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El. knyga: Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

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  • Formatas: PDF+DRM
  • Išleidimo metai: 01-Dec-2013
  • Leidėjas: Birkhauser Boston Inc
  • Kalba: eng
  • ISBN-13: 9781461253143
Kitos knygos pagal šią temą:
Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta FunctionsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 01-Dec-2013
  • Leidėjas: Birkhauser Boston Inc
  • Kalba: eng
  • ISBN-13: 9781461253143
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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.

Recenzijos

"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 reviewers 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)



 



 

Overview ix
Introduction 1(6)
Complex Dimensions of Ordinary Fractal Strings
7(16)
The Geometry of a Fractal String
7(6)
The Multiplicity of the Lengths
10(1)
Example: The Cantor String
11(2)
The Geometric Zeta Function of a Fractal String
13(4)
The Screen and the Window
14(2)
The Cantor String (Continued)
16(1)
The Frequencies of a Fractal String and the Spectral Zeta Function
17(2)
Higher-Dimensional Analogue: Fractal Sprays
19(4)
Complex Dimensions of Self-Similar Fractal Strings
23(32)
The Geometric Zeta Function of a Self-Similar String
23(5)
Dynamical Interpretation, Euler Product
26(2)
Examples of Complex Dimensions of Self-Similar Strings
28(6)
The Cantor String
28(1)
The Fibonacci String
28(2)
A String with Multiple Poles
30(1)
Two Nonlattice Examples
31(3)
The Lattice and Nonlattice Case
34(3)
Generic Nonlattice Strings
36(1)
The Structure of the Complex Dimensions
37(5)
The Density of the Poles in the Nonlattice Case
42(5)
Nevanlinna Theory
42(1)
Complex Zeros of Dirichlet Polynomials
43(4)
Approximating a Fractal String and Its Complex Dimensions
47(8)
Approximating a Nonlattice String by Lattice Strings
49(6)
Generalized Fractal Strings Viewed as Measures
55(16)
Generalized Fractal Strings
55(5)
Examples of Generalized Fractal Strings
58(2)
The Frequencies of a Generalized Fractal String
60(4)
Generalized Fractal Sprays
64(1)
The Measure of a Self-Similar String
65(6)
Measures with a Self-Similarity Property
67(4)
Explicit Formulas for Generalized Fractal Strings
71(40)
Introduction
71(5)
Outline of the Proof
73(1)
Examples
74(2)
Preliminaries: The Heaviside Function
76(3)
The Pointwise Explicit Formulas
79(11)
The Order of the Sum over the Complex Dimensions
89(1)
The Distributional Explicit Formulas
90(16)
Alternate Proof of Theorem 4.12
95(1)
Extension to More General Test Functions
95(4)
The Order of the Distributional Error Term
99(7)
Example: The Prime Number Theorem
106(5)
The Riemann---von Mangoldt Formula
108(3)
The Geometry and the Spectrum of Fractal Strings
111(32)
The Local Terms in the Explicit Formulas
112(3)
The Geometric Local Terms
112(1)
The Spectral Local Terms
113(1)
The Weyl Term
114(1)
The Distribution xw logmx
114(1)
Explicit Formulas for Lengths and Frequencies
115(4)
The Geometric Counting Function of a Fractal String
115(1)
The Spectral Counting Function of a Fractal String
116(2)
The Geometric and Spectral Partition Functions
118(1)
The Direct Spectral Problem for Fractal Strings
119(2)
The Density of Geometric and Spectral States
119(2)
The Spectral Operator
121(1)
Self-Similar Strings
121(11)
Lattice Strings
122(4)
Nonlattice Strings
126(1)
The Spectrum of a Self-Similar String
127(3)
The Prime Number Theorem for Suspended Flows
130(2)
Examples of Non-Self-Similar Strings
132(4)
The a-String
133(3)
The Spectrum of the Harmonic String
136(1)
Fractal Sprays
136(7)
The Sierpinski Drum
138(3)
The Spectrum of a Self-Similar Spray
141(2)
Tubular Neighborhoods and Minkowski Measurability
143(20)
Explicit Formula for the Volume of a Tubular Neighborhood
144(4)
Analogy with Riemannian Geometry
147(1)
Minkowski Measurability and Complex Dimensions
148(5)
Examples
153(10)
Self-Similar Strings
154(6)
The a-String
160(3)
The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena
163(10)
The Inverse Spectral Problem
163(4)
Complex Dimensions of Fractal Strings and the Riemann Hypothesis
167(3)
Fractal Sprays and the Generalized Riemann Hypothesis
170(3)
Generalized Cantor Strings and their Oscillations
173(8)
The Geometry of a Generalized Cantor String
173(3)
The Spectrum of a Generalized Cantor String
176(5)
Integral Cantor Strings: a-adic Analysis of the Geometric and Spectral Oscillations
176(3)
Nonintegral Cantor Strings: Analysis of the Jumps in the Spectral Counting Function
179(2)
The Critical Zeros of Zeta Functions
181(16)
The Riemann Zeta Function: No Critical Zeros in an Arithmetic Progression
182(2)
Extension to Other Zeta Functions
184(4)
Density of Nonzeros on Vertical Lines
186(1)
Almost Arithmetic Progressions of Zeros
187(1)
Extension to L-Series
188(1)
Zeta Functions of Curves Over Finite Fields
189(8)
Concluding Comments
197(38)
Conjectures about Zeros of Dirichlet Series
198(3)
A New Definition of Fractality
201(7)
Comparison with Other Definitions of Fractality
205(1)
Possible Connections with the Notion of Lacunarity
206(2)
Fractality and Self-Similarity
208(4)
The Spectrum of a Fractal Drum
212(7)
The Weyl--Berry Conjecture
212(2)
The Spectrum of a Self-Similar Drum
214(3)
Spectrum and Periodic Orbits
217(2)
The Complex Dimensions as Geometric Invariants
219(2)
Appendices
A Zeta Functions in Number Theory
221(6)
A.1 The Dedekind Zeta Function
221(1)
A.2 Characters and Hecke L-series
222(1)
A.3 Completion of L-Series, Functional Equation
223(1)
A.4 Epstein Zeta Functions
224(1)
A.5 Other Zeta Functions in Number Theory
225(2)
B Zeta Functions of Laplacians and Spectral Asymptotics
227(8)
B.1 Weyl's Asymptotic Formula
227(2)
B.2 Heat Asymptotic Expansion
229(2)
B.3 The Spectral Zeta Function and Its Poles
231(1)
B.4 Extensions
232(1)
B.4.1 Monotonic Second Term
233(2)
References 235(18)
Conventions 253(1)
Symbol Index 254(3)
Index 257(8)
List of Figures 265(2)
Acknowledgements 267
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