Mathematics Commons^™

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Countering Deficit Narratives Through A Blackcrit Lens By Examining Case Studies Of Black College Mathematics Students’ Classroom Experiences, Akhenaton Wilbourn

All Dissertations

This dissertation examines the lived experiences of 11 Black college mathematics students at their predominantly white institution (PWI) through the lens of Black Critical Theory (BlackCrit) to counter deficit narratives about the racial achievement gap (RAG) in mathematics education courses. Deficit narratives often attribute disparities between Black and white students’ mathematical performance to assumed deficiencies in Black students’ abilities rather than systemic inequities, historical marginalization, and ongoing anti-Blackness in educational settings (Cobb & Russell, 2015; Kress, 2021). Using multiple case studies, this study explores how anti-Blackness manifests in mathematics classrooms, how students navigate tensions between multiculturalism, neoliberal educational policies, and …

Development Of Saddlepoint Methodologies For Sparse Sample Multiple Parameter Generalized Linear Models And Correlated Data Scenarios, Christopher Johnson

All Graduate Theses and Dissertations, Fall 2023 to Present

Parameter estimation using maximum likelihood techniques may be biased when sample sizes are small, event rates are low, or otherwise sparse counts exist in a parametric model. This in turn may lead researchers to draw invalid statistical conclusions when conventional methods are utilized. The saddlepoint approximation has potential to lessen the degree of bias in sparse data conditions through its use of moments beyond the mean and variance, which allows for more accurate approximations using a smaller number of observations. We propose two novel saddlepoint methods for use in practical analysis scenarios, as an alternative to maximum likelihood estimation. First, …

Generalized Sierpiński Numbers And New Irreducibility Criteria On Legendre Polynomials, Robert Scottfield Groth

Theses and Dissertations

A Sierpiński number is an odd positive integer k such that k·2n +1 is composite for all n ∈ Z+. Similarly, a Riesel number is an odd positive integer k such that k · 2n − 1 is composite for all n ∈ Z+, and a Brier number is a number that is both Sierpiński and Riesel. In this dissertation, we consider a generalization of a Sierpiński number and establish their existence by showing, for every fixed A ≥ 2, there are infinitely many odd positive integers k such that the k · an + 1 is composite for every …

Supershift Properties For Nonanalytic Signals, Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

The phenomenon of superoscillations is of great interest in microscopy, antenna design, and material sciences. This phenomenon has been generalized and has given rise to the concept of supershift, which is a far reaching extension that applies to functions that may present discontinuous derivatives. From this perspective, this is a notion that might have significant applications. This paper will provide an up to date report on the complex connections between the concept of supershift and that of analyticity.

Analysis And Derivation Of The Compressible Euler System With Nonlinear Velocity Alignment: An Investigation Into Collective Behavior, Mckenzie Meredith Black

Theses and Dissertations

Collective behavior refers to the observable patterns, actions, or movements that emerge within a group of entities, such as individuals, animals, or particles, when they interact with each other. In this study, we concentrate on the pressureless compressible Euler system, expanding it to incorporate a family of nonlinear velocity alignment. This extension represents a nonlinear iteration of the Euler-alignment system within collective dynamics, revealing intriguing asymptotic emergent phenomena such as alignment and flocking. Our investigation explores different types of nonlinearity and nonlocal communication protocols, unveiling a diverse spectrum of asymptotic behaviors within the system.

In focusing on the derivation, we …

Generation And Dimension For Derived Categories, Pat Lank

Theses and Dissertations

This work focuses on problems regarding derived categories arising in algebraic geometry and commutative algebra. There are six chapters regarding generation in the bounded derived category of coherent sheaves over a Noetherian scheme. The targeted topics involve explicitly identifying generators, proving existence results thereof, and producing estimates on Rouquier dimension.

Symmetries Of Del Pezzo Surfaces, Jonathan Smith

Theses and Dissertations

For each del Pezzo surface X of degree d, the action of the automorphism group Aut(X) on the exceptional curves of X induces a map ρ : Aut(X) → W(R_d), where W(R_d) is the Weyl group of a root system R_d dependent on the degree of X. The image of ρ is well defined up to conjugacy in W(R_d), so we say that a group G acts by automorphisms on a del Pezzo surface X of degree d if a representative of the …

On Finite Groups With Essential Dimension 2 Over ℚ, Michael Huggins

Theses and Dissertations

We determine all finite subgroups of Aut(X) which have essential dimension 2 over ℚ, where X is a minimal Del Pezzo surface of degree at least 5. Additionally, finite groups with essential dimension 2 whose order is divisible only by the prime numbers 2 and 3 are classified with the exception of the groups dic ₁₂ , C₁₂, and Dih₂₄, whose essential dimensions remain unknown, but may possibly be equal to 2.

An Investigation Into Continued Fraction Expansions Of √ N, Sara R. Ladik, Abdulkadir Hassen

STEM Student Research Symposium Posters

A continued fraction represents a number using an integer part and a sequence of positive integers in the denominators, with all numerators equal to one. This research examines the infinite continued fraction expansions of square roots of non-perfect squares. We observed and proved patterns in these expansions.

