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Paradoxes of Measures and Dimensions Originating in Felix Hausdorff's Ideas

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Product Details

  • ISBN-13: 9789810201890
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 1/14/1994
  • Pages: 764

Table of Contents

Acknowledgments
Preface
Ch. 0 Biographical sketches 1
0.1 Curriculum vitae 1
0.2 Felix Hausdorff: mathematician, philosopher, poet 5
0.3 Author's reflections 24
References: the works of Felix Hausdorff 37
References to the biographical notes 43
Ch. I The paradox of the sphere 45
1.1 Hausdorff's decomposition of the sphere 46
1.2 The Banach-Tarski paradox 58
1.3 Non-measurable sets in R[superscript 1] and R[superscript 2] 61
1.4 Exotic measures and the problem of Ruziewicz 69
1.5 Group theoretic implications of the Hausdorff theorem 71
1.6 A fixed point view on paradoxical decompositions 80
Ch. II Inaccessible numbers and the hierarchal structure of set theory 94
2.1 The infinite number of Cantor's set theory 95
2.2 Hausdorff's weakly inaccessible cardinals and a certain set-theoretic alpinism 117
2.3 The axioms of Zermelo and Fraenkel and their paradoxical background 127
2.4 The Godel theorem and the cumulative hierarchy of sets 141
2.5 On the continuum hypothesis and related problems 167
2.6 The conditions of the existence and non-existence of non-measurable sets 188
2.7 Cantor, Hausdorff and Godel's pyramid of paradoxes 197
2.8 The Godel theorem and a certain mathematical catastrophism 203
2.9 Hausdorff's intuitions versus present-day mathematics 205
Ch. III The Hausdorff measures, Hausdorff dimensions and fractals 219
3.1 The Hausdorff measure and dimension 221
3.2 The standard example of the Cantor set 231
3.3 Ephemeral sets of strong measure zero 234
3.4 The implications in number theory 236
3.5 The Hausdorff dimension of the Cartesian product of sets 239
3.6 The Hausdorff-Besicovitch dimension versus Hausdorff operations 241
3.7 Metric spaces 245
3.8 The Hausdorff topology and topological stable dimension 250
3.9 The Hausdorff-Besicovitch dimension versus the topological dimension 272
3.10 The coefficient [actual symbol not reproducible], that is, the topological measure of Borsuk 276
3.11 The concept of fractals 278
3.12 One flew over the land of fractals 281
3.13 The phenomenon of self-similarity 367
3.14 Multifractals 375
3.15 Natural fractals 384
3.16 The Olbers paradox and fractal approaches to cosmology 391
3.17 Fractals: illusion, speculation or mathematics? 397
Ch. IV The Baker-Campbell-Hausdorff formula 414
4.1 The Hausdorff series 415
4.2 A continuous Baker-Campbell-Hausdorff problem 433
4.3 The problem of the convergence of the Hausdorff series 439
4.4 Lie algebras, Lie groups and the BCH theorem 443
4.5 Lie superalgebras 463
4.6 The BCH formula for Lie superalgebras 475
4.7 A discussion about Lie supergroups defined via the BCH formula 488
4.8 Examples 497
4.9 What sort of things are superfractals? 511
4.10 Superbundles formed by means of the BCH formula 514
4.11 What is a first super Chern class? 519
4.12 Grassmann structures for "extrinsic" supergeometries 521
4.13 Super Lie groups, that is, supergroups in the sense of Berezin and Alice Rogers 536
4.14 Graded Lie groups, that is, Lie supergroups in the sense of Kostant and Berezin 548
4.15 What sort of supergroups are the best? 553
Ch. V Hausdorff matrices 566
5.1 The Holder, Cesaro and Hausdorff means 567
5.2 The Toeplitz theorem 573
5.3 The regularity and equivalency conditions for Hausdorff matrices 576
5.4 Essential Hausdorff cores of certain infinite sequences 593
5.5 The generalizations of the Hausdorff summation 597
5.6 Solitons and soliton equations 610
5.7 An outline of the inverse scattering method 617
5.8 The direct method of Hirota 623
5.9 Periodic solutions of the KdV equation and related problems 633
5.10 Hausdorff's other results in classical, spectral and harmonic analysis 647
Appendix 667
References 688
Addendum 691
Hints to problems 699
Notation 701
Index of authors and subjects 707
Index of paradoxes 735
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