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Article Citation - WoS: 1Citation - Scopus: 1On the Rate of Convergence for the q-durrmeyer Polynomials in Complex Domains(Walter de Gruyter Gmbh, 2024) Gurel, Ovgu; Ostrovska, Sofiya; Turan, MehmetThe q-Durrmeyer polynomials are one of the popular q-versions of the classical operators of approximation theory. They have been studied from different points of view by a number of researchers. The aim of this work is to estimate the rate of convergence for the sequence of the q-Durrmeyer polynomials in the case 0 < q < 1. It is proved that for any compact set D subset of C, the rate of convergence is O(q(n)) as n -> infinity. The sharpness of the obtained result is demonstrated.Article On the Injectivity With Respect To q of the Lupas q-transform(Taylor & Francis Ltd, 2024) Yilmaz, Ovgue Gurel; Ostrovska, Sofiya; Turan, MehmetThe Lupas q-transform has first appeared in the study of the Lupas q-analogue of the Bernstein operator. Given 0 < q < 1 and f is an element of C[0, 1], the Lupas q-transform is defined by Lambda(q)(f; x) Pi(infinity)(k=0) 1/1 + q(k)x Sigma(k=0)f(1 - q(k))q(k(k-1)/2)x(k)/(1 - q)(1 - q(2)) center dot center dot center dot (1 - q(k)), x >= 0. During the last decades, this transform has been investigated from a variety of angles, including its analytical, geometric features, and properties of its block functions along with their sums. As opposed to the available studies dealing with a fixed value of q, the present work is focused on the injectivity of Lambda(q) with respect to parameter q. More precisely, the conditions on f such that equality Lambda(q)(f; x) = Lambda(r)(f; x); x >= 0 implies q = r have been established.Article Citation - WoS: 2Citation - Scopus: 2On the eigenfunctions of the q-Bernstein operators(Springer Basel Ag, 2023) Ostrovska, Sofiya; Turan, MehmetThe eigenvalue problems for linear operators emerge in various practical applications in physics and engineering. This paper deals with the eigenvalue problems for the q-Bernstein operators, which play an important role in the q-boson theory of modern theoretical physics. The eigenstructure of the classical Bernstein operators was investigated in detail by S. Cooper and S. Waldron back in 2000. Some of their results were extended for other Bernstein-type operators, including the q-Bernstein and the limit q-Bernstein operators. The current study is a pursuit of this research. The aim of the present work is twofold. First, to derive for the q-Bernstein polynomials analogues of the Cooper-Waldron results on zeroes of the eigenfunctions. Next, to present in detail the proof concerning the existence of non-polynomial eigenfunctions for the limit q-Bernstein operator.Article Citation - WoS: 3Citation - Scopus: 3q-stieltjes Classes for Some Families of q-densities(Elsevier Science Bv, 2019) Ostrovska, Sofiya; Turan, MehmetThe Stieltjes classes play a significant role in the moment problem allowing to exhibit explicitly infinite families of probability densities with the same sequence of moments. In this paper, the notion of q-moment determinacy/indeterminacy is proposed and some conditions for a distribution to be either q-moment determinate or indeterminate in terms of its q-density have been obtained. Also, a q-analogue of Stieltjes classes is defined for q-distributions and q-Stieltjes classes have been constructed for a family of q-densities of q-moment indeterminate distributions. (C) 2018 Elsevier B.V. All rights reserved.Article On the Image of the Limit Q-Durrmeyer Operator(Academic Press Inc Elsevier Science, 2026) Ostrovska, Sofiya; Turan, MehmetThe focus of this work is on the properties of the q-Durrmeyer operators Mn,q, n E N, and M infinity,q introduced, for q E (0, 1), by V. Gupta and H. Wang. First, it is shown that, for each f E C[0, 1], the sequence {Mn,q f}nEN converges to M infinity,q f uniformly on [0, 1] with a rate not slower than Cq, fqn, which refines the previously available result by V. Gupta and H. Wang, and implies the possibility of an analytic continuation for M infinity,q f into a neighbourhood of [0, 1]. Further investigation shows that M infinity,q f admits an analytic continuation as an entire function regardless of f E C[0, 1]. Finally, the growth estimates for these functions are received and applied to describe the point spectrum of M infinity,q. The paper also addresses the significant differences between the properties of M infinity,q and the previously (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.Article Citation - WoS: 4Citation - Scopus: 4On the Powers of the Kummer Distribution(Academic Publication Council, 2017) Ostrovska, Sofiya; Turan, Mehmet; MathematicsThe Kummer distribution is a probability distribution, whose density is given by f (x) = cx (alpha-1)(1 + delta x)(-gamma) e(-beta x), X > 0, where alpha, beta, delta > 0, gamma is an element of R and C is a normalizing constant. In this paper, the distributions of random variable X-P, p > 0, where X has the Kummer distribution, are considered with the conditions being IFR/DFR, some properties of moments depending on the parameters and the moment-(in) determinacy. In the case of moment-indeterminacy, exemplary Stieltjes classes are constructed.Article Qualitative results on the convergence of the q-Bernstein polynomials(North Univ Baia Mare, 2015) Ostrovska, Sofiya; Turan, MehmetDespite many common features, the convergence properties of the Bernstein and the q-Bernstein polynomials are not alike. What is more, the cases 0 < q < 1 and q > 1 are not similar to each other in terms of convergence. In this work, new results demonstrating the striking differences which may occur in those convergence properties are presented.Article Citation - WoS: 1Citation - Scopus: 1Shape-Preserving Properties of the Limit q-durrmeyer Operator(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetThe present work aims to establish the shape-preserving properties of the limit q- Durrmeyer operator, D q for 0 < q < 1. It has been proved that the operator is monotonicity- and convexity-preserving. What is more, it maps a function m - convex along {q (j)}(infinity)(j =0) to a function m - convex along any sequence { xq( j )}(infinity)(j =0) , x is an element of (0, 1). (c) 2024 Elsevier Inc. All rights reserved.Article Citation - WoS: 1Citation - Scopus: 1Evaluation and Optimization of Nonlinear Central Pattern Generators for Robotic Locomotion(Romanian Soc Control Tech informatics, 2018) Elbori, Abdalftah; Turan, Mehmet; Arikan, Kutluk Bilge; Department of Mechatronics Engineering; MathematicsWith regard to the optimization of Central Pattern Generators (CPGs) for bipedal locomotion in robots, this paper investigates how the different cases of CPGs such as uncoupled, unidirectional, bidirectional two CPGs are used to produce rhythmic patterns for one leg with two degrees of freedom (DOF). This paper also discusses the stability analysis of CPGs and attempts to utilize genetic algorithms with the hybrid function and adapts the CPGs to robotic systems that perform one-leg movement, by utilizing the bidirectional two CPGs. The results show far greater improvement than in the other cases. CPGs not only enhance movement but also control locomotion without any sensory feedback.Article Citation - WoS: 1Citation - Scopus: 1On the Invariant Manifolds of the Fixed Point of a Second-Order Nonlinear Difference Equation(Springer/plenum Publishers, 2020) Turan, MehmetThis paper addresses the asymptotic approximations of the stable and unstable manifolds for the saddle fixed point and the 2-periodic solutions of the difference equationx(n+ 1)=alpha+beta x(n- 1)+x(n- 1)/x(n), where alpha> 0,0 <=beta<1$0\leqslant \beta and the initial conditionsx(- 1)andx(0)are positive numbers. These manifolds determine completely global dynamics of this equation. The theoretical results are supported by some numerical examples.
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