Einstein's Model Of "The Movement Of Small Particles In A Stationary Liquid" Revisited: Finite Propagation Speed, 2023 TÜBİTAK

#### Einstein's Model Of "The Movement Of Small Particles In A Stationary Liquid" Revisited: Finite Propagation Speed, Akif Ibragimov, Zeev Sobol, Isanka Hevage

*Turkish Journal of Mathematics*

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is dependent on the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentrations. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality, and the Ladyzhenskaya-Ural'tseva iteration lemma.

Studying A Doubly Nonlinear Model Of Slightly Compressible Forchheimer Flows In Rotating Porous Media, 2023 TÜBİTAK

#### Studying A Doubly Nonlinear Model Of Slightly Compressible Forchheimer Flows In Rotating Porous Media, Emi̇ne Çeli̇k, Luan Hoang, Thinh Kieu

*Turkish Journal of Mathematics*

We study the generalized Forchheimer flows of slightly compressible fluids in rotating porous media. In the problem's model, the varying density in the Coriolis force is fully accounted for without any simplifications. It results in a doubly nonlinear parabolic equation for the density. We derive a priori estimates for the solutions in terms of the initial, boundary data and physical parameters, emphasizing on the case of unbounded data. Weighted Poincare-Sobolev inequalities suitable to the equation's nonlinearity, adapted Moser's iteration, and maximum principle are used and combined to obtain different types of estimates.

Attractors For Semigroups With Multi-Dimensional Time And Pdes In Unbounded Domains, 2023 TÜBİTAK

#### Attractors For Semigroups With Multi-Dimensional Time And Pdes In Unbounded Domains, Anna Kostianko, Sergey Zelik

*Turkish Journal of Mathematics*

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous works in this direction our theory does not require the underlying domain to be cylindrical or cone-like or to be shift semiinvariant with respect to some direction. In particular, the theory is applicable to the exterior domains.

Convergence Of A Linearly Regularized Nonlinear Wave Equation To The P-System, 2023 TÜBİTAK

#### Convergence Of A Linearly Regularized Nonlinear Wave Equation To The P-System, Hüsnü Ata Erbay, Saadet Erbay, Albert Kohen Erki̇p

*Turkish Journal of Mathematics*

We consider a second-order nonlinear wave equation with a linear convolution term. When the convolution operator is taken as the identity operator, our equation reduces to the classical elasticity equation which can be written as a $p$-system of first-order differential equations. We first establish the local well-posedness of the Cauchy problem. We then investigate the behavior of solutions to the Cauchy problem in the limit as the kernel function of the convolution integral approaches to the Dirac delta function, that is, in the vanishing dispersion limit. We consider two different types of the vanishing dispersion limit behaviors for the convolution …

Discontinuous Galerkin Method For Blow-Up Solutions Of Nonlinear Wave Equations, 2023 TÜBİTAK

#### Discontinuous Galerkin Method For Blow-Up Solutions Of Nonlinear Wave Equations, Asma Azaiez, Mondher Benjemaa, Aida Jrajria, Hatem Zaag

*Turkish Journal of Mathematics*

e develop and study an explicit time-space discrete discontinuous Galerkin finite element method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several examples and benchmarks.

Blow-Up Of Solutions For Wave Equation With Multiple ?(X)-Laplacian And Variable Exponent Nonlinearities, 2023 TÜBİTAK

#### Blow-Up Of Solutions For Wave Equation With Multiple ?(X)-Laplacian And Variable Exponent Nonlinearities, Aya Khaldi, Amar Ouaoua, Messaoud Maouni

*Turkish Journal of Mathematics*

We consider an initial value problem related to the equation \begin{equation*} u_{tt}-{div}\left( \left\vert \nabla u\right\vert ^{m\left( x\right) -2}\nabla u\right) -{div}\left( \left\vert \nabla u_{t}\right\vert ^{r\left( x\right) -2}\nabla u_{t}\right) -\gamma \Delta u_{t}=\left\vert u\right\vert ^{p\left( x\right) -2}u, \end{equation*} with homogeneous Dirichlet boundary condition in a bounded domain $\Omega $. Under suitable conditions on variable-exponent $m\left( .\right) ,$ $r\left( .\right), $ and $p\left( .\right) ,$ we prove a blow-up of solutions with negative initial energy.

