Applied Mathematics Commons^™

Inertial Dynamics Of Non-Spherical Particles In Fluid Flows, Takashi Yashiro

Theses, Dissertations and Culminating Projects

We investigate the dynamics of small inertial spherical and non-spherical particles in fluid flows. We consider the Maxey-Riley-Gatignol (MRG) equation, which models well the motion of spherical inertial particles in low Reynolds number flows. To study how shape affects the dynamics, we implement a corrective factor on the Stokes drag term in the MRG equation. This corrective factor, or shape factor, is based on the geometric properties of the particles. The Basset-Boussinesq history term in the MRG equation is often neglected to simplify analytical and computational studies involving the equation. We include this history term and implement a multi-step integration …

An Exploration Of The Autorotating Pendulum Model, Vlad Nita

Theses, Dissertations and Culminating Projects

Autorotation is the spontaneous rotation of an object, usually caused by an external fluid flow. The study of autorotation has many physical applications, such as in the design of wind/water turbines. In this thesis, we explore a nonlinear pendulum ordinary differential equation (ODE) which is used to model rotating plates in a fluid and has the capacity to reveal autorotation. In the context of an ODE, autorotation emerges as a bifurcation past oscillations, when the initial velocity of the system crosses a particular threshold. In his classic study from 1983, Lugt [14] utilizes this equation to capture experimental autorotation. Copeland’s …

Advances In Computational Methods For Sparsity-Promoting Linear Inverse Problems, Jonathan Lindbloom

Dartmouth College Ph.D Dissertations

Inverse problems arise throughout science and engineering, where indirect, incomplete, and noisy observations are used to recover unknown parameters of interest. In these applications, the corresponding forward or measurement models are often ill-conditioned or underdetermined, so direct inversion is unstable and regularization is required. This thesis develops computational methods for linear inverse problems in which the unknown is assumed to be approximately sparse in a transformed domain defined by a linear, possibly rank-deficient operator, such as a finite-difference matrix, with particular emphasis on large-scale problems.

The thesis makes three main contributions. First, it generalizes hierarchical Bayesian maximum a posteriori estimation …

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 …

Multivariate Time-Series Forecasting Of 24-Hour Ambulatory Blood Pressure Using Long Short-Term Memory And Temporal Fusion Transformers, Sebastian Alejos Torres

Theses and Dissertations

Time series play a central role in healthcare by enabling continuous patient monitoring and forecasting of physiological and clinical measurements. Traditional models, such as autoregressive integrated moving average (ARIMA), are limited in capturing nonlinear dynamics and irregular sampling. In this study, we develop and evaluate Long Short-Term Memory (LSTM) networks and Temporal Fusion Transformers (TFTs) to forecast 24-hour ambulatory systolic and diastolic blood pressure (SBP and DBP) time series enriched with demographic and clinical features. Mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), and R² were used to evaluate predictive accuracy and temporal pattern …

Stochastic Universal Differential Equations For Epidemiological Modeling: Uncertainty Quantification In Disease Transmission Dynamics, Alice Menaya Armah-Bonney

Electronic Theses and Dissertations

Epidemic forecasting requires not only predictions of expected case counts, but also quantification of uncertainty, although existing surrogate modeling frameworks for agent-based models remain fundamentally deterministic. In this thesis a Stochastic Universal Differential Equation framework is presented that extends the deterministic Universal Differential Equation approach by incorporating a learnable stochastic diffusion term, enabling calibrated probabilistic forecasts while preserving the mechanistic interpretability and computational efficiency of the deterministic baseline. In doing so, a two-phase training algorithm is introduced to ensure stable convergence and the framework is validated against the ensemble output from ExaEpi, an exascale agent-based model of a COVID-19 outbreak …

Adaptive Artificial Potential Field Guidance And Control For Autonomous Docking With Uncooperative And Unknown Spacecraft, Steven Holmberg

Theses and Dissertations

The increasing demand for on-orbit servicing (OOS), active debris removal (ADR), and space domain awareness (SDA) missions has increased the need for autonomous spacecraft rendezvous and proximity operations (RPO) with uncooperative and unknown targets. Traditional guidance and control methods are typically designed for cooperative systems with known geometry and state information. This work builds on previous research to develop and evaluate an artificial potential field (APF)-based control framework capable of autonomous operation with minimal prior target knowledge and applicability to both relatively static and tumbling spacecraft.

