A Tale Of Two Fractals

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A Tale of Two Fractals

A.A. Kirillov
A Tale of Two Fractals Book PDF

Author : A.A. Kirillov
Publisher : Springer Science & Business Media
Page : 148 pages
File Size : 42,5 Mb
Release : 2013-04-23
Category : Mathematics
ISBN : 9780817683825

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A Tale of Two Fractals by A.A. Kirillov Pdf

Since Benoit Mandelbrot's pioneering work in the late 1970s, scores of research articles and books have been published on the topic of fractals. Despite the volume of literature in the field, the general level of theoretical understanding has remained low; most work is aimed either at too mainstream an audience to achieve any depth or at too specialized a community to achieve widespread use. Written by celebrated mathematician and educator A.A. Kirillov, A Tale of Two Fractals is intended to help bridge this gap, providing an original treatment of fractals that is at once accessible to beginners and sufficiently rigorous for serious mathematicians. The work is designed to give young, non-specialist mathematicians a solid foundation in the theory of fractals, and, in the process, to equip them with exposure to a variety of geometric, analytical, and algebraic tools with applications across other areas.

The Butterfly in the Quantum World

Indubala I Satija
The Butterfly in the Quantum World Book PDF

Author : Indubala I Satija
Publisher : Morgan & Claypool Publishers
Page : 350 pages
File Size : 50,5 Mb
Release : 2016-09-06
Category : Science
ISBN : 9781681741178

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The Butterfly in the Quantum World by Indubala I Satija Pdf

Butterfly in the Quantum World by Indu Satija, with contributions by Douglas Hofstadter, is the first book ever to tell the story of the "Hofstadter butterfly", a beautiful and fascinating graph lying at the heart of the quantum theory of matter. The butterfly came out of a simple-sounding question: What happens if you immerse a crystal in a magnetic field? What energies can the electrons take on? From 1930 onwards, physicists struggled to answer this question, until 1974, when graduate student Douglas Hofstadter discovered that the answer was a graph consisting of nothing but copies of itself nested down infinitely many times. This wild mathematical object caught the physics world totally by surprise, and it continues to mesmerize physicists and mathematicians today. The butterfly plot is intimately related to many other important phenomena in number theory and physics, including Apollonian gaskets, the Foucault pendulum, quasicrystals, the quantum Hall effect, and many more. Its story reflects the magic, the mystery, and the simplicity of the laws of nature, and Indu Satija, in a wonderfully personal style, relates this story, enriching it with a vast number of lively historical anecdotes, many photographs, beautiful visual images, and even poems, making her book a great feast, for the eyes, for the mind and for the soul.

Geometry of Möbius Transformations

Vladimir V. Kisil
Geometry of M bius Transformations Book PDF

Author : Vladimir V. Kisil
Publisher : World Scientific
Page : 207 pages
File Size : 46,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9781848168589

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Geometry of Möbius Transformations by Vladimir V. Kisil Pdf

This book is a unique exposition of rich and inspiring geometries associated with Möbius transformations of the hypercomplex plane. The presentation is self-contained and based on the structural properties of the group SL2(R). Starting from elementary facts in group theory, the author unveils surprising new results about the geometry of circles, parabolas and hyperbolas, using an approach based on the Erlangen programme of F Klein, who defined geometry as a study of invariants under a transitive group action.The treatment of elliptic, parabolic and hyperbolic Möbius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers which represent all non-isomorphic commutative associative two-dimensional algebras with unit. The hypercomplex numbers are in perfect correspondence with the three types of geometries concerned. Furthermore, connections with the physics of Minkowski and Galilean space-time are considered.

