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2020

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Articles 991 - 1020 of 1327

Full-Text Articles in Mathematics

Fourth Order Differential Operators With Distributional Potentials, Eki̇n Uğurlu, Elgi̇z Bairamov Jan 2020

Fourth Order Differential Operators With Distributional Potentials, Eki̇n Uğurlu, Elgi̇z Bairamov

Turkish Journal of Mathematics

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 …

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Existence And Uniqueness Of Solutions For Nonlinear Caputo Fractional Difference Equations, Churong Chen, Martin Bohner, Baoguo Jia Jan 2020

Existence And Uniqueness Of Solutions For Nonlinear Caputo Fractional Difference Equations, Churong Chen, Martin Bohner, Baoguo Jia

Turkish Journal of Mathematics

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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On Bishop Frame Of A Pseudo Null Curve In Minkowski Space-Time, Jelena Djordjevic, Emilija Nesovic Jan 2020

On Bishop Frame Of A Pseudo Null Curve In Minkowski Space-Time, Jelena Djordjevic, Emilija Nesovic

Turkish Journal of Mathematics

In this paper, we introduce the Bishop frame of a pseudo null curve $\alpha$ in Minkowski space-time. We obtain the Bishop frame's equations and the relation between the Frenet frame and the Bishop frame. We find the third order nonlinear differential equation whose particular solutions determine the form of the Bishop curvatures. By using space-time geometric algebra, we derive the Darboux bivectors $D$ and $\tilde{D}$ of the Frenet and the Bishop frame of $\alpha$, respectively. We give geometric interpretations of the Frenet and the Bishop curvatures of $\alpha$ in terms of areas of the projections of the corresponding Darboux bivectors …

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Bertrand And Mannheim Curves Of Framed Curves In The 3-Dimensional Euclidean Space, Shun'ichi Honda, Masatomo Takahashi Jan 2020

Bertrand And Mannheim Curves Of Framed Curves In The 3-Dimensional Euclidean Space, Shun'ichi Honda, Masatomo Takahashi

Turkish Journal of Mathematics

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 …

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Inverse Problems For Differential Operators With Two Delays Larger Than Half The Length Of The Interval And Dirichlet Conditions, Biljana Vojvodic, Milenko Pikula, Vladimir Vladicic, Fatma Ayça Çeti̇nkaya Jan 2020

Inverse Problems For Differential Operators With Two Delays Larger Than Half The Length Of The Interval And Dirichlet Conditions, Biljana Vojvodic, Milenko Pikula, Vladimir Vladicic, Fatma Ayça Çeti̇nkaya

Turkish Journal of Mathematics

This paper deals with nonself-adjoint second-order differential operators with two constant delays from $\left[\frac{\pi}{2}, \pi\right)$ and two real-valued potentials from $L_{2} [0,\pi]$. An inverse spectral problem of recovering operators from the spectra of four boundary value problems is studied.

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Solution Of Nonlinear Ordinary Differential Equations With Quadratic And Cubic Terms By Morgan-Voyce Matrix-Collocation Method, Mehmet Tarakçi, Mustafa Özel, Mehmet Sezer Jan 2020

Solution Of Nonlinear Ordinary Differential Equations With Quadratic And Cubic Terms By Morgan-Voyce Matrix-Collocation Method, Mehmet Tarakçi, Mustafa Özel, Mehmet Sezer

Turkish Journal of Mathematics

Nonlinear differential equations have many applications in different science and engineering disciplines. However, a nonlinear differential equation cannot be solved analytically and so must be solved numerically. Thus, we aim to develop a novel numerical algorithm based on Morgan-Voyce polynomials with collocation points and operational matrix method to solve nonlinear differential equations. In the our proposed method, the nonlinear differential equations including quadratic and cubic terms having the initial conditions are converted to a matrix equation. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB. The solution of this method for …

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A Simple Derivation Of The Refined Sphere Packing Bound Under Certain Symmetry Hypotheses, Bariş Naki̇boğlu Jan 2020

A Simple Derivation Of The Refined Sphere Packing Bound Under Certain Symmetry Hypotheses, Bariş Naki̇boğlu

Turkish Journal of Mathematics

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 $\mathit{\Omega}\left(n^{-0.5(1-E_{sp}'(R))}\right)$ 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 …

