# Showing papers in "Turkish Journal of Mathematics in 2020"

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TL;DR: This work introduces and studies the concepts of $T-soft subset and $T$-soft equality relations, and utilizes them to define the concepts for arbitrary family of soft sets and investigates new types of soft linear equations with respect to some soft equality relations.

Abstract: The desire of generalizing some set-theoretic properties to the soft set theory motivated many researchers to define various types of soft operators. For example, they redefined the complement of a soft set, and soft union and intersection between two soft sets in a way that satisfies De Morgan's laws. In this paper, we introduce and study the concepts of $T$-soft subset and $T$-soft equality relations. Then, we utilize them to define the concepts of $T$-soft union and $T$-soft intersection for arbitrary family of soft sets. By $T$-soft union, we successfully keep some classical properties via soft set theory. We conclude this work by giving and investigating new types of soft linear equations with respect to some soft equality relations. Illustrative examples are provided to elucidate main obtained results.

47 citations

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TL;DR: In this article, the authors prove new fixed-circle resp. fixed-disc results using the bilateral type contractions on a metric space using the Jaggi-type bilateral contraction and the Dass-Gupta type bilateral contraction.

Abstract: In this paper, we prove new fixed-circle resp. fixed-disc results using the bilateral type contractions on a metric space. To do this, we modify some known contractive conditions called the Jaggi-type bilateral contraction and the Dass-Gupta type bilateral contraction. We give some examples to show the validity of our obtained results. Also, we construct an application to rectified linear units activation functions used in the neural networks. This application shows the importance of studying "fixed-circle problem".

20 citations

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TL;DR: In this paper, the authors studied Horadam hybrid numbers and gave the exponential generating function, Poisson generator, generating matrix, Vajda's, Catalan's, Cassini's, and d'Ocagne's identities.

Abstract: In this paper, we study Horadam hybrid numbers. For these numbers, we give the exponential generating function, Poisson generating function, generating matrix, Vajda's, Catalan's, Cassini's, and d'Ocagne's identities. In addition, we offer Honsberger formula, general bilinear formula, and some summation formulas for these numbers.

18 citations

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TL;DR: In this article, the Bertrand and Mannheim curves of framed curves are investigated and the existence conditions of the Bertran and Mannhen curves of the moving frame of a curve with singular points are clarified.

Abstract: A Bertrand curve is a space curve whose principal normal line is the same as the principal normal line of another curve. On the other hand, a Mannheim curve is a space curve whose principal normal line is the same as the binormal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the principal normal vector. Even if that is the regular case, the existence conditions of the Bertrand and Mannheim curves seem to be wrong in some previous research. Moreover, parallel curves may have singular points. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curves. We clarify that the Bertrand and Mannheim curves of framed curves are dependent on the moving frame.

18 citations

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TL;DR: In this article, the concept of the st-uτ -convergence was introduced and the notions of closed subset, Cauchy sequence, continuous and complete locally solid vector lattice were introduced.

Abstract: A sequence xn in a locally solid Riesz space E, τ is said to be statistically unbounded τ -convergent to x ∈ E if, for every zero neighborhood U , 1 n {k ≤ n : |xk − x| ∧ u /∈ U} → 0 as n → ∞. In this paper, we introduce the concept of the st-uτ -convergence and give the notions of st-uτ -closed subset, st-uτ -Cauchy sequence, st-uτ -continuous and st-uτ -complete locally solid vector lattice. Also, we give some relations between the order convergence and the st-uτ -convergence.

14 citations

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TL;DR: In this paper, it was shown that for any prime exponent $p>B_K$ the Fermat type equation does not have certain type of solutions, and the main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Abstract: Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

14 citations

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TL;DR: In this paper, the existence and uniqueness results for the two initial value problems were studied through Banach's contraction principle and Krasnoselskii's fixed point theorem.

Abstract: This research article deals with novel two species of initial value problems, one of them, the fractional neutral functional integrodifferential equations, and the other one, the coupled system of fractional neutral functional integrodifferential equations, with finite delay and involving a ψ–Caputo fractional operator. The existence and uniqueness results are studied through Banach’s contraction principle and Krasnoselskii’s fixed point theorem. We also establish two various kinds of Ulam stability results for the proposed problems. Further, two pertinent examples are presented to demonstrate the reported results.

14 citations

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12 citations

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TL;DR: In this article, a class of second order differential equations with iterative source term is considered and the main results are obtained by virtue of a Krasnoselskii fixed point theorem and some useful properties of a Green's function which allows to prove the existence of positive periodic solutions.

