Applied Mathematics Commons^™

The Dynamics Of Educational Change: A Complex Systems Perspective, Preethi Nanjundan, Lijo Thomas, Abith K. Sunil

Northeast Journal of Complex Systems (NEJCS)

The perception of university teachers toward educational reforms plays an important role in determining the success of changes introduced in the education sector. This study focuses on teachers’ attitudes toward change, their emotional responses, and their overall views on educational reforms. Across the world, many educational reforms have failed to achieve their expected outcomes in improving teaching practices and student learning. As education systems are highly complex, the approach toward implementing reforms has also changed over time. Some reforms are introduced gradually, while others involve major innovations within the system. Complexity theory provides useful insights and tools that help educators …

Gamma Belief Functions And Fuzzy Sets And Application To Combining Predictive Models, Liping Liu

University Research

Extending classic finite frameworks to continuous settings, this paper proposes the concept of gamma belief functions and gamma fuzzy sets. It shows that both the combination of gamma belief functions and the intersection of gamma fuzzy sets remain within the gamma family, enabling their application in combining gamma probability judgments in decision making and gamma regression models in ensemble learning. Using both simulated and real datasets, and under both constant and varying dispersion assumptions, experimental results show that the combined gamma regression models closely approximate the reference models learned from the full datasets, aligning with the objectives of bootstrapping. Notably, …

Bias, Structure, And Inference In Applied Network Analysis, Anna Vasenina

Dartmouth College Ph.D Dissertations

This dissertation develops mathematical and statistical methods for extracting reliable information from network data across biological applications, with an emphasis on understanding what observed network structure can and cannot resolve. The first study leverages protein–protein interaction network topology in the c-di-GMP signaling system of Pseudomonas fluorescens, showing that node centrality measures accurately classify protein domain types and that physical interaction structure contributes statistically significant predictive power for biofilm formation phenotypes across nearly 200 environments, while gene expression does not. The second study examines sampling bias in lemur-plant trophic interaction networks in Madagascar, demonstrating that differential detection of diurnal versus …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

The Method Of Periodic Averaging Applied To Reduced Coupled Mode Theory Models For Fiber Laser Amplifiers, Rebecca Nicole Bryant

Dissertations and Theses

Fiber laser amplifier (FLA) models are often implemented without rigorous mathematical justification or thorough numerical validation. Without a proper theoretical basis for assumptions and approximations, or a technical analysis of model performance, there is significant uncertainty about the limitations of any given reduced model and its suitability for an application. This research aims to address the lack of comprehensive assessment of FLA models by directly comparing distinct models and recommending a mathematical alternative to replace heuristic model-reduction techniques. The work in this dissertation is divided into two projects: a comparative study that uses existing FLA models to assess the validity …

Decoding Stock Market Movements: The Role Of Unemployment And Volatility, Bipllab Roy, Suparna Bhattacharjee

Northeast Journal of Complex Systems (NEJCS)

This study investigates how unemployment and market volatility interact with stock prices in the Indian context, framing the stock–labour–volatility nexus as a complex adaptive system (CAS) rather than a set of linear, time-invariant relationships. Using verified secondary data on unemployment, India VIX, and NSE stock indices for 2013–2023, we first apply simple and multiple regression as a descriptive baseline. Results show a strong negative association between unemployment and stock prices (R ≈ 0.824, R² ≈ 0.68, p < 0.05), consistent with Keynesian demand-side channels, while the linear VIX–stock relationship is weak and statistically insignificant (R² ≈ 0.07, p > 0.05), consistent with the expectation that volatility operates through non-linear, regime-dependent mechanisms not captured by OLS.

Importantly, we document and transparently disclose critical …

Mathematical Analysis Of Within-Host Models: Viral–Immune Dynamics, Bifurcations, And Disease Severity, Nazia Afrin

Doctoral Dissertations

My research develops and analyzes ODE-based within-host models at multiple scales. Using dynamical systems theory and numerical methods, I study host–pathogen interactions and immune responses, providing insights into disease dynamics and control. The first model describes the complex dynamics of Hepatitis B virus (HBV) infection and addresses the question: what mechanisms determine whether the infection is cleared during the acute phase or progresses to a chronic state? A key feature of this model is the assumption that all classes of liver cells (uninfected, infected, and protected from reinfection) proliferate at different rates. The findings provide insight into two important aspects …

A Bifurcation Theorem And Its Application To Discrete-Time Models In Ecology And Epidemiology, Jenita Jahangir

Doctoral Dissertations

Matrix models are useful for modeling populations or diseases that involve discrete developmental stages, multiple stages of infection, and interactions among species. To study the coexistence dynamics in matrix models, we extend a bifurcation theorem for resident-invader host-parasitoid type populations by allowing every block of the projection matrix, depending on the bifurcation parameter and the off-diagonal blocks, to be nonzero. As an application, in the first part of the dissertation, we propose a discrete-time host-parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria. …

Strang-Type Exponential Integrators For Stiff Reaction-Diffusion Systems, Saburi Tolulope Rasheed

Doctoral Dissertations

Reaction-diffusion systems, as examples of semilinear parabolic partial differential equations, have played significant roles in the mathematical modeling of physical, chemical, and biological processes. Several reaction-diffusion systems typically do not have exact solutions in closed form, and numerically solving them also comes with challenges due to the presence of the nonlinear local interaction/chemical reaction dynamics representing the reaction term, coupling between components, multidimensionality of the diffusion operator, and stiffness of the diffusion and/or reaction terms. Wederive and analyze several second-order accurate exponential integrators of the Strang type for the time discretization of stiff reaction-diffusion systems. We utilize the finite difference …

Generating Live Heatmaps Of Edr Data Through A Spatiotemporal Weighting, Joel T. Williams, C. Sean Bohun, Alberto Fornaci, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

This paper introduces a novel method for generating live heatmaps of eddy dissipation rate (EDR) data through a spatiotemporal weighting designed to enhance turbulence visualization in aviation. As more flight data become available, approaches relying solely on in-flight EDR measurements have the potential to accurately nowcast and visualize turbulence with low computational cost. The proposed method significantly improves the turbulence visualization capabilities of common commercial aircraft. This is particularly valuable for pilot decision-making and trip planning, enhancing flight safety and operational efficiency. This approach also incorporates an innovative uncertainty threshold, which refrains from predicting when there are insufficient data, thereby …

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 …