Techniques in Fractal Geometry

Author: Kenneth Falconer

Publisher: Wiley

ISBN: 9780471957249

Category: Mathematics

Page: 274

View: 4862

Following on from the success of Fractal Geometry: Mathematical Foundations and Applications, this new sequel presents a variety of techniques in current use for studying the mathematics of fractals. Much of the material presented in this book has come to the fore in recent years. This includes methods for studying dimensions and other parameters of fractal sets and measures, as well as more sophisticated techniques such as thermodynamic formalism and tangent measures. In addition to general theory, many examples and applications are described, in areas such as differential equations and harmonic analysis. This book is mathematically precise, but aims to give an intuitive feel for the subject, with underlying concepts described in a clear and accessible manner. The reader is assumed to be familiar with material from Fractal Geometry, but the main ideas and notation are reviewed in the first two chapters. Each chapter ends with brief notes on the development and current state of the subject. Exercises are included to reinforce the concepts. The author's clear style and up-to-date coverage of the subject make this book essential reading for all those who with to develop their understanding of fractal geometry.

Fractal Geometry

Mathematical Foundations and Applications

Author: Kenneth Falconer

Publisher: John Wiley & Sons

ISBN: 0470299452

Category: Mathematics

Page: 366

View: 4909

Since its original publication in 1990, Kenneth Falconer's Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general mathematical theory and applications of fractals in a way that is accessible to students from a wide range of disciplines. This new edition has been extensively revised and updated. It features much new material, many additional exercises, notes and references, and an extended bibliography that reflects the development of the subject since the first edition. * Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals. * Each topic is carefully explained and illustrated by examples and figures. * Includes all necessary mathematical background material. * Includes notes and references to enable the reader to pursue individual topics. * Features a wide selection of exercises, enabling the reader to develop their understanding of the theory. * Supported by a Web site featuring solutions to exercises, and additional material for students and lecturers. Fractal Geometry: Mathematical Foundations and Applications is aimed at undergraduate and graduate students studying courses in fractal geometry. The book also provides an excellent source of reference for researchers who encounter fractals in mathematics, physics, engineering, and the applied sciences. Also by Kenneth Falconer and available from Wiley: Techniques in Fractal Geometry ISBN 0-471-95724-0 Please click here to download solutions to exercises found within this title:

Fractal Geometry in Digital Imaging

Author: Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews

Publisher: Academic Press

ISBN: 9780127039701

Category: Mathematics

Page: 328

View: 5902

This book presents the analysis of textured images using fractal geometry, and discusses its application to imaging science and computer vision when modeling natural objects. The authors explore the methods which can be used to simulate, analyze, and interpret coherent images, and demonstrate a new approach which segments each image into regions of similarity that can be characterized by a random fractal with a given fractal dimension. Fractal Geometry in Digital Imaging is based on a research project, but has been written with a broad coverage and user friendly math to make the book accessible to a wider audience. It includes real world experiences and applications using the techniques described. * Discusses the analysis of textured images using fractal geometry * Explores the methods used to simulate, analyze, and interpret coherent images * Contains coverage of real world experiences and applications * Written in a user friendly style

Fractals and Spectra

Related to Fourier Analysis and Function Spaces

Author: Hans Triebel

Publisher: Springer Science & Business Media

ISBN: 3034800347

Category: Mathematics

Page: 272

View: 1586

This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators as it has emerged quite recently. Most of the presented material is published here for the first time.

Fraktalgeometrie: Selbstähnlichkeit und fraktale Dimension –

Author: Jutta Otterbein,Christina Sawatzki

Publisher: GRIN Verlag

ISBN: 3656262365

Category: Mathematics

Page: 20

View: 2604

Studienarbeit aus dem Jahr 2010 im Fachbereich Mathematik - Geometrie, Note: 1,3, Universität Kassel (Institut für Mathematik), Veranstaltung: Fachwissenschaftliches Seminar, Sprache: Deutsch, Abstract: In der nachfolgenden Arbeit soll die Selbstähnlichkeit und fraktale Dimension, Teil 1 behandelt werden. Vorab wird der Begriff „Fraktale“ im Allgemeinen beschrieben und erklärt. Zur Verdeutlichung des Begriffs wird ferner auf die unterschiedlichen Eigenschaften der Fraktale, die das Grundgerüst der Fraktalgeometrie und den Schwerpunkt der Arbeit bilden, eingegangen. Des Weiteren wird die Selbstähnlichkeit dargestellt, die sich unter anderem zwischen der exakten und der statistischen Selbstähnlichkeit unterscheiden lässt. Einige Beispiele sollen diesen Unterschied deutlich machen und herauskristallisieren. Darauf aufbauend wird die Selbstähnlichkeitsdimension allgemein definiert sowie die Formel zu ihrer Berechnung abgeleitet. Anschließend wird sich den mathematischen Fraktalen zugewandt. Im Mittelpunkt stehen die Cantor-Drittelmenge und das Sierpinski-Dreieck, bei denen jeweils die Selbstähnlichkeit sowie deren Dimension beschrieben und vertiefend erklärt wird. Abschließend werden unterschiedliche Wischaktivi-täten in der Ebene und im Raum anhand zahlreicher Beispiele skizziert und diese miteinander verglichen.

