study Of Knots And Quandles,
2025
College of the Holy Cross
A Study Of Knots And Quandles, Zhaoqi Wu
Math and Computer Science Honors Theses
We explore the mathematical theory of knots through the lens of algebraic structures known as kei and quandles. We begin by introducing classical knot invariants and then study the fundamental kei of a knot as a tool for distinguishing knot types. We generalize this approach using various kinds of quandles, including Alexander and dihedral quandles, and investigate their associated polynomial invariants. We also examine the connection between quandles and group theory, as well as their algebraic representations in quandle rings. Moreover, we analyze idempotent elements in quandle rings over finite fields, providing both general results and specific examples.
Modules Of Finite Projective Dimension And Singularities,
2025
University of Nebraska-Lincoln
Modules Of Finite Projective Dimension And Singularities, Nawaj Kc
Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–
In the first part of this thesis, we study liftings of modules of finite projective dimension. We introduce a notion of “Serre liftable” modules and deduce applications to multiplicity conjectures in local algebra. In the second part, we introduce and study a module of mixed K\"ahler differentials for finite algebras over ramified discrete valuation rings of mixed characteristic. We state and prove a Jacobian criterion for computing the singular loci of such algebras.
Advisors: Jack Jeffries and Mark Walker
Using Permutation Groups To Identify Family Of Capacity Achieving Codes,
2025
University of Nebraska-Lincoln
Using Permutation Groups To Identify Family Of Capacity Achieving Codes, Daniel Joseph Welchons
Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–
When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.
This this thesis seeks to …
Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors,
2025
University of Nebraska-Lincoln
Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors, Johan Benedict A. Cristobal
Department of Mathematics: Dissertations, Theses, and Student Research
Why do instructors teach the way that they do? This question is key in understanding how and why asset-based or deficit-based practices are enacted in the classroom. In consequence, this question also gives insight into how the environment where students learn is shaped. Mathematics education research, in particular, has been concerned with how instructors respond to students’ contributions in the classroom and how these ways of responding can (dis)empower students.
One way to answer this question is to understand how instructors frame their own teaching and their students’ learning. Frames are mental constructs which help individuals filter details, interpret information, …
Maximal Independent Set Algorithms Within Procedural Planar Maps: A Large-Scale Evaluation,
2025
Murray State University
Maximal Independent Set Algorithms Within Procedural Planar Maps: A Large-Scale Evaluation, Chaucer Ihrig
Honors College Theses
Analysis of a childhood game has led us to the problem of maximum independent sets in planar graphs. We wrote a graph creation utility using R to generate a random planar map and its dual graph. This utility then finds a graph’s maximal independent set using a variety of six algorithms. We investigate statistical connections between graph structure, colorability, and the maximal independent sets found using these algorithms over an incredibly large and procedurally generated dataset. We find one can always win the coloring game if the resultant graph is two-colorable. The algorithms perform statistically and practically significantly better on …
Functional Data Analysis Of Weekly Confirmed Covid-19 Cases Per Million In Relation To Unemployment Rates And Emergency Declaration Responses,
2025
Boise State University
Functional Data Analysis Of Weekly Confirmed Covid-19 Cases Per Million In Relation To Unemployment Rates And Emergency Declaration Responses, Elaina Kay Mcgovern
Boise State University Theses and Dissertations
At the onset of the COVID-19 pandemic, U.S. states implemented policies that attempted to mitigate the spread and effects of the disease, such as a state of emergency declaration. This study explores the application of functional data analysis (FDA) as a means to investigate the impact of U.S. state-level emergency declarations on weekly confirmed COVID-19 cases per million during the first year of the pandemic. We focus on the duration and extent to which proactive states showed significantly different patterns of COVID-19 spread than those that implemented policies reactively. Additionally, we examine patterns in monthly state-level unemployment rates over the …
Leading Digits Of Some P-Adic Numbers,
2025
Boise State University
Leading Digits Of Some P-Adic Numbers, Jinha Park
Boise State University Theses and Dissertations
I investigated the leading digits of some sequences in the p-adic numbers Q_p, hoping to find some sequences following Benford's law, or following something completely different. We found that the sequence of partial sums Sum^N_{n=1}\frac{1}{n^s}, where s\in\N is fixed, shows uneven distribution of leading digits, except for s\equiv 0 (mod{p,/em>-1}), where the distribution of leading digits is eventually even. We were able to characterize the distribution of this partial sum for almost all s.
