Applied Mathematics Commons^™

Inverse Modeling The Geo-Spread Of Covid-19 In Brazil, Cameron Keith Mills

ETDs from 2020-2029

In 2021 Fitzgibbon, Morgan, Webb, and Wu used a modified SEIR (susceptible, exposed, infected, and recovered) model to predict how COVID-19 spread through Brazil [12]. For their model, six constant coefficients were used that were fitted, referenced, or assumed. In this thesis, we geo-spatially modify their SEIR model and formulate an inverse problem to recover the now spatial coefficients of the model. We first show there exists a unique solution to the modified model. To solve the inverse problem, we modify an inverse method [17] that focused on minimizing convex functionals. These recovered spatial coefficients can be used with Matlab’s …

Investigating An Energy-Preserving Spectral Finite Difference Numerical Method For A Wave Equation On The Metric Graph, Tue Minh Tran

University Honors Theses

To investigate the accuracy and long-term energy conservation of a spectral finite difference numerical method for a wave equation on metric graphs. In conservative systems, numerical methods should preserve total energy. However, explicit finite difference methods require impractically small space steps and exhibit energy drift at end points. To address these limitations, a spectral finite difference method is implemented using a Fourier transformation. This semi-spectral method improves stability at endpoints while maintaining second-order accuracy, achieving an overall error of O(∆t²). We implement the semi-spectral method on the IEEE14 metric graph and provide visuals showing the initial condition …

Geometric Characterization Of Ideals In Bipolar Semigroups, Kittipong Laipaporn, Rasimate Maungchang, David M. Cook, Prathomjit Khachorncharoenkul

Research outputs 2022 to 2026

This paper develops a geometric framework for analyzing the ideal structure of the bipolar semigroup ��={(−��,��)∣��,��∈ℝ+0} under coordinate-wise addition. Subsets of B are interpreted as planar regions, allowing ideals to be described in terms of boundary behavior. In particular, we prove that the complement of a simply connected region is an ideal of the commutative additive semigroup (��,+) if and only if its boundary contains no strictly decreasing segment. This provides a direct and visually verifiable criterion for ideality, linking algebraic structure to geometric shape. Each ideal can be written as a union of translates of the form ��+��, with …

Developing A Framework For Mooc Dropout: The Role Of Utilitarian And Hedonic Values In Student Retention, Bipllab Roy, Mave D'Souza, Madhura Laghane

Northeast Journal of Complex Systems (NEJCS)

Massive Open Online Courses (MOOCs) have expanded access to higher education but continue to face persistently high dropout rates, raising concerns about their long‑term effectiveness and sustainability. This study develops and empirically tests a structural framework that links utilitarian values (perceived usefulness, certificate value, time flexibility), hedonic values (enjoyment, variety and novelty, personal interest alignment), and individual characteristics (goal orientation, self‑efficacy, motivation type) to MOOC student retention. Data were collected through a structured questionnaire administered to 200 MOOC learners from Christ University, Lavasa Campus, and analyzed using Structural Equation Modelling (SEM) in AMOS. The results show that goal orientation, self‑efficacy …

Modeling The Impact Of Treatment During An Ongoing Tuberculosis Outbreak, Bach Le, Anna Robicheaux, Jude Shive, Alana Wells, Erin N. Bodine

Spora: A Journal of Biomathematics

In response to a significant tuberculosis outbreak in Wyandotte and Johnson Counties, Kansas, this study presents a compartmental model of ordinary differential equations to evaluate the impact of standard antibiotic treatment. The model incorporates latent, active, and treated disease states. Parameter values were informed by epidemiological data and uncertain parameter value ranges were explored systematically through uncertainty analysis using constrained Latin hypercube sampling. Cumulative infections and deaths, and the basic reproduction number, were computed over a five-year simulation period. Sensitivity analyses using partial rank correlation coefficients identified symptomatic treatment rate and transmission rate as primary drivers of cumulative infections and …

Predicting The Outcome Of Ischemic Hepatitis With Real-Patient Data Using Machine Learning Tools, Christiana Beard, Madison Utterback, Olcay Akman, Priya Kohli, William M. Lee, Aditi Ghosh

Spora: A Journal of Biomathematics

Ischemic hepatitis (IH) results from shock-related conditions that impair oxygenated blood flow to the liver, causing hepatocyte death. Diagnosis relies largely on clinical history due to the absence of specific diagnostic tests and limited ability to predict outcomes. This study applies machine learning methods to real-world IH patient data to improve outcome prediction. Biomedical indicators analyzed include creatinine, international normalized ratio (INR), aspartate aminotransferase (AST), alanine transaminase (ALT), and bilirubin. Data were collected from multiple U.S. centers through the Acute Liver Failure Study Group (ALFSG), a multicenter network focused on this rare condition. We implemented logistic regression, regression tree methods …

Traveling Wave Fronts For Sir Epidemic Models With Nonlocal Dispersal And Delayed Effects, Ashley Evette Embry