Fractal Geometry in Biological Systems

An Analytical Approach

Author: Philip M. Iannaccone,Mustafa Khokha

Publisher: CRC Press

ISBN: 9780849376368

Category: Science

Page: 384

View: 7903

Fractal Geometry in Biological Systems was written by the leading experts in the field of mathematics and the biological sciences together. It is intended to inform researchers in the bringing about the fundamental nature of fractals and their widespread appearance in biological systems. The chapters explain how the presence of fractal geometry can be used in an analytical way to predict outcomes in systems, to generate hypotheses, and to help design experiments. The authors make the mathematics accessible to a wide audience and do not assume prior experience in this area.

Fractal Geometry in Architecture and Design

Author: Carl Bovill

Publisher: Springer Science & Business Media

ISBN: 1461208432

Category: Mathematics

Page: 195

View: 2925

na broad sense Design Science is the grammar of a language of images Irather than of words. Modern communication techniques enable us to transmit and reconstitute images without needing to know a specific verbal sequence language such as the Morse code or Hungarian. International traffic signs use international image symbols which are not specific to any particular verbal language. An image language differs from a verbal one in that the latter uses a linear string of symbols, whereas the former is multi dimensional. Architectural renderings commonly show projections onto three mutual ly perpendicular planes, or consist of cross sections at different altitudes capa ble of being stacked and representing different floor plans. Such renderings make it difficult to imagine buildings comprising ramps and other features which disguise the separation between floors, and consequently limit the cre ative process of the architect. Analogously, we tend to analyze natural struc tures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures. Perception is a complex process. Our senses record; they are analogous to audio or video devices. We cannot, however, claim that such devices perceive.

Fractal Architecture

Organic Design Philosophy in Theory and Practice

Author: James Harris

Publisher: UNM Press

ISBN: 0826352022

Category: Architecture

Page: 420

View: 6664

Throughout history, nature has served as an inspiration for architecture and designers have tried to incorporate the harmonies and patterns of nature into architectural form. Alberti, Charles Renee Macintosh, Frank Lloyd Wright, and Le Courbusier are just a few of the well- known figures who have taken this approach and written on this theme. With the development of fractal geometry--the study of intricate and interesting self- similar mathematical patterns--in the last part of the twentieth century, the quest to replicate nature’s creative code took a stunning new turn. Using computers, it is now possible to model and create the organic, self-similar forms of nature in a way never previously realized. In Fractal Architecture, architect James Harris presents a definitive, lavishly illustrated guide that explains both the “how” and “why” of incorporating fractal geometry into architectural design.

Methods of Fracture Mechanics: Solid Matter Physics

Author: G.P. Cherepanov

Publisher: Springer Science & Business Media

ISBN: 9780792344087

Category: Science

Page: 322

View: 4199

Modern fracture mechanics considers phenomena at many levels, macro and micro; it is therefore inextricably linked to methods of theoretical and mathematical physics. This book introduces these sophisticated methods in a straightforward manner. The methods are applied to several important phenomena of solid state physics which impinge on fracture mechanics: adhesion, defect nucleation and growth, dislocation emission, sintering, the electron beam effect and fractal cracks. The book shows how the mathematical models for such processes may be set up, and how the equations so formulated may be solved and interpreted. The many open problems which are encountered will provide topics for MSc and PhD theses in fracture mechanics, and in theoretical and experimental physics. As a supplementary text, the book can be used in graduate level courses on fracture mechanics, solid matter physics, and mechanics of solids, or in a special course on the application of fracture mechanics methods in solid matter physics.

Modern Mathematical Tools and Techniques in Capturing Complexity

Author: Leandro Pardo,Narayanaswamy Balakrishnan,Maria Angeles Gil

Publisher: Springer Science & Business Media

ISBN: 3642208525

Category: Technology & Engineering

Page: 514

View: 7140

Real-life problems are often quite complicated in form and nature and, for centuries, many different mathematical concepts, ideas and tools have been developed to formulate these problems theoretically and then to solve them either exactly or approximately. This book aims to gather a collection of papers dealing with several different problems arising from many disciplines and some modern mathematical approaches to handle them. In this respect, the book offers a wide overview on many of the current trends in Mathematics as valuable formal techniques in capturing and exploiting the complexity involved in real-world situations. Several researchers, colleagues, friends and students of Professor María Luisa Menéndez have contributed to this volume to pay tribute to her and to recognize the diverse contributions she had made to the fields of Mathematics and Statistics and to the profession in general. She had a sweet and strong personality, and instilled great values and work ethics in her students through her dedication to teaching and research. Even though the academic community lost her prematurely, she would continue to provide inspiration to many students and researchers worldwide through her published work.

