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Kolmogorov's Heritage in Mathematics

:	Éric Charpentier, Annick Lesne, Nikolaï K. Nikolski
:	Eric Charpentier, Annick LESNE, Nikolaï K. Nikolski
:	Kolmogorov's Heritage in Mathematics
:	Springer-Verlag
:	9783540363514
:	1
:	CHF 74.40
:

:	Grundlagen
:	English

:	318
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

In this book, several world experts present (one part of) the mathematical heritage of Kolmogorov. Each chapter treats one of his research themes or a subject invented as a consequence of his discoveries. The authors present his contributions, his methods, the perspectives he opened to us, and the way in which this research has evolved up to now. Coverage also includes examples of recent applications and a presentation of the modern prospects.

	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