It is shown how a pure background tensor formalism provides a concise but explicit and highly flexible machinery for the generalised curvature analysis of individual embedded surfaces and foliations such as arise in the theory of topological defects in cosmological and other physical contexts. The unified treatment provided here shows how the relevant extension of the Raychaudhuri identity is r...

Added the 2 new JPAII activities in WP5 to the Conclusions References amalgamated, sorted into alphabetical order and moved to end of document. Additional web links added.

in this paper, we introduce a class of $j$-quasipolar rings. let $r$ be a ring with identity. an element $a$ of a ring $r$ is called {it weakly $j$-quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $a-p$ are contained in $j(r)$ and the ring $r$ is called {it weakly $j$-quasipolar} if every element of $r$ is weakly $j$-quasipolar. we give many characterizations and investiga...

In the spirit of Diffie Hellman the concept of a protocol algebra is introduced using certain amalgamated free product of Braid group (B) and Thompson group (T ) together with a nilpotent subgroup H of index 2