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Discrete Fractional Integrals, Lattice Points On Short Arcs, And Diophantine Approximation, Faruk Temur

Turkish Journal of Mathematics

Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate quadratic polynomials. We achieve this in part by establishing connections to problems on concentration of lattice points on short arcs of conics, whence we study discrete fractional integrals and lattice point concentration from a unified perspective via tools of sieving and diophantine approximation, and prove theorems that are of interest to researchers in both subjects.

Some Asymptotic Results For The Continued Fraction Expansions With Odd Partial Quotients, Gabriela Ileana Sebe, Dan Lascu

Turkish Journal of Mathematics

We present and develop different approaches to study the asymptotic behavior of the distribution functions in the odd continued fractions case. Firstly, by considering the transition operator of the Markov chain associated with these expansions on a certain Banach space of complex-valued functions of bounded variation, we make a brief survey of the solution in the Gauss-Kuzmin-type problem. Secondly, we use the method of Szüsz to obtain a similar asymptotic result and to give a good estimate of the convergence rate involved.

Some Recent Results In Plastic Structure On Riemannian Manifold, Akbar Dehghan Nezhad, Zohreh Aral

Turkish Journal of Mathematics

The plastic ratio is a fascinating topic that continually generates new ideas. The purpose of this paper is to point out and find some applications of the plastic ratio in the differential manifold. Precisely, we say that an $(1,1)$-tensor field $P$ on a $m$-dimensional Riemannian manifold $(M, g)$ is a plastic structure if it satisfies the equation $ P^3 = P + I $, where $ I $ is the identity. We establish several properties of the plastic structure. Then we show that a plastic structure induces on every invariant submanifold a plastic structure, too.

On The Behaviour Of Solutions To A Kind Of Third Order Nonlinear Neutral Differential Equation With Delay, Adeleke Ademola, Peter Arawomo, Olufemi Adesina, Samuel Okoya

Turkish Journal of Mathematics

This paper presents a novel class of third order nonlinear nonautonomous neutral differential equation with delay. The third order neutral differential equation is cut down to a system of first order, a suitable complete Lyapunov-Krasovskii's functional is constructed and used, to obtain standard conditions on the nonlinear functions to ensure stability and uniform asymptotic stability of the trivial solutions, the existence of a unique periodic solution, uniform boundedness and uniform ultimate boundedness of solutions when the forcing term is nonzero. The obtained results are new and include many prominent results on neutral and nonneutral delay differential equations in literature. Finally, …

Constant Angle Surfaces In The Lorentzian Warped Product Manifold $I \Times_{F} \Mathbb E^2_1$, Uğur Dursun

Turkish Journal of Mathematics

Let $I \times_{f} \mathbb E^2_1$ be a 3-dimensional Lorentzian warped product manifold with the metric $\tilde g = dt^2 + f^2(t) (dx^2 - dy^2)$, where $I$ is an open interval, $f$ is a strictly positive smooth function on $I,$ and $\mathbb E^2_1$ is the Minkowski 2-plane. In this work, we give a classification of all space-like and time-like constant angle surfaces in $I \times_{f} \mathbb E^2_1$ with nonnull principal direction when the surface is time-like. In this classification, we obtain space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we have some …

Novel Exact Solutions To Navier-Stokes Momentum Equations Describing An Incompressible Fluid, Yahya Öz

Turkish Journal of Mathematics

An analytical solution to the incompressible Navier-Stokes momentum equations for a divergence-free flow $\boldsymbol{\nabla}\cdot \vec u\left(\vec x,t\right)=0$ with time-dependent dynamic viscosity $\mu\left(t\right)$ is presented. The demonstrated methodology holds for the physically relevent three dimensions. The constructed flow velocities $\vec u\left(\vec x,t\right)$ are eigenvectors of the vector operator curl. Moreover, vortex $\vec \omega\left(\vec x,t\right)$, helicity $H\left(\vec x,t\right)$, enstrophy $\mathcal{E}\left(t\right)$ and enstrophy evolution $\frac{\mathrm{d}\mathcal{E}\left(t\right)}{\mathrm{d}t}$ are explicitly determined.

