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lapidus michel l.; van frankenhuijsen machiel - fractal geometry, complex dimensions and zeta functions

FRACTAL GEOMETRY, COMPLEX DIMENSIONS AND ZETA FUNCTIONS GEOMETRY AND SPECTRA OF FRACTAL STRINGS

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Genere:Libro
Lingua: Inglese
Editore:

Springer US

Pubblicazione: 09/2006
Edizione: 1





Trama

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.
Key Features:
- The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings
- Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra
- Explicit formulas are extended to apply to the geometric, spectral, and dynamic zeta functions associated with a fractal
- Examples of such formulas include Prime Orbit Theorem with error term for self-similar flows, and a tube formula
- The method of diophantine approximation is used to study self-similar strings and flows
- Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions
Throughout new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts.
The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. TOC:List of Figures.- Preface.- Overview.- Introduction.- Complex Dimensions of Ordinary Fractal Strings.- Complex Dimensions of Self-Similar Fractal Strings.- Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation.- Generalized Fractal Strings Viewed as Measures.- Explicit Formulas for Generalized Fractal Strings.- The Geometry and the Spectrum of Fractal Strings.- Periodic Orbits of Self-Similar Flows.- Tubular Neighborhoods and Minkowski Measurability.- The Riemann Hypothesis and Inverse Spectral Problems.- Generalized Cantor Strings and their Oscillations.- The Critical Zeros of Zeta Functions.- Concluding Comments, Open Problems, and Perspectives.- Appendices.- A. Zeta Functions in Number Theory.- B. Zeta Functions of Laplacians and Spectral Asymptotics.- C. An Application of Nevanlinna Theory.- Bibliography.- Acknolwedgements.- Conventions.- Index of Symbols.- Author Index.- Subject Index.










Altre Informazioni

ISBN:

9780387332857

Condizione: Nuovo
Collana: Springer Monographs in Mathematics
Dimensioni: 242 x 32 x 160 mm
Pagine Arabe: 460


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