Mathematics Commons^™

Spatial Prediction Under Uncertainty: Methodological And Computational Advances In Bayesian Maximum Entropy, Kinspride K. Duah

All Graduate Theses and Dissertations, Fall 2023 to Present

Environmental decisions such as infrastructure design, water management, and snow load estimation depend on spatial data that are often incomplete or uncertain. In many cases, measurements are not exact values but ranges, reflecting limitations in data collection methods. Traditional mapping techniques typically simplify these uncertain measurements, which can lead to less accurate predictions. This dissertation introduces improved statistical tools for making spatial predictions when data are uncertain or partially known. By utilizing a framework called Bayesian Maximum Entropy (BME), this research demonstrates how exact measurements and range-based data can be combined in a mathematically consistent way. The work demonstrates that …

Learning Latent Structure In High-Dimensional Data Via Geometry And Graphs, Haozhe Chen

All Graduate Theses and Dissertations, Fall 2023 to Present

Modern datasets often contain many measured variables for each observation, such as gene-expression levels, brain activity signals, or features in tabular data. These data are also often noisy, meaning that useful patterns are mixed with measurement error or irrelevant variation. Although such datasets can appear complex, they are frequently represented by simpler hidden structures, such as trajectories, clusters, or relationships between observations. This dissertation develops methods for uncovering these hidden structures by learning geometric and graph-based representations directly from data. The first part introduces Functional Information Geometry, which represents local patterns in high-dimensional data using functional features and constructs a …

Unifying And Expanding Global And Local Variable Importance Methods For Explainable Machine Learning, Kelvyn K. Bladen

All Graduate Theses and Dissertations, Fall 2023 to Present

Machine learning methods are powerful analytical tools used across all scientific disciplines and many other fields of investigation for prediction and inference from diverse data sources. Despite their broad applicability, machine learning methods are often highly complex and difficult to interpret. Developing a greater understanding of which variables most influence a response is essential for increasing the interpretability of these models and supporting informed decision-making. This research focuses on improving how we evaluate the importance of these variables.

One common approach is to shuffle the values of a variable and see how much the model accuracy gets worse. Another approach …

Applications Of Machine Learning To Gas Plume Analysis In Longwave Infrared Hyperspectral Images, Scout C. Jarman

All Graduate Theses and Dissertations, Fall 2023 to Present

Each pixel from a hyperspectral camera measures the intensity of light over a continuous range of wavelengths, which is in contrast to traditional color cameras, which just measure the intensity of red, green, and blue wavelengths of light. Longwave infrared hyperspectral images can be used to detect gases from a distance by measuring how different materials emit and absorb heat. This makes them useful for applications such as monitoring industrial emissions or locating hazardous gas leaks. In practice, however, gas signatures in these hyperspectral images are often weak and easily obscured by variations in the background scene, making reliable identification …

Mapler: An R Package For Estimating The Impact Of Climate Change On Maple Syrup Production, Matthew T. White

All Graduate Theses and Dissertations, Fall 2023 to Present

Successful maple sap tapping depends on the freeze/thaw cycle (i.e., temperatures fluctuating above/below freezing) during the winter and spring. Climate change threatens to alter the timing and duration of the tapping season. This necessitates research into how maple sap tapping will be impacted by climate change in order to help maple syrup producers prepare for the future. We define a sap day as a day where the freeze/thaw cycle occurred. Using information climate scientists use to predict future temperatures, we calculate how many sap days could occur each year. We develop software to analyze these sap day calculations to determine …

Rainbow Dominating Sets Of Graphs, Samuel L. Powell

All Graduate Theses and Dissertations, Fall 2023 to Present

The content of this thesis may be compared to a stack of plates with a common design that are broken, one at a time. If the plates are broken into large enough pieces you would be able to choose one fragment from each broken plate to discover the entire design that the plates share. For example, if you were still missing the center of the design after taking a piece from some plates, you could look for the piece of the next broken plate with the region in question.

We study the question of how many broken plates might guarantee …

Gradient Based Optimization Methods For Robust Learning And Biomedical Signal Modeling, Jarrod Mau

All Graduate Theses and Dissertations, Fall 2023 to Present

This dissertation explores how modern artificial intelligence techniques can be used to better understand complex biological data. Specifically, it develops new machine learning based methods and applies them to two important biomedical problems: analyzing brain signals and studying protein behavior.

