Digital Commons Network^™

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 …

Sequences That Do Frame Reconstruction, Chad Berner

Mathematics and Statistics Faculty Research & Creative Works

Frames allow all elements of a Hilbert space to be reconstructed by inner product data in a stable manner. Recently, there is interest in relaxing the definition of frames to understand the implications for stable signal recovery. In this paper, we relax the definition of a frame by allowing the operator in the frame decomposition formula to not be invertible. We provide a complete classification of sequences that allow this decomposition via a type of frame operator. In addition, we provide several examples of sequences that allow this reconstruction property that are not frames and illustrate in which ways they …

Maddenlite, Sergio Pena

Presentations - 2026

Problem •“What If” scenarios impossible to test accurately •Commercial games rely on arcade physics •Spreadsheets lack visual engagement

Motivation •Passion for football analytics •Desire to simulate cross-era matchups •Apply math models to real-world sports data

Solution •Python based simulation engine using historical play-by-play data •Simulates outcomes based on probability

Deep Learning Frameworks For Biological Data Integration And Generation, Alexa Beachum

Statistical Science Theses and Dissertations

Data integration represents a key area of research for analyzing the rapidly growing volume of high-dimensional biological data across sources, stages, and modalities. To model and understand these complex, often non-linear relationships, deep learning has become an increasingly powerful tool. Here, we present two novel deep learning frameworks that address distinct but complementary integration challenges. The first framework aligns single-cell omics data across temporal stages, and the second bridges imaging and omics modalities to generate patient-level molecular profiles.

In Chapter 1, we briefly summarize existing approaches---both statistical and deep learning-based---for single-cell omics data integration and discuss their limitations for handling …

Next-Generation Democratic Cyber Statecraft - Balancing The Signal: Shutdown Shocks And Democratic Digital Governance, Scott M. Di Panni

School of Public Policy Capstones

This paper develops Next-Generation Democratic Cyber Statecraft (NG-DCS), a unified strategic doctrine for democratic governments to contest the cognitive domain against authoritarian adversaries. Drawing on twenty-six years of cross-national panel data (1999–2024) spanning 213 countries, game-theoretic modeling, and qualitative case analysis, the paper establishes three interconnected empirical and theoretical foundations. First, cross-national OLS regression across 160+ countries demonstrates that regime type is the dominant structural determinant of internet freedom (R²=0.615, β=2.513, p< 0.001), explaining more than twice the variance attributable to per-capita wealth (R²=0.268). Democratic governance, not economic development, produces open digital environments. Second, a two-way fixed effects (TWFE) difference-in-differences study exploiting government-ordered internet shutdowns as discrete policy interventions finds that digital restrictions causally degrade V-Dem governance quality by 0.21–0.38 standard deviations (p< 0.001 across all specifications). Treatment effects are immediate (β=−0.302 at k=0) and persist through five post-treatment years (β=−0.246 at k=+5), indicating structural rather than transitory governance damage. Parallel trends validation (p=0.352) and Callaway–Sant’Anna heterogeneity-robust estimation (ATT=−0.230, SE=0.077) support causal identification. Instrumental variable triangulation (2SLS β=−0.949, p=0.005) confirms that simultaneity was attenuating, not inflating, the primary estimates. Third, formal game-theoretic analysis reveals that the current U.S.–adversary equilibrium is (Restrain, Escalate)—the risk-dominant but Pareto-inferior outcome of a Stag Hunt structure. China, Russia, North Korea, and Venezuela each occupy structurally distinct positions (Stackelberg commitment, asymmetric two-level, autarky, and reactive trigger, respectively), requiring differentiated doctrinal responses rather than a uniform strategic playbook. Generative AI and algorithmic governance are shown to accelerate cognitive vulnerability by collapsing influence operation costs and exploiting engagement-optimized platform architectures that systematically degrade deliberative capacity in democratic populations.

