Applied Mathematics Commons^™

Potential Pitfalls In Visual Models Of Tipping Points - And How To Fix Them, Jonathan Dechert, Svetlana Gurevich, Stefan Heusler

Northeast Journal of Complex Systems (NEJCS)

Visual models play a crucial role in both science and science communication. However, the distinction between mere analogies and mathematically sound graphical representations is not easy and can be misunderstood not only by laypeople but also within academic literature itself. Moreover, even when the graphical representation exactly corresponds to the mathematical model, its interpretation is often far from obvious. In this paper we discuss the potential landscape visualization commonly used for tipping points in the context of nonlinear dynamics and reveal potential pitfalls, in particular when distinguishing bifurcation induced tipping (B-tipping) from noise-induced tipping (N-tipping).

We propose new visualization techniques …

Project-Based Learning With Odes: Modeling Straw Rocket Motion With Air Resistance, Viktoria Savatorova, Ethan Dyer, Aleksei Talonov

CODEE Journal

This paper presents a hands-on project that guides students through building and validating a mathematical model of projectile motion. The project starts with the idealized case of motion under gravity without air resistance and then introduces air drag : first as a linear force, and then as a nonlinear quadratic force, with the Reynolds number providing the justification for the quadratic model. Students perform experiments with vertical and angled launches, capturing and analyzing motion data using video analysis software. Vertical launch data allows parameter estimation via least squares fitting of the nonlinear drag model, yielding values for initial velocity and …

Faster, Higher, Stronger: Modern Methodologies For The Calibration, Exploration, And Utilization Of Agent-Based Models, David O'Gara

McKelvey School of Engineering Graduate Student Theses & Dissertations

Increases in computing power and availability have led to the widespread use and adoption of agent-based models (ABMs) in the physical, biological, and social sciences. At their core, ABMs represent a novel paradigm for scientific inquiry: their micro-level specification allows for the simulation of the individual components of a system which yields a deeper understanding of dynamic processes such as contagion and adaptive decision-making when used appropriately. However, several hurdles exist when attempting to use ABMs to their full potential. In this thesis, we will explore and address several challenges in designing and utilizing ABMs, with a focus on applying …

Topological And Information-Theoretic Analysis Of Climate-Driven Indonesian Throughflow Dynamics, Sandy H. S. Herho, Katarina E.P. Herho, Iwan P. Anwar, Rusmawan Suwarman

Northeast Journal of Complex Systems (NEJCS)

The Indonesian Throughflow (ITF) represents the sole tropical pathway connecting Pacific and Indian Oceans, yet quantitative understanding of climate mode influences on its variability remains incomplete. We applied information-theoretic and topological frameworks to analyze 34 years (1984-2017) of observational ITF transport data alongside ENSO and IOD indices. Bootstrap analysis revealed pronounced ITF seasonality with 13.28 Sv amplitude peaking in September, contrasting with negligible climate index annual cycles, indicating scale separation in forcing mechanisms. Multi-method extrema detection identified 36-41 extreme events per variable, with 23.1% coincidence between ENSO and IOD high extrema confirming known co-occurrence patterns. Ensemble information-theoretic metrics demonstrated ENSO …

Mathematical Models Of Disease Transmission In Long-Term Care Facilities, Brittany Stephenson, Cara Sulyok

Engineering, Computing and Mathematical Sciences Faculty Conferences

Clostridioides difficile, also known as C.difficile, is a prevalent cause of infectious diarrhea in United States healthcare facilities. Spread through the fecal-oral route and primarily through contact with spores on contaminated surfaces, C. difficile can cause severe diarrhea, stomach pain, and colitis. Most individuals can mount an effective immune response, but older populations, immunocompromised individuals, and those taking antibiotics have an increased risk of being colonized by C. difficile. While extensive research has been conducted in hospital-based settings to improve understanding of the transmission of this bacteria, few studies apply mathematical models in the context of long-term …

Building Countably Many Disjoint Stat Sets In Ω1, Jordan Mills

Summer Undergraduate Research Program (SURP) Symposium

The first uncountable ordinal is ω₁. Here, we will describe the process for partitioning ω1 into countably many disjoint stationary (stat) sets. We first partition ω₁ into 2 disjoint stat sets, and then describe the process for continuing to partition ω₁ into infinitely many disjoint stat sets.

