Reflection and uniqueness theorems for harmonic functions

Suppose that is harmonic on an open half-ball in such that the origin 0 is the centre of the flat part of the boundary . If has non-negative lower limit at each point of and tends to 0 sufficiently rapidly on the normal to at 0, then has a harmonic continuation by reflection across . Under somewhat stronger hypotheses, the conclusion is that . These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set can be replaced by a spherical surface, it cannot in general be replaced by a smooth -dimensional manifold.

     

On semisimple Hopf algebras of dimension

We consider the problem of the classification of semisimple Hopf algebras of dimension where are two prime numbers. First we prove that the order of the group of grouplike elements of is not , and that if it is , then . We use it to prove that if and its dual Hopf algebra are of Frobenius type, then is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension , and a partial classification in dimensions and .

     

An extension of a non-commutative Choquet-Deny theorem

Let be a discrete group, and let be a normal subgroup of . Then the quotient map induces a group algebra homomorphism . It is shown that the kernel of this map may be decomposed as , where is a closed right ideal with a bounded left approximate identity and is a closed left ideal with a bounded right approximate identity. It follows from this fact that, if is a closed two-sided ideal in , then is closed in . This answers a question of Reiter.

     

Oscillating global continua of positive solutions of semilinear elliptic problems

Let be a bounded domain in , , with boundary , and consider the semilinear elliptic boundary value problem

where is a uniformly elliptic operator on , , is strictly positive in , and the function is continuously differentiable, with , . A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue of the linear problem. We show that under certain oscillation conditions on the nonlinearity , this continuum oscillates about , in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each in an open interval containing .

     

Noncontinuity of spectrum for the adjoint of an operator

This paper deals with the connection between continuity of spectrum at an element of the Banach algebra of all bounded linear operators on a Banach space and at the adjoint of . In particular, we show that, if is not reflexive, the spectrum function may be continuous at and discontinuous at .

     

Inequalities for the gamma function

We prove the following two theorems:

(i) Let be the th power mean of and . The inequality

holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.-L. Qiu (1997).

     

No submaximal topology on a countable set is -complementary

Two -topologies and given on the same set , are called transversal if their union generates the discrete topology on . The topologies and are -complementary if they are transversal and their intersection is the cofinite topology on . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no -complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).

     

Involutions with

Let be a smooth involution on a closed -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each component of the fixed point set of vanish in positive dimension. In this paper, we estimate the least possible lower bound of dim if does not bound.

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

On Tate-Shafarevich groups of abelian varieties

Let be a finite Galois extension of number fields with Galois group , let be an abelian variety defined over , and let and denote, respectively, the Tate-Shafarevich groups of over and of over . Assuming that these groups are finite, we derive, under certain restrictions on and , a formula for the order of the subgroup of of -invariant elements. As a corollary, we obtain a simple formula relating the orders of , and when is a quadratic extension and is the twist of by the non-trivial character of .

     

Invertible completions of upper triangular operator matrices

In this note we prove that if

is a upper triangular operator matrix acting on the Banach space , then is invertible for some if and only if and satisfy the following conditions:

(i)

is left invertible;

(ii)

is right invertible;

(iii)

.

Furthermore we show that , where is the union of certain of the holes in which happen to be subsets of .

     

Tunnel numbers of small knots do not go down under connected sum

Let and be two knots in and , the tunnel numbers of them. In this paper, we show that if both and are small, then . Moreover we show that for any small knots .

     

A criterion for splitting -algebras in tensor products

The goal of the paper is to prove the following theorem: if , are unital -algebras, simple and nuclear, then any -subalgebra of the -tensor product of and , which contains the tensor product of with the scalar multiples of the unit of , splits in the -tensor product of with some -subalgebra of .

     

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

A bound on the reduction number of a primary ideal

Let be a local ring of positive dimension and let be an -primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is Cohen-Macaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.

     

Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

Extreme points of the unit ball of the Fourier-Stieltjes algebra

Let be a locally compact group. Among other things, we proved in this paper that for an IN-group , the extreme points of the unit ball of the Fourier-Stieltjes algebra are not in the Fourier algebra if and only if is non-compact, or equivalently, there is no irreducible representation of which is quasi-equivalent to a subrepresentation of the left regular representation of if and only if is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group , the extreme points of the unit ball of the measure algebra are not in the group algebra if and only if is non-discrete. On the other hand, we also showed that if has the RNP, then there are extreme points of the unit ball of that are in . Since it is well known there are non-compact locally compact group for which has the RNP, there exist non-compact locally compact groups where extreme points of the unit ball of can be in . This shows that the condition be an IN-group cannot be entirely removed.

     

An existence theorem for systems of boundary value problems

We prove an existence theorem for , , in , using the shooting method. The function is supposed to be asymptotically linear.

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.