Mathematics Commons^™

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 …

Observations On Recurrent Loss In The Neural Network Model Of A Partial Differential Equation: The Advection–Diffusion Equation, Jonah A. Reeger

Faculty Publications

A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic exploration of the stability of the solutions generated by these ML approaches. Here, a recurrent network is introduced that matches precisely the evaluation of a multi-step method paired with a collocation method for approximating spatial derivatives in the advection–diffusion equation. This allows for two things: (1) the use of traditional tools for analyzing the stability of a numerical method for solving PDEs and (2) bringing to bear …

Multiscale Computational Modeling Of The Cardiopulmonary Consequences Of Postnatal Hyperoxia With Implications For Preterm-Born Children, Salla M. Kim, Filip Jezek, Pim J. A. Oomen, Gregory P. Barton, Feng Gu, Daniel A. Beard, Kara N. Goss, Mitchel J. Colebank, Naomi C. Chesler

Faculty Publications

Moderate to extreme preterm birth (<  32 weeks gestation) affects cardiopulmonary structure and function and is associated with increased risk of heart failure through adulthood. The rat hyperoxia (Hx) model (term born; postnatal Hx exposure) captures biventricular changes, including at the cell and organ scale, and pulmonary vascular remodeling seen in preterm humans. However, synthesizing these measures across scales and organ systems is challenging. We hypothesized that in silico modeling of biventricular mitochondrial, myofiber, and organ-scale function plus circulatory function could capture key features of cardiopulmonary abnormalities due to preterm birth. Therefore, we calibrated a multiscale model to subject-specific biventricular pressure–volume data previously obtained from Hx rats alongside normoxic (Nx) controls to investigate the abnormalities in cardiopulmonary function at multiple scales in this animal model of human preterm birth. The calibrated model demonstrates excellent agreement with the data and captures the expected pulmonary vascular changes and right ventricular dilation seen in preterm-born children. Our multiscale modeling approach captures cardiopulmonary abnormalities across spatial scales and provides an innovative approach to explore the consequences of preterm birth beyond preclinical experimental data alone. This is a foundational step in understanding the impact of preterm birth on cardiopulmonary disease in childhood as well as adulthood.

Optimal Error Estimates Of The Diffuse Domain Method For Second Order Parabolic Equations, Wenrui Hao, Lili Ju, Yuejin Xu

Faculty Publications

In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM employs a phase-field function to extend the original parabolic problem to a similar but slightly modified problem defined over a larger rectangular domain that contains the target physical domain. Based on the weighted Sobolev spaces, we rigorously establish the convergence of the diffuse domain solution to the original solution as the interface thickness parameter goes to zero, together with the corresponding optimal error estimates under …

Efficient Energy-Stable Discontinuous Galerkin Schemefor The Non-Isothermal Cahn–Hilliard–Navier–Stokestwo-Phase Fluid Flow System, Guang-An Zou, Meiting Wang, Kejia Pan, Yin Yang, Xiaofeng Yang

Faculty Publications

In this article, we propose a novel numerical framework for the non-isothermal Cahn–Hilliard–Navier–Stokes two-phase flow system, which couples the incompressible Navier–Stokes equations, the Cahn–Hilliard phase-field equation, and the heat transport equation to capture temperature-dependent two-phase flow dynamics. The pro-posed scheme achieves three major advances: (i) unconditional energy stability through a combined scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) approach, (ii) linearity and full decoupling of all variables while using a second-order temporal discretization, and (iii) efficient implementation via discontinuous Galerkin (DG) spa-tial discretization together with a second-order projection method for the Navier–Stokes equations. We rigorously prove the unconditional energy stability …

Congruence Properties Modulo Prime Powers For A Class Of Partition Functions, Matthew Boylan, Swati .

Faculty Publications

Let p be prime, and let p_[1,p](n) denote the function whose generating function is

∏ _n≥1 (1 − qⁿ)⁻¹ (1 − q^pn)⁻¹.

This function and its generalizations p_[c^r,d^m](n) are the subject of study in several recent papers. Let ℓ ≥ 5, let j ≥ 1, and let p ∈ {2, 3, 5}. In this paper, we prove that the generating function for p_[1,p](n) in the progression β_{p,ℓ,j …}

Limit Theorems For Andrews’ Restricted Overpartitions, Tapas Bhowmik, Alex Cao, Jack Frew, John Lehman, Wei-Lun Tsai

Faculty Publications

The study of overpartitions in recent years has been used to great effect in various fields, including hypergeometric series, q-series identities, and mathematical physics. We investigate the limiting distributions of the number of parts in a family of overpartitions of n,  introduced by Andrews, where parts are counted with two different weights. Using Andrews’ identities and the saddle-point method, we establish two central limit theorems (CLTs) for the number of parts as n → ∞, corresponding to these weightings. We also derive explicit formulas for the mean and variance in each case.

