Mathematics Commons^™

The Uncertainty Principles, Lee Michael Felicetti

Mathematics & Statistics ETDs

The Heisenberg uncertainty principle is a central aspect of quantum mechanics, but also illustrates an essential quality of the Fourier transform. After Heisenberg, a variety of uncertainty inequalities emerged in the fields of physics and mathematics. In this thesis we will analyze the Heisenberg uncertainty principle in both the setting of quantum mechanics and Fourier analysis. We will then look at how the work of Heisenberg has been expanded upon in both physics and mathematics. Particularity, we will see how uncertainty principles can be applied to signal recovery and explore current research in this field.

Boundary Integral Method For A Modified Mullins–Sekerka System, Ly Le

Mathematics & Statistics ETDs

This thesis develops a boundary integral method for a modified Mullins–Sekerka system arising as the sharp-interface limit of a nonreciprocal Cahn–Hilliard model. Nonreciprocal coupling changes the classical Cahn–Hilliard structure by introducing an additional conserved field, leading to coupled elliptic and parabolic dynamics at the interface. Using matched asymptotic expansions, we formally derive the modified Mullins–Sekerka model and then apply the boundary integral method to rewrite it on the moving interface. The elliptic component is represented using the periodic Green’s function for the Laplace equation, while the parabolic component is represented using the periodic heat kernel. This method reduces the bulk …

Nerve Constructions And Mapper, Alexander Bram Fritschi

Mathematics & Statistics ETDs

Mapper is a data visualization tool commonly used in topological data analysis to study large, often high-dimensional datasets. Mapper operates through the selection of a lens function, a clustering algorithm, and a cover. The Mapper graph is constructed using the nerve of the cover after the clustering algorithm is performed; it is therefore useful to study nerves to better understand Mapper. In this thesis, we will utilize the properties of nerves to find the minimal point set that produces a given graph. We will then extend this to Mapper to determine what Mapper graphs may be constructed over a given …

Interpretable Case-Control Inference Through Log-Linear General Location Models, Zacharia Stuart

Mathematics & Statistics ETDs

This dissertation analyzes one of the few publicly available NFL injury datasets to study field type and non-contact lower-limb injuries. Field type is studied jointly with other risk factors to understand how these factors interact to affect injury risk. The data were gathered through a case-control sampling scheme, which limits direct inference on absolute injury probabilities. While not the most common approach for case-control data, this dissertation models the retrospective distribution directly through Log-Linear General Location Models (Log-Linear GLOMs). Through a log-linear structure placed on a log-odds-ratio reparameterization, the model provides directly interpretable marginal and interaction contributions to injury log-odds …

Bayesian Designs For Two-Arm Clinical Trials With Time-To-Event Endpoints: Incorporating Historical Data Through Power Priors, Sara Hajraf H. Almutiri

Mathematics & Statistics ETDs

Bayesian methods provide a flexible framework for time-to-event analysis by incorporating prior information. The power prior offers a systematic way to borrow information from historical data. This approach is especially valuable in clinical research, where historical data can enhance inference in early-phase trials with limited sample sizes. This dissertation develops Bayesian approaches for two-arm survival studies using both closed-form and simulation-based methods. The closed-form inference is derived under exponential and Weibull survival models. Under the proportional hazards framework, the posterior is derived through a normal approximation to the log hazard ratio, allowing inference on the treatment effect when the variance …

Kakeya Sets Over The Heisenberg Group, Gabriel Jacob Gress

Mathematics & Statistics ETDs

The Hausdorff dimension of Kakeya sets is an interesting problem where the two-dimensional case can be proven directly, but even obtaining bounds on the higher dimension analogues can require highly technical machinery. The difficulty of the general case has inspired analysts to look at Kakeya sets from non-Euclidean viewpoints.

