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.
Rezensionen / Stimmen
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)
Reihe
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Research
Produkt-Hinweis
Illustrationen
54 s/w Abbildungen
1, black & white illustrations
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 26 mm
Gewicht
ISBN-13
978-0-387-33285-7 (9780387332857)
DOI
10.1007/978-0-387-35208-4
Schweitzer Klassifikation
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.
