Table of contents

CONTENTS Chapter I. Topological invariants of cascades satisfying axiom A § 1. Morse-Smale diffeomorphisms 1.1. Existence conditions for Morse-Smale diffeomorphisms 1.2. The topological classification of gradient-like diffeomorphisms and of diffeomorphisms with orientable heteroclinic sets § 2. The topological classification of basic sets Chapter II. The problem of realizing cascades with given topological properties § 3. Nielsen's theory of the homotopy classification of homeomorphisms of two-dimensional manifolds § 4. The topological classification of cascades with a pair of invariant transverse foliations References

An incomplete survey of hypergeometric series is given, and some of Ramanujan's work on hypergeometric and basic hypergeometric series is put into the general framework as we understand it now.

CONTENTS Introduction § 1. The function and its properties § 2. Generalized Taylor series for certain classes of infinitely differentiable functions § 3. Approximation by means of the function § 4. Application of solutions of an FDE with compact support and generalized Taylor series in the theory of FDE's § 5. Generalizations and unsolved problems References

CONTENTS Introduction Chapter I. The main notions and facts § 1.1. The notion of randomness depends on a given probability distribution § 1.2. Three faces of randomness: stochasticness, chaoticness, typicalness § 1.3. Typical, chaotic and stochastic sequences: ways to a mathematical definition 1.3.1. Typicalness 1.3.2. Chaoticness 1.3.3. Stochasticness 1.3.4. Comments § 1.4. Typical and chaotic sequences: basic definitions (for the case of the uniform Bernoulli distribution) 1.4.1. Typicalness 1.4.2. Chaoticness Chapter II. Effectively null sets, constructive support, and typical sequences § 2.1. Effectively null sets, computable distributions, and the statement of Martin-Löf's theorem § 2.2. Proof of Martin-Löf's theorem § 2.3. Different versions of the definition of the notion of typicalness 2.3.1. Schorr's definition of typicalness 2.3.2. Solovay's criterion for typicalness 2.3.3. The axiomatic approach to the definition of typicalness Chapter III. Complexity, entropy, and chaotic sequences § 3.1. Computable mappings § 3.2. Kolmogorov's theorem. Monotone entropy § 3.3. Chaotic sequences Chapter IV. What is a random sequence? § 4.1. The proof of the Levin-Schorr theorem for the uniform Bernoulli distribution § 4.2. The case of an arbitrary probability distribution § 4.3. The proofs of the lemmas Chapter V. Probabilistic machines, a priori probability, and randomness § 5.1. Probabilistic machines § 5.2. A priori probability § 5.3. A priori probability and entropy § 5.4. A priori probability and randomness Chapter VI. The frequency approach to the definition of a random sequence § 6.1. Von Mises' approach. The Church and Kolmogorov-Loveland definitions § 6.2. Relations between different definitions. Ville's construction. Muchnik's theorem. Lambalgen's example 6.2.1. Relations between different definitions 6.2.2. Ville's example 6.2.3. Muchnik's theorem 6.2.4. Lambalgen's example § 6.3. A game-theoretic criterion for typicalness Addendum. A timid criticism regarding probability theory References