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Fractal stochastic processes: Clusters and intermittancies

Conference Lectures

Part of the Lecture Notes in Mathematics book series (LNM,volume 1035)

Abstract

Many scaling relations for complex systems in the physical sciences involve non-integer exponents. We list several examples all of which may not be well known. We interpret non-integer exponents as indicating singularities arising from a long tailed probability distribution governing the physical observables. If the first appropriate moment of the probability distribution diverges, then no scale exists in which to qauge measurements and phenomena occur on all scales. Self-similar fractals, non-differentiability, and also non-integer exponents will arise. Random walk examples are presented where the above characteristics appear simply and naturally. The analysis provides a generalization of Weierstrass' continuous, but nowhere differentiable function. Lastly, the Riemann Hypothesis is recast in a random walk framework.

Keywords

  • Fractal Dimension
  • Random Walk
  • Structure Function
  • Riemann Hypothesis
  • High Velocity Impact

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1983 Springer-Verlag

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Shlesinger, M.F., Montroll, E.W. (1983). Fractal stochastic processes: Clusters and intermittancies. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073257

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  • DOI: https://doi.org/10.1007/BFb0073257

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12707-9

  • Online ISBN: 978-3-540-38693-3

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