in the present paper, the concepts of module (uniform) approximate amenability and contractibility of banach algebras that are modules over another banach algebra, are introduced. the general theory is developed and some hereditary properties are given. in analogy with the banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same ...

In this paper, we generalize the definitions of approximately inner $C_0$-groups and their ground states to the two- parameter case and study necessary and sufficient conditions for a state to be ground state. Also we prove that any approximately inner two- parameter $C_0$--group must have at least one ground state. Finally some applications are given.

In this paper, a new notion of learnability is introduced, referred to as learnability with prior information (w.p.i.). This notion is weaker than the standard notion of probably approximately correct (PAC) learnability which has been much studied during recent years. A property called “dispersability” is introduced, and it is shown that dispersability plays a key role in the study of learnabil...