Efficacy Analysis In Clinical Trials: A Comprehensive Review Of Statistical And Machine Learning Approaches, Dhrubajyoti Ghosh, Samhita Pal

Faculty Articles

Efficacy testing is a cornerstone of clinical trials, ensuring that medical interventions achieve their intended therapeutic effects. Over the decades, a wide range of statistical methodologies have been developed to address the complexities of clinical trial data, including parametric, nonparametric, Bayesian, and machine learning approaches. Parametric methods, such as t-tests, ANOVA, and LMMs, have traditionally been the foundation of efficacy testing due to their efficiency under well-defined assumptions. Nonparametric techniques, including the Friedman test, Brunner-Munzel test, and modern extensions like nparLD, have emerged as robust alternatives, particularly for skewed, ordinal, or non-normal data. Bayesian methodologies have enabled the incorporation of …

Bayesball : A Comprehensive Framework For Predicting Ucl Injury, Brady M. Pinter, Will Best Ph.D.

SPARK Symposium Presentations

Ulnar Collateral Ligament (UCL) reconstruction, commonly referred to as Tommy John Surgery, has seen a significant rise among Major League Baseball (MLB) pitchers, prompting growing interest in identifying the mechanical and performance-based factors that contribute to injury risk. While previous studies have examined these relationships using traditional frequentist approaches separately, this study combines multiple different model techniques to present a broad framework for finding significant predictors of UCL Surgery. These models include Lasso and Ridge Regression,  Principal Component Regression (PCR) , Partial Least Squares Regression (PLS) , Random Forest, Multiple Linear Regression, and a Bayesian Statistical Model. Using these models, …

Implementing A Clustering Framework On Distributional Bee Data, Roheel Nawaz

Research and Creativity Symposium

Understanding spatial patterns in biodiversity is important for identifying how ecological communities are structured across landscapes. Clustering methods provide a useful way to detect non-random species distributions and reveal potential environmental or geographic influences. In this study, we applied a clustering framework to a large long-term bee abundance dataset from the Mid-Atlantic United States to evaluate community structure across regions of Maryland, Delaware, and Washington, D.C.

The dataset included over 1,100 sites and more than 3,000 transects. Because sampling methods were not fully standardized, abundance data were converted to presence/absence and analyzed using the Jaccard index to measure site similarity. …

Factors Associated With Lapses In Care Among People Living With Hiv In South Carolina, Xueying Yang Ph.D., Fanghui Shi, Shujie Chen, Gavi Samuel, Bankole Olatosi Ph.D., Sharon Weissman, Xiaoming Li Ph.D., Jiajia Zhang Ph.D.

Faculty Publications

Utilizing statewide electronic health records (EHR) data, this study aims to assess the factors determining the occurrence of lapses in HIV care among people with HIV (PWH) in South Carolina (SC). All adult (≥ 18 years old) PWH who were diagnosed with HIV between 2006 and 2018 with at least two HIV care encounters and at least 1-year follow-up record were included in the analysis. The outcome, a lapse in care, was defined as a repeated measure of HIV care encounter that occurs over a year following the previous visit. Generalized Estimation Equation models were employed. The study cohort had …

Implementation And Costs Of A Food Insecurity Resource Navigation Program For Primary Care Patients With Diabetes And Hypertension In South Carolina, Deeksha Gupta, Darin Thomas, Stella Coker Watson Self Ph.D., Ms, Edward A. Frongillo Jr. Ph.D., Alain H. Litwin, Joseph A. Ewing, Lynnette Ramos-Gonzalez, Lynnette Ramos-Gonzalez

Faculty Publications

Objective

To examine food insecurity resource navigation program costs and how navigation intensity relates to clinical outcomes, healthcare costs, and quality of life (QOL) for diabetes and/or hypertension patients.

