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.     

On multicolour noncomplete Ramsey graphs of star graphs

Given graphs , where k≥2, the notation

     

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.     

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.     

A new construction of vertex algebras and quasi-modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal subset(space) of for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal subsets of . A notion of Γ-vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C^× of nonzero complex numbers. It is proved that any maximal quasilocal subspace of is naturally a Γ-vertex algebra and that any quasilocal subset of generates a Γ-vertex algebra. It is also proved that a Γ-vertex algebra exactly amounts to a vertex algebra equipped with a Γ-module structure which satisfies a certain compatibility condition. Finally, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.     

On the commutator of the Marcinkiewicz integral

boundedness is considered for the commutator of higher-dimensional Marcinkiewicz integral. Some conditions implying the and the boundedness for the commutator of the Marcinkiewicz integral are obtained.     

Equivalent formulations for Monge-Kantorovich equation

In this paper, we study some equivalent formulations in divergence form for the optimization problem where and k>0 in Ω. This is the so called dual equation of Monge-Kantorovich problem.     

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<∞.     

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.     

Lifting Bailey pairs to WP-Bailey pairs

A pair of sequences such that and