Fractal geometries
著者
書誌事項
Fractal geometries
CRC, 1991
- タイトル別名
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Les Géometries fractales
- 統一タイトル
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Géometries fractales
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注記
Translation of: Les Géometries fractales
Includes bibliographical references and index
内容説明・目次
内容説明
Fractal geometry, based on recursive mathematical schemas, provides a means for modeling a great number of natural phenomena. For this reason, it is of increasing interest to physicists, chemists, biologists, and geographers, among others. A major quality of fractality is that it not only unifies phenomena previously thought to be anomalous or disparate in a single theoretical framework, but it also promotes a return to graphical treatment, which had been almost completely banished from scientific thought in favor of analysis.
This book casts a new, lively light on scientific territories still not fully explored. It is designed for research workers, engineers, and experimentalists faced with problems of measure and action in heterogenous materials and environments. Several color plates illustrate the implications and consequences of this theory for most of the questions raised by the taking into consideration of time in a fractal space.
目次
- Historical Landmarks: Key Dates. Notation: Important Symbols. THE DISCOVERY OF FRACTAL GEOMETRY. THE SURVEYOR'S TASK: RECTIFIABLE CURVES, MEASURING BY ARC LENGTHS. A JOURNEY INTO PATHOLOGY: THE SEARCH FOR LOST RECTIFIABILITY. WHAT IS A MEASURE? FROM LINE TO SURFACE. A SIMPLE EXPRESSION OF FRACTALITY: SELF-SIMILARITY. FRACTAL CLOSED CURVES (LOOPS): PERIMETER, AREA DENSITY
- FRACTAL MASS DIMENSION
- THE CONCEPT OF CO-DIMENSION. SCALING LAWS WITH VARIABLE RATIOS. FROM SELF-SIMILARITY TO SELF-AFFINESS. MEASURES OF DIMENSION
- TIME IN FRACTAL GEOMETRY. PRACTICAL METHODS: DIFFERENT MEASURES OF DIMENSION. The Hausdorff Dimension. The Mindkowski-Bouligand Dimension. The Packing Dimension. TWO METHODS FOR MEASURING THE FRACTAL DIMENSION. The Boxes Method in R2. Variations Method. RELATION BETWEEN TIME AND MEASURE
- PARAMETRISATION OF FRACTAL CURVES. Measure and Length. Length of a Rectifiable Curve. The Case of Non-Rectifiable Curves. Fractal Parametrisation of a Self-Affine Graph. CASE WHERE THE GEOMETRY IS A TRANSFER FUNCTION
- PHYSICAL MEASURE OF THE GEOMETRY. DERIVATIVES OF NON-INTEGRAL ORDER. DEFINITION OF DERIVATION OF NON-INTEGRAL ORDER. Derivative of tp. The Euler gamma G Function. Derivative of ept. An Inconsistency. Generalisation: The Riemann-Liouville Integral. Definition of the Integro-Differential Operator. Bessel Functions. SPECTRAL ANALYSIS AND NON-INTEGRAL DERIVATION. Fractional Derivation and the Spectrum. AN IRREGULARITY PARAMETER FOR CONTINUOUS NON-DERIVABLE FUNCTIONS: THE MAXIMUM-ORDER OF DERIVATION. Existence of the Derivatives. Extremum for the Order of Derivation, and the Fractal Dimension. COMPOSITION OF FRACTAL GEOMETRIES. STATISTICAL ASPECTS: MULTIFRACTALITY - THE COMBINATION OF SEVERAL FRACTAL DIMENSIONS. Distortion of the Scaling Laws. Multiplicative Binomial Processes. Fractal Sub-Set. The Lipschitz-Hoelder Exponent. Dimension and Entropy. Distribution of the Mass Exponents t(q). HYPERFRACTALITY - COMBINATION OF DERIVATIONS OF DIFFERENT ORDERS. From Brownian Motion to Fractional Brownian Motion. Composition of Fractional Derivative Indices: Hyperfractality. APPLICATIONS. MEASURE AND UNCERTAINTY. Distribution and Measure: The Mathematical Microscope. Dimension of the Weierstrasse Distribution. Local Hoelderian Properties: Geometrical Meaning of Derivatives of Non-Integral Order. The NOISE Transformation. Wavelet Transform of a Fractal Object. FRACTAL MORPHOGENESIS. Fractal Structure of Manganese Dioxide. Diffraction by a Fractal Object. Growth of Aggregates. Hydrodynamical Fingering. Electrical Discharge Fingering. Combinatorial Tree Structures and Stochastic Matrices. Trees in Speech Studies: A Graphical Tool for Language. Creation of a Fractal Object by Diffusion. Tribology or Geography? FRACTAL GEOMETRY AND IRREVERSABILITY. Adsorption on a Fractal Object: Chromatography. Thermodynamics of Curves, Entropy and Dimension. Chemical Potential, Entropy and Dimension. Time in Fractal Geometry. Energy Distribution and Fractional Derivation. Continued Fractions, Scale Laws and Electrical Circuits. Fractional Differential Equations, Laplace Equations. Fractional Derivation and Diffusion. Motive Power and Energy Yield in a Fractal Medium. An Example of Hyper-Fractality. Use of Non-Integral Derivation in Control Engineering: The CRONE Method. COMPLEXITY. Multi-Fractal Electrical Network. Fractal Molecules. Julia and Mandelbrot Sets, Time and Stability Considerations for Iterated Functions in the Complex Plan. Propagation of Electromagnetic Waves in a Multi-Layer Structure. Time and Morphogenesis: Interated Affine and Quadralic Transformations. BY WAY OF EPILOGUE. CONCLUSION. BIBLIOGRAPHY. INDEX.
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