In this paper, we explore a construction of the Kakeya set in the first Heisenberg group H^1. By utilizing the sub-Riemannian manifold of H^1 we can apply tools in geometric measure theory which at this time cannot be applied in R^3. We restate and provide a detailed proof for a sharp …

On Sharpest Tail Bounds For Functions Of Tail Bounded Random Variables, Stephen Harrison

Mathematics & Statistics ETDs

Consider n real/complex, independent/dependent random variables with respective tail bounds and g a measurable function of the r.v.’s. Consider f the “sharpest” tail bound of g (sharpest in the sense, if f were any less, then for some X1, ..., Xn satisfying the conditions, g(X1, ..., Xn) would not satisfy the tail f). Significant research has been done to approximate f often with high accuracy. These results are often of the form, for g in this family, and tail bounds of Xk in this family, f is bounded by some f′ with high accuracy. However, the question “what would it …

Algebraic Multigrid Methods For Nonsymmetric And Indefinite Problems: Theory And Applications, Ahsan Ali

Mathematics & Statistics ETDs

Algebraic multigrid (AMG) is a well-established and highly efficient solver for symmetric positive definite (SPD) systems arising from elliptic and parabolic PDEs, while nonsymmetric systems from hyperbolic PDEs remain a significant challenge. This dissertation develops AMG methods and theory for nonsymmetric problems. First, we develop a novel approach combining mode constraints from energy-minimization AMG with local approximations of ideal restriction in $\ell$AIR, resulting in constrained $\ell$AIR (C$\ell$AIR), which demonstrates scalable convergence across advective and diffusive problems. Second, we extend optimal AMG theory by deriving spectral radius estimates for the two-grid error transfer operator using matrix-induced orthogonality, enabling convergence predictions for …

A Data-Driven Approach To Time Series Forecasting And Clustering Of U.S. Regional Drug Overdose Mortality, Koshali Hamy Muthunama Gonnage

Mathematics & Statistics ETDs

The increasing rate of drug overdose deaths in the United States poses a critical public health challenge, particularly due to the surge in synthetic opioids and other high-risk substances. This study presents a data-driven framework that integrates time series forecasting and clustering techniques. Monthly mortality data for five key drug types: cocaine, fentanyl, heroin, methamphetamine, and oxycodone were analyzed using four time series forecasting models: ARIMA, ETS, TBATS, and NNAR. These models were evaluated using standard accuracy metrics RMSE, MAPE, and MAE to assess predictive performance. Signal decomposition approach based on Singular Value Decomposition and subspace modeling was employed to …

Minimal Error Functions On Irregular Subsets Of The Real Line, Robert Michael Dukes

Mathematics & Statistics ETDs

Chebyshev Polynomials, those that minimize the maximal error on a compact set, are one of the most practical tools for approximating smooth functions. The classical results are on the set [-1, 1]; in this paper, we extend to more complicated subsets of the real line. We demonstrate some classical results and then take the result from [2] on regular Parreau-Widom Sets and extend it to semi-regular sets, defined as sets whose regular part is closed. We introduce the Regularity Coefficient as a series formed by evaluating the Green’s Function at irregular points. This new machinery is applied to the lower …

Derivation Of Adjoint Based Error Estimates For Nonlinear Ordinary Differential Equations With Application To Multistage Sir Models With Demographics, Daniel Alcala

Mathematics & Statistics ETDs

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error …

Toward Simulating 2d Cell Surfaces In A Disk, Myriam Allred

Mathematics & Statistics ETDs

Certain evolution models of cell surfaces (treated in two-dimensions) involve the solution of the Helmholtz equation with jump conditions enforced on an immersed closed curve. This thesis presents a sparse, modal spectral method for solving such Helmholtz problems. The solution is required to be continuous across the curve, but with a jump discontinuity in the normal derivative proportional to the planar curvature. The method relies on classical Fourier-Chebyshev basis functions, with the application of modal Chebyshev integration matrices to achieve sparse, banded approximations of the Helmholtz equation. The method achieves spectral convergence, despite the inherent low regularity of the relevant …

Evaluating The Performance Of Bayesian Removal Models For Estimating Population Density And Detecting Trends With Variable Detection Probability, David R. Stewart