The first part of the work introduces a new machine learning approach designed to improve how computers classify structured data. Traditional neural networks are powerful but can sometimes generalize poorly. This research proposes a method that combines the flexibility of neural networks with the reliability of ensemble techniques, leading to more robust and accurate predictions across different types of datasets. …

Constraint-Aware Metaheuristic Optimization For Experimental Design, Benjamin N. Fuller

All Graduate Theses and Dissertations, Fall 2023 to Present

Designing experiments becomes much more challenging when many variables and strict constraints are involved, as is common in modern science and engineering. This thesis introduces a new computational and mathematical framework that efficiently searches for optimal experiments in complex, high-dimensional spaces where traditional methods fail. By combining geometric techniques with flexible optimization algorithms like particle swarm optimization, our methods handle difficult constraints while scaling to real-world problems. Built in the high-performance Julia programming language and released as open-source software, this work bridges advanced theory with practical tools, offering researchers a powerful and accessible way to design better experiments under realistic …

From The Hopf Fibration To Instantons: Geometry In Gauge Theory, Emily D. Wessman

Undergraduate Honors Capstone Projects

This paper explores the relationship between topology, differential geometry, and gauge theory through the study of Yang-Mills theory and its solutions, known as instantons. Beginning with the Hopf fibration, we show how principal fiber bundles encode topological information and appear in physical contexts such as electromagnetism. In particular, we consider how the fibration of S³ over CP¹ ≅ S² represents the Dirac magnetic monopole, and how the Chern number associated with the bundle is exactly the winding number for the monopole.

We then develop the framework of gauge theory, focusing on connections on principal SU(2) bundles …

Power Approximations With Non-Normal Data In Generalized Linear Mixed Models In R Using Steep Priors On Variance Components, Carlie Prinster, John Stevens

Mathematics and Statistics Student Research and Class Projects

ENAR Spring 2026 Conference presentation on Power Approximations with Non-Normal Data in Generalized Linear Mixed Models in R Using Steep Priors on Variance Components

Predicting Tart Cherry Stem Water Potential Using Uav Multispectral Imagery And Environmental Data Via Symbolic Regression, Anderson L. S. Safre, Alfonso Torres-Rua, Kurt Wedegaertner, Brent Black, Brennan Bean, Burdette Barker, Matt Yost

Civil and Environmental Engineering Faculty Publications

Tart cherry is an important fruit crop in Utah, where irrigation is essential due to arid conditions. Precision irrigation requires reliable indicators of plant water status, and stem water potential (Ψ_stem), is among the most sensitive though labor-intensive and spatially limited. This study develops Ψ_stem estimation models using high-resolution multispectral Unmanned Aerial Vehicle (UAV) imagery combined with meteorological and soil moisture data, applying Symbolic Regression (SR). Results show a stronger correlation between optical bands and Ψ_stem during the pre-harvest period. Among 85 vegetation indices, the Red Chromatic Coordinate (RCC) index performed best (R² = 0.67). …

Compilation Of A Nationwide River Image Dataset For Identifying River Channels And River Rapids Via Deep Learning, Nicholas Brimhall, Kelvyn K. Bladen, Thomas Kerby, Carl J. Legleiter, Cameron Swapp, Hannah Fluckiger, Julie Bahr, Makenna Roberts, Kaden Hart, Christina L. Stegman, Brennan L. Bean, Kevin R. Moon

Mathematics and Statistics Student Research and Class Projects

Remote sensing enables large-scale, image-based assessments of river dynamics, offering new opportunities for hydrological monitoring. We present a publicly available dataset consisting of 281,024 satellite and aerial images of U.S. rivers, constructed using an Application Programming Interface (API) and the U.S. Geological Survey’s National Hydrography Dataset. The dataset includes images, primary keys, and ancillary geospatial information. We use a manually labeled subset of the images to train models for detecting rapids, defined as areas where high velocity and turbulence lead to a wavy, rough, or even broken water surface visible in the imagery. To demonstrate the utility of this dataset, …

Collaborative Research: Cif: Medium: Foundations Of Robust Deep Learning Via Data Geometry And Dyadic Structure, Kevin Moon

Funded Research Records

No abstract provided.

Representation Theory In Unoriented And Non-Semisimple Physics, Matthew Bruce Young

Funded Research Records

No abstract provided.

