If N ³ 2, then there exist finitely many rotations of the sphere SN such that the set of the corresponding rotation operators on Lp(SN) determines the norm topology for 1<p£¥. For N = 1 the situation is different: the norm topology of L2(S1) cannot be determined by the set of operators corresponding to the rotations by elements of any ‘thin’ set of rotations of S1.

In this paper, we consider the width invariant trapezoidal and triangular approximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserve the expected value too.

In this paper, we give an extension of the Wendel's theorem on KPC-hypergroups. We also show that every translation invariant mapping is corresponding with a unique positive measure on the KPC-hypergroup.

In this paper, we discuss the periodicity problems in the , finitely generated algebraic structures and exhibit their natural sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the imposing edifice of group theory. One of the deepest and important results of the related theory of , finite groups is a complete classification of all periodic groups, that is, the finite groups with periodic properties.

In this paper we de ne a metric on the collection of all trans-lation invariant spaces on a locally compact abelian group, and we study some topological properties of the metric space.