{"product_id":"fractal-geometry-isbn-9781119942399","title":"Fractal Geometry","description":"\u003cp\u003e\u003cb\u003eThe seminal text on fractal geometry for students and researchers:\u003c\/b\u003e extensively revised and updated with new material, notes and references that reflect recent directions.\u003c\/p\u003e \u003cp\u003eInterest 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 \u003ci\u003eFractal Geometry: Mathematical Foundations and Applications\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eFractal Geometry: Mathematical Foundations and Applications\u003c\/i\u003e 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.\u003c\/p\u003e \u003cul\u003e \u003cli\u003eProvides a comprehensive and accessible introduction to the mathematical theory and     applications of fractals\u003c\/li\u003e \u003cli\u003eCarefully explains each topic using illustrative examples and diagrams\u003c\/li\u003e \u003cli\u003eIncludes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics\u003c\/li\u003e \u003cli\u003eFeatures a wide range of exercises, enabling readers to consolidate their understanding\u003c\/li\u003e \u003cli\u003eSupported by a website with solutions to exercises and additional material \u003cb\u003ewww.wileyeurope.com\/fractal\u003c\/b\u003e\n\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eLeads onto the more advanced sequel \u003ci\u003eTechniques in Fractal Geometry\u003c\/i\u003e (also by Kenneth Falconer and available from Wiley)\u003c\/p\u003e \u003cp\u003ePreface to the first edition ix\u003c\/p\u003e \u003cp\u003ePreface to the second edition xiii\u003c\/p\u003e \u003cp\u003ePreface to the third edition xv\u003c\/p\u003e \u003cp\u003eCourse suggestions xvii\u003c\/p\u003e \u003cp\u003eIntroduction xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I Foundations 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Mathematical background 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Basic set theory 3\u003c\/p\u003e \u003cp\u003e1.2 Functions and limits 7\u003c\/p\u003e \u003cp\u003e1.3 Measures and mass distributions 11\u003c\/p\u003e \u003cp\u003e1.4 Notes on probability theory 17\u003c\/p\u003e \u003cp\u003e1.5 Notes and references 24\u003c\/p\u003e \u003cp\u003eExercises 24\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Box-counting dimension 27\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Box-counting dimensions 27\u003c\/p\u003e \u003cp\u003e2.2 Properties and problems of box-counting dimension 34\u003c\/p\u003e \u003cp\u003e2.3 Modified box-counting dimensions 38\u003c\/p\u003e \u003cp\u003e2.4 Some other definitions of dimension 40\u003c\/p\u003e \u003cp\u003e2.5 Notes and references 41\u003c\/p\u003e \u003cp\u003eExercises 42\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Hausdorff and packing measures and dimensions 44\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Hausdorff measure 44\u003c\/p\u003e \u003cp\u003e3.2 Hausdorff dimension 47\u003c\/p\u003e \u003cp\u003e3.3 Calculation of Hausdorff dimension – simple examples 51\u003c\/p\u003e \u003cp\u003e3.4 Equivalent definitions of Hausdorff dimension 53\u003c\/p\u003e \u003cp\u003e3.5 Packing measure and dimensions 54\u003c\/p\u003e \u003cp\u003e3.6 Finer definitions of dimension 57\u003c\/p\u003e \u003cp\u003e3.7 Dimension prints 58\u003c\/p\u003e \u003cp\u003e3.8 Porosity 60\u003c\/p\u003e \u003cp\u003e3.9 Notes and references 63\u003c\/p\u003e \u003cp\u003eExercises 64\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Techniques for calculating dimensions 66\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Basic methods 66\u003c\/p\u003e \u003cp\u003e4.2 Subsets of finite measure 75\u003c\/p\u003e \u003cp\u003e4.3 Potential theoretic methods 77\u003c\/p\u003e \u003cp\u003e4.4 Fourier transform methods 80\u003c\/p\u003e \u003cp\u003e4.5 Notes and references 81\u003c\/p\u003e \u003cp\u003eExercises 81\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Local structure of fractals 83\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Densities 84\u003c\/p\u003e \u003cp\u003e5.2 Structure of 1-sets 87\u003c\/p\u003e \u003cp\u003e5.3 Tangents to s-sets 92\u003c\/p\u003e \u003cp\u003e5.4 Notes and references 96\u003c\/p\u003e \u003cp\u003eExercises 96\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Projections of fractals 98\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Projections of arbitrary sets 98\u003c\/p\u003e \u003cp\u003e6.2 Projections of s-sets of integral dimension 101\u003c\/p\u003e \u003cp\u003e6.3 Projections of arbitrary sets of integral dimension 103\u003c\/p\u003e \u003cp\u003e6.4 Notes and references 105\u003c\/p\u003e \u003cp\u003eExercises 106\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Products of fractals 108\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Product formulae 108\u003c\/p\u003e \u003cp\u003e7.2 Notes and references 116\u003c\/p\u003e \u003cp\u003eExercises 116\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Intersections of fractals 118\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Intersection formulae for fractals 119\u003c\/p\u003e \u003cp\u003e8.2 Sets with large intersection 122\u003c\/p\u003e \u003cp\u003e8.3 Notes and references 128\u003c\/p\u003e \u003cp\u003eExercises 128\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II Applications and Examples 131\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Iterated function systems – self-similar and self-affine sets 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Iterated function systems 133\u003c\/p\u003e \u003cp\u003e9.2 Dimensions of self-similar sets 139\u003c\/p\u003e \u003cp\u003e9.3 Some variations 143\u003c\/p\u003e \u003cp\u003e9.4 Self-affine sets 149\u003c\/p\u003e \u003cp\u003e9.5 Applications to encoding images 155\u003c\/p\u003e \u003cp\u003e9.6 Zeta functions and complex dimensions 158\u003c\/p\u003e \u003cp\u003e9.7 Notes and references 167\u003c\/p\u003e \u003cp\u003eExercises 167\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Examples from number theory 169\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Distribution of digits of numbers 169\u003c\/p\u003e \u003cp\u003e10.2 Continued fractions 171\u003c\/p\u003e \u003cp\u003e10.3 Diophantine approximation 172\u003c\/p\u003e \u003cp\u003e10.4 Notes and references 176\u003c\/p\u003e \u003cp\u003eExercises 176\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Graphs of functions 178\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Dimensions of graphs 178\u003c\/p\u003e \u003cp\u003e11.2 Autocorrelation of fractal functions 188\u003c\/p\u003e \u003cp\u003e11.3 Notes and references 192\u003c\/p\u003e \u003cp\u003eExercises 192\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Examples from pure mathematics 195\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Duality and the Kakeya problem 195\u003c\/p\u003e \u003cp\u003e12.2 Vitushkin’s conjecture 198\u003c\/p\u003e \u003cp\u003e12.3 Convex functions 200\u003c\/p\u003e \u003cp\u003e12.4 Fractal groups and rings 201\u003c\/p\u003e \u003cp\u003e12.5 Notes and references 204\u003c\/p\u003e \u003cp\u003eExercises 204\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Dynamical systems 206\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Repellers and iterated function systems 208\u003c\/p\u003e \u003cp\u003e13.2 The logistic map 209\u003c\/p\u003e \u003cp\u003e13.3 Stretching and folding transformations 213\u003c\/p\u003e \u003cp\u003e13.4 The solenoid 217\u003c\/p\u003e \u003cp\u003e13.5 Continuous dynamical systems 220\u003c\/p\u003e \u003cp\u003e13.6 Small divisor theory 225\u003c\/p\u003e \u003cp\u003e13.7 Lyapunov exponents and entropies 228\u003c\/p\u003e \u003cp\u003e13.8 Notes and references 231\u003c\/p\u003e \u003cp\u003eExercises 232\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Iteration of complex functions – Julia sets and the Mandelbrot set 235\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 General theory of Julia sets 235\u003c\/p\u003e \u003cp\u003e14.2 Quadratic functions – the Mandelbrot set 243\u003c\/p\u003e \u003cp\u003e14.3 Julia sets of quadratic functions 