Journal of Linear and Topological Algebra
Volume:13 Issue: 1, Winter 2024

‎Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero identity‎, ‎$S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ be a graded $R$-module‎. ‎A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-strongly prime if there exists $s\in S$ such that whenever $((N+Rx_{g}):_{R}M)y_{h}\subseteq N$‎, ‎then $sx_{g}\in N$ or $sy_{h}\in N$ for all $x_{g},y_{h}\in h(M)$‎. ‎The aim of this paper is to introduce and investigate some basic properties of the notion of graded $S$-strongly prime submodules‎, ‎especially in graded multiplication modules‎. ‎Moreover‎, ‎we investigate the behaviour of this structure under graded module homomorphisms‎, ‎localizations of graded modules‎, ‎quotient graded modules‎, ‎Cartesian product‎.

New norm and numerical radius inequalities for operators on Hilbert space are given. Among other inequalities, we prove that if $ A, B \in B(H) $, then \[\Vert A \Vert - \frac{3 \Vert A-B^* \Vert }{2} \leq \omega\left(\left[\begin{array}{cc} 0 & A \\ B & 0 \end{array}\right]\right).\] Moreover, $\omega(AB) \leq \frac{3}{2} \Vert Im(A) \Vert \Vert B \Vert + D_{B}\; \omega(A) $. In particular, if $ A $ is self-adjointable, then $\omega(AB) \leq D_{B} \Vert A \Vert$, where $D_{B}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf}}\,\left\| B-\lambda I \right\|$.

‎The underlying aim of this paper is first to state the cyclic‎‎version of $\mathcal{J}$-integral Banach type contractive mappings introduced by Fallahi‎, ‎Ghahramani and Soleimani Rad‎‎[Integral type contractions in partially ordered metric spaces and best proximity point‎, ‎Iran‎. ‎J‎. ‎Sci‎. ‎Technol‎. ‎Trans‎. ‎Sci‎. ‎44 (2020)‎, ‎177-183]‎ ‎and second to show the existence of best proximity points for such contractive mappings in a metric space with a graph‎, ‎which can entail a large number of former best proximity point results‎. ‎One fundamental issue that can be distinguished between this work and previous researches is that it can also involve all of results stated by taking comparable and $\vartheta$-close elements‎.

The main goal of this paper is to compare the proof of the existence and uniqueness of fixed points for $\mathcal{T}$-contractive mappings in various metric spaces and different distances regarding some techniques in mathematical analysis. Also, several comparisons to show the efficiency of the obtained result will be considered.‎

In this paper, we study some categorical structures of the category $\mathbf{CoarsePro}$, whose objects are coarse proximity spaces and whose morphisms are coarse proximity maps. We investigate the structure of initial, final, embedding and quotient morphisms in the construct $\mathbf{CoarsePro}$. A special attention is paid to investigate quotients by introducing some conditions that they exist. Also, it is shown that bimorphisms are exactly bijective coarse proximity maps, but not isomorphisms. Consequently, $\mathbf{CoarsePro}$ is not balanced.