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, …

From Solitons To Chaos: Nonlinear Dynamics Of The Time-Fractional Modified Regularized Long-Wave Equation, Samsun Nahar Ananna, Chunhui Lu, Abdulla Al Mamun, Md Al Amin Meia, Nur Hasan Mahmud Shahen

School of Mathematical & Statistical Sciences Faculty Publications

This study investigates the nonlinear wave structures and complex dynamical behavior of the time-fractional modified regularized long-wave (mRLW) equation through an integrated analytical, numerical, and dynamical framework, with particular emphasis on connecting localized wave formation with perturbation-induced nonlinear dynamics. Within the adopted fractional framework, a Hirota-type bilinear approach is employed to construct one-wave, two-wave, three-wave, lump, one-lump–one-soliton, and one-lump–one-stripe structures, whose numerical consistency is subsequently examined using the Modified Variational Iteration (MVI) method through graphical comparisons and L 2 and L ∞ error norms. The periodically perturbed mRLW system is further investigated using phase portraits, Lyapunov exponents, bifurcation diagrams, Poincaré …

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 …

Representational Power As A Framework For Fairness With Applications To Redistricting, Jeffrey Barton

Mathematics Sciences: Faculty Publications

We introduce a framework for assessing fairness in electoral outcomes based on a new notion of representational power that determines a group's power as a combination of its size and how advantageously its voters are situated given the electoral rules. The framework is adaptable to a variety of contexts, and it produces intuitive results. When applied to redistricting, we call the measure the geographic representational baseline. The new measure is deterministic, accessible to non-experts, and relatively simple to calculate. The results generated by the new method are overall mutually reinforcing with those of a variety of ensemble methods for partisan …

Geometric Puzzles On A Circular Track: A Recreational Introduction To Quantization, Sayandip Pandit, Amit Priyadarshi, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Imagine a circular running track with several water stations placed around it. If only a few stations may remain open, or if a few new stations may be built, where should they be located so that runners travel the shortest possible distance to reach one? This simple question leads to a collection of recreational geometry puzzles involving distance, symmetry, optimization, Voronoi regions, and quantization. Quantization is the mathematical problem of replacing a large collection of points by a smaller set of representative points while minimizing the approximation error. Using only elementary mathematics, we investigate several configurations of points on a …

Progress Towards An Analog Of The Darboux-Griffiths Converse Of Abel's Theorem In Characteristic P, John B. Little

Mathematics and Computer Science Department Faculty Scholarship

In this working paper, we report our recent efforts concerning a possible characteristic p analog of the so-called converse of Abel's theorem discussed by Darboux and Griffiths.  The characteristic zero version is used in one proof of the Lie-Wirtinger theorem on double translation manifolds and also in related aspects of the geometry of webs.  In informal terms, our results indicate that, apart from some isolated counterexamples, formal power series Abel relations over an algebraically closed field of characteristic p essentially only arise in the (generalized) Jacobians of algebraic curves.  The ultimate goal is to complete thevwork started in the author's …

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 …

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 …

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 …

Deflation-Based Preconditioning For Parallel Time Stepping Algorithms, Marina Nacianceno, Josef A. Sifuentes

School of Mathematical & Statistical Sciences Faculty Publications

Often, in solving large systems of differential equations using a time-stepping algorithm, one must solve multiple linear systems where the coefficient matrix is unchanged or shifted by a constant multiple of the identity. This can be computationally expensive when the coefficient matrix involved is very large. Where direct methods like Gaussian elimination are infeasible, iterative methods like GMRES are often used. It is especially useful when the matrices involved are sparse or allow for fast matrix multiplication algorithms. In this paper, we propose preconditioning strategies that glean information from previous runs of GMRES to build an effective preconditioner for subsequent …

Emergence Of Purely Elastoplastic Turbulence In Shear Flows, Muhammad Abdullah, Shravan Pradeep, Doug J. Jerolmack, Becca Thomases, Paulo E. Arratia

Mathematics Sciences: Faculty Publications

Yield-stress fluids only flow above a critical stress. Here, we observe the emergence of a distinct, elasticity-driven flow state in a yield-stress fluid in the absence of inertia. Numerical simulations show that this elastoplastic turbulent state is characterized by non-Gaussian bimodal velocity distribution and a broad spectrum of velocity and stress fluctuations. Results show a nonmonotonic relationship between the volume fraction of the unyielded flow and plasticity. Surprisingly, the system can fluidize near the emergence of the elastoplastic turbulent state. Our results reveal the complex interplay between elasticity and plasticity in simple shear flows, indicating that plasticity can mobilize the …

Generalized Apostol-Bernoulli Functions And The Evaluation Of Character Sums, Brad Isaacson

Publications and Research

We introduce two types of generalized Apostol–Bernoulli functions attached to Dirichlet characters and express them by generalized Apostol–Bernoulli numbers. As a corollary, we obtain formulas expressing two families of character sums by generalized Apostol–Bernoulli numbers. These sums are twisted generalizations of sums introduced and studied by Arakawa, Berndt, Ibukiyama, Kaneko, Kim, Ramanujan, and Zaharescu in the context of modular forms and theta function identities.

