We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a resu...

A composition operator Tbf = f o b is completely continuous on H1 if and only if \b\ < 1 a.e. If the adjoint operator Tg is completely continuous on VMOA , then 7¿ is completely continuous on Hl . Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of ce...

A boolean algebra is shown to be completely representable if and only if it is atomic whereas it is shown that the class of completely representable relation algebras is not elementary.

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H ≤ G. In this note, we give a notion of G-complete reducibility – G-cr for short – for Lie subalgebras of Lie(G), and we show that if the smooth subgroup H < G is G-cr, then Lie(H) is G-cr as well.

Cryptography the art of secure communications, has been developed at least over 2500 years. Still at present, no perfectly secure as well as practically suitable classical or quantum cryptosystems exist. Statistically encoding the individual bit, here we present a practical key distribution technique which is absolutely secure both for classical and quantum keys. To achieve perfect security, no...

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both posi...

Assume that there is no quasi-measurable cardinal not greater than 2. We show that for a c.c.c. σ-ideal I with a Borel base of subsets of an uncountable Polish space, if A is a point-finite family of subsets from I then there is a subfamily of A whose union is completely nonmeasurable i.e. its intersection with every nonsmall Borel set does not belong to the σ-field generated by Borel sets and ...