Paradoxes of measures and dimensions originating in Felix Hausdorff's ideas
In this book, many ideas by Felix Hausdorff are described and contemporary mathematical theories stemming from them are sketched
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| Format: | Electronic Book |
| Language: | English |
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Singapore ; River Edge, N.J. :
World Scientific Pub. Co.,
c1994
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Table of Contents:
- ch. 0. Biographical sketches. 0.1. Curriculum vitae
- 0.2. Felix Hausdorff: mathematician, philosopher, poet
- 0.3. Author's reflections
- References: the works of Felix Hausdorff
- References to the biographical notes
- ch. I. The paradox of the sphere. 1.1. Hausdorff s decomposition of the sphere
- 1.2. The Banach-Tarski paradox
- 1.3. Non-measurable sets in [symbol] and [symbol]
- 1.4. Exotic measures and the problem of Ruziewicz
- 1.5. Group theoretic implications of the Hausdorff theorem
- 1.6. A fixed point view on paradoxical decompositions
- References
- ch. II. Inaccessible numbers and the hierarchal structure of set theory. 2.1. The infinite number of Cantor's set theory
- 2.2. Hausdorffs weakly inaccessible cardinals and a certain set-theoretic alpinism
- 2.3. The axioms of Zermelo and Fraenkel and their paradoxical background
- 2.4. The Godel theorem and the cumulative hierarchy of sets
- 2.5. On the continuum hypothesis and related problems
- 2.6. The conditions of the existence and non-existence of non-measurable sets
- 2.7. Cantor, Hausdorff and Godel's pyramid of paradoxes
- 2.8. The Godel theorem and a certain mathematical catastrophism
- 2.9. Hausdorff's intuitions versus present-day mathematics
- References
- ch. III. The Hausdorff measures, Hausdorff dimensions and fractals. 3.1. The Hausdorff measure and dimension
- 3.2. The standard example of the Cantor set
- 3.3. Ephemeral sets of strong measure zero
- 3.4. The implications in number theory
- 3.5. The Hausdorff dimension of the Cartesian product of sets
- 3.6. The Hausdorff-Besicovitch dimension versus Hausdorff operations
- 3.7. Metric spaces
- 3.8. The Hausdorff topology and topological stable dimension
- 3.9. The Hausdorff-Besicovitch dimension versus the topological dimension
- 3.10. The coefficient [symbol](X), that is, the topological measure of Borsuk
- 3.11. The concept of fractals
- 3.12. One flew over the land of fractals
- 3.13. The phenomenon of self-similarity
- 3.14. Multifractals
- 3.15. Natural fractals
- 3.16. The Olbers paradox and fractal approaches to cosmology
- 3.17. Fractals: illusion, speculation or mathematics?
- References
- ch. IV. The Baker-Campbell-Hausdorff formula. 4.1. The Hausdorff series
- 4.2. A continuous Baker-Campbell-Hausdorff problem
- 4.3. The problem of the convergence of the Hausdorff series
- 4.4. Lie algebras, Lie groups and the BCH theorem
- 4.5. Lie superalgebras
- 4.6. The BCH formula for Lie superalgebras
- 4.7. A discussion about Lie supergroups defined via the BCH formula
- 4.8. Examples
- 4.9. What sort of things are superfractals?
- 4.10. Superbundles formed by means of the BCH formula
- 4.11. What is a first super Chern class?
- 4.12. Grassmann structures for "extrinsic" supergeometries
- 4.13. Super Lie groups, that is, supergroups in the sense of Berezin and Alice Rogers
- 4.14. Graded Lie groups, that is Lie supergroups in the sense of Kostant and Berezin
- 4.15. What sort of supergroups are the best?
- References
- ch. V. Hausdorff matrices. 5.1. The Holder, Cesaro and Hausdorff means
- 5.2. The Toeplitz theorem
- 5.3. The regularity and equivalency conditions for Hausdorff matrices
- 5.4. Essential Hausdorff cores of certain infinite sequences
- 5.5. The generalizations of the Hausdorff summation
- 5.6. Solitons and soliton equations
- 5.7. An outline of the inverse scattering method
- 5.8. The direct method of Hirota
- 5.9. Periodic solutions of the KdV equation and related problems
- 5.10. Hausdorffs other results in classical, spectral and harmonic analysis
- References