Fractal Geometry and Stochastics VI

Uta Freiberg,Ben Hambly,Michael Hinz,Steffen Winter
Fractal Geometry and Stochastics VI Book PDF

Author : Uta Freiberg,Ben Hambly,Michael Hinz,Steffen Winter
Publisher : Springer Nature
Page : 307 pages
File Size : 48,5 Mb
Release : 2021-03-23
Category : Mathematics
ISBN : 9783030596491

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Fractal Geometry and Stochastics VI by Uta Freiberg,Ben Hambly,Michael Hinz,Steffen Winter Pdf

This collection of contributions originates from the well-established conference series "Fractal Geometry and Stochastics" which brings together researchers from different fields using concepts and methods from fractal geometry. Carefully selected papers from keynote and invited speakers are included, both discussing exciting new trends and results and giving a gentle introduction to some recent developments. The topics covered include Assouad dimensions and their connection to analysis, multifractal properties of functions and measures, renewal theorems in dynamics, dimensions and topology of random discrete structures, self-similar trees, p-hyperbolicity, phase transitions from continuous to discrete scale invariance, scaling limits of stochastic processes, stemi-stable distributions and fractional differential equations, and diffusion limited aggregation. Representing a rich source of ideas and a good starting point for more advanced topics in fractal geometry, the volume will appeal to both established experts and newcomers.

Advances in Applied Analysis

Sergei V. Rogosin,Anna A. Koroleva
Advances in Applied Analysis Book PDF

Author : Sergei V. Rogosin,Anna A. Koroleva
Publisher : Springer Science & Business Media
Page : 260 pages
File Size : 45,5 Mb
Release : 2012-08-21
Category : Mathematics
ISBN : 9783034804172

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Advances in Applied Analysis by Sergei V. Rogosin,Anna A. Koroleva Pdf

This book contains survey papers based on the lectures presented at the 3rd International Winter School “Modern Problems of Mathematics and Mechanics” held in January 2010 at the Belarusian State University, Minsk. These lectures are devoted to different problems of modern analysis and its applications. An extended presentation of modern problems of applied analysis will enable the reader to get familiar with new approaches of mostly interdisciplinary character. The results discussed are application oriented and present new insight into applied problems of growing importance such as applications to composite materials, anomalous diffusion, and fluid dynamics.

Topics in Clifford Analysis

Swanhild Bernstein
Topics in Clifford Analysis Book PDF

Author : Swanhild Bernstein
Publisher : Springer Nature
Page : 503 pages
File Size : 41,8 Mb
Release : 2019-10-15
Category : Mathematics
ISBN : 9783030238544

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Topics in Clifford Analysis by Swanhild Bernstein Pdf

Quaternionic and Clifford analysis are an extension of complex analysis into higher dimensions. The unique starting point of Wolfgang Sprößig’s work was the application of quaternionic analysis to elliptic differential equations and boundary value problems. Over the years, Clifford analysis has become a broad-based theory with a variety of applications both inside and outside of mathematics, such as higher-dimensional function theory, algebraic structures, generalized polynomials, applications of elliptic boundary value problems, wavelets, image processing, numerical and discrete analysis. The aim of this volume is to provide an essential overview of modern topics in Clifford analysis, presented by specialists in the field, and to honor the valued contributions to Clifford analysis made by Wolfgang Sprößig throughout his career.

A Concise Introduction to Hypercomplex Fractals

Andrzej Katunin
A Concise Introduction to Hypercomplex Fractals Book PDF

Author : Andrzej Katunin
Publisher : CRC Press
Page : 91 pages
File Size : 50,5 Mb
Release : 2017-10-05
Category : Computers
ISBN : 9781351801218

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A Concise Introduction to Hypercomplex Fractals by Andrzej Katunin Pdf

This book presents concisely the full story on complex and hypercomplex fractals, starting from the very first steps in complex dynamics and resulting complex fractal sets, through the generalizations of Julia and Mandelbrot sets on a complex plane and the Holy Grail of the fractal geometry – a 3D Mandelbrot set, and ending with hypercomplex, multicomplex and multihypercomplex fractal sets which are still under consideration of scientists. I tried to write this book in a possibly simple way in order to make it understandable to most people whose math knowledge covers the fundamentals of complex numbers only. Moreover, the book is full of illustrations of generated fractals and stories concerned with great mathematicians, number spaces and related fractals. In the most cases only information required for proper understanding of a nature of a given vector space or a construction of a given fractal set is provided, nevertheless a more advanced reader may treat this book as a fundamental compendium on hypercomplex fractals with references to purely scientific issues like dynamics and stability of hypercomplex systems.

