From Classical Analysis To Analysis On Fractals

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From Classical Analysis to Analysis on Fractals

Patricia Alonso Ruiz,Michael Hinz,Kasso A. Okoudjou,Luke G. Rogers,Alexander Teplyaev
From Classical Analysis to Analysis on Fractals Book PDF

Author : Patricia Alonso Ruiz,Michael Hinz,Kasso A. Okoudjou,Luke G. Rogers,Alexander Teplyaev
Publisher : Springer Nature
Page : 294 pages
File Size : 54,5 Mb
Release : 2023-11-25
Category : Mathematics
ISBN : 9783031378003

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From Classical Analysis to Analysis on Fractals by Patricia Alonso Ruiz,Michael Hinz,Kasso A. Okoudjou,Luke G. Rogers,Alexander Teplyaev Pdf

Over the course of his distinguished career, Robert Strichartz (1943-2021) had a substantial impact on the field of analysis with his deep, original results in classical harmonic, functional, and spectral analysis, and in the newly developed analysis on fractals. This is the first volume of a tribute to his work and legacy, featuring chapters that reflect his mathematical interests, written by his colleagues and friends. An introductory chapter summarizes his broad and varied mathematical work and highlights his profound contributions as a mathematical mentor. The remaining articles are grouped into three sections – functional and harmonic analysis on Euclidean spaces, analysis on manifolds, and analysis on fractals – and explore Strichartz’ contributions to these areas, as well as some of the latest developments.

Analysis, Probability And Mathematical Physics On Fractals

Patricia Alonso Ruiz,Joe Po-chou Chen,Luke G Rogers,Alexander Teplyaev
Analysis Probability And Mathematical Physics On Fractals Book PDF

Author : Patricia Alonso Ruiz,Joe Po-chou Chen,Luke G Rogers,Alexander Teplyaev
Publisher : World Scientific
Page : 594 pages
File Size : 55,9 Mb
Release : 2020-02-26
Category : Mathematics
ISBN : 9789811215544

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Analysis, Probability And Mathematical Physics On Fractals by Patricia Alonso Ruiz,Joe Po-chou Chen,Luke G Rogers,Alexander Teplyaev Pdf

In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results.

Fractals in Probability and Analysis

Christopher J. Bishop,Yuval Peres
Fractals in Probability and Analysis Book PDF

Author : Christopher J. Bishop,Yuval Peres
Publisher : Cambridge University Press
Page : 415 pages
File Size : 55,9 Mb
Release : 2017
Category : Mathematics
ISBN : 9781107134119

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Fractals in Probability and Analysis by Christopher J. Bishop,Yuval Peres Pdf

A mathematically rigorous introduction to fractals, emphasizing examples and fundamental ideas while minimizing technicalities.

Recent Developments in Fractals and Related Fields

Julien Barral,Stéphane Seuret
Recent Developments in Fractals and Related Fields Book PDF

Author : Julien Barral,Stéphane Seuret
Publisher : Birkhäuser
Page : 312 pages
File Size : 41,7 Mb
Release : 2017-08-23
Category : Mathematics
ISBN : 9783319578057

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Recent Developments in Fractals and Related Fields by Julien Barral,Stéphane Seuret Pdf

This contributed volume provides readers with an overview of the most recent developments in the mathematical fields related to fractals, including both original research contributions, as well as surveys from many of the leading experts on modern fractal theory and applications. It is an outgrowth of the Conference of Fractals and Related Fields III, that was held on September 19-25, 2015 in île de Porquerolles, France. Chapters cover fields related to fractals such as harmonic analysis, multifractal analysis, geometric measure theory, ergodic theory and dynamical systems, probability theory, number theory, wavelets, potential theory, partial differential equations, fractal tilings, combinatorics, and signal and image processing. The book is aimed at pure and applied mathematicians in these areas, as well as other researchers interested in discovering the fractal domain.

Frontiers of Fractal Analysis

Santo Banerjee,A. Gowrisankar
Frontiers of Fractal Analysis Book PDF

Author : Santo Banerjee,A. Gowrisankar
Publisher : CRC Press
Page : 182 pages
File Size : 41,8 Mb
Release : 2022-07-07
Category : Mathematics
ISBN : 9781000625875

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Frontiers of Fractal Analysis by Santo Banerjee,A. Gowrisankar Pdf

The history of describing natural objects using geometry is as old as the advent of science itself, in which traditional shapes are the basis of our intuitive understanding of geometry. However, nature is not restricted to such Euclidean objects which are only characterized typically by integer dimensions. Hence, the conventional geometric approach cannot meet the requirements of solving or analysing nonlinear problems which are related with natural phenomena, therefore, the fractal theory has been born, which aims to understand complexity and provide an innovative way to recognize irregularity and complex systems. Although the concepts of fractal geometry have found wide applications in many forefront areas of science, engineering and societal issues, they also have interesting implications of a more practical nature for the older classical areas of science. Since its discovery, there has been a surge of research activities in using this powerful concept in almost every branch of scientific disciplines to gain deep insights into many unresolved problems. This book includes eight chapters which focus on gathering cutting-edge research and proposing application of fractals features in both traditional scientific disciplines and in applied fields.

