219 research outputs found
Structure and evolution of strange attractors in non-elastic triangular billiards
We study pinball billiard dynamics in an equilateral triangular table. In
such dynamics, collisions with the walls are non-elastic: the outgoing angle
with the normal vector to the boundary is a uniform factor
smaller than the incoming angle. This leads to contraction in phase space for
the discrete-time dynamics between consecutive collisions, and hence to
attractors of zero Lebesgue measure, which are almost always fractal strange
attractors with chaotic dynamics, due to the presence of an expansion
mechanism. We study the structure of these strange attractors and their
evolution as the contraction parameter is varied. For in
the interval (0, 1/3), we prove rigorously that the attractor has the structure
of a Cantor set times an interval, whereas for larger values of the
billiard dynamics gives rise to nonaccessible regions in phase space. For
close to 1, the attractor splits into three transitive components,
the basins of attraction of which have fractal basin boundaries.Comment: 12 pages, 10 figures; submitted for publication. One video file
available at http://sistemas.fciencias.unam.mx/~dsanders
A theory of non-local linear drift wave transport
Transport events in turbulent tokamak plasmas often exhibit non-local or
non-diffusive action at a distance features that so far have eluded a
conclusive theoretical description. In this paper a theory of non-local
transport is investigated through a Fokker-Planck equation with fractional
velocity derivatives. A dispersion relation for density gradient driven linear
drift modes is derived including the effects of the fractional velocity
derivative in the Fokker-Planck equation. It is found that a small deviation (a
few percent) from the Maxwellian distribution function alters the dispersion
relation such that the growth rates are substantially increased and thereby may
cause enhanced levels of transport.Comment: 22 pages, 2 figures. Manuscript submitted to Physics of Plasma
A Fractional Fokker-Planck Model for Anomalous Diffusion
In this paper we present a study of anomalous diffusion using a Fokker-Planck
description with fractional velocity derivatives. The distribution functions
are found using numerical means for varying degree of fractionality observing
the transition from a Gaussian distribution to a L\'evy distribution. The
statistical properties of the distribution functions are assessed by a
generalized expectation measure and entropy in terms of Tsallis statistical
mechanics. We find that the ratio of the generalized entropy and expectation is
increasing with decreasing fractionality towards the well known so-called
sub-diffusive domain, indicating a self-organising behavior.Comment: 22 pages, 14 figure
Rational approximation and arithmetic progressions
A reasonably complete theory of the approximation of an irrational by
rational fractions whose numerators and denominators lie in prescribed
arithmetic progressions is developed in this paper. Results are both, on the
one hand, from a metrical and a non-metrical point of view and, on the other
hand, from an asymptotic and also a uniform point of view. The principal
novelty is a Khintchine type theorem for uniform approximation in this context.
Some applications of this theory are also discussed
The Duffin-Schaeffer Conjecture with extra divergence II
This paper takes a new step in the direction of proving the Duffin-Schaeffer
Conjecture for measures arbitrarily close to Lebesgue. The main result is that
under a mild `extra divergence' hypothesis, the conjecture is true.Comment: 7 page
Levy statistics and anomalous transport in quantum-dot arrays
A novel model of transport is proposed to explain power law current
transients and memory phenomena observed in partially ordered arrays of
semiconducting nanocrystals. The model describes electron transport by a
stationary Levy process of transmission events and thereby requires no time
dependence of system properties. The waiting time distribution with a
characteristic long tail gives rise to a nonstationary response in the presence
of a voltage pulse. We report on noise measurements that agree well with the
predicted non-Poissonian fluctuations in current, and discuss possible
mechanisms leading to this behavior.Comment: 7 pages, 2 figure
An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation
The classical law of the iterated logarithm (LIL for short)as fundamental
limit theorems in probability theory play an important role in the development
of probability theory and its applications. Strassen (1964) extended LIL to
large classes of functional random variables, it is well known as the
invariance principle for LIL which provide an extremely powerful tool in
probability and statistical inference. But recently many phenomena show that
the linearity of probability is a limit for applications, for example in
finance, statistics. As while a nonlinear expectation--- G-expectation has
attracted extensive attentions of mathematicians and economists, more and more
people began to study the nature of the G-expectation space. A natural question
is: Can the classical invariance principle for LIL be generalized under
G-expectation space? This paper gives a positive answer. We present the
invariance principle of G-Brownian motion for the law of the iterated logarithm
under G-expectation
Generic Continuous Spectrum for Ergodic Schr"odinger Operators
We consider discrete Schr"odinger operators on the line with potentials
generated by a minimal homeomorphism on a compact metric space and a continuous
sampling function. We introduce the concepts of topological and metric
repetition property. Assuming that the underlying dynamical system satisfies
one of these repetition properties, we show using Gordon's Lemma that for a
generic continuous sampling function, the associated Schr"odinger operators
have no eigenvalues in a topological or metric sense, respectively. We present
a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page
Ultrametric Logarithm Laws, II
We prove positive characteristic versions of the logarithm laws of Sullivan
and Kleinbock-Margulis and obtain related results in Metric Diophantine
Approximation.Comment: submitted to Montasefte Fur Mathemati
An Introduction to Data Analysis in Asteroseismology
A practical guide is presented to some of the main data analysis concepts and
techniques employed contemporarily in the asteroseismic study of stars
exhibiting solar-like oscillations. The subjects of digital signal processing
and spectral analysis are introduced first. These concern the acquisition of
continuous physical signals to be subsequently digitally analyzed. A number of
specific concepts and techniques relevant to asteroseismology are then
presented as we follow the typical workflow of the data analysis process,
namely, the extraction of global asteroseismic parameters and individual mode
parameters (also known as peak-bagging) from the oscillation spectrum.Comment: Lecture presented at the IVth Azores International Advanced School in
Space Sciences on "Asteroseismology and Exoplanets: Listening to the Stars
and Searching for New Worlds" (arXiv:1709.00645), which took place in Horta,
Azores Islands, Portugal in July 201
- …