Applied Mathematics Commons^™

Some New Three-Term Conjugate Gradient Methods For Riemannian Optimization With Application To The Gough-Stewart Platform, Nasiru Salihu, Poom Kumam, Lin Wang, Sani Salisu

Mathematical Modelling and Numerical Simulation with Applications

Three-term conjugate gradient (TTCG) methods have been extensively studied for optimization within Euclidean geometry, with some inherently satisfying the sufficient descent property, thus enhancing their theoretical superiority. However, other TTCG methods have been developed that incorporate restarting the search direction, simplify the steepest descent approach, or rely on the convexity assumptions of f to establish their convergence results. This paper introduces the Riemannian three-term conjugate gradient (RTTCG) methods. The search direction in these RTTCG methods consistently satisfies the sufficient descent condition, regardless of the line search employed, and does so without requiring a restart mechanism. By utilizing retraction and vector …

Memoir On A General Property Of A Very Extensive Class Of Transcendental Functions, Niels Henrik Abel 1802--1829, John Little

Mathematics and Computer Science Department Faculty Scholarship

We present this new commentary and translation anticipating the 200th anniversary of the work, commonly known as Abel's ``Paris memoir.'' This is recognized today as one of Abel's most original and influential works. It is significant mostly because it marked the first appearance of a form of a result in the theory of algebraic curves and Riemann surfaces that has come to be known as ``Abel's theorem.'' However, Abel's original understanding of the meaning and context of his result was quite different from the typical modern formulation and the development of the modern understanding has been a long and tortuous …

Constructions Of Compact Dupin Hypersurfaces With Non-Constant Lie Curvatures, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface M in the unit sphere Sⁿ ⊂ Rⁿ⁺¹ is Dupin if along each curvature surface of M, the corresponding principal curvature is constant. If the number g of distinct principal curvatures is constant on M, then M is called proper Dupin. In this expository paper, we give a detailed description of two important types of constructions of compact proper Dupin hypersurfaces in Sⁿ. One construction was published in 1989 by Pinkall and Thorbergsson [35], and the second was published in 1989 by Miyaoka and Ozawa [26]. Both types of examples have the …

Quantum General Relativity, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

We construct a quantum general relativity model of spin 2 Yang-Mills fields, complete at the level of theoretical physics with complete mathematics not included. The construction includes a cosmological model. Existing Type Ia supernova data analysis used for this cosmology shows zero cosmological constant and zero dark energy.

Investigating Potential Alzheimer's Disease Therapies Through An Agent-Based Model Of Impaired Microglia Metabolismmodeling Impaired Microglia Metabolism, Cheyenne Ty, Abigail Penland, John Gerving, Martin Mendoza-Ceja, Megan Pratt, Kamila Larripa

Spora: A Journal of Biomathematics

Alzheimer’s disease is the leading cause of dementia globally and is marked by amyloid-β plaque accumulation, decreased brain pH, disrupted metabolism, and increased permeability of the blood-brain barrier. Microglia, the resident immune cells of the central nervous system, are essential for maintaining brain homeostasis and are critically involved in the progression of AD. While microglia can initiate protective immune responses to injury and disease, dysregulated microglial metabolism may contribute to the exacerbation of Alzheimer's pathology. To investigate potential therapeutic strategies, we developed an agent-based model simulating the effects of exercise and metabolic enhancement on dysfunctional microglia. Our findings suggest that …

Imaging A Digital Stock Exchange, Mrigank Mrigank

All ETDs from UAB

In 2021, Rama Cont and Marvin M¨uller proposed a stochastic partial differential equation (SPDE) that models the density of orders in the limit order book at a given distance from the mid-price. In this thesis, we begin with a generalized version of the Cont–M¨uller SPDE and formulate an associated inverse problem. We present two alternative approaches to framing this inverse problem and establish, under appropriate assumptions, that it is conditionally well-posed. To solve the inverse problem, specifically, to recover the coefficients in the generalized Cont–M¨uller model, we adapt a variational approach involving the minimization of convex functionals. The recovered coefficients …

Analytical And Numerical Approaches To Parameter Estimation In Damped Oscillatory Systems, Gracie Crooks, F. Ayça Çetinkaya

CODEE Journal

We investigate the inverse problem of identifying damping and stiffness parameters in one-dimensional damped oscillatory systems governed by second-order differential equations. Focusing on mass–spring–damper models, we analyze the qualitative behavior of solutions across underdamped, critically damped, and overdamped regimes, and derive explicit conditions for parameter recovery based on time-domain observations such as equilibrium crossings and turnaround points. Two numerical estimation methods are developed and compared: a finite-difference least-squares approach based on central difference approximations, and a finite element formulation derived from a variational framework using piecewise linear basis functions. Computational experiments using synthetic data assess the accuracy, stability, and noise …

Greedy Algorithm For Neural Networks For Indefinite Elliptic Problems, Qingguo Hong, Jiwei Jia, Young Ju Lee, Ziqian Li

Mathematics and Statistics Faculty Research & Creative Works

The paper presents a priori error analysis of the shallow neural network approximation to the solution to the indefinite elliptic equation and a cutting-edge implementation of the Orthogonal Greedy Algorithm (OGA) tailored to overcome the challenges of indefinite elliptic problems, which is a domain where conventional approaches often struggle due to the lack of coerciveness. A rigorous priori error analysis that shows the neural network's ability to approximate the solution of indefinite problems is confirmed numerically by OGA. We also present the error analysis of the relevant numerical quadrature. In particular, massive numerical implementations are conducted to justify the theory, …

(Si15-068) Advanced Numerical Methods For The Solution Of Nonlinear Fisher Equation, Vikash Vimal, Richa Kumari, Ashish Awasthi

Applications and Applied Mathematics: An International Journal (AAM)

This paper examines the use of advanced numerical techniques to approximate solutions of the Fisher equation with higher-order accuracy. This technique integrates the method of lines with a strong stability-preserving Runge–Kutta scheme of orders four and five stages (SSPRK-54) for the numerical formulation. This scheme is then tested on two examples and the results show that it is more efficient than existing methods and requires less computing power. These equations are widely used across scientific and engineering disciplines, with particular relevance in biomedical studies, such as estimating the boundary size of tumors. The difficulties arising from their nonlinear nature are …

The Pure Yang-Mills Field, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

We construct a pure Yang-Mills quantum gauge field theory. The construction is based on the axial gauge, ghost states, the BRST framework and the entropy production maximizing Gribov extension of the Hamiltonian, with a loop expansion to all finite orders for the dynamics. The construction is established by renormalized perturbation theory to finite order. Reflection positivity and the reconstruction theorem are established. Hilbert space convergence of renormalized perturbation theory is a consequence of the reconstruction theorem. All Osterwalder-Schrader axioms are verified. A two-sided mass gap exists in the energy spectrum for the weak coupling limit of low temperature. The conditions …

A Dual-Model Machine Learning Framework For Predictive Maintenance Of Industrial Digital Press Components, Richmond Darko, Emmanuel Adabor

African Conference on Information Systems and Technology

This study presents a novel dual-model predictive maintenance framework designed to improve maintenance scheduling for components in industrial digital presses. The framework integrates two complementary approaches: a Threshold-Based Maintenance Approach (TBMA) for components operating within acceptable usage limits, and an Overdue Severity-Based Maintenance Approach (OSBMA) for those that have exceeded their expected lifespans or show signs of critical degradation. This study uses real-world operational data from a Konica Minolta C6000 press. It applies advanced machine learning models, including Gradient Boosting Machines and Random Forest for classification, and Generalized Additive Models (GAM) for Remaining Useful Life (RUL) prediction. The goal is …

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 …

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 …

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.

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 …

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 …