Duality Of Quadrilaterals Via Quaternions, Parker Sjomeling

Summer Undergraduate Research Program (SURP) Symposium

A quaternion is number which is hyper-complex, it has the form,

Where a₁, a₂, a₃, and a₄ are real “scalars”. Then i, j, and k are imaginary numbers which have the following properties:

q = a₁ +a₂i + a₃j + a₃k

Relevant operations on these quaternions include taking the conjugate and the norm (or commonly described as the distance).

The norm of q is equal to:

(a₁)² + (a₂)² + (a₃)² + (a₄)²

The conjugate of q is:

¯q = a …

Simultaneous Application Of Multiple Process Control Rules, Tran B. Ngo

Electronic Theses and Dissertations

Statistical Process Control (SPC) charts are tools used in quality control to monitor and analyze the stability of a process over time. This study evaluates the effectiveness of eight individual Western Electric rules, also known as WECO rules, and the various combinations of these rules with Shewhart rule (or WECO rule 1) to SPC charts. As more rules are added to a process control scheme with Rule 1, there is a trade-off: a higher false out-of-control signal rate but an increase in sensitivity, that is the ability of a specified process control scheme to capture a true out-of-control signal. This …

Mass Effects On Energy Transfer Paths In Nonlinear Vibrating Systems, Manal Mustafa

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

This dissertation examines the role of mass in nonlinear systems, uncovering its role in enabling passive energy redistribution and robust vibration control in both idealized and real-world structures. Focusing on a strongly nonlinear two-degree-of-freedom system, it investigates how changes in mass ratio influence the dynamics of energy transfer, nonlinear normal modes (NNMs), and dissipation behavior.

A number of significant contributions are introduced in this work beginning with the introduction of the frequency-energy-peaks (FE-pks) plot, a novel tool that visualizes how energy flows through the system, revealing transient resonance orbits, internal resonance effects, and effectively capturing the different nonlinear phenomena with …

Simultaneous Selection Of Inflations And Variables In Multiple Inflations Poisson Model (Mip), John Koomson

Open Access Theses & Dissertations

Count data frequently arise in biomedical, economic, and social science research and are often characterized by structural excesses at specific count levels. To accommodate such patterns, Su et al. (2013), among others, introduced the Multiple-Inflation Poisson (MIP) model, which allows for multiple inflated counts within the distribution. However, two critical challenges remain in modeling such data: (i) identifying the true inflation points where excess counts occur, and (ii) selecting the relevant covariates that explain variation in the inflation and count process. This dissertation addresses these issues by advancing the MIP model through a novel methodology that enables the simultaneous selection …

Accurate Temporal Integration Schemes For Nonlinear Adsorption Problems, Evan D. Butterworth

All Dissertations

We consider a nonlinear transport problem to model the chromatography process of high-capacity multimodal membranes. Robust and efficient algorithms that simulate these bioseparation processes are critical to developing therapeutics for various chronic illnesses and infectious diseases. However, much of the current methodology focuses on stabilization and linearization techniques, often implementing low-order time-discretizations and linearized adsorption, resulting in inefficiencies and inaccuracies in the numerical solution. Utilizing Rothe's method, we develop various time-discretization schemes coupled with the finite element method to solve the fully implicit problems. Stability and solvability results are presented for several methods. Through multiple high-level software implementations paired with …

Advancing Real-World Implementation Of The Well Optimized Linear Finder (Wolf) High-Speed Atmospheric Turbulence Compensation Method, Timothy Evan Coon

Theses and Dissertations

This dissertation advances the real-world implementation of the Well Optimized Linear Finder (WOLF) method for high-speed Atmospheric Turbulence Compensation (ATC). Atmospheric turbulence introduces phase aberrations into optical wavefronts and degrades image quality in terrestrial imaging systems. Traditional phase diversity methods are computationally intensive and poorly suited to real-time operation. The WOLF method addresses these limitations through a novel, point-wise formulation of the optical transfer function (OTF) as a structured autocorrelation of the generalized pupil function (GPF). This formulation enables the estimation of phase aberrations at individual spatial coordinates with distributed computational complexity.

