An R-module M is called a 2-regular module if every submodule N of M is 2-pure submodule, where a submodule N of M is 2-pure in M if for every ideal I of R, I2MN = I2N, [1]. This paper is a continuation of [1]. We give some conditions to characterize this class of modules, also many relationships with other related concepts are introduced.

     The -s-extending modules will be purpose of this paper, a module M  is -s-extending if each submodule in M is essential in submodule has a supplement that is direct summand. Initially, we give relation between this concept with weakly supplement extending modules and -supplemented modules. In fact, we gives the following implications:

Lifting modules   -supplemented modules   -s-extending modules  weakly supplement extending modules.

It is also we give examples show that, the converse of this result is not true. Moreover, we study when the converse of this result is true.

In This paper generalized spline method and Caputo differential operator is applied to solve linear fractional integro-differential equations of the second kind. Comparison of the applied method with exact solutions reveals that the method is tremendously effective.

  Let M be an R-module, where R is commutative ring with unity. In this paper we study the behavior of strongly hollow and quasi hollow submodule in the class of strongly comultiplication modules. Beside this we give the relationships between strongly hollow and quasi hollow submodules with V-coprime, coprime, bi-hollow submodules.

 In the present work a theoretical analysis depending on the new higher order . element in shear deformation theory for simply supported cross-ply laminated plate is developed. The new displacement field of the middle surface expanded as a combination of exponential and trigonometric function of thickness coordinate with the transverse displacement taken to be constant through the thickness. The governing equations are derived using Hamilton’s principle and solved using Navier solution method to obtain the deflection and stresses under uniform sinusoidal load. The effect of many design parameters such as number of laminates, aspect ratio and thickness ratio on static behavior of the laminated composite plate has been studied. The

The aim of this paper is introducing the concept of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal. Some properties of (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal have been studied and another characterizations have been given. The relationship of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal that states,  a B- -module Ӽ is (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal  , if and only if  for any two ɱ-element sub-sets and of Ӽ^ɳ, if , for each j = 1, …, ɱ,  i = 1,…, ɳ  and   implies _Ạɳ( ) _Ạɳ(  have been proved..

This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method.

   In this paper we show that the function , () p fLI α ∈ ,0<p<1 where I=[-1,1] can be approximated by an algebraic polynomial with an error not exceeding , 1 ( , , ) kp ft n Ï• αω      where
,
1 ( , , ) kp ft n ϕ αω is the Ditizian–Totik modules of smoothness of unbounded function in , () p LI

In this work we study gamma modules which are implying full stability or implying by full stability. A gamma module is fully stable if for each gamma submodule of and each homomorphism of into . Many properties and characterizations of these classes of gamma modules are considered. We extend some results from the module to the gamma module theories.

In this thesis, we introduce eight types of topologies on a finite digraphs and state the implication between these topologies. Also we studied some pawlak's concepts and generalization rough set theory, we introduce a new types for approximation rough digraphs depending on supra open digraphs. In addition, we present two various standpoints to define generalized membership relations, and state the implication between it, to classify the digraphs and help for measure exactness and roughness of digraphs. On the other hand, we define several kinds of fuzzy digraphs. We also introduce a topological space, which is induced by reflexive graph and tolerance graphs, such that the graph may be infinite. Furthermore, we offered some properties of th