Mathematics Commons^™

Regularity, Uniqueness And The Relative Size Of Small And Large Scales In Sqg Flows, Z. Akridge, Z. Bradshaw

Mathematical Sciences Faculty Publications and Presentations

The problems of regularity and uniqueness are open for the supercritically dissipative surface quasi-geostrophic equations in certain classes. In this note we examine the extent to which small or large scales are necessarily active both for the temperature in a hypothetical blow-up scenario and for the error in hypothetical non-uniqueness scenarios, the latter understood within the class of Marchand's solutions. This extends prior work for the 3D Navier-Stokes equations. The extension is complicated by the fact that mild solution techniques are unavailable for supercritical SQG. This forces us to develop a new approach using energy methods and Littlewood-Paley theory.

On Uses For And Constructions Of Differential Graded Algebras, Henry Potts-Rubin

Dissertations - ALL

This thesis is comprised of two projects (in reverse chronological order) with the unifying theme of differential graded (dg) algebras. The first is a collaborative effort with Hugh Geller and Desiree Martin, in which we completely classify the trees and cycles whose edge ideals are minimally resolved by dg algebras. To do this, we pull from discrete Morse theory and homological algebra to give descriptions of the dg algebra structures at play. We also use a "pruning" technique due to Boocher to give a combinatorial obstruction to the existence of dg algebra structure on minimal free resolutions of edge ideals …

Applications Of Contracting Self-Similar Groups To Cryptography And Scale Groups, Arsalan Akram Malik

USF Tampa Graduate Theses and Dissertations

Given their peculiar properties, self-similar groups are of great interest both from applications and theoretical standpoints. In this work we study the scope of their applications in post-quantum cryptography and in constructing scale groups via lifting maps.

We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). This class of groups admits fast polynomial-time algorithms for the word problem and element multiplication that can be used for effective encryption and decryption of messages. It contains extraordinary examples like the Grigorchuk group, which is known to be non-linear, thus making some of existing …

On Some Hypergraph Coloring Algorithms, Boyoon Lee

USF Tampa Graduate Theses and Dissertations

Of a given bipartite graph G = (V, E), it is elementary to construct a bipartition in timeO(|V |+|E|). For a given k-graph H = H(k) with k ≥ 3 fixed, Lovász proved that deciding whether H is bipartite is NP-complete. In this thesis, we consider the average running time of this problem. For that, let B_n = (B^(k))_n be the collection (family) of all bipartite k-graphs H on the fixed vertex set [n] = {1, ...., n}. We construct, of a given H ∈ B_n, a bipartition in time averaging O(n^k) over the class B_n. We provide two proofs …

The Only Constant Is Change: The Role Of Genetic Diversification In Cancer And Beyond, Malgorzata Tyczynska Weh

USF Tampa Graduate Theses and Dissertations

Genetic diversification, the process by which genetic variation arises in a population, is fundamental to evolution by natural selection. Mutation, a central diversification process, fuels adaptation across species, including in cancer. Yet most new mutations are neutral or deleterious on cell fitness, raising the question of how mutation-driven adaptation persists. To explore this paradox, I developed a spatial agent-based model (ABM) in which population fitness emerges from individual cells acquiring mutations at varied rates and with diverse fitness effects. I evaluated model behavior across adaptive states, mutation rates, and distributions of fitness effects. The results show that high mutation rates …

Probabilities With Values In Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of a probabilistic measure which takes values in t-scaled hyperbolic numbers for t ∈ R, with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set Dt of all t-scaled hyperbolic numbers for arbitrarily fixed t ∈ R.

Machine Learning: Neural Networking With Relu And Optimization, Aidan Redmond Brownell

Undergraduate Theses, Capstones, and Recitals

At its core, learning is an algorithmic process: it begins with input data, undergoes a series of transformations or computations, and yields an output intended to solve a specific task. This output is then compared against a target or desired result, and the internal mechanisms are updated based on how well the output aligns with expectations. While this feedback-driven process occurs almost effortlessly in humans, it is a far more structured, deliberate, and computationally intensive undertaking for machines.

Basic Theory And Implementations Of Quantum Error Correction, Derek Rodriguez

Undergraduate Theses, Capstones, and Recitals

The introduction of quantum computing has presented algorithmic solutions to computationally difficult challenges that are far more efficient than those of classical computers. These algorithms leverage the properties of quantum mechanics to manipulate the quantum properties of subatomic particles, requiring immense precision and stability. Current quantum hardware, however, is too noisy and introduces too many errors for these algorithms to be useful in practice, necessitating the use of error correction algorithms. This field survey seeks to introduce various principles of quantum mechanics relevant to quantum computing and quantum error correction (QEC), detail the implementation and motivations of a basic QEC …

Simplicial Decomposition And Realization, Matthew Ellison

Dartmouth College Ph.D Dissertations

In simplicial decomposition, we define two invariants --- V_Z and V_Q --- which represent notions of integral and rational volume of a certain class of simplicial complexes. We prove V_Z and V_Q are additive under disjoint union and connected sum, and investigate `integrality gaps' between the two quantities. We apply the theory to establish a conjecture of Sleator, Thurston, and Tarjan on tetrahedral fillings, and, as a corollary, obtain a new proof of Pournin's 2012 result on the diameter of the associahedron. In simplicial realization, we provide practical sufficient conditions and computer code to prove the existence of Euclidean embeddings …

