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An analysis of the normal forms to which functions can be reduced in neighbourhoods of degenerate critical points shows that many of them are quasihomogeneous or semiquasihomogeneous. A semiquasihomogeneous function is a sum of a quasihomogeneous (or weighted homogeneous) polynomial with an isolated critical point and summands of a higher degree of quasihomogeneity. The normal form to which a semiquasihomogeneous function can be reduced is described in terms of the local ring of the gradient mapping given by the quasihomogeneous part of the function. The number of parameters in this normal form is called the inner modality of the quasihomogeneous part. A classification is given of all quasihomogeneous critical points of inner modality 1: up to stable equivalence they are exhausted by three one-parameter families of parabolic singularities and 14 exceptional polynomials, 8 of which are functions of two variables, and 6 functions of three variables.

CONTENTS Introduction § 1. Optimal time in a stochastic control § 2. Optimal time and optimal design § 3. Stationary quasi-variational inequalities References

First integrals are constructed for non-linear parabolic systems (in the sense of Petrovskii) of differential equations with periodic boundary conditions; these are functionals taking a constant value with respect to on any solution of the original system: . First integrals are looked for as solutions of a certain first order partial differential equation in infinitely many variables. It is proved that the Cauchy problem for this equation in the case of analytic initial values has a unique solution that is analytic in  and defined in a neighbourhood of zero of the corresponding function space. The result is used for the construction of moment functions and the characteristic functional of a statistical solution of the original parabolic system. All the results of this article are valid also for the Navier-Stokes system.

This article is devoted to a study of the behaviour of the solution to the Cauchy problem for the quasihyperbolic equation (1) (defined below in § 1). For such equations, as we shall show, certain regions inside the base of the characteristic cone can turn out to be lacunae or weak lacunae (defined in § 1). Next we show that each quasihyperbolic equation (1) can be regarded as the limit for some hyperbolic equation whose coefficients in the series of higher derivatives in tend to zero. We establish a connection between fundamental solutions to the Cauchy problem for both equations. The statements of the main results have been published in [1].

In this paper we give necessary conditions for the local solubility of linear differential and pseudodifferential equations with smooth coefficients. Our results generalize those of H. Lewi, Hörmander, Nirenberg, Treves, Egorov, and others, described in the survey paper [16] (see also [17], [18]).

CONTENTS Introduction § 1. Microlocalization § 2. Equations with simple real characteristics § 3. Equations with simple real characteristics (continued) § 4. Higher but constant multiplicity of the characteristics § 5. Discrete phenomena in the ramified case References

The aim of this article is to give an account of results on the mathematical study of the statistical evolution of solutions of the Navier-Stokes equations (see [6], [7], and especially [8] and [9]) that can serve as a rigorous mathematical basis of the theory of turbulence of a fluid bounded by surfaces. The presentation is in abstract functional language (prompted by [7]).