Stability Analysis & Analytic Solutions To Unforced Kinematic Property Equations, James Turbett

STEM Student Research Symposium Posters

We present analytic solutions and a stability analysis for a system of nonlinear coupled differential equations that describe the time evolution of kinematic properties of geophysical fluids. Solution curves in phase space are interestingly hyperbolic, elliptical, or parabolic in nature, depending on the initial conditions we input and have interesting geometrical properties.

Fixed-Size Ledger Of Fungible Tokens, Sky P. Waterpeace, Nick Ivanov

STEM Student Research Symposium Posters

Hypotheses:

1. Using a fixed number of fixed-size tokens, the storage of transaction history can be outsourced to the owners of the tokens or to the network.

2. With distributed transaction histories, disjunctive (targeted) proof becomes possible.

Enabling Localized Execution Of Smart Contracts Via Distributed Event Beacons, Rakesh Vasa, Sky P. Waterpeace, Nick Ivanov

STEM Student Research Symposium Posters

Modern smart contracts save their state transitions on a distributed ledger, such as blockchain [3, 4] or a particularly configured cloud [1]. These approaches necessitate storage and synchronization of large amounts of distributed data, leading to scalability deficiencies in decentralized permissionless smart contract platforms. In this poster, we introduce a novel concept of local execution of smart contracts which only required synchronizing the transaction data between the members of the private channel, without the need to storing state transitions on the distributed storage. To achieve this, we design a distributed storage-free decentralized beacon network that allows to perform provenance signatures …

An Analysis On The Time Effectiveness Of Speeding While Driving A Vehicle, Joy Skaggs

ATU Scholars Symposium

Speeding is responsible for roughly a third of all crashes, often coming at a deadly cost. The National Highway Traffic Safety Administration reports many negative consequences, such as a greater likelihood of loss of vehicle control, reduced protection equipment effectiveness, increased stopping distance, crash severity, and fuel consumption (National Highway Traffic Safety Administration, n.d.). Yet despite these costs, drivers continue to speed. In fact, in our fast paced world, speeding is so normalized that it’s uncommon to encounter a driver following the speed limit. Hawaiian psychology professor Leon James explains it best: people have twisted their definitions of speeding to …

The Dialogue Dilemma: The Role Of Patient-Clinician Communication For Low-Income People Who Smoke And Manage Multiple Conditions, Monique T. Cano, Michael R. Lindstrom, Ricardo F. Muñoz

School of Mathematical & Statistical Sciences Faculty Publications

Introduction: Adults from low-income backgrounds who smoke face significant health disparities related to tobacco use, often at disproportionately high rates. These individuals are more likely to endure multiple mental and physical (MP) health conditions, which can negatively influence their self-rated health (SRH). The quality and effectiveness of patient-clinician communication (PCC) can influence how patients perceive their own health. Understanding how PCC influences SRH among low-income adults who smoke and suffer from multiple MP conditions is essential for clinical care as multimorbidity is on the rise. This study examines how PCC may influence the health perceptions of low-income adults who smoke …

Quantization For A Condensation System, Shivam Dubey, Mrinal Kanti Roychowdhury, Saurabh Verma

School of Mathematical & Statistical Sciences Faculty Publications

For a given r∈(0,+∞), the quantization dimension of order r, if it exists, denoted by Dr(μ), represents the rate at which the nth quantization error of order r approaches zero as the number of elements n in an optimal set of n-means for μ tends to infinity. If Dr(μ) does not exist, we define D−−r(μ) and ¯¯¯Dr(μ) as the lower and the upper quantization dimensions of μ of order r, respectively. In this paper, we investigate the quantization dimension of the condensation measure μ associated with a condensation system ({Sj}Nj=1, (pj)Nj=0,ν). We provide two examples: one where ν is an …

Construction Of Stock Portfolios By Machine Learning Methods, Alfan Gehad Abulehia

Thesis/ Dissertation Defenses

We review the theoretical foundations and empirical workings of Long Short-Term Memory Networks (LSTM) and Random Forest for stock market prediction. To be precise, our work initiates with a deep-dive exploration in the mathematics of these algorithms which covers topics like gradient descent, automatic differentiation, and a bit on recurrent neural networks as well.
An application creates stock portfolios through LSTM and Random Forest rules- based strategies, using either technical or fundamental indicators as input variables. The process involves predicting returns and ranking stocks accordingly to build portfolios with a certain level of diversification and annual rebalancing over a defined …

Does Better Firm Performance Imply Better Management And Esg Scores?, Shawn Lisann

Research Symposium

No abstract provided.

Prime Theta-Curves From Torus Knots, Sam Nieman, Jack Calcut

Research Symposium

Classical knot theory is the placement problem for a circle in Euclidean 3-space or in the 3-sphere. That problem has been generalized in numerous ways. Knotted graph theory replaces the circle with a graph. Knotting of certain graphs has been well-studied. That includes the theta-graph consisting of two vertices connected by three edges---a copy of the letter theta. A theta-curve is an embedding of the theta-graph in 3-space or in the 3-sphere. Connected sum of knots yields the notion of a prime knot. Similarly, one may sum two theta-curves at a vertex and one may sum a theta-curve and a …

Study Habits And Review Sessions, Lillian Gottstine

Fisher Showcase 2025

Seeing the outcomes of using and teaching different studying habits in a review session to see the effects on student preparedness for unit exams.