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, …

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 …

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 …

Untangling Neural Complexity, Kyle Cahill

Mathematics Colloquium Series

It’s been said that “neurons that fire together wire together”—but this is only part of the story. Learning involves neuroplastic refinement of neural circuits and networks, promoting effective coordination while weakening or pruning redundant connections. How can mathematical models help us understand this phenomenon? Using visual simulations of the Kuramoto model of synchronization, we will explore a central feature of neural complexity: how interactions among individual neurons give rise to coordinated activity across a network. I will introduce ideas from Dynamic Resource Theory to help us physically constrain this model through the distribution and availability of finite energetic resources. Finally, …

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, …

Constructing Chaotic Functions On Nonorientable Surfaces Via Interval Maps, Ebru Usluca, Fati̇h Uçan, Nedi̇m Deği̇rmenci̇

Turkish Journal of Mathematics

We construct chaotic functions on the nonorientable surfaces known as the Möbius strip, Klein bottle, and projective plane, which are well-known objects in topology. As is customary, these surfaces are obtained as quotient spaces of the unit square [0,1]×[0,1]. To construct chaotic functions on these surfaces, we first obtain chaotic functions on the unit square using the product method. In the product method, we utilize well-known chaotic maps on the unit interval, namely the logistic and tent maps. We then observe that each Cartesian product map induces a well-defined chaotic function on the corresponding surfaces. We also point out that …

Stability Of The Fractional (P,Q)-Difference Duffing Equation Formulated By The Postquantum Operators, Sina Etemad, Shahram Rezapour

Turkish Journal of Mathematics

Modeling different physical phenomena plays an important role when their qualitative behaviors are analyzed from a mathematical point of view. In this paper, we focus on the stability analysis of a fractional initial value problem formulated by (p,q)-operators. More precisely, on the basis of the standard model of a Duffing equation, we reformulate it as a fractional Caputo-type (p,q)-difference Duffing equation equipped with two initial conditions. Indeed, (p,q) derivatives are used for the first time to model the fractional version of the Duffing equation. In this direction, we …

Dynamical Behavior In A Three-Dimensional Rational Discrete-Time System, Zafar Saydimaxmudovich Boxonov

Turkish Journal of Mathematics

In this paper, we investigate the dynamic behavior of a three-dimensional rational discrete system with nine parameters. We analyze the system's local and global dynamics, determining conditions for parameter values to ensure positivity of trajectories. We prove the absence of periodic points and conduct global stability analysis using invariant sets. Furthermore, we give biological interpretations of our results.

Interpolative Ideals Of Bloch Mappings, David Ruiz-Casternado

Turkish Journal of Mathematics

For 0 ≤ α < 1, we present a generalization of Matter's interpolation procedure for generating Banach normalized Bloch ideals. As a main example, given 1 ≤ p,q,r < ∞ and 0 ≤ α,β < 1 verifying 1/r + (1−α)/p + (1−β)/q = 1, we introduce and study the class of (p,α,q,β)-dominated Bloch maps from the open complex unit disc D into a complex Banach space X, denoted by D^ℬ_(p,α,q,β)(D,X). We show this subclass of holomorphic mappings constitutes a Möbius-invariant injective Banach normalized Bloch ideal under a suitable norm D^ℬ_{(p,α …}

Spectral Properties Of The Cauchy Transform On Modified Bergman Spaces, Khaled Chbichib, Noureddine Ghiloufi, Safa Snoun

Turkish Journal of Mathematics

In this paper, we determine the singular values s_n(T_α,β) and s_n(R_α,β) of the operators T_α,β = CP_α,β and R_α,β = P_α,βCP_α,β, where C is the integral Cauchy transform and P_α,β is the orthogonal projection from L²(D,μ_α,β) onto the modified Bergman space A²(D,μ_α,β). These singular values are …

A Study Of Random Fuzzy Fractional Differential Equations Under The $\Psi$-Caputo Fractional Derivative, Ho Vu, Osman Tunç

Turkish Journal of Mathematics

In this paper, we study a class of random fuzzy fractional initial value problems (RFFIVPs) that involve the Ψ–Caputo fractional derivative (Ψ–Cdf). We establish a theorem on the existence and uniqueness of d-monotone solutions for this problem by employing a method of successive approximations. Additionally, we derive explicit a priori estimates for the solution and show its continuous dependence on both the initial conditions and the nonlinear terms. Furthermore, by applying an appropriate Grönwall-type inequality adapted to the Ψ-kernel, we demonstrate that this problem exhibits ℌ–U stability.