Comparative Study Of Non-Iterative Seqential Method For Biot's Poroelasticity Model,
2025
University of Texas at El Paso
Comparative Study Of Non-Iterative Seqential Method For Biot's Poroelasticity Model, Jeff Frimpong Amponsah
Open Access Theses & Dissertations
Linear poroelasticity theory describes the interaction between the motion of fluids and deformation of porous media. The theory serves various applications in a wide range of science and engineering fields, such as soil mechanics, oil reservoir modeling, and bio-medical applications. Partial differential equations are used to model the complex interaction between fluid flow and solid deformation in porous media. Since analytical solutions to these equations are rarely available for realistic problems, we resort to numerical solutions. One major advantage is their ability to provide accurate approximations of the solutions. Various numerical methods have been proposed to solve Biot's poroelasticity system. …
An Algebraic-Combinatorial Proof Of A Bézout-Type Inequality For Mixed Volumes Of Three-Dimensional Zonoids,
2025
BTU Cottbus Senftenberg
An Algebraic-Combinatorial Proof Of A Bézout-Type Inequality For Mixed Volumes Of Three-Dimensional Zonoids, Gennadiy Averkov, Ivan Soprunov
Mathematics and Statistics Faculty Publications
We present a new algebraic-combinatorial approach to proving a Bézout-type inequality for zonoids in dimension three, which has recently been established by Fradelizi, Madiman, Meyer, and Zvavitch. Our approach hints at connections between inequalities for mixed volumes of zonoids and real algebra and matroid theory.
A Multi-Scale Compartmental Model For Glucose Regulation And Diabetes Treatment,
2025
University of Richmond
A Multi-Scale Compartmental Model For Glucose Regulation And Diabetes Treatment, Andrew M. Watts
Honors Theses
Glucose is a fundamental energy source for cellular function, and its regulation is critical for maintaining metabolic stability in the human body. Glucose homeostasis is governed by a network of biochemical processes involving multiple organ systems that coordinate glucose production, storage, and uptake. Disruptions in this regulation contribute to metabolic disorders such as diabetes mellitus, underscoring the need for mathematical models that provide a mechanistic understanding of systemic glucose dynamics. Herein, we report a multi-time scale discrete-time dynamical systems model using compartmental difference equations to describe glucose concentrations across key physiological compartments. By quantitatively modeling glucose transport and metabolism, this …
Small Scale Magnetohydrodynamic Thruster,
2025
College of Saint Benedict and Saint John's University
Small Scale Magnetohydrodynamic Thruster, Samuel Rabaey
Celebrating Scholarship and Creativity Day (2018-)
This paper explores the performance of a self-constructed, small-scale magnetohydrodynamic (MHD) thruster using saltwater as the conductive fluid. The MHD drive generates propulsion through Lorentz force by applying an electric current across electrodes in the presence of an orthogonal magnetic field. Two experiments were conducted: one measuring static water displacement to determine force output, and another using fluid velocity in a closed loop to evaluate both force and efficiency at varying magnetic field strengths. Theoretical predictions based on MHD and fluid dynamics principles were compared with experimental results. While the measured outputs were consistently lower than theoretical approximations, likely due …
Using Permutation Groups To Identify Families Of Capacity Achieving Codes,
2025
University of Nebraska-Lincoln
Using Permutation Groups To Identify Families Of Capacity Achieving Codes, Daniel Welchons
Department of Mathematics: Dissertations, Theses, and Student Research
When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory, but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.
This thesis seeks to apply the …
Course Insight Portfolio: Math 1060 Calculus Of One Variable I,
2025
Clemson University
Course Insight Portfolio: Math 1060 Calculus Of One Variable I, Julianne Barnhart, Rayna Elizabeth Maleki, Antsa Rakotondrafara
Publications
We presented this Course Insight Portfolio project as a final assignment for the class Teaching Undergraduate Math MATH 9700 in Summer 2025. It covers the Clemson class MATH 1060 Calculus of One Variable I and is a fruit of the collaborative work of Julianne Barnhart, Rayna Maleki, and Antsa Rakotondrafara.
