Digital Commons Network^™

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 …

Music Genre Classification By Convolutional Neural Networks, Usame Suud

Mathematics & Statistics ETDs

In today’s world, deep learning models are widely used in a variety of fields. Audio

applications include speech recognition, audio classification, and music information

retrieval. In this paper, we will focus on the classification of music genres using an

artificial neural network. The development of audio machine learning techniques has

created an independence from traditional, more time-consuming signal processing

techniques. Starting with raw audio data, we will gain an understanding of what

audio is and its digital representation. Then, the focus will be on obtaining frequency

information from audio signals through the use of spectrograms. Transforming the

spectrograms into the …

Determining The Idealizers Of Principal Monomial Ideals Over A Rational Normal Curve, Perla A. Maldonado Cortez

Mathematics & Statistics ETDs

Given an ideal J generated by an element of the form sm1 tm2 , where
m1 ≥ 2 and m2 ≥ 0, we illustrate how to compute the idealizer I(J) over the ring
of the rational normal curve of degree n and we give a formula for it using the
graded pieces of the sets of differential operators.

Machine Learning Model Comparison And Arma Simulation Of Exhaled Breath Signals Classifying Covid-19 Patients, Aaron Christopher Segura

Mathematics & Statistics ETDs

This study compared the performance of machine learning models in classifying COVID-19 patients using exhaled breath signals and simulated datasets. Ground truth classification was determined by the gold standard Polymerase Chain Reaction (PCR) test results. A residual bootstrapped method generated the simulated datasets by fitting signal data to Autoregressive Moving Average (ARMA) models. Classification models included neural networks, k-nearest neighbors, naïve Bayes, random forest, and support vector machines. A Recursive Feature Elimination (RFE) study was performed to determine if reducing signal features would improve the classification models performance using Gini Importance scoring for the two classes. The top 25% of …

Robust Uncertainty Quantification With Analysis Of Error In Standard And Non-Standard Quantities Of Interest, Zachary Stevens

Mathematics & Statistics ETDs

This thesis derives two Uncertainty Quantification (UQ) methods for differential equations that depend on random parameters: (\textbf{i}) error bounds for a computed cumulative distribution function (\textbf{ii}) a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria. Both UQ approaches utilize adjoint-based \textit{a posteriori} error analysis in order to accurately estimate the error in samples of numerically approximated quantities of interest. The adaptive MLMC algorithm developed in this thesis relies on the adjoint-based error analysis to adaptively create meshes and accurately monitor a stopping criteria. This is in contrast to classical MLMC algorithms which employ either a …

Effect Of Phylogeny Misestimation On Estimating Trait Evolution Parameters, Tabytha Ariel Perez

Mathematics & Statistics ETDs

Biologists are interested in estimating trait evolution models from phylogenies. However, phylogenies are imperfectly estimated, generally from DNA sequence data. In this study, true phylogenies are simulated to understand whether errors in phylogeny estimation affect inference of the trait evolution model. Given the tree, DNA sequences as well as traits are then simulated from the true phylogeny; both are simulated independently from the other. After the simulation, the DNA sequences were used to estimate trees using the UPGMA method without utilizing the trait information. The estimated trees combined with the traits are used to infer the evolutionary trait models, specifically …

The Obata First Eigenvalue Theorem On A Seven Dimensional Quaternionic Contact Manifold, Abdelrahman Mohamed

Mathematics & Statistics ETDs

We prove an Obata-type rigidity result for the first eigenvalue of the sub-Laplacian on a compact seven dimensional quaternionic contact (QC) manifold which satisfies a Lichnerowicz-type bound on its QC-Ricci tensor, and has a non-negative Paneitz P -function. In particular, under the stated conditions, the lowest possible eigenvalue of the sub-Laplacian is achieved if and only if the manifold is QC-equivalent to the standard 3-Sasakian sphere.

