Techniques in Fractal Geometry

Author: Kenneth Falconer

Publisher: Wiley

ISBN: 9780471957249

Category: Mathematics

Page: 274

View: 2994

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: 111876286X

Category: Mathematics

Page: 400

View: 6592

The seminal text on fractal geometry for students andresearchers: extensively revised and updated with new material,notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both asa subject that is fascinating in its own right and as a conceptthat is central to many areas of mathematics, science andscientific research. Since its initial publication in 1990Fractal Geometry: Mathematical Foundations and Applicationshas become a seminal text on the mathematics of fractals. Thebook introduces and develops the general theory and applications offractals in a way that is accessible to students and researchersfrom a wide range of disciplines. Fractal Geometry: Mathematical Foundations andApplications is an excellent course book for undergraduate andgraduate students studying fractal geometry, with suggestions formaterial appropriate for a first course indicated. The book alsoprovides an invaluable foundation and reference for researchers whoencounter fractals not only in mathematics but also in other areasacross physics, engineering and the applied sciences. Provides a comprehensive and accessible introduction to themathematical theory and applications offractals Carefully explains each topic using illustrative examples anddiagrams Includes the necessary mathematical background material, alongwith notes and references to enable the reader to pursue individualtopics Features a wide range of exercises, enabling readers toconsolidate their understanding Supported by a website with solutions to exercises andadditional material ahref="" Leads onto the more advanced sequel Techniques in FractalGeometry (also by Kenneth Falconer and available from Wiley)

Fractals and Spectra

Related to Fourier Analysis and Function Spaces

Author: Hans Triebel

Publisher: Springer Science & Business Media

ISBN: 3034800339

Category: Mathematics

Page: 272

View: 9944

This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators. It investigates distributions of eigenvalues of fractal (pseudo)differential operators.

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: 307

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

Fraktalgeometrie: Selbstähnlichkeit und fraktale Dimension –

Author: Jutta Otterbein,Christina Sawatzki

Publisher: GRIN Verlag

ISBN: 3656262365

Category: Mathematics

Page: 20

View: 2747

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 Architecture and Design

Author: Carl Bovill

Publisher: Springer Science & Business Media

ISBN: 1461208432

Category: Mathematics

Page: 195

View: 3649

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 Geometry in Biological Systems

An Analytical Approach

Author: Philip M. Iannaccone,Mustafa Khokha

Publisher: CRC Press

ISBN: 9780849376368

Category: Science

Page: 384

View: 9807

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.

Fractals in Multimedia

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

Publisher: Springer Science & Business Media

ISBN: 9780387955216

Category: Computers

Page: 270

View: 9376

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.

Fractals and Fractional Calculus in Continuum Mechanics

Author: Alberto Carpinteri,Francesco Mainardi

Publisher: Springer

ISBN: 3709126649

Category: Technology & Engineering

Page: 348

View: 553

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.

Fractal Architecture

Organic Design Philosophy in Theory and Practice

Author: James Harris

Publisher: UNM Press

ISBN: 0826352022

Category: Architecture

Page: 420

View: 5428

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: 9401722625

Category: Science

Page: 322

View: 529

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.

The Fractal Geometry of the Brain

Author: Antonio Di Ieva

Publisher: Springer

ISBN: 1493939955

Category: Medical

Page: 585

View: 5952

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.

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: 4219

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.

Fractal Speech Processing

Author: Marwan Al-Akaidi

Publisher: Cambridge University Press

ISBN: 9780521814584

Category: Computers

Page: 214

View: 1462

Although widely employed in image processing, the use of fractal techniques and the fractal dimension for speech characterisation and recognition is a relatively new concept which is now receiving serious attention. This book represents the fruit of research carried out to develop novel fractal-based techniques for speech and audio signal processing. Much of this work is finding its way into practical commercial applications with Nokia Communications and other key organisations. The book starts with an introduction to speech processing and fractal geometry, setting the scene for the heart of the book where fractal techniques are described in detail with numerous applications and examples, and concluding with a chapter summing up the advantages and potential of these new techniques over conventional processing methods. A valuable reference for researchers, academics and practising engineers working in the field of audio signal processing and communications.

Fractals and Chaos

An illustrated course

Author: Paul S. Addison

Publisher: CRC Press

ISBN: 9780750304009

Category: Science

Page: 256

View: 6350

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.

Wissenschaftliches Rechnen mit MATLAB

Author: Alfio Quarteroni,Fausto Saleri

Publisher: Springer-Verlag

ISBN: 3540293078

Category: Mathematics

Page: 269

View: 3484

Aus den Rezensionen der englischen Auflage: Dieses Lehrbuch ist eine Einführung in das Wissenschaftliche Rechnen und diskutiert Algorithmen und deren mathematischen Hintergrund. Angesprochen werden im Detail nichtlineare Gleichungen, Approximationsverfahren, numerische Integration und Differentiation, numerische Lineare Algebra, gewöhnliche Differentialgleichungen und Randwertprobleme. Zu den einzelnen Themen werden viele Beispiele und Übungsaufgaben sowie deren Lösung präsentiert, die durchweg in MATLAB formuliert sind. Der Leser findet daher nicht nur die graue Theorie sondern auch deren Umsetzung in numerischen, in MATLAB formulierten Code. MATLAB select 2003, Issue 2, p. 50. [Die Autoren] haben ein ausgezeichnetes Werk vorgelegt, das MATLAB vorstellt und eine sehr nützliche Sammlung von MATLAB Funktionen für die Lösung fortgeschrittener mathematischer und naturwissenschaftlicher Probleme bietet. [...] Die Präsentation des Stoffs ist durchgängig gut und leicht verständlich und beinhaltet Lösungen für die Übungen am Ende jedes Kapitels. Als exzellenter Neuzugang für Universitätsbibliotheken- und Buchhandlungen wird dieses Buch sowohl beim Selbststudium als auch als Ergänzung zu anderen MATLAB-basierten Büchern von großem Nutzen sein. Alles in allem: Sehr empfehlenswert. Für Studenten im Erstsemester wie für Experten gleichermassen. S.T. Karris, University of California, Berkeley, Choice 2003.

Lectures on Fractal Geometry and Dynamical Systems

Author: Ya. B. Pesin,Vaughn Climenhaga

Publisher: American Mathematical Soc.

ISBN: 0821848895

Category: Mathematics

Page: 314

View: 3753

Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular 'chaotic' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory - Cantor sets, Hausdorff dimension, box dimension - using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science - the FitzHugh - Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.


Mathematical Perspective and Fractal Geometry in Art

Author: Marc Frantz,Annalisa Crannell

Publisher: Princeton University Press

ISBN: 9781400839056

Category: Mathematics

Page: 248

View: 4239

An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students. Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery. Classroom-tested activities and problem solving Accessible problems that move beyond regular art school curriculum Multiple solutions of varying difficulty and applicability Appropriate for students of all mathematics and art levels Original and exclusive essays by contemporary artists Forthcoming: Instructor's manual (available only to teachers)