Applied Mathematics Commons^™

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 …

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 …

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.

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 …

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 …

Studies Of Pathways To T2d And Interventions Through A Dynamical System Model., Rafiqul Islam

Electronic Theses and Dissertations

Existing mathematical models investigating the progression of type 2 diabetes (T2D) over time primarily focus on glucose, insulin, β-cell mass, and other related factors, while often omitting fatty acids (FA) as an explicit variable—despite FA being a major energy source for the body. There exists a complex network of dynamical interactions among glucose, insulin, FA, and β-cell mass. To gain deeper insights into the metabolic dynamics and pathophysiology of T2D, it is essential to incorporate FA into such models. In this study, we extend the classic Topp’s GIβ model by explicitly incorporating FA and exploring its interactions with glucose, insulin, …

Modeling Dna Repair In Escherichia Coli Using A Boolean And Stochastic Framework, Gabrianne Ivey

All Theses

DNA can be damaged through both internal and external sources. Therefore, cells have created methods to repair DNA damage. In Escherichia coli, the system responsible for DNA repair is termed the SOS response. This system consists of more than 50 genes and contains three main repair pathways: nucleotide excision repair, translesion synthesis, and homologous recombination. The response is initiated when DNA lesions result in the accumulation of single-stranded DNA (ssDNA). The protein RecA is activated by binding to ssDNA and is then denoted RecA*. RecA* assists in the auto-cleavage of LexA which is the primary repressor protein involved in …

Improving Research Software Engineering In Mathematics, Abram Miller

Honors Theses

Research Software Engineering is critical to modern mathematical research, enabling the creation, maintenance, and dissemination of computational tools that bridge theory and practice. However, the field faces systemic challenges, including insufficient funding, lack of institutional recognition, and gaps in training and infrastructure. This thesis investigates these challenges through two approaches: (1) a comparative survey study focused on mathematicians and (2) hands-on contributions to an open-source research software project.

The Improving Research Software Engineering in Mathematics survey, conducted from September 2024 to January 2025, adapts the survey framework developed by Carver et al. in A survey of the state of the …

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.

A Spectroscopic Survey Of Atmospheres Of Super-Earths, Luke Brust

Honors Theses

Recent research has made it possible to use spectroscopy to analyze the composition of the atmospheres of exoplanets, planets that orbit other stars. Most projects thus far have focused on the atmospheres of gas giants, as it is less challenging to observe them with available equipment. This project studies the atmospheres of four Super- Earths, planets that have a mass greater than the Earth but smaller than Neptune. This is accomplished by processing the raw spectroscopic data from the Hubble Space Telescope and modeling the atmosphere using the program 𝝉-Rex3. This survey found clear results from two of the selected …

Globally Adaptive Exponential Integrators For Stiff Systems Of Odes, Anzhelika Vasilyeva

Honors Theses

This thesis introduces a novel method for solving systems of Ordinary Differential Equations (ODEs) resulting from the spatial discretization of Partial Differential Equations (PDEs). The proposed approach builds upon an existing technique that employs Krylov projection, which requires evaluating a matrix function at each timestep. The innovation of the new method lies in its reuse strategy, which shifts the perspective from direct matrix function evaluation to polynomial interpolation. Numerical experiments conducted on constant and variable coefficient heat equations, with both smooth and discontinuous initial data, demonstrate the computational time advantage of the new approach. The results indicate that this method …

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 …

Solution Of Preconditioned Nonsymmetric Saddle Point Systems Through Modified Conjugate Gradient Iteration, Samson Ayo

Dissertations

In this dissertation, we present an iterative method (Preconditioned Nonsymmetric Saddle Point Conjugate Gradient) for simultaneously solving forward ($A{\bf x}={\bf b}$) and adjoint ($A^T{\bf y}={\bf g}$) linear systems. Our approach involves constructing an augmented nonsymmetric saddle point matrix that has a real positive spectrum and developing a conjugate gradient-like iteration for this matrix. We investigate the use of Schur Complement preconditioners with block-diagonal factorization computed by an incomplete QR factorization of $A$ to speed up the convergence of our method and compare the results to the preconditioned generalized least squares residual (GLSQR) and quasi-minimal residual (QMR) methods. We develop quadrature …

The Numerical Method For Finding All The Zeros Of A Function F (X) On An Interval [A,B], Franissa Simon

Honors Theses

In this study, we develop a simple mathematical method for finding all the roots of a function on a specified interval. Existing classical numerical methods, including the Bisection Method, Secant Method, and Newton Method, cannot find all the zeros of a function f (x) on an interval [a,b]. Popular numerical root solvers like Matlab’s ‘fzero’, Maple’s ‘fsolve’, and SageMath’s ‘find_root’ typically yield only a single zero. This thesis explores a proposed interval computation bisection method, a systematic approach based on the traditional Bisection Method and interval computation. Unlike traditional bisection, which relies on the intermediate value theorem, this approach uses …

Analysing Bell Experiments Through Test Factors: Applications To Randomness And Strength Of Nonlocality, Soumyadip Patra

LSU New Orleans Theses and Dissertations

This work presents practical tools to analyse Bell experiments---experiments demonstrating correlations that defy classical explanations and proving that nature violates local realism. We begin by showing that in the Bell scenario specified by n parties with each party having a choice of m binary-outcome measurements---the (n,m,2) scenario---projecting weakly-signalling settings-conditional outcome distributions onto the smallest-dimensional affine subspace (containing the no-signalling set) via an L^2-distance-minimising map preserves correlators. This result ensures that Bell inequalities written in terms of correlators remain invariant under such projections, and we provide an efficient construction method for the projection operator that avoids computationally costly steps such as …

Decoding The Algorithm: The Mathematics Behind Tiktok’S Short-Form Content Success, Ashley N. Lynch

Honors Scholar Theses

Within the realm of social networks, TikTok has become the central hub for short-form video content. The network’s unique ability to capture individual preferences using predictive analytics has greatly contributed to its massive success, allowing the company to optimize its performance and content personalization. In an age where digital media have such a significant influence on society, it is essential that users develop an understanding of how social network algorithms function to make more informed online decisions. Although TikTok’s technological system is primarily undisclosed, the platform certifiably leverages several key mathematical principles within its algorithm to achieve its core goals …