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Fractal Geometry and Number Theory : Complex Dimensions of Fractal Strings and Zeros of Zeta Functions - Michel L. Lapidus

Fractal Geometry and Number Theory

Complex Dimensions of Fractal Strings and Zeros of Zeta Functions


Published: 10th December 1999
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A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo­ metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di­ mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref­ erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap­ pendix B. ) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex­ tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.

"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 First Edition)

"It is the reviewer'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 First Edition)

"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 First Edition)

Introductionp. 1
Complex Dimensions of Ordinary Fractal Stringsp. 7
Complex Dimensions of Self-Similar Fractal Stringsp. 23
Generalized Fractal Strings Viewed as Measuresp. 55
Explicit Formulas for Generalized Fractal Stringsp. 71
The Geometry and the Spectrum of Fractal Stringsp. 111
Tubular Neighbourhoods and Minkowski Measurabilityp. 143
The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomenap. 163
Generalized Cantor Strings and their Oscillationsp. 173
The Critical Zeros of Zeta Functionsp. 181
Concluding Commentsp. 197
Zeta Functions in Number Theoryp. 221
Zeta Functions of Laplacians and Spectral Asymptoticsp. 227
Referencesp. 235
Conventionsp. 253
Symbol Indexp. 254
Indexp. 257
List of Figuresp. 265
Acknowledgementsp. 267
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780817640989
ISBN-10: 0817640983
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 268
Published: 10th December 1999
Country of Publication: US
Dimensions (cm): 24.77 x 15.88  x 1.91
Weight (kg): 0.57