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El. knyga: Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

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Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Numerous theorems, examples, remarks and illustrations enrich the text.

Number theory, spectral geometry, and fractal geometry are interlinked in this study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings, and the book offers explicit formulas extended to apply to the geometric, spectral and dynamic zeta functions associated with a fractal.

Recenzijos

Review of the First Edition: "In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH "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 previous book by authors) "It is the reviewera (TM)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 previous book by authors) "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 previous book by authors)

Preface xi
List of Figures xv
List of Tables xix
Overview xxi
Introduction 1(278)
1 Complex Dimensions of Ordinary Fractal Strings
9(24)
1.1 The Geometry of a Fractal String
9(7)
1.1.1 The Multiplicity of the Lengths
12(1)
1.1.2 Example: The Cantor String
13(3)
1.2 The Geometric Zeta Function of a Fractal String
16(7)
1.2.1 The Screen and the Window
18(4)
1.2.2 The Cantor String (continued)
22(1)
1.3 The Frequencies of a Fractal String and the Spectral Zeta Function
23(3)
1.4 Higher-Dimensional Analogue: Fractal Sprays
26(3)
1.5 Notes
29(4)
2 Complex Dimensions of Self-Similar Fractal Strings
33(30)
2.1 Construction of a Self-Similar Fractal String
33(5)
2.1.1 Relation with Self-Similar Sets
35(3)
2.2 The Geometric Zeta Function of a Self-Similar String
38(3)
2.2.1 Self-Similar Strings with a Single Gap
40(1)
2.3 Examples of Complex Dimensions of Self-Similar Strings
41(10)
2.3.1 The Cantor String
41(2)
2.3.2 The Fibonacci String
43(3)
2.3.3 The Modified Cantor and Fibonacci Strings
46(1)
2.3.4 A String with Multiple Poles
47(1)
2.3.5 Two Nonlattice Examples: the Two-Three String and the Golden String
47(4)
2.4 The Lattice and Nonlattice Case
51(3)
2.5 The Structure of the Complex Dimensions
54(7)
2.6 The Asymptotic Density of the Poles in the Nonlattice Case
61(1)
2.7 Notes
62(1)
3 Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation
63(52)
3.1 Dirichlet Polynomial Equations
64(2)
3.1.1 The Generic Nonlattice Case
65(1)
3.2 Examples of Dirichlet Polynomial Equations
66(5)
3.2.1 Generic and Nongeneric Nonlattice Equations
66(5)
3.2.2 The Complex Roots of the Golden Plus Equation
71(1)
3.3 The Structure of the Complex Roots
71(7)
3.4 Approximating a Nonlattice Equation by Lattice Equations
78(12)
3.4.1 Diophantine Approximation
81(3)
3.4.2 The Quasiperiodic Pattern of the Complex Dimensions
84(3)
3.4.3 Application to Nonlattice Strings
87(3)
3.5 Complex Roots of a Nonlattice Dirichlet Polynomial
90(11)
3.5.1 Continued Fractions
90(2)
3.5.2 Two Generators
92(6)
3.5.3 More than Two Generators
98(3)
3.6 Dimension-Free Regions
101(7)
3.7 The Dimensions of Fractality of a Nonlattice String
108(11)
3.7.1 The Density of the Real Parts
108(11)
3.8 A Note on the Computations
119
4 Generalized Fractal Strings Viewed as Measures
115(18)