248\u003c\/p\u003e \u003cp\u003e14.4 Characterisation of quasi-circles by dimension 256\u003c\/p\u003e \u003cp\u003e14.5 Newton’s method for solving polynomial equations 258\u003c\/p\u003e \u003cp\u003e14.6 Notes and references 262\u003c\/p\u003e \u003cp\u003eExercises 262\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Random fractals 265\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 A random Cantor set 266\u003c\/p\u003e \u003cp\u003e15.2 Fractal percolation 272\u003c\/p\u003e \u003cp\u003e15.3 Notes and references 277\u003c\/p\u003e \u003cp\u003eExercises 277\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Brownian motion and Brownian surfaces 279\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Brownian motion in R 279\u003c\/p\u003e \u003cp\u003e16.2 Brownian motion in R n 285\u003c\/p\u003e \u003cp\u003e16.3 Fractional Brownian motion 289\u003c\/p\u003e \u003cp\u003e16.4 Fractional Brownian surfaces 294\u003c\/p\u003e \u003cp\u003e16.5 Lévy stable processes 296\u003c\/p\u003e \u003cp\u003e16.6 Notes and references 299\u003c\/p\u003e \u003cp\u003eExercises 299\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Multifractal measures 301\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Coarse multifractal analysis 302\u003c\/p\u003e \u003cp\u003e17.2 Fine multifractal analysis 307\u003c\/p\u003e \u003cp\u003e17.3 Self-similar multifractals 310\u003c\/p\u003e \u003cp\u003e17.4 Notes and references 320\u003c\/p\u003e \u003cp\u003eExercises 320\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Physical applications 323\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Fractal fingering 325\u003c\/p\u003e \u003cp\u003e18.2 Singularities of electrostatic and gravitational potentials 330\u003c\/p\u003e \u003cp\u003e18.3 Fluid dynamics and turbulence 332\u003c\/p\u003e \u003cp\u003e18.4 Fractal antennas 334\u003c\/p\u003e \u003cp\u003e18.5 Fractals in finance 336\u003c\/p\u003e \u003cp\u003e18.6 Notes and references 340\u003c\/p\u003e \u003cp\u003eExercises 341\u003c\/p\u003e \u003cp\u003eReferences 342\u003c\/p\u003e \u003cp\u003eIndex 357\u003c\/p\u003e  \u003cp\u003e“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!.”  (\u003ci\u003eMathematical Association of America\u003c\/i\u003e, 11 June 2014)\u003c\/p\u003e \u003cb\u003eKenneth Falconer\u003c\/b\u003e, University of St Andrews, UK. \u003cp\u003e\u003cb\u003eThe seminal text on fractal geometry for students and researchers:\u003c\/b\u003e extensively revised and updated with new material, notes and references that reflect recent directions.\u003c\/p\u003e \u003cp\u003eInterest 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 \u003ci\u003eFractal Geometry: Mathematical Foundations and Applications\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eFractal Geometry: Mathematical Foundations and Applications\u003c\/i\u003e 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.\u003c\/p\u003e \u003cul\u003e \u003cli\u003eProvides a comprehensive and accessible introduction to the mathematical theory and     applications of fractals\u003c\/li\u003e \u003cli\u003eCarefully explains each topic using illustrative examples and diagrams\u003c\/li\u003e \u003cli\u003eIncludes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics\u003c\/li\u003e \u003cli\u003eFeatures a wide range of exercises, enabling readers to consolidate their understanding\u003c\/li\u003e \u003cli\u003eSupported by a website with solutions to exercises and additional material \u003cb\u003ewww.wileyeurope.com\/fractal\u003c\/b\u003e\n\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eLeads onto the more advanced sequel \u003ci\u003eTechniques in Fractal Geometry\u003c\/i\u003e (also by Kenneth Falconer and available from Wiley)\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989240201445,"sku":"NP9781119942399","price":54.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119942399.jpg?v=1761783336","url":"https:\/\/k12savings.com\/es\/products\/fractal-geometry-isbn-9781119942399","provider":"K12savings","version":"1.0","type":"link"}