A one-to-one correspondence between colorings and stable sets

Given a graph G, we construct an auxiliary graph with vertices such that the set of all stable sets of is in one-to-one correspondence with the set of all colorings of G. Then, we show that the Max-Coloring problem in G reduces to the Maximum Weighted Stable set problem in .     

Majorization of regular measures and weights with finite and positive critical exponent

For the sets , 1?p<∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp(R,dμ), we establish a majorization principle of their “boundaries,” i.e. for every there exists such that dμ/dν?1. A corresponding principle holds for the sets , p>0, of non-negative upper semi-continuous on R functions (weights) w such that P is dense in the space : For every there exists such that w?ω.     

Estimate of the L-Fourier transform norm on nilpotent Lie groups

Let 1<p?2 and q be such that . It is well known that the norm of the L^p-Fourier transform of the additive group is , where . For a nilpotent Lie group G, we obtain the estimate , where m is the maximal dimension of the coadjoint orbits. Such a result was known only for some particular cases.     

Estimates of weighted Hardy-Littlewood averages on the p-adic vector space

In the p-adic vector space , we characterize those non-negative functions ψ defined on for which the weighted Hardy-Littlewood average is bounded on (1?r?∞), and on . Also, in each case, we find the corresponding operator norm ‖Uψ‖.     

The Cayley transform and uniformly bounded representations

Let G be a simple Lie group of real rank one, with Iwasawa decomposition and Bruhat big cell . Then the space may be (almost) identified with N and with K/M, and these identifications induce the (generalised) Cayley transform . We show that is a conformal map of Carnot-Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplication by appropriate powers of the Jacobian, induces isomorphisms of Sobolev spaces and . We use this to construct uniformly bounded and slowly growing representations of G.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

Kolmogorov and linear widths of weighted Besov classes

We study the Kolmogorov m-widths and the linear m-widths of the weighted Besov classes on [−1,1], where Lq,μ, 1?q?∞, denotes the Lq space on [−1,1] with respect to the measure , μ>0. Optimal asymptotic orders of and as m→∞ are obtained for all 1?p,τ?∞. It turns out that in many cases, the orders of are significantly smaller than the corresponding orders of the best m-term approximation by ultraspherical polynomials, which is somewhat surprising.     

Edge-recognizable domination numbers

For any undirected graph G, let be the collection of edge-deleted subgraphs. It is always possible to construct a graph H from so that . The general edge-reconstruction conjecture states that G and H must be isomorphic if they have at least four edges. A graphical invariant that must be identical for all graphs that can be constructed from the given collection is said to be edge-recognizable. Here we show that the domination number and many of its common variations are edge-recognizable.     

Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ≥9, then , and there are forests G with . It is also proved that for an outerplanar graph G of maximum degree Δ, , and for a planar graph G of maximum degree Δ, .     

APPLICATION OF TWO OPERATORS TRANSFORM FROM CLASS A OPERATOR TO THE CLASS OF HYPONORMAL OPERATOR

In this article, we give an operator transform T (*) from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ(T ) = σ( T (*)) and σa(T )\{0} = σa( T (*))\{0}, in case T belongs to class A. Next, we obtain some relations between T and T (9).     

Weighted norm inequalities of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.