YU.  A.Brailov, Moscow State University, Russia

Integrable systems generated by consistent Poisson brackets on semi-simple Lie Algebras are related to the structure of ground algebra.  The most important properties of Liouvilles foliation are expressed in algebraic term.  The exact relation is also established between the singularities of momentum mapping and degenerations of the corresponding spectral curve.

V.L.Golo, Dept of Mathematics and Mechanics, Moscow University, Russia

This review discusses the dynamics of DNA determined by its elasticity and hydrogen bonds between base pairs.  The author considers models that take into account the interaction of elastic modes and tunneling of protons inside the hydrogen bonds and also considers possible chaotic regimes of tunneling.

V.Ostrovskyi and Yu. Samoilenko

Institute of Mathematics, Kiev, Ukraine

This volume provides the fundamentals of representations of finitely presented *-algebras by bounded operators. The theory is illustrated with numerous examples of *-algebra. The examples, in particular, include *-algebra with two self-adjoint generators that satisfy a quadratic or a more general relation, *-algebra that arise from one- and many-dimensional discrete dynamical systems, Wick *-algebra, various *-wild algebras.

O.I.Mokhov

L.D.Landau Institute for Theoretical Physics, Moscow, Russia and

University of Paderborn, Germany

This review is devoted to the differential-geometric theory of homogenous forms and other different homogenous structures (mainly Poisson and symplectic structures) on loop spaces of smooth manifolds, their natural generalizations and applications in mathematical physics and field theory.

The possibility of approximating functions of a sufficiently general form by relatively simple functions and therefore more convenient for analysis and computation is the initial reason for approximation theory. The basic fact of this theory is the Weierstrasse theorem. In the first part of this book the authors consider the problem of description of complete power systems. In the second part, the authors study the problem on bases in Sobolev spaces, in their subspaces and also in weighted spaces and finally consider continuous analogs of basis expansions (the generalized sine and cosine Fourier transforms).

The aim of this volume is to review the results on isometric immersions of locally Euclidean metrics into Euclidean spaces along with the description of the extrinsic geometry of these immersions. The review begins with the consideration of a problem specific only for constant curvature metrics, namely the problem of “natural” realization of locally Euclidean metrics by Euclidean-space domains of corresponding dimension with the standard Euclidean metric and then studies their isometric immersions into Euclidean spaces of greater dimension with emphasis on the problems of smoothness. The author presents many results with their original proofs, and also formulates a large number of unsolved problems of theory.

Multidimensional Monge-Ampère Equation by A.V.Pogorelov presents a detailed exposition of the results concerning the existence and uniqueness of the solutions of the general Monge-Ampère multidimensional equations of elliptic type. This division of the theory of partial differential equations is closely connected with geometry. This review is intended for students, postgraduates and researchers in geometry and differential equations.