Some aspects of the Liouville equation in mathematical physics and statistical mechanics

This paper presents some mathematical aspects of Classical Liouville theorem and we have noted some mathematical theorems about its initial value problem. Furthermore, we have implied on the formal frame work of Stochastic Liouville equation (SLE)

Isotopic Grand Unification with the Inclusion of Gravity (revised version)

We introduce a dual lifting of unified gauge theories, the first characterized by the isotopies, which are axiom- preserving maps into broader structures with positive-definite generalized units used for the representation of matter under the isotopies of the Poincare' symmetry, and the second characterized by the isodualities, which are anti-isomorphic maps with negative-definite generalized units used for the representation of antimatter under the isodualities of the Poincare' symmetry. We then submit, apparently for the first time, a novel grand unification with the inclusion of gravity for matter embedded in the generalized positive-definite units of unified gauge theories while gravity for antimatter is embedded in the isodual isounit. We then show that the proposed grand unification provides realistic possibilities for a resolution of the axiomatic incompatibilities between gravitation and electroweak interactions due to curvature, antimatter and the fundamental space-time symmetries.Comment: 20 pages, Latex, revised in various details and with added reference