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A Tale of Two Fractals

A Tale of Two Fractals
Author: A.A. Kirillov
Publisher: Springer Science & Business Media
Total Pages: 148
Release: 2013-04-23
Genre: Mathematics
ISBN: 0817683828
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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

The Butterfly in the Quantum World
Author: Indubala I Satija
Publisher: Morgan & Claypool Publishers
Total Pages: 356
Release: 2016-09-06
Genre: Science
ISBN: 1681741814
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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

Geometry of Möbius Transformations
Author: Vladimir V. Kisil
Publisher: World Scientific
Total Pages: 207
Release: 2012
Genre: Mathematics
ISBN: 1848168586
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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

Fractal Geometry and Stochastics VI
Author: Uta Freiberg
Publisher: Springer Nature
Total Pages: 307
Release: 2021-03-23
Genre: Mathematics
ISBN: 3030596494
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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

Advances in Applied Analysis
Author: Sergei V. Rogosin
Publisher: Springer Science & Business Media
Total Pages: 260
Release: 2012-08-21
Genre: Mathematics
ISBN: 3034804172
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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

Topics in Clifford Analysis
Author: Swanhild Bernstein
Publisher: Springer Nature
Total Pages: 503
Release: 2019-10-15
Genre: Mathematics
ISBN: 3030238547
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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.

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry
Author: Jonathan M. Fraser
Publisher: Cambridge University Press
Total Pages: 287
Release: 2020-10-29
Genre: Mathematics
ISBN: 1108478654
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The first thorough treatment of the Assouad dimension in fractal geometry, with applications to many fields within pure mathematics.

Fractals for the Classroom

Fractals for the Classroom
Author: Heinz-Otto Peitgen
Publisher: Springer Science & Business Media
Total Pages: 528
Release: 1992-08-26
Genre: Mathematics
ISBN: 9780387977225
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Published in cooperation with the National Council of Teachers of Mathematics

The Fractalist

The Fractalist
Author: Benoit Mandelbrot
Publisher: Vintage
Total Pages: 362
Release: 2014-01-14
Genre: Biography & Autobiography
ISBN: 030738991X
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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.

Strange Attractors

Strange Attractors
Author: Julien C. Sprott
Publisher: M & T Books
Total Pages: 426
Release: 1993
Genre: Computers
ISBN: 9781558512986
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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.