Table of contents

CONTENTS § 1. The growing role of finite mathematics § 2. Information theory § 3. The definition of "complexity" § 4. Regularity and randomness § 5. The stability of frequencies § 6. Infinite random sequences § 7. Relative complexity and quantity of information § 8. Barzdin's theorem § 9. Conclusion References

CONTENTS § 1. Introduction § 2. Definitions and auxiliary results § 3. Kolmogorov's example of a trigonometric Fourier series that diverges almost everywhere § 4. Further results on divergent Fourier series § 5. Kolmogorov's theorem on the divergence of Fourier series of class L² in a rearranged trigonometric system and some generalizations of it § 6. The Kolmogorov-Men'shov theorem on divergent Fourier series in an orthonormal system of sign functions References

CONTENTS Introduction § 1. Statistical hydromechanics § 2. The equations for the first and second moments § 3. Turbulence in stratified media § 4. Two-dimensional turbulence § 5. Geostrophic turbulence References

CONTENTS Introduction § 1. Maximal attractors of semigroups generated by evolution equations § 2. Examples of parabolic equations and systems having a maximal attractor § 3. The Hausdorff dimension of invariant sets § 4. Estimate of the change in volume under the action of shift operators generated by linear evolution equations § 5. An upper bound for the Hausdorff dimension of attractors of semigroups corresponding to evolution equations § 6. A lower bound for the dimension of an attractor § 7. Differentiability of shift operators § 8. Estimates of the Hausdorff dimension of an attractor of a two-dimensional Navier-Stokes system § 9. Upper and lower bounds for the Hausdorff dimension of attractors of parabolic equations and parabolic systems § 10. Attractors of semigroups having a global Lyapunov function § 11. Regular attractors of semigroups having a Lyapunov function References

CONTENTS Introduction § 1. Rayleigh-Schrödinger series for perturbations of a simple eigenvalue § 2. Perturbations of a multiple eigenvalue § 3. Widths of forbidden zones of even Hill's equations of Mathieu type § 4. Formulae for the denominators of terms in the perturbation theory series § 5. Conclusion of the investigation of forbidden zones § 6. Widths of resonance zones for mappings of the circle References

CONTENTS § 1. Introduction, statement and discussion of the results § 2. Construction of the ground states for the model (1) § 3. Proof of Theorem 2 § 4. Investigation of the measure and Hausdorff dimension of the Cantor set in Theorem 3 § 5. Concluding remarks References

CONTENTS § 1. Introduction § 2. Boundary-value problems and the central limit theorem § 3. The rate of convergence in boundary-value problems § 4. Special case (rectilinear boundaries) § 5. Large deviations in boundary-value problems § 6. The principle of invariance § 7. The principle of invariance and large deviations § 8. Generalizations to the many-dimensional case References

CONTENTS Introduction § 1. The three-series theorem § 2. Stochastic semigroups § 3. Limit theorems for products of independent operators in a series scheme § 4. The asymptotic behaviour of a product of independent identically distributed operators References