Smooth Manifolds and Observables

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Springer Science & Business Media, 2003 - Computers - 222 pages
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The author isvery pleased that hisbook, ?rst publishedinRussian in2000 by MCCME Publishers, is now appearing in English under the auspices of such a truly classical publishing house as Springer-Verlag. In this edition several pertinent remarks by the referees (to whom the author expresses his gratitude) were taken into account, and new exercises were added (mostly) to the ?rst half of the book, thus achieving a better balancewiththesecond half.Besides, sometyposandminorerrors,noticed in the Russian edition, were corrected. Weare extremely grateful to all our readers whoassisted usinthistiresomebughunt.Weareespeciallygrateful toA.DeParis,I.S.Krasil’schik,andA.M.Verbovetski,whodemonstrated their acute eyesight, truly of degli Lincei standards. The English translation was carried out by A.B. Sossinsky (Chapters 1–8), I.S. Krasil’schik(Chapter 9), and S.V. Duzhin (Chapters 10–11)and reduced to a common denominator by the ?rst of them; A.M. Astashov prepared new versions of the ?gures; all the T X-nical work was done by E M.M. Vinogradov. In the process of preparing this edition, the author was supported bythe Istituto Nazionaledi Fisica Nucleare and the Istituto Italianoper gliStudi Filoso?ci. It is only thanks to these institutions, and to the e?cient help of Springer-Verlag, that the process successfully came to its end in such a short period of time. Jet Nestruev Moscow–Salerno April 2002 Preface The limits of my language are the limits of my world.
  

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Contents

Introduction
1
Cutoff and Other Special Smooth Functions on R
13
Algebras and Points
21
Smooth Manifolds Algebraic Definition
37
Charts and Atlases
53
Smooth Maps
65
Equivalence of Coordinate and Algebraic Definitions
77
Spectra and Ghosts
85
The Differential Calculus as a Part of Commutative Algebra
95
Smooth Bundles
143
Vector Bundles and Projective Modules
163
Observability Principle Set Theory and the Foundations of Mathematics
209
Copyright

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