17,230 research outputs found
A new lower bound for the Towers of Hanoi problem
More than a century after its proposal, the Towers of Hanoi puzzle with 4
pegs was solved by Thierry Bousch in a breakthrough paper in 2014. The general
problem with p pegs is still open, with the best lower bound on the minimum
number of moves due to Chen and Shen. We use some of Bousch's new ideas to
obtain an asymptotic improvement on this bound for all p >= 5
Dynamical systems approach to G2 cosmology
In this paper we present a new approach for studying the dynamics of
spatially inhomogeneous cosmological models with one spatial degree of freedom.
By introducing suitable scale-invariant dependent variables we write the
evolution equations of the Einstein field equations as a system of autonomous
partial differential equations in first-order symmetric hyperbolic format,
whose explicit form depends on the choice of gauge. As a first application, we
show that the asymptotic behaviour near the cosmological initial singularity
can be given a simple geometrical description in terms of the local past
attractor on the boundary of the scale-invariant dynamical state space. The
analysis suggests the name ``asymptotic silence'' to describe the evolution of
the gravitational field near the cosmological initial singularity.Comment: 28 pages, 3 tables, 1 *.eps figure, LaTeX2e (10pt), matches version
accepted for publication by Classical and Quantum Gravit
Asymptotic dynamics of the exceptional Bianchi cosmologies
In this paper we give, for the first time, a qualitative description of the
asymptotic dynamics of a class of non-tilted spatially homogeneous (SH)
cosmologies, the so-called exceptional Bianchi cosmologies, which are of
Bianchi type VI. This class is of interest for two reasons. Firstly,
it is generic within the class of non-tilted SH cosmologies, being of the same
generality as the models of Bianchi types VIII and IX. Secondly, it is the SH
limit of a generic class of spatially inhomogeneous cosmologies.
Using the orthonormal frame formalism and Hubble-normalized variables, we
show that the exceptional Bianchi cosmologies differ from the non-exceptional
Bianchi cosmologies of type VI in two significant ways. Firstly, the
models exhibit an oscillatory approach to the initial singularity and hence are
not asymptotically self-similar. Secondly, at late times, although the models
are asymptotically self-similar, the future attractor for the vacuum-dominated
models is the so-called Robinson-Trautman SH model instead of the vacuum SH
plane wave models.Comment: 15 pages, 6 figures, submitted to Class. Quantum Gra
General formulation of general-relativistic higher-order gauge-invariant perturbation theory
Gauge-invariant treatments of general-relativistic higher-order perturbations
on generic background spacetime is proposed. After reviewing the general
framework of the second-order gauge-invariant perturbation theory, we show the
fact that the linear-order metric perturbation is decomposed into
gauge-invariant and gauge-variant parts, which was the important premis of this
general framework. This means that the development the higher-order
gauge-invariant perturbation theory on generic background spacetime is
possible. A remaining issue to be resolve is also disscussed.Comment: 4 pages, no figure. (v3) some explanations are added and a reference
is adde
Future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI
Using the methods developed for the Bianchi I case we have shown that a
boostrap argument is also suitable to treat the future non-linear stability for
reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types
II and VI. These solutions are asymptotic to the Collins-Stewart solution
with dust and the Ellis-MacCallum solution respectively. We have thus
generalized the results obtained by Rendall and Uggla in the case of locally
rotationally symmetric Bianchi II spacetimes to the reflection symmetric case.
However we needed to assume small data. For Bianchi VI there is no
analogous previous result.Comment: 30 page
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