The proposed APF formulation incorporates established safety constructs from cooperative docking systems, including an …

Analysis Of Collective Behavior In Living And Nonliving Systems, Kaitlyn Cohan

Theses, Dissertations and Culminating Projects

This thesis aims at understanding the phenomenon of of self-organization in complex dissipative systems, living and nonliving. Dissipative systems are characterized by their search for energy, interactions with their surroundings and the production of entropy, all of which result in the creation of stable structures or patterns, which persist as long as the initial environmental conditions are maintained. The two specific models that we chose to study here are (a) Futbol (or Soccer) and (b) a chemical system involving free-floating menthol crystals floating on a fluid surface to represent nonliving systems. Using experiments and mathematical models, we will try to …

Pattern Dynamics And Stochasticity Of Brain Rhythms And Spike Trains In A Tauopathy Mouse Model Of Alzheimer’S Disease, Clarissa M. Hoffman

Dissertations and Theses (Open Access)

Systems neuroscience posits that every aspect of perceived physical reality, every aspect of animal and human behavior, and every cognitive phenomenon emerges from patterns of neuronal activity. While most researchers embrace this idea, there are major difficulties in describing and analyzing these complex neuronal dynamics—spike flows produced by cells ensembles, synchronized extracellular field oscillations, and other patterns—which limits our understanding of how the activity of individual neurons and the whole-animal cognition and behavior might be connected. In particular, we lack the approaches and even the semantics for connecting the individual cell outputs and the integrated results of their activity. Current …

Safe Control Design For Quadruped Locomotion In Unstructured Environments Using Linear Transfer Operators, Sriram Sundar Krishnamoorthy Shankara Narayanan

All Dissertations

Deploying quadruped robots in unstructured, obstacle-rich environments requires control and planning methods that remain safe and reliable despite complex terrain geometry, limited sensing, and inevitable modeling errors. This thesis develops operator-theoretic tools for safe control design of robotic systems using linear transfer operators, with a focus on quadruped locomotion in unstructured environments. The central goal is to develop a unified operator-theoretic framework for safe control design based on the Perron–Frobenius (P–F) and Koopman operators. In particular, the thesis leverages \emph{density functions} to develop safe navigation frameworks in the dual space of densities. In the operator-theoretic perspective, the P–F operator governs …

Discrete-Event Simulation: A Leslie System Model For Transportation Demography, Md Zobaer Ahammad

Electronic Theses and Dissertations

Transportation systems are influenced by demographic change, household formation,and patterns of vehicle ownership. These factors affect long-run transportation demand and congestion levels within urban infrastructure. Understanding how demographic dynamics interact with transportation behavior is therefore important for analyzing the long-term evolution of transportation systems. This thesis develops a modeling framework that integrates discrete-event simulation with a Leslie-type matrix model to study transportation–demography interactions.Simulation outputs are aggregated to construct a transition matrix describing changes in transportation states. This matrix acts as a linear (or affine) transformation on the transportation state vector, allowing the system to be analyzed as a discrete linear …

Log Anomaly Detection With Parameter-Efficient Tiny Language Models: From Centralized Fine-Tuning To Privacy-Preserving Federated Learning, Isaiah Thompson Thompson Ocansey

Open Access Theses & Dissertations

System logs are a primary source of information for identifying faults, misconfigurations, and security incidents in computing infrastructure. As these logs grow in volume and complexity, manual inspection becomes impractical, and automated detection methods are needed. This thesis investigates how compact language models can be applied to the task of log anomaly detection under two operational conditions: when log data is centralized and when it is located at different data sites.

In the first part, we propose LogTinyLLM, which applies parameter-efficient fine-tuning through Low-Rank Adaptation and adapter-based techniques to adapt tiny language models for identifying contextual anomalies in log sequences. …

A Penalty-Free Runge-Kutta Discontinuous Galerkin Method For \\[12pt] Time-Dependent Fourth-Order Partial Differential Equations, Jose Armando Perez Becerra

Open Access Theses & Dissertations

Time-dependent fourth-order partial differential equations arise in a wide range of applications in applied mathematics, physics, and engineering, including thin structure models, phase separation, and pattern formation. Their numerical approximation is challenging because the presence of fourth-order spatial derivatives typically requires high-regularity discretizations. A useful alternative is to reformulate the original problem as a coupled second-order system and approximate it by mixed discontinuous Galerkin (DG) methods.

