255 research outputs found
Entropy and Poincar\'e recurrence from a geometrical viewpoint
We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove
that the metric entropy is given by the exponential growth rate of return times
to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss
theorem. Moreover, we show that minimal return times to dynamical balls grow
linearly with respect to its length. Finally, some interesting relations
between recurrence, dimension, entropy and Lyapunov exponents of ergodic
measures are given.Comment: 11 pages, revised versio
A "metric" complexity for weakly chaotic systems
We consider the number of Bowen sets which are necessary to cover a large
measure subset of the phase space. This introduce some complexity indicator
characterizing different kind of (weakly) chaotic dynamics. Since in many
systems its value is given by a sort of local entropy, this indicator is quite
simple to be calculated. We give some example of calculation in nontrivial
systems (interval exchanges, piecewise isometries e.g.) and a formula similar
to the Ruelle-Pesin one, relating the complexity indicator to some initial
condition sensitivity indicators playing the role of positive Lyapunov
exponents.Comment: 15 pages, no figures. Articl
High purity nanoparticles exceed stoichiometry limits in rebox chemistry: the nano way to cleaner water
A potentially cheaper and more effective way of cleaning wastewater has been discovered by scientists
at Nazarbayev University and the University of Brighton researching nanotechnology [1]. It is well
established that when particles are reduced to the nanoscale unexpected effects occur. Silver, for example,
interacts with mercury ions in a fixed ratio of atoms (stoichiometry), typically 2:1, which presents a limit
that has never been exceeded. In this project we used an alternative chemical procedure based on modified
quartz sand to immobilise silver nanoparticles (NPs) with control over their size. We found that when the
size of the silver NPs decreased below 35 nm the amount of mercury ions reacting with silver increased
beyond the long-held limit and rose to a maximum of 1:1.2 for 10 nm sized silver
High purity nanoparticles exceed stoichiometry limits in rebox chemistry: the nano way to cleaner water
A potentially cheaper and more effective way of cleaning wastewater has been discovered by scientists
at Nazarbayev University and the University of Brighton researching nanotechnology [1]. It is well
established that when particles are reduced to the nanoscale unexpected effects occur. Silver, for example,
interacts with mercury ions in a fixed ratio of atoms (stoichiometry), typically 2:1, which presents a limit
that has never been exceeded. In this project we used an alternative chemical procedure based on modified
quartz sand to immobilise silver nanoparticles (NPs) with control over their size. We found that when the
size of the silver NPs decreased below 35 nm the amount of mercury ions reacting with silver increased
beyond the long-held limit and rose to a maximum of 1:1.2 for 10 nm sized silver
A Poincar\'e section for the general heavy rigid body
A general recipe is developed for the study of rigid body dynamics in terms
of Poincar\'e surfaces of section. A section condition is chosen which captures
every trajectory on a given energy surface. The possible topological types of
the corresponding surfaces of section are determined, and their 1:1 projection
to a conveniently defined torus is proposed for graphical rendering.Comment: 25 pages, 10 figure
Cross sections for geodesic flows and \alpha-continued fractions
We adjust Arnoux's coding, in terms of regular continued fractions, of the
geodesic flow on the modular surface to give a cross section on which the
return map is a double cover of the natural extension for the \alpha-continued
fractions, for each in (0,1]. The argument is sufficiently robust to
apply to the Rosen continued fractions and their recently introduced
\alpha-variants.Comment: 20 pages, 2 figure
Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry
We consider a triality between the Zermelo navigation problem, the geodesic
flow on a Finslerian geometry of Randers type, and spacetimes in one dimension
higher admitting a timelike conformal Killing vector field. From the latter
viewpoint, the data of the Zermelo problem are encoded in a (conformally)
Painleve-Gullstrand form of the spacetime metric, whereas the data of the
Randers problem are encoded in a stationary generalisation of the usual optical
metric. We discuss how the spacetime viewpoint gives a simple and physical
perspective on various issues, including how Finsler geometries with constant
flag curvature always map to conformally flat spacetimes and that the Finsler
condition maps to either a causality condition or it breaks down at an
ergo-surface in the spacetime picture. The gauge equivalence in this network of
relations is considered as well as the connection to analogue models and the
viewpoint of magnetic flows. We provide a variety of examples.Comment: 37 pages, 6 figure
Cosmological measurements, time and observables in (2+1)-dimensional gravity
We investigate the relation between measurements and the physical observables
for vacuum spacetimes with compact spatial surfaces in (2+1)-gravity with
vanishing cosmological constant. By considering an observer who emits lightrays
that return to him at a later time, we obtain explicit expressions for several
measurable quantities as functions on the physical phase space of the theory:
the eigentime elapsed between the emission of a lightray and its return to the
observer, the angles between the directions into which the light has to be
emitted to return to the observer and the relative frequencies of the lightrays
at their emission and return. This provides a framework in which conceptual
questions about time, observables and measurements can be addressed. We analyse
the properties of these measurements and their geometrical interpretation and
show how they allow an observer to determine the values of the Wilson loop
observables that parametrise the physical phase space of (2+1)-gravity. We
discuss the role of time in the theory and demonstrate that the specification
of an observer with respect to the spacetime's geometry amounts to a gauge
fixing procedure yielding Dirac observables.Comment: 38 pages, 11 eps figures, typos corrected, references update
Periodic orbits of period 3 in the disc
Let f be an orientation preserving homeomorphism of the disc D2 which
possesses a periodic point of period 3. Then either f is isotopic, relative the
periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi
/3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page
Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system
We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic
critical manifold of a Hamiltonian system. Using this
result, trajectories with small energy shadowing chains of homoclinic
orbits to are represented as extremals of a discrete variational problem,
and their existence is proved. This paper is motivated by applications to the
Poincar\'e second species solutions of the 3 body problem with 2 masses small
of order . As , double collisions of small bodies correspond to
a symplectic critical manifold of the regularized Hamiltonian system
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