Methods

This retrospective study included patients receiving resource navigation (July 12, 2021-December 31, 2022 with twelve-month follow-up) across three primary care practices in South Carolina's largest health system. Participants were 18+ years old (from electronic medical records/Epic), had food insecurity (from Hunger Vital Sign™), and diabetes and/or hypertension (from Epic registries). Matched controls came from food insecurity screening-only practices. Patients in each group (n= 219) had diabetes (9.13%), hypertension …

Information Theory Analysis Of The Solar Wind Magnetic Structures For Space Weather Prediction, Katherine Holland

Doctoral Dissertations and Master's Theses

Forecasting space weather at Earth is highly complicated, because of the limited measurements of the dynamic processes in the Sun that span multiple temporal, spatial, and energy-scales. The solar wind is a highly structured, multi-scale, evolving plasma and consists of coronal mass ejections (CMEs), stream interaction regions (SIRs), expanding flux tubes (Borovsky, 2008), and interplanetary magnetic field (IMF) discontinuities and fluctuations. The aim of this research is to improve our understanding of the evolution and dissipation of different scale-size solar wind magnetic structures as they move from the Sun-Earth Lagrange point 1 (L1) to Earth's bow shock and, ultimately, to …

Traveling Waves In Reaction-Diffusion Equations With Delay In The Diffusion Term, William Barker, Van Minh Nguyen

Faculty Scholarship

We investigate the existence of traveling wave solutions for the reaction–diffusion equation ∂u(x,t)∂t=Δu(x,t−τ1)+f(u(x,t),u(x,t−τ2)), where τ1, τ2 > 0. This model is motivated by ecological applications in which migration rates incorporate historical effects and reproduction/death processes are subject to time delays at a given location. To address such systems under standard monotonicity assumptions, we extend the classical monotone iteration method. A key step involves a thorough investigation of the Green function associated with the functional equation x″(t)−ax′(t+r)−bx(t+r)=f(t), where a ≠ 0 and b > 0. Building on the resulting framework, we construct quasi-upper and lower solutions for the Belousov–Zhabotinski equations, thereby demonstrating the …

Saturated Hierarchical Atomic Incremental Learning (Shail): A Behavioral Learning Perspective On Staged Mastery And Saturation, Ernest Fokoue

Articles

We introduce Saturated Hierarchical Atomic Incremental Learning (sHAIL), a learning paradigm in which complex tasks are approached through a sequence of simpler atomic subtasks, each mastered to saturation before progression. The central mechanism is a saturation criterion that detects when learning dynamics enter a plateau region, triggering consolidation and subsequent ascent to a higher level of task complexity. We develop a theoretical framework for sHAIL and show that it naturally gives rise to \emph{staircased convergence}: alternating phases of rapid improvement and genuine plateau. Within each level, classical convergence guarantees apply under standard smoothness conditions, while the hierarchical transitions are driven …

No Intelligence Without Statistics: The Invisible Backbone Of Artificial Intelligence, Ernest Fokoue

Articles

The rapid ascent of artificial intelligence (AI) is often portrayed as a revolution born from computer science and engineering. This narrative, however, obscures a fundamental truth: the theoretical and methodological core of AI is, and has always been, statistical. This paper systematically argues that the field of statistics provides the indispensable foundation for machine learning and modern AI. We deconstruct AI into nine foundational pillars—Inference, Density Estimation, Sequential Learning, Generalization, Representation Learning, Interpretability, Causality, Optimization, and Unification—demonstrating that each is built upon century-old statistical principles. From the inferential frameworks of hypothesis testing and estimation that underpin model evaluation, to the …

Decorrelation, Diversity, And Emergent Intelligence: The Isomorphism Between Social Insect Colonies And Ensemble Machine Learning, Ernest Fokoue, Gregory Babbitt, Yuval Levental

Articles

Social insect colonies and ensemble machine learning methods represent two of the most successful examples of decentralized information processing in nature and computation respectively. Here we develop a rigorous mathematical framework demonstrating that ant colony decision-making and random forest learning are isomorphic under a common formalism of stochastic ensemble intelligence. We show that the mechanisms by which genetically identical ants achieve functional differentiation— through stochastic response to local cues and positive feedback—map precisely onto the bootstrap aggregation and random feature subsampling that decorrelate decision trees. Using tools from Bayesian inference, multi-armed bandit theory, and statistical learning theory, we prove that …

