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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. Key features include: 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; and 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.
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Subject categories
- Academic > Mathematics > Geometry. Trigonometry.Topology > Global analysis (Mathematics)
- Academic > Mathematics > Geometry. Trigonometry.Topology > Differentiable dynamical systems
- Academic > Mathematics > Geometry. Trigonometry.Topology > Fractals
- Academic > Mathematics > Tables
- Mathematics > Number Theory
ISBNs
0387352082
9780387332857
9780387352084