https://jims.ims.ir/ Journal of the Iranian Mathematical Society en daily 1 Sat, 01 Aug 2026 00:00:00 +0330 Sat, 01 Aug 2026 00:00:00 +0330 https://jims.ims.ir/article_247715.html In this article‎, ‎we show that if $(M=G/H,g)$ is a naturally reductive homogeneous Riemannian manifold and $\frak{g}' \subset \frak{h}+\frak{m}_1$‎, ‎then the $(\alpha_1,\alpha_2)$-metric $F$ induced by $g$ and the decomposition $\frak{m}=\frak{m}_1+\frak{m}_2$ is also naturally reductive‎. ‎To establish this‎, ‎we provide an explicit formula for the Cartan tensor of homogeneous $(\alpha_1,\alpha_2)$-metrics‎. ‎We further examine the relationship between geodesic vector fields on homogeneous Riemannian spaces and their counterparts on homogeneous $(\alpha_1,\alpha_2)$-spaces‎. ‎Finally‎, ‎we construct left-invariant $(\alpha_1,\alpha_2)$-metrics on the tangent bundle of Lie groups from left-invariant Randers metrics on the base Lie group and study their geometric relations‎. https://jims.ims.ir/article_247839.html Let $R$ be a prime ring with char $(R)\neq 2, 3$ and $f(x_{1}, \ldots, x_{n})$ be a non-central multilinear polynomial over $C(=Z(U))$, where $U$ be the Utumi ring of quotients of $R$. Let $d\neq 0$ be a derivation of $R$. Suppose that $F, G$ are two generalized derivations of $R$. Let $A=\{[(F^2+G)(f(x_1,\ldots,x_n)),f(x_1,\ldots,x_n)] : x_1,\ldots,x_n\in R\}$. In the present article, we determine the nature of the maps $F$ and $G$, when $d(A)=0$. Moreover, it is proved that if $A\neq 0$, then $C_R(A)=Z(R)$, where $C_R (A)=\{a\in R : [a,x]=0 ~~~ \forall x\in A\}$ is the centralizer of $A$. https://jims.ims.ir/article_250584.html This study explores a collection of fixed point theorems and the existence of fixed points for large enriched contractions within convex probabilistic b-metric spaces. We identify the necessary conditions that guarantee the presence of fixed points for such mappings. The research employs the extends Manns iteration algorithm to this context. Several intricate examples are presented, and our principal findings are applied to the investigation of free mechanical oscillations, which are described by second-order differential equations.