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, …

Quasi-Poisson Analogs Of Regular Poisson Varieties, Casen Thompson

All Graduate Theses and Dissertations, Fall 2023 to Present

Let G be a complex semisimple linear algebraic group with 𝐶 a conjugacy class of parabolic subgroups of G. In previous work, Dr. Crooks defines 𝑢_𝐶→ B_𝐶 , a universal flat family of a fine Hamiltonian Lagrangian G-bundles over 𝐶, and shows that 𝑢_𝐶 is a family of regular Poisson varieties. This manuscript introduces a natural candidate for a quasi-Poisson analog of Dr. Crooks’s construction, 𝑢_𝐶, using the framework of quasi-Poisson geometry established by Alekseev, Kosmann-Scwarzbach, and Meinrenken. It will be shown that this proposed quasi-Poisson analog indeed has a valid quasi-Poisson structure and …

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 …

Shadowable Diffeomorphisms With Dominated Splitting, Naoya Sumi

Journal of Stochastic Analysis

We study the shadowability for diffeomorphisms with dominated

splitting. In this paper we show that if a diffeomorphism f has a periodic

saddle q whose unstable manifold is compatible with dominated splitting in

some sense and if f is shadowable, then any other periodic point p has a large

local stable set. Our result does not require the hyperbolicity of p.

Data-Driven Discovery Of Canonical Models In Undergraduate Differential Equations Education, Devon L. Garls, Raina Majumder, Scott A. Strong

CODEE Journal

Roughly a decade ago, Brunton, Proctor, and Kutz built upon the re-

sults of classical control theory, compressed sensing, and symbolic regression

to create a method for Sparse Identification of Nonlinear Dynamics (SINDy),

allowing one to infer a dynamical system from time series and bring mathe-

matical dynamics into the modern era of large data. This paper is designed for

a reader concurrently enrolled in a first course in ordinary differential equa-

tions and begins with a technical overview of least-squares regression and

finite-difference approximations to derivatives, which is enough to perform

an example hand-calculation on three data points. The …

A Biased Variant Of The Ladybug Clock Problem, Jacob Vogelpohl, Russell May

Communications on Number Theory and Combinatorial Theory

Instead of the common assumption of uniform probabilities for steps in a random walk on a graph, we generalize the steps to be biased and compute several statistics related to biased random walks on a cycle graph, more fancifully known as the ``ladybug clock problem." We obtain new explicit formulas for the distribution of probabilities for a walk to end on a particular node, the expected cover time for the cycle, and other similar statistics. We carry out these computations solely with generating functions related to the classic gambler's ruin problem.

Computing The Greatest Common Divisor Of Binomial Coefficients C(Mn, Mk), Chai Wah Wu

Communications on Number Theory and Combinatorial Theory

The greatest common divisor (GCD) of C(2n,2k) for 1 ≤ k < n is known to be some power of 2 times the product of all odd primes p such that 2n = pⁱ + p^j. We complete the analysis of this GCD by showing that its 2-adic valuation is at most 1 and relates it to Mersenne primes. We also show how to efficiently compute GCD {C(mn,mk) : 1 ≤ k < n} when n and m satisfy certain conditions.

On A Remark Of De Gennes Concerning Three-Dimensional Polyelectrolytes, Shiquan Li, Carl Mueller

Journal of Stochastic Analysis

This work is inspired by a remark of de Gennes [3] about polyelectrolytes, which are charged polymers. A common model for a polymer is a self-avoiding or self-repelling random walk or Brownian motion. For polyelectrolytes, the repelling potential is the Coulomb potential arising from pairs of charged particles. We show that in the continuous case of a Brownian motion in three dimensions, the spread of the polymer, in particular the radius of gyration of a polyelectrolyte of length T grows linearly with T, up to logarithmic corrections.

How Wide Is The Screening Window For Aggressive Non-Small Cell Lung Cancers? The Cure Threshold Metric And Its Implications For Lung Cancer Screening Guidelines, Deborah Goldwasser

Mathematics Colloquium Series

The entire NSU community, including students at all levels of mathematics, is invited and encouraged to attend. A reception with coffee and snacks will take place in the Department of Mathematics Suite Parker 320 from 12:00-12:30 just prior to the colloquium talk. Abstract: Lung cancer screening by low-dose computed tomography (LDCT) has demonstrated a mortality reduction in two large randomized controlled trials. However, LDCT also frequently identifies benign pulmonary nodules in high-risk but otherwise healthy individuals. The management of small pulmonary nodules is guided primarily by malignancy probability.  However, the surgical cure threshold, the maximum tumor size at which surgical …

A Note On The Voronoi Congruences And The Residue Of The Fermat Quotient, Claire Levaillant

Communications on Number Theory and Combinatorial Theory

We prove a congruence on the residue of the Fermat quotient in base a which arises from a generalization of the Voronoi congruences and other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we recover a congruence for the generalized harmonic number H_2,(p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known for the harmonic number H_(p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruences that is different from the one of …

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 …

On Graphs With Chromatic Number And Maximum Degree Both Equal To Nine, Rachel Galindo, Jessica Mcdonald

Theory & Applications of Graphs

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with $\chi = \Delta = 9$ contains $K_3 \lor E_6$ as a subgraph. Here we prove several results in support of this conjecture, where vertex-criticality and forbidden substructure conditions get us either close or all the way to containing $K_3 \lor E_6$.

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 …

Expressing Matrix Powers With Pell Numbers, Walby Encarnacion Cena

Rose-Hulman Undergraduate Mathematics Journal

Abstract. Motivated by the theory of Cayley hash functions, we aim to write a matrix as a product of another set of given matrices. Riley gave such an algorithm using Fibonacci numbers. We extend Riley’s work to the Pell numbers. We compare our algorithm to Riley’s method and find that it is an improvement in certain cases.

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 …

Beyond Intent: Teacher Capacity, Opportunity To Learn, And The Persistence Of Math Equity Gaps In Tennessee (2000-2025), Soso Dede

The Journal of the Research Association of Minority Professors

For a quarter-century (2000–2025), Tennessee pursued mathematics reforms aiming for equity, yet stark racial and socioeconomic achievement gaps endure. Framed by Opportunity to Learn (OTL) and Mathematical Knowledge for Teaching (MKT), this study investigates the intersection of equity, teacher capacity, and policy. By synthesizing assessment data, program evaluations, policy documents, and research literature, this study reveals that systemic OTL inequities—disparate access to quality instruction, adequate resources, and teachers possessing strong MKT—are primary drivers of these enduring gaps. Such inequities deny marginalized students a fair chance at mathematical proficiency. The interplay between OTL and MKT is critical: inadequate OTL constrains knowledgeable …