Kolmogorov's Heritage in Mathematics (eBook)

Contents 6
List of Contributors 8
Introduction 10
Acknowledgements 15
1 The youth of Andrei Nikolaevich and Fourier series 16
1.1 Convergence and divergence of Fourier series 16
1.2 Harmonic conjugates and Fourier series 18
1.3 Fourier series, integration and probability 19
1.4 The descendants of the articles of young Kolmogorov 21
Appendix: Two other aspects of Kolmogorov’s results concerning harmonic conjugates 24
I The A-integral of Kolmogorov 24
II On the weak (1,1) type of the harmonic conjugate 25
References 25
2 Kolmogorov’s contribution to intuitionistic logic 28
2.1 The first paper (1925). Formalization of intuitionistic logic 28
2.2 Classical and intuitionistic mathematics 35
2.3 Refinements of Kolmogorov’s result 38
2.4 A calculus of problems 39
2.5 Some recent developments 41
2.6 A calculus of problems for classical logic? 43
References 45
3 Some aspects of the probabilistic work 50
3.1 Introduction 50
3.2 The axiomatization of probability calculus 51
3.3 Limit theorems and series of independent random variables 55
3.4 Processes in continuous time 60
References 73
4 Infinite-dimensional Kolmogorov equations 76
4.1 Introduction and setting of the problem 76
4.2 The Ornstein-Uhlenbeck semigroup 87
4.3 Regular nonlinearities 94
4.4 Some Kolmogorov equations arising in the applications 95
References 102
5 From Kolmogorov’s theorem on empirical distribution to number theory 106
5.1 Introduction 106
5.2 New estimates for uniform order statistics 109
5.3 Number theory applications 112
Acknowledgement 115
References 115
6 Kolmogorov’s e-entropy and the problem of statistical estimation 118
6.1 Overview of the problem: parametric and non- parametric statistics 118
6.2 Notations and definitions 120
6.3 The Kullback-Leibler distance and the maximum likelihood estimator 123
6.4 The entropy of a partition and Fano’s inequality 124
6.5 The lower bound for the minimax risk 130
6.6 Consistency of the estimation 137
6.7 The estimator of the minimal distance 138
6.8 Using entropy to estimate a density 141
Conclusion 144
References 144
7 Kolmogorov and topology 148
7.1 Prelude 148
7.2 The main topological results of A. N. Kolmogorov 151
7.3 A topological idea of Kolmogorov 156
References 157
8 Geometry and approximation theory in A. N. Kolmogorov’s works 160
8.1 Geometric motives in Kolmogorov’s works 160
8.2 Kolmogorov’s works on the approximation theory 176
References 184
9 Kolmogorov and population dynamics 186
9.1 Introduction 186
9.2 From Volterra equations to Gause equations 187
9.3 The Kolmogorov equations 188
9.4 Technical aspects 190
9.5 The impact 191
References 193
10 Resonances and small divisors 196
10.1 A periodic world 196
10.2 Kepler, Newton. . . 200
10.3 An almost periodic world 203
10.4 Lagrange and Laplace: the almost periodic world 203
10.5 Poincar´e and chaos 207
10.6 A “toy model” of the theory of perturbations 208
10.7 Solution to the stability problem “in the toy model” 214
10.8 Are the irrational diophantine numbers rare or abundant? 216
10.9 A statement of the theorem of Kolmogorov- Arnold- Moser 217
10.10 Is the KAM theorem useful in our solar system? 219
References 221
11 The KAM Theorem 224
11.1 The solar system 225
11.2 The forced pendulum 233
11.3 Summary of Hamiltonian mechanics 236
11.4 A precise statement of Kolmogorov’s theorem 241
11.5 Strategy of the proof 242
References 246
12 From Kolmogorov’s work on entropy of dynamical systems to non- uniformly hyperbolic dynamics 248
12.1 General dynamical systems 248
12.2 Kolmogorov’s paper on entropy 249
12.3 The notion of hyperbolicity 251
12.4 Lorenz system 252
12.5 Hyperbolicity in one-dimensional systems 253
12.6 Two-dimensional systems 255
12.7 Conservative systems 256
12.8 Conclusion 258
References 258
13 From Hilbert’s 13th Problem to the theory of neural networks: constructive aspects of Kolmogorov’s Superposition Theorem 262
13.1 Hilbert’s 13th problem 262
13.2 Kolmogorov’s Superposition Theorem 264
13.3 Computability of Sprecher’s function 266
13.4 A computable Kolmogorov Superposition Theorem 274
13.5 Aspects of dimension 277
13.6 Aspects of constructivity 279
13.7 Applications to feedforward neural networks 281
13.8 Conclusion 284
13.9 Acknowledgement 285
References 285
14 Kolmogorov complexity 290
14.1 Algorithms 290
14.2 Descriptions and sizes 294
14.3 Gödel’s theorem 297
14.4 Definition of randomness 301
Acknowledgement 307
References 307
15 Algorithmic chaos and the incompressibility method 310
15.1 Introduction 310
15.2 Kolmogorov complexity 313
15.3 Algorithmic chaos theory 318
Acknowledgement 325
References 325