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The Statistically Unbounded $\Tau$-Convergence On Locally Solid Riesz Spaces, Abdullah Aydin Jan 2020

The Statistically Unbounded $\Tau$-Convergence On Locally Solid Riesz Spaces, Abdullah Aydin

Turkish Journal of Mathematics

A sequence $(x_n)$ in a locally solid Riesz space $(E,\tau)$ is said to be statistically unbounded $\tau$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce the concept of the $st$-$u_\tau$-convergence and give the notions of $st$-$u_\tau$-closed subset, $st$-$u_\tau$-Cauchy sequence, $st$-$u_\tau$-continuous and $st$-$u_\tau$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_\tau$-convergence.

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Erratum To "Study On Quasi-$\Gamma$-Hyperideals In $\Gamma$-Semihypergroups", Niovi Kehayopulu Jan 2020

Erratum To "Study On Quasi-$\Gamma$-Hyperideals In $\Gamma$-Semihypergroups", Niovi Kehayopulu

Turkish Journal of Mathematics

We wrote this note to show that the definition of $\Gamma$-hypersemigroups in [2] should be corrected, and that it is not enough to replace the hyperoperation $\circ$ of the hypersemigroup by $\Gamma$ to pass from a hypersemigroup to a $\Gamma$-hypersemigroup. Care should be taken about the definitions of $(m,n)$-quasi-$\Gamma$-hyperideal, the $m$-left $\Gamma$-hyperideal and the $n$-right $\Gamma$-hyperideal as well.

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Global Existence And Blow-Up Of Solutions Of The Time-Fractional Space-Involution Reaction-Diffusion Equation, Rami̇z Tapdigoğlu, Berikbol Torebek Jan 2020

Global Existence And Blow-Up Of Solutions Of The Time-Fractional Space-Involution Reaction-Diffusion Equation, Rami̇z Tapdigoğlu, Berikbol Torebek

Turkish Journal of Mathematics

A time-fractional space-nonlocal reaction-diffusion equation in a bounded domain is considered. First, the existence of a unique local mild solution is proved. Applying Poincaré inequality it is obtained the existence and boundedness of global classical solution for small initial data. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time.

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New Multiple Solutions For A Schrödinger-Poisson System Involving Concave-Convex Nonlinearities, Chun-Yu Lei, Gao-Sheng Liu, Chang-Mu Chu, Hong-Min Suo Jan 2020

New Multiple Solutions For A Schrödinger-Poisson System Involving Concave-Convex Nonlinearities, Chun-Yu Lei, Gao-Sheng Liu, Chang-Mu Chu, Hong-Min Suo

Turkish Journal of Mathematics

In this paper, we study the following critical growth Schrödinger-Poisson system with concave-convex nonlinearities term $\left\{\begin{array} -\Delta u + u + \eta\varphi u = \lambda f(x) u^{q-1} + u^5, in R^3, \\ -\Delta \varphi = u^2, in R^3,\end{array}\right. $ where $1 < q < 2, \eta\in \mathbb{R}, \lambda > 0$ is a real parameter and $f \in L^{\frac{6}{6-q}} (\mathbb{R}^3)$ is a nonzero nonnegative function. Using the variational method, we obtain that there exists a positive constant $\lambda_* > 0$ such that for all $\lambda \in (0,\lambda_*)$, the system has at least two positive solutions.

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Degree Of Approximation By Means Of Hexagonal Fourier Series, Ali̇ Güven Jan 2020

Degree Of Approximation By Means Of Hexagonal Fourier Series, Ali̇ Güven

Turkish Journal of Mathematics

Let $f$ be a continuous function which is periodic with respect to the hexagon lattice, and let $A$ be a lower triangular infinite matrix of nonnegative real numbers with nonincreasing rows. The degree of approximation of the function $f$ by matrix means $T_{n}^{\left( A\right) }\left( f\right) $ of its hexagonal Fourier series is estimated in terms of the modulus of continuity of $f.$

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Korovkin-Type Theorems And Their Statistical Versions In Grand Lebesgue Spaces, Yusuf Zeren, Miqdad Ismailov, Cemi̇l Karaçam Jan 2020

Korovkin-Type Theorems And Their Statistical Versions In Grand Lebesgue Spaces, Yusuf Zeren, Miqdad Ismailov, Cemi̇l Karaçam