The Generalized Lyapunov Function As Ao’S Potential Function: Existence In Dimensions 1 And 2, 2023 Missouri University of Science and Technology

#### The Generalized Lyapunov Function As Ao’S Potential Function: Existence In Dimensions 1 And 2, Haoyu Wang, Wenqing Hu, Xiaoliang Gan, Ping Ao

*Mathematics and Statistics Faculty Research & Creative Works*

By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collabora-tors recently. We show that this potential function agrees with the generalized Lyapunov function of the deterministic part of the stochastic dynamical sys-tem. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also …

Biological Oscillator Synchronization With The Cellular Potts Model, 2023 Western Washington University

#### Biological Oscillator Synchronization With The Cellular Potts Model, Rose Una

*WWU Honors College Senior Projects*

Similar to how neurons synchronize their firing in the brain, individual cells of certain single-celled species can synchronize their internal oscillatory molecular clocks to those of their neighboring cells. This study develops and analyzes an abstract, discrete agent-based computational model to investigate the movement and synchronization of internal oscillators in biological cells. We adapt a Cellular Potts Model to explore this oscillator synchronization process with two-dimensional cells on a square lattice. Model assumptions are motivated by behavior in single-celled species of slime mold (Dictyostelium discoideum) and slime bacteria (myxobacteria). The effects of the spatial attraction parameter and the neighboring clock …

Complete Study Of An Original Power-Exponential Transformation Approach For Generalizing Probability Distributions, 2023 King Saud University

#### Complete Study Of An Original Power-Exponential Transformation Approach For Generalizing Probability Distributions, Mustafa S. Shama, Farid El Ktaibi, Jamal N. Al Abbasi, Christophe Chesneau, Ahmed Z. Afify

*All Works*

In this paper, we propose a flexible and general family of distributions based on an original power-exponential transformation approach. We call it the modified generalized-G (MGG) family. The elegance and significance of this family lie in the ability to modify the standard distributions by changing their functional forms without adding new parameters, by compounding two distributions, or by adding one or two shape parameters. The aim of this modification is to provide flexible shapes for the corresponding probability functions. In particular, the distributions of the MGG family can possess increasing, constant, decreasing, “unimodal”, or “bathtub-shaped“ hazard rate functions, which are …

Higher Spanier Groups, 2023 West Chester University

#### Higher Spanier Groups, Johnny Aceti

*West Chester University Master’s Theses*

When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …

On The Measure Of Noncompactness In $L_P(\Mathbb{R}^+)$ And Applications To A Product Of $N$-Integral Equations, 2023 TÜBİTAK

#### On The Measure Of Noncompactness In $L_P(\Mathbb{R}^+)$ And Applications To A Product Of $N$-Integral Equations, Mohamed M. A. Metwali, Vishnu Narayan Mishra

*Turkish Journal of Mathematics*

In this article, we prove a new compactness criterion in the Lebesgue spaces $L_p({\mathbb{R}}^+), 1 \leq p < \infty$ and use such criteria to construct a measure of noncompactness in the mentioned spaces. The conjunction of that measure with the Hausdroff measure of noncompactness is proved on sets that are compact in finite measure. We apply such measure with a modified version of Darbo fixed point theorem in proving the existence of monotonic integrable solutions for a product of $n$-Hammerstein integral equations $n\geq 2$.

On The Uniqueness Of Continuation Of A Partially Defined Metric, 2023 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine

#### On The Uniqueness Of Continuation Of A Partially Defined Metric, Evgeniy Petrov

*Theory and Applications of Graphs*

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\colon E\to\mathbb R^+$ on the graph $G$ has a unique continuation to a metric $d\colon V\times V\to\mathbb R^+$ is found.

On Rainbow Cycles And Proper Edge Colorings Of Generalized Polygons, 2023 Middle Georgia State University

#### On Rainbow Cycles And Proper Edge Colorings Of Generalized Polygons, Matt Noble

*Theory and Applications of Graphs*

An edge coloring of a simple graph G is said to be *proper rainbow-cycle-forbidding* (PRCF, for short) if no two incident edges receive the same color and for any cycle in G, at least two edges of that cycle receive the same color. A graph G is defined to be *PRCF-good* if it admits a PRCF edge coloring, and G is deemed *PRCF-bad* otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than …

