Let R be an associative ring with identity and M a non – zero unitary R-module.In this paper we introduce the definition of purely co-Hopfian module, where an R-module M is said to be purely co-Hopfian if for any monomorphism f Ë› End (M), Imf is pure in M and we give  some properties of this kind of modules.

Let R be associative ring with identity and M is a non- zero unitary left module over R. M is called M- hollow if every maximal submodule of M is small submodule of M. In this paper we study the properties of this kind of modules.

     In this work, we introduce a new generalization of both Rationally extending and Goldie extending which is Goldie Rationally extending module which is known as follows: if for any submodule K of an R-module M there is a direct summand U of M (denoted by  U⊆_⊕ M) such that K β_r  U. A β_r  is a relation of K⊆M and U⊆M, which defined as  K β_r  U if and only if  K ⋂U⊆_r K and K⋂U⊆_r U.

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

In this paper, we introduce a type of modules, namely S-K-nonsingular modules, which is a generalization of K-nonsingular modules. A comprehensive study of these classes of modules is given.

During more than (50) years past, India has achieved considerable social and economic progress. It is also generally assumed that the future progress will be even more rapid and that India will be an important player in the global market. India has only (2.5) percent of global land whereas it has to provide home for one-sixth of world's population .On examining the past trends of India's population ,it may be observed that during the latter half of the twentieth century ,about (650) million populations were added to the country ,thus living in a country with a high population density and high growth rate , India in need a transition from high fertility high mortality to a low fertility low mortality and towards stable population situatio

Sufficient conditions for boundary controllability of nonlinear system in quasi-Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and some techniques of nonlinear functional analysis, such as, fixed point theorem and quasi-Banach contraction principle theorem. Moreover, we given an example which is provided to illustrate the theory.

On Goldie 

Let R be a ring with identity and let M be a left R-module. M is called µ-lifting modulei f for every sub module A of M, There exists a direct summand D of M such that M = D D', for some sub module D' of M such that A≤D and A D'<<_µ D'. The aim of this paper is to introduce properties of µ-lifting modules. Especially, we give characterizations of µ-lifting modules. On the other hand, the notion of amply µ-supplemented iis studied as a generalization of amply supplemented modules, we show that if M is amply µ-supplemented such that every µ-supplement sub module of M

In this paper, several combination algorithms between Partial Update LMS (PU LMS) methods and previously proposed algorithm (New Variable Length LMS (NVLLMS)) have been developed. Then, the new sets of proposed algorithms were applied to an Acoustic Echo Cancellation system (AEC) in order to decrease the filter coefficients, decrease the convergence time, and enhance its performance in terms of Mean Square Error (MSE) and Echo Return Loss Enhancement (ERLE). These proposed algorithms will use the Echo Return Loss Enhancement (ERLE) to control the operation of filter's coefficient length variation. In addition, the time-varying step size is used.The total number of coefficients required was reduced by about 18% , 10% , 6%