rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Fractal Geometry, Complex Dimensions and Zeta Functions
    sub title: Geometry and Spectra of Fractal Strings
  name:
    name part: Lapidus, Michel L.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: Frankenhuijsen, Machiel van.
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: xxu
    date issued: 2006
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XXIV, 460 p. 54 illus. online resource.
  abstract: 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 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…
  table of contents: 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.
  note: by Michel L. Lapidus, Machiel van Frankenhuijsen.
  subject:
    topic: Mathematics
  subject:
    topic: Dynamics
  subject:
    topic: Ergodic theory
  subject:
    topic: Global analysis (Mathematics)
  subject:
    topic: Manifolds (Mathematics)
  subject:
    topic: Measure theory
  subject:
    topic: Partial differential equations
  subject:
    topic: Number theory
  subject:
    topic: Topology
  subject:
    topic: Mathematics
  subject:
    topic: Topology
  subject:
    topic: Number Theory
  subject:
    topic: Measure and Integration
  subject:
    topic: Partial Differential Equations
  subject:
    topic: Dynamical Systems and Ergodic Theory
  subject:
    topic: Global Analysis and Analysis on Manifolds
  classification: QA611-614.97
  classification: 514
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Springer Monographs in Mathematics
  identifier: 9780387352084
  identifier: http://dx.doi.org/10.1007/978-0-387-35208-4
  record info:
    record creation date: 100301
    record change date: 20151204182455.0
    record identifier: 978-0-387-35208-4