Applied Mathematics Commons^™

Modeling French Language Preservation: An Optimal Control Problem, Charlotte Blasi

Scripps Senior Theses

The French language is one of the most globally spoken languages, with over 300 million speakers and designated the official language of 29 nations. French continues to prosper as a result of its titular nations' imperialist history that emphasized linguistic diffusion as well as the support of international organizations dedicated to promoting both the language and the culture of French-speaking nations. In this thesis, we explore the nuanced history of French-speaking countries and one institution dedicated  to promoting the French language, the OIF. We further examine the media mechanism of the OIF and construct a system of Ordinary Differential Equations …

Computationally Modelling Nmda Blockages Within A Neural Network, Anya Raetsch

UNH URC Open (2026 and after)

The N-Methyl-D-Aspartate (NMDA) Receptor is fundamentally important to memory formation within the brain due to its control of calcium entry into the cell.  In recent years, there has been an increased interest in long-term effects of NMDA blockages on the brain, due to the “re-wiring” of communication channels (synapses) between neurons. This project models the effects of NMDA blockages due to drugs such as Ketamine, and how the blocking of NMDA receptors affects firing rates, which can then be applied to studying long-term plasticity within the neural hierarchies. Using the Nest Online Simulator, a 50x50 grid of neurons was created …

Uncertainty Quantification, Propagation & Conjunction Assessment In Orbital Mechanics Using Generalized Polynomial Chaos Expansion & 2-Dimensional Conjunction Plane Analysis Techniques, Monalisa Karim

Mechanical and Aerospace Engineering Theses

Uncertainties, that are inherent to dynamic models, can be associated with state initial conditions, force modelling errors, navigation and actuation errors. In system modelling stochastic differential equations are used to represent dynamic phenomena with uncertainties, for which the solutions are probability density functions of quantities of interest characterizing the realization of the stochastic processes. In Polynomial Chaos Expansion (PCE) propagation, these solutions are represented as weighted sums of multivariate spectral polynomials that are functions of the input random variables. Generalized polynomial chaos expansion (gPC) is an extension to the original homogenous PCE which projects the random solution onto a basis …

Liutex - A Fluid Vortex, Oscar Alvarez

Mathematics Dissertations

Fluid vortices are found everywhere in our universe. A vortex can take the form of almost anything - from the classical spiral vortex to chaotic plumes. Defining a vortex physically and mathematically is absolutely necessary if we desire to study vortices and their interactions with each other as well as our physical world. Fluid vortices are incredibly important in the study of turbulent flows. From determining wear, optimizing design for better flow, efficiency, etc., to even predicting the weather on Earth or other planets, having the ability to measure vortices in fluid flow is invaluable. In this study, I investigate …

Using Provided Guided Notes In Coordinated Introductory First-Year Mathematics Courses, Jennifer L. Huber

Mathematics Dissertations

The goal of this study is to investigate how standardized guided notes shape instructional practices and student engagement in coordinated introductory first-year college mathematics courses at a large public university. The researcher explored three multi-section introductory mathematics courses with overlapping learning objectives. Each course required students to purchase a student workbook as part of the instructional materials for the class. The instructors taught primarily from the workbook containing guided notes created by a former coordinator of the course. The researcher used a mixed-methods approach. Instructors and students participated in surveys, class observations and provided class meeting notes. Instructors shared additional …

Mathematical Model Of Graphene, Douglas M. Sanor

Williams Honors College, Honors Research Projects

Graphene, a single-atom-thick layer of carbon arranged in a hexagonal lattice, exhibits exceptional mechanical, electrical, and thermal properties that make it a promising material for a wide range of engineering applications. This paper presents a mathematical framework for modeling the mechanical behavior of graphene, with a focus on atomistic-to-continuum approaches. We begin with a onedimensional Frenkel-Kontorova model that represents graphene as a discrete chain of particles interacting with both their nearest neighbors through harmonic spring potentials and an underlying substrate through van der Waals forces. Numerical simulations of this discrete model demonstrate the commensurate-toincommensurate phase transition, revealing how geometric mismatch …

Investigating The Connection Between Als Through The Mutation R522s In The Rna Binding Protein, Dennia Estrella-Vargas, Lydia Uptain

Mathematics

Amyotrophic lateral sclerosis (ALS) is a fatal disease that causes the deterioration of motor neurons , death is usually due to respiratory paralysis. The variant R522S was chosen because it is near a hot spot of pathogenic variants. It is an arginine-to-serine swap, this swap is present in pathogenic variants near the 522 position, such as R514S, R521S, R524S. Recent evidence suggests that arginine-deficiency can influence disease progression.

