On the implementation of Sμ U(2) action in the irreducible representations of twisted canonical commutation relations

The natural S U(2) action on the creation and annihilation operators satisfying twisted canonical commutation relations (TCCR) is investigated. It is shown that the Fock representation is the only covariant irreducible representation of TCCR.Partially supported by the program RP.1.10.     

Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

A new quantum group associated with a ‘nonstandard’ braid group representation

A new quantum group is derived from a nonstandard braid group representation by employing the Faddeev-Reshetikhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like x ²–y ²=0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one- and two-dimensional ones.     

Quantum Berezinian and the classical capelli identity

We study a superanalogue of the Yangian of the Lie algebra gl _m . We apply our constructions to invariant theory.     

A remark on compact matrix quantum groups

A simpler set of axioms of the theory of compact matrix quantum groups (pseudogroups) is found.Supported by Japan Society for Promoting Science. On leave from the Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland.     

On coherent states for the simplest quantum groups

The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU _q (2) and the discrete series of representations of SU _q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on , a sphere, and the Lobatchevsky plane are discussed.     

Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L ^m/2( ^m ) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|^– , (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.     

Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations

Summary We discuss the application of a class of spline collocation methods to first-order Volterra integro-differential equations (VIDEs) which contain a weakly singular kernel (t–s)^– with 0<<1. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes. The grading exponents of graded meshes used are not greater thanm (the polynomial degree) which is independent of . This is in contrast to the theories of spline collocation methods for Volterra (or Fredholm) integral equation of the second kind. Numerical examples are given to illustrate the theoretical results.     

Oscillations of certain second order nonlinear differential equations

A second order nonlinear differential equation

Sur un théorème de Choquet

Résumé Le but de ce travail est de caractériser les éspaces topologiques qui possèdent la propriété suivant: pour toute famille(f _t ) _{t I} de fonctions numériques définies dansX il existe une partie dénombrableI ₀ de l'ensemble d'indicesI telle que les fonctions inf _tI f _i et inf _tI0 f _i ont la meme régularisation inférieurement semi-continue.On démontre que l'espace topologiqueX possède cette propriété si et seulement si pour tout sous-ensembleA deX il existe une sous-ensemble dénombrableA ₀,A ₀ A tel queA ₀.On généralise un résultat topologique de G. Choquet.

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