Mathematics & Statistics ETDs

Removal models have long been used to estimate population abundance by progressively capturing and removing individuals from a closed population. These models provide a valuable tool for ecological monitoring, but their accuracy depends heavily on assumptions about detection probability, which may decline over successive sampling passes. Traditional removal models assume constant detection probabilities, an assumption that is often violated in real-world applications. This thesis aims to advance hierarchical Bayesian models by accounting for variable detection probabilities, improving the reliability of abundance estimates and trend detection. By integrating simulation-based analyses with empirical data from Lahontan Cutthroat Trout (Oncorhynchus clarkia henshawi …

Unraveling The Impact Of Curricular Complexity On Graduation Time: A Causal Analysis In Higher Education, Ameer Slim

Mathematics & Statistics ETDs

This study examines the causal relationship between program complexity and graduation time at UNM. While program complexity is recognized as a factor influencing student outcomes, its precise impact on graduation timelines remains underexplored. Using comprehensive cohort data, this study employs causal inference methods, including generalized propensity scores, to estimate the effect of complexity on time-to-degree. Findings reveal that higher program complexity extends graduation timelines, even after controlling for demographics and academic preparedness. Socioeconomic factors also play a role. Specifically, programs with more Pell Grant recipients and lower median high school GPAs tend to have lower complexity levels. These results provide …

Operations On Submodules With The Multiplicative And Quotient Properties, Jiekai Pang

Mathematics & Statistics ETDs

Inspired by the works of Petro, Epstein, Vassilev, and Morre, this thesis aims to study the generalized definitions of the semiprime operation, weakly prime operation, and standard closure operation on rings, that is, the multiplicative operation, weakly multiplicative operation, and standardly multiplicative operation on submodules. Then, we will use Matlis duality to induce the dual notions of these definitions on submodules of Matlis-dualizable Artinian modules. In order to understand the dual notion of the standardly multiplicative operation, that is, the standardly quotient operation, we will classify the operations on the injective hull of residue field of the ring K[[x,y]]/(xy) which …

Visualization Of Species Tree Likelihood Under The Multispecies Coalescent Model, Jaimasan Sutton

Mathematics & Statistics ETDs

A commonly used tool for evolutionary biologists is a phylogenetic tree that represents the ancestry of a set of species and the evolution of traits. Statistical models can be used to predict the probabilities of gene trees which represent ancestral relationships of genes sampled from species. Because of this, we are able to represent the likelihood of a species tree, which represents the evolutionary history of a set of species, as a function of the counts of gene tree topologies, where each gene tree represents the ancestry of a specific genetic locus for multiple species. Because we can represent these …

Parallel Multigrid In Time For Chaotic Dynamical Systems, David Alan Vargas

Mathematics & Statistics ETDs

Despite the fact that Parallel-in-Time (PinT) methods are predicted to become necessary to fully utilize next-generation exa- and zettascale machines, there are currently no known practical methods which scale well with the length of the time-domain for chaotic problems, due to exponential dependence of the condition number on the fastest chaotic timescale. I present modifications to the coarse-grid equations along with a novel rediscretization approach which together greatly improve convergence of the multigrid reduction in time (MGRIT) algorithm and allow the first known PinT speedup for a chaotic PDE. The novel Local Shadowing Relaxation (LSR) is presented as an alternative …

Improved Rational Approximation Of Near-To-Far Propagation Kernels For The Wave Equation, Sampson Owusu

Mathematics & Statistics ETDs

The 3-space, 1-time dimensional scalar wave equation, or 3+1 wave equation, describes the propagation of scalar or acoustic waves. The unforced homogeneous equation admits a class of outgoing solutions relative to a chosen fixed center, so called “multipole” solutions. This thesis examines near-to-far signal propagation in the context of these multipole solutions. Given a time-series (history of values) for a multipole solution recorded at a radius r1, near- to-far signal propagation recovers the corresponding time-series at larger radius r2 ≫ r1. This propagation takes into account both the appropriate time delay r2 − r1 and corrections to the wave shape. …

On The Limitations And Restrictions Of The Hardy-Littlewood Circle Method, Daniel W. Havens

Mathematics & Statistics ETDs

We discuss herein the history, layout, and philosophy of the Hardy-Littlewood Circle method, as well as the more modern renditions thereof. The limitations and scope of each method presented is discussed in detail, providing examples of cases where the failure of the circle method is of relevance. We include a summary of famous problems which have been resolved using each methodology, as well as what limitations each methodology showcases.

