Mathematics Commons^™

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 …

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 …

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 …

Exploratory Study Of Semiconductor Nanomembranes In Em Applications, Grant D. Heileman

Electrical and Computer Engineering ETDs

Antenna systems are a cornerstone of modern technologies, playing an increasingly vital role in their advancement. As demand for compact, high-performance, and adaptable communication platforms grows reconfigurable antenna technologies are becoming essential. This research explores a novel front-end reconfigurable antenna system (FERAS) architecture that leverages the mechanical flexibility and photoconductive behavior of semiconductor nanomembrane (SNM) devices. By exploiting the emergent properties of ultra-thin silicon (Si) or gallium arsenide (GaAs) nanomaterials and optically exciting these samples using vertical-cavity surface-emitting laser (VCSEL) arrays, this study develops lightweight, low-cost, deployable antenna structures for satellite communications, remote sensing, GPS, and radar. Despite their significant …

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 …

Algorithms To Estimate Contours: Two Applications Of Analytical Tools In Differential Geometry And Topology, Mohammad Abirul Islam

Computer Science ETDs

We develop distributed robotics algorithms with analytical tools needed to define and analyze angle turned and distance traversed by robots executing geometric algorithms. We then use these analytical tools to obtain information, via sensor measurements, about an a priori unknown surface. Our contributions are threefold. First, we develop the Sketch Algorithm, which estimates the boundary of any unknown contour and is asymptotically optimal in terms of distance traversed and angle turned. Second, we present experimental field work that validates the Sketch Algorithm. Finally, we propose an approach to find multiple sources of a surface with potential applications to approximate that …

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 …

Module 5 - Project, Karon Klipple, Cheryl Hedden

Module 5 Assessments

For this project, you will create a video in which you TEACH someone how to find the surface area and volume of a box.

Module 5 - Student Workbook, Karon Klipple, Cheryl Hedden

Module 5 Workbook

No abstract provided.

Module 3 - Exam, Karon Klipple, Cheryl Hedden

Module 3 Assessments

Use only class notes and assignments and Use of calculator, computer and internet is forbidden. Do not get help from others and just take it by yourself.

Module 4 - Project, Karon Klipple, Cheryl Hedden

Module 4 Assessments

Project 4 – Exponential Functions & Properties of Exponents

Module 4 - Project 4.2 - Mutual Funds, Karon Klipple, Cheryl Hedden

Module 4 Assessments

Mutual funds are another type of investment used by many people. A mutual fund is a company that brings together money from many people and invests it in stocks, bonds, or other assets. Each investor in the fund owns shares, which represent a part of these holdings. Mutual funds are an attractive investment to many people for several reasons.

Module 4 - Solar System Activity Cards, Karon Klipple, Cheryl Hedden

Module 4 Workbook

No abstract provided.

Module 3 - 3.3 Activity Cards, Karon Klipple, Cheryl Hedden

Module 3 Workbook

No abstract provided.

Module 3 - Student Workbook, Karon Klipple, Cheryl Hedden

Module 3 Workbook

No abstract provided.

Module 4 - Student Workbook, Karon Klipple, Cheryl Hedden

Module 4 Workbook

No abstract provided.

Module 3 Barbie Bungie Contest, Karon Klipple, Cheryl Hedden

Module 3 Assessments

Your goal is to have Barbie or Ken safely bungie jump from 218cm above ground.

Whichever group gets their Barbie or Ken closest to the ground, without hitting their head, wins!

Module 4 - Rules Of Exponents Game, Karon Klipple, Cheryl Hedden

Module 4 Workbook

No abstract provided.

Module 2 Check Ups, Karon Klipple, Cheryl Hedden

Module 2 Assessments

Table : Matching Division Expressions

Module 2 Exam Questions, Karon Klipple, Cheryl Hedden

Module 2 Assessments

Use only class notes and assignments and NOT computers, Internet and calculator.

Activity 1.4 Cards, Karon Klipple, Cheryl Hedden

Module 1 Workbook

No abstract provided.

Module 1 Student Workbook, Karon Klipple, Cheryl Hedden

Module 1 Workbook

No abstract provided.

Module 1 Project I, Karon Klipple, Cheryl Hedden

Module 1 Assessments

Your instructor will give you a variety of news clippings that include percentages and a set of questions associated with each. Select one that you find the most interesting.

Module 1 Exam, Karon Klipple, Cheryl Hedden

Module 1 Assessments

Directions:

- Your exam must be completed by you and you alone.

- No resources - books, websites, people - can be used, except a calculator.

- You will have to sign a statement affirming that you followed these rules.

- Once you start the exam, you must finish it.

- The exam includes 11 questions.

- It will probably take you about an hour, but Canvas will allow you up to 4 hours to complete it.

- The exam is due NO LATER THAN Thursday February 15th at 12:45pm. NO LATE EXAMS WILL BE ACCEPTED. I encourage you to …

Module 2 Fall 2024 Project Sample, Karon Klipple, Cheryl Hedden

Module 2 Assessments

Individual Tax Rates for New Mexico