Fractal Geometry and Number Theory - Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Hardcover, 1999 ed.)

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.

General

Imprint:	Birkhauser Boston
Country of origin:	United States
Release date:	December 1999
First published:	2000
Authors:	Michel L Lapidus • Machiel Van Frankenhuysen
Dimensions:	235 x 155 x 29mm (L x W x T)
Format:	Hardcover
Pages:	268
Edition:	1999 ed.
ISBN-13:	978-0-8176-4098-9
Categories:	Books > Science & Mathematics > Mathematics > Geometry > General Books > Science & Mathematics > Mathematics > Applied mathematics > Fractals
LSN:	0-8176-4098-3
Barcode:	9780817640989

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