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 …

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 …

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 …

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 …

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 …

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%), …

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 …

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 …

A Stochastic Nonlinear Dynamical System For Smoothing Noisy Eye Gaze Data, Thoa Thieu, Roderick Melnik

School of Mathematical & Statistical Sciences Faculty Publications

In this study, we address the challenges associated with accurately determining gaze location on a screen, which is often compromised by noise from factors such as eye tracker limitations, calibration drift, ambient lighting changes, and eye blinks. We propose the use of an extended Kalman filter (EKF) to smooth the gaze data collected during eye-tracking experiments and systematically explore the interaction of different system parameters. Our results demonstrate that the EKF significantly reduces noise, leading to a marked improvement in tracking accuracy. Furthermore, we show that our proposed stochastic nonlinear dynamical model aligns well with real experimental data and holds …

Curious Patterns And Identities Involving Consecutive Integers, Carlo Cattani, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Consecutive integers exhibit numerous surprising patterns and elegant identities. In this article, we investigate several classical properties involving products, sums, divisibility, perfect squares, cubes, pronic numbers, triangular numbers, sums of squares, and Pythagorean triples. Using only elementary algebra, symmetry arguments, and mathematical induction, we establish a collection of identities that reveal rich connections among these classical topics. The exposition is intended for students, teachers, and recreational mathematicians, illustrating how simple observations about consecutive integers can lead to beautiful mathematical patterns and deeper insights into elementary number theory.

The Mathematics Of Calendar Patterns: Day-Of-Week Phenomena, Number Patterns, And Recreational Investigations, Vijender Nallapu, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Calendars are among the most familiar mathematical structures encountered in everyday life. Beyond their practical use in organizing dates and measuring time, calendars contain a rich collection of numerical patterns, symmetries, and arithmetic relationships. Many of these patterns can be explored using only elementary mathematics, making calendars an attractive subject for recreational investigation and mathematical enrichment.

In this article we examine several mathematical properties arising from the structure of monthly calendars. We investigate arithmetic patterns associated with rows, columns, and rectangular blocks of dates, establish properties of calendar squares, and explore connections with modular arithmetic through day-of-week calculations. We also …

Vedic Mathematics And Mental Computation: Fast Multiplication Techniques, Algebraic Foundations, And Historical Perspectives, Rakesh Bhatia, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Mental computation has fascinated mathematicians, educators, and students for centuries. Various computational techniques allow arithmetic calculations to be performed rapidly without the use of calculators or computers. Among these methods, several popular techniques are associated with what is commonly known as Vedic Mathematics. In this article, we examine a collection of fast multiplication techniques, including multiplication near powers of ten, multiplication by eleven, squaring numbers ending in five, and the vertical-and-crosswise method. For each technique, we provide elementary mathematical explanations based on algebraic identities and place-value representations.

Hidden Symmetries Of The Classical 3×3 Magic Square, Vijender Nallapu, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The classical 3×3 magic square is one of the most famous objects in recreational mathematics. Its defining feature is that every row, every column, and the two main diagonals have the same sum. However, beyond this well-known property, the square contains several hidden symmetries involving rotations, reflections, complementary pairs, parity, and balance around the central entry. In this article, we present an elementary exploration of these symmetries and show how the classical magic square provides a beautiful introduction to mathematical structure and pattern recognition.

The 2025 Measles Outbreak In Texas, Tamer Oraby, Martial L. Ndeffo-Mbah

School of Mathematical & Statistical Sciences Faculty Publications

Background: In 2025, Texas experienced its largest measles outbreak in decades, reporting 762 cases by mid-August. Measles is a highly contagious but vaccine-preventable infection transmitted mainly among unvaccinated individuals and capable of causing severe outcomes.

Methods: We investigate counterfactual measles control scenarios based on daycare and school closures and reactive vaccination of infants and children, including mixed interventions. We analyze the 2025 Texas outbreak using an age-structured multi-stage SEIR model formulated as a system of ordinary differential equations. The model is fit to case data using Bayesian inference to estimate the effective reproduction number and generate posterior predictive trajectories under …

A Finite Element Model To Analyze Crack-Tip Fields In A Transversely Isotropic Strain-Limiting Elastic Solid, Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

School of Mathematical & Statistical Sciences Faculty Publications

This paper presents a finite element model for the analysis of crack-tip fields in a transversely isotropic strain-limiting elastic body. A nonlinear constitutive relationship between stress and linearized strain characterizes the material response. This algebraically nonlinear relationship is critical as it mitigates the physically inconsistent strain singularities that arise at crack tips. These strain-limiting relationships ensure that strains remain bounded near the crack tip, representing a significant advancement in the formulation of boundary value problems (BVPs) within the context of first-order approximate constitutive models. For a transversely isotropic elastic material containing a crack, the equilibrium equation, derived from the balance …

Bayesian Variable Selection In High-Dimensional Ordinal Quantile Regression Models, Mai Dao, Md. Sakhawat Hossain, Zhuanzhuan Ma

School of Mathematical & Statistical Sciences Faculty Publications

Quantile regression (QR) provides a flexible statistical framework for modeling the entire conditional distribution of the response variable, making it useful for analysis in various fields. Despite its advantages, existing methods for QR often encounter numerical challenges in high-dimensional settings, especially for those with ordinal responses. In this paper, we use a latent-response framework to construct a Bayesian hierarchical model to conduct parameter estimation and variable selection for ordinal QR. Using the asymmetric Laplace working likelihood and the horseshoe prior for the regression coefficients, we obtain the posterior samples to be screened by the sequential two-means clustering process to identify …

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified by using numbers in the ring of integers of non-Archimedean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.