Explicit Motion Planning In Digital Projective Product Spaces, Seher Fi̇şekci̇, İsmet Karaca

Turkish Journal of Mathematics

We introduce digital projective product spaces based on Davis' projective product spaces. We determine an upper bound for the digital LS-category of digital projective product spaces. In addition, we obtain an upper bound for the digital topological complexity of these spaces through an explicit motion planning construction, which shows digital perspective validity of results given by S. Fişekci and L. Vandembroucq. We apply our outcomes on specific spaces in order to be more clear.

From Ordered Semigroups To Ordered $\Gamma$-Hypersemigroups, Niovi Kehayopulu

Turkish Journal of Mathematics

In an attempt to show the way we pass from ordered semigroups to ordered $\Gamma$-hypersemigroups, we examine the results of Semigroup Forum (1992; 46: 341-346) for an ordered $\Gamma$-hypersemigroup. It has been shown that the concept of semisimple ordered $\Gamma$-hypersemigroup $S$ is identical with the concept "the ideals of $S$ are idempotent" and the ideals of $S$ are idempotent if and only if for all ideals $A, B$ of $S$, we have $A\cap B=(A\Gamma B]$. The main results of the paper are the following: The ideals of an ordered $\Gamma$-hypersemigroup $S$ are weakly prime if and only if they form …

There From The Beginning: The Women Of Los Alamos National Laboratory Supporting National And International Nuclear Security, Olga Martin, Laura Mcclellan, Octavio Ramos, Heather Quinn

International Journal of Nuclear Security

From the beginning of the Manhattan Project in the early 1940s, the women of what would become Los Alamos National Laboratory (LANL) worked in technical positions alongside their male counterparts, played a key role as computers, and worked in administrative jobs as secretaries, phone operators, bookkeepers, and on behalf of the U.S. Army in the Women’s Army Corps.

Throughout the history of the Laboratory, women experts at LANL helped establish and lead important national and international security programs, with careers in science, technology, engineering, and mathematics. Over time, the women of Los Alamos have come together under various Employee Resource …

Global Differential Invariants Of Nondegenerate Hypersurfaces, Yasemi̇n Sağiroğlu, Uğur Gözütok

Turkish Journal of Mathematics

Let $\{g_{ij}(x)\}_{i, j=1}^n$ and $\{L_{ij}(x)\}_{i, j=1}^n$ be the sets of all coefficients of the first and second fundamental forms of a hypersurface $x$ in $R^{n+1}$. For a connected open subset $U\subset R^{n}$ and a $C^{\infty }$-mapping $x:U\rightarrow R^{n+1}$ the hypersurface $x$ is said to be $d$-\textit{nondegenerate}, where $d\in \left\{1, 2, \ldots n\right\}$, if $L_{dd}(x)\neq 0$ for all $u\in U$. Let $M(n)=\{F:R^{n}\longrightarrow R^{n}\mid Fx=gx+b, \; g\in O(n), \; b\in R^{n}\}$, where $O(n)$ is the group of all real orthogonal $n\times n$-matrices, and $SM(n)=\{F\in M(n)\mid g\in SO(n)\}$, where $SO(n)=\left\{g\in O(n)\mid \det(g)=1\right\}$. In the present paper, it is proved that the set $\left\{g_{ij}(x), …

