Showing papers in "Russian Mathematical Surveys in 1989"

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

Abstract: CONTENTS Introduction Chapter I. Hamiltonian theory of systems of hydrodynamic type § 1. General properties of Poisson brackets § 2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry § 3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation § 4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices § 5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems § 6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging § 7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave § 8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies § 9. Influence of small viscosity on the evolution of the oscillatory zone References

Lyapunov indices of a product of random matrices

Abstract: CONTENTS Introduction § 1. The multiplicative ergodic theorem § 2. Quasi-projective transformations § 3. Contraction property of a semigroup § 4. Invariant measure and lemmas on convergence § 5. Fundamental theorem on Lyapunov indices § 6. Generalization of the fundamental theorem and the multiplicity of indices. Kotani's result § 7. Example: Lyapunov indices of a difference matrix Schrodinger equation Appendix References

Spectral theory of two-dimensional periodic operators and its applications

Abstract: CONTENTS Introduction Chapter I. The spectral theory of the non-stationary Schrodinger operator § 1. The perturbation theory for formal Bloch solutions § 2. The structure of the Riemann surface of Bloch functions § 3. The approximation theorem § 4. The spectral theory of finite-gap non-stationary Schrodinger operators § 5. The completeness theorem for products of Bloch functions Chapter II. The periodic problem for equations of Kadomtsev-Petviashvili type § 1. Necessary information on finite-gap solutions § 2. The perturbation theory for finite-gap solutions of the Kadomtsev-Petviashvili –2 equation § 3. Whitham equations for space two-dimensional "integrable systems" § 4. The construction of exact solutions of Whitham equations § 5. The quasi-classical limit of two-dimensional integrable equations. The Khokhlov-Zabolotskaya equationChapter III. The spectral theory of the two-dimensional periodic Schrodinger operator for one energy level § 1. The perturbation theory for formal Bloch solutions § 2. The structure of complex "Fermi-curves" § 3. The spectral theory of "finite-gap operators with respect to the level E0" and two-dimensional periodic Schrodinger operators References

Two decades of fuzzy topology: basic ideas, notions, and results

Abstract: CONTENTS Introduction ??0. Preliminaries: fuzzy sets ??1. Fuzzy topological spaces: the basic categories of fuzzy topology ??2. Fundamental interrelations between the category Top of topological spaces and the categories of fuzzy topology ??3. Local structure of fuzzy topological spaces ??4. Convergence structures in fuzzy spaces ??5. Separation in fuzzy spaces ??6. Normality and complete regularity type properties in fuzzy topology ??7. Compactness in fuzzy topology ??8. Connectedness in fuzzy spaces ??9. Fuzzy metric spaces and metrization of fuzzy spaces ??10. The fuzzy real line and its subspaces ??11. Fuzzy modification of a linearly ordered space ??12. Fuzzy probabilistic modification of a topological space ??13. The interval fuzzy real line ??14. On hyperspaces of fuzzy sets ??15. Another view of the subject of fuzzy topology and certain categorical aspects of it Conclusion: some reflections on the role and significance of fuzzy topology References

Smoothness of conjugacies of diffeomorphisms of the circle with rotations

Abstract: CONTENTS § 1. Introduction § 2. Decompositions of the circle and Denjoy's theorem § 3. Density of an absolutely continuous invariant measure § 4. Proof of the main theorem § 5. Discussion of the results and some generalizations References

The symplectic topology of completely integrable Hamiltonian systems

Abstract: CONTENTS § 1. Introduction § 2. Necessary definitions § 3. The symplectic topology of Hamiltonian systems (on four-dimensional symplectic manifolds) that are integrable in the class of Bott integrals § 4. The multi-dimensional case. The topological classification of bifurcations of Liouville tori in systems with Bott integrals § 5. Topological invariants of systems that are integrable by means of Bott integrals § 6. The general topological properties of Hamiltonian systems of differential equations that are integrable by means of Bott integrals References

Brownian local time

Abstract: The local time of a Brownian motion occupies a special place in the theory of local times of stochastic processes. Among the reasons for this are, first, the fact that it allows the construction of a theory very rich in content and, second, that in its example we can see features of the behavior of local times of more general processes, in particular, stable processes and diffusions. The study of different properties of local time is often based on knowledge of distributions of various functionals of its sample paths.

The theory of difference analogues of special functions of hypergeometric type

Abstract: CONTENTS Introduction ??1. Preliminary information ??2. Construction of particular solutions for a difference equation of hypergeometric type on non-uniform lattices ??3. Some properties of difference functions of hypergeometric type ??4. Classical orthogonal polynomials of a discrete variable on non-uniform lattices ??5. Functions of a discrete variable of the second kind ??6. Some solutions of a difference equation of hypergeometric type on non-uniform lattices Conclusion Appendix. A transformation of the original equation References

Geometric aspects of averaging

Abstract: CONTENTS Introduction ??1. Asymptotic expansion of Laplace's variational integrals ??2. Computation of dispersive media ??3. Extremal property of the hexagonal distribution of discs ??4. Random chess structure ??5. Asymptotic expansion of the effective conductivity for a small concentration of the non-conducting cellsConcluding remarks References

