A continuous operator $T$ between two Banach lattices $E$ and     $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$,  $T(A)$ is a relatively weakly compact subset of $F$. We show that the positive operator $T$ from  $E$ into a  Dedekind complete Banach lattice $F$  is almost order-weakly compact iff  $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of operators and its relationships with the others known classes of operators.

Let G = A B be the product of two groups A and B. In this paper we show that if the groups A and B are almost central, then G is almost soluble.We remind that N.S. Cernikov in 1981 gave a very condensed proof in short note, but here we present a completely different approach, which will be understandable for the interested readers.

The aim of the present work is to introduce the concept of $\lambda _{r}$-almost convergence of sequences. We define the spaces $f\left( \lambda _{r}\right) $ and $f_{0}\left( \lambda _{r}\right) $ of $ \lambda _{r}$-almost convergent and $\lambda _{r}$-almost null sequences. We investigate some inclusion relations concerning those spaces with examples and we determine the $\beta $- and $\gamma $-duals of the space $f\left( \lambda _{r}\right) $. Finally, we give the characterization of some matrix classes.

In this paper, we will focus our attention on the structure of -almost Ricci solitons on complex hyperbolic quadric. We will prove non-existence a contact real hypersurface in the complex hyperbolic quadric , admitting the gradient almost Ricci soliton. Moreover, the gradient almost Ricci soliton function is trivial.

‎‎Let $R$ be a commutative ring with non-zero identity, $S\subseteq R$ be a multiplicatively closed subset of $R$ and let $M$ be an $R$-module.  A submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be almost $S$-prime, if there exists an $s\in S$ such that whenever $rm\in N-(N:_{R}M)N$, then $sm\in N$ or $sr\in (N:_{R}M)$ for each $r\in R$, $m\in M$. The aim of this paper is to introduce and investigate some properties of the notion of almost $S$-prime submodules, especially in multiplication modules. Moreover, we investigate the behaviour of this structure under module homomorphisms, localizations, quotient modules, Cartesian product. Finally, we state two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are almost $S$-prime.