Assouad Dimension and Fractal Geometry

Jonathan M. Fraser
Assouad Dimension and Fractal Geometry Book PDF

Author : Jonathan M. Fraser
Publisher : Cambridge University Press
Page : 287 pages
File Size : 46,7 Mb
Release : 2020-10-29
Category : Mathematics
ISBN : 9781108478656

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Assouad Dimension and Fractal Geometry by Jonathan M. Fraser Pdf

The first thorough treatment of the Assouad dimension in fractal geometry, with applications to many fields within pure mathematics.

Introduction to Fractals and Chaos

Richard M. Crownover
Introduction to Fractals and Chaos Book PDF

Author : Richard M. Crownover
Publisher : Jones & Bartlett Publishers
Page : 328 pages
File Size : 52,8 Mb
Release : 1995
Category : Mathematics
ISBN : UOM:39015039020527

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Introduction to Fractals and Chaos by Richard M. Crownover Pdf

Physics / Quantum Physics

Fractals and Chaos

A.J. Crilly,Rae Earnshaw,Huw Jones
Fractals and Chaos Book PDF

Author : A.J. Crilly,Rae Earnshaw,Huw Jones
Publisher : Springer Science & Business Media
Page : 278 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461230342

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Fractals and Chaos by A.J. Crilly,Rae Earnshaw,Huw Jones Pdf

This volume is based upon the presentations made at an international conference in London on the subject of 'Fractals and Chaos'. The objective of the conference was to bring together some of the leading practitioners and exponents in the overlapping fields of fractal geometry and chaos theory, with a view to exploring some of the relationships between the two domains. Based on this initial conference and subsequent exchanges between the editors and the authors, revised and updated papers were produced. These papers are contained in the present volume. We thank all those who contributed to this effort by way of planning and organisation, and also all those who helped in the production of this volume. In particular, we wish to express our appreciation to Gerhard Rossbach, Computer Science Editor, Craig Van Dyck, Production Director, and Nancy A. Rogers, who did the typesetting. A. J. Crilly R. A. Earnshaw H. Jones 1 March 1990 Introduction Fractals and Chaos The word 'fractal' was coined by Benoit Mandelbrot in the late 1970s, but objects now defined as fractal in form have been known to artists and mathematicians for centuries. Mandelbrot's definition-"a set whose Hausdorff dimension is not an integer" -is clear in mathematical terms. In addition, related concepts are those of self-similarity and sub-divisibility. A fractal object is self-similar in that subsections of the object are similar in some sense to the whole object.

Women's International Thought: A New History

Patricia Owens,Katharina Rietzler
Women s International Thought A New History Book PDF

Author : Patricia Owens,Katharina Rietzler
Publisher : Cambridge University Press
Page : 371 pages
File Size : 42,6 Mb
Release : 2021-01-07
Category : History
ISBN : 9781108494694

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Women's International Thought: A New History by Patricia Owens,Katharina Rietzler Pdf

The first cross-disciplinary history of women's international thought, analysing leading international thinkers of the twentieth century.

Strange Attractors

Julien C. Sprott
Strange Attractors Book PDF

Author : Julien C. Sprott
Publisher : M & T Books
Page : 426 pages
File Size : 49,9 Mb
Release : 1993
Category : Computers
ISBN : 1558512985

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Strange Attractors by Julien C. Sprott Pdf

Chaos and fractals are new mathematical ideas that have revolutionized our view of the world. They have application in virtually every academic discipline. This book shows examples of the artistic beauty that can arise from very simple equations, and teaches the reader how to produce an endless variety of such patterns. Disk includes a full working version of the program.