Rearrangements of functions and embedding theorems

Abstract: CONTENTS Introduction § 1. Rearrangements § 2. Differential properties of rearrangements and the Polya-Szego law § 3. Moduli of continuity of rearrangements § 4. Embedding of classes § 5. On embedding of Sobolev spaces § 6. Addenda References

Small perturbations of chaotic dynamical systems

Abstract: CONTENTS Introduction 1. Chaotic dynamical systems 1.1. A stochastic attractor. Hyperbolic systems. The method of symbolic dynamics 1.2. Dynamical systems with singularities. Piecewise stretching maps. The operator approach 1.3. The Li-Yorke chaos 2. Random perturbations of stochastic attractors 2.1. Random perturbations of hyperbolic systems 2.2. Random perturbations of systems with singularities 2.3. Stabilization of unstable invariant measures 2.4. The "typical" ("generic") property and properties of e- and e-a-trajectories 2.5. The most probable trajectories 3. Space discretization in chaotic systems 3.1. Definitions and basic examples 3.2. Properties of the periodic trajectories under discretization 3.3. Statistical probability under discretizations 3.4. Stability of stochastic attractors under space discretizations 3.5. Chaos under a partial space discretization 4. Time discretization in dynamical systems 4.1. Chaos under time discretizations References

Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems

Abstract: CONTENTS Introduction § 1. Hyperbolic problems without dispersion. Asymptotic decomposition of a small amplitude solution into simple waves § 2. The method of averaging in hyperbolic problems without dispersion § 3. The method of matching in hyperbolic problems without dispersion § 4. Long waves in problems with weak dissipation § 5. Long waves in problems with small dispersion § 6. Small amplitude solutions in problems with strong dispersion § 7. Continuum limits of discrete equations on small amplitude solutions § 8. Asymptotic passages in non-one-dimensional waves § 9. Comments, problems References

Boundary values of solutions of operator-differential equations

Abstract: CONTENTS Introduction § 1. Spaces of infinitely-differentiable and generalized vectors of a closed operator § 2. Representation of solutions of operator-differential equations and investigation of their boundary values § 3. Behaviour at infinity of solutions of operator-differential equations § 4. Theory of boundary values for partial differential equations References

Normalization of a Hamiltonian system near an invariant cycle or torus

Abstract: CONTENTS Introduction § 1. Normal form of a linear Hamiltonian system with periodic coefficients § 2. Normal form in the neighbourhood of a periodic solution § 3. Neighbourhood of an invariant torus § 4. A system with two degrees of freedom § 5. Remarks References

Asymptotics of the distribution of the spectrum of random matrices

Abstract: CONTENTS Introduction § 1. A new inversion formula for the Stieltjes transforms of spectral functions § 2. An inequality for the difference of the Stieltjes transforms of spectral functions § 3. An estimate of the deviation of sums of smoothed densities of distributions of eigenvalues from the semicircle law § 4. An estimate of the deviation of sums of smoothed densities of distributions of eigenvalues from the Marchenko-Pastur density § 5. The asymptotics of sums of smoothed densities of eigenvalue distributions of symmetric random matrices whose diagonal elements have non-zero mathematical expectations § 6. The asymptotics of Stieltjes transforms of spectral functions of symmetric random matrices whose elements have different dispersions § 7. The asymptotics of Stieltjes transforms of spectral functions of empirical covariance matrices § 8. Limit theorems for boundary eigenvalues of symmetric random matrices § 9. Limit theorems for boundary singular eigenvalues of random matrices References

The theory of n-shapes

Abstract: CONTENTS Introduction § 1. n-homotopy § 2. n-shape § 3. n-shape and UVn-maps § 4. n-cohomology groups § 5. n-homology and n-homotopy groups References

Algebraic geometric aspects of smooth structure. I. The Donaldson polynomials

Abstract: CONTENTS Introduction ??1. From homotopy type to smooth structure ??2. Sheaves and vector bundles over a surface ??3. Connections in vector bundles and metrics over 4-manifolds ??4. The Donaldson polynomials ??5. Riemann relations and smooth invariance of the canonical class ??6. Concluding remarks References

Factorization theory for single-valued analytic functions on compact Riemann surfaces with boundary

Abstract: CONTENTS Introduction § 1. Green-Stieltjes integrals § 2. Some classes of analytic functions § 3. Periods of harmonic functions conjugate to Green-Stieltjes integrals § 4. Construction of non-singular period matrices § 5. The Schwarz kernel § 6. The Blaschke factor § 7. The Blaschke product and its convergence § 8. Boundary behaviour of the Blaschke product § 9. Division by the Blaschke product § 10. Factorization References

A.N. Kolmogorov and approximation theory

Abstract: CONTENTS § 1. Kolmogorov's work on approximation theory § 2. Exact estimates for approximations to classes § 3. Diameters § 4. Inequalities for derivatives § 5. Diameters of finite-dimensional sets § 6. Criteria for the closest elements § 7. Smoothness and approximation § 8. Proofs References