Physical Sciences and Mathematics | Open Access Articles

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.

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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.

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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 $p$.

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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.

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Curvature Identities For Einstein Manifolds Of Dimensions 5 And 6, Yunhee Euh, Jihun Kim, Jeonghyeong Park

Turkish Journal of Mathematics

Patterson discussed the curvature identities on Riemannian manifolds based on the skew-symmetric properties of the generalized Kronecker delta, and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem. In this paper, we provide the explicit formulae of Patterson's curvature identity that holds on 5-dimensional and 6-dimensional Einstein manifolds. We confirm that the curvature identities on the Einstein manifold derived from the Chern-Gauss-Bonnet Theorem are the same as the curvature identities deduced from Patterson's result. We also provide examples that support the theorems.

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Existence Of Fixed Points In Conical Shells Of A Banach Space For Sum Of Two Operators And Application In Odes, Amirouche Mouhous, Karima Mebarki

Turkish Journal of Mathematics

In this work a new functional expansion-compression fixed point theorem of Leggett--Williams type is developed for a class of mappings of the form $T+F,$ where $(I-T)$ is Lipschitz invertible map and $F$ is a $k$-set contraction. The arguments are based upon recent fixed point index theory in cones of Banach spaces for this class of mappings. As application, our approach is applied to prove the existence of nontrivial nonnegative solutions for three-point boundary value problem.

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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.

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A Fredholm Theory On Krein Spaces And Its Application To Weyl-Type Theorems And Homogeneous Equations, Danilo Polo Ojito, Jose Sanabria, Yina Ospino Buelvas

Turkish Journal of Mathematics

In this paper, we review the approach presented by An and Heo on the study of Weyl-type theorems for self-adjoint operators on Krein spaces and show that this approach is not appropriate due to a fallacy. Motivated by this fact, we define a new modification of the kernel of a bounded linear operator on a Krein space, namely $J$-kernel, which allows us to successfully introduce a Fredholm theory in this context and study some variations of Weyl-type theorems for bounded linear operators defined on these spaces. In addition, we will describe the $J$-index in terms of solution sets of homogeneous …

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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.

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A Characterization Of Abelian Group Codes In Terms Of Their Parameters, Fatma Altunbulak Aksu, İpek Tuvay

Turkish Journal of Mathematics

In 1979, Miller proved that for a group $G$ of odd order, two minimal group codes in $\mathbb{F}_2G$ are $G$-equivalent if and only if they have identical weight distribution. In 2014, Ferraz-Guerreiro-Polcino Milies disproved Miller's result by giving an example of two non-$G$-equivalent minimal codes with identical weight distribution. In this paper, we give a characterization of finite abelian groups so that over a specific set of group codes, equality of important parameters of two codes implies the $G$-equivalence of these two codes. As a corollary, we prove that two minimal codes with the same weight distribution are $G$-equivalent if …

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On Parabolic And Elliptic Elements Of The Modular Group, Bi̇lal Demi̇r, Özden Koruoğlu

Turkish Journal of Mathematics

The modular group $\Gamma=PSL(2, \mathbf{Z})$ is isomorphic to the free product of two cyclic groups of orders $2$ and $3$. In this paper, we give a necessary and sufficient condition for the existence of elliptic and parabolic elements in $\Gamma$ with a given cusp point. Then we give an algorithm to obtain such elements in words of generators using continued fractions and paths in the Farey graph.

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Adjunction Identity To Hypersemigroups, Niovi Kehayopulu

Turkish Journal of Mathematics

It is shown that some embedding problems on hypersemigroups are actually problems of adjunction. According to the theorem of this paper, for every hypersemigroup $S$ which does not have identity element, an hypersemigroup $T$ having identity element can be constructed in such a way that $S$ is an ideal of $T$. Moreover, if $S$ is regular, intra-regular, right (left) regular, right (left) quasi-regular or semisimple, then so is $T$. If $A$ is an ideal, subidempotent bi-ideal or quasi-ideal of $S$, then it is an ideal, bi-ideal, quasi-ideal of $T$ as well. Illustrative examples are given.

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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.

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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.

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Nonic B-Spline Algorithms For Numerical Solution Of The Kawahara Equation, Meli̇s Zorşahi̇n Görgülü

Turkish Journal of Mathematics

In this paper, the nonic (9th order) B-spline functions which have not been used before for the numerical solutions of the partial differential equations by finite element methods are used to solve numerically the Kawahara equation. These approaches involve the collocation and Galerkin finite element methods based on the nonic B-spline functions in space discretization and second order scheme (Crank-Nicolson method) in time discretization. To see the accuracy of the proposed methods three test problems are demonstrated and the obtained numerical results for both of the methods are compared with the exact solution of the Kawahara equation.

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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.