On A Class Of Nonlocal Porous Medium Equations Of Kirchhoff Type, Uğur Sert

Turkish Journal of Mathematics

We study the Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type \[ u_{t}-a\left(\ u\ _{L^{p}(\Omega)}^{p}\right)\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p-2}D_{i}u\right) +b\left( x,t,u\right)=f\left( x,t\right) \quad \text{in $Q_T=\Omega \times (0,T)$}, \] where $p\geq2$, $T>0$, $\Omega \subset \mathbb{R}^{n}$, $n\geq 2$, is a smooth bounded domain. The coefficient $a(\cdot)$ is real-valued function defined on $\mathbb{R}_+$ and $b(\cdot,\cdot,\tau)$ is a measurable function with variable nonlinearity in $\tau$. We prove existence of weak solutions of the considered problem under appropriate and general conditions on $a$ and $b$. Sufficient conditions for uniqueness are found and in the case $f\equiv0$ the decay rates for $\ u\ …

On Properties Of The Reeb Vector Field Of $(\Alpha,\Beta)$ Trans-Sasakian Structure, Alexander Yampolsky

Turkish Journal of Mathematics

The paper focused on the mean curvature and totally geodesic property of the Reeb vector field $\xi$ on $(\alpha,\beta)$ trans-Sasakian manifold $M$ of dimension $(2n+1)$ as a submanifold in the unit tangent bundle $T_1M$ with Sasaki metric $g_S$. We give an explicit formula for the norm of mean curvature vector of the submanifold $\xi(M)\subset (T_1M,g_S)$. As a byproduct, for the Reeb vector field, we get some known results concerning its minimality, harmonicity and the property to define a harmonic map. We prove that on connected proper trans-Sasakian manifold the Reeb vector field does not give rise to totally geodesic submanifold …

On A Generalization Of Szasz-Mirakyan Operators Including Dunkl-Appell Polynomials, Serdal Yazici, Fatma Taşdelen Yeşi̇ldal, Bayram Çeki̇m

Turkish Journal of Mathematics

In this study, we have introduced a generalization of Szasz-Mirakyan operators including Dunkl-Appell polynomials with help of sequences satisfying certain conditions and have derived some approximation properties of this generalization.

Equicontinuity And Sensitivity On Countable Amenable Semigroup, Nader Asadi Karam, Mohammad Kbari Tootkaboni, Abbas Sahleh

Turkish Journal of Mathematics

In this paper, we obtain the classification of topological dynamical systems with a discrete action. The equicontinuity and sensitivity for amenable discrete countable semigroup action are shown by the left Følner sequence. We consider the notion of uniquely ergodic and mean equicontinuous on amenable discrete countable semigroup action and develop the notion of density with respect to the Følner sequence on equicontinuous and sensitivity.

Semisymmetric Hypersurfaces In Complex Hyperbolic Two-Plane Grassmannians, Doo Hyun Hwang, Changhwa Woo

Turkish Journal of Mathematics

In this paper, we introduce new notions of symmetric operators such as semisymmetric shape operator and structure Jacobi operator in complex hyperbolic two-plane Grassmannians. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with such notions.

$(P,Q)$-Chebyshev Polynomials For The Families Of Biunivalent Function Associating A New Integral Operator With $(P,Q)$-Hurwitz Zeta Function, Sarem H. Hadi, Maslina Darus

Turkish Journal of Mathematics

In the present article, making use of the $(p,q)$-Hurwitz zeta function, we provide and investigate a new integral operator. Also, we define two families ${\mathcal{S}\mathcal{M}}_{p,q}\left(\xi ,\zeta,\delta,u,\tau \right)$ and ${\mathcal{S}\mathcal{C}}_{p,q}\left(\lambda, \zeta,\vartheta,u,\tau \right)$ of biunivalent and holomorphic functions in the unit disc connected with $(p,q)$-Chebyshev Polynomials. Then we find coefficient estimates $\left a_2\right $ and $\left a_3\right .$ Finally, we obtain Fekete-Szeg$\ddot{\mathrm{o}}$ inequalities for these families.

Inverse Nodal Problems For Dirac Type Integro Differential System With A Nonlocal Boundary Condition, Baki̇ Keski̇n

Turkish Journal of Mathematics

In this paper, the Dirac type integro differential system\ with a nonlocal integral boundary condition is considered. First, we derive the asymptotic expressions for the solutions and large eigenvalues. Second, we provide asymptotic expressions for the nodal points and prove that a dense subset of nodal points uniquely determines the boundary condition parameter and the potential function of the considered differential system. We also provide an effective procedure for solving the inverse nodal problem.

