These are the Lorentz gas, a particle in a Euclidean space with (not necessarily convex) scattering domains, and the Boltzmann–Gibbs gas, a system of small identical balls in a rectangular box which collide elastically with one another and the walls of the box.
In particular, the additional decay conditions are removed in the case where a resonance is present at the edge of the continuous spectrum. A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems.In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter.The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem.The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view.In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered.
The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method.Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results.Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self-adjoint evolution equations, and higher-order elliptic operators. The problem of conditions ensuring the existence of first integrals that are polynomials in the momenta (velocities) is considered for certain multidimensional billiard systems which play an important role in non-equilibrium statistical mechanics. Until the last issue of 1997 the journal was published jointly by the London Mathematical Society and the British Library.Starting from the first issue of 1998 the journal is published jointly by the London Mathematical Society, Turpion Ltd, and the Russian Academy of Sciences.The English language version is a cover-to-cover translation of all the material: that is, the survey articles, the Communications of the Moscow Mathematical Society, and the biographical material. All these articles are free to read until 31 March 2017.