2,433 research outputs found
Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view.Comment: 24 page
The Fractal Geometry of the Cosmic Web and its Formation
The cosmic web structure is studied with the concepts and methods of fractal
geometry, employing the adhesion model of cosmological dynamics as a basic
reference. The structures of matter clusters and cosmic voids in cosmological
N-body simulations or the Sloan Digital Sky Survey are elucidated by means of
multifractal geometry. A non-lacunar multifractal geometry can encompass three
fundamental descriptions of the cosmic structure, namely, the web structure,
hierarchical clustering, and halo distributions. Furthermore, it explains our
present knowledge of cosmic voids. In this way, a unified theory of the
large-scale structure of the universe seems to emerge. The multifractal
spectrum that we obtain significantly differs from the one of the adhesion
model and conforms better to the laws of gravity. The formation of the cosmic
web is best modeled as a type of turbulent dynamics, generalizing the known
methods of Burgers turbulence.Comment: 35 pages, 8 figures; corrected typos, added references; further
discussion of cosmic voids; accepted by Advances in Astronom
Brownian motion: a paradigm of soft matter and biological physics
This is a pedagogical introduction to Brownian motion on the occasion of the
100th anniversary of Einstein's 1905 paper on the subject. After briefly
reviewing Einstein's work in its contemporary context, we pursue some lines of
further developments and applications in soft condensed matter and biology.
Over the last century Brownian motion became promoted from an odd curiosity of
marginal scientific interest to a guiding theme pervading all of the modern
(live) sciences.Comment: 30 pages, revie
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