Fractal-Based Methods in Analysis

Herb Kunze,Davide La Torre,Franklin Mendivil,Edward R. Vrscay
Fractal Based Methods in Analysis Book PDF

Author : Herb Kunze,Davide La Torre,Franklin Mendivil,Edward R. Vrscay
Publisher : Springer Science & Business Media
Page : 417 pages
File Size : 46,7 Mb
Release : 2011-11-18
Category : Mathematics
ISBN : 9781461418917

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Fractal-Based Methods in Analysis by Herb Kunze,Davide La Torre,Franklin Mendivil,Edward R. Vrscay Pdf

The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems. "Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters. For each topic the theoretical framework is carefully explained using examples and applications. The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises. This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences. Herb Kunze is a professor of mathematics at the University of Guelph in Ontario. Davide La Torre is an associate professor of mathematics in the Department of Economics, Management and Quantitative Methods of the University of Milan. Franklin Mendivil is a professor of mathematics at Acadia University in Nova Scotia. Edward Vrscay is a professor in the department of Applied Mathematics at the University of Waterloo in Ontario. The major focus of their research is on fractals and the applications of fractals.

The Beauty of Fractals

Denny Gulick,Jon Scott
The Beauty of Fractals Book PDF

Author : Denny Gulick,Jon Scott
Publisher : MAA
Page : 107 pages
File Size : 41,8 Mb
Release : 2010
Category : Mathematics
ISBN : 9780883859711

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The Beauty of Fractals by Denny Gulick,Jon Scott Pdf

The Beauty of Fractals includes six essays related to fractals, with perspectives different enough to give you a taste of the breadth of the subject. Each essay is self-contained and expository. Moreover, each of the essays is intended to be accessible to a broad audience that includes college teachers, high school teachers, advanced undergraduate students, and others who wish to learn or teach about topics in fractals that are not regularly in textbooks on fractals.

Fractal Analysis

Fernando Brambila
Fractal Analysis Book PDF

Author : Fernando Brambila
Publisher : BoD – Books on Demand
Page : 296 pages
File Size : 54,9 Mb
Release : 2017-06-14
Category : Mathematics
ISBN : 9789535131915

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Fractal Analysis by Fernando Brambila Pdf

Fractal analysis has entered a new era. The applications to different areas of knowledge have been surprising. Let us begin with the fractional calculus-fractal geometry relationship, which allows for modeling with extreme precision of phenomena such as diffusion in porous media with fractional partial differential equations in fractal objects. Where the order of the equation is the same as the fractal dimension, this allows us to make calculations with enormous precision in diffusion phenomena-particularly in the oil industry, for new spillage prevention. Main applications to industry, design of fractal antennas to receive all frequencies and that is used in all cell phones, spacecraft, radars, image processing, measure, porosity, turbulence, scattering theory. Benoit Mandelbrot, creator of fractal geometry, would have been surprised by the use of fractal analysis presented in this book: "Part I: Petroleum Industry and Numerical Analysis"; "Part II: Fractal Antennas, Spacecraft, Radars, Image Processing, and Measure"; and "Part III: Scattering Theory, Porosity, and Turbulence." It's impossible to picture today's research without fractal analysis.

Recent Developments in Fractals and Related Fields

Julien Barral,Stéphane Seuret
Recent Developments in Fractals and Related Fields Book PDF

Author : Julien Barral,Stéphane Seuret
Publisher : Birkhäuser
Page : 419 pages
File Size : 48,5 Mb
Release : 2010-08-12
Category : Mathematics
ISBN : 0817648879

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Recent Developments in Fractals and Related Fields by Julien Barral,Stéphane Seuret Pdf

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scienti?c communities with s- ni?cant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series re?ects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the int- leaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has ?o- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilizationwith diverse areas. The intricate and f- damental relationship between harmonic analysis and ?elds such as signal processing, partial di?erential equations (PDEs), and image processing is - ?ected in our state-of-the-art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.