Non-Gradient Quaternion Training Matrix Modifications For Color-Image Distillation, Tahsin Shahnewaz, Megdam Ahmed Chowdhury, Nikolay Metodiev Sirakov

Student Publications

This paper develops a new multi-stage image distillation method that combines two well known techniques. In the first stage, our method creates a matrix from all training images. In the next stage, it adapts a modified principal component analysis (M-PCA) approach to transform the training matrix. In the third stage, Singular Value Decomposition (SVD) fur ther refines the training-image matrix through low-rank reconstruction and controlled row selection. In the fourth stage, rotation of small 2 ×2 matrix blocks on the entire left singular matrix is conducted. The upper m (user-selected number) rows of the reconstructed matrix are selected and transformed …

Continuous-Time Quantum Markov Chains And Discretizations Of P-Adic Schrödinger Equations: Comparisons And Simulations, Wilson A. Zuniga-Galindo, L. F. Chacón-Cortés

School of Mathematical & Statistical Sciences Faculty Publications

Continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. Unlike classical continuous-time Markov chains (CTMCs), the CTQMCs studied here do not satisfy the classical Chapman–Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in …

Describing Preservice Secondary Mathematics Teachers’ Critical Statistical Literacy Habits Of Mind Enactment, Nina G. Bailey, Allison W. Mcculloch

Department of Mathematics Faculty Scholarship and Creative Works

Globally, school standards documents increasingly emphasize statistical literacy from the consumer orientation (making sense of real-world data representations). This orientation requires the adoption of a Critical lens (e.g., focus on power and equity) because how statistics are used (e.g., decision making) cannot be separated from the fact that social issues operate within systems of marginalization, privilege, and power. This study aims to explore how preservice secondary mathematics teachers enact critical statistical literacy habits of mind (CSLHM) when presented with data representations from the media. Findings reveal that preservice mathematics teachers typically emergently enact CSLHM, and some robustly. There is no …

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 …

Using Approximations Of Practice To Support Calculus Instructors’ Enactment Of Equitable Active Learning Teaching Practices, Kaitlyn Stephens Serbin, Karina Uhing

School of Mathematical & Statistical Sciences Faculty Publications

We implemented a semester-long professional development workshop series to help Calculus I and II instructors implement active learning and equitable teaching practices. The workshops actively engaged instructors in approximations of practice (Grossman et al., 2009) in which they learned to enact various teaching practices like designing tasks; eliciting, noticing, and responding to evidence of student thinking; orchestrating class discussions; fostering inclusive social norms; and equitably managing student participation. After they participated in the workshops, we interviewed the instructors about their decisions to implement the teaching practices from the approximations of practice in their teaching. We analyzed the instructors’ resources, orientations, …

Health-Related Quality-Of-Life Profiles In Adults With Tinnitus, Preeti Pandey, Hansapani Rodrigo, Megan Kruger, Eldré W. Beukes, De Wet Swanepoel, Anu Sharma, Gerhard Andersson, Vinaya Manchaiah

School of Mathematical & Statistical Sciences Faculty Publications

Purpose: The study aimed to characterize the multidimensional burden of tinnitus on health-related quality of life (HRQoL) using the EuroQol 5-Dimension 5-Level (EQ-5D-5L) dimensions (mobility, self-care, usual activities, pain/discomfort, and anxiety/depression) moving beyond symptom severity alone.

Method: A cross-sectional analysis was conducted using baseline data (N = 428, Mage = 54.9 years, 51.9% female) from several clinical trials examining the efficacy of internet-based cognitive behavioral therapy for tinnitus. Data were collected using a series of online questionnaires including demographic, audiological, and clinical variables. HRQoL was assessed using the EQ-5D-5L profile dimensions.