A General Weighting Theory For Ensemble Learning: Beyond Variance Reduction Via Spectral And Geometric Structure, Ernest Fokoue

Articles

Ensemble learning is traditionally justified as a variance-reduction strategy, explaining its strong performance for unstable predictors such as decision trees. This explanation, however, does not account for ensembles constructed from intrinsically stable estimators-including smoothing splines, kernel ridge regression, Gaussian process regression, and other regularized reproducing kernel Hilbert space (RKHS) methods whose variance is already tightly controlled by regularization and spectral shrinkage. This paper develops a general weighting theory for ensemble learning that moves beyond classical variance-reduction arguments. We formalize ensembles as linear operators acting on a hypothesis space and endow the space of weighting sequences with geometric and spectral constraints. …

On The Scientific Stature Of Data Science: The Epistemological Unicorn, Ernest Fokoue

Articles

Data Science has ignited unprecedented academic, industrial, and pedagogical fervor, yet its status as a \textit{science} in the classical sense---comparable to physics or biology---remains profoundly unsettled. This article interrogates the epistemological foundations of Data Science by examining its hybrid theoretical lineage, from the Universal Approximation Theorem to the No-Free-Lunch Theorems, with special emphasis on the fundamental Bayesian optimality results for both regression and classification. We argue that Data Science is in a vigorous \textit{gestational period}, characterized not by an absence of principles but by a creative tension between empirical pragmatism and deep mathematical theory. The Cross-Validation score emerges as the …

On Fibonacci Ensembles: An Alternative Approach To Ensemble Learning Inspired By The Timeless Architecture Of The Golden Ratio, Ernest Fokoue

Articles

Nature rarely reveals her secrets bluntly, yet in the Fibonacci sequence she grants us a glimpse of her quiet architecture of growth, harmony, and recursive stability \citep{Koshy2001Fibonacci, Livio2002GoldenRatio}. From spiral galaxies to the unfolding of leaves, this humble sequence reflects a universal grammar of balance. In this work, we introduce \emph{Fibonacci Ensembles}, a mathematically principled yet philosophically inspired framework for ensemble learning that complements and extends classical aggregation schemes such as bagging, boosting, and random forests \citep{Breiman1996Bagging, Breiman2001RandomForests, Friedman2001GBM, Zhou2012Ensemble, HastieTibshiraniFriedman2009ESL}. Two intertwined formulations unfold: (1) the use of normalized Fibonacci weights -- tempered through orthogonalization and Rao--Blackwell optimization -- …

Learning Ordinal Geometry: Semantic–Aware Kernels For Ordered Categorical Data, Ernest Fokoue

Articles

Ordinal data arise ubiquitously in survey research, psychology, medicine, economics, and recommender systems, yet kernel methods for such data typically rely on either nominal encodings or arbitrary numeric codings. The former discards order information; the lat- ter imposes a fictitious metric structure. This paper develops a principled framework for kernel design on ordinal scales and introduces a new class of Semantic–Aware Ordinal Ker- nels (SAOK) that simultaneously capture ordinal order and semantic proximity between categories. We begin by formalizing order–preserving embeddings of finite chains and characterizing a broad family of chain distances that are conditionally negative definite. Through Schoen- berg …

Measuring Market Risk Through Entropic Var, Dragomir Nedeltchev, Tsvetelin Zaevski

Mathematical Modelling and Numerical Simulation with Applications

The article aims to measure the market risk beyond the basic risk measures like the Value-at-Risk (VaR) and the Expected Shortfall (ES). The Entropic Value-at-Risk is selected among the available measures based on its advantages -- it is the coherent upper bound of the VaR and ES. This risk measure is applied to the classical Black-Scholes model as well as to some more realistic ones, such as the exponential tempered stable model (the log-returns are presented by a tempered stable L\'evy process), the stochastic volatility model of Heston, its jump extension of Bates, and another stochastic volatility model but with …