Conceptual Mathematics In Society, 2023 Northern Illinois University

#### Conceptual Mathematics In Society, Ricela Feliciano-Semidei

*Books, Book Chapters, & Supplemental Materials*

This textbook is a compilation of chapters with educational purposes for the course MATH 103 in Spring 2023. The first part (Chapters 1 and 2) includes logic and critical thinking. Understanding the thinking process and strategies for solving problems in an effective way will provide students with skills that will be required to succeed in all college math courses. The second part (Chapters 3 & 4) is an opportunity to develop numbers sense through strengthening conceptual understanding of fractions and algebraic thinking. This will help develop foundational mathematical knowledge for college mathematics courses. The third and fourth parts of this …

The Merchant And The Mathematician: Commerce And Accounting, 2023 Università di Firenze

#### The Merchant And The Mathematician: Commerce And Accounting, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

In this article we describe the invention of double-entry bookkeeping (or *partita doppia*as it was called in Italian), as a fertile intersection between mathematics and early commerce. We focus our attention on this seemingly simple technique that requires only minimal mathematical expertise, but whose discovery is clearly the result of a mathematical way of thinking, in order to make a conceptual point about the role of mathematics as the humus from which disciplines as different as operations research, computer science, and data science have evolved.

Long-Time Asymptotics Of A Complex Cubic Camassa-Holm Equation, 2023 The University of Texas Rio Grande Valley

#### Long-Time Asymptotics Of A Complex Cubic Camassa-Holm Equation, Hongyi Zhang, Yufeng Zhang, Zhijun Qiao

*School of Mathematical and Statistical Sciences Faculty Publications and Presentations*

n this paper, we study the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation mt=bux+12[m(|u|2−|ux|2)]x−12m(uu¯x−uxu¯),m=u−uxx,

where b is an arbitrary real constant. %By applying ∂¯-steepest descent method, l Long-time asymptotics of the equation is obtained through the ∂¯-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed %can be expressed by %the solution of via solving the corresponding Riemann-Hilbert problem (RHP). Then, we present %obtain different long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ=y/t. The …

Hart, Robert W., Papers, 1924-1966, 2023 Pittsburg State University

#### Hart, Robert W., Papers, 1924-1966, Special Collections, Leonard H. Axe Library

*Finding Aids*

Personal documents, correspondence, and financial records of Robert W. Hart, former professor at Pittsburg State University.

Robert William Hart was born in 1896 to William and Sarah Hart and grew up in Crawford County, Kansas. In 1916 he graduated from the State Manual Training Normal School (now Pittsburg State University), and found early work as an engineer. Robert served as an ensign in the navy during World War I. He taught in the public schools in Fort Scott and Pittsburg from 1916-1918; 1921-1923. In 1920 he married Retta Statler and they had two children. In 1923 he was hired as …

On Solvability Of Homogeneous Riemann Boundary Value Problems In Hardy-Orlicz Classes, 2023 TÜBİTAK

#### On Solvability Of Homogeneous Riemann Boundary Value Problems In Hardy-Orlicz Classes, Yusuf Zeren, Fi̇dan A. Ali̇zadeh, Feyza Eli̇f Dal

*Turkish Journal of Mathematics*

This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on …

On Max-Min Solutions Of Fuzzy Games With Nonlinear Memberships Functions, 2023 TÜBİTAK

#### On Max-Min Solutions Of Fuzzy Games With Nonlinear Memberships Functions, Adem Cengi̇z Çevi̇kel

*Turkish Journal of Mathematics*

In this paper, we deal with two-person zero-sum games with fuzzy goals. We investigated the cases where the membership functions of the players are nonlinear. We examined how the solutions should be if the membership functions of players were exponential functions. In case players' membership functions are exponential, we developed a new method for the maximin solution according to a degree of attainment of the fuzzy goals. An application was made to show the effectiveness of the method.

Geodesics And Natural Complex Magnetic Trajectories On Tangent Bundles, 2023 TÜBİTAK

#### Geodesics And Natural Complex Magnetic Trajectories On Tangent Bundles, Mohamed Tahar Kadaoui Abbassi, Noura Amri

*Turkish Journal of Mathematics*

In this paper, we investigate geodesics of the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ endowed with an arbitrary pseudo-Riemannian $g$-natural metric of Kaluza-Klein type. Then considering a class of naturally defined almost complex structures on $TM$, constructed by V. Oproiu, we construct a class of magnetic fields and we characterize the corresponding magnetic curves on $TM$, when $(M,g)$ is a space form.