Formulas For Special Numbers And Polynomials Derived From Functional Equations Of Their Generating Functions, Nesli̇han Kilar

Turkish Journal of Mathematics

The main purpose of this paper is to investigate various formulas, identities and relations involving Apostol type numbers and parametric type polynomials. By using generating functions and their functional equations, we give many relations among the certain family of combinatorial numbers, the Vieta polynomials, the two-parametric types of the Apostol-Euler polynomials, the Apostol-Bernoulli polynomials, the Apostol-Genocchi polynomials, the Fibonacci and Lucas numbers, the Chebyshev polynomials, and other special numbers and polynomials. Moreover, we give some formulas related to trigonometric functions, special numbers and special polynomials. Finally, some remarks and observations on the results of this paper are given.

On The Solutions Of Fractional Integro-Differential Equations Involving Ulam-Hyers-Rassias Stability Results Via $\Psi$-Fractional Derivative With Boundary Value Conditions, Kulandhivel Karthikeyan, Gobi Selvaraj Murugapandian, Özgür Ege

Turkish Journal of Mathematics

In this paper, we study boundary value problems for the impulsive integro-differential equations via $\psi$-fractional derivative. The contraction mapping concept and Schaefer's fixed point theorem are used to produce the main results. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam-Hyers-Rassias stability of the solution to the proposed system.

Explicit Examples Of Constant Curvature Surfaces In The Supersymmetric ${C}P^{2}$ Sigma Model, İsmet Yurduşen

Turkish Journal of Mathematics

The surfaces constructed from the holomorphic solutions of the supersymmetric (susy) ${C}P^{N-1}$ sigma model are studied. By obtaining compact general expansion formulae having neat forms due to the properties of the superspace in which this model is described, the explicit expressions for the components of the radius vector as well as the elements of the metric and the Gaussian curvature are given in a rather natural manner. Several examples of constant curvature surfaces for the susy ${C}P^{2}$ sigma model are presented.

The Dual Spaces Of Variable Anisotropic Hardy-Lorentz Spaces And Continuity Of A Class Of Linear Operators, Wenhua Wang, Aiting Wang

Turkish Journal of Mathematics

In this paper, the authors obtain the continuity of a class of linear operators on variable anisotropic Hardy--Lorentz spaces. In addition, the authors also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type spaces with variable exponents. This result is still new even when the exponent function $p(\cdot)$ is $p$.

A Fixed Point Theorem Using Condensing Operators And Its Applications To Erdelyi--Kober Bivariate Fractional Integral Equations, Anupam Das, Mohsen Rabbani, Bipan Hazarika, Sumati Kumari Panda

Turkish Journal of Mathematics

The primary aim of this article is to discuss and prove fixed point results using the operator type condensing map, and to obtain the existence of solution of Erdelyi-Kober bivariate fractional integral equation in a Banach space. An instance is given to explain the results obtained, and we construct an iterative algorithm by sinc interpolation to find an approximate solution of the problem with acceptable accuracy.

On The $2$-Class Group Of Some Number Fields Of $2$-Power Degree, Idriss Jerrari, Abdelmalek Azizi

Turkish Journal of Mathematics

Let $K$ be an imaginary cyclic quartic number field whose $2$-class group is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, and let $K^*$ denote the genus field of $K$. In this paper, we compute the rank of the $2$-class group of $K^*_n$ the $n$-th layer of the cyclotomic $Z_2$-extension of $K^*$.

Bernstein-Walsh-Type Inequalities For Derivatives Of Algebraic Polynomials On The Regions Of Complex Plane, Naci̇ye Peli̇n Özkartepe, Cevahi̇r Doğanay Gün, Fahreddi̇n Abdullayev

Turkish Journal of Mathematics

In this paper, we study Bernstein-Walsh-type estimates for the derivatives of an arbitrary algebraic polynomial on some general regions of the complex plane.

