Statistics and Probability Commons^™

Revisiting Ulam Stability For Boundary Value Problems, Martin Bohner, Snezhana Hristova, Agnieszka B. Malinowska, Ewa Girejko

Mathematics and Statistics Faculty Research & Creative Works

The main goal of this paper is to apply Ulam stability theory to boundary value problems for dynamic equations, while addressing several common misconceptions found in the existing literature. We identify the key issues that arise when applying Ulam stability to such problems and propose three distinct approaches to overcome them. To enhance clarity and accessibility, we begin with nonlinear ordinary differential equations and subsequently extend the analysis to nonlinear dynamic equations on time scales. Since a time scale is defined as any nonempty closed subset of the real numbers, our results are applicable to dynamic equations on continuous, discrete, …

Adaptive Rational Approximation In Dynamic Economic Models: A Novel Application Of The Aaa Algorithm To Economic Growth, Adaye Sosthene Yvan N'Guettia

Mathematics, Statistics, and Computer Science Honors Projects

I study adaptive rational approximation for fixed points that arise in infinite-horizon dynamic programming. I integrate the Adaptive Antoulas–Anderson (AAA) algorithm into Bellman- and Euler-based fixed-point solvers by recomputing a barycentric rational interpolant at each update. In addition to standard AAA, which selects support points from interpolation residuals, I study a residual-weighted variant in which Bellman,Euler, or KKT diagnostics act as secondary weights on the greedy pivot rule. This alignment of approximation adaptivity with the underlying equilibrium conditions can concentrate degrees of freedom in regions of steep curvature, sharp transitions in localbehavior, and other localized features that typically degrade polynomial …

An Analysis Of The Effects And Implementations Of The Early Literacy Grant In Arizona, Alicia Severiano Perez

Mathematics, Statistics, and Computer Science Honors Projects

Over the years, states have implemented Science of Reading (SoR) frameworks to address low literacy levels. The Early Literacy Grant (ELG) in Arizona funds and supports such frameworks for schools serving low-income students. This paper is the first to explore the grant through interrupted time series modeling to evaluate effectiveness and text analysis to understand its implementation. We do not find clear evidence of positive effects caused by the grant, other than some cases, such as Yuma County. Schools typically allocate funds toward salaries and hiring instructors. These findings raise questions about whether its allocations should be closely monitored.

Level Sets For Lehmer Codes Of Pattern Avoiding Permutations, Avery Sinclair

Mathematics, Statistics, and Computer Science Honors Projects

We study the poset structures for two families of pattern avoiding permutations. An n-permutation is a list of the numbers [n]={1,2,...,n}. A permutation is 321-avoiding when it does not contain a decreasing subsequence of length 3. A poset (partially ordered set) is a set such that some elements can be compared with one another. Using Lehmer codes, we define a poset for 321-avoiding permutations. We then fully describe the six lowest levels of this poset. We then consider the analogous poset for 123-avoiding permutations (which don't contain an increasing subsequence of length 3) and fully describe the three lowest levels.

A Compact Representation Of Oscillatory Limits Via Asymptotic Value Distributions, Ibrahim Arnous, Eric M. Rodarte, Chirag Kumar

Discovery Day - Daytona Beach

Classical limits describe asymptotic behavior through convergence to a single value, but many important oscillatory functions do not converge in this sense. Standard examples such as sin(𝑥) as 𝑥→∞ and sin(1/x) as x→0 instead display stable distributions of values over time. This project examines how such behavior can be described using a measure-theoretic framework, particularly through occupation measures and, in sequence-based settings, Young measures. The objective is to present this perspective in a clear and accessible way while introducing the Ansatz representation, a compact notation for recording the support and density of an asymptotic value distribution when the limiting measure …

Algebraic And Topological Methods In Computational Neuroscience, Trong-Thuc Trang

