Let R be a commutative ring with unity and an R-submodule N is called semimaximal if and only if

 the sufficient conditions of F-submodules to be semimaximal .Also the concepts of (simple , semisimple) F- submodules and quotient F- modules are  introduced and given some  properties .

This study aimed to extract, purify, and characterize the protease of local Okra Abelmoschus esculentus pods. The extraction process was conducted using ten extraction solutions with different pH and ionic strength values. Phosphate buffer solution with (pH 7, 0.05M, containing 2% sodium chloride) gave the highest activity which was (7.2 Unit/ml) as compared to other solutions, which ranged from 0.8-5.9 Unit/ml. The extracted enzyme purified by several stages. Being, precipitation by gradual addition of Ammonium sulphate from 20 to 85% saturation, then the precipitated enzyme was dialyzed and fractionated through DEAE-Cellulose (22X1.1cm), the enzymic fractions were pooled. The specific activity, purification fold and the enzyme yield value

In this work, we give an identity that leads to establishing the operator . Also, we introduce the polynomials . In addition, we provide Operator proof for the generating function with its extension and the Rogers formula for . The generating function with its extension and the Rogers formula for the bivariate Rogers-Szegö polynomials  are deduced. The Rogers formula for  allows to obtain the inverse linearization formula for , which allows to deduce the inverse linearization formula for . A solution to a q-difference equation is introduced and the solution is expressed in terms of the operators . The q-difference method is used to recover an identity of the operator  and the generating function for the polynomials

This article discusses a discussion of trends and patterns of understanding and application of the concept of metaphor to various subjects that may interfere with the perspective of metaphors in translation theory, an attempt was made to use the principles and characteristics of metaphors and their fundamental tradition in translation theory, and to uncover the perspective of considering metaphor as a conceptual process. presenting its merits, since it is still considered an eccentric expression of linguistics.

Let  be an n-Banach space, M be a nonempty closed convex subset of , and S:M→M be a mapping that belongs to the class  mapping. The purpose of this paper is to study the stability and data dependence results of a Mann iteration scheme on n-Banach space

This paper proposes a new algorithm (F2SE) and algorithm (Alg(n – 1)) for solving the
two-machine flow shop problem with the objective of minimizing total earliness. This
complexity result leads us to use an enumeration solution approach for the algorithm (F2SE)
and (DM) is more effective than algorithm Alg( n – 1) to obtain approximate solution.

This paper deals with the continuous classical optimal control problem for triple partial differential equations of parabolic type with initial and boundary conditions; the Galerkin method is used to prove the existence and uniqueness theorem of the state vector solution for given continuous classical control vector. The proof of the existence theorem of a continuous classical optimal control vector associated with the triple linear partial differential equations of parabolic type is given. The derivation of the Fréchet derivative for the cost function is obtained. At the end, the theorem of the necessary conditions for optimality of this problem is stated and is proved.

 Computer theoretical study has been carried out on the design of five electrode immersion electrostatic lens used in electron gun application. The finite element method (FEM) is used in the solution of the Poisson's equation fro determine axial potential distribution, the electron trajectory under Zero magnification condition .      The optical  properties : focal length ,spherical and chromatic aberrations are calculated,From studying the properties of the designed electron gun. we have good futures for these  electron gun where are abeam current 4*10-4A    can be supplied by using cathode tip of radius 100 nm.

  Numerous integral and local electron density’s topological parameters of significant metal-metal and metal-ligand bonding interactions in a trinuclear tetrahydrido cluster [(Cp* Ir) (Cp Ru)₂ (μ₃-H) (μ-H)₃]1 (Cp = η⁵ -C₅Me₅), (Cp* = η⁵ -C₅Me₄Et) were calculated and interpreted by using the quantum theory of atoms in molecules (QTAIM). The properties of bond critical points such as the delocalization indices δ (A, B), the electron density ρ(r), the local kinetic energy density G(r), the Laplacian of the electron density ∇²ρ(r), the local energy density