Results: Participants primarily reported no problems with mobility (76.6%), …

Braids On The Stranded Cellular Automata Model, Alexa Renner

Mathematical Sciences Technical Reports (MSTR)

The Stranded Cellular Automata (SCA) model is a grid of cells such that each cell can contain 0, 1, or 2 strands, together with two cellular automata that control when and how strands turn and cross. It was developed to study patterns occurring in fiber arts. We define a notion of what it means for a braid, in the sense of an element of a braid group, to be represented by an SCA pattern, and provide several algorithms to determine when a braid has an SCA representation with certain additional properties.

Pseudo-Traveling Waves And Bumps In Quantum And Classical Hierarchical Cellular Neural Networks, Wilson A. Zuniga-Galindo, B. A. Zambrano-Luna, Chayapuntika Indoung

School of Mathematical & Statistical Sciences Faculty Publications

We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks (CNNs) defined over the ring of p-adic integers Zp. The first type is a p-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The p-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs; these networks have a tree-like hierarchical architecture with infinitely many cells and hidden layers. The states are governed by integro-differential equations on Zp. The p-adic traveling waves behave fundamentally differently from their Archimedean counterparts. A traveling wave restricted …

Generalizations Of The Dirichlet Problem For Bianalytic Functions, William L. Blair

Math Faculty Publications and Presentations

We prove the Dirichlet problem for second-order iterated Vekua equations, a natural generalization of the Bitsadze equation, is well-posed when the boundary condition is defined as a product of an exponential function and a polynomial on a non-degenerate conic that is not a circumference. Also, we extend this result, as well as other related results for the Dirichlet problem for polyanalytic and generalized analytic functions from the literature, to their analogues for bicomplex differential equations.

Pedagogical Moves For Provoking Prospective Teachers’ Intellectual Needs While Connecting Abstract Algebra And Secondary Mathematics, Kaitlyn Stephens Serbin, Estrella Johnson, Sthefanía Espinosa

School of Mathematical & Statistical Sciences Faculty Publications

Researchers have exhibited ways in which prospective secondary mathematics teachers can connect content from abstract algebra with secondary mathematics in ways that reshape their understanding of secondary mathematics and thereby support their teaching of that content. However, there has been very little research done on what mathematics teacher educators can do in the classroom to help prospective teachers make connections between abstract algebra and secondary algebra in ways that support such reshaping. This study focuses on a mathematics teacher educator who guided a class of prospective teachers to connect properties of algebraic structures with equation-solving procedures by directing their focus …

Adaptive Rational Approximation In Dynamic Economic Models: A Novel Application Of The Aaa Algorithm To Economic Growth, Adaye Sosthene Yvan N'Guettia

Mathematics, Statistics, and Computer Science Honors Projects

I study adaptive rational approximation for fixed points that arise in infinite-horizon dynamic programming. I integrate the Adaptive Antoulas–Anderson (AAA) algorithm into Bellman- and Euler-based fixed-point solvers by recomputing a barycentric rational interpolant at each update. In addition to standard AAA, which selects support points from interpolation residuals, I study a residual-weighted variant in which Bellman,Euler, or KKT diagnostics act as secondary weights on the greedy pivot rule. This alignment of approximation adaptivity with the underlying equilibrium conditions can concentrate degrees of freedom in regions of steep curvature, sharp transitions in localbehavior, and other localized features that typically degrade polynomial …

An Analysis Of The Effects And Implementations Of The Early Literacy Grant In Arizona, Alicia Severiano Perez

Mathematics, Statistics, and Computer Science Honors Projects

Over the years, states have implemented Science of Reading (SoR) frameworks to address low literacy levels. The Early Literacy Grant (ELG) in Arizona funds and supports such frameworks for schools serving low-income students. This paper is the first to explore the grant through interrupted time series modeling to evaluate effectiveness and text analysis to understand its implementation. We do not find clear evidence of positive effects caused by the grant, other than some cases, such as Yuma County. Schools typically allocate funds toward salaries and hiring instructors. These findings raise questions about whether its allocations should be closely monitored.

Level Sets For Lehmer Codes Of Pattern Avoiding Permutations, Avery Sinclair

Mathematics, Statistics, and Computer Science Honors Projects

We study the poset structures for two families of pattern avoiding permutations. An n-permutation is a list of the numbers [n]={1,2,...,n}. A permutation is 321-avoiding when it does not contain a decreasing subsequence of length 3. A poset (partially ordered set) is a set such that some elements can be compared with one another. Using Lehmer codes, we define a poset for 321-avoiding permutations. We then fully describe the six lowest levels of this poset. We then consider the analogous poset for 123-avoiding permutations (which don't contain an increasing subsequence of length 3) and fully describe the three lowest levels.

Ngram Data Social Media, William Zywiak

Data and Datasets

Data set for a NSF STEM grant.