Electronic Theses and Dissertations

Neural data is incredibly rich in combinatorial, topological, and geometrical information, reflecting the intricate shape and connectivity of neural firing patterns. To decipher these structures, neuroscience increasingly relies on advanced mathematical tools to analyze neural activity. Here we study (1) neural codes within the poset P_Code of neural codes and (2) the connectivity of neural population activity within the insular cortex when responding to interoceptive information. In (1), we establish combinatorial constructions for all upward covering relations based on what we call “isolated subsets” with supporting theorems and give a slight modification of the existing downward covering relations. We …

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 …

Machine Learning For Predictive Energy And Emissions Modeling Of Vehicles And Power Grids In The United States, S M Tanvir Faysal Alam Chowdhoury

Dissertations

The environmental benefits of electric vehicle (EV) adoption depend on more than replacing internal combustion engine vehicles with electric powertrains. EV adoption reshapes electricity demand, interacts with regional generation mixes, and influences travel behavior and congestion, creating a coupled transportation-energy system in which vehicle and power-plant emissions must be evaluated together. This dissertation develops machine-learning frameworks for predicting energy consumption and emissions from vehicles and power grids under rising EV adoption. The first component forecasts grid emissions from EV charging. Using simulation data from NREL's Cambium database, a Prophet-based time-series framework predicts carbon dioxide, nitrous oxide, and methane emission rates …

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 …

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 …

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 …

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.

Existence And Uniqueness Of Positive Solutions For Hilfer–Hadamard-Type Fractional Differential Equations With Γ-Concave And Sub-Homogeneous Operators, Hasan Rasouli, Hojjat Afshari, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

In this research, we present necessary and sufficient conditions for the existence and uniqueness of positive solutions for a class of Hilfer–Hadamard-type fractional differential equations with boundary value problems, including those with integral boundary conditions. The obtained results are conditional on a specific set of strong assumptions, which substantially narrow the class of admissible nonlinearities, coefficients, and boundary data. Thus, the present work extends the Hadamard-type framework to the Hilfer–Hadamard setting only within this restrictive regime, rather than providing a full extension to all Hilfer–Hadamard systems. We utilize the properties of (Formula presented.) -concave and sub-homogeneous operators along with two …

Bias, Structure, And Inference In Applied Network Analysis, Anna Vasenina

Dartmouth College Ph.D Dissertations

This dissertation develops mathematical and statistical methods for extracting reliable information from network data across biological applications, with an emphasis on understanding what observed network structure can and cannot resolve. The first study leverages protein–protein interaction network topology in the c-di-GMP signaling system of Pseudomonas fluorescens, showing that node centrality measures accurately classify protein domain types and that physical interaction structure contributes statistically significant predictive power for biofilm formation phenotypes across nearly 200 environments, while gene expression does not. The second study examines sampling bias in lemur-plant trophic interaction networks in Madagascar, demonstrating that differential detection of diurnal versus …

Computational Insights Into Nucleosome Dynamics In Epigenetics Using Molecular Dynamics Simulations, Rutika Patel

Dissertations, Theses, and Capstone Projects

Nucleosome core particles (NCP) are the building blocks that form a highly organized and compact chromatin structure. Nucleosomes package DNA in the nucleus of eukaryotic cells. The NCP consists of about 147 base pairs of DNA wrapped around the histone octamer, with 1.65 superhelical turns in a left-handed manner. The histone octamer is composed of two copies of H3, H4, H2A, and H2B. Together with histone H1 and linker DNA, they further assemble into a higher-order chromatin structure. The nucleosome complex is stabilized by electrostatic interactions between positively charged histone residues and the negatively charged DNA backbone. To effectively access …

A Copula-Based Framework For Multivariate Count Time Series With Mixed Marginal Distributions, Dimuthu Fernando, Yuxin Wen, Wimarsha Jayanetti