Applications Of Machine Learning Algorithms In Materials Science And Bioinformatics, Mohammed Quazi

Mathematics & Statistics ETDs

The piezoelectric response has been a measure of interest in density functional theory (DFT) for micro-electromechanical systems (MEMS) since the inception of MEMS technology. Piezoelectric-based MEMS devices find wide applications in automobiles, mobile phones, healthcare devices, and silicon chips for computers, to name a few. Piezoelectric properties of doped aluminum nitride (AlN) have been under investigation in materials science for piezoelectric thin films because of its wide range of device applicability. In this research using rigorous DFT calculations, high throughput ab-initio simulations for 23 AlN alloys are generated.

This research is the first to report strong enhancements of piezoelectric properties …

Sparse Spectral-Tau Method For The Two-Dimensional Helmholtz Problem Posed On A Rectangular Domain, Gabriella M. Dalton

Mathematics & Statistics ETDs

Within recent decades, spectral methods have become an important technique in numerical computing for solving partial differential equations. This is due to their superior accuracy when compared to finite difference and finite element methods. For such spectral approximations, the convergence rate is solely dependent on the smoothness of the solution yielding the potential to achieve spectral accuracy. We present an iterative approach for solving the two-dimensional Helmholtz problem posed on a rectangular domain subject to Dirichlet boundary conditions that is well-conditioned, low in memory, and of sub-quadratic complexity. The proposed approach spectrally approximates the partial differential equation by means of …

Eigenfunction Restriction Estimates For Curves With Nonvanishing Geodesic Curvatures In Compact Riemannian Surfaces With Nonpositive Sectional Curvatures, Chamsol Park

Mathematics & Statistics ETDs

For 2 ≤ p < 4, we study the Lp norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2-dimensional Riemannian manifolds. Burq, G\´erard, and Tzvetkov [12], and Hu [21] found eigenfunction restriction estimates for a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p = 4, we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [37], Sogge [33], and Bourgain [10]. At the end of this dissertation, we will talk about a future work, which is a follow up study for higher dimensional analogues of the above curve cases.

Sparse Domination Of The Martingale Transform, Michael Scott Kutzler

Mathematics & Statistics ETDs

Linear operators are of huge importance in modern harmonic analysis. Many operators can be dominated by finitely many sparse operators. The main result in this thesis is showing a toy operator, namely the Martingale Transform is dominated by a single sparse operator. Sparse operators are based on a sparse family which is simply a subset of a dyadic grid. We also show the A2 conjecture for the Martingale Transform which follows from the sparse domination of the Martingale Transform and the A2 conjecture for sparse operators.

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Multiple Baseline Interrupted Time Series: Describing Changes In New Mexico Medicaid Behavioral Health Home Patients’ Care, Jessica Reno

Mathematics & Statistics ETDs

In 2016, the CareLink New Mexico behavioral health homes program began enrolling Medicaid recipients with the goal of increasing care coordination, improving access to services, and decreasing long-term costs of care for adults with serious mental illness (SMI) and children with severe emotional disturbance (SED). To evaluate these aims, a retrospective interrupted time series study using Medicaid claims data was designed. First, a comparable subset of non-enrolled individuals was selected from the pool of Medicaid recipients with SMI or SED using propensity score matching. Then, segmented regression was applied to three outcomes: total Medicaid charges, number of outpatient behavioral health …

Optimal Transport Driven Bayesian Inversion With Application To Signal Processing, Elijah F. Perez

Mathematics & Statistics ETDs

This paper will outline a Debiased Sinkhorn Divergence driven Bayesian inversion framework. Conventionally, a Gaussian Driven Bayesian framework is used when performing Bayesian inversion. A major issue with this Gaussian framework is that the Gaussian likelihood, driven by the L2 norm, is not affected by phase shift in a given signal. This issue has been addressed in [1] using a Wasserstein framework. However, the Wasserstein framework still has an issue because it assumes statistical independence when multidimensional signals are analyzed. This assumption of statistical independence cannot always be made when analyzing signals where multiple detectors are recording one event, say …

