Applied Mathematics Commons^™

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 …

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.

Global Weak Solutions Of Optical Variational Wave System, Shahrazad Hamed Alnafie

Graduate Theses, Dissertations, and Problem Reports (ETD)

The coupling of a variational wave equation with Maxwell’s equations gives rise to the optical variational wave system, a hyperbolic PDE system that models the director field of the nematic liquid crystals. This system presents unique analytical challenges that have not been addressed in the existing literature. In this dissertation, we study the one-dimensional case of this system.

We establish the global existence of conservative weak solutions to the associated Cauchy problem. The hyperbolic system is derived using the energy variational method. Through a sequence of suitable coordinate transformations, we reformulate the system while preserving its hyperbolicity. Due to degeneracy …

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 …

Type Ii Diabetes Treatment Comparison Via Compartment Modeling, Abigail M. Collins

Theses and Dissertations

Type II diabetes mellitus affects one in ten adults worldwide, yet the effects of treatment type and adherence level on developing complications and quality of life have not been well characterized at the population level, and mathematical modeling offers a structured way to examine these dynamics. This thesis adapts the Boutayeb et al. (2004) model to incorporate dynamic treatment types and levels of adherence, producing nine scenarios in which complication development rate and complication recovery rate differed, to compare peak complications and quality of life across treatment and adherence conditions. Using a system of ordinary differential equations and compartment modeling, …

Dataless Neural Networks For Boolean Satisfiability And Network Optimization, Andrew Evan Gautier

Graduate Theses, Dissertations, and Problem Reports (ETD)

Combinatorial optimization problems (COPs) require searching over a finite solution space subject to constraints, with the goal of satisfying an objective function. They arise in operations research, scheduling, resource allocation, circuit design, and many other fields. Many problems in combinatorial optimization (including satisfiability and network design) are NP-hard. Traditionally, researchers have built approximate solvers that return near- optimal solutions efficiently by developing increasingly sophisticated heuristics and meta- heuristics. Deep learning has provided new opportunities for improving combinatorial solvers by leveraging neural guidance to prune the search space. Traditional neural networks have distinct drawbacks in this context: separate training and …

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 …

Applications Of Machine Learning For Evaluating Downward-Coupled Stratosphere-Troposphere Interactions And Subseasonal Forecasts Of Opportunity, Elena M. Fernandez

Electronic Theses & Dissertations (2024 - present)

Wintertime stratospheric dynamics provide key information for understanding atmospheric teleconnections and improving subseasonal-to-seasonal (S2S) predictions on timescales of two weeks to two months. Periods of enhanced predictability, often referred to as forecasts of opportunity, arise from large-scale teleconnected variability, within which the stratosphere serves as an important precursor for tropospheric states, such as near-surface temperatures. While traditional diagnostics of downward coupled stratosphere-troposphere interactions typically rely on zonal-mean representations of wind and geopotential height, this dissertation presents an alternative vortex-centric framework through metrics that capture the daily geometric and dynamical evolution of the stratospheric polar vortex. The proposed stratospheric …

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 …

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 …

Machine Learning-Based Intrusion Detection System For Iot Networks Using The Rt-Iot 2022 Dataset, Bukunmi Ebenezer Afolabi

Theses, Dissertations and Capstones

The rapid expansion of the Internet of Things (IoT) has transformed modern computing by enabling seamless connectivity among heterogeneous devices across diverse application domains. However, this increased interconnectivity has significantly enlarged the attack surface of IoT networks, exposing them to a wide range of sophisticated cyber threats. Conventional security mechanisms often lack the capability to detect emerging attacks in real time, thereby necessitating the development of intelligent Intrusion Detection Systems (IDS) capable of accurately identifying malicious network activities. This study developed and evaluated a machine learning-based intrusion detection framework for multiclass IoT attack detection using the RT-IoT2022 dataset. The dataset …

Quantum Mechanics As A Framework For Data Assimilation And Its Application To Atmospheric Parameterization, David Freeman

Dartmouth College Ph.D Dissertations

Quantum mechanics, as a mathematical system, can be understood as a generalization of classical probability theory. Quantum Mechanical Data Assimilation (QMDA) is a method in which classical dynamical systems are embedded into a quantum mechanical setting, with an associated data assimilation scheme leveraging the operator algebraic setting. In this dissertation, the algebraic structure underlying the operator theoretic formulation of QMDA is discussed. A procedure for closure of dynamical systems based on QMDA, known as Quantum Mechanical Closure (QMCl), is then constructed, and the procedures for constructing the quantum embeddings and implementing QMCl in practice are laid out and implemented for …

The Yang-Mills Field: The Fermion Sign Problem, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A mathematical resolution of the Fermion sign problem is accomplished based on the theory of homology groups and thimbles. The original contribution of this paper is the mathematicly rigorous convergence of residual states. The residual states remain after use of a rigorous Picard-Lefschetz saddle point optimization.

The proofs depend on an assumed principle of a maximum rate of entropy production.

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

The Pure Yang-Mills Field: The Mass Gap; Axioms Minus One, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A strictly positive gap at low temperatures is demonstrated in the energy spectrum of the previously constructed pure Yang-Mills quantum gauge field theories. Axioms other than the cluster axiom are verified.

General Relativity: Existence, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A mathematically rigorous renormalized perturbative expansion, truncated to all finite orders, establishes the existence of quantum general relativity theories with Fermion matter fields. The expansion is initialized with the selection of a vacuum state.

Distinct infrared (IR) and ultraviolet (UV) vacuum states are considered. The IR vacuum state defines a cosmology model. The UV vacuum state defines a black hole model.

The vacuum state is initialized with a $\mathfrak{o}^3$ gauge Lie algebra defining a quark-gluon plasma. With this choice, string theory is avoided.

Finite order renormalized perturbation theory, defined using Bosons to avoid Fermion sign cancellation, is convergent to all …

The Pure Yang-Mills Field: I: Fixed Time Existence, James Glimm, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The convergence of renormalized perturbation theory to all finite orders is defined and shown to be valid for the fixed time perturbation theory of a pure Yang-Mills field.

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 depends on an assumed principle of a maximum rate of entropy production.

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 …

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 …