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Fractals - forms - point fields. Methods of geometry-statistics. (Fraktale - Formen - Punktfelder. Methoden der Geometrie-Statistik.) (German) Zbl 0763.60005

Berlin: Akademie-Verlag. 394 S. (1992).
The book is designed mainly for non-mathematicians, in particular for biologists and geologists. This strongly affects the style of the book: the authors explain the intuitions behind all the introduced notions and avoid more complicated reasoning. They give a rich list of references concerning basic knowledge and most recent results, as well as applications in biology, geology, environment, and so on.
The book is divided into three parts. Part I: Fractals and methods of calculating Hausdorff dimension. The authors introduce the Hausdorff measure and dimension and discuss related notions, as local Hausdorff dimension, fractal dimensions. They consider deterministic and random fractals and present methods of calculating fractal dimensions. Part II: The shape statistics. It consists of five chapters: 1. Preliminaries, 2. Describing contours, 3. Set-theoretical approach to shape, 4. Describing points, 5. Examples. (The title of Chapter 3 is misleading, because that chapter concerns also topological notions (e.g. compactness) and metric notions (e.g. convexity, volume)). Part III: Statistics of point fields. The lists of references do not contain the book by I. Saxl, Stereology of objects with internal structure (Academia Prague, 1989 (co- published with Elsevier Science Publishers)), which I would strongly recommend.

MSC:

60D05 Geometric probability and stochastic geometry
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
28A78 Hausdorff and packing measures
28A80 Fractals
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