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Fractal Geometry : Mathematical Foundations and Applications
| Fractal Geometry : Mathematical Foundations and Applications |
| Autore | Falconer Kenneth |
| Edizione | [3rd ed.] |
| Pubbl/distr/stampa | New York : , : John Wiley & Sons, Incorporated, , 2014 |
| Descrizione fisica | 1 online resource (400 pages) |
| Disciplina | 514/.742 |
| Altri autori (Persone) | FalconerKenneth |
| Soggetto topico |
Fractals
Dimension theory (Topology) |
| Soggetto genere / forma | Electronic books. |
| ISBN |
9781118762851
9781119942399 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Cover -- Title Page -- Copyright -- Contents -- Preface to the first edition -- Preface to the second edition -- Preface to the third edition -- Course suggestions -- Introduction -- Part I Foundations -- Chapter 1 Mathematical background -- 1.1 Basic set theory -- 1.2 Functions and limits -- 1.3 Measures and mass distributions -- 1.4 Notes on probability theory -- 1.5 Notes and references -- Exercises -- Chapter 2 Box-counting dimension -- 2.1 Box-counting dimensions -- 2.2 Properties and problems of box-counting dimension -- 2.3 Modified box-counting dimensions -- 2.4 Some other definitions of dimension -- 2.5 Notes and references -- Exercises -- Chapter 3 Hausdorff and packing measures and dimensions -- 3.1 Hausdorff measure -- 3.2 Hausdorff dimension -- 3.3 Calculation of Hausdorff dimension-simple examples -- 3.4 Equivalent definitions of Hausdorff dimension -- 3.5 Packing measure and dimensions -- 3.6 Finer definitions of dimension -- 3.7 Dimension prints -- 3.8 Porosity -- 3.9 Notes and references -- Exercises -- Chapter 4 Techniques for calculating dimensions -- 4.1 Basic methods -- 4.2 Subsets of finite measure -- 4.3 Potential theoretic methods -- 4.4 Fourier transform methods -- 4.5 Notes and references -- Exercises -- Chapter 5 Local structure of fractals -- 5.1 Densities -- 5.2 Structure of 1-sets -- 5.3 Tangents to s-sets -- 5.4 Notes and references -- Exercises -- Chapter 6 Projections of fractals -- 6.1 Projections of arbitrary sets -- 6.2 Projections of s-sets of integral dimension -- 6.3 Projections of arbitrary sets of integral dimension -- 6.4 Notes and references -- Exercises -- Chapter 7 Products of fractals -- 7.1 Product formulae -- 7.2 Notes and references -- Exercises -- Chapter 8 Intersections of fractals -- 8.1 Intersection formulae for fractals -- 8.2 Sets with large intersection -- 8.3 Notes and references.
Exercises -- Part II Applications and Examples -- Chapter 9 Iterated function systems-self-similar and self-affine sets -- 9.1 Iterated function systems -- 9.2 Dimensions of self-similar sets -- 9.3 Some variations -- 9.4 Self-affine sets -- 9.5 Applications to encoding images -- 9.6 Zeta functions and complex dimensions -- 9.7 Notes and references -- Exercises -- Chapter 10 Examples from number theory -- 10.1 Distribution of digits of numbers -- 10.2 Continued fractions -- 10.3 Diophantine approximation -- 10.4 Notes and references -- Exercises -- Chapter 11 Graphs of functions -- 11.1 Dimensions of graphs -- 11.2 Autocorrelation of fractal functions -- 11.3 Notes and references -- Exercises -- Chapter 12 Examples from pure mathematics -- 12.1 Duality and the Kakeya problem -- 12.2 Vitushkin's conjecture -- 12.3 Convex functions -- 12.4 Fractal groups and rings -- 12.5 Notes and references -- Exercises -- Chapter 13 Dynamical systems -- 13.1 Repellers and iterated function systems -- 13.2 The logistic map -- 13.3 Stretching and folding transformations -- 13.4 The solenoid -- 13.5 Continuous dynamical systems -- 13.6 Small divisor theory -- 13.7 Lyapunov exponents and entropies -- 13.8 Notes and references -- Exercises -- Chapter 14 Iteration of complex functions-Julia sets and the Mandelbrot set -- 14.1 General theory of Julia sets -- 14.2 Quadratic functions-the Mandelbrot set -- 14.3 Julia sets of quadratic functions -- 14.4 Characterisation of quasi-circles by dimension -- 14.5 Newton's method for solving polynomial equations -- 14.6 Notes and references -- Exercises -- Chapter 15 Random fractals -- 15.1 A random Cantor set -- 15.2 Fractal percolation -- 15.3 Notes and references -- Exercises -- Chapter 16 Brownian motion and Brownian surfaces -- 16.1 Brownian motion in R -- 16.2 Brownian motion in Rn -- 16.3 Fractional Brownian motion. 16.4 Fractional Brownian surfaces -- 16.5 Lévy stable processes -- 16.6 Notes and references -- Exercises -- Chapter 17 Multifractal measures -- 17.1 Coarse multifractal analysis -- 17.2 Fine multifractal analysis -- 17.3 Self-similar multifractals -- 17.4 Notes and references -- Exercises -- Chapter 18 Physical applications -- 18.1 Fractal fingering -- 18.2 Singularities of electrostatic and gravitational potentials -- 18.3 Fluid dynamics and turbulence -- 18.4 Fractal antennas -- 18.5 Fractals in finance -- 18.6 Notes and references -- Exercises -- References -- Index. |
| Record Nr. | UNINA-9910795832003321 |
Falconer Kenneth
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| New York : , : John Wiley & Sons, Incorporated, , 2014 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Fractal Geometry and Stochastics V / / edited by Christoph Bandt, Kenneth Falconer, Martina Zähle
| Fractal Geometry and Stochastics V / / edited by Christoph Bandt, Kenneth Falconer, Martina Zähle |
| Edizione | [1st ed. 2015.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2015 |
| Descrizione fisica | 1 online resource (339 p.) |
| Disciplina | 514.742 |
| Collana | Progress in Probability |
| Soggetto topico |
Probabilities
Geometry Measure theory Probability Theory and Stochastic Processes Measure and Integration |
| ISBN | 3-319-18660-4 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Preface -- Introduction -- Part 1: Geometric Measure Theory -- Sixty Years of Fractal Projections -- Scenery flow, conical densities, and rectifiability -- The Shape of Anisotropic Fractals: Scaling of Minkowski Functionals -- Projections of self-similar and related fractals: a survey of recent developments -- Part 2: Self-similar Fractals and Recurrent Structures -- Dimension of the graphs of the Weierstrass-type functions -- Tiling Z2 by a set of four elements -- Some recent developments in quantization of fractal measures -- Apollonian Circle Packings -- Entropy of Lyapunov-optimizing measures of some matrix cocycles -- Part 3: Analysis and Algebra on Fractals -- Poincaré functional equations, harmonic measures on Julia sets, and fractal zeta functions -- From self-similar groups to self-similar sets and spectra -- Finite energy coordinates and vector analysis on fractals -- Fractal zeta functions and complex dimensions: A general higher-dimensional theory -- Part 4: Multifractal Theory -- Inverse problems in multifractal analysis -- Multifractal analysis based on p-exponents and lacunarity exponents -- Part 5: Random Constructions -- Dimensions of Random Covering Sets -- Expected lifetime and capacity. |
| Record Nr. | UNINA-9910299761803321 |
| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2015 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||