Applications Of Evidence Theory To High-Consequence Systems Safety, Christina Marie Deffenbaugh

Mathematics & Statistics ETDs

Issues linked to abnormal environments (like high-consequence systems safety, e.g., nuclear weapon components, bridges, apartment buildings, etc.) may have insufficient information to use either classical statistical methods or Bayesian approaches for calculating associated probabilistic risks, so there is often a requirement for another method that can deal with a low-information situation to obtain a risk assessment. Belief/plausibility measures of uncertainty from A. P. Dempster and G. Shafer’s Evidence Theory is one such method. This thesis has two goals. First, a brief discussion on belief/plausibility measures as an application of Evidence Theory will familiarize the audience with its history and how …

An Analysis Of Growth Of The Community Integration Psychological Score In An Ethnically Diverse Population Experiencing Homelessness In A Permanent Supportive Housing Program Using Hierarchical Mixed Modeling, Leah Hollis Puglisi

Mathematics & Statistics ETDs

Hierarchical models are becoming increasingly common in epidemiological and psychological research. When analyzing data from such studies, the nested structure of the data must be taken into account. Mixed modeling in conjunction with hierarchical mixed modeling allows researchers to ask broad questions about the population of interest. Modeling under restricted maximum likelihood estimation (REML), as opposed to full maximum likelihood estimation (ML), increases the accuracy of estimates for the random effects in the model. We use hierarchical mixed modeling under REML estimation to analyze which factors increase “community integration”, a concept and a construct developed and used in the mental …

Numerical Simulations Of Nonlinear Waves And Their Stability: Stokes Waves And Nonlinear Schroedinger Equation, Anastassiya Semenova

Mathematics & Statistics ETDs

The present work offers an investigation of dynamics and stability of nonlinear waves in Hamiltonian systems. The first part of the manuscript discusses the classical problem of water waves on the surface of an ideal fluid in 2D. We demonstrate how to construct the Stokes waves, and how to apply a continuation method to find waves in close vicinity to the limiting Stokes wave. We provide new insight into the stability of the Stokes waves by identifying previously inaccessible branches of instability in the equations of motion for the fluid. We provide numerical evidence that pairs of unstable eigenvalues of …

From Wave Propagation To Spin Dynamics: Mathematical And Computational Aspects, Oleksii Beznosov

Mathematics & Statistics ETDs

In this work we concentrate on two separate topics which pose certain numerical challenges. The first topic is the spin dynamics of electrons in high-energy circular accelerators. We introduce a stochastic differential equation framework to study spin depolarization and spin equilibrium. This framework allows the mathematical study of known equations and new equations modelling the spin distribution of an electron bunch. A spin distribution is governed by a so-called Bloch equation, which is a linear Fokker-Planck type PDE, in general posed in six dimensions. We propose three approaches to approximate solutions, using analytical and modern numerical techniques. We also present …

Intrinsic Curvature For Schemes, Pat Lank

Mathematics & Statistics ETDs

This thesis develops an algebraic analog of psuedo-Riemannian geometry for relative schemes whose cotangent sheaf is finite locally free. It is a generalization of the algebraic differential calculus proposed by Dr. Ernst Kunz in an unpublished manuscript to the non-affine case. These analogs include the psuedo-Riemannian metric, Levi-Civit´a connection, curvature, and various existence theorems.

Lattice Of Maximal-Primary Ideals In Quadratic Orders, Ryan Bridges

Mathematics & Statistics ETDs

An order is a subring of the ring of integers of an algebraic extension, Peruginelli and Zanardo classified the lattices of orders with prime index inside te ring of integers of quadratic extensions of the rational numbers. The lattices are quite striking and have different layered structure depending on whether the prime is inert, split, or ramified. This thesis considers the orders which have prime power index inside the Gaussian integers. This is a nice generalization of the work of Peruginelli and Zanardo, and succeeds in a few classifications of specific instances of orders derived from inert primes.