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Fractal Geometry and Number Theory ebook

by Machiel van Frankenhuysen,Michel L. Lapidus


In this book, we restrict ourselves to the one-dimensional case of fractal . A Zeta Functions in Number Theory.

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. 221. B Zeta Functions of Laplacians and Spectral Asymptotics. Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings.

ISBN-13: 978-0817640989.

Includes a geometric reformulation of the Riemann hypothesis and a new final chapter on recent topics and results.

Van Frankenhuijsen, Machiel, The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators, London Mathematical Society Student Texts (Book 80), Cambridge University Press (January 9, 2014), 162 pp.

eBook n/a. ISBN 978-1-4612-5314-3.

This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.

Usually ships within 4 to 5 days.

Fractal Geometry and Number Theory. Number theory and fractal geometry are combined in this study of the vibrations of fractal strings. The book centres around a notion of complex dimension, originally developed for the proof of th. More).

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.
Fractal Geometry and Number Theory ebook
Author:
Machiel van Frankenhuysen,Michel L. Lapidus
Category:
Mathematics
Subcat:
EPUB size:
1153 kb
FB2 size:
1504 kb
DJVU size:
1958 kb
Language:
Publisher:
Birkhäuser; 1999 edition (December 10, 1999)
Pages:
280 pages
Rating:
4.7
Other formats:
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