Quotients, Equivalence Relations, And Normality In Non-Associative Algebra With Regards To Loops And Quasigroups,
2025
Northern Michigan University
Quotients, Equivalence Relations, And Normality In Non-Associative Algebra With Regards To Loops And Quasigroups, Matthew L. Mulholland
All NMU Master's Theses
This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be …
Stability Analysis Of Turbulent Fluid Flow,
2025
Macalester College
Stability Analysis Of Turbulent Fluid Flow, Adam D. Schroeder
Mathematics, Statistics, and Computer Science Honors Projects
Hydrodynamic stability refers to the study of when and how laminar flows transition to turbulence. This includes investigations of the mechanisms of transition, as well as the classification of known flow configurations as either stable or unstable and the identification of critical values of flow parameters at which this bifurcation occurs. In this thesis, we introduce the mathematical theory behind continuum mechanics and fluid dynamics as well as some tools from the study of dynamical systems. We apply these concepts to the linear stability analysis of zero pressure gradient flat plate flow via numerical simulations in OpenFOAM, discussing both the …
P-Adic Quantum Mechanics, Infinite Potential Wells, And Continuous-Time Quantum Walks,
2025
The University of Texas Rio Grande Valley
P-Adic Quantum Mechanics, Infinite Potential Wells, And Continuous-Time Quantum Walks, Nathaniel P. Mayes
Theses and Dissertations
In this thesis, we introduce a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. …
The Herzog-Takayama Resolution Over A Skew Polynomial Ring,
2025
The University of Texas Rio Grande Valley
The Herzog-Takayama Resolution Over A Skew Polynomial Ring, Linoy Utkina
Theses and Dissertations
Let k be a field, and let I be a monomial ideal in the polynomial ring R = k[x1,..., xn]. In her thesis, Taylor introduced a complex that yields a finite free resolution of R/I as an R-module. Building on Taylor’s work, Ferraro, Martin, and Moore extended this construction to monomial ideals in skew polynomial rings. Because the Taylor resolution is typically not minimal, subsequent research efforts went into identifying specific classes of ideals whose minimal free resolutions can be constructed more simply. In 1990, Eliahou and Kervaire devised an approach for handling minimal resolutions of …
Camassa-Holm Type Equations And Multi-Peakon Solutions,
2025
The University of Texas Rio Grande Valley
Camassa-Holm Type Equations And Multi-Peakon Solutions, Yonghong Chen
Theses and Dissertations
This thesis investigates the multi-peakon solutions of a class of Camassa-Holm type equations with quadratic nonlinearities, specifically focusing on the differential equation mt +θmu+ amxu+bmux = 0, where m = u−uxx, and θ, a, b are arbitrary constants. We derive the nonlocal form of the equation and explore its peakon dynamics, including the well-known Camassa–Holm, Degasperis–Procesi, and Holm–Staley b-family equations. By employing a practical computational approach, we analyze the N-peakon solutions and present the corresponding dynamical systems, particularly focusing on the two-peakon case. The results will …
Congruences In Arithmetic Progression For Coefficients Of Gaussian Polynomials And Crank Statistics,
2025
The University of Texas Rio Grande Valley
Congruences In Arithmetic Progression For Coefficients Of Gaussian Polynomials And Crank Statistics, Joselyne Aniceto
Theses and Dissertations
The study of partition congruences, inspired by Ramanujan’s discoveries for ��(��) over a century ago, remains a central topic in this field. This dissertation examines congruence properties in two restricted partition functions: ��(��,��), which counts partitions of �� into at most �� parts, and ��(��,��,��), which further limits the size of the largest part to be at most ��. Building on Kronholm’s 2007 result, now known as the Interval Theorem, and a recent result by Eichhorn, Engle, and Kronholm, we establish new infinite families of congruences for ��(��,��,��). This dissertation extends not only the recent results of Eichhorn, Engle, …
Relaxation Maximum-Based Iteration Method For Solving The Generalized Absolute Value Equation,
2025
The University of Texas Rio Grande Valley
Relaxation Maximum-Based Iteration Method For Solving The Generalized Absolute Value Equation, Ximing Fang, Zhidong Wang, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
The generalized absolute value equation (GAVE) has wide applications in scientific computing. Establishing a high performance computing method to solve the GAVE is a hot research topic in recent years. In this paper, with the aid of the maximum function, the GAVE is decomposed of two equations, and then we present the relaxation maximum-based (RM) iteration method. To see the feasibility of the method, we discuss the necessary and sufficient conditions for the GAVE to have a unique solution. Next, the convergence analysis of the RM iteration is discussed under some convergence conditions. Moreover, some numerical examples of low and …
