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 .     

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 .

     

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 .

     

The distribution of sequences in residue classes

We prove that any set of integers with lies in at least many residue classes modulo most primes . (Here is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below which are multiplicatively generated by the coprime integers (i.e. whose counting function is also ) lie in at least residue classes, modulo most small primes , where as .

     

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.

     

Fourier asymptotics of Cantor type measures at infinity

Let be an integer and let . In this note we prove that for all ; if is odd and if is even This improves a classical result of Wiener and Wintner. We also give a necessary and sufficient condition for the product to approach zero at infinity.

     

Finite homological dimension and primes associated to integrally closed ideals

Let be an integrally closed ideal in a commutative Noetherian ring . Then the local ring is regular (resp. Gorenstein) for every if the projective dimension of is finite (resp. the Gorenstein dimension of is finite and satisfies Serre's condition (S)).

     

P.I. envelopes of classical simple Lie superalgebras

Let be a classical simple Lie superalgebra. We describe the prime ideals in the enveloping algebra such that satisfies a polynomial identity. If the factor algebra is not artinian, then it is an order in a matrix algebra over .     

On a theorem of R. Steinberg on rings of coinvariants

Let be a representation of a finite group over the field . Denote by the algebra of polynomial functions on the vector space . The group acts on and hence also on . The algebra of coinvariants is , where is the ideal generated by all the homogeneous -invariant forms of strictly positive degree. If the field has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that is a Poincaré duality algebra if and only if is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case and give a counterexample in the modular case when .

     

Character degree sets that do not bound the class of a -group

Suppose that we are given a set of powers of a prime and that . A technique is presented that enables the construction of a -group of specified nilpotence class such that its set of irreducible character degrees is exactly . If , then this can be done for and if , then the only requirement is .