Engineering Faculty Articles and Research

We developed a class of multivariate integer-valued time series models using copula theory. Each count time series is modeled as a Markov chain, with serial dependence characterized through copula-based transition probabilities for Poisson and negative binomial marginals. Cross-sectional dependence is modeled via a trivariate Gaussian or a “t-copula”, allowing for both positive and negative correlations and providing a flexible dependence structure. Model parameters are estimated using likelihood-based inference, where the trivariate Gaussian or t-copula integrals are evaluated through standard randomized Monte Carlo methods. Simulation results, along with an analysis of annual counts of major hurricanes (Category 3+) across the North …

Measuring Stock Market Inefficiency Using A Multilayer Composite Efficiency Index: A Case Of The Egyptian Exchange, Patrick K. Owido, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

Financial markets play a critical role in resource allocation. Their performance depends on the decisions of millions of independent investors constantly reacting to one another. Their informational efficiency remains a subject of debate across economic systems. When informational efficiency is present at the weak form, historical price information should not consistently predict future returns. Several empirical tests of this hypothesis often focus on the behavior of aggregate market indices, and use individual efficiency proxies such as autocorrelation, GARCH-type volatility, or entropy-based measures to measure efficiency. This has often yielded mixed results, particularly in emerging markets. Here we show that testing …

Conditional Product Sampling For Gaussian Process Implicit Surfaces, Song Shi

Dartmouth College Master’s Theses

Gaussian Process Implicit Surfaces (GPISes) provide a powerful and unified stochastic geometry representation for rendering surfaces, volumes, and the rich continuum between them. Recent work has shown that GPISes can model a broad space of visual appearances under a unified light transport framework. However, practical rendering with GPISes remains challenging: existing estimators can become inefficient for particular correlation structures, and highly anisotropic or heightfield-like GPISes require specialized treatment to obtain robust variance reduction.

This thesis extends recent work on GPIS rendering by introducing a new next-event estimation (NEE) technique for anisotropic GPISes.We show that standard NEE provides diminishing benefits as …

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 …

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 …

Analyzing The Evolution Of Science: Topological Cycles And Community Detection In Knowledge Networks, Frances C. Mcconnell

Mathematics, Statistics, and Computer Science Honors Projects

How scientific knowledge grows and organizes itself is a central question in the study of science. This thesis uses tools from topology and network science to detect and characterize knowledge gaps—places in a field’s literature where related concepts do not co-occur. We develop a metric to quantify the degree of interdisciplinarity of each gap, using the community structure of the underlying network as a proxy for subfields. Across a wide range of fields, gaps reliably span multiple subfields and evolve in recognizable temporal patterns, highlighting new insights into how scientific fields are structured and their stage of development.

A New Approach To Generate Combinatorial Patterns In Logical Analysis Of Data And Its Application To Predict College Retention, Salihah Ahmed E. Jaafari

Theses and Dissertations

Student retention and degree completion remain central challenges for higher-education institutions, with significant implications for student success, institutional effectiveness, and public accountability. While advances in predictive analytics have enabled earlier identification of students at risk of withdrawal, many commonly used machine learning approaches suffer from limited interpretability, constraining their practical usefulness for advising, intervention, and policy decision making. This dissertation addresses the problem of predicting student persistence by developing and evaluating optimization based, interpretable classification models within the Logical Analysis of Data (LAD) framework. Building on existing LAD formulations, this research introduces two novel pattern generation models, the Best Term …

A Computer Vision Approach To Analyzing Taxane Effects On Prostate Cancer Cells, Diana Elizabeth Dancea

Electronic Theses and Dissertations

Actin is a family of proteins that help create the structure of the cytoskeleton, which gives shape to the cell. In many chemotherapy treatments, researchers target actin because it controls the cell division process. Therefore, if they are able to understand the actin fibers, that may help in formulating methods to stop or slow down cancer cells from reproducing. Another important protein is PAK6, which regulates actin. In our research, a collaborative effort with Prof. Michael Lu’s lab at Florida Atlantic University, we use machine learning techniques to analyze cells which had their PAK6 protein knocked out, and compare them …

