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 …

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.

The Little Diagram That Could: Geometric Properties And Statistical Applications Of Persistence Diagrams In Topological Data Analysis, Eugene Kler

McKelvey School of Engineering Graduate Student Theses & Dissertations

Topological Data Analysis (TDA) is a collection of techniques for data analysis that leverages topological invariants of spaces formed from data points. These methods excel at extracting useful information from noisy or sparse data, making them attractive to many mathematicians, statisticians, and scientists. In this thesis, we explore TDA on three fronts: algebraic foundations, statistical applications, and metric properties. Throughout, the central object of study is the Persistence Diagram (PD), a summary of the changes in homology that occur as one builds simplicial complexes from the data by increasing a parameter.

Using Gaussian Process Regression To Learn Thermodynamic Equations Of State With Uncertainty Quantification, Austen T. Lee

Chemical Engineering Undergraduate Honors Theses

This study investigates the use of derivative-informed Gaussian Process (GP) models to estimate thermodynamic behavior across temperature and density by building a Helmholtz-based equation of state. Argon, a stable monatomic gas, was chosen as a case study within the vapor region. The GP model was trained using values of experimentally measurable properties found by taking first and second derivatives of the original potential function. Results show that while the GP model offered uncertainty quantification and informed thermodynamic behavior, it predicted values that deviated from the ground truth depending on the property. The model exhibited high confidence in regions with substantial …

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 …

A Profile Wald Test In M-Estimation, Reagan Kesseku

Open Access Theses & Dissertations

Despite the growing popularity of machine learning-based inference, classical statistical inference remains highly relevant in modern data science due to its interpretability and theoretical rigor. Among its core tools, the likelihood ratio test, Wald test, and score test are foundational methods for hypothesis testing within the maximum likelihood framework. Although these tests are asymptotically equivalent under regularity conditions, each offers distinct advantages depending on the context, computational demands, and the availability of parameter estimates. In this dissertation, we introduce a fourth method, the Profile Wald Test (PWT), within the broader M-estimation framework. The PWT is based on profile estimators of …

Decomposition And Coordination For Multiobjective Optimization: A Framework And Methodology, Philip J. De Castro

All Dissertations

In this work, we consider finding Pareto efficient solutions for complex multiobjective optimization problems (MOPs). Complex MOPs are unique in the literature because they have many more objective functions than is typically considered. In fact, such complex MOPs will have 30+ objective functions. This large problem size presents computational and coginitive difficulties. Computationally, standard techniques for solving MOPs are often ineffective and cognitively it is difficult for a decision maker (DM) to handle all of the information provided in such a large problem. To address these challenges, we develop a decomposition and coordination framework. This framework will allow us to …

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 …

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 …

Simulations Of Richtmyer-Meshkov Instability Using High Order Weno Methods, Ryan Holley

Graduate Theses and Dissertations

Turbulent mixing due to hydrodynamic instabilities occurs in a broad spectrum of engineering, astrophysical and geophysical applications. Theory, experiment, and numerical simulation help us to understand the dynamics of interface instabilities between two fluids. This thesis presents an increasingly accurate and robust front tracking method for the numerical simulations of shock-induced turbulent mixing known as Richtmyer-Meshkov Instability (RMI). Front tracking is an adaptive computational method, where the interface instability is explicitly represented as lower dimensional manifolds moving through a rectangular grid. All the cell-center states (density, velocity and pressure) are updated using higher order weighted essentially non-oscillatory (WENO) scheme. Performance …

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 …

Sliding Window Method For Simulating Action Potentials In Axons, Hayden Reed

Honors Theses

ABSTRACT Hodgkin and Huxley’s nonlinear partial differential equations model the excitation and propagation of action potentials in neurons, and there have been numerous attempts at finding the best numerical solution method. This thesis proposes a novel approach to solving these equations: the Sliding Window method, in which a fixed sub-interval is found through capturing the signal’s head and tail. The system is then solved on the sub-interval instead of the entire interval. Using the Sliding Window technique also involves implementing the backward and forward Euler methods and the finite difference method. It will be demonstrated that, in utilizing the Sliding …

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 …

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 …

Evaluating Risk And Return Of Corn And Soybean Marketing Strategies Using Monte Carlo Simulation Methods, Johanna Ilves

Department of Agricultural Economics: Dissertations, Theses, and Student Research

This study evaluates the performance of pre-harvest marketing strategies for corn and soybeans, using Monte Carlo simulation techniques. Given the increasing volatility in commodity prices and the evolving landscape of agricultural markets, producers face growing challenges in developing effective marketing plans that manage risk and enhance profitability. This thesis examines thirteen marketing strategies from 2008 to 2024, including benchmark harvest-only sales and various pre-harvest futures contract approaches. Historical futures price data for December corn and November soybean contracts were used to simulate 1,000 marketing outcomes per strategy per year, capturing a broad range of market conditions. Key performance indicators such …

Physics Embedded Neural Network: A Novel Data-Free Numerical Method For Solving Computational Physics Problems, Pawan Gaire

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

A novel approach for solving partial differential equations (PDEs) using neural networks for scientific computing is introduced. The proposed approach, referred to as physics-embedded neural network (PENN), features a unique architecture that incorporates the PDE and boundary conditions information directly within the final fully-connected layer of the feed-forward neural network (NN). The key aspect of PENN is the parallel numerical embedding of a differential equation associated with physical problems within the activation function of the network’s final layer. This integration leads to a new class of computational solvers competitive with classical methods like the Finite Element Method (FEM) and capable …

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 …

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 …

Car Price Prediction Using Machine Learning: Analyzing The Dvm-Car Dataset, Yaman Abu Ghareebaih

Electronic Theses and Dissertations

The objective of this study is to predict car prices using machine learning models and the DVM-CAR dataset, which includes over 1.4 million images and car specifi- cations from 899 car models. Key factors such as mileage, engine power, and year of registration were analyzed for their correlation with car prices. Extensive data cleaning was performed, including filling missing values, identifying outliers, and normalizing numerical variables. Discrete variables like car make and body type were encoded using one-hot encoding. Linear relationships were analyzed with Multiple Logistic Regression, and Random Forest models were used for nonlinear patterns. Model performance was evaluated …