Robust Prediction Of Charpy Toughness Of Additively Manufactured Kovar Using Deep Convolutional Neural Networks, Nathan R. Bianco

Mathematics & Statistics ETDs

Understanding the reason for mechanical failures of manufactured parts in their operating environments is critical to prevention of future failures. However, in-situ post-mortem evaluation of physical properties, such as fracture toughness, is time consuming and alters the condition of the material, leading to potentially misleading findings. In this study, additively manufactured test coupons were produced over a wide range of process conditions to test the impact toughness of a material. The Charpy V-Notch toughness was measured on over 200 samples alongside corresponding optical images of both sides of the fracture surface. Convolutional neural network models were trained to correlate fracture …

On Properties Of Pair Operations, Sarah Jane Poiani

Mathematics & Statistics ETDs

For any closure operation $\cl$ and interior operation $\ri$ on a class of $R$-modules, we develop the theory of $\cl$-prereductions and $\ri$-postexpansions. A pair operation is a generalization of closure and interior operations. Using Epstein, R.G. and Vassilev's duality \cite{ERGV-nonres}, we show that these notions are in fact dual to each other. We discuss the relationship between the core and hull and prereductions and postexpansions. We further the thematic notion of duality and seek to understand how it arises in the context of properties pair operations can be endowed with and focus on inner product spaces and properties demonstrated by …

Mathematically Rigorous Deep Learning Paradigms For Data-Driven Scientific Modeling, Owen Nicholas Davis

Mathematics & Statistics ETDs

This dissertation explores the crucial role of data-driven modeling in science and engineering, with a focus on developing surrogate models to accelerate large-scale computational tasks, aiding in both outer-loop functions like uncertainty quantification and expensive inner-loop tasks within broader computational frameworks. Challenges arise with increased problem dimension and sparse, noisy training data, particularly significant when constructing surrogates for very expensive computational models where acquiring sufficient high-fidelity training data is unfeasible. In such scenarios, training surrogates from an ensemble of multifidelity information sources of varying accuracy and cost becomes essential. We emphasize neural network-based modeling paradigms, which are flexible in integrating …

Probabilistic Modeling Of Social Media Networks, Distinguishing Phylogenetic Networks From Trees, And Fairness In Service Queues, Md Rashidul Hasan

Mathematics & Statistics ETDs

In this dissertation, three primary issues are explored. The first subject exposes who-saw-from-whom pathways in post-specific dissemination networks in social media platforms. We describe a network-based approach for temporal, textual, and post-diffusion network inference. The conditional point process method discovers the most probable diffusion network. The tool is capable of meaningful analysis of hundreds of post shares. Inferred diffusion networks demonstrate disparities in information distribution between user groups (confirmed versus unverified, conservative versus liberal) and local communities (political, entrepreneurial, etc.). A promising approach for quantifying post-impact, we observe discrepancies in inferred networks that indicate the disproportionate amount of automated bots. …

Modified Geometries, Clifford Algebras And Graphs: Their Impact On Discreteness, Locality And Symmetr, Roman Sverdlov

Mathematics & Statistics ETDs

In this dissertation I will explore the question whether various entities commonly used in quantum field theory can be “constructed". In particular, can spacetime be “constructed" out of building blocks, and can Berezin integral be “constructed" in terms of Riemann integrals.