4.1 Generalized Fractal Strings
116(5)
4.1.1 Examples of Generalized Fractal Strings
119(2)
4.2 The Frequencies of a Generalized Fractal String
121(5)
4.2.1 Completion of the Harmonic String: Euler Product
124(2)
4.3 Generalized Fractal Sprays
126(1)
4.4 The Measure of a Self-Similar String
126(5)
4.4.1 Measures with a Self-Similarity Property
128(3)
4.5 Notes
131(2)
5 Explicit Formulas for Generalized Fractal Strings
133(42)
5.1 Introduction
133(5)
5.1.1 Outline of the Proof
135(1)
5.1.2 Examples
136(2)
5.2 Preliminaries: The Heaviside Function
138(4)
5.3 Pointwise Explicit Formulas
142(12)
5.3.1 The Order of the Sum over the Complex Dimensions
153(1)
5.4 Distributional Explicit Formulas
154(16)
5.4.1 Extension to More General Test Functions
159(4)
5.4.2 The Order of the Distributional Error Term
163(7)
5.5 Example: The Prime Number Theorem
170(3)
5.5.1 The Riemann-von Mangoldt Formula
172(1)
5.6 Notes
173(2)
6 The Geometry and the Spectrum of Fractal Strings
175(34)
6.1 The Local Terms in the Explicit Formulas
176(4)
6.1.1 The Geometric Local Terms
176(2)
6.1.2 The Spectral Local Terms
178(1)
6.1.3 The Weyl Term
179(1)
6.1.4 The Distribution xw logm x
179(1)
6.2 Explicit Formulas for Lengths and Frequencies
180(4)
6.2.1 The Geometric Counting Function of a Fractal String
180(1)
6.2.2 The Spectral Counting Function of a Fractal String
181(1)
6.2.3 The Geometric and Spectral Partition Functions
182(2)
6.3 The Direct Spectral Problem for Fractal Strings
184(5)
6.3.1 The Density of Geometric and Spectral States
184(2)
6.3.2 The Spectral Operator and its Euler Product
186(3)
6.4 Self-Similar Strings
189(9)
6.4.1 Lattice Strings
190(3)
6.4.2 Nonlattice Strings
193(2)
6.4.3 The Spectrum of a Self-Similar String
195(3)
6.5 Examples of Non-Self-Similar Strings
198(4)
6.5.1 The a-String
198(3)
6.5.2 The Spectrum of the Harmonic String
201(1)
6.6 Fractal Sprays
202(7)
6.6.1 The Sierpinski Drum
203(3)
6.6.2 The Spectrum of a Self-Similar Spray
206(3)
7 Periodic Orbits of Self-Similar Flows
209(24)
7.1 Suspended Flows
210(2)
7.1.1 The Zeta Function of a Dynamical System
211(1)
7.2 Periodic Orbits, Euler Product
212(3)
7.3 Self-Similar Flows
215(6)
7.3.1 Examples of Self-Similar Flows
218(2)
7.3.2 The Lattice and Nonlattice Case
220(1)
7.4 The Prime Orbit Theorem for Suspended Flows
221(5)
7.4.1 The Prime Orbit Theorem for Self-Similar Flows
223(1)
7.4.2 Lattice Flows
224(1)
7.4.3 Nonlattice Flows
225(1)
7.5 The Error Term in the Nonlattice Case
226(4)
7.5.1 Two Generators
226(1)
7.5.2 More Than Two Generators
227(3)
7.6 Notes
230(3)
8 Tubular Neighborhoods and Minkowski Measurability
233(34)
8.1 Explicit Formulas for the Volume of Tubular Neighborhoods
234(9)
8.1.1 The Pointwisc Tube Formula
239(3)
8.1.2 Example: The a-String
242(1)
8.2 Analogy with Riemannian Geometry
243(1)
8.3 Minkowski Measurability and Complex Dimensions
244(5)
8.4 Tube Formulas for Self-Similar Strings
249(15)
8.4.1 Generalized Cantor Strings
249(3)
8.4.2 Lattice Self-Similar Strings
252(5)
8.4.3 The Average Minkowski Content
257(3)
8.4.4 Nonlattice Self-Similar Strings
260(4)
8.5 Notes
264(3)
9 The Riemann Hypothesis and Inverse Spectral Problems
267(12)
9.1 The Inverse Spectral Problem
268(3)
