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Number 5, October 2002
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R M Absava, E A Nadaraya and L E Nadaraya
N Ya Amburg
I V Baskakov
V Grujic
A V Inshakov
G Yu Kokarev
V O Manturov
O I Mokhov
I A Pushkar'
L D Pustyl'nikov
S Yu Shorina
S P Suetin
A A Tarasov
M A Vsemirnov and M G Rzhevskii
Yu F Borisov
V V Zhuk
A A Borovkov
M S Agranovich
Spectral boundary-value problems with discrete spectrum are considered for second-order strongly elliptic systems of partial differential equations in a domain whose boundary is compact and may be , , or Lipschitz. The principal part of the system is assumed to be Hermitian and to satisfy an additional condition ensuring that the Neumann problem is coercive. The spectral parameter occurs either in the system (then is assumed to be bounded) or in a first-order boundary condition. Also considered are transmission problems in with spectral parameter in the transmission condition on . The corresponding operators in or are self-adjoint operators or weak perturbations of self-adjoint ones. Under some additional conditions a discussion is given of the smoothness, completeness, and basis properties of eigenfunctions or root functions in the Sobolev -spaces or of non-zero order as well as of localization and the asymptotic behaviour of the eigenvalues. The case of Coulomb singularities in the zero-order term of the system is also covered.
V A Gritsenko and V V Nikulin
The general theory of Lorentzian Kac-Moody algebras is considered. This theory must serve as a hyperbolic analogue of the classical theories of finite-dimensional semisimple Lie algebras and affine Kac-Moody algebras. The first examples of Lorentzian Kac-Moody algebras were found by Borcherds. Here general finiteness results for the set of Lorentzian Kac-Moody algebras of rank ≥3 are considered along with the classification problem for these algebras. As an example, a classification is given for Lorentzian Kac-Moody algebras of rank 3 with hyperbolic root lattice , symmetry lattice , and symmetry group , , where and are given by
and is trivial on , is the extended paramodular group. This is perhaps the first example in which a large class of Lorentzian Kac-Moody algebras has been classified.
F A Bogomolov