This thesis builds upon the penalty-free mixed DG framework developed by Liu and Yin for time-dependent fourth-order problems. That framework avoids the use of interior penalty parameters, preserves the symmetry of the associated bilinear …

A Numerical Method For The Phase Field Crystal Model, Patrick Ameyaw Tabiri

Open Access Theses & Dissertations

The Phase Field Crystal (PFC) model is a continuum-based framework used to study theevolution of crystalline materials while preserving microscopic structural features over diffusive time scales. It captures complex phenomena such as phase transitions, defect dynamics, and microstructure formation via a nonlinear sixth-order partial differential equation derived from a free-energy functional. In this context, the phrase "sixth-order" indicates that the highest spatial derivative appearing in the equation is of order six. In this work, we study the mathematical formulation and numerical approximation of solutions to the PFC model. Due to the high-order spatial derivatives and nonlinearity in the governing partial …

Bayesian Deep Learning For Photovoltaic Power Forecasting: A Probabilistic Framework For Uncertainty Quantification And Grid Reliability Optimization, Pablo Abraham Bustamante

Open Access Theses & Dissertations

The global energy landscape is undergoing a profound transformation, driven by the urgent need to decarbonize power systems, enhance energy security, and meet growing electricity demands. Solar photovoltaic (PV) power has emerged as a critical component of future energy infrastructure due to its abundance, scalability, and cost-effectiveness. However, PV generation is inherently variable and weather-dependent, introducing significant uncertainty into grid operations and complicating the task of balancing supply and demand. Accurate forecasting of PV power generation-particularly on day-ahead and hour-ahead horizons-has become a strategic necessity for grid stability, economic efficiency, and environmental sustainability. PV output is influenced by numerous factors, …

Decision Making For Large-Scale Problems Under Uncertainty And Conflict, Benjamin J. Hamlin

All Dissertations

Large-scale decision-making problems appear in many areas including long-range forecasting such as energy generation forecasting. Many such problems are subject to conflicting objectives and uncertain data, and can be modeled as linear optimization problems. We study novel theoretical results and algorithms for large-scale linear decision problems under conflict and uncertainty. First, we propose a parametric Benders decomposition algorithm for solving large-scale linear optimization problems with multiple objectives or deterministically uncertain objectives. Second, we extend the parametric Benders decomposition to a multi-stage setting, developing a parametric stochastic dual dynamic programming algorithm, which enables decision-making when conflicts and uncertainty have planning impacts …

Fractsynth: Exploring Glitch Timbres With Chaos Theory, Aidan Roach

The Transdisciplinary STEAM+ Journal

In this paper, I explore how chaos theory can be used to design a new kind of synthesizer with the primary focus of producing glitchy, unpredictable sounds. Glitch music embraces abstract sound design, malfunctioning electronics, and randomness as the main compositional elements. However, most synthesizers rely on stable, repetitive oscillators that often sound too controlled. To challenge this, I developed FractSynth, a real-time synthesizer that uses chaotic attractors–including the Logistic Map, Henon Map, and Lorenz System–as modulation sources for frequency, amplitude, and tone. The software also features real-time Lyapunov Exponent Tracking, which gives users a direct visual of how …

Consistency-Based Computing Of Fuzzy Eigenvalues And Fuzzy Eigenvectors: Method And Application, Kamala. R. Aliyeva Prof., Nihad Mehdiyev, Shamil Mehdi

Chemical Technology, Control and Management

The computation of eigenvalues and eigenvectors under uncertainty is a fundamental problem in fuzzy linear algebra and decision analysis. When matrix elements are represented by fuzzy numbers, classical spectral methods cannot be directly applied due to nonlinearity, ambiguity in ordering, and the propagation of uncertainty. Moreover, in many practical applications, particularly those involving pairwise comparison matrices, the reliability of eigenvalue-based results strongly depends on the consistency of the underlying data. This paper proposes a consistency-based framework for computing fuzzy eigenvalues and fuzzy eigenvectors that explicitly integrates consistency analysis into the spectral derivation process. The proposed method preserves the fuzzy structure …

Intermediate-Scale Outflow Dynamics Of Eta Carinae, Edmund J. Garcia, Matthew C. Fleenor

Departmental Honors & Graduate Capstone Projects

η Carinae (η Car) is a binary system, with the larger star being an extremely massive, luminous blue variable (LBV) beyond the Eddington Limit. Surrounding the η Car system, numerous multi-wavelength imaging campaigns reveal axisymmetric structures with the expanding bipolar Homunculus Nebula (¡1 pc). In combination with the episodic eruptive history of the η Car system, our initial intermediate-scale imaging revealed further axisym- metric structures (1-5 pc). To gain a more expansive view of how the small scale structure connects to panoramic imaging of the η Car region, we constructed a deep, optical, narrowband mosaic of 189 images utilizing the …

Theory And Simulations Of Delayed Stochastic And Deterministic Models Of Prion Diseases, Gangadhara Boregowda, Omar Sharif, Daniel Gutierrez Iii, Allegra Simmons, Laurent Pujo-Menjouet, Tamer Oraby, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Neurodegenerative diseases (NDs), such as Alzheimer’s, Parkinson’s, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein (PrPC), produced by neurons, misfolds into a toxic form, known as scrapie prion protein (PrPSc). PrPSc induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial …