Mathematics Commons^™

Toward Mapping Multiphase Multicomponent Mixtures With Neural Networks, Kristen L. Hallas, Melissa De Jesus, Christine J. Wu, Jianzhi Li, Jason Bernstein, Philip C. Myint

School of Mathematical & Statistical Sciences Faculty Publications

Equation of state (EOS) tables are commonly used in hydrodynamic simulations of high-pressure, high-temperature phenomena in fields like planetary science, astrophysics, and high-energy-density science. However, generating and storing EOS tables for multiphase, multicomponent mixtures over a wide range of pressures and temperatures is computationally infeasible due to their memory-intensive nature. To address this issue, we have developed a neural network-based machine learning model to predict new EOS tables for binary mixtures. In particular, a deep feedforward neural network trained on a set of ten EOS tables at particular mixture compositions is able to predict nine new (hold-out) EOS tables at …

Revisiting Ulam Stability For Boundary Value Problems, Martin Bohner, Snezhana Hristova, Agnieszka B. Malinowska, Ewa Girejko

Mathematics and Statistics Faculty Research & Creative Works

The main goal of this paper is to apply Ulam stability theory to boundary value problems for dynamic equations, while addressing several common misconceptions found in the existing literature. We identify the key issues that arise when applying Ulam stability to such problems and propose three distinct approaches to overcome them. To enhance clarity and accessibility, we begin with nonlinear ordinary differential equations and subsequently extend the analysis to nonlinear dynamic equations on time scales. Since a time scale is defined as any nonempty closed subset of the real numbers, our results are applicable to dynamic equations on continuous, discrete, …

Peakon Solutions And Analytical Properties For The Camassa–Holm-Type Equations With Quadratic Nonlinearities, Yonghong Chen, Zhijun Qiao, Mingxuan Zhu

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we derive the multi-peakon dynamical system of a class of Camassa-Holm-type equations with quadratic nonlinearities. We also consider the analytical properties for the Cauchy problem. Firstly, we establish local well-posedness of solutions in Besov spaces and then provide the blow-up criteria. Subsequently, we impose appropriate sufficient conditions on the initial data to guarantee that the corresponding solution either exists globally or blows up in a finite time. Finally, we prove the ill-posedness in the Besov space B2,∞^3/2 by utilizing the non-traveling wave solutions.

A Finite Element Model For Thermomechanical Stress-Strain Fields In Transversely Isotropic Strain-Limiting Materials, Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

School of Mathematical & Statistical Sciences Faculty Publications

This paper presents a comprehensive computational framework for investigating thermo-elastic fracture in transversely isotropic materials, where classical linear elasticity fails to predict physically realistic behavior near stress concentrations. We address the challenge of unphysical strain singularities at crack tips by employing a strain-limiting theory of elasticity. This theory is characterized by an algebraically nonlinear constitutive relationship between stress and strain, which intrinsically enforces a limit on the norm of the strain tensor. This approach allows the development of very large stresses, as expected near a crack tip, while ensuring that the corresponding strains remain physically bounded. A loosely coupled system …

Pattern Formation In Quantum Hierarchical Cellular Neural Networks, Wilson A. Zuniga-Galindo, B. A. Zambrano-Luna, Chayapuntika Indoung

School of Mathematical & Statistical Sciences Faculty Publications

We present a new class of quantum neural networks (QNNs) whose states are solutions of p -adic Schrödinger equations with a non-local potential that controls the interaction between the neurons. These equations are obtained as Wick rotations of the state equations of p -adic cellular neural networks (CNNs). The p -adic CNNs arise as continuous limits of large discrete hierarchical neural networks (NNs). The CNNs are bio-inspired by the Wilson–Cowan model, which describes the macroscopic dynamics of large populations of neurons. We provide a detailed study of the discretization of the new p -adic Schrödinger equations, which allows the construction …