The research begins by developing a MATLAB-based simulation …

Analysis Of Multi Grade Deep Learning, Ronglong Fang

Mathematics & Statistics Theses & Dissertations

Multi-Grade Deep Learning (MGDL) is a training framework that incrementally builds deep neural networks. It does this by dividing the training process into multiple “grades,” where each grade sequentially trains a shallow neural network to learn the residue from the previous one, using the outputs of prior grades as input. This approach progresses from shallow to deep architectures. This dissertation offers a comprehensive theoretical and numerical analysis of the MGDL methodology.

We first demonstrate that MGDL can effectively learn target functions within the sum-composition learning format. In this context, MGDL approximates high-frequency components by composing multiple low-frequency functions. This unique …

Cami: A Counselor Agent Supporting Motivational Interviewing Through State Inference And Topic Exploration, Yizhe Yang, Palakorn Achananuparp, Heyan Huang, Jing Jiang, Phey Ling Kit, Nicholas Gabriel Lim, Cameron Shi Ern Tan, Ee-Peng Lim

Research Collection School Of Computing and Information Systems

Conversational counselor agents have become essential tools for addressing the rising demand for scalable and accessible mental health support. This paper introduces CAMI, a novel automated counselor agent grounded in Motivational Interviewing (MI) – a client-centered counseling approach designed to address ambivalence and facilitate behavior change. CAMI employs a novel STAR framework, consisting of client’s state inference, motivation topic exploration, and response generation modules, leveraging large language models (LLMs). These components work together to evoke change talk, aligning with MI principles and improving counseling outcomes for diverse clients. We evaluate CAMI’s performance through both automated and expert evaluations, utilizing simulated …

Two Analytic Issues Arising From Models In Probability And Materials Science, Giang Vu Thanh Nguyen

Mathematics & Statistics Theses & Dissertations

This dissertation explores two distinct topics centered on mathematical models in probability theory and materials science. The first part investigates a series of functions derived from an adaptive algorithm designed to address the score-based secretary problem, a classic challenge in probability theory. This problem involves making immediate decisions to select the best candidate from a sequence of interviews. The algorithm aims to maximize the probability of selecting the optimal candidate based on observed scores. We prove two fundamental analytic properties of this sequence of functions as a theoretic support of the algorithm: first, the functions in the sequence each possess …

A Strict Physicality-Preserving Scheme For A 2d Q-Tensor Flow With A Singular Potential, Md Mashud Parvez

Mathematics & Statistics Theses & Dissertations

Nematic liquid crystals are a state of matter that exhibit properties between those of conventional liquids and solid crystals. Their unique ability to align molecules in specific directions makes them essential in various applications, including display technologies and advanced materials. To model their complex behavior, mathematical frameworks such as the Q-tensor model are used to describe the orientation and degree of molecular order. In this work, we introduce a numerical scheme for a two-dimensional (2D) dynamic Q-tensor model, which is formulated as an L²-gradient flow driven by the liquid crystal free energy and incorporates a singular potential to …

Algebraic Multigrid Methods For Nonsymmetric And Indefinite Problems: Theory And Applications, Ahsan Ali

Mathematics & Statistics ETDs

Algebraic multigrid (AMG) is a well-established and highly efficient solver for symmetric positive definite (SPD) systems arising from elliptic and parabolic PDEs, while nonsymmetric systems from hyperbolic PDEs remain a significant challenge. This dissertation develops AMG methods and theory for nonsymmetric problems. First, we develop a novel approach combining mode constraints from energy-minimization AMG with local approximations of ideal restriction in $\ell$AIR, resulting in constrained $\ell$AIR (C$\ell$AIR), which demonstrates scalable convergence across advective and diffusive problems. Second, we extend optimal AMG theory by deriving spectral radius estimates for the two-grid error transfer operator using matrix-induced orthogonality, enabling convergence predictions for …

Minimal Error Functions On Irregular Subsets Of The Real Line, Robert Michael Dukes

Mathematics & Statistics ETDs

Chebyshev Polynomials, those that minimize the maximal error on a compact set, are one of the most practical tools for approximating smooth functions. The classical results are on the set [-1, 1]; in this paper, we extend to more complicated subsets of the real line. We demonstrate some classical results and then take the result from [2] on regular Parreau-Widom Sets and extend it to semi-regular sets, defined as sets whose regular part is closed. We introduce the Regularity Coefficient as a series formed by evaluating the Green’s Function at irregular points. This new machinery is applied to the lower …

