---

In this article, we have solved the Fokker-Planck Equation(FPE) by numerical
method. For the approximate solution of this problem, we used of polynomial basis functions
and the least squares method. The least squares method together with the satisfier function is
used to transform the the FPE to the solution of equation systems. Also we debate the con-
vergence of the presented technique. Then we consider illustrative examples to represent the
applicability and validity of the this method.

---

Let $mathcal{A}$ and $mathcal{B}$ be two standard operator algebra on  Banach spaces $mathcal{X}$ and $mathcal{Y}$, respectively. In this paper, we determine the forms of the surjective maps from $mathcal{A}$ onto $mathcal{B}$ such that completely preserve zero triple Jordan product in both directions.

---

In this paper we introduce a new sequence of mappings in connection to Hermite-Hadamard type inequality. Some bounds and refinements of Hermite-Hadamard inequality for convex functions via this  sequence  are given

---

Let A and B be Banach algebras with preduals A∗ and B∗ respectively, and
Θ : B → A be an algebraic homomorphism. In this paper, we derive some specific results
concerning the characterizations of module Connes amenability of certain Banach algebras.
Indeed, we investigate and give necessary and sufficient conditions for module Connes
amenability of projective tensor product Ab⊗B. Moreover, we characterize the module (ψ, θ)-
Connes amenability of Θ-Lau product A×Θ B, which ψ and θ are homomorphisms in A∗ and
B∗, respectively.

---

Let $(pi,,H)$ be a unitary representation of $G$. Our interest to us here is the Banach space $WAP(pi)$; in particular,
we state some relations between $pi$ and any subrepresentation of $pi$ in this regard.
 

---

The existence of infinitely many solutions for a class of impulsive
fractional boundary value problems is established.
Our approach is based on recent variational methods for smooth
functionals defined on reflexive Banach spaces. Some recent results are extended and improved. One example is
given in this paper to illustrate the main results.

---

In this paper we introduced direct product of intuitionistic fuzzy multigroups of G under norms(IFMSN(G)) and we prove that it will be also IFMSN(G). Next we shall study some important properties and theorems for them. On the other hand we shall give the definition of the identity element, strong upper- lower and weak upper- lowerof them and study the main theorem for this. We shall also give new results on this subject. Also we define the concepts of conjugate and commutative of IFMSN(G) and investigate them under direct product. Finally, we organize them under group homomorphisms and we prove that the image and preimage of direct product of IFMSN(G) will be also IFMSN(G).
 

---

In this paper, we get some results about $alpha$-duals of g-frames and fusion frames
in Hilbert spaces. Especially, the direct sums and tensor products for $alpha$-duals of g-frames and fusion
frames are considered and some of the obtained results for duals are generalized to $alpha$-duals.

---

The notion of a K-frame in n-Hilbert space is presented and some of their characterizations are given. We establish stability condition of K-frame in n-Hilbert space under some perturbations. We verify that sum of two K-frames is also a K-frame in n-Hilbert space. Also, the concept of tight K-frame in n-Hilbert space is described and some properties of its are going to be established.

---

In this article the Caputo time- and Riesz space-fractional Fokker-Planck equation (TSFFPE) is solved by the stable Gaussian radial basis function (RBF) method. By a spatial discretization and using the Riesz fractional derivative of the stable Gaussian radial basis function interpolants computed in [23], the computations of TSFFPE reduced to a system of fractional ODEs. A high order finite difference method is presented for this system of ODEs, and the computations are converted to a system of linear or nonlinear algebraic equations, in each time step. In the nonlinear case, these systems can be easily solved by the Newton iterative method. Numerical illustrations are performed to confirm the accuracy and efficiency of the presented method. Some comparisons are made with the results in other literature.

---

This paper presents a study of $R$-quadratic Finsler spaces and a new class of Finsler metrics called $bar{D}$-metrics. The non-Riemannian curvatures of $R$-quadratic Finsler spaces and their special case, the $R$-quadratic generalized $(alpha, beta)$-metrics, are analyzed to gain insights into their behavior. The paper then introduces the $bar{D}$-metrics, which are shown to be a proper subset of the class of $GDW$-metrics and contain the class of Douglas metrics. This paper contributes to the understanding of $R$-quadratic Finsler spaces and their properties, and presents a novel class of Finsler metrics with potential applications in the field.

---

In this paper, we first extend the well-known inequalities to the case of sector matrices. We also explore the adjointness of operator inequalities with binary operations for sector matrices. As a result of our exploration, we establish four distinct inequalities: a matrix inequality, a unitarily invariant norm inequality, a singular value inequality, and a determinant inequality. For example, we demonstrate that if $sigma_{1}$ and $ sigma_{2} $ are non-zero connections, and if $A$, $B$, and $C$ belong to $mathcal{S}_{alpha}$,  such that

	begin{equation*}

	mathcal{R}left(A  sigma_{1}  (B  sigma_{2}  C)right) leq cos^{4}(alpha)  mathcal{R}left((A  sigma_{1}  B)  sigma_{2}  (A  sigma_{1}  C)right),

	end{equation*}

	then

	begin{equation*}

	mathcal{R}left(A  sigma_{1}^* (B  sigma_{2}^*  C)right)geq cos^{4}(alpha)  mathcal{R}left((A sigma_{1}^*   B)  sigma_{2}^*  (A  sigma_{1}^*  C)right).

	end{equation*}

---

In this paper, we study Cartan torsion, mean Cartan torsion and mean Landsberg
curvature of 4-dimensional Finsler metrics. First, we find the necessary and sufficient
condition under which a 4-dimensional Finsler manifold has bounded Cartan torsion and
mean Cartan torsion. Then, we show that a 4-dimensional Finsler manifold has relatively
isotropic mean Landsberg curvature if and only if it is Riemannian or the main scalars of
Finsler metric satisfy the certain conditions.

---

In this paper, we study Kropina spaces whose geodesics are the orbits of one-parameter subgroup of the group of isometries. Also, we study Kropina g.o. metrics on ho-mogeneous spaces with two isotropy summands and we will investigate Kropina g.o. metrics
on compact homogeneous spaces with two isotropy summands. A complete characterization
of navigation data of non-Riemannian Kropina g.o. metrics is given.

---

The ease with which digital images can be manipulated with readily available editing tools highlights the critical need for robust copyright protection mechanisms. Digital watermarking tackles this challenge by embedding imperceptible ownership information within images. However, striking a balance between transparency (invisibility of the watermark) and robustness against attacks remains a significant hurdle. This paper proposes a watermarking method that uses the combined strengths of Finite Ridgelet Transform (FRIT) and Bidiagonal Singular Value Decomposition (BSVD). Our approach first pre-processes the image using FRIT to extract prominent features. Subsequently, the watermark is imperceptibly embedded into the singular values obtained from the FRIT coefficients. Our evaluation using standard sample images confirms high visual quality and robustness to attacks for the proposed method.
 

---

In this paper, we  study the Landsberg and mean Landsberg curvatures of two-dimensional Finsler manifolds. First, we prove that a two-dimensional Finsler metric  is a generalized Landsberg metric if and only if it is a stretch metric. Then, we study Finsler surfaces with  isotropic main scalar and find the necessary and sufficient condition under which these spaces has vanishing ${bf tilde J}$-curvature.