Getting Acquainted with Fractals

Gilbert Helmberg
Getting Acquainted with Fractals Book PDF

Author : Gilbert Helmberg
Publisher : Walter de Gruyter
Page : 190 pages
File Size : 47,5 Mb
Release : 2007
Category : Mathematics
ISBN : 3110190923

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Getting Acquainted with Fractals by Gilbert Helmberg Pdf

The first instance of pre-computer fractals was noted by the French mathematician Gaston Julia. He wondered what a complex polynomial function would look like, such as the ones named after him (in the form of z2 + c, where c is a complex constant with real and imaginary parts). The idea behind this formula is that one takes the x and y coordinates of a point z, and plug them into z in the form of x + i*y, where i is the square root of -1, square this number, and then add c, a constant. Then plug the resulting pair of real and imaginary numbers back into z, run the operation again, and keep doing that until the result is greater than some number. The number of times you have to run the equations to get out of an 'orbit' not specified here can be assigned a colour and then the pixel (x, y) gets turned that colour, unless those coordinates can't get out of their orbit, in which case they are made black. Later it was Benoit Mandelbrot who used computers to produce fractals. A basic property of fractals is that they contain a large degree of self similarity, i.e., they usually contain little copies within the original, and these copies also have infinite detail. That means the more you zoom in on a fractal, the more detail you get, and this keeps going on forever and ever. The well-written book 'Getting acquainted with fractals' by Gilbert Helmberg provides a mathematically oriented introduction to fractals, with a focus upon three types of fractals: fractals of curves, attractors for iterative function systems in the plane, and Julia sets. The presentation is on an undergraduate level, with an ample presentation of the corresponding mathematical background, e.g., linear algebra, calculus, algebra, geometry, topology, measure theory and complex analysis. The book contains over 170 color illustrations.

The Fractalist

Benoit Mandelbrot
The Fractalist Book PDF

Author : Benoit Mandelbrot
Publisher : Vintage
Page : 362 pages
File Size : 53,9 Mb
Release : 2014-01-14
Category : Biography & Autobiography
ISBN : 9780307389916

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The Fractalist by Benoit Mandelbrot Pdf

Here is the remarkable life story of Benoit Mandelbrot, the creator of fractal geometry, and his unparalleled contributions to science mathematics, the financial world, and the arts. Mandelbrot recounts his early years in Warsaw and in Paris, where he was mentored by an eminent mathematician uncle, through his days evading the Nazis in occupied France, to his education at Caltech, Princeton, and MIT, and his illustrious career at the IBM Thomas J. Watson Research Center. An outside to mainstream scientific research, he managed to do what others had thought impossible: develop a new geometry that combines revelatory beauty with a radical way of unfolding formerly hidden scientific laws. In the process he was able to use geometry to solve fresh, real-world problems. With exuberance and an eloquent fluency, Benoit Mandelbrot recounts the high points of his fascinating life, offering us a glimpse into the evolution of his extraordinary mind. With full-color inserts and black-and-white photographs throughout.

Classics On Fractals

Gerald A. Edgar
Classics On Fractals Book PDF

Author : Gerald A. Edgar
Publisher : CRC Press
Page : 384 pages
File Size : 40,8 Mb
Release : 2019-03-08
Category : Science
ISBN : 9780429711237

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Classics On Fractals by Gerald A. Edgar Pdf

Read the masters! Experience has shown that this is good advice for the serious mathematics student. This book contains a selection of the classical mathematical papers related to fractal geometry. For the convenience of the student or scholar wishing to learn about fractal geometry, nineteen of these papers are collected here in one place. Twelve of the nineteen have been translated into English from German, French, or Russian. In many branches of science, the work of previous generations is of interest only for historical reasons. This is much less so in mathematics.1 Modern-day mathematicians can learn (and even find good ideas) by reading the best of the papers of bygone years. In preparing this volume, I was surprised by many of the ideas that come up.