Caspian Journal of Mathematical Sciences
Volume:12 Issue: 2, Summer Autumn 2023

In this paper the generalization of the converse of Baer's theorem for two-nilpotent variety of class row $(n,m)$. is carried out. Baer proved that finiteness of $G/Z_n(G)$ implies that $\gamma_{n+1}(G)$ is finite. Hekster proved the converse of the Baer's theorem with the assumption that $G$ can be finitely generated. The Baer's theorem can be considered as a result of a classical theorem by Schur denoting that finiteness of $G/Z(G)$ leads to the finiteness of $G'$.The converse of the Baer's theorem has been proved conditionally by Taghavi et al. (2019), as well.In the Main Theorem, we prove that, if $\gamma_{m,n}(G)\cap Z_{n,m}(G)=1$ and $\gamma_{m,n+i}(G)$ is finite for some $n,i,m \geq 0$. Then $G/Z_{n,m}(G)$ is finite.In this article some other results are attained by the converse of the Baer's theorem. It is also concluded that when $n=m=1$. Similar results are obtained for variety of the soluble groups. In addition, the converse of the Schur's theorem which proved by Halasi and Podoski is concluded in this paper, for two-nilpotent variety. We have also obtained some similar results of Chakaneh et al. (2019) for $(n,m)$-isoclinic family of groups and $(1,m)$-stem groups.

In this paper, we consider invariant (α, β)- metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces. We first study geodesic vectors and investigates the set of all homogeneous geodesics of (α, β)- metrics. Then we study the geometry of simply connected two-step nilpotent Lie groups of dimension five equipped with a left invariant (α, β)- metrics and we examine Lie algebras with 1-dimensional center, 2-dimensional center and 3-dimensional center.

In this study, we examine biorthogonal wavelets that are tailored to a specific discrete pseudo-differential equation ‎‎‎of the form $T_{\sigma}u = f$, where $T_{\sigma}$ is an invertible discrete pseudo-differential operator defined on ‎‎‎the lattice $\mathbb{Z}^{n}$ for every $f\in\ell^{2}(\mathbb{Z}^{n})‎$‎. Our focus is on computing Galerkin approximations ‎‎‎of the solution to this problem using an adaptive algorithm.‎

This paper aims to study the oscillatory nature of solutions for fourth-order neutral differential equations with mixed deviating arguments and improved oscillation conditions obtained in various cases. We reduced the problem to first-order differential inequality by using suitable substitutions that enabled us to use comparison theorems. Further, we discuss the asymptotic nature of solutions, and in the end, an example is given to validate the results.

In this article, we study a conformable fractional heat conduction equation. Applying the method of separation variables to this problem, we get a conformable fractional Sturm–Liouville eigenvalue problem. Later, we prove the existence of a countably infinite set of eigenvalues and eigenfunctions. Finally, we establish uniformly convergent expansions in the eigenfunctions.

In recent years, the hesitant fuzzy set as a proper generalization of the fuzzy set theory, which can imply in situations that the decision maker hesitates in determining the membership of parameters, has been introduced. Several applications of such sets have been revealed in the multi-criteria decision-making, graph theory and clustering methods, but there is little research on hesitant fuzzy programming problems and solving process for them in the literature. However, recently some research has been carried out in the field of linear programming under hesitant fuzzy information. However, less research can be found that has developed this perspective in nonlinear mode and especially for linear fractional programming under hesitant fuzzy data. Hence, our main focus is to propose the modeling of the linear fractional programming problem with hesitant fuzzy parameters along with the introduction of a method to solve this type of structure. For this aim, two kinds of linear fractional programming with hesitancy in different values are introduced. Then, a novel method is suggested to determine the optimal solutions for them. Some numerical examples show the reliability and validity of the models.

This paper is concerned with the existence and uniqueness of solutions for a class of frictional antiplane contact problems of p(x)-Kirchhoff type on a bounded domain of R2. Using an abstract Lagrange multiplier technique and the Schauder fixed point theorem we establish the existence of weak solutions. Imposingsome suitable monotonicity conditions on the datum f1 the uniqueness of the solution is obtained.

In this paper, the degree of convergence of Newton’s method has been increased from two to four using two function evaluations. For this purpose,the weakness of Newton’s method, derivative calculation has been eliminated with a proper approximation of the previous data. Then, by entering two selfaccelerating parameters, the family new with-memory methods with Steffensen-Like memory with convergence orders of 2.41, 2.61, 2.73, 3.56, 3.90, 3.97, and 4 are made. This goal has been achieved by approximating the self-accelerator parameters by using the secant method and Newton interpolation polynomials.Finally, we have examined the dynamic behavior of the proposed methods for solving polynomial equations.

In this work‎, ‎we generalize the equalities from ‎ g‎-frame and g‎-Bessel sequences to continuous g‎-frame and continuous g‎-Bessel sequences in Hilbert spaces‎. ‎This generalization enables us to obtain the relationship between continuous g-frames and their alternate dual on a measure space‎.