Scaling. (English) Zbl 1094.00006

Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (ISBN 0-521-53394-5/pbk; 0-521-82657-8/hbk). xiv, 171 p. (2003).
Self-similarity, that is, a similar structure recurring over different length or time scales, is one of the most astonishing properties which can be found in many branches of the sciences. A typical example, also treated in the book under review, is the occurrence of fractal sets in chaotic dynamics. If self-similarity is properly recognized and taken care of it leads to a great simplification in the treatment of specific problems or in the understanding of complicated phenomena by introducing so-called scaling laws. How they can be discovered for various different problems is the central theme of the book. Whereas simple similarity arguments and dimensional analysis, which are widely known nowadays, are also given a new treatment, the author, one of the few living masters of applied mathematics educated in the Russian mathematics tradition, also deals with quite complicated subjects like the idea of incomplete similarity and its connection to the renormalization group.
The present book is the result of notes for classes taught at UC Berkeley over a period of seven years and hence has been optimized concerning clarity and content. All mathematical ideas are motivated by physical examples, mostly taken from fluid dynamics with, for example, profound explanations of the structure of a class of turbulent flows. The book is a superb introduction into the subject, well readable already for advanced undergraduates, explaining how to construct and understand self-similar solutions of various physical problems and as it is written in the Foreword, does not admit any excuse for ignorance of this important and fascinating subject.
Reviewer: Hans Troger (Wien)


00A73 Dimensional analysis (MSC2010)
76F30 Renormalization and other field-theoretical methods for turbulence
00-02 Research exposition (monographs, survey articles) pertaining to mathematics in general
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
28A80 Fractals
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology