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.

Triangular norms and conorms on [0, 1] as well as on  nite chains are characterized by 4 independent properties, namely by the associativity, commutativity, monotonicity and neutral element being one of extremal points of the considered domain (top element for t-norms, bottom element for t-conorms). In the case of [0, 1] domain, earlier results of Mostert and Shields on I-semigroups can be used to relax the latest three properties signi cantly, once the continuity of the underlying t-norm or t-conorm is considered. The aim of this short note is to show a similar result for  nite chains, we signi cantly relax 3 basic properties of t-norms and t-conorms (up to the associativity) when the divisibility of a t-norm or of a t-conorm is considered.

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Separation between mental conceptions and analysis them, is an important issue, concerning about the nature consideration. Some Islamic scholars accept the actual plurality of it, but other believe to intellectual and subjective multiplicity of them. In this paper, in critical-analytical method, we have shown that the distinction between this conceptions is merely intellectual and subjective, our view can reduce the conflict between supporters and opponents of the multiplicity of this credits, plurality of this conceptions in mind has scientific and educational purposes and can not prove real divergence between them abroad.

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.

In this paper we defined the concept of module amenability of Banach algebras and module connes amenability of module dual Banach algebras. Also we assert the concept of module Arens regularity that is different with [1] and investigate the relation between module amenability of Banach algebras and connes module amenability of module second dual Banach algebras. In the following we study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra. The notation of amenability of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-bimodule is inner, equivalently if H(A; X) = 0 for any Banach A-bimodule X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak amenability on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak amenability for arbitrary Banach algebras.