Applied Mathematics Commons^™

Identifiability, Sequentiality And Infinity, Jose L. Menaldi

Mathematics Faculty Research Publications

Abstract: A definition of identifiable-sets is used with sequential analysis to establish a realm of mathematics. Within this imaginary world, a specific consonant between infinite sets and sequentiality is reached.  This consonant allows some mathematical constructions to model pieces of the reality, based on dual philosophy and physics itself.  There is an effort made to render this understandable for the scientific community.

A Mathematical & Computational Study Of Voting Power In Social Choice Systems, Madison T. Gambon

Undergraduate Honors Theses

This thesis develops a computational framework for measuring the vulnerability of voting rules to coordinated strategic manipulation. While classical results show that most voting systems are theoretically manipulable, less is known about the magnitude of coordination required to alter outcomes or how that magnitude varies across institutional designs. I define the minimal manipulating coalition size k* as the smallest number of voters whose strategic ballot changes can overturn a sincere election outcome under a given rule. To enable cross-election comparison, I introduce the normalized manipulation threshold θ = k*/n . Using algorithmic search procedures and Monte Carlo simulation, I estimate …

Momentum Space Algorithm For Electronic Structure Of Double-Incommensurate Trilayer Graphene, Kenneth Silver Beard

LSU Doctoral Dissertations

Numerical algorithms for computing the electronic structure of incommensurate 2D-materials using ab initio models are critical for predicting material properties and guiding experiments. For bilayers, momentum space and continuum models have been introduced to approximate observables of ab initio tight-binding models using a momentum description, despite the lack of periodicity in the tight-binding model required for Bloch theory. A similar structure has been introduced for double-incommensurate trilayers using a continuum model, where the three lattices are mutually incommensurate. However, this description leads to a four-dimensional lattice space, and numerical convergence of the density of states has been observed to be …

Largest 2-Regular Subgraphs In Complete S-Partite Graphs, Yiyang Jiang

McKelvey School of Engineering Graduate Student Theses & Dissertations

In this thesis, we focus on the class of complete $S$-partite graphs, for $S$ an undirected graph possibly with self-loops, and address the problem of finding largest $2$-regular subgraphs of these graphs, which can be formulated as an integer linear program. Roughly speaking, a complete $S$-partite graph is obtained by replacing every single node of $S$ with a number of nodes, preserving the edge/non-edge relations of $S$. Our motivation in studying largest $2$-regular subgraphs is rooted in the structural systems theory, particularly in the problem of finding largest subnetworks that can sustain controllability or asymptotic stability of the corresponding subsystems. …

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

Phenological Overlap In Obligate Plant-Pollinator Mutualism, Austin J. Carlson

2026 Spring Honors Capstones Projects

Plant-pollinator mutualisms require temporal overlap between flowering and pollinator activity, so climate-driven timing shifts can weaken the interaction and, in severe cases, destabilize the system. This work investigates how reduced overlap affects persistence in an obligate plant-pollinator pair using a coupled differential equation model in which a phenological overlap factor scales the saturating mutualistic benefit. Simplification with a constant overlap enables closed-form equilibrium and stability analysis, revealing that below a critical overlap threshold, coexistence is no longer maintained. Rescaling reduces the parameter space from ten quantities to seven dimensionless groups, and sensitivity analysis identifies the degree of species dependence and …

Machine Learning For Handwritten Character Recognition, Hannah Freitag

Honors Capstones

Handwritten character recognition remains a challenging problem in machine learning due to the high variability of handwriting across individuals and the visual similarity between certain character classes. This project explores whether Singular Value Decomposition (SVD)-based dimensionality reduction can serve as an effective preprocessing step for a fully connected neural network trained on the EMNIST Balanced dataset, a 47-class benchmark of handwritten digits and letters. By projecting 784- dimensional pixel inputs onto the top 70 principal components, approximately 90% of the total variance is preserved while reducing input dimensionality by 91%. The resulting SVD-based model achieves approximately 94% test accuracy, outperforming …

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 …

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 …

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 …

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 …

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 …

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. …

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 …

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 …