Convexity numbers of closed sets in

For 2$"> let be the -ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each 2$"> a set is constructed in so that the -ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover -- the convexity number of .

For every non-increasing function \aleph_0\}$"> we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and .     

Stability of disjointness preserving mappings

Let and be compact Hausdorff spaces and let . A linear mapping is called -disjointness preserving if implies that . If is a continuous or surjective -disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping satisfying . We also prove that every unbounded -disjointness preserving linear functional on is disjointness preserving.

     

Vector measure Banach spaces containing a complemented copy of

Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

Applications of a theorem of H. Cramér to the Selberg class

We prove two results on the nature of the Dirichlet coefficients of the -functions in the extended Selberg class . The first result asserts that if for some entire function of order 1 and finite type, then is constant. The second result states, roughly, that if are still the coefficients of some -function from , then with and . The proofs are based on an old result by Cramér and on the characterization of the functions of degree 1 of .

     

Subgroup growth in some pro- groups

For a group let be the number of subgroups of index and let be the number of normal subgroups of index . We show that for 2$">. If and does not divide or if and or , we show that for all sufficiently large . On the other hand if and divides , then is not even bounded as a function of .

     

Sur une question de capitulation

Let and be prime numbers such that and . Let , , and let be the 2-Hilbert class field of , the 2-Hilbert class field of and the Galois group of . The 2-part of the class group of is of type , so contains three extensions . Our goal is to study the problem of capitulation of the 2-classes of in , and to determine the structure of .

RSESUM´E. Soient et deux nombres premiers tels que et , , , , le 2-corps de classes de Hilbert de , le 2-corps de classes de Hilbert de et le groupe de Galois de . La 2-partie du groupe de classes de est de type , par suite contient trois extensions . On s'intéresse au problème de capitulation des 2-classes de dans , et à déterminer la structure de .

     

On one problem of uniqueness of meromorphic functions concerning small functions

In this paper, we show that if two non-constant meromorphic functions and satisfy for , where are five distinct small functions with respect to and , and is a positive integer or with , then . As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.

     

On a problem of J. P. Williams

Let be the algebra of all bounded operators on a Hilbert space . Let be a continuous function on the closed disk and let

for some 0,$"> for all and all with . Then is differentiable on . The paper shows that the function may have a discontinuous derivative.

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

On representable linearly compact modules

For a flat module we prove that is a functor from the category of linearly compact modules to itself and is exact. Moreover, is representable when is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization of a linearly compact representable module with respect to a multiplicative set in is described.

     

On the predictability of discrete dynamical systems

Let be a metric space. A function is said to be non-sensitive at a point if for every 0$"> there is a 0$"> such that for any choice of points , , , we have that for every . Let be the set of all homeomorphisms from onto endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces , ``most' functions in are non-sensitive at ``most' points of .

     

Basic characters of the unitriangular group (for arbitrary primes)

Let denote the (upper) unitriangular group of degree over the finite field with elements. In this paper we consider the basic (complex) characters of and we prove that every irreducible (complex) character of is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption , where is the characteristic of the field .

     

A note concerning the index of the shift

Let be a finite, positive Borel measure with support in such that - the closure of the polynomials in - is irreducible and each point in is a bounded point evaluation for . We show that if 0$">and there is a nontrivial subarc of such that

-\infty,\end{displaymath}">

then for each nontrivial closed invariant subspace for the shift on .

     

Quadratic initial ideals of root systems

Let be one of the root systems , , and and write for the set of positive roots of together with the origin of . Let denote the Laurent polynomial ring over a field and write for the affine semigroup ring which is generated by those monomials with , where if . Let denote the polynomial ring over and write for the toric ideal of . Thus is the kernel of the surjective homomorphism defined by setting for all . In their combinatorial study of hypergeometric functions associated with root systems, Gelfand, Graev and Postnikov discovered a quadratic initial ideal of the toric ideal of . The purpose of the present paper is to show the existence of a reverse lexicographic (squarefree) quadratic initial ideal of the toric ideal of each of , and . It then follows that the convex polytope of the convex hull of each of , and possesses a regular unimodular triangulation arising from a flag complex, and that each of the affine semigroup rings , and is Koszul.     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

A proof of Weinberg's conjecture on lattice-ordered matrix algebras

Let be a subfield of the field of real numbers and let () be the matrix algebra over . It is shown that if is a lattice-ordered algebra over in which the identity matrix 1 is positive, then is isomorphic to the lattice-ordered algebra with the usual lattice order. In particular, Weinberg's conjecture is true.

     

Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

On the absolute continuity of a class of invariant measures

Let be a compact connected subset of , let , be contractive self-conformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigen-equation

for some 0$">. We prove that if the attractor is an -set and is absolutely continuous with respect to , the Hausdorff -dimensional measure restricted on the attractor , then is absolutely continuous with respect to (i.e., they are equivalent). A special case of the result was considered by Mauldin and Simon (1998). In another direction, we also consider the -property of the Radon-Nikodym derivative of and give a condition for which is unbounded.

     

Integral representation for a class of vector valued operators

Let be a compact space and let , be a (real, for simplicity) Banach space. We consider the space of all continuous -valued functions on , with the supremum norm .

We prove in this paper a Bochner integral representation theorem for bounded linear operators

which satisfy the following condition:

where is the conjugate space of . In the particular case where , this condition is obviously satisfied by every bounded linear operator

and the result reduces to the classical Riesz representation theorem.

If the dimension of is greater than , we show by a simple example that not every bounded linear admits an integral representation of the type above, proving that the situation is different from the one dimensional case.

Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.