The Blurred Boundaries Of Care: A Qualitative Analysis Of Nurse-Doula Role Conflict, Curisa Mae Tucker Phd, Rn, Maria Mcclam, Kelly Russin Dnp, Rnc-Ob, Chse, Nansi Boghossian, Rachel Gulczinski, Lauren Workman

Faculty Publications

Introduction:

Methods:

Association Between The Dietary Index For Gut Microbiota (Di-Gm) And Colorectal Cancer In The Plco Cohort, Bezawit E. Kase, Angela D. Liese Ph.D., Jiajia Zhang Ph.D., Elizabeth Angela Murphy, Susan E. Steck Phd, Mph, Rd

Faculty Publications

Objectives: The study aimed to examine the association between a dietary index for gut microbiota (DI-GM) and the risk of incident colorectal cancer (CRC). Clarifying the role of diet-induced alterations in the composition and function of gut microbiota on the development of CRC can contribute to prevention efforts. Methods: Participants from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening trial enrolled in the intervention arm and who completed baseline assessments were included in the analysis (n = 55,685). The DI-GM is a literature-derived index used to score diet quality in terms of maintaining healthy gut microbiota. A time-dependent Cox model …

Association Between The Dietary Index For Gut Microbiota (Di-Gm) And Colorectal Cancer In The Plco Cohort, Bezawit E. Kase, Angela D. Liese, Jiajia Zhang, Elizabeth Angela Murphy, Susan E. Steck

Faculty Publications

Objectives: The study aimed to examine the association between a dietary index for gut microbiota (DI-GM) and the risk of incident colorectal cancer (CRC). Clarifying the role of diet-induced alterations in the composition and function of gut microbiota on the development of CRC can contribute to prevention efforts. Methods: Participants from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening trial enrolled in the intervention arm and who completed baseline assessments were included in the analysis (n = 55,685). The DI-GM is a literature-derived index used to score diet quality in terms of maintaining healthy gut microbiota. A time-dependent Cox model …

Dietary Inflammatory Index And Objective Disease Activity In Ibd: No Association Found, Rúbia Moresi Vianna De Oliveira, Ana Carolina Junqueira Vasques, Stefhani Andrioli Romero, Nitin Shivappa Mbbs, Mph, Ph.D., Michael David Wirth, James Hébert Scd, Glaucia Fernanda Soares Ruppert Reis, Cristiane Kibune Nagasako

Faculty Publications

Background

Inflammatory Bowel Disease (IBD) involves genetic and environmental factors, but the relationship between disease activity, adiposity, and diet remains unclear.

Objective

To investigate the association between endoscopic/radiological activity of IBD, body adiposity, and the Dietary Inflammatory Index with or without adjustment for energy density (E-DII or DII).

Method

An observational, cross-sectional study was carried out. Endoscopic activity was defined by an endoscopic Mayo score >2, Crohn’s Disease Endoscopic Index of Severity (CDEIS) > 5, and/or the presence of a deep ulcer in any intestinal segment. Body adiposity was estimated using the body mass index, waist circumference, and waist-hip ratio (WHR). …

Circulating Levels Of Insulin-Like Growth Factor I (Igf-I) And Risk Of Multiple Myeloma: An Observational And Mendelian Randomisation Study, Yolanda Benavente, Sara Hermosa, Nikos Papadimitriou, Alyssa I. Clay-Gilmour Ph.D., Elizabeth E. Brown, Jonathan N. Hofmann, Nathaniel Rothman, Qin Lan, Sonja I. Berndt, Demetrius Albanes, Mark Purdue, Mitchell J. Machiela, Stephen J. Chanock, Parveen Bhatti, Wendy Cozen, Aaron Norman, Susan L. Slager, James R. Cerhan, Vincent Rajkumar, Et. Al.

