A basic fact of life Let G be a connected reductive group over Q. Set G∞ = G(R), and let K∞ ⊂ G∞ be a maximal compactmod-center subgroup, so D∞ = G∞/K∞ is the usual symmetric space for G. Following the notation in Borel and Wallach’s book, we set l0 = rank(G∞) − rank(K∞) and q0 = 1 2 (dimD∞ − l0). These are both nonnegative integers. For any open compact subgroup K ⊂ G(Af ) we have the usual lo...

The definition of a D-admissible fuzzy subset for an operator domain D on a group G is modified to obtain new kinds of (∈,∈ ∨q)-fuzzy subgroups such as an (∈,∈ ∨q)-fuzzy normal subgroup, an (∈,∈ ∨q)-fuzzy characteristic subgroup, an (∈,∈ ∨q)fuzzy fully invariant subgroup which are invariant under D. As results, some of the fundamental properties of such (∈,∈ ∨q)−fuzzy subgroups are obtained.

We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer conjecture.

Let K be a field of characteristic 6= 2, let Br(K)2 be the 2primary part of its Brauer group, and let GK(2) = Gal(K(2)/K) be the maximal pro-2 Galois group of K. We show that Br(k)2 is a finite elementary abelian 2-group (Z/2Z), r ∈ N, if and only if GK(2) is a free pro-2 product of a closed subgroup H which is generated by involutions and of a free pro-2 group. Thus, the fixed field of H in K(...

a necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.

In this paper, we introduce a new class of set-valued contractions and obtain a xed point theoremfor such mappings in complete metric spaces. Our main result generalizes and improves many well-known xed point theorems in the literature.