Patient-Specific Lumped-Parameter Model For Quantifying Vessel-Specific Remodeling And Predicting Right Ventricular Function In Pulmonary Hypertension, Christopher G. Lechuga, Amirreza Kachabi, Mitchel J. Colebank, Claudia E. Korcarz, Farhan Raza

Faculty Publications

Purpose: Pulmonary hypertension (PH) is a heterogeneous disease with patient-specific variability and vessel-specific remodeling, which eventually lead to right ventricular (RV) failure. The gold standard for RV assessment—pressure–volume (PV) loop acquisition—is invasive and limited to specialized settings. This study aims to develop a patient-specific lumped-parameter model that quantifies vessel-specific remodeling and simulates RV PV loops across PH phenotypes using routine clinical data.

Methods: A lumped-parameter model was calibrated using right heart catheterization and echocardiography data. Model agreement was assessed by R2 values for pressure and flow goodness-of-fit, and model-derived hemodynamic metrics were comparedwith clinical values. A dimensionality reduction approach was …

High Frobenius Pushforwards Generate The Bounded Derived Category, Matthew R. Ballard, Srikanth B. Iyengar, Pat Lank, Alapan Mukhopadhyay, Josh Pollitz

Faculty Publications

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of D(X) the Frobenius pushforward F^e_*G generates the bounded derived category whenever p^e is larger than the codepth of X, an invariant that is a measure of the singularity of X. The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take G = …

Wave Intensity Analysis With Exercise Identifies Impairments In Pulmonary Hypertension, Christopher G. Lechuga, Farhan Raza, Mitchel J. Colebank, Claudia E. Korcarz, Jens C. Eickhoff, Naomi C. Chesler

Faculty Publications

Wave intensity analysis provides a novel approach to understanding the dynamic interactions between the right ventricle and pulmonary vasculature, particularly in pulmonary hypertension, a condition characterized by elevated pulmonary arterial pressures and vascular remodeling. This prospective study used wave intensity analysis to evaluate right ventricular and pulmonary vascular mechanics in 22 participants with pulmonary hypertension (including precapillary, isolated postcapillary, and combined pre/postcapillary pulmonary hypertension), and three without pulmonary hypertension. Forward and backward compression and decompression waves were quantified at rest and during incremental exercise (25, 50, and 75 W). Relationships between metrics of wave intensity analysis, hemodynamics, right ventricular function, …

How Multi-Scale Modeling Can Help Examine Social Determinants Of Health And Resulting Disparities, Kyoko Yoshida, Elsje Pienaar, Shalanda A. Bynum, Naomi Chesler, Mitchel J. Colebank, Jessie Heneghan, Nadra Tyus, Jasmine Miller-Kleinhenz, Bruce Y. Lee

Faculty Publications

Social determinants of health (SDOH) are the conditions in which people live, work, and play, and the wider set of factors (e.g., social and economic systems and policies) that shape a person’s daily life. SDOH can differ significantly across communities and populations, having positive impacts for some and negative impacts for others. Ultimately, this results in differences in health and disease distribution, that are known as health disparities. Despite the known impacts of SDOH and calls to characterize, address, reduce, and eliminate health disparities, they persist and, in some cases, have worsened. To address this challenge, a session at the …

Let’S Teach More Accurate And Inclusive History! The Case Of Islamic Contributions To Mathematics And Science, Noah (Nuh) Aydin

Faculty Publications

Schools commonly teach a history of mathematics and science that is Eurocentric. This selective version of the history of science, called Classical Narrative by some, is rooted in colonial times and mindset, and has reinforced certain negative opinions about other cultures. It ignores or downplays contributions to mathematics and sciences from non-western civilizations, and falsely attributes many scientific discoveries to European scholars. Medieval Islamic Civilization, which had strong connections to Renaissance Europe, serves as a clear example of how this narrative is distorted. Based on research on primary sources since the middle of the 20th century, we now know that …

Time-Dependent Shear Flows And Their Applications In Parabolic–Parabolic Patlak–Keller–Segel Systems*, Siming He

Faculty Publications

In this study, we investigate the behavior of three-dimensional parabolic–parabolic Patlak–Keller–Segel systems in the presence of ambient shear flows. Our findings demonstrate that when the total mass of the cell density is below a specific threshold, the solution remains globally regular as long as the flow is sufficiently strong. The primary difficulty in our analysis stems from the fast creation of chemical gradients due to strong shear advection.

