000			05244nam a22005895i 4500
001			978-0-387-35208-4
003			DE-He213
005			20161121231026.0
007			cr nn 008mamaa
008			100301s2006 xxu| s |||| 0|eng d
020			_a9780387352084 _9978-0-387-35208-4
024	7		_a10.1007/978-0-387-35208-4 _2doi
050		4	_aQA611-614.97
072		7	_aPBP _2bicssc
072		7	_aMAT038000 _2bisacsh
082	0	4	_a514 _223
100	1		_aLapidus, Michel L. _eauthor.
245	1	0	_aFractal Geometry, Complex Dimensions and Zeta Functions _h[electronic resource] : _bGeometry and Spectra of Fractal Strings / _cby Michel L. Lapidus, Machiel van Frankenhuijsen.
264		1	_aNew York, NY : _bSpringer New York, _c2006.
300			_aXXIV, 460 p. 54 illus. _bonline resource.
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	1		_aSpringer Monographs in Mathematics, _x1439-7382
505	0		_aComplex 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.
520			_aNumber 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 dynamical zeta functions associated with a fractal Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric 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, and discusses several open problems and extensions. 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. From Reviews of Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel Lapidus and Machiel van Frankenhuysen, Birkhäuser Boston Inc., 2000. "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 "It is the reviewer’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.
650		0	_aMathematics.
650		0	_aDynamics.
650		0	_aErgodic theory.
650		0	_aGlobal analysis (Mathematics).
650		0	_aManifolds (Mathematics).
650		0	_aMeasure theory.
650		0	_aPartial differential equations.
650		0	_aNumber theory.
650		0	_aTopology.
650	1	4	_aMathematics.
650	2	4	_aTopology.
650	2	4	_aNumber Theory.
650	2	4	_aMeasure and Integration.
650	2	4	_aPartial Differential Equations.
650	2	4	_aDynamical Systems and Ergodic Theory.
650	2	4	_aGlobal Analysis and Analysis on Manifolds.
700	1		_aFrankenhuijsen, Machiel van. _eauthor.
710	2		_aSpringerLink (Online service)
773	0		_tSpringer eBooks
776	0	8	_iPrinted edition: _z9780387332857
830		0	_aSpringer Monographs in Mathematics, _x1439-7382
856	4	0	_uhttp://dx.doi.org/10.1007/978-0-387-35208-4
912			_aZDB-2-SMA
950			_aMathematics and Statistics (Springer-11649)
999			_c507473 _d507473