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Solving A Class Of Ordinary Differential Equations And Fractional Differential Equations With Conformable Derivative By Fractional Laplace Transform, Mohammad Molaei, Farhad Dastmalchi Saei, Mohammad Javidi, Yaghoub Mahmoudi

Turkish Journal of Mathematics

In this paper, we use the fractional Laplace transform to solve a class of second-order ordinary differential equations (ODEs), as well as some conformable fractional differential equations (CFDEs), including the Laguerre conformable fractional differential equation. Specifically, we apply the transform to convert the differential equations into first-order, linear differential equations. This is done by using the fractional Laplace transform of order $\alpha+\beta$ or $\alpha+\beta+\gamma$. Also, we investigate some more results on the fractional Laplace transform, obtained by Abdeljawad.

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Oscillation Of Second Order Mixed Functional Differential Equations With Sublinear And Superlinear Neutral Terms, Shan Shi, Zhenlai Han

Turkish Journal of Mathematics

In this paper, we shall establish some new oscillation theorems for the functional differential equations with sublinear and superlinear neutral terms of the form $$ (r(t)(z'(t))^\alpha)'=q(t)x^\alpha(\tau(t)), $$ where $z(t)=x(t)+p_1(t)x^\beta(\sigma(t))-p_2(t)x^\gamma(\sigma(t))$ with $0

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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.

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Oscillation Of Third-Order Neutral Differential Equations With Oscillatory Operator, Miroslav Bartusek

Turkish Journal of Mathematics

A third-order damped neutral sublinear differential equation for which its differential operator is oscillatory is studied. Sufficient conditions are given under which every solution is either oscillatory or the derivative of its neutral term is oscillatory (or it tends to zero).

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Continuous Wavelet Transform On Triebel-Lizorkin Spaces, Antonio L. Baisón Olmo, Víctor A. Cruz Barriguete, Jaime Navarro

Turkish Journal of Mathematics

The continuous wavelet transform in higher dimensions is used to prove the regularity of weak solutions $u \in L^p(\mathbb R^n)$ under $Qu = f$ where $f$ belongs to the Triebel-Lizorkin space $F^{r,q}_p(\mathbb R^n)$ where $1 < p,q < \infty$, $0< r 0$ with positive constant coefficients $c_{\beta}$.

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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 …

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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.

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The Fourier Spectral Method For Determining A Heat Capacity Coefficient In A Parabolic Equation, Durdimurod Durdiev, Dilshod Durdiev

Turkish Journal of Mathematics

In this paper, the comparison of finite difference and Fourier spectral numerical methods for an inverse problem of simultaneously determining an unknown coefficient in a parabolic equation with the usual initial and boundary conditions is proposed. We represent the detailed description of the methods and their algorithms. The research work conducted in this paper shows that the Fourier spectral method is highly accurate.

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$3d$-Flows Generated By The Curl Of A Vector Potential & Maurer-Cartan Equations, Oğul Esen, Partha Guha, Hasan Gümral

Turkish Journal of Mathematics

We examine $3D$ flows $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ admitting vector identity $M\mathbf{v} = \nabla \times \mathbf{A}$ for a multiplier $M$ and a potential field $\mathbf{A}$. It is established that, for those systems, one can complete the vector field $\mathbf{v}$ into a basis fitting an $\mathfrak{sl}(2)$-algebra. Accordingly, in terms of covariant quantities, the structure equations determine a set of equations in Maurer-Cartan form. This realization permits one to obtain the potential field as well as to investigate the (bi-)Hamiltonian character of the system. The latter occurs if the system has a time-independent first integral. In order to exhibit the theoretical results on some …

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A Matrix-Collocation Method For Solutions Of Singularly Perturbed Differential Equations Via Euler Polynomials, Deni̇z Elmaci, Şuayi̇p Yüzbaşi, Nurcan Baykuş Savaşaneri̇l

Turkish Journal of Mathematics

In this paper, a matrix-collocation method which uses the Euler polynomials is introduced to find the approximate solutions of singularly perturbed two-point boundary-value problems (BVPs). A system of algebraic equations is obtained by converting the boundary value problem with the aid of the collocation points. After this algebraic system, the coefficients of the approximate solution are determined. This error analysis includes two theorems which consist of an upper bound of errors and an error estimation technique. The present method and error analysis are applied to three numerical examples of singularly perturbed two-point BVPs. Numerical examples and comparisons with other methods …

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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 …

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Bessel Equation And Bessel Function On $\Mathbb{T}_{(Q,H)}$, Ahmet Yantir, Burcu Si̇li̇ndi̇r Yantir, Zehra Tuncer

Turkish Journal of Mathematics

This article is devoted to present nabla $(q, h)$-analogues of Bessel equation and Bessel function. In order to construct series solution of nabla $(q, h)$-Bessel equation, we present nabla $(q, h)$-analysis regarding nabla generalized quantum binomial, nabla $(q,h)$-analogues of Taylor's formula, Gauss's binomial formula, Taylor series, analytic functions, analytic exponential function with its fundamental properties, analytic trigonometric and hyperbolic functions. We emphasize that nabla $(q, h)$-Bessel equation recovers classical, $h$- and $q$-discrete Bessel equations. In addition, we establish nabla $(q, h)$-Bessel function which is expressed in terms of an absolutely convergent series in nabla generalized quantum binomials and intimately demonstrate …

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Notes On The Quadraticity Of Linear Combinations Of A Cubic Matrix And A Quadratic Matrix That Commute, Tuğba Peti̇k, Hali̇m Özdemi̇r, Burak Tufan Gökmen