Turkish Journal of Mathematics

The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace $G^{p)} (-\pi ;\pi )$ of grand Lebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions is dense in $G^{p)}(-\pi ;\pi )$. The analogs of Korovkin theorems are proved in $G^{p)} (-\pi ;\pi )$. These results are established in $G^{p)} (-\pi ;\pi )$ 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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The Fekete-Szegö Inequality For Subclasses Of Analytic Functions Related To Modified Sigmoid Functions, Muhammet Kamali̇, Hali̇t Orhan, Murat Çağlar Jan 2020

The Fekete-Szegö Inequality For Subclasses Of Analytic Functions Related To Modified Sigmoid Functions, Muhammet Kamali̇, Hali̇t Orhan, Murat Çağlar

Turkish Journal of Mathematics

In this paper, the authors investigate the initial coefficient bounds for a new generalized subclass of analytic functions related to Sigmoid functions. Also, the relevant connections with the famous classical Fekete?Szegö inequality for these classes are discussed.

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The Homogenization Of Diffusion-Convection Equations In Non-Periodic Structures, Anvarbek Meirmanov, Oleg Galtsev Jan 2020

The Homogenization Of Diffusion-Convection Equations In Non-Periodic Structures, Anvarbek Meirmanov, Oleg Galtsev

Turkish Journal of Mathematics

We consider the homogenization of diffusion-convective problems with given divergence-free velocities in nonperiodic structures defined by sequences of characteristic functions(the first sequence). These quence of concentration (the second sequence)is uniformly bounded in the space of square-summable functions with square-summable derivatives with respect to spatial variables. At the same time, the sequence of time-derivative of product of these concentrations on the characteristic functions, that define a nonperiodic structure, is bounded in the space of square-summable functions from time interval into the conjugated space of functions depending on spatial variables, withsquare-summable derivatives. We prove the strong compactness of the second sequences in …

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An Improved Trudinger--Moser Inequality And Its Extremal Functions Involving $L^P$-Norm In $\Mathbb{R}^2$, Xiaomeng Li Jan 2020

An Improved Trudinger--Moser Inequality And Its Extremal Functions Involving $L^P$-Norm In $\Mathbb{R}^2$, Xiaomeng Li

Turkish Journal of Mathematics

Let $W^{1,2}(\mathbb{R}^2)$ be the standard Sobolev space. Denote for any real number $p>2$ \begin{align*}\lambda_{p}=\inf\limits_{u\in W^{1,2}(\mathbb{R}^2),u\not\equiv0}\frac{\int_{\mathbb{R}^{2}}( \nabla u ^2+ u ^2)dx}{(\int_{\mathbb{R}^{2}} u ^pdx)^{2/p}}. \end{align*} Define a norm in $W^{1,2}(\mathbb{R}^2)$ by \begin{align*}\ u\ _{\alpha,p}=\left(\int_{\mathbb{R}^{2}}( \nabla u ^2+ u ^2)dx-\alpha(\int_{\mathbb{R}^{2}} u ^pdx)^{2/p}\right)^{1/2}\end{align*} where $0\leq\alpha2$ and $0\leq\alpha

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Gradient Weyl-Ricci Soliton, Cornelia-Livia Bejan, Şemsi̇ Eken Meri̇ç, Erol Kiliç Jan 2020

Gradient Weyl-Ricci Soliton, Cornelia-Livia Bejan, Şemsi̇ Eken Meri̇ç, Erol Kiliç

Turkish Journal of Mathematics

The classical notion of gradient Ricci soliton is extended here to the gradient Weyl-Ricci soliton. A Weyl structureofthebasemanifold M is lifted to its tangent bundle TM, by using the Sasaki metric. We give some necessary and sufficient conditions such that the Weyl structure on TM to be a gradient Weyl-Ricci soliton.

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A New Solution To The Discontinuity Problem On Metric Spaces, Ufuk Çeli̇k, Ni̇hal Özgür Jan 2020

A New Solution To The Discontinuity Problem On Metric Spaces, Ufuk Çeli̇k, Ni̇hal Özgür

Turkish Journal of Mathematics

We study on the Rhoades' question concerning the discontinuity problem at fixed point for a self-mapping T of a metric space. We obtain a new solution to this question. Our result generalizes some recent theorems existing in the literature and implies the uniqueness of the fixed point. However, there are also cases where the fixed point set of a self-mapping contains more than 1 element. Therefore, by a geometric point of view, we extend the Rhoades' question to the case where the fixed point set is a circle. We also give a solution to this extended version.

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An Extragradient Algorithm For Split Generalized Equilibrium Problem And The Set Of Fixed Points Ofquasi-$\Phi $-Nonexpansive Mappings In Banach Spaces, Olawale Oyewole, Oluwatosin Mewomo, Lateef Jolaoso, Safeer Hussain Khan Jan 2020

An Extragradient Algorithm For Split Generalized Equilibrium Problem And The Set Of Fixed Points Ofquasi-$\Phi $-Nonexpansive Mappings In Banach Spaces, Olawale Oyewole, Oluwatosin Mewomo, Lateef Jolaoso, Safeer Hussain Khan

Turkish Journal of Mathematics

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 …

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A Study On Horadam Hybrid Numbers, Tuncay Deni̇z Şentürk, Göksal Bi̇lgi̇ci̇, Ahmet Daşdemi̇r, Zafer Ünal Jan 2020

A Study On Horadam Hybrid Numbers, Tuncay Deni̇z Şentürk, Göksal Bi̇lgi̇ci̇, Ahmet Daşdemi̇r, Zafer Ünal

Turkish Journal of Mathematics

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.

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Abstract Korovkin-Type Theorems In The Filter Setting With Respect To Relative Uniform Convergence, Antonio Boccuto, Kami̇l Demi̇rci̇, Sevda Yildiz Jan 2020

Abstract Korovkin-Type Theorems In The Filter Setting With Respect To Relative Uniform Convergence, Antonio Boccuto, Kami̇l Demi̇rci̇, Sevda Yildiz

Turkish Journal of Mathematics

We prove a Korovkin-type approximation theorem using abstract relative uniform filter convergence of a net of functions with respect to another fixed filter, a particular case of which is that of all neighborhoods of a point, belonging to the domain of the involved functions. We give some examples, in which we show that our results are strict generalizations of the classical ones.

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A Semi-Markovian Renewal Reward Process With $\Gamma(G)$ Distributed Demand, Asli Bektaş Kamişlik, Büşra Alakoç, Tülay Kesemen, Tahi̇r Khani̇yev Jan 2020

A Semi-Markovian Renewal Reward Process With $\Gamma(G)$ Distributed Demand, Asli Bektaş Kamişlik, Büşra Alakoç, Tülay Kesemen, Tahi̇r Khani̇yev

Turkish Journal of Mathematics

We consider a classical semi-Markovian stochastic model of type $(s,S)$ with Logistic distributed demand random variables. Logistic distribution is a member of special distribution class known as $\Gamma(g)$ that encounters in many real-life applications involving extreme value theory. The objective of this study is to observe some major characteristics of a stochastic process $X(t)$ which represents semi-Markovian renewal reward process of type $(s,S)$. We used new approximation results for renewal function that allow us to obtain three-term asymptotic expansion for ergodic distribution function and for $n^{th}$ order moments of ergodic distribution of the process $X(t)$.

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On A Three-Dimensional Solvable System Of Difference Equations, Yacine Halim, Massaoud Berkal, Amira Khelifa Jan 2020

On A Three-Dimensional Solvable System Of Difference Equations, Yacine Halim, Massaoud Berkal, Amira Khelifa

Turkish Journal of Mathematics

In this paper we solve the following system of difference equations \begin{equation*} x_{n+1}=\dfrac{z_{n-1}}{a+by_nz_{n-1}},\quad y_{n+1}=\dfrac{x_{n-1}}{a+bz_nx_{n-1}},\quad z_{n+1}=\dfrac{y_{n-1}}{a+bx_ny_{n-1}},\quad n\in \mathbb{N}_{0} \end{equation*} where parameters $a, b$ and initial values $x_{-1},x_{0},y_{-1},y_{0},z_{-1},z_{0}$ are nonzero real numbers, and give a representation of its general solution in terms of a specially chosen solutions to homogeneous linear difference equation with constant coefficients associated to the system.

