Applied Mathematics Commons^™

Toward Conditions For Consensus Of Noisy Bounded-Confidence Models, Madeline Reeve

HMC Senior Theses

Models of opinion dynamics aim to describe the spread of opinions over time in a social network. However, many canonical models are deterministic and thus may fail to capture uncertainty present in social interactions. This work investigates the effect of adding noise into bounded-confidence models, a class of opinion dynamics models where agents are more likely to be influenced by opinions close to their own. In particular, we propose a noisy modification of the Deffuant–Weisbuch bounded-confidence model. We prove that in this model all agents eventually adopt the same opinion—that is, our model converges to consensus. In doing so, we …

Eigenvalue Bound Preservation: Numerical Experiments On The 2d Q-Tensor Flow, Marcel C. Deguzman

Knowledge and Creativity Expo

We study the evolution of nematic liquid crystals in two dimensions using the Q-tensor model, a continuum framework that describes the orientational order of rod-like molecules via symmetric, traceless matrices. Focusing on the Landau-de Gennes energy and its associated gradient flow, we consider a reduced two-dimensional formulation in which the Q-tensor is fully described by two scalar functions. This reduction simplifies the system to a nonlinear, coupled PDE for the scalars, while preserving essential physical features. A key question is whether the eigenvalues of the Q-tensor remain within the physically admissible range under this flow. Building on a theoretical result …

Asymptotic Profiles And Disease Prevalence At The Steady State For An Sis Patch Model, Daozhou Gao, Xin Li

Mathematics and Statistics Faculty Publications

Infected individuals often display mobility patterns that differ significantly from those of healthy individuals-traveling less frequently, covering shorter distances, visiting fewer destinations, and altering their timing and modes of movement. In this paper, to explore the influence of changes in travel frequency and destination on the spatial spread of infectious diseases, we propose a susceptible-infectious-susceptible patch model in which susceptible and infected populations have different dispersal rates and connectivity matrices. We first establish the threshold dynamics in terms of the basic reproduction number R-0 and show the existence and uniqueness of endemic equilibrium (EE) when R-0>1. Then we examine …

Comparative Machine Learning Models For Disease Risk Prediction, Mercy Mawusi Agbley

Theses, Dissertations and Capstones

Accurate prediction of disease outcomes is crucial for improving clinical decision-making and enabling early intervention. This study compares the performance of various statistical and machine learning models for clinical risk prediction using two healthcare datasets: diabetic retinopathy and heart disease. The models assessed include Logistic Regression, LASSO, k-Nearest Neighbors (KNN), Support Vector Machines (SVM), Neural Networks, Random Forests, Gradient Boosting Machines (GBM), and a stacked ensemble model. Prior to modeling, datasets were split into train and test sets. Standardization was applied to numeric features whilst categorical features were one-hot encoded. These transformations were later applied to the test set. Principal …

Detecting And Repairing Conflicting Constraints In Co-Trained Physics-Informed Neural Networks For Composite Curing Processes, Cooper J. Evans

Dissertations, Master's Theses and Master's Reports

Composite materials have become a critical component of modern manufacturing, especially in the automotive and aerospace industries. The curing process for these composites has been modeled using a variety of partial differential equations representing the heat transfer and composite curing kinetics. Optimizing the applied temperature profile is critical for maximizing the efficiency and capacity of composite part manufacturers. Constraints must be placed on the inputs and outputs of the model, including but not limited to, the applied temperature profile, part temperature, and final degree of cure. Conflicting sets of constraints are easy to unknowingly impose due to the highly coupled …

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, …

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 …

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 …

Modeling Landslide-Generated Tsunami For Hazard Analysis: A Case-Study Of Grewingk, Ak, Cassidy J. Deer

All Master's Theses

Landslide-generated tsunamis pose an increasing hazard in glacial and paraglacial environments, where glacial retreat results in slope instability and the potential for mass movement events into a body of water. This thesis analyzes a landslide–generated tsunami at Grewingk Lake in October 1967 by integrating numerical simulations from the D-Claw and GeoClaw models, which are both derived from the Clawpack framework for solving hyperbolic conservation laws. By modeling landslide dynamics and wave propagation in a coupled manner, this study seeks to improve understanding of the hazard, risks and spatial impact at Grewingk. The D-Claw and GeoClaw modeling approaches complement each other …

Mathematicly Rigorous Quantum General Relativity, I: Pure, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A pure quantum general relativity field is one lacking in matter. Such a field has a Lorentzian space-time geometry. A renormalized perturbation expansion, truncated to all finite orders, establishes  the existence of pure quantum general relativity with full mathematical rigor

Quantum General Relativity V2, James Glimm, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

We construct a quantum general relativity theory.. The construction includes specification of a cosmological model. Based on type Ia supernova data, the model has zero dark energy and zero cosmological constant.

The analysis is at the level of theoretical physics, with some mathematical details omitted.

The Pure Yang-Mills Field, \, Iia: The Axioms, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A previously constructed pure Yang-Mills quantum gauge field theory is shown to satisfy all Osterwalder Schrader axioms.

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 …

Mathematicly Rigorous Quantum General Relativity, I: Pure, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A pure general relativity (quantum or classical) is one lacking in matter. Such a field has a Lorentzian space time geometry. A renormalization perturbative expansion, truncated to all finite orders, establishes both the quantum and the general relativity theory with full mathematical rigor.

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

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.

The Boson Yang-Mills Field: The Loop Expansion, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

This paper demonstrates convergence of the loop expansion for Yang-Mills fields.

The loop construction of perturbation theory is based on the axial gauge, ghost states, the BRST framework and the Gribov extension of the Hamiltonian, with the loop expansion perturbatively renormalizable.

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

Two distinct Yang-Mills 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.

The paper has sufficient generality to include existence for all cases that …