-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

Free

It has been conjectured that every free algebraic action of the additive group of complex numbers on complex affine three space is conjugate to a global translation. The main result lends support to this conjecture by showing that the morphism to the variety defined by the ring of invariants of the associated action on the coordinate ring is smooth. As a consequence, the graph morphism is an open immersion, and simple proofs of certain cases of the conjecture are obtained.

     

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.

     

Products on -modules

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of -modules such as , and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over that are concentrated in degrees divisible by ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising -local -modules as -modules.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Sur les -corps

Here we give new examples of fields in characteristic whose -invariant and -invariant are different: or . These fields are also -fields.

RSUM. Nous donnons ici de nouveaux exemples de corps en caractéristique dont le -invariant et le -invariant diffèrent. Plus précisément: et ou . Ces corps sont aussi des -corps.

     

Order -adic field

Let be an algebraically closed field of characteristic the ring of Witt vectors and a complete discrete valuation ring dominating and containing a primitive -th root of unity. Let denote a uniformizing parameter for We study order automorphisms of the formal power series ring which are defined by a series

The set of fixed points of is denoted by and we suppose that they are -rational and that for Let be the minimal semi-stable model of the -adic open disc over in which specializes to distinct smooth points. We study the differential data that can be associated to each irreducible component of the special fibre of Using this data we show that if , then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order automorphisms in which are not the identity

     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

New facts

We use ``iterated square sequences' to show that there is an -definable partition such that if is an inner model not containing :

(a)

For some is stationary.

(b)

For each there is a generic extension of in which does not exist and is non-stationary.

This result is then applied to show that if is an inner model without , then some sentence not true in can be forced over .

     

The Pontryagin -form

The unit -planes on which the first Pontryagin form of the Grassmann manifolds achieves its maximum are determined. This is a shorter and unified proof of results first obtained in 1995 by H. Gluck et al. and in 1998 by W. Gu.

     

Willmore Functionals

We prove some integral inequalities for immersed tori in the three sphere. The functionals considered are generalizations of the Willmore functional.

     

Surfaces with pg=q=3

 In this paper it is proved that a complete algebraic surface of general type with p _g =q=3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p _g =q=3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface. Received 14 August 2001 / Revised version: 17 January 2002     

Newton's formula for

This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra of the Lie algebra : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.

     

groups and quasi-equivalence

A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

     

The Diophantine equation

If and are given positive integers with , then we show that the equation of the title possesses at most one solution in positive integers . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.

     

Left-symmetric algebras for

We study the classification problem for left-symmetric algebras with commutation Lie algebra in characteristic . The problem is equivalent to the classification of étale affine representations of . Algebraic invariant theory is used to characterize those modules for the algebraic group which belong to affine étale representations of . From the classification of these modules we obtain the solution of the classification problem for . As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.