Magnetic Curves In Tangent Sphere Bundles Iii, Jun-Ichi Inoguchi, Marian Ioan Munteanu

Turkish Journal of Mathematics

We investigate contact magnetic curves in the unit tangent sphere bundle of the hyperbolic plane under appropriate conservation law. We prove that normal contact magnetic curves in UH²(−c²) making constant angle with fibers are tangential Gauss maps of Kähler magnetic trajectories in H²(−c²).

Power Series Statistical Riemann And Lebesgue Integrability With Applications To Korovkin Theorems, Bidu Bhusan Jena, Sevda Yildiz, Ni̇lay Şahi̇n Bayram

Turkish Journal of Mathematics

This study investigates the statistical convergence and summability of sequences of real-valued functions under a generalized power series method. Fundamental concepts of Riemann integrals for sequences of functions are revisited, leading to the formulation of statistical Riemann summability and integrability via the proposed power series framework. The scope of this research is subsequently expanded to analyze statistical Lebesgue summability and integrability through the lens of the introduced method. Based on these summability techniques, we construct and prove Korovkin-type approximation theorems concerning sequences of real-valued functions. Moreover, a computational and geometrical interpretation of the results is presented with the help of …

Chen-Type Inequalities Involving Mutual Curvatures On Submanifolds Of Riemannian 3-Product And Statistical Manifolds, Aliya Naaz Siddiqui, Mehmet Gülbahar, Fatemah Mofarreh

Turkish Journal of Mathematics

The aim of this work is to establish several fundamental relations involving mutual curvatures on submanifolds of Riemannian and statistical manifolds. In the first part, we investigate mutual curvatures on Riemannian 3-product manifolds and establish the main relations involving mutual curvatures on submanifolds of a Riemannian 3-product space. In the second part, we investigate mutual curvatures on statistical manifolds. We introduce the concept of mutual curvature for statistical manifolds endowed with pairwise orthogonal, but noncomplementary, distributions of the tangent bundle. We establish a geometric inequality for statistical submanifolds that relates mutual-curvature-based invariants to the squared norm of the mean curvature …

Generalized Rectifying Ruled Surfaces According To Quasi-Frame, Onur Kaya

Turkish Journal of Mathematics

In this study, we define generalized rectifying ruled surfaces according to the quasi-frame and determine their singular points. We then investigate when such surfaces are developable or minimal. We give the fundamental forms and Weingarten map. Furthermore, we determine the cases when curves lying on the surface are lines of curvature, asymptotic curves, or parabolic curves. We present relations between generalized rectifying ruled surfaces and some other special ruled surfaces. Finally, we illustrate the obtained results with examples.

Concept Of A Trace And Shauder-Type Estimate In Banach–Sobolev Spaces, Eminaga Mamedov, Yonca Sezer, Yeter M. Farhadova

Turkish Journal of Mathematics

The results of this paper are a continuation and generalization of those obtained in previous works [4, 6, 24]. The main aim is to establish a Schauder-type estimate up to the boundary in Banach–Sobolev spaces generated by rearrangement-invariant spaces. We present a more abstract method rather than the classical one, using the interior Schauder-type estimate established in [24], extendability properties of the continuous functions, and elliptic operators defined on closed domains. Note that Lebesgue spaces L^p(Ω), grand-Lebesgue spaces L^p)(Ω), Marcinkiewicz spaces, weak L_w^p spaces, Orlicz spaces, Lorentz spaces, etc. are covered.