13,776 research outputs found
On the fractal characteristics of a stabilised Newton method
In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed
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
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
We show that the uncertainty in distance and time measurements found by the
heuristic combination of quantum mechanics and general relativity is reproduced
in a purely classical and flat multi-fractal spacetime whose geometry changes
with the probed scale (dimensional flow) and has non-zero imaginary dimension,
corresponding to a discrete scale invariance at short distances. Thus,
dimensional flow can manifest itself as an intrinsic measurement uncertainty
and, conversely, measurement-uncertainty estimates are generally valid because
they rely on this universal property of quantum geometries. These general
results affect multi-fractional theories, a recent proposal related to quantum
gravity, in two ways: they can fix two parameters previously left free (in
particular, the value of the spacetime dimension at short scales) and point
towards a reinterpretation of the ultraviolet structure of geometry as a
stochastic foam or fuzziness. This is also confirmed by a correspondence we
establish between Nottale scale relativity and the stochastic geometry of
multi-fractional models.Comment: 25 pages. v2: minor typos corrected, references adde
The geometry of fractal percolation
A well studied family of random fractals called fractal percolation is
discussed. We focus on the projections of fractal percolation on the plane. Our
goal is to present stronger versions of the classical Marstrand theorem, valid
for almost every realization of fractal percolation. The extensions go in three
directions: {itemize} the statements work for all directions, not almost all,
the statements are true for more general projections, for example radial
projections onto a circle, in the case , each projection has not
only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc
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