Analyzing Label Structure And Regional Similarity In Watershed Data Via Spectral Clustering, Yifan Luo

25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026)

This project focuses on identifying patterns of seasonal transitions and nutrient salt fluctuations within the watershed environment. To capture the complex relationships between multiple sampling sites and environmental variables, we represent the watershed dataset as a weighted graph, where nodes correspond to water samples and edge weights reflect similarity in environmental conditions or nutrient concentrations. Using the Gaussian kernel function, we encode the connectivity structure of this network and quantify how similar different nodes are. We then perform spectral embedding by projecting the high-dimensional graph into a lower-dimensional space using the eigenvectors of the Laplacian matrix. This approach preserves the …

The “How Many” Routine As A Catalyst For Computational Fluency And Student Participation, Lillian Iden, Ella Williams, Jen Munson, Sarah Larison, Leslie Yuqui

25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026)

It is not uncommon to hear adults claim they are bad at math or always disliked the subject. This negativity often stems from early mathematical experiences which ranged from boring to highly discouraging and embarrassing. Thus a new emphasis in mathematics education recommendations is to change the narrative and help students develop joy, wonder, and curiosity about mathematics (MAISA & GELN, 2023). This is reflected in defining computational fluency (skill in carrying out arithmetic procedures like addition or multiplication) as comprised of flexibility, accuracy, efficiency, and appropriate strategy use (NRC, 2001). The “How Many” Routine, in which a carefully-designed image …

Investigations In Bertrand’S Paradox, Mary Moore, Hope Weeda, Annika Cunill Krones

25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026)

Bertrand’s paradox is a classic problem that highlights how different notions of randomness can lead to different outcomes, even in a simple geometric setting. It concerns the lengths of chords chosen “at random” in a circle. In this talk, we begin by reviewing the three original methods Bertrand proposed for generating random chords, along with several related distributions that have been studied since. We then turn to a geometric application, examining triangles formed by two random chords that share a common endpoint. By joining the remaining endpoints, we obtain a random triangle and compute the probability that it is acute. …

Irreversible K-Threshold Dynamics On Corona And Base-B Corona Product Graphs, Eric J. Moon, Soumya Bhoumik, Paul Flesher

SACAD: Scholarly Activities

This poster studies the irreversible k-threshold process on corona-type graph products, where a vertex becomes colored once at least k of its neighbors are colored and then remains colored permanently. We focus on corona, double corona, and base-b corona product graphs built from cycles and complete graphs, with particular attention to how graph structure affects complete activation from a minimum seed set.

A generalized reduction lemma is used to relate threshold dynamics on layered corona graphs to smaller residual graphs, yielding explicit formulas for the irreversible k-threshold conversion number on both corona and double corona families. The …

Sex-Specific Differences In Lung Mitochondrial Function And Injury In Rats Exposed To Hyperoxia, Taheri Pardis, Abraham G. Taye, Devanshi D. Dave, Elizabeth R. Jacobs, Guru Prasad Sharma, Anne V. Clough, Ranjan K. Dash, Said H. Audi

Mathematical and Statistical Science Faculty Research and Publications

Hyperoxia is both an essential therapy and a contributor to lung injury in acute respiratory distress syndrome. We hypothesized that adult female rats are relatively protected from hyperoxia-induced acute lung injury (HALI) compared with males and that this protection is associated with sex-dependent differences in lung mitochondrial bioenergetics and H₂O₂ production. Adult rats were exposed to room air (normoxia) or hyperoxia (>95% O₂) for up to 60 h. Lung injury was assessed by pleural effusion, lung wet weight, pulmonary vascular filtration coefficient (K_f), histologic injury scores, and cleaved caspase-3 (CC3) staining. …

The Kaczmarz Algorithm In Hilbert C∗-Modules, Daniel Alpay, Chad Berner, Eric S. Weber

Mathematics and Statistics Faculty Research & Creative Works

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C^∗-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz algorithm and …