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Fractal geometries: theory and applications. Transl. from the French by J. Howlett. (English) Zbl 0805.58002

Boca Raton, FL: CRC Press. x, 181 p. (1991).
This is another book about fractals and their geometry, this time written by a practicing engineer who is quite enthusiastic about the possibilities this theory has opened up in understanding complicated phenomena in nature. The book is divided into two parts: “Theory and technique” and “Applications”, both consisting of four chapters. The first chapters treat rather standard themes such as nonintegral dimension, Hausdorff measure and dimension, capacity, packing dimension, box counting methods and so on. In Chapter 3 the author discusses differentiation of nonintegral order and its relation to fractal curves. Chapter 4 is on multifractals and their description in terms of the singularity spectrum. Part II starts in Chapter 5 with an introduction to the theory of distributions and test functions and their possible role in examining fractal sets. Finally, Chapters 6-8 are reserved for several examples of systems in nature where fractal objects in one form or another seem to exist.
The style of the book is rather uncommon. For instance, several times the author praises the work of people without giving the corresponding literature. He uses rather cryptic language (at least in the present English translation), as witness the following phrase from page 69: “The variable speed distribution leads to a mixing of information of different types and to some confusing of the transfer function or state function at the interface that has to be taken into account in all linear dissipative processes that involve geometry.” The meaning of this and similar sentences did not become clear to the reviewer even after repeated reading.

MSC:

58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N99 Applications of dynamical systems
28A80 Fractals
82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium)
82D99 Applications of statistical mechanics to specific types of physical systems
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