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On Geometric Applications Of Quaternions, Burcu Bektaş Demi̇rci̇, Nazim Aghayev Jan 2020

On Geometric Applications Of Quaternions, Burcu Bektaş Demi̇rci̇, Nazim Aghayev

Turkish Journal of Mathematics

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 …

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Solvability And Maximal Regularity Results For A Differential Equation With Diffusion Coefficient, Kordan Ospanov, Adilet Yesbayev Jan 2020

Solvability And Maximal Regularity Results For A Differential Equation With Diffusion Coefficient, Kordan Ospanov, Adilet Yesbayev

Turkish Journal of Mathematics

We consider a second-order differential equation with rapidly growing intermediate coefficients. We obtain a solvability result in the cases that the diffusion coefficient of equation is unbounded or it tends to zero at the infinity. Under additional conditions, we prove the $L_p - $ maximal regularity estimate for the solution of this equation.

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Bilateral-Type Solutions To The Fixed-Circle Problem With Rectified Linear Units Application, Ni̇hal Taş Jan 2020

Bilateral-Type Solutions To The Fixed-Circle Problem With Rectified Linear Units Application, Ni̇hal Taş

Turkish Journal of Mathematics

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".

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Faber Polynomial Coefficients For Certain Subclasses Of Analytic And Biunivalent Functions, Abdel Moneim Lashin, Fatma Elemam Jan 2020

Faber Polynomial Coefficients For Certain Subclasses Of Analytic And Biunivalent Functions, Abdel Moneim Lashin, Fatma Elemam

Turkish Journal of Mathematics

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions defined in the open unit disc. We use the Faber polynomial expansions to find upper bounds for the $n$th$~(n\geq 3)$ Taylor-Maclaurin coefficients $\left\vert a_{n}\right\vert $ of functions belong to these new subclasses with $a_{k}=0$ for $2\leq k\leq n-1$, also we find non-sharp estimates on the first two coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $. The results, which are presented in this paper, would generalize those in related earlier works of several authors.

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On Lyapunov-Type Inequalities For Boundary Value Problems Of Fractional Caputo-Fabrizio Derivative, Şuayi̇p Toprakseven Jan 2020

On Lyapunov-Type Inequalities For Boundary Value Problems Of Fractional Caputo-Fabrizio Derivative, Şuayi̇p Toprakseven

Turkish Journal of Mathematics

In this study, Lyapunov-type inequalities for fractional boundary value problems involving the fractional Caputo Fabrizio differential equation with mixed boundary conditions when the fractional order of $\beta \in (1,2]$ and Dirichlet-type boundary condition when the fractional order of $\sigma \in (2,3]$ have been derived. Some consequences of the results related to the fractional Sturm?Liouville eigenvalue problems have also been given. Additionally, we examine the nonexistence of the solution of the fractional boundary value problem.

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Remarks On The One-Dimensional Sloshing Problem Involving The $P$-Laplacian Operator, Wei-Chuan Chen, Yanhsiou Cheng Jan 2020

Remarks On The One-Dimensional Sloshing Problem Involving The $P$-Laplacian Operator, Wei-Chuan Chen, Yanhsiou Cheng

Turkish Journal of Mathematics

In this paper, we study the inverse nodal problem and the eigenvalue gap for the one-dimensional sloshing problem with the $p$-Laplacian operator. By applying the Prüfer substitution, we first derive the reconstruction formula of the depth function by using the information of the nodal data. Furthermore, we employ the Tikhonov regularization method to consider how to reconstruct the depth function using only zeros of one eigenfunction. Finally, we investigate the eigenvalue gap under the restriction of symmetric single-well depth functions. We show the gap attains its minimum when the depth function is constant.

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On Holomorphic Poly-Norden Manifolds, Çağri Karaman, Zühre Topuz Jan 2020

On Holomorphic Poly-Norden Manifolds, Çağri Karaman, Zühre Topuz

Turkish Journal of Mathematics

In this paper, we investigated a new manifold with a poly-Norden structure, which is inspired by the positive root of the equation $x^{2}-mx-1=0$. We call this new manifold as holomorphic poly-Norden manifolds. We examine some properties of the Riemann curvature tensor and give an example of this manifold. Then, we define a different connection on this manifold which is named the semisymmetric metric poly F-connection and study some properties of the curvature and torsion tensor field according to this connection.

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