Managing Multi-Drug Resistance: An Evolutionary Game Theory And Optimal Control Approach, Shukhrat Nasrulloev

Theses and Dissertations

Multi-drug resistance is an evolutionary process in which treatment eliminates sensitive cells, allowing resistant clones to dominate. This thesis investigates this process using a framework integrating population dynamics, evolutionary game theory, and optimal control theory. We develop a two-population logistic growth model describing competition between drug-sensitive and drug-resistant cells under treatment, construct dose-dependent payoff matrices and replicator dynamics to characterize evolutionary competition, and derive a critical drug level D_crit = (r_S - rR)/(dS - d_R) at which resistant cells gain a fitness advantage. An optimal control problem is formulated via Pontryagin's Maximum Principle to identify schedules …

Data-Driven Partitioning In Distributed Optimization For Networked Systems, Prosper Azameti

Theses and Dissertations

The convergence behavior of distributed optimal power flow (OPF) depends strongly on how the power network is partitioned into regions. Classical graph-based methods such as METIS are widely used, but they rely mainly on static topological criteria and do not explicitly incorporate operating-point-dependent information that may affect distributed optimization performance. This thesis develops a data-driven partitioning framework for distributed OPF using graph neural networks (GNNs). Each OPF scenario is represented as a graph in which buses are nodes and transmission lines are edges. Node and edge features capture both structural and operational characteristics of the network. Partition prediction is formulated …

Inverse Problems For The Radiative Transport Equation In Local And Non-Convex Geometries, Faith E. Hensley

Theses and Dissertations--Mathematics

Inverse problems for the radiative transport equation (RTE) arise in a wide range of imaging applications, including optical tomography and problems motivated by non-line-of-sight imaging. Classical reconstruction methods rely heavily on ballistic, or unscattered, photons and typically require full boundary access, leading to severe instability and limited applicability in geometrically constrained settings. This dissertation investigates inverse radiative transport problems with restricted boundary data and develops reconstruction techniques based on scattered photons. The central focus of this work is the analysis and isolation of the single-collision term in the collision expansion of solutions to the RTE. By exploiting its distinct analytical …

Design And Analysis Of Modern Quantum Neural Network Architectures For Intelligent Systems, Lakshmi Chandrakanth Kasireddy, Prabhakara Rao Kapula, Dineshkumar Rajendran, Neha Bharani, Srikanth Pulipeti, Islombek Khushvaktov

Computer Science Faculty Publications

Quantum neural networks (QNNs) offer a principled pathway for integrating quantum computation with machine learning through superposition- and entanglement-based representations. This chapter proposes an architecture-aware design and evaluation framework for modern QNNs, emphasizing robustness and system feasibility alongside predictive performance. Multiple architectures variational QNNs, quantum convolutional neural networks, tensor-network hybrids, and fully quantum models—are assessed under a unified protocol. Experimental analysis shows that the proposed architecture-search–guided QNN achieves 91.8% classification accuracy and an F1-score of 0.914, outperforming fixed-template variational QNNs by approximately 5.6 percentage points. Under depolarizing noise with probability p = 0.10, the proposed model retains 85.3% accuracy, whereas …

Stochastic Fractional-Order Memristive Fuzzy Bam Neural Networks With Time Delays And Leakage Term For Finite-Time Stability Analysis, J. Kumar, M. Syed Ali, Sumaya Sanober, Mohammad Yarish, Abeer M. Alotaibi, Tarek F. Ibrahim