Theses and Dissertations

This dissertation investigates traveling wave solutions for two classes of delayed nonlocal dispersal susceptible--infected--recovered (SIR) epidemic models incorporating biologically realistic mechanisms such as delayed infectivity, delayed dispersal, nonlocal transmission, demographic turnover, and nonlinear incidence effects. These models extend classical spatial epidemic frameworks by allowing long-range population movement through nonlocal dispersal operators and incorporating temporal memory into both diffusion and transmission processes. The primary objective is to establish the existence of traveling wave solutions connecting disease-free equilibria to endemic states and to characterize threshold conditions governing epidemic propagation. The simultaneous presence of nonlocal dispersal, multiple delays, and non-monotone nonlinear incidence terms …

Distributed Self-Control Of Dynamical Networks By Adaptive Link Weight Adjustments, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

Conventional control theory considers controlling the behavior of a dynamical system toward a desired state by injecting externally designed inputs into the system. Meanwhile, most complex systems exhibit self-organization through local information exchanges among individual dynamical components that are embedded within a complex network of interactions. The self-organizing dynamics of those systems are realized in a highly distributed manner using locally available information only, and therefore, their behaviors have not been discussed much from a control theoretic viewpoint. Meanwhile, adaptive networks, i.e., dynamical networks whose states and topologies coevolve at similar time scales, can offer a promising theoretical framework in …

Exploring Maximal Length Cellular Automata To Generate Primitive Polynomials In Gf(2), Sumit Adak, Subhrajit Deb, Anurag Ghosh, Angshuman Roy, Souvik Roy

Northeast Journal of Complex Systems (NEJCS)

We present a simple method that uses cellular automata (CAs) to find primitive polynomials over GF(2). We used maximal length CAs as tools to generate primitive polynomials. It is usually very difficult to find maximal length CAs or primitive polynomials since they require exponential time, and there is no linear time method. However, in our work, given an n-size specific sequence of CA with reasonable probability, our technique computes a cycle of length at most 2^n-1 (maximal length) in O(n) time. The characteristic polynomials of synthesized maximal length CAs are claimed to be primitive since it was previously established that …

Evaluating Optimal Capacity And Investment Strategies For Renewable Energy Projects: A Combined Technical And Financial Approach, Helen Josephine, Indhumathi Shanmugasundaram, Sharad Gupta, Manjari Sharma

Northeast Journal of Complex Systems (NEJCS)

The global shift toward clean energy is accelerating, and by 2050 renewable sources are expected to supply more than 85% of the world’s electricity. This transition, however, introduces new layers of complexity. Wind and solar energy behave as interconnected subsystems whose output fluctuates with weather, season, and geography. Their interaction with fixed hourly demand, capital-intensive investments, and financing structures creates a multi-dimensional system in which small changes can trigger significant operational and economic consequences.  This study presents a simulation-driven framework designed to understand and optimize this complex behaviour. The framework models hourly wind and solar generation alongside projected demand to …

Decision Making At Triage Classification Using Svm With Smote Technique, Mehanas Shahul, Pushpalatha Kp

Northeast Journal of Complex Systems (NEJCS)

The efficient functioning of triage gates in overcrowded emergency departments (EDs) occurs in the context of the complex adaptive system (CAS) framework, where diverse system elements – patients, medical personnel, resources, patients’ inflow patterns, and patients themselves – simultaneously and dynamically influence the decision process. This study addresses the automated incorporation of machine learning triage algorithms as part of the system triage process to support automated classified risk-level recognition based on a limited set of vital signs. Patients are dynamically subsumed under high and low-risk categories enhanced by sensitivity, which enables optimal diagnosis and triage response to the critical clinician …

Uncovering Discrete States From Multimodal Psychophysiological Data Using Gaussian Latent Dirichlet Allocation (Glda), Congyu Wu, Aaron Fisher, David Schnyer

Northeast Journal of Complex Systems (NEJCS)

In this article we explore and validate the utility of an unsupervised probabilistic model, Gaussian Latent Dirichlet Allocation (GLDA), for discovering discrete states from repeated, multimodal psychophysiological samples collected from multiple individuals. Psychology and medical research heavily involves measuring potentially related but individually inconclusive variables from a cohort of participants to derive diagnosis, necessitating clustering analysis for state identification. Traditional probabilistic clustering models such as Gaussian Mixture Model (GMM) assume a global mixture of component distributions, which may not be realistic for observations from different patients. The GLDA model borrows the individual-specific mixture structure from a popular topic model Latent …

Asymmetric Opinion Formation Of Emotional Excitable Agents, Irene Ferri, Emanuele Cozzo, Aleix Nicolás-Olivé, Albert Díaz-Guilera, Luce Prignano

Northeast Journal of Complex Systems (NEJCS)

The bounded confidence model represents a widely adopted framework for modeling opinion dynamics wherein actors have a continuous-valued opinion and interact and approach their positions in the opinion space only if their opinions are within a specified confidence threshold. Here, we propose a novel framework where the confidence bound is determined by a decreasing function of their emotional arousal, an additional independent variable distinct from the opinion value. Additionally, our framework accounts for agents' ability to broadcast messages, with interactions influencing the timing of each other's message emissions. Our findings underscore the significant role of synchronization in shaping consensus formation. …

Conditional Product Sampling For Gaussian Process Implicit Surfaces, Song Shi

Dartmouth College Master’s Theses

Gaussian Process Implicit Surfaces (GPISes) provide a powerful and unified stochastic geometry representation for rendering surfaces, volumes, and the rich continuum between them. Recent work has shown that GPISes can model a broad space of visual appearances under a unified light transport framework. However, practical rendering with GPISes remains challenging: existing estimators can become inefficient for particular correlation structures, and highly anisotropic or heightfield-like GPISes require specialized treatment to obtain robust variance reduction.