Finite Mathematics: A Course Guide, Angela West Dixon, Hillary Dosser

Faculty Publications

Finite Mathematics: A Course Guide is a supplement to Mathematics for Business and Social Sciences by Kathryn Bollinger and Vanessa Coffelt. The authors of this Course Guide, Angela Dixon and Hilary Dosser, are instructors of Mathematics at Stephen F. Austin State University in Nacogdoches, Texas. This Course Guide was developed in response to adapting MATH 1324, Finite Mathematics to a zero-cost course, using an Open Educational Resource (OER) implemented in the Fall semester of 2026. According to the Texas Higher Education Coordinating Board’s Academic Course Guide Manual (ACGM), MATH 1324 Mathematics for Business and Social Sciences concerns the application of …

Quantum Mechanics, Non-Locality, And The Space Discreteness Hypothesis, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space X as a model for the physical space at short distances. However, we consider the time as a real variable. In this framework, the Dirac–von Neumann formalism can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose R×(R×X)3 as …

Study Of A Nonlinear Delayed Parabolic Model For Prion Disease Dynamics With The Unfolded Protein Response, Gangadhara Boregowda, Laurent Pujo-Menjouet, Zhaosheng Feng, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Prion diseases are neurodegenerative disorders characterized by the dynamic spread of misfolded toxic proteins in the brain. In this process, the normal cellular prion protein (PrPC) produced by neurons misfolds into a toxic form known as scrapie prion protein (PrPSc). These misfolded proteins propagate through the brain by converting healthy prions into their toxic form. This biological mechanism can be modeled by a system of nonlinear parabolic partial differential equations, accompanied by a nonlinear delayed integral boundary condition. Our primary objective is to establish the existence of nonnegative classical solutions to this system. Furthermore, we derive a priori estimates for …

Differential Equation Modeling For Sustainable Resource Management: A Steady-State Optimal Harvesting Approach, Iordanka N. Panayotova, Aleksei Talonov

CODEE Journal

Mathematical models based on differential equations provide a powerful framework for connecting real-world data to informed decision-making. In this work, we present a student-accessible project that uses an optimal-control framework to study the sustainable management of biological resources.

Motivated by fisheries management, we examine a predator--prey system in which harvesting decisions must balance ecological and economic considerations. The model is formulated as an optimal control problem that seeks to maximize the total discounted net revenue from harvesting. Rather than solving for the complete time-dependent harvesting trajectory, we restrict the analysis to positive controlled coexistence equilibria and characterize an interior stationary …

How Euler Could Have Done It: Euler And An Integral Of Ramanujan, Alexander Aycock

Euleriana

We present a derivation of a definite integral discovered by Ramanujan (1887–1920) in his 1915 paper "Some Definite Integrals," from formulas and ideas already developed by Euler (1707–1783). Additionally, it is argued why Euler did not discover Ramanujan's integral himself, although he had all tools required for this task at his disposal.

A Note On Euler's Definition Of The Sum Of A Divergent Series, Alexander Aycock

Euleriana

We outline Euler's theory of divergent series and present another approach he could have taken to sum his ``factorial series". Finally, we address the contradictions that arise from his concept of a sum of a divergent series.

Euler's Explorations Of Extremal Ellipses, Jonathan David Evans

Euleriana

In the 1770s, Euler wrote a series of papers (E563, E691 and E692) about finding the ellipse with minimal area or perimeter in the family of all ellipses passing through a fixed set of points. This is a translation of all three papers from the original Latin, together with a commentary which discusses Euler's results and an appendix which addresses a question from E691 which Euler left for others to consider.