Computer Science Faculty Publications

In this study, a finite-time stability analysis with time delays and a leakage term is conducted on stochastic fractional-order memristive fuzzy BAM neural networks. FOMFBAMNNs are developed using set-valued map theories as well as differential inclusion. We obtained several significant adequate criteria of uniform stability in the mean square of such networks by using analytical methods and inequality approaches, such as Cauchy–Schwarz inequality and Burkholder–Davis–Gundy inequality. In addition to examining two different fractional-order derivatives between the U-layer and V-layer synchronously with fractional order, the existence, uniqueness, and stability of its equilibrium point are also shown ½ ≤ α ≤ 1. …

Robust Deep Learning One-Class Classification, Shahd Alnofaie

Graduate Studies Theses and Dissertations 2026

One-Class Classification (OCC) focuses on learning the characteristics of normal data and identifying observations that deviate from this learned pattern as anomalies. It is commonly used in applications such as medical diagnosis, cybersecurity, industrial monitoring, and fraud detection, where abnormal examples are often rare or unavailable during training. Classical approaches such as SVDD and LS-SVDD describe normal data using a hypersphere. While effective in some settings, these methods rely on shallow representations and can be sensitive to noise and contaminated observations. To address these limitations, this dissertation introduces a Deep LS-SVDD framework that combines hypersphere-based data description with deep neural …

Using Ai To Analyze Survey Data, Sara Martucci

Open Educational Resources

This assignment in Methodology in Sociology/Criminology engages students in the full research process by guiding them through variable selection, data analysis, interpretation, and critical reflection on AI-assisted decision-making. Using a class-generated survey dataset (or an existing dataset), students develop a research question, identify independent and dependent variables, and formulate a hypothesis. They then compare their selections with those suggested by an AI tool, analyzing differences in reasoning and variable choice. Through SPSS, students generate frequency tables, charts, and scatterplots to examine relationships between variables, including potential intervening factors. The assignment culminates in a group presentation and reflective analysis on the …

Beyond Full Fine-Tuning: The New Playbook For Adapting Deep Neural Networks, Cristian S. Mcgee

Honors Undergraduate Theses

Fine-tuning is the process of teaching and specializing a pre-trained neural network on a downstream task. Fine-tuning is a rapidly growing topic in artificial intelligence domains; however, many fine-tuning endeavors are highly specialized without a coherent framework connecting them. This work presents a unified perspective on fine-tuning methods and performance metrics. Our perspective organizes the methods in terms of how they are applied to fine-tuning. This framework showcases methods that (i) update effective subspaces of the pre-trained model, (ii) change the adaptation optimization procedure, and (iii) alter the representations of the embedded input. Additionally, we present unconventional metrics such as …

Inequality In The Urban Forest: Modeling Tree Canopy Dynamics Through Demographics And Restoration Strategies, Eve Johansson

Honors Theses

Urban tree canopies play an important role in environmental quality, public health, and neighborhood livability, yet their distribution is highly uneven and often reflects historical patterns of inequality. In Brooklyn, long-term processes such as redlining, uneven development, and demographic change have contributed to persistent disparities in access to green space.

This thesis examines how urban tree canopy evolves across space and time in Brooklyn and how different restoration strategies affect long-run outcomes. The analysis uses demographic and canopy data from 1990-2020, considering race, income, employment, and educational attainment. Among these, education is the most consistent predictor of canopy coverage, with …

The Pure Yang-Mills Field. I: Eistence, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

Two pure Yang-Mills quantum gauge field theories are constructed, one based on short distance asymptotics and the other based on long distance asymptotics.

The construction is based on the axial gauge, ghost states, the BRST framework and Gribov extension of the Hamiltonian, with a loop expansion cutoff to all finite orders for the dynamics.

The construction is established by renormalized perturbation theory to all finite orders.

The construction depends on an assumed principle of a maximum rate of entropy production

Using L1-Magic For Feature Enhancement And Reduced Redundancy In Hyperspectral Data, Ashley Alfred

Mathematics Dissertations

Hyperspectral imaging offers detailed spectral information, but achieving high spatial resolution typically requires large and expensive equipment. This study explores an alternative approach: enhancing low-quality hyperspectral bands using an L1-norm minimization technique known as L1-magic. The goal is to improve the utility of low-cost hardware by preserving discriminative features, promoting sparsity, and reducing spectral redundancy. We apply L1-magic to enhance low-quality bands and hypothesize that this method selectively amplifies key features while suppressing redundant information. Experimental results indicate that the enhanced bands approach the quality of high-resolution data, enabling robust feature extraction without reliance on high-end hyperspectral cameras.

