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Fractal Geometry: Mathematical Foundations and Applications 3rd edition [Hardback]

4.26/5 (56 ratings by Goodreads)
(University of St Andrews, UK)
  • Formāts: Hardback, 400 pages, height x width x depth: 229x155x25 mm, weight: 635 g
  • Izdošanas datums: 24-Jan-2014
  • Izdevniecība: John Wiley & Sons Inc
  • ISBN-10: 111994239X
  • ISBN-13: 9781119942399
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Fractal Geometry: Mathematical Foundations and Applications 3rd editionPalielināt
  • Formāts: Hardback, 400 pages, height x width x depth: 229x155x25 mm, weight: 635 g
  • Izdošanas datums: 24-Jan-2014
  • Izdevniecība: John Wiley & Sons Inc
  • ISBN-10: 111994239X
  • ISBN-13: 9781119942399
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781119942399.html
Keywords:
"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. New research developments are highlighted, such as porosity, while covering other much more sophisticated topics, such as fractal aspects of conformal invariance, complex dimensions, and non-commutative fractal geometry. The book emphasizes dimension in its various forms, but other notions of fractality are also prominent"--

"This comprehensive, accessible and very popular textbook presents fractal geometry at a level accessible to a final year undergraduate mathematician or physicist whilst also providing a useful primer or reference for the research mathematician or scientist"--



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)

Recenzijas

Falconers book is excellent in many respects and the reviewer strongly recommends it. May every university library own a copy, or three! And if youre a student reading this, go check it out today!.  (Mathematical Association of America, 11 June 2014)

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(130)
1 Mathematical background
3(24)
1.1 Basic set theory
3(4)
1.2 Functions and limits
7(4)
1.3 Measures and mass distributions
11(6)
1.4 Notes on probability theory
17(7)
1.5 Notes and references
24(3)
Exercises
24(3)
2 Box-counting dimension
27(17)
2.1 Box-counting dimensions
27(7)
2.2 Properties and problems of box-counting dimension
34(4)
*2.3 Modified box-counting dimensions
38(2)
2.4 Some other definitions of dimension
40(1)
2.5 Notes and references
41(3)
Exercises
42(2)
3 Hausdorff and packing measures and dimensions
44(22)
3.1 Hausdorff measure
44(3)
3.2 Hausdorff dimension
47(4)
3.3 Calculation of Hausdorff dimension -- simple examples
51(2)
3.4 Equivalent definitions of Hausdorff dimension
53(1)
*3.5 Packing measure and dimensions
54(3)
*3.6 Finer definitions of dimension
57(1)
*3.7 Dimension prints
58(2)
*3.8 Porosity
60(3)
3.9 Notes and references
63(3)
Exercises
64(2)
4 Techniques for calculating dimensions
66(17)
4.1 Basic methods
66(9)
4.2 Subsets of finite measure
75(2)
4.3 Potential theoretic methods
77(3)
*4.4 Fourier transform methods
80(1)
4.5 Notes and references
81(2)
Exercises
81(2)
5 Local structure of fractals
83(15)
5.1 Densities
84(3)
5.2 Structure of 1-sets
87(5)
5.3 Tangents to s-sets
92(4)
5.4 Notes and references
96(2)
Exercises
96(2)
6 Projections of fractals
98(10)
6.1 Projections of arbitrary sets
98(3)
6.2 Projections of s-sets of integral dimension
101(2)
6.3 Projections of arbitrary sets of integral dimension
103(2)
6.4 Notes and references
105(3)
Exercises
106(2)
7 Products of fractals
108(10)
7.1 Product formulae
108(8)
7.2 Notes and references
116(2)
Exercises
116(2)
8 Intersections of fractals
118(13)
8.1 Intersection formulae for fractals
119(3)
*8.2 Sets with large intersection
122(6)
8.3 Notes and references
128(3)
Exercises
128(3)
PART II APPLICATIONS AND EXAMPLES
131(211)
9 Iterated function systems -- self-similar and self-affine sets
133(36)
9.1 Iterated function systems
133(6)
9.2 Dimensions of self-similar sets
139(4)
9.3 Some variations
143(6)
9.4 Self-affine sets
149(6)
9.5 Applications to encoding images
155(3)
*9.6 Zeta functions and complex dimensions
158(9)
9.7 Notes and references
167(2)
Exercises
167(2)
10 Examples from number theory
169(9)
10.1 Distribution of digits of numbers
169(2)
10.2 Continued fractions
171(1)
10.3 Diophantine approximation
172(4)
10.4 Notes and references
176(2)
Exercises
176(2)
11 Graphs of functions
178(17)
11.1 Dimensions of graphs
178(10)
*11.2 Autocorrelation of fractal functions
188(4)
11.3 Notes and references
192(3)
Exercises
192(3)
12 Examples from pure mathematics
195(11)
12.1 Duality and the Kakeya problem
195(3)
12.2 Vitushkin's conjecture
198(2)
12.3 Convex functions
200(1)
12.4 Fractal groups and rings
201(3)
12.5 Notes and references
204(2)
Exercises
204(2)
13 Dynamical systems
206(29)
13.1 Repellers and iterated function systems
208(1)
13.2 The logistic map
209(4)
13.3 Stretching and folding transformations
213(4)
13.4 The solenoid
217(3)
13.5 Continuous dynamical systems
220(5)
*13.6 Small divisor theory
225(3)
*13.7 Lyapunov exponents and entropies
228(3)
13.8 Notes and references
231(4)
Exercises
232(3)
14 Iteration of complex functions -- Julia sets and the Mandelbrot set
235(30)
14.1 General theory of Julia sets
235(8)
14.2 Quadratic functions -- the Mandelbrot set
243(5)
14.3 Julia sets of quadratic functions
248(8)
14.4 Characterisation of quasi-circles by dimension
256(2)
14.5 Newton's method for solving polynomial equations
258(4)
14.6 Notes and references
262(3)
Exercises
262(3)
15 Random fractals
265(14)
15.1 A random Cantor set
266(6)
15.2 Fractal percolation
272(5)
15.3 Notes and references
277(2)
Exercises
277(2)
16 Brownian motion and Brownian surfaces
279(22)
16.1 Brownian motion in R
279(6)
16.2 Brownian motion in Rn
285(4)
16.3 Fractional Brownian motion
289(5)
16.4 Fractional Brownian surfaces
294(2)
16.5 Levy stable processes
296(3)
16.6 Notes and references
299(2)
Exercises
299(2)
17 Multifractal measures
301(22)
17.1 Coarse multifractal analysis
302(5)
17.2 Fine multifractal analysis
307(3)
17.3 Self-similar multifractals
310(10)
17.4 Notes and references
320(3)
Exercises
320(3)
18 Physical applications
323(19)
18.1 Fractal fingering
325(5)
18.2 Singularities of electrostatic and gravitational potentials
330(2)
18.3 Fluid dynamics and turbulence
332(2)
18.4 Fractal antennas
334(2)
18.5 Fractals in finance
336(4)
18.6 Notes and references
340(2)
Exercises
341(1)
References 342(15)
Index 357
Kenneth Falconer, University of St Andrews, UK.