Error Reduction Methodology And Data Simulation For Interval Data, Ranik Christopher Jelinek

Undergraduate Honors Capstone Projects

Chronic kidney disease (CKD) is a progressive condition affecting hundreds of millions of individuals worldwide. However, clinical datasets often record continuous laboratory measurements as categorical intervals rather than precise numerical values. This interval-censored structure presents methodological challenges for standard regression-based classifiers. This study compares three strategies for handling interval-valued predictors prior to fitting a logistic LASSO model: (1) midpoint imputation, which replaces each interval with its arithmetic center; (2) ordinal encoding, which maps intervals to integer ranks; and (3) a Monte Carlo simulation approach, which repeatedly samples uniformly from each observed interval and averages predictions across replications. Using a 10-fold …

Self-Assessment In Calculus I, Bertha Naa Dei Neequaye

All Graduate Theses and Dissertations, Fall 2023 to Present

The purpose of this research investigate how undergraduate students in Calculus I at Utah State University learn and reflect on their level of understanding of Calculus I. In the study, self-assessment practices refer to students reflecting on what they have learned, rating their level of understanding and providing examples that illustrate their level of understanding. The structured and unstructured self-assessment rubrics were the two types of self-assessment rubrics used in the study. The structured self-assessment rubric provides a clear list of objectives of Calculus I topics for students to rate themselves on, while the unstructured self-assessment rubric is an open …

Fast Computation Of The Friction Stir Welding Process With Model Order Reduction, Joshua Kay

All Graduate Theses and Dissertations, Fall 2023 to Present

Friction stir welding (FSW) is a manufacturing process used to join materials through intense heat and pressure. To better understand and optimize this FSW process, mathematical models incorporating non-Newtonian Navier-stokes equations for large plastic deformation and heat equation for the heat transfer are used to simulate the FSW process. However, solving these coupled and nonlinear equations with high accuracy requires significant computational power and time. This thesis has applied model order reduction, more precisely the Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM), to efficiently solve the FSW system in low-dimensional space. Slight modifications are made regarding the …

Implications Of The Attenuated Allee Effect On Population Dynamics, Dana Strong

All Graduate Theses and Dissertations, Fall 2023 to Present

The Allee effect is an ecological phenomenon characterized by a per capita growth rate that increases as the population size increases in the context of low population density. The Allee effect can lead to rapid extinction events. Consequently, ecologists can attempt to control the strength of the Allee effect to help increase the population (in the case of endangered species) or decrease the population (in the case of parasites or pests).

This research aimed to study not the strength of the Allee effect, but the intensity of the Allee effect, or how rapidly cooperation between individuals causes the per capita …

Machine Learning Applications: Cell Tracking And Nonparametric Estimation Of Non-Smooth Divergences, Mina Mahbub Hossain

All Graduate Theses and Dissertations, Fall 2023 to Present

This thesis brings together two important research directions: how to compare different sets of data more accurately, and how to better understand how brain cancer cells move and change shape.

In the first part, we look at a problem in statistics: measuring how different two data sources are from each other. Traditional methods often make strong assumptions, which may not always hold in real situations. Our approach avoids those assumptions by using an ensemble method, a way of combining many weak estimators into one stronger result. This makes the method more flexible and reliable, especially when dealing with complex or …

The Impact Of Transmission Thresholds Across Multiple Scales On The Spread Of Chronic Wasting Disease In Wisconsin, Jen Mcclure

All Graduate Theses and Dissertations, Fall 2023 to Present

Wildlife diseases can be difficult to control once they are established. This is especially true when they spread through contact with infectious material left in the environment. One such disease is chronic wasting disease (CWD), a fatal illness affecting deer and related species in North America and other regions. CWD is caused by prions, misfolded proteins that can remain infectious for years after shedding by infected hosts.

Recent research shows that CWD infection does not always follow from the gradual accumulation of prions through small contact events. Rather, an individual may need to encounter a certain prion dose all at …

Site Specific Reliability-Targeted Snow Loads And Winter Wind Parameters Across The World, Brennan Bean, Nicholas Brimhall, Bikram Bhusal, Marc Maguire, Maha Moussa

Mathematics and Statistics Faculty Publications

This report presents a framework for estimating Reliability Targeted Snow Loads (RTSLs) and Winter Wind Parameters (WWPs) at locations outside the Conterminous United States (OCONUS). The methodology integrates ground-based in-situmeasurements with gridded global climate products to generate spatially continuous RTSL estimates for nearly any location in OCONUS. RTSLs are computed at qualifying in-situ stations, and predictive models relating gridded climate products to in-situRTSLs enable estimation at locations lacking direct measurements. This report describes the development of the in-situ annual maximum snow load database, the construction of global gridded climate summaries, the estimation of site-specific RTSLs and Service Targeted …

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.

