Let R be a ring and V be a matrix valuation on R. It is shown that, there exists a correspondence between matrix valuations on R and some special subsets ?(MVPR) of the set of all square matrices over R, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. Furthermore, based on Malcolmson’s localization, an alternative proof for t...

This paper introduces an implicit scheme for a   continuous representation of nonexpansive mappings on a closed convex subset of a Hilbert space with respect to a   sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup.   The main result is to    prove the strong convergence of the proposed implicit scheme to the unique solutio...

A bstract The O ( d, d ) invariant worldsheet theory for bosonic string with abelian isometries is employed to compute the beta functions and Weyl anomaly at one-loop. We show that vanishing of coefficients implies equations motion Maharana-Schwarz action. give a self-contained introduction into required techniques, including functions, two-dimensional sigma models background field method. This...