Fractals in Multimedia

Author: Michael F. Barnsley,Dietmar Saupe,Edward R. Vrscay

Publisher: Springer Science & Business Media

ISBN: 1468492446

Category: Computers

Page: 270

View: 5802

This IMA Volume in Mathematics and its Applications FRACTALS IN MULTIMEDIA is a result of a very successful three-day minisymposium on the same title. The event was an integral part of the IMA annual program on Mathemat ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley (Department of Mathematics and Statistics, University of Melbourne), Di etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward R. Vrscay (Department of Applied Mathematics, University of Waterloo) for their excellent work as organizers of the meeting and for editing the proceedings. We take this opportunity to thank the National Science Foundation for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume grew out of a meeting on Fractals in Multimedia held at the IMA in January 2001. The meeting was an exciting and intense one, focused on fractal image compression, analysis, and synthesis, iterated function systems and fractals in education. The central concerns of the meeting were to establish within these areas where we are now and to develop a vision for the future.

The Fractal Geometry of the Brain

Author: Antonio Di Ieva

Publisher: Springer

ISBN: 1493939955

Category: Medical

Page: 585

View: 7648

Reviews the most intriguing applications of fractal analysis in neuroscience with a focus on current and future potential, limits, advantages, and disadvantages. Will bring an understanding of fractals to clinicians and researchers also if they do not have a mathematical background, and will serve as a good tool for teaching the translational applications of computational models to students and scholars of different disciplines. This comprehensive collection is organized in four parts: (1) Basics of fractal analysis; (2) Applications of fractals to the basic neurosciences; (3) Applications of fractals to the clinical neurosciences; (4) Analysis software, modeling and methodology.

Fractals and Chaos

An illustrated course

Author: Paul S. Addison

Publisher: CRC Press

ISBN: 9780750304009

Category: Science

Page: 256

View: 9330

Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.

Advances in Ecological Research

Author: N.A

Publisher: Academic Press

ISBN: 9780080567143

Category: Science

Page: 430

View: 5818

The six reviews in this latest issue of Advances in Ecological Research cover a broad spectrum of ecology, from micro-patterns and processes, to the ecophysiology of the individual organism, to forest-scale processes. Topics covered include the possible evolutionary forces that have shaped particular strategies, and the potential and limitations for techniques in ecology, such as fractal geometry, field experiments and eddy co-variance measures. Despite this diversity of topics, there are plenty of points of contact and cross-reference.

Fractal Geometry

Mathematical Methods, Algorithms, Applications

Author: J M Blackledge,A K Evans,M J Turner

Publisher: Elsevier

ISBN: 0857099590

Category: Mathematics

Page: 244

View: 1262

International authorities from Canada, Denmark, England, Germany, Russia and South Africa focus on research on fractal geometry and the best practices in software, theoretical mathematical algorithms, and analysis. They address the rich panoply of manifold applications of fractal geometry available for study and research in science and industry: i.e., remote sensing, mapping, texture creations, pattern recognition, image compression, aeromechanical systems, cryptography and financial analysis. Economically priced, this important and authoritative reference source for research and study cites over 230 references to the literature, copiously illustrated with over 320 diagrams and photographs. The book is published for The Institute of Mathematics and its Applications, co-sponsored with The Institute of Physics and The Institution of Electrical Engineers. Outlines research on fractal geometry and the best practices in software, theoretical mathematical algorithms, and analysis International authorities from around the world address the rich panoply of manifold applications of fractal geometry available for study and research in science and industry Addresses applications in key research fields of remote sensing, mapping, texture creations, pattern recognition, image compression, aeromechanical systems, cryptography and financial analysis

Fractal Geometry and Stochastics II

Author: Christoph Bandt,Siegfried Graf,Martina Zähle

Publisher: Springer Science & Business Media

ISBN: 9783764362157

Category: Mathematics

Page: 292

View: 2167

A collection of contributions by outstanding mathematicians, highlighting the principal directions of research on the combination of fractal geometry and stochastic methods. Clear expositions introduce the most recent results and problems on these subjects and give an overview of their historical development.

Fractals and Fractional Calculus in Continuum Mechanics

Author: Alberto Carpinteri,Francesco Mainardi

Publisher: Springer

ISBN: 3709126649

Category: Technology & Engineering

Page: 348

View: 5197

The book is characterized by the illustration of cases of fractal, self-similar and multi-scale structures taken from the mechanics of solid and porous materials, which have a technical interest. In addition, an accessible and self-consistent treatment of the mathematical technique of fractional calculus is provided, avoiding useless complications.

Fractals in Biology and Medicine

Author: Gabriele A. Losa,Danilo Merlini,Theo F. Nonnenmacher,Ewald R. Weibel

Publisher: Springer Science & Business Media

ISBN: 9783764371722

Category: Mathematics

Page: 314

View: 5623

This volume is number four in a series of proceedings volumes from the International Symposia on Fractals in Biology and Medicine in Ascona, Switzerland which have been inspired by the work of Benoît Mandelbrot seeking to extend the concepts towards the life sciences. It highlights the potential that fractal geometry offers for elucidating and explaining the complex make-up of cells, tissues and biological organisms either in normal or in pathological conditions.