http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
complex dimensions of fractal strings and zeros of zeta functions, with 26 illustrations
1 online resource (x, 268 pages) : illustrations
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 Weyls 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.
Michel L. Lapidus, Machiel van Frankenhuysen.
Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002. http://purl.oclc.org/DLF/benchrepro0212 MiAaHDL
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Includes bibliographical references and indexes.