Abstract: In this paper, we consider a class of second order differential equations with iterative source term. The main results are obtained by virtue of a Krasnoselskii fixed point theorem and some useful properties of a Green's function which allows us to prove the existence of positive periodic solutions. Finally, an example is included to illustrate the correctness of our results.

12 citations

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TL;DR: In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically solve higher order linear Fredholm-Volterra integro differential equations FVIDE.

Abstract: In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically solve higher order linear Fredholm-Volterra integro differential equations FVIDE . The approximate solutions are assumed in form of the truncated Pell-Lucas polynomial series. By using Pell-Lucas polynomials and relations of their derivatives, the solution form and its derivatives are brought to matrix forms. By applying the collocation method based on equally spaced collocation points, the method reduces the problem to a system of linear algebraic equations. Solution of this system determines the coefficients of assumed solution. Error estimation is made and also a method with the help of the obtained approximate solution is developed that finds approximate solution with better results. Then, the applications are made on five examples to show that the method is successful. In addition, the results are supported by tables and graphs and the comparisons are made with other methods available in the literature. All calculations in this study have been made using codes written in Matlab.

11 citations

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TL;DR: In this article, the problem of finding a common solution to split generalized mixed equilibrium problem and fixed point problem for quasi-$% \phi $-none-expansive mappings in 2-uniformly convex and uniformly smooth Banach space was studied.

Abstract: In this paper, we study the problem of finding a common solution to split generalized mixed equilibrium problem and fixed point problem for quasi-$% \phi $-nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach space $E_1$ and a smooth, strictly convex, and reflexive Banach space $% E_2$. An iterative algorithm with Armijo linesearch rule for solving the problem is presented and its strong convergence theorem is established. The convergence result is obtained without using the hybrid method which is mostly used when strong convergence is desired. Finally, numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm in comparison with other existing algorithms in the literature. Our results extend and improve many recent results in this direction.

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TL;DR: The analogs of Korovkin theorems in grand-Lebesgue spaces are proved in this article, where the subspace G p −π; π of grand Lebesgue space is defined using shift operator.

Abstract: The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace G p −π; π of grand Lebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions is dense in G p −π; π . The analogs of Korovkin theorems are proved in G p −π; π . These results are established in G p −π; π in the sense of statistical convergence. The obtained results are applied to a sequence of operators generated by the Kantorovich polynomials, to Fejer and Abel-Poisson convolution operators.

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TL;DR: In this paper, the existence and uniqueness results for fractional Caputo difference equations were derived using a Banach fixed pointtheorem, and the existence results for delta fractional caputo difference equation were obtained in Chen and Zhou.

Abstract: We study two cases of nabla fractional Caputo difference equations. Our main tool used is a Banach fixed pointtheorem, which allows us to give some existence and uniqueness theorems of solutions for discrete fractional Caputo equations. In addition, we develop the existence results for delta fractional Caputo difference equations, which correct ones obtained in Chen and Zhou. We present two examples to illustrate our main results.

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TL;DR: In this paper, an inverse spectral problem of recovering operators from the spectra of four boundary value problems is studied for nonself-adjoint second-order differential operators with two constant delays from [ π 2, π and two real-valued potentials from L 2[0, π].

Abstract: This paper deals with nonself-adjoint second-order differential operators with two constant delays from [ π 2 , π and two real-valued potentials from L2[0, π]. An inverse spectral problem of recovering operators from the spectra of four boundary value problems is studied.

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TL;DR: In this paper, the authors construct generating functions for new classes of Catalan-type numbers and polynomials and give various new identities and relations involving these numbers, including the Bernoulli numbers, the Stirling numbers of the second kind, the Catalan numbers and other classes of special numbers.

Abstract: Abstract: The aim of this paper is to construct generating functions for new classes of Catalan-type numbers and polynomials. Using these functions and their functional equations, we give various new identities and relations involving these numbers and polynomials, the Bernoulli numbers and polynomials, the Stirling numbers of the second kind, the Catalan numbers and other classes of special numbers, polynomials and functions. Some infinite series representations, including the Catalan-type numbers and combinatorial numbers, are investigated. Moreover, some recurrence relations and computational algorithms for these numbers and polynomials are provided. By implementing these algorithms in the Python programming language, we illustrate the Catalan-type numbers and polynomials with their plots under the special conditions. We also give some derivative formulas for these polynomials. Applying the Riemann integral, contour integral, Volkenborn (bosonic p -adic) integral and fermionic p -adic integral to these polynomials, we also derive some integral formulas. With the help of these integral formulas, we give some identities and relations associated with some classes of special numbers and also the Cauchy-type numbers.

