We consider maximal operators associated to singular averages along finite subsets Σ of the Grassmannian Gr(d,n) d-dimensional subspaces Rn. The well studied d=1 case corresponds directional function with respect arbitrary Gr(1,n)=Sn−1. provide a systematic study all cases 1≤d<n and prove essentially sharp L2(Rn) bounds for subspace averaging operator in terms cardinality Σ, no assumption on st...

Let G be a locally compact group. Then for every G-space X the maximal G-proximity βG can characterized by topological proximity β as follows:AβG‾B⇔∃V∈NeVAβ‾VB. Here, βG:X→βGX is G-compactification of (which an embedding classical result J. de Vries), V neighbourhood e and AβG‾B means that closures A B do not meet in βGX. Note local compactness essential. This theorem comes corollary general ab...

the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

In this paper, we study the minimality of the boundary of a Coxeter system. We show that for a Coxeter system (W,S) if there exist a maximal spherical subset T of S and an element s0 ∈ S such that m(s0, t) ≥ 3 for each t ∈ T and m(s0, t0) = ∞ for some t0 ∈ T , then every orbit Wα is dense in the boundary ∂Σ(W,S) of the Coxeter system (W,S), hence ∂Σ(W,S) is minimal, where m(s0, t) is the order ...

Recently, the authors studied the connection between each maximal monotone operator T and a family H(T ) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a uniq...

We are concerned with the poset P(n)=P([1, 2, ..., n]). This is the power set of [n]=[1, 2, ..., n], ordered by inclusion. A set system is simply a subset of P(n). A set system is an antichain if no two of its members are comparable. Conversely a chain is a totally ordered set system. We shall often consider maximal chains; those chains which cannot be extended. In particular such chains contai...