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Fractal Geometry and Number Theory : Complex Dimensions of Fractal Strings and Zeros of Zeta Functions Paperback
by Michel L. Lapidus, Machiel van Frankenhuysen
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Description
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.
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- Format:Paperback
- Pages:268 pages, biography
- Publisher:Springer-Verlag New York Inc.
- Publication Date:13/06/2012
- Category:
- ISBN:9781461253167
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Information
-
Unavailable
- Format:Paperback
- Pages:268 pages, biography
- Publisher:Springer-Verlag New York Inc.
- Publication Date:13/06/2012
- Category:
- ISBN:9781461253167