Approximation By Sampling Kantorovich Series In Weighted Spaces Of Functions, Tuncer Acar, Osman Alagöz, Ali̇ Aral, Dani̇lo Costarelli̇, Meti̇n Turgay, Gianluca Vinti

Turkish Journal of Mathematics

This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for functions that are not necessarily uniformly continuous and bounded on $\mathbb{R}$. In that context we also prove quantitative estimates for the rate of convergence of the family of the above operators in terms of weighted modulus of continuity. Finally, pointwise convergence results in quantitative form by means of Voronovskaja type theorems have been also established.

Spinor Representation Of Framed Mannheim Curves, Bahar Doğan Yazici, Zehra İşbi̇li̇r, Murat Tosun

Turkish Journal of Mathematics

In this paper, we obtain spinor with two complex components representations of Mannheim curves of framed curves. Firstly, we give the spinor formulas of the frame corresponding to framed Mannheim curve. Later, we obtain the spinor formulas of the frame corresponding to framed Mannheim partner curve. Moreover, we explain the relationships between spinors corresponding to framed Mannheim pairs and their geometric interpretations. Finally, we present some geometrical results of spinor representations of framed Mannheim curves.

Initial Value Problem For Elastic System In Transversely Isotropic Inhomogeneous Media, Meltem Altunkaynak

Turkish Journal of Mathematics

In this paper, we consider an initial value problem (IVP) for three dimensional elasticity system in a transversely isotropic inhomogeneous media. We will rewrite the problem in the form of Fourier images by means of Fourier transform method. After some arrangements, the problem is reduced to integral equations in the vector form. Using the properties of the vector integral equation and successive approximations method, an explicit formula for the solution of the IVP in transversely isotropic inhomogeneous media is constructed, and existence and uniqueness of the solution is stated. By a computational example, we illustrate the robustness of the method.

On The Convergence And Stability Analysis Of Finite-Difference Methods For The Fractional Newell-Whitehead-Segel Equations, İnci̇ Çi̇li̇ngi̇r Süngü, Emre Aydin

Turkish Journal of Mathematics

In this study, standard and non-standard finite-difference methods are proposed for numerical solutions of the time-spatial fractional generalized Newell-Whitehead-Segel equations describing the dynamical behavior near the bifurcation point of the Rayleigh-Benard convection of binary fluid mixtures. The numerical solutions have been found for high values of $p$ which shows the degree of nonlinear terms in the equations. The stability and convergence conditions of the obtained difference schemes are determined for each value of $p$. Errors of methods for various values of $p$ are given in tables. The compatibility of exact solutions and numerical solutions and the effectiveness of the methods …

Some Convergence, Stability, And Data Dependence Results For $K^{\Ast }$ Iterative Method Of Quasi-Strictly Contractive Mappings, Ruken Çeli̇k, Neci̇p Şi̇mşek

Turkish Journal of Mathematics

In a recent paper, Yu et al. obtained convergence and stability results of the $K^{\ast }$ iterative method for quasi-strictly contractive mappings [An iteration process for a general class of contractive-like operators: Convergence, stability and polynomiography. AIMS Mathematics 2021; 6 (7): 6699-6714.]. To guarantee these convergence and stability results, the authors imposed some strong conditions on parametric control sequences which are used in the $K^{\ast }$ iterative method. The aim of the presented work is twofold: (a) to recapture the aforementioned results without any restrictions imposed on the mentioned parametric control sequences (b) to complete the work of Yu et …

Minimal Legendrian Submanifolds Of $\Mathbb S^{9}$ With Nonnegative Sectional Curvature, Shujie Zhai, Heng Zhang

Turkish Journal of Mathematics

In this paper, we established a complete classification of 4-dimensional compact minimal Legendrian submanifolds with nonnegative sectional curvature in the 9-dimensional unit sphere.