TY - BOOK ID - 14308683 TI - Fractal Geometry, Complex Dimensions and Zeta Functions : Geometry and Spectra of Fractal Strings AU - Lapidus, Michel L. AU - van Frankenhuijsen, Machiel. PY - 2013 SN - 14397382 SN - 1461421756 1461421764 1283909553 1489988386 PB - New York, NY : Springer New York : Imprint: Springer, DB - UniCat KW - Fractals. KW - Functions, Zeta. KW - Geometry, Riemannian. KW - Mathematics. KW - Fractals KW - Functions, Zeta KW - Geometry, Riemannian KW - Mathematics KW - Physical Sciences & Mathematics KW - Algebra KW - Geometry KW - Number theory. KW - Zeta functions KW - Fractal geometry KW - Fractal sets KW - Geometry, Fractal KW - Sets, Fractal KW - Sets of fractional dimension KW - Number study KW - Numbers, Theory of KW - Riemann geometry KW - Riemannian geometry KW - Dynamics. KW - Ergodic theory. KW - Functional analysis. KW - Global analysis (Mathematics). KW - Manifolds (Mathematics). KW - Measure theory. KW - Partial differential equations. KW - Number Theory. KW - Measure and Integration. KW - Partial Differential Equations. KW - Dynamical Systems and Ergodic Theory. KW - Global Analysis and Analysis on Manifolds. KW - Functional Analysis. KW - Analysis, Global (Mathematics) KW - Differential topology KW - Functions of complex variables KW - Geometry, Algebraic KW - Functional calculus KW - Calculus of variations KW - Functional equations KW - Integral equations KW - Ergodic transformations KW - Continuous groups KW - Mathematical physics KW - Measure theory KW - Transformations (Mathematics) KW - Partial differential equations KW - Lebesgue measure KW - Measurable sets KW - Measure of a set KW - Algebraic topology KW - Integrals, Generalized KW - Measure algebras KW - Rings (Algebra) KW - Geometry, Differential KW - Topology KW - Dynamical systems KW - Kinetics KW - Mechanics, Analytic KW - Force and energy KW - Mechanics KW - Physics KW - Statics KW - Math KW - Science KW - Dimension theory (Topology) KW - Generalized spaces KW - Geometry, Non-Euclidean KW - Semi-Riemannian geometry KW - Differential equations, partial. KW - Differentiable dynamical systems. KW - Global analysis. KW - Differential dynamical systems KW - Dynamical systems, Differentiable KW - Dynamics, Differentiable KW - Differential equations KW - Global analysis (Mathematics) KW - Topological dynamics UR - https://www.unicat.be/uniCat?func=search&query=sysid:14308683 AB - 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. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. 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 are studied in detail, and 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 The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. ER -