A Study On Fuzzy Time-Series And Its Applications To Stock Price Forecasting, Takeshi Stormer

University Honors Theses

Fuzzy mathematics looks to incorporate the vagueness that exists within the real world, specifically regarding imprecise classes, or non-numerical information expressed as "linguistic" variables. Since most traditional mathematical theories do not have the ability to be applied with the exactness that is otherwise seen in mathematics. As such, there had been many applications of fuzzy mathematics throughout many different fields of mathematics, including that of forecasting. By exploring the fundamentals of fuzzy mathematics, including fuzzy sets, operations of fuzzy sets, the surface level introduction to fuzzy logic, fuzzy relations, operations of fuzzy relations, and fuzzy time-series, this work looks to …

Some Results In Thermodynamic Formalism, C. Evans Hedges

Electronic Theses and Dissertations

This dissertation investigates several key questions at the intersection of dynamical systems, computability theory, and thermodynamic formalism. In the symbolic setting, we establish novel results regarding the statistical properties of equilibrium states, deriving bounds on probabilities of configurations and relating these bounds to the Gibbs property through the homoclinic relation. Additionally, we examine the computability of thermodynamic quantities such as pressure, ground state energy, and residual entropy. We show that topological pressure is computable from above for general subshifts and computable for strongly irreducible shifts, with similar results extending to ground state energy and residual entropy.

Extending beyond subshifts, we …

A Spatial Modeling Framework For Identifying Multidimensional Risk Factors Of Colorectal Cancer, Johnna K. Berryhill

Student Theses and Dissertations

Colorectal cancer (CRC) is one of the most common and preventable cancers, yet significant disparities in screening, incidence, and late-stage diagnosis persist across different geographic and socioeconomic contexts. This thesis presents a spatial modeling framework for identifying multidimensional risk factors associated with CRC outcomes across various geographic scales. Using county- and census-tract-level data from national and state sources, the study integrates demographic, socioeconomic, environmental, built environment, and behavioral predictors to analyze geographic variability in CRC risk factors. Advanced spatial modeling techniques, including Geographically Weighted Regression (GWR) and Geographically Weighted Random Forest (GW-RF), are employed to capture spatial heterogeneity in these …

Notes On The Invariance Of Tautness Under Lie Sphere Transformations, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

An embedding ϕ : V → Sⁿ of a compact, connected manifold V into the unit sphere Sⁿ ⊂ Rⁿ⁺¹ is said to be taut, if every nondegenerate spherical distance function d_p, p ∈ Sⁿ, is a perfect Morse function on V , i.e., it has the minimum number of critical points on V required by the Morse inequalities. In these notes, we give an exposition of the proof of the invariance of tautness under Lie sphere transformations due to ´Alvarez Paiva. First we extend the definition of tautness of submanifolds of S …

Class Groups And Selmer Groups In Special Families, Debanjana Kundu, Abhishek _

School of Mathematical & Statistical Sciences Faculty Publications

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

Temporal Modeling And Forecasting Of Blood Glucose Dynamics In Individuals With Diabetes Mellitus, Mj Ruff

University Honors Theses

People living with Diabetes Mellitus face significant health risks, including an increased likelihood of heart disease, stroke, and fluctuations in blood glucose levels. The unpredictable nature of glucose levels can lead to dangerous conditions such as ketoacidosis and hypoglycemia. This study employs advanced time series analysis tools to forecast the glucose levels for an individual diagnosed with Type 1 Diabetes Mellitus.

A Proposal To Explore Geometry With Geogebra: Graphical Exploration And Formal Demonstration Of The Collinearity Of The Barycenters Of A Polygon, Saulo Mosquera Lopez, Marlio Paredes, Walter Castro

School of Mathematical & Statistical Sciences Faculty Publications

This paper illustrates an example of a mathematical activity that teachers and students can replicate to create an experience that resembles professional mathematical activity. We extend the property “Consider a triangle ABC, any straight line and let A’, B’, C’ be the reflections of the points A, B, C on the straight line then the barycenters of the triangles ABC, A’BC, AB’C and ABC’ are collinear and the line of collinearity is perpendicular to the straight line” for any quadrilateral. It is proved that there are four additional triangles, for a total of eight, whose barycenters are collinear and that …

On Excursions Associated With A Certain Local Time Of Simple Symmetric Random Walks, With Applications, Takahiko Fujita, Naohiro Yoshida

Journal of Stochastic Analysis

In this note, some applications of excursions associated with a certain local time of simple symmetric random walks are presented. Specifically, the excursions are applied to calculate some probability distributions of interest regarding the random walks. Furthermore, a solution of the Skorokhod embedding problem for random walks is obtained through the excursions.