Derivation Of Adjoint Based Error Estimates For Nonlinear Ordinary Differential Equations With Application To Multistage Sir Models With Demographics, Daniel Alcala

Mathematics & Statistics ETDs

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error …

Toward Simulating 2d Cell Surfaces In A Disk, Myriam Allred

Mathematics & Statistics ETDs

Certain evolution models of cell surfaces (treated in two-dimensions) involve the solution of the Helmholtz equation with jump conditions enforced on an immersed closed curve. This thesis presents a sparse, modal spectral method for solving such Helmholtz problems. The solution is required to be continuous across the curve, but with a jump discontinuity in the normal derivative proportional to the planar curvature. The method relies on classical Fourier-Chebyshev basis functions, with the application of modal Chebyshev integration matrices to achieve sparse, banded approximations of the Helmholtz equation. The method achieves spectral convergence, despite the inherent low regularity of the relevant …

Correlated Sampling Of Parallel Partial And Linked Emulators For Geophysical Applications, Joseph Lyon

Dissertations (1934 -)

Computer models, or simulators, are mathematical representations of real world phenomena that, to run at new input settings, are often computationally intensive and prohibitively slow. Surrogate models, or emulators, provide a method of rapidly predicting simulator outputs with uncertainty at untested configurations by treating the computer model output as a single realization of a stochastic process, specifically a Gaussian Process (GP).In this dissertation, we consider three projects: Correlated Linked GP Emulation, Spatially Correlated Parallel Partial Emulation, and Optimizing the Zero-censored Gaussian Process with Spatially Correlated Sampling. First, the Correlated Linked Emulator, focuses on developing a model for the covariance of …

Quasistatic Peridynamics, Existence Of Unique Solution In The Presence Of Damage, Nuwanthi N. Samarawickrama

LSU Doctoral Dissertations

A mathematical model for damage propagation based on nonlocal potentials is developed within the framework of peridynamics. This model is applied to simulate damage evolution in cyclically loaded structures. By neglecting inertial effects, a well-posed quasistatic formulation for cyclic loading is obtained.\\ The resulting equation is expressed as a nonlocal and nonlinear integral operator that couples damage evolution to the deformation field.\\ This coupling occurs through the product of a damage factor and the derivative of a force potential. The damage factor ranges between zero and one, where one represents undamaged material and zero indicates complete damage.\\ It serves to …

Theoretical And Experimental Investigation Of Liquid-Liquid Phase Separation: Characterizing Elastin-Like-Polypeptides, Adam D. Quintana

Chemical and Biological Engineering ETDs

This dissertation develops and validates a semi-empirical Flory–Huggins-based interaction model, combined with Cahn–Hilliard simulations, for predicting multi-component liquid–liquid phase separation (LLPS) in elastin-like polypeptide (ELP) systems. Equilibrium droplet compositions, measured using a PDMS-based microfluidic device, enabled direct parameterization of interaction coefficients. The model was applied to generate phase diagrams and assess composition dependence in ternary mixtures. Cahn–Hilliard simulations were conducted to explore potential phase morphologies under different interfacial conditions. Multi-component Lattice Boltzmann simulations were implemented to model droplet morphology evolution under varying interfacial and diffusive parameters, reproducing experimentally relevant morphologies. A three-phase wetting study revealed conditions for selective wetting and …

Gain Threshold Optimization Using Fano Resonance, Alina Oktiabrskaia

LSU Doctoral Dissertations

The study of resonances in electromagnetics plays a critical role in the design of optical systems. This dissertation investigates the interaction between resonance and gain in optical structures to establish a universal principle for achieving ultra-low-threshold lasing. Through the analysis of geometric symmetries, material properties, and coupling mechanisms, this research develops prototype structures applicable to a wide range of optical and electromagnetic systems. A range of models is considered, starting from a simple onedimensional string-resonator system (based on the model of H. Lamb), then advancing to two- and three-dimensional waveguide models, and culminating with a realistic high-contrast model in open …

Modifications To The Spiral Array: A Computational Approach To Music Analysis, Rose Bittle