As far as “constructing" spacetime out of building blocks, it has been attempted by multiple scientific communities and various models were proposed. But the common downfall is they break the principles of relativity. I will explore the ways of doing so in such a way that principles of relativity are respected. One of my approaches is to replace points …

Multilevel Optimization With Dropout For Neural Networks, Gary Joseph Saavedra

Mathematics & Statistics ETDs

Large neural networks have become ubiquitous in machine learning. Despite their widespread use, the optimization process for training a neural network remains com-putationally expensive and does not necessarily create networks that generalize well to unseen data. In addition, the difficulty of training increases as the size of the neural network grows. In this thesis, we introduce the novel MGDrop and SMGDrop algorithms which use a multigrid optimization scheme with a dropout coarsening operator to train neural networks. In contrast to other standard neural network training schemes, MGDrop explicitly utilizes information from smaller sub-networks which act as approximations of the full …

Using Physics-Informed Neural Networks For Multigrid In Time Coarse Grid Equations, Jonathan P. Gutierrez

Mathematics & Statistics ETDs

For parallel-in-time integration methods, the multigrid-reduction-in-time (MGRIT) method has shown promising results in both improved convergence and increased computational speeds when solving evolution problems. However, one problem the MGRIT algorithm currently faces is it struggles solving hyperbolic problems efficiently. In particular, hyperbolic problems are generally solved using explicit methods and this causes issues on the coarser multigrid levels, where larger (coarser) time step sizes can violate the stability condition. In this thesis, physics-informed neural networks (PINNs) are used to evaluate the coarse grid equations in the MGRIT algorithm with the goal to improve convergence for problems with hyperbolic behavior, as …

Mitigation Impact Of Statewide Non-Pharmaceutical Policies On Covid-19: An Application Of Infectious Disease Transmission Model And Partially Observed Markov Process To New Mexico, Xingya Ma

Mathematics & Statistics ETDs

This thesis is an application of epidemiological models for infectious disease transmission and the use of partially observed Markov process (POMP) for model fitting. It focuses on COVID-19 pandemic in the state of New Mexico. The analysis covered March 2020 to June 2021. Daily data of COVID19 cases and deaths and a daily index of eleven statewide government non-pharmaceutical intervention (NPI) policies were collected from six public sources and were validated. These data were integrated through the Susceptible-Exposed-Infected-Removed (SEIR) model. Estimated daily transmission rates between the model compartments quantify the impact of the mitigation policies, and show that transmission rates …

Convexity Of Regularized Optimal Transport Dissimilarity Measures For Signed Signals, Christian P. Fowler

Mathematics & Statistics ETDs

Debiased Sinkhorn divergence (DS divergence) is a distance function of

regularized optimal transport that measures the dissimilarity between two

probability measures of optimal transport. This thesis analyzes the advantages of

using DS divergence when compared to the more computationally expensive

Wasserstein distance as well as the classical Euclidean norm. Specifically, theory

and numerical experiments are used to show that Debiased Sinkhorn divergence

has geometrically desirable properties such as maintained convexity after data

normalization. Data normalization is often needed to calculate Sinkhorn

divergence as well as Wasserstein distance, as these formulas only accept

probability distributions as inputs and do not directly …

Statistical Methods For Differential Gene Expression Analysis Under The Case-Cohort Design, Lidong Wang

Mathematics & Statistics ETDs

Differential gene expression analysis has the potential to discover candidate biomarkers, therapeutic targets, and gene signatures. How to save money when using an unaffordable sample is a practical question. The case-cohort (CCH) study design can blend the economy of case-control studies with the advantages of cohort studies. But it has not been seen in the medical research literature where high-throughput genomic data were involved.

A score test does not need to fit the Cox PH model iteratively; hence, it can save computing time and avoid potential convergence issues. We developed a score test under the CCH design to identify DEGs …

Functional Data Analysis Of Covid-19, Nichole L. Fluke

Mathematics & Statistics ETDs

This thesis deals with Functional Data Analysis (FDA) on COVID data. The Data involves counts for new COVID cases, hospitalized COVID patients, and new COVID deaths. The data used is for all the states and regions in the United States. The data starts in March 1^st, 2020 and goes through March 31^st, 2021. The FDA smooths the data and looks to see if there are similarities or differences between the states and regions in the data. The data also shows which states and regions stand out from the others and which ones are similar. Also shown …