9.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis
271(3)
9.3 Fractal Sprays and the Generalized Riemann Hypothesis
274(2)
9.4 Notes
276(3)
10 Generalized Cantor Strings and their Oscillations 279(14)
10.1 The Geometry of a Generalized Cantor String
279(3)
10.2 The Spectrum of a Generalized Cantor String
282(5)
10.2.1 Integral Cantor Strings: α-adic Analysis of the Geometric and Spectral Oscillations
284(3)
10.2.2 Nonintegral Cantor Strings: Analysis of the Jumps in the Spectral Counting Function
287(1)
10.3 The Truncated Cantor String
287(4)
10.3.1 The Spectrum of the Truncated Cantor String
290(1)
10.4 Notes
291(2)
11 The Critical Zeros of Zeta Functions 293(32)
11.1 The Riemann Zeta. Function: No Critical Zeros in Arithmetic Progression
294(9)
11.1.1 Finite Arithmetic Progressions of Zeros
297(6)
11.2 Extension to Other Zeta Functions
303(2)
11.3 Density of Nonzeros on Vertical Lines
305(2)
11.3.1 Almost Arithmetic Progressions of Zeros
306(1)
11.4 Extension to L-Series
307(9)
11.4.1 Finite Arithmetic Progressions of Zeros of L-Series
308(8)
11.5 Zeta Functions of Curves Over Finite Fields
316(9)
12 Concluding Comments, Open Problems, and Perspectives 325(62)
12.1 Conjectures about Zeros of Dirichlet Series
327(3)
12.2 A New Definition of Fractality
330(10)
12.2.1 Fractal Geometers' Intuition of Fractality
331(3)
12.2.2 Our Definition of Fractality
334(4)
12.2.3 Possible Connections with the Notion of Lacunarity
338(2)
12.3 Fractality and Self-Similarity
340(14)
12.3.1 Complex Dimensions and Tube Formula for the Koch Snowflake Curve
342(7)
12.3.2 Towards a Higher-Dimensional Theory of Complex Dimensions
349(5)
12.4 Random and Quantized Fractal Strings
354(15)
12.4.1 Random Fractal Strings and their Zeta Functions
354(7)
12.4.2 Fractal Membranes: Quantized Fractal Strings
361(8)
12.5 The Spectrum of a Fractal Drum
369(9)
12.5.1 The Weyb-Berry Conjecture
369(2)
12.5.2 The Spectrum of a Self-Similar Drum
371(4)
12.5.3 Spectrum and Periodic Orbits
375(3)
12.6 The Complex Dimensions as the Spectrum of Shifts
378(1)
12.7 The Complex Dimensions as Geometric Invariants
378(6)
12.7.1 Connection with Varieties over Finite Fields
380(2)
12.7.2 Complex Cohomology of Self-Similar Strings
382(2)
12.8 Notes
384(3)
A Zeta Functions in Number Theory 387(12)
A.1 The Dedekind Zeta Function
387(1)
A.2 Characters and Hecke L-series
388(1)
A.3 Completion of L-Series. Functional Equation
389(1)
A.4 Epstein Zeta Functions
390(1)
A.5 Two-Variable Zeta Functions
391(5)
A.5.1 The Zeta Function of Pellikaan
392(2)
A.5.2 The Zeta Function of School' and van der Geer
394(2)
A.6 Other Zeta Functions in Number Theory
396(3)
B Zeta Functions of Laplacians and Spectral Asymptotics 399(8)
B.1 Weyl's Asymptotic Formula
399(2)
B.2 Heat Asymptotic Expansion
401(1)
B.3 The Spectral Zeta Function and its Poles
402
B.4 Extensions
401(5)
B.4.1 Monotonic Second Term
405(1)
B.5 Notes
406(1)
C An Application of Nevanlinna Theory 407(6)
C.1 Nevanlinna Height
408(1)
C.2 Complex Zeros of Dirichlet Polynomials
409(4)
Bibliography 413(26)
Acknowledgements 439(4)
Conventions 443(2)
Index of Symbols 445(4)
Author Index 449(2)
Subject Index 451
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