The aim of this note is to characterize a class of distributions whose characteristic functions f (t) for all a > 0 satisfy the relation; " tخ R where H and H΄ are positive real constants. First, it is observed that these distributions turn out to belong to the well-known class of infinitely divisible distributions. More specifically, they are strictly stable and thus absolutely continuous, self-decomposable and unimodal. But they are not strongly unimodal except for the case of H=2H. Then some representations are obtained for f (t). Finally, some characterizations are arrived at for the Cauchy and N (0, s 2); 0< s 2 < ¥ ; distributions.

There exist several versions of discrete counterpart of continuity in the framework of finite chains, e.g., the smoothness, the divisibility, intermediate-value property and the 1-Lipschitz property. In this paper, we first discuss the relationships among the smoothness, divisibility, intermediate-value property and 1-Lipschitz property. Second, we present complete characterizations of divisible associative aggregation operations on finite chains.

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Let A be a Banach algebra and E be a Banach A -bimodule then S=A ÅE, the l1-direct sum of A and E becomes a module extension Banach algebra when equipped with the algebras product (a, x): (a', x')=(aa', a.x'+x.a'). In this paper, we investigate D-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the module extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -module if and only if l1 (ES) is amenable as Banach algebra.