Permutations Minimizing The Number Of Collinear Triples, Joshua Cooper, Jack Hyatt

Faculty Publications

We characterize the permutations of Fq whose graph minimizes the number of collinear triples and describe the lexicographically-least one, confirming a conjecture of Cooper-Solymosi. This question is connected to Dudeney’s No-3-in-a-Line problem, the Heilbronn triangle problem, and the structure of finite plane Kakeya sets. We discuss a connection with complete sets of mutually orthogonal latin squares and state a few open problems primarily about general finite affine planes.

A Comparison Of Gaussian Processes And Polynomial Chaos Emulators In The Context Of Haemodynamic Pulse–Wave Propagation Modelling, L. Mihaela Paun, Mitchel J. Colebank, Dirk Husmeier

Faculty Publications

Computational modelling of the cardiovascular system is a promising future direction for patient-specific healthcare. However, the computational cost of these simulators is a bottleneck for their practical use in clinic for real-time digital twins. Emulation can overcome this, yet an extensive investigation into cardiovascular emulators is warranted. In this study, we emulate two one-dimensional haemodynamics models of the pulmonary circulation and compare two common emulation strategies: Gaussian processes (GPs) and polynomial chaos expansions (PCEs). We start by reducing the parameter space of the models through global sensitivity analysis, and then compare both emulation strategies using a multivariate, time-series output …

Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, Benjamin F. Akers, Kristina O. F. Williams

Faculty Publications

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data …

A Generalization Of The Graham-Pollak Tree Theorem To Even-Order Steiner Distance, Joshua N. Cooper, Gabrielle Tauscheck

Faculty Publications

Graham and Pollak showed in 1971 that the determinant of a tree’s distance matrix depends only on its number of vertices, and, in particular, it is always nonzero. The Steiner distance of a collection of 𝑘 vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for 𝑘 =2 , this reduces to the ordinary definition of graphical distance. Here we show that the hyperdeterminant of the 𝑘 th order Steiner distance hypermatrix is always nonzero if 𝑘 is even, extending their result beyond 𝑘 =2 . Previously, the authors showed that the …

On The Asymptotics Of Cubic Fields Ordered By General Invariants, Arul Shankar, Frank Thorne

Faculty Publications

In this article, we introduce a class of invariants of cubic fields termed “generalized discriminants”. We then obtain asymptotics for the families of cubic fields ordered by these invariants. In addition, we determine which of these families satisfy the Malle–Bhargava heuristic.

Deterministic Global 3d Fractal Cloud Model For Synthetic Scene Generation, Aaron M. Schinder, Shannon R. Young, Bryan J. Steward, Michael L. Dexter, Andrew Kondrath, Stephen Hinton, Ricardo Davila

Faculty Publications

This paper describes the creation of a fast, deterministic, 3D fractal cloud renderer for the AFIT Sensor and Scene Emulation Tool (ASSET). The renderer generates 3D clouds by ray marching through a volume and sampling the level-set of a fractal function. The fractal function is distorted by a displacement map, which is generated using horizontal wind data from a Global Forecast System (GFS) weather file. The vertical windspeed and relative humidity are used to mask the creation of clouds to match realistic large-scale weather patterns over the Earth. Small-scale detail is provided by the fractal functions which are tuned to …

Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox

Faculty Publications

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface …

Note On The Spectra Of Steiner Distance Hypermatrices, Joshua Cooper, Zhibin Du

Faculty Publications

The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-k Steiner distance hypermatrix of a graph is the n-dimensional array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of k = 2, this reduces to the classical distance matrix of a graph. Graham and Pollak showed in 1971 that the determinant of the distance matrix of a tree only depends on its number n of vertices. Here, we show that the hyperdeterminant of the Steiner distance …

A Generalization Of The Graham-Pollak Tree Theorem To Even-Order Steiner Distance, Joshua Cooper, Gabrielle Tauscheck

Faculty Publications

Graham and Pollak showed in 1971 that the determinant of a tree’s distance matrix depends only on its number of vertices, and, in particular, it is always nonzero. The Steiner distance of a collection of k vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for k = 2, this reduces to the ordinary definition of graphical distance. Here we show that the hyperdeterminant of the kth order Steiner distance hypermatrix is always nonzero if k is even, extending their result beyond k = 2. Previously, the authors showed that the k-Steiner …

Extending Natural Mates In Euclidean 3-Space And Applications To Bertrand Pairs, Yun Myung Oh, Alexander Navarro

Faculty Publications

In Euclidean 3-space, a family of curves, the co-successor, is motivated and then introduced in relation to the natural mate. A complete characterization of co-successors is proved, followed by an application of the co-successor towards describing Bertrand curves and their mates.

