The Structure of the (Outer) Automorphism Group of a Bieberbach Group

Referring to Tits alternative, we develop a necessary and sufficient condition to decide whether the normalizer of a finite group of integral matrices is polycyclic-by-finite or is containing a non-Abelian free group. This result is of fundamental importance to conclude whether the (outer) automorphism group of a Bieberbach group is polycyclic-by-finite or has a non-cyclic free subgroup.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

Support problem for the intermediate Jacobians of l-adic representations

We consider the support problem of Erdös in the context of l-adic representations of the absolute Galois group of a number field. Main applications of the results of the paper concern Galois cohomology of the Tate module of abelian varieties with real and complex multiplications, the algebraic K-theory groups of number fields and the integral homology of the general linear group of rings of integers. We answer the question of Corrales-Rodrigáñez and Schoof concerning the support problem for higher dimensional abelian varieties.     

Chern numbers for two families of noncommutative Hopf fibrations

We consider noncommutative line bundles associated with the Hopf fibrations of SU_q(2) over all Podle? spheres and with a locally trivial Hopf fibration of S³_pq. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SU_q(2) and S³_pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Correlation in calculations of density of states in thin films

The density of electronic states and its spatial distribution in thin films are considered including the electron correlations in the coherent phase approximation applied to the intra-atomic interaction. This new approach allows us to derive the line shape for electronic levels which is important from the point of view of the convergency for the density function in the case of systems with restricted dimensions. Detailed calculations are performed for 4s copper film functions with the fcc lattice and various crystallographic orientations of the surfaces.The authors are very grateful to Professor L. Valenta (Prague) for his kind interest in this paper and to Dr. J. Mizia (Cracow) for helpful discussions. We would like to express our gratitude to Professor M. Wonicki and Mrs M. Firszt (Toru) for their help in obtaining the computer program procedure for two-particle interaction integrals calculations.The paper has been done in the framework of the Problem M.R I-11.     

Ground-state correlation properties of charged bosons trapped in strongly anisotropic harmonic potentials

We study systems of a few charged bosons contained within a strongly anisotropic harmonic trap. A detailed examination of the ground-state correlation properties of two-, three-, and four-particle systems is carried out within the framework of the single-mode approximation of the transverse components. The linear correlation entropy of the quasi-1D systems is discussed in dependence on the confinement anisotropy and compared with a strictly 1D limit. Only at weak interaction the correlation properties depend strongly on the anisotropy parameter.