Applied Mathematics Commons^™

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 …

Adaptive Volatility Forecasting Models, Jeffrey K. Tan

Master's Theses

In finance, risk is often quantified by volatility, and computing accurate volatility forecasts — while vital to financial decision making — remains one of the most challenging tasks in financial modeling. This thesis, motivated in part by the Black-Scholes-Merton Model and its limitations, adopts a statistical approach to volatility forecasting. The two main models of interest are the Exponential Weighted Moving Average (EWMA) model and the GARCH(1,1) model. Specifically, this work expands upon a potential adaptive lambda algorithm for EWMA models first proposed by Bernard Bollen (2014), and this work also utilizes the Momentum of Predictability (MoP) to generate adaptive …

Equiangularity From Compatible Orthobiangularity, Tyler J. Myers

Theses and Dissertations

An equiangular tight frame (ETF) is an equal norm sequence of vectors in a Hilbert space whose coherence achieves equality in the Welch bound. Such sequences necessarily have minimal coherence and thus are, in some sense, as "spread out" in space as possible. ETFs have a variety of applications, such as compressed sensing and waveform design. The main problem in the study of ETFs is determining the pairs (D, N) for which an ETF with N vectors in a D-dimensional space exists. Real ETFs are moreover equivalent to a special subset of a well-studied class of graphs known as strongly …

Machine Learning And Optimization For Intelligent Decision-Making, Elson Cibaku

Dissertations

This dissertation presents a series of innovative machine learning and optimization model designs that address complex operational challenges across logistics and power systems. By integrating advanced neural architectures with robust optimization techniques, the work delivers scalable solutions designed to improve efficiency, reliability, and decision-making in dynamic and real-world environments. The first study introduces a two-stage approach to effective vaccine distribution. This framework tackles the capacitated vehicle routing problem by combining adaptive clustering techniques with reinforcement learning and a simulated annealing pickup policy. Through extensive computational experiments, the approach demonstrates substantial improvements in routing efficiency, reducing both computational time and logistical …

From Neural Networks To Large Language Models: Innovations In Financial Ai, Mathematical Reasoning, And Structured Data Representation, Junyi Ye

Dissertations

This dissertation explores the evolution and application of artificial intelligence techniques across three critical domains: financial modeling, mathematical reasoning, and structured data analysis. The dissertation presents seven research projects that chart a progression from specialized neural architectures to sophisticated large language models (LLMs), contributing novel methodologies and frameworks at each stage.

In the financial domain, the research first introduces TS-Mixer, a MLP-based architecture for time-series forecasting that captures both feature relationships and temporal dependencies through a simple yet effective design, outperforming more complex models in S&P500 index prediction. The dissertation then presents DySTAGE, a dynamic graph representation learning framework that …

Setting Up Students For Success: Analysis Of Effectiveness Of Mathematics Placement, Jeff Carvell, Jason Ho, David Klanderman, Sarah Klanderman

University Faculty Publications and Creative Works

How do we e!ectively and equitably place students into math classes in a way that provides them the best chance of success? As many higher education institutions veer away from placement based on standardized testing, many departments are seeking placement alternatives that will properly support students. Additionally, math placement determines not only a student’s mathematics courses but also influences their progress in related fields, including physics, chemistry, engineering, and more. This paper will describe three di!erent existing placement systems at each of our liberal arts institutions as well as the a!ordances and constraints of each approach. Further, we analyze data …

Nonlinear Phenomena Of Discrete Wave Systems With Nonlocal Coupling, Austin E. Marstaller

Mathematics Theses and Dissertations

There is a long and rich story of nonlinear dynamics in discrete lattices. A particular well studied model is the discrete Nonlinear Schr\"odinger Equation, with many applications, most notably in nonlinear optics.

Recent studies have explored dynamics in scenarios where coupling amongst the array elements is nonlocal. The interest in particular is that in the longwave approximation, it leads to fractional diffraction. In this contribution, we consider two long range interaction lattice wave models, the fractional discrete nonlinear Schr\"odinger equation and the Long Range Ablowitz-Ladik system. By use of perturbation methods, asymptotics and numerical simulations, we present results on modulational …

Constructing The Soliton Wave Structure And Stability Analysis To Generalized Calogero–Bogoyavlenskii–Schiff Equation Using Improved Simple Equation Method, Mina Fahim, Hamdy Mohamed Ahmed, Islam Samir Soliman, Mohamed Elsaid, Kamal Hassan Eldib

Basic Science Engineering

In this work, we investigated the (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation, which models long wave propagation in shallow water and plays a significant role in fluid mechanics and plasma physics. Using the improved simple equations method, we obtained various solutions, including dark, bright, and singular solitons, and combinations of singular periodic solutions and exponential rational solutions. Additionally, we performed a linear stability analysis to examine the stability properties of these wave solutions. To further illustrate their characteristics during propagation, we provided 3D and contour plots for some opted wave solutions.