Faculty Publications

No abstract provided.

Flexible Modeling Of Non-Gaussian Longitudinal Data: Some Approaches Using Copula, Subhajit Chattopadhyay

Doctoral Theses

Longitudinal data are common in medical and biological sciences, where measurements are gathered from subjects over time to explore relationships with explanatory variables (covariates) and to uncover the underlying mechanisms of dependence among these measurements. The responses observed at each instance can be either discrete or continuous. One of the primary challenges in longitudinal data analysis lies in the non-Gaussian nature of the response variables. As a result, there are relatively few multivariate models in the literature that effectively address the specific characteristics observed in such datasets. In this dissertation, we address four problems concerning longitudinal data analysis by developing …

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

Associational Inference With Many Potential Covariates: Bayesian Information Criterion Elastic Net, Farideh Bagherzadeh Khiabani, I-Chan Huang, Jose Miguel Martinez Martinez, Shizue Izumi, Sedigheh Mirzaei, Tiange Zheng, Irina Dinu, Yutaka Yasui

COBRA Preprint Series

Background: An emerging feature in modern biomedical research is collecting and analyzing numerous variables. In the presence of many potential covariates, inference becomes challenging requiring both distinguishing a set of covariates truly associated with an outcome and estimating their corresponding regression coefficients consistently. Traditional statistical inference typically focuses on estimating coefficients assuming a pre-specified set of covariates. Further, advanced machine/statistical learning methods performing both selection and estimation predominantly focus on outcome prediction rather than association inference.

Methods: Motivated by our epidemiological research on long-term childhood cancer survivors, where we aimed to investigate associations between a large pool of longitudinal symptom …

Database-Driven Revelations In Epilepsy: Patient Reporting Accuracy And The Circadian Timing Of Seizures, Ithay Biton

Theses and Dissertations

For over thirty years, patients have been visiting the Arkansas Epilepsy Program for diagnosis and treatment for seizures and seizure-like episodes. As part of their clinical evaluation, patients often undergo ambulatory EEG (electroencephalogram) monitoring. This routine process produces valuable data for treating the patient. In this study, over 2000 ambulatory EEG reports from 1998 to 2016 were reviewed. A large database of seizures was created from the reports, with information on 407 patients, 1611 EEG-confirmed seizures, and 1726 patient-reported seizures (IRB Protocol #17-093). The database was used to address two important issues in epilepsy. The first topic of the study …

Ulk4 And Cdkn2a Polymorphisms Influence The Risk Of Developing Monoclonal Gammopathy Of Undetermined Significance, José Manuuel Sánchez-Maldonado, Angelica Macauda, Antonio José Cabrera-Serrano, Hauke Thomsen, Murat Güler, Rob Ter Horst, Bethany Van Guelpen, Pavel Vodicka, Stefano Landi, Subhayan Chattopadhyay, Pelin Ünal, Lucía Ruiz-Durán, Delphine Casabonne, Hartmut Goldschmidt, Istemi Serin, María Carretero-Fernández, Elena Cabezudo, Fernando Reyes-Zurita, Alyssa I. Clay-Gilmour, Et. Al.

Faculty Publications

Monoclonal gammopathy of undetermined significance (MGUS) is a necessary precursor condition to multiple myeloma (MM). Given the role of autophagy in modulating MM risk, we investigated whether genetic variation in autophagy-related genes influences susceptibility to MGUS. We analyzed the association of 34,042 common autophagy-related single nucleotide polymorphisms (SNPs) with MGUS across six independent cohorts, five from Europe and one from North America, comprising 2317 MGUS cases and 282,358 controls. We also assessed their impact on immune parameters, including absolute counts of 91 blood-derived immune cell subsets and 103 circulating immunological proteins. Meta-analysis revealed a genome-wide significant association between the ULK4 …