Shock-Wave Tolerant Phase Reconstruction Algorithm For Shack–Hartmann Wavefront Sensor Data, Thomas E. Defoor, Matthew Kalensky, Matthew R. Kemnetz, Timothy J. Bukowski, Mark F. Spencer

Faculty Publications

We develop a phase reconstruction algorithm for the Shack–Hartmann wavefront sensor (SHWFS) that is tolerant to phase discontinuities, such as the ones imposed by shock waves. In practice, this algorithm identifies SHWFS locations where the resultant tilt information is affected by the shock and improves the tilt information in these locations using the local SHWFS observation-plane irradiance patterns. The algorithm was shown to work well over the range of conditions tested with both simulated and experimental data. In turn, the reconstruction algorithm will enable robust wavefront sensing in transonic, supersonic, and hypersonic environments.

On Determining The Equation Of A Salkowski Curve Satisfying Tau/Kappa=1/S, Yun Myung Oh, Devin Garcia-Roblero

Faculty Publications

In this paper, we determine the equation of a Salkowski curve whose ratio of torsion to curvature is given by 1/s, where s is the arc length of the curve. The Frenet-Serret equations provide the third-order vector differential equation for the unit tangent vector T(s) and the general (series) solution was obtained. In the end, the series solution is entirely determined by the given initial conditions.

Legendre Pairs Of Lengths ℓ ≡ 0 (Mod 5), Ilias S. Kotsireas, Christopher Koutschan, Dursun A. Bulutoglu, David M. Arquette, Jonathan S. Turner, Kenneth J. Ryan

Faculty Publications

By assuming a type of balance for length ℓ = 87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length ℓ = 85 , we find LPs of these lengths. We then study the power spectral density (PSD) values of m compressions of LPs of length 5 m . We also formulate a conjecture for LPs of lengths ℓ ≡ 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range ≤ 200 for …

Anomaly Detection In The Molecular Structure Of Gallium Arsenide Using Convolutional Neural Networks, Timothy Roche, Aihua W. Wood, Philip Cho, Chancellor Johnstone

Faculty Publications

This paper concerns the development of a machine learning tool to detect anomalies in the molecular structure of Gallium Arsenide. We employ a combination of a CNN and a PCA reconstruction to create the model, using real images taken with an electron microscope in training and testing. The methodology developed allows for the creation of a defect detection model, without any labeled images of defects being required for training. The model performed well on all tests under the established assumptions, allowing for reliable anomaly detection. To the best of our knowledge, such methods are not currently available in the open …

A Comparison Of Quaternion Neural Network Backpropagation Algorithms, Jeremiah Bill, Bruce A. Cox, Lance Champaign

Faculty Publications

This research paper focuses on quaternion neural networks (QNNs) - a type of neural network wherein the weights, biases, and input values are all represented as quaternion numbers. Previous studies have shown that QNNs outperform real-valued neural networks in basic tasks and have potential in high-dimensional problem spaces. However, research on QNNs has been fragmented, with contributions from different mathematical and engineering domains leading to unintentional overlap in QNN literature. This work aims to unify existing research by evaluating four distinct QNN backpropagation algorithms, including the novel GHR-calculus backpropagation algorithm, and providing concise, scalable implementations of each algorithm using a …

Numerical Simulation Of The Korteweg–De Vries Equation With Machine Learning, Kristina O. F. Williams, Benjamin F. Akers

Faculty Publications

A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. This method trains stencil weights of a discretization of the equation, with the truncation error of the scheme as the objective function for training. The method uses centered finite differences to initialize the optimization routine and a second-order implicit-explicit time solver as a framework. Symmetry conditions are enforced on the learned operator to ensure a stable method. The procedure is applied to the Korteweg–de Vries equation. It is observed to be more accurate than finite difference or spectral methods on coarse …

A Bit-Parallel Tabu Search Algorithm For Finding Es2 -Optimal And Minimax-Optimal Supersaturated Designs, Luis B. Morales, Dursun A. Bulutoglu

Faculty Publications

We prove the equivalence of two-symbol supersaturated designs (SSDs) with N (even) rows, m columns, s_max=4t+i, where i ∈ {0,2}, t ∈ Z^≥0 and resolvable incomplete block designs (RIBDs) whose any two blocks intersect in at most (N+4t+i)/4 points. Using this equivalence, we formulate the search for two-symbol E(s²)-optimal and minimax-optimal SSDs with s_max ∈ {2,4,6} as a search for RIBDs whose blocks intersect accordingly. This allows developing a bit-parallel tabu search (TS) algorithm. The TS algorithm found E(s²)-optimal and minimax-optimal SSDs achieving the sharpest known E(s²) lower bound with …