The study of cohomology groups is one of the most intensive and exciting researches that arises from algebraic topology. Particularly, the dimension of cohomology groups is a highly useful invariant which plays a rigorous role in the geometric classification of associative algebras. This work focuses on the applications of low dimensional cohomology groups. In this regards, the cohomology groups of degree zero and degree one of nilpotent associative algebras in dimension four are described in matrix form.

Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ? P ? (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.

Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ? P ? (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.

     Given an exterior algebra  over a finite dimension vector space v, and let   , where  is a graded ideal in . The relation between the algebra  and    regarding to -quadratic and - quadratic will be investigated. We show that the algebra   is - quadratic if and only if   is - quadratic. Furthermore, it has been shown that the algebra  is - quadratic if and only if  is - quadratic.

The main purpose of this article is to study the soft LC-spaces as soft spaces in which every soft Lindelöf subset of  is soft closed. Also, we study the weak  forms of soft LC-spaces and we discussed their relationships with soft LC-spaces as well as among themselves.

 Let  be a metric space and  be a continuous map. The notion of the  -average shadowing property ( ASP )  for a continuous map on  â€“space is introduced  and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if  has ASP, then   has ASP for every . We prove that if a map  be pseudo-equivariant with dense set of periodic points and has the ASP,  then  is weakly mixing. We also show that if   is a expansive pseudo-equivariant homeomorphism that has the ASP and  is topologically mixing,  then  has a  -specification. We obtained that the identity map  on  has the ASP  if and only if th

    In this paper, the concept of soft closed groups is presented using the soft ideal pre-generalized open and soft pre-open, which are -ᶅ- - -closed sets " -closed", Which illustrating several characteristics of these groups.  We also use some games and   -  Separation Axiom, such as (Æ®_0, Ó¼, ᶅ) that use many tables and charts to illustrate this. Also, we put some proposals to study the relationship between these games and give some examples.

A new class of generalized open sets in a topological space, called G-open sets, is introduced and studied. This class contains all semi-open, preopen, b-open and semi-preopen sets. It is proved that the topology generated by G-open sets contains the topology generated by preopen,b-open and semi-preopen sets respectively.

Modify Multi-Connect Architecture (MMCA) associative memory‏