Microgravity-Induced Alterations In Left Atrial Hemodynamics And Thrombogenic Risk: Insights From Healthy And Atrial Fibrillation Models, Grace M. Hoeppner

Dissertations, Master's Theses and Master's Reports

Background: Microgravity exposure alters cardiovascular loading, yet its impact on left atrial flow dynamics and thrombotic risk remains poorly understood. This study investigates how spaceflight-relevant microgravity-induced changes in cardiac outflow affect left atrial hemodynamics in healthy individuals and patients with atrial fibrillation.

Methods: Patient-specific left atrial models were generated for three healthy individuals and three AF patients. Computational fluid dynamics (CFD) simulations were performed using each patient’s baseline mitral outflow waveform and two modified waveforms representing short- and long-duration post-flight cardiac loading changes derived from echocardiographic observations. Hemodynamic metrics included left atrial velocity, time averaged wall shear stress, oscillatory shear …

Ma 250 – Evaluating & Creating With Genai, Mohamed Ben Zid

Open Educational Resources

In this assignment, students use Excel and ChatGPT to design, analyze, and interpret a regression model. They create visualizations, calculate the regression equation manually, and make predictions before consulting AI-generated feedback on their model’s strengths and limitations. Students then compare their own interpretation with ChatGPT’s insights, summarize their findings, and critically assess the model’s accuracy and real-world usefulness. The exercise develops quantitative reasoning, practical AI application, and reflective evaluation skills.

Arrangements Of N Planes Resulting In One Bounded Tetrahedral Chamber, Ava Knight

Williams Honors College, Honors Research Projects

This paper investigates the combinatorial geometry of plane arrangements in three-dimensional space, focusing on configurations that produce exactly one bounded tetrahedral chamber. We define T(n) as the number of face-combinatorial equivalence classes of arrangements of n planes in ℝ³ containing exactly one bounded tetrahedral chamber. Known values — T(3) = 0, T(4) = 1, and T(5) = 2 — are established through direct construction, while T(6) remains an open problem. This paper contributes experimental evidence toward resolving T(6) by systematically extending the two valid 5-plane arrangements and verifying, through a plane removal argument, that each yields a valid plane configuration …

A 1d Symmetric Interior Penalty Discontinuous Galerkin Solver In Rust, William Aey

Williams Honors College, Honors Research Projects

This honors project will build a 1D Symmetric Interior Discontinuous Galerkin (SIPDG) solver in Rust for Stum-Liouville type problems such as the Poisson equation, with Robin, Dirichlet, and Neumann boundary conditions. The work will cover the full pipeline: starting from the strong form of the PDE, deriving the DG weak form, implementing element and interface operators, and assembling or apply the discrete operator. Rust's safety and concurrency (e.g, via Rayon) will be used to explore serial and parallel performance. A test-driven development approach will be used to maintain a strong suite of tests. The project will result in a documented …

A Bayesian Late-Fusion Supportability Framework For Rare-Disease Severity Prediction In Glut1 Deficiency Syndrome, Jordan M. Rodriguez

Mathematics Dissertations

Glucose transporter type 1 deficiency syndrome (GLUT1-DS) is a rare neurometabolic disorder with heterogeneous neurological and developmental severity. Because patient-level severity is not observed as a single validated outcome, this dissertation develops a Bayesian late-fusion supportability framework for constructing and predicting an ordered latent severity phenotype from clinical, genetic, and EEG-derived evidence. The primary target was constructed in a larger clinical cohort using age-5 symptom burden and learning cognition, then assigned to an aligned multimodal prediction cohort. Target-defining variables were excluded from supervised predictors, and models were evaluated using patient-exclusive cross-validation with training-fold preprocessing and fold-wise EEG PCA.