Arithmetic Theorems, Demonstrated By A New Method: An English Translation Of E271, Brendon Lasell

Euleriana

After defending in general terms the value of arithmetic demonstrations, where, according to Euler, the force of genius shines more brightly than in any other kind of demonstration, he goes on to let the force of his own genius shine forth in demonstrating the fact that, for any pair of relatively prime numbers a and N, a^ϕ(N) − 1 is always divisible by N, where we anachronistically denote by ϕ(N) the number of numbers less than N that are relatively prime to it. He approaches this theorem by considering the remainders that result when the numbers in an arithmetic …

On Double Integral Formulas: An English Translation Of E391, Thomas W. Polaski

Euleriana

In this paper, Euler investigates the theory of double intergrals to uses them to find areas, volumes and surface areas. He considers transformations of variables in double integrals. After showing that simple multiplication of the transformed differentials is not appropriate, he derives the formula for a change of variables in double integrals. Euler then uses such changes of variables to investigate areas and volumes, and then turns his attention to the so-called ``Florentine problem.'' This problem requires one to remove four equal windows from a hemispherical surface so that the remaining area is equal to the area of a square. …

Eulerian Problem Solving, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Volume 6 (Issue 2) of Euleriana.

Taut And Dupin Submanifolds (Updated Version), Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an updated version of the paper [29] by the author which originally appeared in 1997. The original paper was a survey of the closely related fields of taut and Dupin submanifolds of Euclidean space, and this updated version includes many results in the field that have appeared since the publication of the original version. The emphasis is on stating results in their proper context and noting areas for future research, and relatively few proofs are given. The important class of isoparametric hypersurfaces is surveyed in detail, as is the relationship between the two concepts of taut and Dupin. …

Fast Computation And Model Order Reduction Of The Friction Stir Welding Process With Pod-Deim, Joshua Kay, Zilong Song

Mathematics and Statistics Student Research and Class Projects

Friction stir welding (FSW) is a solid-state manufacturing process widely used in joining aluminum and other metal workpieces. The FSW process can be modeled by a coupled system of non-Newtonian Navier–Stokes and heat-transfer equations. However, solving this non-linear system with high accuracy requires significant computational power. This work refines the system by introducing corrected coefficients and new treatments for boundary conditions near the tool. Then, model order reduction, including the Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM), is applied to efficiently solve the FSW system in a low-dimensional space. To enhance accuracy and effectiveness, two novel treatments …

On The Existence, Uniqueness And Stability Of Solutions Of Sdes With State-Dependent Variable Exponent, Mustafa Avci

Journal of Stochastic Analysis

We study a time-inhomogeneous nonlinear SDE with drift and diffusion governed by state-dependent variable exponents. This framework generalizes models like the geometric Brownian motion (GBM) and the constant elasticity of variance (CEV), offering flexibility to capture complex dynamics while posing analytical challenges. Using a fixed-point approach, we prove existence and uniqueness, analyze higher-order moments, derive asymptotic estimates, and assess stability. Finally, we illustrate an application where Poisson’s equation admits a probabilistic representation via a timehomogeneous nonlinear SDE with state-dependent variable exponents.

Companion Matrices Associated To Stochastic Matrices, Andreas Boukas, Philip Feinsilver

Journal of Stochastic Analysis

Starting with a stochastic matrix, we study the behavior of powers of an associated companion matrix, which has the same characteristic polynomial as the original matrix. In general the companion matrix will have negative entries while maintaining rowsums equal to 1. We will find the growth rate even if the Ces`aro limit of the sums of the companion matrix diverge. Surprisingly, in the irreducible aperiodic case the powers of the companion matrix will converge even though the norm of the matrix exceeds 1 and it has possibly negative entries.

Z+-Valued Additive Processes In Law With Nonnegative Increments, Nadjib Bouzar

Journal of Stochastic Analysis

The goal of this article is to study in depth the subclass of Z+- valued additive processes in law with nonnegative increments. We establish key distributional properties of these processes and obtain some existence results under a variety of conditions. We show that they form a subclass of the family of inhomogeneous Markov chains with spatial homogeneity. We extend the notion of factoring introduced by Sato (2004, [9]) for Rd-valued additive processes in law to their Z+-valued counterparts with nonnegative increments. We give a sufficient condition for the existence of a factoring for these processes. Lastly, we obtain their representation …