This thesis extends recent work on GPIS rendering by introducing a new next-event estimation (NEE) technique for anisotropic GPISes.We show that standard NEE provides diminishing benefits as …

Machine Learning For Modeling In An Elementary Differential Equations Class, Nathan Albin, Andrew G. Bennett, Abhinav Chand

CODEE Journal

Mixing machine learning with modeling is an area of increasing importance. This paper presents a lesson where students model a spring-mass system both using traditional analysis with linear damping and using machine learning to learn the damping from real data. The machine learning is implemented in a Jupyter notebook hosted on Google Colab, allowing students to train the neural network without requiring the students to carry out coding. Students get experience with how machine learning can fail, how it can work, and the time and data requirements for machine learning to succeed, and are asked to apply this knowledge to …

Geometric Structure In High-Dimensional Representations: Theory And Applications To Language, Jiayi Chen

Dartmouth College Ph.D Dissertations

This thesis develops a geometric perspective on high-dimensional representations, motivated by applications to language. Rather than treating representations solely as inputs to predictive models, we view them as structured objects whose geometry encodes meaningful information. In particular, we argue that such representations exhibit organization at multiple scales: at a global level, metric and clustering structure capture relationships such as genre, authorship, and discourse; at a local level, geometric quantities such as intrinsic dimension and curvature describe how these relationships vary across the space.

To study these phenomena, we combine empirical analysis with theoretical development. On the empirical side, we examine …

An Analytical Framework For Quantifying Urban And Community Resilience To Natural Hazards From Cell-Phone Gps-Location And Traffic-Flow Data, Georgios Chatzikyriakidis

Civil and Environmental Engineering Theses and Dissertations

Urban areas are increasingly exposed to natural hazards while accommodating a growing share of the global population, yet a consistent science-based framework for quantifying urban and community resilience remains lacking. This dissertation develops a physics-based analytical framework grounded in statistical mechanics and the quantitative theory of Brownian motion. A city is conceptualized as a complex medium in which citizens move analogously to Brownian particles within a viscoelastic environment, influenced by socioeconomic interactions and infrastructure functionality.

A central premise is that urban resilience, interpreted as engineering resilience (an outcome), can be quantified through a single metric: the mean-square displacement MSD=⟨r²(t)⟩, of …

Stochastic Derivative-Free Deep Learning Methods For Solving High Dimensional Partial Differential Equations, Qing He Mr.

Mathematics Theses and Dissertations

Solving high-dimensional partial differential equations (PDEs) is a fundamental challenge in scientific computing, with applications ranging from quantum chemistry and computational finance to statistical physics and stochastic optimal control.  Classical numerical methods such as finite element or finite difference schemes suffer from the curse of dimensionality, rendering them computationally infeasible when the dimension $d$ exceeds a handful. Physics-informed neural network (PINN) methods alleviate this by embedding the PDE residual directly into a loss function, but they require computing derivatives of the network with respect to its spatial inputs---an operation that scales poorly in high dimensions and demands that the approximate …

Bayesian Designs For Two-Arm Clinical Trials With Time-To-Event Endpoints: Incorporating Historical Data Through Power Priors, Sara Hajraf H. Almutiri

Mathematics & Statistics ETDs

Bayesian methods provide a flexible framework for time-to-event analysis by incorporating prior information. The power prior offers a systematic way to borrow information from historical data. This approach is especially valuable in clinical research, where historical data can enhance inference in early-phase trials with limited sample sizes. This dissertation develops Bayesian approaches for two-arm survival studies using both closed-form and simulation-based methods. The closed-form inference is derived under exponential and Weibull survival models. Under the proportional hazards framework, the posterior is derived through a normal approximation to the log hazard ratio, allowing inference on the treatment effect when the variance …

Welcome Tilly Norwood: Forecasting Hollywood’S Ai Policy Futures, Samuel P. Rooker

Senior Honors Projects, 2020-current

In late 2025, weekly trade publication Variety Magazine reported on the announcement of a new acting talent in Hollywood: Tilly Norwood. Norwood is an industry outsider and the pet project of Eline Van der Velden, who unveiled the actress’ existence to the world at the Zurich Film Festival. The announcement quickly gained media coverage while Van der Velden has since faced cyclical backlash from Hollywood trade unions, which does not seem entirely without reason. Tilly Norwood is a digital persona, a generative artificial intelligence (GenAI) program, designed by Van der Velden’s novel AI talent studio, Xicoia, to become the next …