DePaul Discoveries

The Spiral Array is a geometric model of musical tonality and exists as a tool in computer-aided music analysis. The model was first published in 2000 by Elaine Chew, PhD, in her thesis, Towards a Mathematical Model of Tonality. This project aimed to restructure the Spiral Array, limiting user ambiguity and optimizing the application of musical key finding. The existing model defines pitch, chord, and key location to form a series of spirals dependent on a set of flexible weights. In our research we were able to identify potential issues with the pitch definitions themselves and experiment with methods …

Modeling, Analysis, And Prediction Of Covid-19 Dynamics With Interacting Subpopulations And Implicit Behavior Using Physics-Informed Neural Networks, Naima Aubry-Romero, Alonso Ogueda-Oliva, Padmanabhan Seshaiyer

Spora: A Journal of Biomathematics

In this paper, we consider an extended SEIR compartmental model that incorporates young and old interacting subpopulations, allowing for cross-group transmission dynamics. Implicit behavioral changes are included to determine the influence of social behavior on coronavirus transmission dynamics. The basic reproduction number, the average number of secondary cases of infection produced by a single primary case, is derived for both the explicit and implicit model using the next-generation matrix method. We solve the associated differential equation systems and estimate useful parameters in the explicit model using physics-informed neural networks (PINNs). Our results point to how the PINNs approach offers an …

Statistical Power To Detect Simultaneous Violation Of Proportionality In Hazards And Additive Assumption In Cox Regression Model, Lawrence Mensah Agbota

Theses and Dissertations

The Cox proportional hazards regression model is a widely employed semi-parametric tool in epidemiological and medical research for analyzing time-to-event data and assessing the relationship between patient survival times and one or more predictors. This method involves regression analyses necessitating a meticulous approach to careful examination of the covariates and the relationship among covariates included in the model through a series of critical decisions and steps. Violation of the additivity of the effects and the proportionality in hazards (PH) assumption can lead to biased results and misleading scientific findings. We conducted a Monte Carlo simulation study to assess the performance …

Predicting Enzyme-Substrate Association Using Heterogeneous Knowledge Graph, Jannatul Ferdaus

Theses and Dissertations

Phosphorylation and dephosphorylation are dynamic processes that control many aspects of cellular activity, such as metabolic pathways, cell cycle progression, and signal transduction. Protein activity and interactions are modulated by the reversible addition or removal of phosphate groups, which allows cells to react abruptly to evolving conditions. Although kinase-specific phosphorylation site prediction has advanced, phosphatase-specific dephosphorylation site computational prediction is still a major obstacle that prevents us from fully comprehending the extent of cellular regulation. In this study, we constructed a knowledge graph for the prediction of enzymes (kinases and phosphatases) and their associated substrates with specific phosphosites. As part …

Sequential Data Modeling Of Influenza A Via Traditional, Dwt-Gpr Hybrid, And Deep Learning Architectures, Edmund Fosu Agyemang

Theses and Dissertations

Influenza A is responsible for 290,000 to 650,000 respiratory deaths a year, though this estimate is an improvement from years past due to improved sanitation, healthcare practices, and vaccination programs. In this study, we perform a comparative analysis of traditional, deep-learning and discrete wavelet (DWT)-Gaussian Process (GP) hybrid models to predict Influenza A outbreaks. Using historical data from January 2009 to December 2023, we compared the performance of traditional ARIMA and ETS models, four variants of DWT-GPR models and six distinct deep learning architectures: Simple RNN, LSTM, GRU, BiLSTM, BiGRU and Transformer. The results reveal a clear superiority of all …

Fuzzy Logic-Based Decision Support System For Adoption Of Industry 4.0 Predictive Maintenance By Manufacturing Industries, Fred E. Peter

Tanzania Journal of Engineering and Technology (TJET)

In the context of Industry 4.0, predictive maintenance enhances operational efficiency by optimizing processes, minimizing downtime, and improving cost-effectiveness. However, implementing predictive maintenance requires a systematic approach due to its complexity. This study collected expert input from 15 food and beverage manufacturing industries located in Dar es Salaam, Tanzania, using a purposive sampling technique. Six representatives were selected from each industry, and their opinions were analyzed using MATLAB 7.6 through a fuzzy logic inference system. The analysis focused on key factors influencing Industry 4.0 technology adoption for predictive maintenance, including adoption intention (strategic decision, equipment data, perceived benefit) and perceived …