The primary …

Mathematical Models With Clinical Applications For Improving Health Outcomes, Helen Harris

Theses and Dissertations

In clinical settings, patients are exposed to many risks and stressors that could result in adverse health outcomes. Here we present mathematical models that seek to address these risks. First, we present a Markov Chain model to investigate the effect of medication reconciliation (MR) completion on patient health outcomes in the intensive care unit. Using this model, we simulate the annual incidence of adverse drug events (ADEs) for three different ADE rates. Based on the simulated results, we conduct a cost-benefit analysis for various levels of compliance to determine the financial implications of increasing MR completion depending on the baseline …

The Effects Of Overwash On Barrier Island Evolution And Building A Barrier Island Measure Of Resistance, Beth Thomas

Theses and Dissertations

Barrier islands are critical for coastal communities, as they serve as a natural buffer against storm surge, waves, and the effects of rising sea levels, protecting life and property. These islands continuously evolve due to both normal and severe environmental conditions; global warming makes it increasingly difficult to predict the evolution of these islands due to increases in storm frequency and intensity. We present a cellular model of barrier island evolution consisting of biotic and abiotic processes including the effects of vegetation, wind, ocean currents, and gravity. The model is used to predict the future evolution of barrier islands off …

Eigenvalue Spacing Distributions And The Weak Disorder Limit For Random Schrodinger Operators, Kyle E. Hammer

Theses and Dissertations--Mathematics

We study a collection of discrete Schrodinger Operators with random potentials through the lens of global and local eigenvalue spacings. We discuss the three models: the standard scaled disorder Anderson Model, the Anderson-Bernoulli Polymer Model, and the Discrete Fractional Laplacian Anderson Model. First, we discuss the scaled disorder case using the invariant measure and its application to the density of states in the weak disorder limit. We also prove the limit of the local and global eigenvalue spacings in the non random case, and demonstrate numerically how randomness affects the eigenvalue spacings. We then discuss a special family of random …

Entropic Foundation Of Finance And Physics: Securities Price Dynamics And Quantum Theory, Mohammad Abedi

Electronic Theses & Dissertations (2024 - present)

In many scientific and financial contexts, we must reason and make predictions under conditions of incomplete information. This dissertation develops Entropic Dynamics (ED) as a unified framework for deriving dynamical laws directly from principles of inference. Within this approach, probability distributions represent states of knowledge, and their evolution is determined through entropy maximization subject to relevant constraints. This leads to a novel concept of entropic time and a formulation of dynamics as an inferential process. In this talk, I will present how ED provides a common foundation across multiple domains. In physics, quantum dynamics for particles and scalar fields in …

The Ellipsoidal Principal Semi-Axis Geometry Of The Solution To An Ivp For A Matrix Diffusion Pde, Brandon M. Fox

Electronic Theses & Dissertations (2024 - present)

We analyze the underlying geometry of the solution to an IVP for a matrix diffusion PDE. We first derive the fundamental solution to the PDE. We then determine the unique solution to the IVP. From there, we begin analyzing its underlying geometry. We first observe that the geometry exhibits an ellipsoidal nature. Furthermore, we observe that it is described by the principal semi-axis geometry of the ellipsoids associated with the solution. This conclusion follows from applying the Principal Axis Theorem to the associated ellipsoids to establish their principal semi-axis geometry, as governed by the eigenstructure of the matrix. This thesis …

Brst Conserved Moments For Fluids, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

This note adds a missing proof for an assertion in the original paper Gauge Invariance and Repeated Time Isolated Solution Discontinuities.

The assertion is that under weak limits and within a constant energy isosurface, enstrophy is conserved up to losses due to viscous dissipation.

According to the BRST theory for quantum Yang-Mills fields, the conserved moments are exactly those with infinite vacuum expectation values. Application of the BRST theory to fluids is explored. The BRST restriction to the loop expansion (fixed space) is justified by the reqirement that a perturbative expansion can be constructed. In this case the theory predicts …

Weak Solutions Of The Navier-Stokes Equation And Their Euler Limits, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The existence of a weak solution of the incompressible isothermal Navier-Stokes equation with given initial conditions in the Sobolev space $\mathcal{H}_{-2}$ for energy fluctuations and in $\mathcal{H}_{-3}$ for enstrophy fluctuations is assumed. The existence is uniform with respect to the Euler limit of zero viscosity $\nu$. Thus, existence of weak solutions of the Euler equation with given initial conditions is established, and these Euler solutions are the zero viscosity limit of Navier-Stokes solutions, provided the Navier-Stokes solutions exist.