Quantifying The Sensitivity Of Land Use Land Cover Metrics Through Simulation Techniques, Haley Burger

All Graduate Theses and Dissertations, Fall 2023 to Present

As human activities and climate change continue to reshape our landscape, understanding how land use changes over time is becoming increasingly important. Accurate ways to track and analyze these changes are essential for governments, businesses, and communities to make informed decisions. Monitoring agricultural land is particularly critical, as shifts in land use can impact food production and environmental pollutants. One of the primary tools used in the United States to monitor agricultural land is the Cropland Data Layer (CDL), an annual map created by the United States Department of Agriculture (USDA) from satellite images. While the CDL is highly accurate, …

A Study Of Complex Analysis After Whittaker And Watson, Crystal Steed

All Graduate Reports and Creative Projects, Fall 2023 to Present

The goal of this report is to provide solutions to the exercises found in chapter five of the book titled, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions with an Account of the Principal Transcendental Functions by E.T. Whittaker and G.N. Watson. The fifth chapter is titled, "The Fundamental Properties of Analytic Functions; Taylor's, Laurent's and Liouville's Theorems." This report solves the end-of-chapter exercises in addition to providing details for some in-chapter exercises, which are left to the reader. Many of these exercises are results from famous mathematicians.

Twisted Equivariant Matrix Factorizations, Jan-Luca Spellmann

All Graduate Theses and Dissertations, Fall 2023 to Present

We introduce and study categories of twisted equivariant matrix factorizations MF^α_G(R, w), which are categories of matrix factorizations of a potential w over the local ring R = C[x₁, . . . , x_n] together with an action by a finite group G that is twisted via a 2-cocycle α. These categories provide rich examples of Z/2Z-differentially graded categories in the context of non-commutative geometry and come up naturally in the study of boundary conditions of 3d Rozansky–Witten theories. We prove that under certain assumptions on (R, w …

Studies In Number Theory: Reciprocity Laws And Fundamental Domains, C. Xavier Parent

All Graduate Theses and Dissertations, Fall 2023 to Present

This thesis consists of two sections. The first section is an introductory survey of number theory discussing the reciprocity laws with a focus on accessibility. Number Theory has always been a fundamental area of mathematical study, with Gauss calling it “the queen of mathematics”. The reciprocity laws are a classical set of results from number theory which have driven number theory for quite a long time. Unfortunately, these results, while important, have always been very inaccessible to undergraduate students, making it hard to start studying the field. This survey attempts to help bridge that gap, giving a resource for novices …

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Development Of Saddlepoint Methodologies For Sparse Sample Multiple Parameter Generalized Linear Models And Correlated Data Scenarios, Christopher Johnson

All Graduate Theses and Dissertations, Fall 2023 to Present

Parameter estimation using maximum likelihood techniques may be biased when sample sizes are small, event rates are low, or otherwise sparse counts exist in a parametric model. This in turn may lead researchers to draw invalid statistical conclusions when conventional methods are utilized. The saddlepoint approximation has potential to lessen the degree of bias in sparse data conditions through its use of moments beyond the mean and variance, which allows for more accurate approximations using a smaller number of observations. We propose two novel saddlepoint methods for use in practical analysis scenarios, as an alternative to maximum likelihood estimation. First, …

A Survey Of Master's Qualifying Exam Practices And Content In Real Analysis, Zachary M. Coverstone

All Graduate Theses and Dissertations, Fall 2023 to Present

Graduate programs in mathematics are intended to develop experts in mathematics. As part of a graduate program, many students are expected to engage in a qualifying examination that can serve myriad purposes. This dissertation investigates real analysis qualifying examinations at the master's level through a nation-wide survey of institutions in the Association of Public and Land-Grant Universities (APLU). This study reveals the practices in administration of real analysis qualifying exams, institutions reported offering traditional, pencil-and-paper exams and generally asked students to prepare individually using previously administered exams. The survey results show that approximately two in three APLU institutions offering master's …