Applied Mathematics Commons^™

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 …

The Herzog-Takayama Resolution Over A Skew Polynomial Ring, Linoy Utkina

Theses and Dissertations

Let k be a field, and let I be a monomial ideal in the polynomial ring R = k[x1,..., xn]. In her thesis, Taylor introduced a complex that yields a finite free resolution of R/I as an R-module. Building on Taylor’s work, Ferraro, Martin, and Moore extended this construction to monomial ideals in skew polynomial rings. Because the Taylor resolution is typically not minimal, subsequent research efforts went into identifying specific classes of ideals whose minimal free resolutions can be constructed more simply. In 1990, Eliahou and Kervaire devised an approach for handling minimal resolutions of …