‎let $g$ be a group with identity $e.$ let $r$ be a $g$-graded‎ ‎commutative ring and $m$ a graded $r$-module‎. ‎in this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎for example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $m$ are studied‎.‎

let  be a graded ring and  be a graded -module. we define a topology on graded prime spectrum  of the graded -module  which is analogous to that for , and investigate several properties of the topology.

To integrate irreconcilable material properties into a single component, a new class of engineered materials termed “surface composites” has been developed. In this engineered material, the second phase is spatially distributed in the near surface regions, such that the phase composition is linearly graded as a function of distance from the surface. Surface composites are different from existin...

In this paper, vibration analysis of thick functionally graded beam with simply supported boundary condition under constant axial load is studied. The beam has a uniform cross-sectional area and the mechanical properties of the fungtionally graded beam are assumed to be vary through the thickness of the beam. Fundamental relations, the equilibrium and stability equations based on the displaceme...