Table of contents

Decompositions into cycles for random permutations of a large number of elements are very different (in their statistics) from the same decompositions for algebraic permutations (defined by linear or projective transformations of finite sets). This paper presents tables giving both these and other statistics, as well as a comparison of them with the statistics of involutions or permutations with all their cycles of even length. The inclusions of a point in cycles of various lengths turn out to be equiprobable events for random permutations. The number of permutations of elements with all cycles of even length turns out to be the square of an integer (namely, of ). The number of cycles of projective permutations (over a field with an odd prime number of elements) is always even. These and other empirically discovered theorems are proved in the paper.

Bibliography: 6 titles.

In many problems the 'real' spectral data for periodic finite-gap operators (consisting of a Riemann surface with a distingulished 'point at infinity', a local parameter near this point, and a divisor of poles) generate operators with singular real coefficients. These operators are not self-adjoint in an ordinary Hilbert space of functions of a variable (with a positive metric). In particular, this happens for the Lamé operators with elliptic potential , whose wavefunctions were found by Hermite in the nineteenth century. However, ideas in  [1]–[4] suggest that precisely such Baker-Akhiezer functions form a correct analogue of the discrete and continuous Fourier bases on Riemann surfaces. For genus these operators turn out to be symmetric with respect to an indefinite (not positive definite) inner product described in this paper. The analogue of the continuous Fourier transformation is an isometry in this inner product. A description is also given of the image of this Fourier transformation in the space of functions of .

Bibliography: 24 titles.

Electric current is studied in a two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, that is, free flights between collisions are bounded, the resulting current is proportional to the voltage difference , that is, , where is the diffusion matrix of a Lorentz particle moving freely without an electric field (see a mathematical proof in [1]). This formula agrees with Ohm's classical law and the Einstein relation. Here the more difficult model with an infinite horizon is investigated. It is found that infinite corridors between scatterers allow the particles (electrons) to move faster, resulting in an abnormal current (causing 'superconductivity'). More precisely, the current is now given by , where is the 'superdiffusion' matrix of a Lorentz particle moving freely without an electric field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. New results are also obtained for the infinite-horizon Lorentz gas without external fields, complementing recent studies by Szász and Varjú [2].

Bibliography: 31 titles.

A new definition of a chaotic invariant set is given for a continuous semiflow in a metric space. It generalizes the well-known definition due to Devaney and allows one to take into account a special feature occurring in the non-compact infinite-dimensional case: so-called turbulent chaos. The paper consists of two sections. The first contains several well-known facts from chaotic dynamics, together with new definitions and results. The second presents a concrete example demonstrating that our definition of chaos is meaningful. Namely, an infinite-dimensional system of ordinary differential equations is investigated having an attractor that is chaotic in the sense of the new definition but not in the sense of Devaney or Knudsen.

Bibliography: 65 titles.