Harmonic Analysis And Fractal Analysis Over Local Fields And Applications

Su Weiyi
Harmonic Analysis And Fractal Analysis Over Local Fields And Applications Book PDF

Author : Su Weiyi
Publisher : World Scientific
Page : 332 pages
File Size : 50,8 Mb
Release : 2017-08-17
Category : Mathematics
ISBN : 9789813200524

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Harmonic Analysis And Fractal Analysis Over Local Fields And Applications by Su Weiyi Pdf

This book is a monograph on harmonic analysis and fractal analysis over local fields. It can also be used as lecture notes/textbook or as recommended reading for courses on modern harmonic and fractal analysis. It is as reliable as Fourier Analysis on Local Fields published in 1975 which is regarded as the first monograph in this research field.The book is self-contained, with wide scope and deep knowledge, taking modern mathematics (such as modern algebra, point set topology, functional analysis, distribution theory, and so on) as bases. Specially, fractal analysis is studied in the viewpoint of local fields, and fractal calculus is established by pseudo-differential operators over local fields. A frame of fractal PDE is constructed based on fractal calculus instead of classical calculus. On the other hand, the author does his best to make those difficult concepts accessible to readers, illustrate clear comparison between harmonic analysis on Euclidean spaces and that on local fields, and at the same time provide motivations underlying the new concepts and techniques. Overall, it is a high quality, up to date and valuable book for interested readers.

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

Alexander Grigor'yan,Yuhua Sun
Analysis and Partial Differential Equations on Manifolds Fractals and Graphs Book PDF

Author : Alexander Grigor'yan,Yuhua Sun
Publisher : Walter de Gruyter GmbH & Co KG
Page : 337 pages
File Size : 43,6 Mb
Release : 2021-01-18
Category : Mathematics
ISBN : 9783110700855

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Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs by Alexander Grigor'yan,Yuhua Sun Pdf

The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.

Analysis on Fractals

Jun Kigami
Analysis on Fractals Book PDF

Author : Jun Kigami
Publisher : Cambridge University Press
Page : 238 pages
File Size : 44,6 Mb
Release : 2001-06-07
Category : Mathematics
ISBN : 9780521793216

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Analysis on Fractals by Jun Kigami Pdf

This book covers analysis on fractals, a developing area of mathematics which focuses on the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a material with fractal structure. The book provides a self-contained introduction to the subject, starting from the basic geometry of self-similar sets and going on to discuss recent results, including the properties of eigenvalues and eigenfunctions of the Laplacians, and the asymptotical behaviors of heat kernels on self-similar sets. Requiring only a basic knowledge of advanced analysis, general topology and measure theory, this book will be of value to graduate students and researchers in analysis and probability theory. It will also be useful as a supplementary text for graduate courses covering fractals.

Fractal Geometry and Stochastics IV

Christoph Bandt,Peter Mörters,Martina Zähle
Fractal Geometry and Stochastics IV Book PDF

Author : Christoph Bandt,Peter Mörters,Martina Zähle
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 43,8 Mb
Release : 2010-01-08
Category : Mathematics
ISBN : 9783034600309

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Fractal Geometry and Stochastics IV by Christoph Bandt,Peter Mörters,Martina Zähle Pdf

Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling of complicated structures. This process has been forced by problems in these areas related to applications in statistical physics, biomathematics and finance. This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals. The authors were the keynote speakers at the conference "Fractal Geometry and Stochastics IV" at Greifswald in September 2008.

Fractals in Graz 2001

Peter Grabner,Wolfgang Woess
Fractals in Graz 2001 Book PDF

Author : Peter Grabner,Wolfgang Woess
Publisher : Birkhäuser
Page : 288 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034880145

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Fractals in Graz 2001 by Peter Grabner,Wolfgang Woess Pdf

This book contains the proceedings of the conference "Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics" that was held in the second week of June 2001 at Graz University of Technology, in the capital of Styria, southeastern province of Austria. The scientific committee of the meeting consisted of M. Barlow (Vancouver), R. Strichartz (Ithaca), P. Grabner and W. Woess (both Graz), the latter two being the local organizers and editors of this volume. We made an effort to unite in the conference as well as in the present pro ceedings a multitude of different directions of active current work, and to bring together researchers from various countries as well as research fields that all are linked in some way with the modern theory of fractal structures. Although (or because) in Graz there is only a very small group working on fractal structures, consisting of "non-insiders", we hope to have been successful with this program of wide horizons. All papers were written upon explicit invitation by the editors, and we are happy to be able to present this representative panorama of recent work on poten tial theory, random walks, spectral theory, fractal groups, dynamic systems, fractal geometry, and more. The papers presented here underwent a refereeing process.

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot

Michel Laurent Lapidus,Machiel Van Frankenhuysen
Fractal Geometry and Applications A Jubilee of Benoit Mandelbrot Book PDF

Author : Michel Laurent Lapidus,Machiel Van Frankenhuysen
Publisher : American Mathematical Soc.
Page : 534 pages
File Size : 46,7 Mb
Release : 2004
Category : Ergodic theory
ISBN : 9780821836378

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Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot by Michel Laurent Lapidus,Machiel Van Frankenhuysen Pdf

This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.