Cultural Traits May Replace Human Mobility Data In Forecasting Covid-19 Mortality: A Deep Learning Approach, Saif Abbas, Tamer Oraby, Michael G. Tyshenko, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

The COVID-19 pandemic highlighted the need for accurate epidemic forecasting to support public health decision-making. Most existing approaches depend heavily on human mobility data, while largely neglecting population behavior shaped by socio-cultural norms. In this study, we analyze daily COVID-19 mortality and Google mobility data from 72 countries during the first 130 d of the pandemic, a period characterized by high uncertainty and behavioral heterogeneity. In particular, we examine whether Hofstede’s country-level cultural dimensions can serve as latent behavioral forecasters of mortality in lieu of dynamic mobility indicators. Using 100 d for training and 30 d for forecasting, we employ …

A Bayesian-Optimized Ensemble Deep Learning Framework For Automated Detection And Classification Of Retinal Diseases In Ghana Using Oct Images, Gifty Duah, Eric Nyarko, Gideon Nana Amo, Theophilus Dwamena Frimpong, Anani Lotsi

School of Mathematical & Statistical Sciences Faculty Publications

Retinal diseases pose a significant global health challenge due to their potential to cause severe visual impairment and blindness. This study aimed to develop a robust deep learning ensemble framework for the automated detection and classification of retinal diseases from optical coherence tomography (OCT) images. This study used OCT images from WATBORG Eye Services in Ghana, including glaucoma, macular edema, posterior vitreous detachment (PVD), and healthy eyes. The data preprocessing steps included augmentation, resizing, and one-hot encoding. The dataset was divided into training (56%), validation (14%), and testing (30%) sets using stratified sampling. Six convolutional neural network (CNN) architectures, Visual …

Propagation Of Dirac Spherical Waves In The Expanding Universe, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

The explicit formulas for the spherical solutions of the Dirac equation in the expanding universe are given. The initial value of the solution can be, in particular, a wave function of the hydrogen-like atom or a spherical wave in the Minkowski space, that then propagates in the Friedmann-Lemaître-Robertson-Walker space-time, which is expanding with the de Sitter scale factor.

Comparative Analysis Of Traditional And Deep Learning Time Series Architectures For Influenza A Infectious Disease Forecasting, Edmund Fosu Agyemang, Hansapani Rodrigo, Vincent Agbenyeavu

School of Mathematical & Statistical Sciences Faculty Publications

Influenza A remains a major cause of respiratory mortality worldwide, motivating accurate forecasting to support timely preparedness and resource allocation. This study presents a comparative evaluation of two traditional seasonal time series baselines, ARIMA and Holt–Winters exponential smoothing (ETS), and six deep learning (DL) architectures (Simple RNN, LSTM, GRU, BiLSTM, BiGRU, and a Transformer) for forecasting monthly Influenza A case counts in the United States. Data from January 2009 to December 2023 were analyzed, using January 2009 to December 2022 for training and January 2023 to December 2023 for out-of-sample testing. Models were tuned using a validation split and assessed …

Toward A Didactical Phenomenology For The Completeness Axiom, Sean Larsen, Tenchita Alzaga Elizondo, Kristen Vroom, Stephen Strand Ii

School of Mathematical & Statistical Sciences Faculty Publications

The study is part of an instructional design project focused on introductory real analysis. The goal of the project is to develop a theoretically grounded and empirically supported instructional approach that builds on students’ experiences in the calculus sequence to engage them in the reinvention of the rigorous foundations of the calculus. An essential aspect of this foundation is the completeness of the real numbers. Drawing on the didactical phenomenology heuristic from the theory of Realistic Mathematics Education (RME), we conducted an iterative instructional design study focused on the completeness axiom. The work proceeded in two phases. First, we conducted …

Quantization Dimension For A Generalized Inhomogeneous Bi-Lipschitz Iterated Function System, Shivam Dubey, Mrinal Kanti Roychowdhury, Saurabh Verma

School of Mathematical & Statistical Sciences Faculty Publications

For a given r∈(0,+∞), the quantization dimension of order r, if it exists, denoted by Dr(μ), of a Borel probability measure μ on R^d represents the speed how fast the nth quantization error of order r approaches to zero as the number of elements n in an optimal set of n-means for μ tends to infinity. If Dr(μ) does not exists, we call D̲r(μ) and D¯r(μ), the lower and upper quantization dimensions of μ of order r. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems ({fi}i=1^N,(pi)i=0^N,ν), where the …

Centering Culture, Language, And Identity In A History Of Mathematics Course For Preservice Teachers, Luis Miguel Fernández, Tenchita Alzaga Elizondo

School of Mathematical & Statistical Sciences Faculty Publications

This article presents the redesign of an undergraduate History of Mathematics course for preservice teachers at a Hispanic-Serving Institution. Grounded in Community- and Family-Centered Pedagogy (CFCP) and Marshall’s framework for culturally sustaining mathematics teaching, the course intentionally moved beyond Eurocentric narratives to center students’ linguistic, familial, and cultural resources as integral to mathematical meaning-making. Two core assignments anchored the redesign: (1) evolving Padlet reflections that connected global mathematical histories to students’ family and community practices, and (2) bilingual Cuentos Matemáticos that wove mathematical concepts with personal and cultural narratives in English and Spanish. Selected student reflections illustrate how these experiences …