Fractal Geometry and Number Theory - nouveau livre

EAN (ISBN-13): 9781461253143
Date de parution: 2013
Editeur: Birkhäuser Boston

Livre dans la base de données depuis 2016-12-13T15:01:05+01:00 (Zurich)
Page de détail modifiée en dernier sur 2024-03-17T00:11:20+01:00 (Zurich)
ISBN/EAN: 9781461253143

ISBN - Autres types d'écriture:
978-1-4612-5314-3
Autres types d'écriture et termes associés:
Auteur du livre: michel lapidus
Titre du livre: the fractal geometry, fractal geometry and number theory, fractal geometry complex dimensions

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Données de l'éditeur

Auteur: Michel L. Lapidus
Titre: Fractal Geometry and Number Theory - Complex Dimensions of Fractal Strings and Zeros of Zeta Functions
Editeur: Birkhäuser; Birkhäuser Boston
268 Pages
Date de parution: 2013-12-01
Boston; MA; US
Langue: Anglais
84,99 € (DE)

EA; E107; eBook; Nonbooks, PBS / Mathematik/Arithmetik, Algebra; Zahlentheorie; Verstehen; Algebraic geometry; differential geometry; partial differential equations; algebraic geometry; arithmetic; calculus; Dimension; fractal; fractal geometry; number theory; prime number; Riemannian geometry; zeta function; C; Number Theory; Algebraic Geometry; Differential Geometry; Mathematics and Statistics; Algebraische Geometrie; Differentielle und Riemannsche Geometrie; BB

1 Complex Dimensions of Ordinary Fractal Strings.- 1.1 The Geometry of a Fractal String.- 1.2 The Geometric Zeta Function of a Fractal String.- 1.3 The Frequencies of a Fractal String and the Spectral Zeta Function.- 1.4 Higher-Dimensional Analogue: Fractal Sprays.- 2 Complex Dimensions of Self-Similar Fractal Strings.- 2.1 The Geometric Zeta Function of a Self-Similar String.- 2.2 Examples of Complex Dimensions of Self-Similar Strings.- 2.3 The Lattice and Nonlattice Case.- 2.4 The Structure of the Complex Dimensions.- 2.5 The Density of the Poles in the Nonlattice Case.- 2.6 Approximating a Fractal String and Its Complex Dimensions.- 3 Generalized Fractal Strings Viewed as Measures.- 3.1 Generalized Fractal Strings.- 3.2 The Frequencies of a Generalized Fractal String.- 3.3 Generalized Fractal Sprays.- 3.4 The Measure of a Self-Similar String.- 4 Explicit Formulas for Generalized Fractal Strings.- 4.1 Introduction.- 4.2 Preliminaries: The Heaviside Function.- 4.3 The Pointwise Explicit Formulas.- 4.4 The Distributional Explicit Formulas.- 4.5 Example: The Prime Number Theorem.- 5 The Geometry and the Spectrum of Fractal Strings.- 5.1 The Local Terms in the Explicit Formulas.- 5.2 Explicit Formulas for Lengths and Frequencies.- 5.3 The Direct Spectral Problem for Fractal Strings.- 5.4 Self-Similar Strings.- 5.5 Examples of Non-Self-Similar Strings.- 5.6 Fractal Sprays.- 6 Tubular Neighborhoods and Minkowski Measurability.- 6.1 Explicit Formula for the Volume of a Tubular Neighborhood.- 6.2 Minkowski Measurability and Complex Dimensions.- 6.3 Examples.- 7 The Riemann Hypothesis, Inverse Spectral Problems and Oscillatory Phenomena.- 7.1 The Inverse Spectral Problem.- 7.2 Complex Dimensions of Fractal Strings and the Riemann Hypothesis.- 7.3 Fractal Sprays and the Generalized Riemann Hypothesis.- 8 Generalized Cantor Strings and their Oscillations.- 8.1 The Geometry of a Generalized Cantor String.- 8.2 The Spectrum of a Generalized Cantor String.- 9 The Critical Zeros of Zeta Functions.- 9.1 The Riemann Zeta Function: No Critical Zeros in an Arithmetic Progression.- 9.2 Extension to Other Zeta Functions.- 9.3 Extension to L-Series.- 9.4 Zeta Functions of Curves Over Finite Fields.- 10 Concluding Comments.- 10.1 Conjectures about Zeros of Dirichlet Series.- 10.2 A New Definition of Fractality.- 10.3 Fractality and Self-Similarity.- 10.4 The Spectrum of a Fractal Drum.- 10.5 The Complex Dimensions as Geometric Invariants.- Appendices.- A Zeta Functions in Number Theory.- A.l The Dedekind Zeta Function.- A.3 Completion of L-Series, Functional Equation.- A.4 Epstein Zeta Functions.- A.5 Other Zeta Functions in Number Theory.- B Zeta Functions of Laplacians and Spectral Asymptotics.- B.l Weyl’s Asymptotic Formula.- B.2 Heat Asymptotic Expansion.- B.3 The Spectral Zeta Function and Its Poles.- B.4 Extensions.- B.4.1 Monotonic Second Term.- References.- Conventions.- Symbol Index.- List of Figures.- Acknowledgements.