Applications Of The Mathieu Groups And Information Theory In Dna Encoding Functions, Juan C. Nava Jr

Theses and Dissertations

A foundational idea in mathematics lies in breaking down existing components into their bare fundamentals. As evidenced by prime numbers and composites, we learn this idea at an early age. Categorizing these broken-down components into their simplest form allows mathematicians to construct proofs from emergent patterns. John Conway’s Atlas of Finite Groups in the 1990s was particularly concerned with the categorization of structures known as groups. There are certain axioms a group must adhere to, which amount to the retention of symmetry; ultimately a group helps us to better understand symmetric actions performed on a set with a binary operation. …

On The H-Property For Step-Graphons: Residual Case, Wanting Gao

McKelvey School of Engineering Graduate Student Theses & Dissertations

We investigate the H-property for step-graphons. Specifically, we sample graphs Gn on n nodes from a step-graphon and evaluate the probability that Gn has a Hamiltonian decomposition in the asymptotic regime as n → ∞. It has been shown in Belabbas and Chen (2023); Belabbas et al. (2021) that for almost all step-graphons, this probability converges to either zero or one. We focus in this paper on the residual case where the zero-one law does not apply. We show that the limit of the probability still exists and provide an explicit expression of it. We present a complete proof of …

A Modified Sir Model Used To Investigate The Relationship Between Congenital And Adult Syphilis, Kaleesta R. Waysman

Honors Thesis

A complex SIR model integrating the relationship between adult and congenital syphilis was developed. The goal of the project was to determine the specific population(s) or control strategies that should be enforced, altered, or removed to decrease the number of children experiencing clinical sequelae due to congenital syphilis. Early clinical sequelae include hydrops fetalis, preterm birth, central nervous system infection, hepatosplenomegaly, hyperbilirubinemia, cholestasis, hemolytic anemia, snuffles, osteochondritis, and lesions or rashes in the palms and soles. Late clinical sequelae include interstitial keratitis, hearing loss, Hutchinson teeth, saber shins, Clutton joints, mulberry molars, and saddle nose. After implementing real-world data into …

Crystallization Of The Quantized Function Algebras Of Suq(N + 1), Manabendra Giri

Doctoral Theses

The $q$-deformation of a connected, simply connected Lie group $G$ is typically studied through two Hopf algebras associated with it: the quantized universal enveloping algebra $\mathcal{U}_q(\mathfrak{g})$ and the quantized function algebra $\mathcal{O}(G_q)$. If $G$ has a compact real form $K$, one can use the Cartan involution to give a $*$-structure on $\mathcal{O}(G_q)$. The QFA $\mathcal{O}(G_q)$ with this $*$ structure is denoted by $\mathcal{O}(K_q)$ and its $C^*$-completion by $C(K_q)$. Here we study the crystal limits of $\mathcal{O}(SU_q(n+1))$ and $C(SU_q(n+1))$ and classify all irreducible representations of the crystallized algebras. We also prove that the crystallized algebra carries a natural bialgebra structure.

Detection, Analysis, And Modeling Of Time-Scale Separated Modulatory Brain Dynamics: Methods And Applications In Neurocritical Care, Maren Elizabeth Loe

McKelvey School of Engineering Graduate Student Theses & Dissertations

Clinical monitoring of patients with neurological disease generates large volumes of data, including electrophysiology (e.g. EEG), heart rate, blood pressure, and other physiological measures. The use of these data to generate actionable prognostic measures is a long-held goal in clinical neurophysiology. In this regard, the use of engineering theory, including signal processing methods and computational modeling paradigms, is providing new ways of interpreting neurological data and yielding new insights into brain mechanisms and disease pathophysiology. This research focuses on the analysis and modeling of aberrant brain dynamical phenomena that occur over hours-long temporal epochs. Particularly, we consider time-scale separated electrophysiological …

Existence Of Traveling Waves In A Predator-Prey Invasion Model With Nonlocal Dispersal And Delayed Effects In Dispersal, Austin Simms

