Fractal Geometry

This comprehensive and popular textbook makes fractal geometry accessible to final-year undergraduate math or physics majors, while also serving as a reference for research mathematicians or scientists. This up-to-date edition covers introductory multifractal theory, random fractals, and modern applications in finance and science.

Zusatztext "Falconer's book is excellent in many respects and the reviewer strongly recommends it. May every university library own a copy! or three! And if you're a student reading this! go check it out today!." (Mathematical Association of America! 11 June 2014) Informationen zum Autor Kenneth Falconer , University of St Andrews, UK. Klappentext The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions.Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines.Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences.* Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals* Carefully explains each topic using illustrative examples and diagrams* Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics* Features a wide range of exercises, enabling readers to consolidate their understanding* Supported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractalLeads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley) Zusammenfassung This comprehensive and popular textbook makes fractal geometry accessible to final-year undergraduate math or physics majors, while also serving as a reference for research mathematicians or scientists. This up-to-date edition covers introductory multifractal theory, random fractals, and modern applications in finance and science. Inhaltsverzeichnis Preface to the first edition ixPreface to the second edition xiiiPreface to the third edition xvCourse suggestions xviiIntroduction xixPART I FOUNDATIONS 11 Mathematical background 31.1 Basic set theory 31.2 Functions and limits 71.3 Measures and mass distributions 111.4 Notes on probability theory 171.5 Notes and references 24Exercises 242 Box-counting dimension 272.1 Box-counting dimensions 272.2 Properties and problems of box-counting dimension 34*2.3 Modified box-counting dimensions 382.4 Some other definitions of dimension 402.5 Notes and references 41Exercises 423 Hausdorff and packing measures and dimensions 443.1 Hausdorff measure 443.2 Hausdorff dimension 473.3 Calculation of Hausdorff dimension - simple examples 513.4 Equivalent definitions of Hausdorff dimension 53*3.5 Packing measure and dimensions 54*3.6 Finer definitions of dimension 57*3.7 Dimension prints 58*3.8 Porosity 603.9 Notes and references 63Exercises 644 Techniques for calculating dimensions 664.1 Basic methods 664.2 Subsets of finite measure 754.3 Potential theoretic methods 77*4.4 Fourier transform methods 804.5 Notes and references 81Exercises 815 Local structure of fractals 835.1 Densities 845.2 Structure of 1-sets 875.3 Tangents to s-sets 925.4 Notes and references 96Exercises 966 Projections of fractals 986.1 Projections of arbitrary sets 986....

Autorentext
Kenneth Falconer, University of St Andrews, UK

Klappentext
The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines. Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences. * Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals * Carefully explains each topic using illustrative examples and diagrams * Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics * Features a wide range of exercises, enabling readers to consolidate their understanding * Supported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractal Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)

Inhalt
Preface to the first edition ix Preface to the second edition xiii Preface to the third edition xv Course suggestions xvii Introduction xix PART I FOUNDATIONS 1 1 Mathematical background 3 1.1 Basic set theory 3 1.2 Functions and limits 7 1.3 Measures and mass distributions 11 1.4 Notes on probability theory 17 1.5 Notes and references 24 Exercises 24 2 Box-counting dimension 27 2.1 Box-counting dimensions 27 2.2 Properties and problems of box-counting dimension 34 *2.3 Modified box-counting dimensions 38 2.4 Some other definitions of dimension 40 2.5 Notes and references 41 Exercises 42 3 Hausdorff and packing measures and dimensions 44 3.1 Hausdorff measure 44 3.2 Hausdorff dimension 47 3.3 Calculation of Hausdorff dimension - simple examples 51 3.4 Equivalent definitions of Hausdorff dimension 53 *3.5 Packing measure and dimensions 54 *3.6 Finer definitions of dimension 57 *3.7 Dimension prints 58 *3.8 Porosity 60 3.9 Notes and references 63 Exercises 64 4 Techniques for calculating dimensions 66 4.1 Basic methods 66 4.2 Subsets of finite measure 75 4.3 Potential theoretic methods 77 *4.4 Fourier transform methods 80 4.5 Notes and references 81 Exercises 81 5 Local structure of fractals 83 5.1 Densities 84 5.2 Structure of 1-sets 87 5.3 Tangents to s-sets 92 5.4 Notes and references 96 Exercises 96 6 Projections of fractals 98 6.1 Projections of arbitrary sets 98 6.2 Projections of s-sets of integral dimension 101 6.3 Projections of arbitrary sets of integral dimension 103 6.4 Notes and references 105 Exercises 106 7 Products of fractals 108 7.1 Product formulae 108 7.2 Notes and references 116 Exercises 116 8 Intersections of fractals 118 8.1 Intersection formulae for fractals 119 *8.2 Sets with large intersection 122 8.3 Notes and references 128 Exercises 128 PART II APPLICATIONS AND EXAMPLES 131 9 Iterated function systems - self-similar and self-affi…

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