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TL;DR: In this article, a new subclass of analytic bi-univalent functions associated with the Fibonacci numbers is defined by using a relation of subordination, and the bounds of the coefficients for functions in this class are surveyed.

Abstract: In this investigation, by using a relation of subordination, we define a new subclass of analytic bi-univalent functions associated with the Fibonacci numbers. Moreover, we survey the bounds of the coefficients for functions in this class.

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TL;DR: In this paper, the 2-class groups of some multiquadratic number fields of degree 8 and 16 were investigated, including the unit groups, the two class groups, and the 2 class field towers.

Abstract: In this paper, we investigate the unit groups, the 2-class groups, the 2-class field towers and the structures of the second 2-class groups of some multiquadratic number fields of degree 8 and 16.

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TL;DR: In this article, the relationship between the τw-contingent derivative of the Borwein proper perturbation map and the γw-constraint derivative of feasible map in objective space is considered.

Abstract: This paper is concerned with sensitivity analysis in parametric vector optimization problems via τw-contingent derivatives. Firstly, relationships between the τw-contingent derivative of the Borwein proper perturbation map and the τw-contingent derivative of feasible map in objective space are considered. Then, the formulas for estimating the τw-contingent derivative of the Borwein proper perturbation map via the τw-contingent of the constraint map and the Hadamard derivative of the objective map are obtained.

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TL;DR: In this article, it was shown that a weak chain-preserving operator between two ordered Banach spaces is a KB-operator if and only if it is a weak KB operator.

Abstract: We determine that two recent classes of KB-operators and weak KB-operators and the well-known class of b -weakly compact operators, from a Banach lattice into a Banach space, are the same. We extend our study to the ordered Banach space setting by showing that a weak chain-preserving operator between two ordered Banach spaces is a KB-operator if and only if it is a weak KB-operator.

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TL;DR: In this article, the authors construct surfaces possessing an adjoint curve of a given space curve as an asymptotic curve, geodesic or line of curvature, and obtain conditions for ruled surfaces and developable ones.

Abstract: In the present paper, we construct surfaces possessing an adjoint curve of a given space curve as an asymptotic curve, geodesic or line of curvature. We obtain conditions for ruled surfaces and developable ones. Finally, we present illustrative examples to show the validity of the present method.

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TL;DR: Basicity of the system of eigenfunctions of some discontinuous spectral problem for a second order differential equation with spectral parameter in boundary condition for grand-Lebesgue space Lp)(−1; 1) is studied in this paper.

Abstract: Basicity of the system of eigenfunctions of some discontinuous spectral problem for a second order differential equation with spectral parameter in boundary condition for grand-Lebesgue space Lp)(−1; 1) is studied in this work. Since the space is nonseparable, a subspace suitable for the spectral problem is defined. The subspace Gp)(−1; 1) of Lp)(−1; 1) generated by shift operator is considered. Basicity of the system of eigenfunctions for the space Gp)(−1; 1)⊕C , 1 < p < +∞ , is proved. It is shown that the system of eigenfunctions of considered problem forms a basis for Gp)(−1; 1) , 1 < p < +∞ , after removal of any of its even-numbered functions.

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TL;DR: In this paper, the authors developed an approach to understand quaternions multiplication defining subspaces of quaternion and observed the effects of sandwiching maps on the elements of these subspacing.

Abstract: Quaternions have become a popular and powerful tool in various engineering fields, such as robotics, image and signal processing, and computer graphics. However, classical quaternions are mostly used as a representation of rotation of a vector in $3$-dimensions, and connection between its geometric interpretation and algebraic structures is still not well-developed and needs more improvements. In this study, we develop an approach to understand quaternions multiplication defining subspaces of quaternion $\mathbb{H}$, called as $\mbox{Plane} N $ and $\mbox{Line} N $, and then, we observe the effects of sandwiching maps on the elements of these subspaces. Finally, we give representations of some transformations in geometry using quaternion.

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TL;DR: The trade-off between type I and type II error probabilities in the hypothesis testing problem with (possibly non-stationary) independent samples is determined up to some multiplicative constants, assuming that the probabilities of both types of error are decaying exponentially with the number of samples, using the Berry-Esseen theorem.

