Michel L. Lapidus, Machiel Van Frankenhuijsen

eBook, PDF

Fractal Geometry, Complex Dimensions and Zeta Functions (eBook, PDF)

Geometry and Spectra of Fractal Strings

Beschreibung

Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. In this book The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings. The book also offers explicit formulas extended to apply to the geometric, spectral, and dynamic zeta functions associated with a fractal. In addition, numerous theorems, examples, remarks and illustrations enrich the text. Throughout the book new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. The book will further appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics.

Technische Details

• Geräte: PC
• ohne Kopierschutz
• Größe: 3.52MB

Barrierefreiheit

• Text-to-Speech

Produktdetails

• Verlag: Springer US
• Seitenzahl: 460
• Erscheinungstermin: 8. August 2007
• Englisch
• ISBN-13: 9780387352084
• Artikelnr.: 43730215

Rezensionen

"This interesting volume gives a thorough introduction to an active field of research and will be very valuable to graduate students and researchers alike." (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)

"In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. ... The book is written in a self-contained manner the results ... are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory." (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)

"The

authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. ... The book is written in a self-contained manner, the results (including some fundamental ones) are completely proved. ... the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory." (Nicolae-Adrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)

"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH

"This highly original self-contained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields andothers interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." -- Mathematical Reviews (Review of previous book by authors)

"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is self-contained, intelligent and well paced." -- Bulletin of the London Mathematical Society (Review of previous book by authors)

"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics." -- Simulation News Europe (Review of previous book by authors)

From the reviews:

"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH

"This highly original self-contained book will appeal to geometers, fractalists, mathematical physicists and number theorists,

as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." -- Mathematical Reviews (Review of previous book by authors)

"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is self-contained, intelligent and well paced." -- Bulletin of the London Mathematical Society (Review of previous book by authors)

"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics." --  Simulation News Europe (Review of previous book by authors)

From the reviews of the second edition:

"In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. ... The book is written in a self-contained manner the results ... are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory." (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)

"The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. ... The book is written in a self-contained manner, the results (including some fundamental ones) are completely proved. ... the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory." (Nicolae-Adrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)