Table of contents

This paper surveys some recent and classical investigations of geometric progressions of residues that generalize the little Fermat theorem, connect this topic with the theory of dynamical systems, and estimate the degree of chaotic behaviour of systems of residues forming a geometric progression and displaying a distinctive mutual repulsion. As an auxiliary tool, the graphs of squaring operations for the elements of finite groups and rings are studied. For commutative groups the connected components of these graphs turn out to be attracting cycles homogeneously equipped with products of binary rooted trees, the algebra of which is also described in the paper. The equipping with trees turns out to be homogeneous also for the graphs of symmetric groups of permutations, as well as for the groups of even permutations.

In this paper a universal formula is given for the characteristic classes dual to the cycles of multisingularities of holomorphic maps in terms of the so-called residual polynomials. The existence theorem of such a universal formula generalizes the existence theorem for the Thom polynomial to the case of multisingularities. An analogue of this formula for the case of Legendre singularities is given. The residual polynomials of singularities of low codimension are computed. In particular, applications of the formula give generalizations to the case of the classical results of Plücker and Salmon on enumeration of singularities of tangency of a smooth hypersurface in to projective subspaces.

This paper presents a survey of results on computing the small deviation asymptotics for Gaussian measures, that is, the asymptotics of the probabilities

where is a bounded domain in a Banach space (for example, ) and a Gaussian measure on .

The main attention is focused on calculating the values of constants in the exact or logarithmic asymptotics. The survey contains new numerical results; some erroneous assertions in previous papers on this topic are also noted.

The following classes of Gaussian processes and fields are studied in detail: Wiener processes and related processes, Brownian bridges, Bessel processes, vector Wiener processes, Gaussian Markov processes, Gaussian processes with stationary increments, fractional Ornstein-Uhlenbeck processes, -parameter fractional Brownian motion, -parameter Wiener-Chentsov fields, and the Wiener pillow. Results on small deviations are presented in diverse norms, namely, the sup-norm, Hilbert norms, -norms, Hölder norms, Orlicz norms, and weighted sup-norms.

About 30 problems concerned with finding exact constants in asymptotic expressions for small deviations are posed.

The relation to Chung's law of the iterated logarithm is also considered, and a number of other results are presented.