Journal of Linear and Topological Algebra
Volume:11 Issue: 4, Autumn 2022

‎In this paper‎, ‎we prove that chevalley groups $G_2 (q)$‎, ‎where $q\equiv \pm2($mod 5$)$ and $q^2+q+1$ is a prime number‎, ‎can be uniquely determined by the order of the group and the second largest element order‎.

‎A hereditary class on a set $X$ is a nonempty collection of subsets of $X$ which is closed under‎ ‎subsets‎. ‎In this paper‎, ‎we present a new structure of proximity spaces by using a hereditary class‎, ‎called‎ ‎$\mathcal{H}$-proximity spaces‎, ‎as a generalization of Efremovi$\check{c}$ proximity spaces‎, ‎$I$-proximity spaces and‎ ‎coarse proximity spaces‎. ‎Some properties of this proximity structure and generalized topology induced by it are studied‎.

This paper uses the classical Lie method to determine symmetry reductions and exact solutions of the time-dependent Calogero-Bogoyavlenskii-Schiff equation (vCBS). This classical method generates some exact arbitrary solutions and exhibits various qualitative behaviors. Here, we derived the infinitesimal symmetries and six basic combinations of vector fields in the linear forms that can be utilized to transform the given equation into the PDEs with their variables. Further, we obtain comprehensive invariant solutions of the vCBS equation. Next, we apply a direct method to explore conservation laws. Finally, we determine the conservation laws of the vCBS equation via the Bluman-Anco homotopy formula.

‎An analytical fuzzy solution is achieved by means of the fuzzy d'Alembert formula for the fuzzy one-dimensional homogeneous wave‎ ‎equation in a half-line considering the generalized Hukuhara partial differentiability of the solution‎. ‎In the current article‎, ‎the exclusive‎ ‎solution and the stability of the homogeneous fuzzy wave equations are brought into existence‎. ‎Eventually‎, ‎given the various instances represented‎, ‎the efficacy and accuracy of the method are scrutinized‎.

Let $A$ be an $S$-act where $S$ is a monoid. Then $A$ is called lifting if every proper subact $L$ of $A$ lies over a direct summand, that is, $L$ contains a direct summand $K$ of $A$ such that $K\subset L$ is co-small in $A$. In this paper, characterizations of lifting $S$-acts and co-closed subacts are presented. We show that the class of supplemented acts are strictly larger than that of lifting ones.

‎In this note‎, ‎we study the following functional equations‎:‎\begin{align*}‎‎&L(L(p ,r)+L(‎q‎,r)+p + q ,r)+L(L( p, r)+ p , r)+L(q, r )=0,\\‎‎&L(L( p , r )+ p + q+e, r )+L( p, r)=L( p + q , r )+ p L(q , r)‎\end{align*}‎‎and‎ ‎$‎L( p , q )=L(\zeta  p , q), \vert \zeta\vert <1‎$,‎ without any regularity assumption for all $ p , q , r \in A$, where $L:A^2\rightarrow A$ is defined by ‎$‎L( p ,  q ):=g( p + q )-g( p  )-g( q )‎$‎ for all $ p , q\in A$. Also, we find general solutions of the above functional equations on algebras, unital algebras and real numbers, respectively. Finally, we investigate the  stability of those functional equations in algebras and unital algebras, respectively.

‎The main purpose of this paper is to study the class of $e\text{-}\theta$-open sets and explore some of their new properties‎. ‎Also‎, ‎we introduce and study some weak separation axioms by utilizing $e\text{-} \theta$-open sets‎. ‎In addition‎, ‎we define the notion of $e\text{-}\theta\text{-}$kernel and slightly $e\text{-}\theta\text{-}R_0$ spaces‎. ‎Furthermore‎, ‎we apply them to discuss some fundamental properties of the graph functions‎. ‎We obtain not only some characterizations but also many new results‎.