SC - 1007 | Fractal Geometry and Number Theory

A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo- metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays.

We develop a theory of complex di- mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref- erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap- pendix B. ) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function ( c ( 8 ) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex- tension.

2000 | ISBN-13: 978-1-4612-5316-7 | e-ISBN-I3: 978-1-4612-5314-3 | ID: SC - 1007

Read More ...

SC - 1053 | Hard Ball Systems and the Lorentz Gas

Hard Ball Systems and the Lorentz Gas (Encyclopaedia of Mathematical Sciences) by L.A. Bunimovich (Author), D. Burago (Author), N. Chernov (Author), E.G.D. Cohen (Author), C.P. Dettmann (Author), J.R. Dorfman (Author), S. Ferleger (Author), & 15 more

Hard Ball Systems and the Lorentz Gas are fundamental models arising in the theory of Hamiltonian dynamical systems. Moreover, in these models, some key laws of statistical physics can also be tested or even established by mathematically rigorous tools.

The mathematical methods are most beautiful but sometimes quite involved. This collection of surveys written by leading researchers of the fields - mathematicians, physicists or mathematical physicists - treat both mathematically rigourous results, and evolving physical theories where the methods are analytic or computational. Some basic topics: hyperbolicity and ergodicity, correlation decay, Lyapunov exponents, Kolmogorov-Sinai entropy, entropy production, irreversibility. This collection is a unique introduction into the subject for graduate students, postdocs or researchers - in both mathematics and physics - who want to start working in the field.

2000 | ISBN 978-3-642-08711-0 | ISBN 978-3-662-04062-1 | ID: SC - 1053

Read More ...