Abstract: A judicious application of the Berry-Esseen theorem via suitable Augustin information measures is demonstrated to be sufficient for deriving the sphere packing bound with a prefactor that is Ω n −0.5 1−E′ sp R for all codes on certain families of channels including the Gaussian channels and the nonstationary Renyi symmetric channels and for the constant composition codes on stationary memoryless channels. The resulting nonasymptotic bounds have definite approximation error terms. As a preliminary result that might be of interest on its own, the trade-off between type I and type II error probabilities in the hypothesis testing problem with possibly non-stationary independent samples is determined up to some multiplicative constants, assuming that the probabilities of both types of error are decaying exponentially with the number of samples, using the Berry-Esseen theorem.

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TL;DR: In this paper, the authors investigated conditions for a hypersurface of almost poly-Norden Riemannian manifolds to be invariant and totally geodesic.

Abstract: Our aim in the present paper is to initiate the study of submanifolds in an almost poly-Norden Riemannian manifold, which is a new type of manifold first introduced by Sahin [17]. We give fundamental properties of submanifolds equipped with induced structures provided by almost poly-Norden Riemannian structures and find some conditions for such submanifolds to be totally geodesics. We introduce some subclasses of submanifolds in almost poly-Norden Riemannian manifolds such as invariant and antiinvariant submanifolds. We investigate conditions for a hypersurface of almost poly-Norden Riemannian manifolds to be invariant and totally geodesic, respectively, by using the components of the structure induced by the almost poly-Norden Riemannian structure of the ambient manifold. We also obtain some characterizations for totally umbilical hypersurfaces and give some examples of invariant and noninvariant hypersurfaces.

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TL;DR: In this article, a 3-dimensional Riemannian manifold equipped with a tensor structure of type 1,1, whose third power is the identity, is considered, and the manifold and the metric have circulant matrices with respect to some basis.

Abstract: A 3-dimensional Riemannian manifold equipped with a tensor structure of type 1,1 , whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e. these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civitaconnection of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given

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TL;DR: In this article, the existence and uniqueness of solutions of the fourth-order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied, and all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the maximal symmetric operator including direct sum operators are given in the single and direct sum Hilbert spaces.

Abstract: In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.

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TL;DR: In this article, the authors examined the commutativity of multiplicative generalized -skew derivations that satisfy the following conditions: i $F x^{2} +x\\delta x = ǫ 2 +xF x +ǫ 0.

Abstract: Let $R$ be a prime ring with center $Z R $ and an automorphism $\\alpha.$ A mapping $\\delta:R\\to R$ is called multiplicative skew derivation if $\\delta xy =\\delta x y+ \\alpha x \\delta y $ for all $x,y\\in R$ and a mapping $F:R\\to R$ is said to be multiplicative generalized -skew derivation if there exists a unique multiplicative skew derivation $\\delta$ such that $F xy =F x y+\\alpha x \\delta y $ for all $x,y\\in R.$ In this paper, our intent is to examine the commutativity of $R$ involving multiplicative generalized -skew derivations that satisfy the following conditions: i $F x^{2} +x\\delta x =\\delta x^{2} +xF x $, ii $F x\\circ y =\\delta x\\circ y \\pm x\\circ y$, iii $F [x,y] =\\delta [x,y] \\pm [x,y]$, iv $F x^{2} =\\delta x^{2} $, v $F [x,y] =\\pm x^{k}[x,\\delta y ]x^{m}$, vi $F x\\circ y =\\pm x^{k} x\\circ\\delta y x^{m}$, vii $F [x,y] =\\pm x^{k}[\\delta x ,y]x^{m}$, viii $F x\\circ y =\\pm x \\delta x \\circ y x^{m}$ for all $x,y\\in R.$

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TL;DR: In this paper, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order in weighted Orlicz-type spaces.

Abstract: In weighted Orlicz-type spaces Sp,μ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre K-functionals is shown in the spaces Sp,μ.

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TL;DR: In this paper, the strong 3-rainbow index of edge-amalgamation of graphs was studied and an upper bound on the minimum number of colors needed in a strong k -rainbow coloring of a graph was derived.

Abstract: Let G be a nontrivial, connected, and edge-colored graph of order n ≥ 3, where adjacent edges may be colored the same Let k be an integer with 2 ≤ k ≤ n A tree T in G is a rainbow tree if no two edges of T are colored the same For S ⊆ V G , the Steiner distance d S of S is the minimum size of a tree in G containing S An edge-coloring of G is called a strong k -rainbow coloring if for every set S of k vertices of G there exists a rainbow tree of size d S in G containing S The minimum number of colors needed in a strong k -rainbow coloring of G is called the strong k -rainbow index srxk G of G In this paper, we study the strong 3-rainbow index of edge-amalgamation of graphs We provide a sharp upper bound for the srx3 of edge-amalgamation of graphs We also determine the srx3 of edge-amalgamation of some graphs