Forecasting Influenza Rates Using Machine Learning: A Study Of Chatgpt's Predictive Accuracy, Sara Saleh

University Honors Theses

This study evaluates ChatGPT's ability to forecast influenza rates, such as the number of flu cases, hospitalizations, and death during peak season periods using CDC data, and comparing forecasts against actual results to calculate statistical accuracy and consistency. Influenza forecasting is essential for public health planning, but traditional methods may not always provide timely or accurate predictions. In this research study, ChatGPT was utilized to predict the influenza rates for the following week based on the previous week's data obtained from the FluView surveillance system. The predicted rates were compared to the actual influenza rates to assess the model's overall …

Tilings In The 3 Dimensional Lattice With L-Tetrominoes, Ian N. Bridges

Rose-Hulman Undergraduate Mathematics Journal

We consider three dimensional L-tetrominoes. We show that there exists at least one way to tile every three dimensional rectangle whose side lengths are at least $3$ and area is congruent to $1 \pmod 4$ such that one square goes untiled. In addition, we show that every three dimensional rectangle is tileable provided one side has length at least $2$ and the other is a multiple of $4$.

The Other Side Of The Equation: De-Simplification, A Prerequisite For Calculus, Stephen L. Brown

ACMS Conference Proceedings 2005

No abstract provided.

Ceno: Non-Uniform, Segment And Parallel Zero-Knowledge Virtual Machine, Tianyi Liu, Zhenfei Zhang, Yuncong Zhang, Wenqing Hu, Ye Zhang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we explore a novel Zero-knowledge Virtual Machine (zkVM) framework leveraging succinct, non-interactive zero-knowledge proofs for verifiable computation over any code. Our approach divides the proof of program execution into two stages. In the first stage, the process breaks down program execution into segments, identifying and grouping identical sections. These segments are then proved through data-parallel circuits that allow for varying amounts of duplication. In the subsequent stage, the verifier examines these segment proofs, reconstructing the program's control and data flow based on the segments' duplication number and the original program. The second stage can be further attested …

Steiner Coset Partitions Of Groups, Fusun Akman, Papa Sissokho

Faculty Publications – Mathematics

A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture, which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups H₁,...,H_r, is a coset partition of G that contains exactly one coset of each …

Topiarism: The Kernel Embedding Of Distributions Applied To Modern Portfolio Theory, Stephen G. Cook

Master's Theses

The method of kernel embedding of distributions on a set $\Omega$ into a reproducing kernel Hilbert space is a method of studying the space of measures on a set $\Omega$ using Hilbert space geometry. Because positively weighted portfolios can be interpreted as nonnegative probability measures on the space of assets, we are able to apply this technique to portfolio theory. In this thesis, we discuss the theory of "topiarism", the study of positively weighted probability measures on compact sets under the kernel embedding of distributions. Given a specific payoff function $\psi$ on the set of assets $\Omega$, we optimize a …

A Realizability Approach To Constructing Higher Types Via Classifiers, Benjamin Carrick Logsdon

Dartmouth College Ph.D Dissertations

We construct an interpretation of higher types into Peano arithmetic, showing in particular that every model of PA is a model of higher types. This is a reversal of Gödel’s Dialectica construction. We also define the classifier degrees, a degree structure which subsumes the Turing degrees, the enumeration degrees, and the many-one degrees. The classifier degrees boast a rich structure and many well-behaved operations.

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

Axon Initial Segment Plasticity Caused By Auditory Deprivation Degrades Time Difference Sensitivity In A Model Of Neural Responses To Cochlear Implants, Anna Jing , '24, Sylvia Xi , '25, Ivan Fransazov , '24, Joshua H. Goldwyn

Mathematics & Statistics Faculty Works

Synaptic and neural properties can change during periods of auditory deprivation. These changes may disrupt the computations that neurons perform. In the brainstem of chickens, auditory deprivation can lead to changes in the size and biophysics of the axon initial segment (AIS) of neurons in the sound source localization circuit. This is the phenomenon of axon initial segment (AIS) plasticity. Individuals who use cochlear implants (CIs) experience periods of hearing loss, and so we ask whether AIS plasticity in neurons of the medial superior olive (MSO), a key stage of sound location processing, would impact time difference sensitivity in the …

Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses

Master's Theses

This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …

“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 …

Predictors Of Nursing Students' Stress, Anxiety, And Depression During The Covid-19 Pandemic In A Hispanic-Serving University In South Texas: A Cross-Sectional Study, Maria I. Diaz, Eleftherios Gkioulekas, Nancy Nadeau

School of Mathematical & Statistical Sciences Faculty Publications

Background: In nursing education, there have been several studies on the impact of the COVID-19 pandemic on the ability of nursing students to cope while in nursing school.

Purpose statement: The goal of this study is to assess undergraduate nursing students' support mechanisms as predictors of stress, anxiety, and depression during the COVID-19 pandemic within a Hispanic-serving institution in South Texas.

Methods: Across-sectional design was used in this study. An online survey using self-reported questionnaires was used to gather data from an undergraduate nursing student cohort during the Fall 2021 semester. Linear regression was used to identify the predictors of …

Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie

Master's Theses

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …