Mathematics Commons^™

Computational Modeling Of Crack-Tip Fields In Transversely Isotropic Strain-Limiting Solids Subjected To Piecewise Linear Slope Loads, S. M. Mallikarjunaiah, Saugata Ghosh, Dambaru Bhatta

School of Mathematical & Statistical Sciences Faculty Publications

Crack-tip fields within a transversely isotropic strain-limiting elastic body are investigated under the influence of piecewise linear slope boundary loads. The mechanical response is characterized via a nonlinear constitutive framework relating the Cauchy stress to the linearized strain, by which non-physical strain singularities at the crack tip are eliminated. The governing system is formulated as a quasi-linear elliptic boundary value problem in terms of the displacement field and is solved utilizing a continuous Galerkin finite element method coupled with a Picard linearization scheme. Boundary conditions are prescribed such that the vertical displacement varies piecewise linearly along the top and bottom …

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 …

The Paradox Of Sport: Adolescent Athletes Are Highly Sedentary Across Competition, Practice, And Rest Days, Jena Heck Street, Abigail M. Larson, Roman De Guia, Naveen K. Bansal, Sandra K. Hunter, Jacob J. Capin

Mathematical and Statistical Science Faculty Research and Publications

Background

Elite athletes are highly sedentary outside of sports and must stay active after retiring to remain healthy. While sports are often recommended to improve physical activity, the effect of sport training day on daily physical activity levels in adolescent athletes is not well described. The purpose was to assess (1) physical activity (i.e., step counts and activity score [METs]) and (2) sedentary behavior (e.g., total duration and prolonged [> 1 h] bouts) in male and female adolescent athletes across different training days (i.e., competition, practice, and rest days).

Methods

Adolescent basketball and volleyball athletes wore an accelerometer continuously for …

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 �� a conjugacy class of parabolic subgroups of G. In previous work, Dr. Crooks defines ��_��→ B_�� , a universal flat family of a fine Hamiltonian Lagrangian G-bundles over ��, and shows that ��_�� is a family of regular Poisson varieties. This manuscript introduces a natural candidate for a quasi-Poisson analog of Dr. Crooks’s construction, ��_��, 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 …

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 Contextual Introduction To Real Analysis: How We Got From There To Here, Robert Rogers, Eugene Boman

Milne Open Textbooks

A Contextual Introduction to Real Analysis: How We Got From There to Here is an innovative introduction to real analysis that presents the subject through the historical development of its key ideas. Rather than introducing rigorous definitions as isolated abstractions, the text shows how concepts such as limits, continuity, convergence, differentiation, and integration emerged in response to specific mathematical challenges. Historical context is woven into the structure of the book, helping students understand both the motivation and necessity behind modern analysis.

The second edition expands the book's coverage with new material on the Fundamental Theorem of Calculus, the Cauchy, Riemann, …

Computation Of Minimum Numbers Of Tile And Bond-Edge Types For Dna Self-Assembly Of Select Archimedean Graphs, Tabitha Merrithew, Jessica Sorrells

Theory & Applications of Graphs

This project mathematically models the self-assembly of DNA nanostructures in the shape of select Archimedean graphs using the flexible tile model. Under the most strict set of conditions of this model, we employ principles of linear algebra and graph theory to determine the minimum number of different DNA branched molecules needed to construct the desired shapes, theoretically reducing laboratory costs and the waste of biomaterials. We determine exact values for $T_3(G)$, the minimum number of molecule (or ``tile") types needed for all six order 12 and 24 Archimedean graphs. We also determine exact values for $B_3(G)$, the minimum number of …

Codee Expects That Every Mathematician Will Do Their Odes: From The Battle Of Trafalgar To Calculus (Or Nelson To Newton), Steven Miller

CODEE Journal

Using the historical Battle of Trafalgar as a case study, we introduce students to mathematical modeling to describe and explore real-world systems. Starting with the question—how do force sizes and combat effectiveness determine which side wins?—we show how to use a blend of mathematics and computational tools to investigate the answer. Beginning by defining a simple differential equation model for combat dynamics, we analyze its behavior using differential equations and accessible algebraic techniques. We show how computational methods and simulations can support systematic, visual exploration and suggest conjectures that mathematical analysis can then establish. This material could serve as the …

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 …

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 …

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 …