Global Analysis and Discrete Mathematics
Volume:6 Issue: 1, Summer and Autumn 2021

This research tends to focus on proving the results of coupled fixed point in partially ordered G-metric spaces for the sequence of mappings via α-series. At the end of the paper, we give an example to illustrate our results.

textit{Monfared} defined $theta$-Lau product structure $Atimes_{theta} B$ for two Banach algebras $A$ and $B$, where $theta:Brightarrow mathbb{C} $ is a multiplicative linear functional. In this paper, we study the notion of left $phi$-biflatness and left $phi$-biprojectivity for the $theta$ Lau product structure $Atimes_{theta} B$. For a locally compact group $G$, we show that $M(G)times_{theta}M(G)$ is left character biflat (left character biprojective) if and only if $G$ is discrete and amenable ($G$ is finite), respectively. Also we prove that $ell^{1}(Bbb{N}_{vee})times_{theta}ell^{1}(Bbb{N}_{vee})$ is neither $(phi_{Bbb{N}_{vee}}, theta)$-biprojective nor $ (0, phi_{Bbb{N}_{vee}})$-biprojective, where $phi_{Bbb{N}_{vee}}$ is the augmentation character on $ell^{1}(Bbb{N}_{vee}).$ Finally, we give an example among the Lau product structure of matrix algebras which is not left $phi$-biflat.

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The Baer's theorem in the termes of the Lie algebras states that for a Lie algebra $L$ the finiteness of $mathrm{dim}(L/Z_i(L))$ implies the finiteness of $mathrm{dim}(gamma_{i+1}(L))$. Let $(N,L)$ denote a pair of Lie algebras, where $N$ is an ideal of $L$, and $d_i=d_i(L)$ denote the minimal number of generators of $L/Z_i(N, L)$. In this paper we shall consider the pair $(N, L)$ and show that if $d_n$ is finite then the converse of Baer's theorem is true. In fact we shall show that if $d_n$ and $mathrm{dim}(gamma_{i+1}(N, L))$ are finite, where $igeq n$, then $N/Z_i(N, L))$ is finite. In particular, we shall provide an upper bound as following, $$mathrm{dim}(frac{N}{Z_i(N, L)}) leq ((d_n)^nd_nd_{n+1}ldots d_{i-1})mathrm{dim}(gamma_{i+1}(N, L))$$$$leq (d_n)^i(mathrm{dim}gamma_{i+1}(N, L)).$$

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Let $f:Xtimes Ytimes Zlongrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is Aron-Berner regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of tri-linear maps.

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In this paper, a numerical iterative algorithm based on combination of the successive approximations method and the quadrature formula for solving two-dimensional nonlinear Volterra integral equations is proposed. This algorithm uses a trapezoidal quadrature rule for Lipschitzian functions applied at each iterative step. The convergence analysis and error estimate of the method are proved. Finally, two numerical examples are presented to show the accuracy of the proposed method.

In this paper, we use a Milstein scheme to develop a numerical technique for solving Stochastic differential equation which we had its deterministic form in our last article cite{Tepological}, we discuss the existence and uniqueness solution of deterministic and stochastic form, and then we show the advantages of the method with numerical example.

Let V be a variety of groups defined by a set v of laws .Let (N,G) be a pair of groups in which N is a normal subgroup of G. we define the lower and upper V-marginal series of the pair (N,G) and prove some results on the varieties nilpotent pais of groups.Moreover,we extend some properties of the Baer-invariant and isologism of a pairs of groups.

- The goal of 3-D visualization is to provide the user with an intuitive interface which enables him to explore the 3-D data. The rapid development in information technology has immensely contributed to the use of modern approaches for visualizing volumetric data. Consequently, medical volume visualization is increasingly attracting attension towards achieving an effective visualization algorithm for medical diagnosis and pre-treatment planning. Previously, research has been addressing implementation of algorithm that can visualize 2-D images into 3-D. Meanwhile, in medical diagnosis, finding the exact diseases location is an important step of surgery / disease management. For 3-D Medical Data, Magnetic Resonance Images (MRI) have been used to create the 3D model, we used the Direct Volume Rendering technique. This paper proposes a ray casting algorithm for accurate allocation and localization of human abdomen abnormalities using magnetic resonance images (Abdomen MRI) of normal and abnormal patients.

 Let G be simple connected graph with the vertex and edge sets V (G) and E(G), respectively. The Schultz and the Modified Schultz indices of G are defined as Sc(G) = 1 2 ∑ u,v∈V (G) (du+dv)d(u, v), Sc∗ (G) = 1 2 ∑ u,v∈V (G) (du × dv)d(u, v), where d(u, v) is the topological distance between vertices u and v, dv is the degree of vertex v of G. In this paper, computation of the Schultz and the Modified Schultz indices of the Kragujevac trees is proposed. As an application, we obtain an upper bound and a lower bound for the Schultz and the modified Schultz indices of these trees.

In this article, we the concept of soft G-metric space and continuous soft mapping on it introduce. We also the Banach fixed point theorem in complete soft G-metric spaces investigate.

The perfect m-coloring with matrix A = [aij ]i;j2f1;


;mg of a graph G = (V;E) withf1;


;mg color is a vertices coloring of G with m-color so that number of vertex in color j adjacent to a xed vertex in color i is aij , independent of the choice of vertex in color i. The matrix A = [aij ]i;j2f1;


;mg is called the parameter matrix.We study the perfect 4-colorings of the 3-regular graphs of order 10, that is, we determine a list of all color parameter matrices corresponding to perfect colorings of 3-regular graphs of order 10.

In this paper, the fuzzy regression model is considered with crisp inputs and symmetric triangular fuzzy output. This study aims to formulate the fuzzy inference system based on the Sugeno inference model for the fuzzy regression function prediction by the fuzzy least-squares problem-based on Diamond's distance. In this study, the fuzzy least-squares problem is used to optimize consequent parameters, and the results are derived based on the V-fold cross-validation, so that the validity and quality of the proposed method can be guaranteed. The proposed method is used to reduce the bias and the boundary effect of the estimated underlying regression function. Also, a comparative study of the fuzzy nonparametric regression function prediction is carried out between the proposed model and smoothing methods, such as k-nearest neighbor (k-NN), kernel smoothing (KS), and local linear smoothing (LLS). Different approaches are illustrated by some examples and the results are compared. Comparing the results indicates that, among the various prediction models, the proposed model is the best, decreasing the boundary effect significantly. Also, in comparison with different methods, in both one-dimensional and two-dimensional inputs, it may be considered the best candidate for the prediction.