Turkish Journal of Mathematics

Let $A_{1}$ and $A_{2}$ be an $\{\alpha_{1}, \beta_{1}, \gamma_{1}\}$-cubic matrix and an $\{\alpha_{2}, \beta_{2}\}$-quadratic matrix, respectively, with $\alpha_{1} \neq \beta_{1}$, $\beta_{1} \neq \gamma_{1}$, $\alpha_{1} \neq \gamma_{1}$ and $\alpha_{2}\neq \beta_{2}$. In this work, we characterize all situations in which the linear combination $A_{3}=a_{1}A_{1}+a_{2}A_{2}$ with the assumption $A_{1}A_{2}=A_{2}A_{1}$ is an $\{\alpha_{3}, \beta_{3}\}$-quadratic matrix, where $a_{1}$ and $a_{2}$ are unknown nonzero complex numbers.

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Ranks And Presentations For Order-Preserving Transformations With One Fixed Point, Joerg Koppitz, Somnuek Worawiset

Turkish Journal of Mathematics

In the present paper, we consider the semigroup $O_{n,p}$ of all order-preserving full transformations $\alpha $ on an n-elements chain $X_{n}$% , where $p\in X_{n}$ is the only fixed point of $\alpha $. The nilpotent semigroup $O_{n,p}$ was first studied by Ayik et al. in 2011. Moreover, $% O_{n,1}$ is the maximal nilpotent subsemigroup of the Catalan Monoid $C_{n}$. Its rank is the difference of the $(n-1)$th and the $(n-2)$th Catalan number. The aim of the present paper is to provide further fundamental information about the nilpotent semigroup $O_{n,p}$. We will calculate the rank of $O_{n,p}$ for $% p>1$ …

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Full-Text Articles in Physical Sciences and Mathematics

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

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

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

Curvature Identities For Einstein Manifolds Of Dimensions 5 And 6, Yunhee Euh, Jihun Kim, Jeonghyeong Park

Turkish Journal of Mathematics

Existence Of Fixed Points In Conical Shells Of A Banach Space For Sum Of Two Operators And Application In Odes, Amirouche Mouhous, Karima Mebarki

Turkish Journal of Mathematics

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

A Fredholm Theory On Krein Spaces And Its Application To Weyl-Type Theorems And Homogeneous Equations, Danilo Polo Ojito, Jose Sanabria, Yina Ospino Buelvas

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

A Characterization Of Abelian Group Codes In Terms Of Their Parameters, Fatma Altunbulak Aksu, İpek Tuvay

Turkish Journal of Mathematics

On Parabolic And Elliptic Elements Of The Modular Group, Bi̇lal Demi̇r, Özden Koruoğlu

Turkish Journal of Mathematics

Adjunction Identity To Hypersemigroups, Niovi Kehayopulu

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

Nonic B-Spline Algorithms For Numerical Solution Of The Kawahara Equation, Meli̇s Zorşahi̇n Görgülü

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

Solving A Class Of Ordinary Differential Equations And Fractional Differential Equations With Conformable Derivative By Fractional Laplace Transform, Mohammad Molaei, Farhad Dastmalchi Saei, Mohammad Javidi, Yaghoub Mahmoudi

Turkish Journal of Mathematics

Oscillation Of Second Order Mixed Functional Differential Equations With Sublinear And Superlinear Neutral Terms, Shan Shi, Zhenlai Han

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

Oscillation Of Third-Order Neutral Differential Equations With Oscillatory Operator, Miroslav Bartusek

Turkish Journal of Mathematics

Continuous Wavelet Transform On Triebel-Lizorkin Spaces, Antonio L. Baisón Olmo, Víctor A. Cruz Barriguete, Jaime Navarro

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

The Fourier Spectral Method For Determining A Heat Capacity Coefficient In A Parabolic Equation, Durdimurod Durdiev, Dilshod Durdiev

Turkish Journal of Mathematics

$3d$-Flows Generated By The Curl Of A Vector Potential & Maurer-Cartan Equations, Oğul Esen, Partha Guha, Hasan Gümral

Turkish Journal of Mathematics

A Matrix-Collocation Method For Solutions Of Singularly Perturbed Differential Equations Via Euler Polynomials, Deni̇z Elmaci, Şuayi̇p Yüzbaşi, Nurcan Baykuş Savaşaneri̇l

Turkish Journal of Mathematics

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

Turkish Journal of Mathematics

Bessel Equation And Bessel Function On $\Mathbb{T}_{(Q,H)}$, Ahmet Yantir, Burcu Si̇li̇ndi̇r Yantir, Zehra Tuncer

Turkish Journal of Mathematics

Notes On The Quadraticity Of Linear Combinations Of A Cubic Matrix And A Quadratic Matrix That Commute, Tuğba Peti̇k, Hali̇m Özdemi̇r, Burak Tufan Gökmen

Turkish Journal of Mathematics

Ranks And Presentations For Order-Preserving Transformations With One Fixed Point, Joerg Koppitz, Somnuek Worawiset

Turkish Journal of Mathematics