A Cantor-Lebesgue theorem with variable ``coefficients'

If is a lacunary sequence of integers, and if for each , and are trigonometric polynomials of degree then must tend to zero for almost every whenever does. We conjecture that a similar result ought to hold even when the sequence has much slower growth. However, there is a sequence of integers and trigonometric polynomials such that tends to zero everywhere, even though the degree of does not exceed for each . The sequence of trigonometric polynomials tends to zero for almost every , although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree with largest Fourier coefficient equal to , the smallest one ``at' is while the smallest one ``near' is unknown.

     

Invariants of Skew Derivations

If is an automorphism and is a -derivation of a ring , then the subring of invariants is the set The main result of this paper is Theorem. Let be a -derivation of an algebra over a commutative ring such that

for all , where and .

(i)

If , then .

(ii)

If is a -stable left ideal of such that , then .

This theorem generalizes results on the invariants of automorphisms and derivations.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

-bundles over aspherical surfaces and 4-dimensional geometries

Melvin has shown that closed 4-manifolds that arise as -bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries or [depending on whether or ]. Conversely a geometric closed, connected 4-manifold of type or is the total space of an -bundle over a closed, connected aspherical surface precisely when its fundamental group is torsion free. Furthermore the total spaces of -bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold is the total space of an -bundle if and only if where is torsion free.

     

Lifting of generating subgroups

Let be an epimorphism of finite groups. Suppose that is generated by its subgroups and that is generated by its subgroups . Furthermore, suppose that and are conjugate, . We prove that there exist such that generate and , .

     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

     

When do two rational functions have the same Julia set?

It is proved that non-exceptional rational functions and on the Riemann sphere have the same measure of maximal entropy iff there exist iterates of and of and natural numbers such that

If one assumes only that have the same Julia set and no singular or parabolic domains of normality for the iterates, one also proves .

     

On the diophantine equation

One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation . Moreover, we prove that the diophantine equation , , , , , gcd, , has only finitely many solutions, all of which satisfying .

     

Product of distinct simple integrally closed ideals in 2-dimensional regular local rings

Let be a two-dimensional regular local ring and an -primary integrally closed ideal in . In this paper, we give equivalent conditions for to be a product of distinct simple -primary integrally closed ideals (i.e., , where are distinct simple -primary integrally closed ideals of ) in terms of the regularity of for all and in terms of how to choose a minimal generating set for over its minimal reductions.

     

An improved estimate for the highest Lyapunov exponent in the method of freezing

Let and be the eigenvalues of the matrix . The main result of the Method of Freezing states that if , and , then

for the highest exponent of the system, where

The previous best known value and the substantially smaller values of are reduced to the still smaller value.

     

On -summable sequences in the range of a vector measure

Let . Among other results, we prove that a Banach space has the property that every sequence lies inside the range of an -valued measure if and only if, for all sequences in satisfying that the operator is 1-summing, the operator is nuclear, being the conjugate number for . We also prove that, if is an infinite-dimensional -space for , then can't have the above property for any .

     

The rank stable topology of instantons on

Let be the moduli space of based (anti-self-dual) instantons on of charge and rank . There is a natural inclusion . We show that the direct limit space is homotopy equivalent to . Let be a line in the complex projective plane and let be the blow-up at a point away from . can be alternatively described as the moduli space of rank holomorphic bundles on with and and with a fixed holomorphic trivialization on .

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

Eta-quotients and elliptic curves

In this paper we list all the weight newforms that are products and quotients of the Dedekind eta-function

where There are twelve such and we give a model for the strong Weil curve whose Hasse-Weil function is the Mellin transform for each of them. Five of the have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Grössencharacter values. These formulae follow easily from well known series infinite product identities.

     

Cohomological dimension and approximate limits

Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group , . Such systems are more general than (classical) inverse systems. They have limits and structurally have similar properties. In particular, the limit of an approximate system of compacta satisfies the important property of being an approximate resolution. We shall prove herein that if is an abelian group, a compactum is the limit of an approximate system of compacta , , and for each , then .

     

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

     

Extreme points of unit balls in Lipschitz function spaces

We give a new characterization of the set of all extreme points of the unit ball in the Banach space of all Lipschitz functions on a metric space This result is applied to get a total variation characterization of in the particular case when is a convex subset of a Banach space.

     

A Characterization of the Leinert property

Let be a discrete group and denote by its left regular representation on . Denote further by the free group on generators and its left regular representation. In this paper we show that a subset of has the Leinert property if and only if for some real positive coefficients the identity

holds. Using the same method we obtain some metric estimates about abstract unitaries satisfying the similar identity