Theses and Dissertations

We investigate the existence and stability of traveling wave solutions in a predator--prey population model that incorporates both nonlocal dispersal and time-delayed interactions. The problem arises when species distributions and feeding or reproduction processes do not respond instantaneously, but rather exhibit lags in space and time. Our aim is to understand how these delays and nonlocal effects impact the wavefront speed and the eventual establishment of a predator population in previously prey-dominated regions. To accomplish this, we derive a PDE--integro--delay system from ecological considerations of prey self-regulation, predator ratio dependence, and jump-type dispersal kernels. We then employ a traveling wave …

Using A Pharmacokinetic Model To Design And Evaluate An Early Ctdna Biomarker For Response To Targeted Therapy, Aaron Li

Spora: A Journal of Biomathematics

Early prediction of response to therapy or lack thereof can help physicians plan treatment more efficiently. Biomarkers based on circulating tumor DNA (ctDNA) are promising. However, biomarkers beyond direct comparison to baseline have not been thoroughly explored. We develop a model for ctDNA shedding under targeted therapy that incorporates pharmacokinetics. Using a simulated cohort of virtual patients with varied parameters, we define and analyze a biomarker based on ctDNA samples at baseline, 12 hours, and 24 hours after initiation of treatment. The biomarker identified patients who would achieve partial or complete response with high sensitivity and specificity and was able …

Weak Formulation For Solving Inverse Problems In Reproducing Kernel Hilbert Spaces (With Applications To Learning Dynamical Systems), Victor William Rielly

Dissertations and Theses

We combine numerical and machine learning techniques to present a general framework for solving inverse problems using vector valued reproducing kernel Hilbert spaces in a variational formulation. We present this framework in two papers. In the first paper, we present an original state-of-the-art method derived in the context of our general framework for learning dynamical systems. In the second paper, we generalize the method from our first paper to arrive at the framework for solving inverse problems. Then we apply our general framework to the task of learning dynamical systems. In both papers we consider numerous applications of our methods …

Estimating Pedestrian Crossing Times At Scramble Crossings Via Machine Learning And Agent-Based Modeling, Sho Takami

Honors Capstones

Scramble crosswalks differ from conventional crosswalks in their ability for pedestrians to cross diagonally. This research compares the average crossing times and investigates the walking behaviors that pedestrians adopt to produce the speediest times in the two crosswalk configurations. Identification of the most efficient set of walking behaviors is done through an agent-based model, whereas producing polynomials relating crossing times to the most prominent walking behaviors is done through regression algorithms in machine learning. With the combination of these two approaches, it is revealed that pedestrians must adopt a relaxed walking style to make each crosswalk configuration efficient. Additionally, between …

Mathematics-Ai Based Phylogenetic Analysis Of Influenza Virus Mutation Data, Emilee Walden

Mathematical Sciences Undergraduate Honors Theses

The influenza virus is one of the most common viral infections each year and can mutate rapidly. Viral mutations pose significant threats to public health by increasing infectivity and strengthening vaccine resistance. To track these evolving patterns, agencies like the CDC annually evaluate thousands of virus strains to understand viral mutagenesis and evolution in depth. Therefore, a computational method for analyzing high-dimensional, noisy virus data could aid in the rapid identification of antigens essential for an effective influenza vaccine for the upcoming season. Through the integration of genomic analysis, clustering, and dimensionality reduction methods, this study specifically aims to develop …

Under Pressure: A Quantitative Approach To Measuring Clutch Performance In The Nba, Jack Dell'isola

Honors Projects in Information Systems and Analytics

This research investigates the existence and relevance of clutch performance in the 2023-2024 NBA regular season. Players are analyzed both individually and against league averages to determine their clutch performance levels using an original "clutch score formula". This research aims to answer the questions of whether clutch performance is a real phenomenon, how individual player performance is affected in clutch time, and to determine a formula that can effectively predict the winner of the Clutch Player of the Year Award. The findings and formulas developed in this research help to shed light on the complexities of clutch performance, which has …

Efficient Solvers And Anderson Acceleration For The Bingham Equations, Victoria L. Fisher

All Theses

This work studies two techniques utilized to solve the Bingham equations that model viscoplastic flow. An Uzawa-type iterative procedure is first analyzed and tested for poor convergence of velocity and stress. We apply Anderson acceleration (AA) to this method and show improved convergence rates for both velocity and stress. A solver utilizing regularization is introduced, and AA is also applied to display better convergence results. We propose a new method that combines these two solvers and applies Anderson acceleration.