58,407 research outputs found
Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space
We define a new version of modified mean curvature flow (MMCF) in hyperbolic
space , which interestingly turns out to be the natural
negative -gradient flow of the energy functional defined by De Silva and
Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the
MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed
asymptotic boundary at infinity. As an application, we recover the existence
and uniqueness of smooth complete hypersurfaces of constant mean curvature in
hyperbolic space with prescribed asymptotic boundary at infinity, which was
first shown by Guan and Spruck.Comment: 26 pages, 3 figure
The Inverse Square Law of Light
This activity gives an easy way for students to measure the relationship between distance and brightness. Once students discover the relationship, they can begin to understand how astronomers use this knowledge to determine the distances to stars and far away galaxies. Educational levels: Middle school, High school, Middle school, High school
The -invariant massive Laplacian on isoradial graphs
We introduce a one-parameter family of massive Laplacian operators
defined on isoradial graphs, involving elliptic
functions. We prove an explicit formula for the inverse of , the
massive Green function, which has the remarkable property of only depending on
the local geometry of the graph, and compute its asymptotics. We study the
corresponding statistical mechanics model of random rooted spanning forests. We
prove an explicit local formula for an infinite volume Boltzmann measure, and
for the free energy of the model. We show that the model undergoes a second
order phase transition at , thus proving that spanning trees corresponding
to the Laplacian introduced by Kenyon are critical. We prove that the massive
Laplacian operators provide a one-parameter
family of -invariant rooted spanning forest models. When the isoradial graph
is moreover -periodic, we consider the spectral curve of the
characteristic polynomial of the massive Laplacian. We provide an explicit
parametrization of the curve and prove that it is Harnack and has genus . We
further show that every Harnack curve of genus with
symmetry arises from such a massive
Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica
Planetary detection limits taking into account stellar noise. II. Effect of stellar spot groups on radial-velocities
The detection of small mass planets with the radial-velocity technique is now
confronted with the interference of stellar noise. HARPS can now reach a
precision below the meter-per-second, which corresponds to the amplitudes of
different stellar perturbations, such as oscillation, granulation, and
activity. Solar spot groups induced by activity produce a radial-velocity noise
of a few meter-per-second. The aim of this paper is to simulate this activity
and calculate detection limits according to different observational strategies.
Based on Sun observations, we reproduce the evolution of spot groups on the
surface of a rotating star. We then calculate the radial-velocity effect
induced by these spot groups as a function of time. Taking into account
oscillation, granulation, activity, and a HARPS instrumental error of 80 cm/s,
we simulate the effect of different observational strategies in order to
efficiently reduce all sources of noise. Applying three measurements per night
of 10 minutes every three days, 10 nights a month seems the best tested
strategy. Depending on the level of activity considered, from log(R'_HK)= -5 to
-4.75, this strategy would allow us to find planets of 2.5 to 3.5 Earth masses
in the habitable zone of a K1V dwarf. Using Bern's model of planetary
formation, we estimate that for the same range of activity level, 15 to 35 % of
the planets between 1 and 5 Earth masses and with a period between 100 and 200
days should be found with HARPS. A comparison between the performance of HARPS
and ESPRESSO is also emphasized by our simulations. Using the same optimized
strategy, ESPRESSO could find 1.3 Earth mass planets in the habitable zone of
early-K dwarfs. In addition, 80 % of planets with mass between 1 and 5 Earth
masses and with a period between 100 and 200 days could be detected.Comment: 11 pages, 11 figures, accepted for publication in A&
The Fate of Monsters in Anti-de Sitter Spacetime
Black hole entropy remains a deep puzzle: where does such enormous amount of
entropy come from? Curiously, there exist gravitational configurations that
possess even larger entropy than a black hole of the same mass, in fact,
arbitrarily high entropy. These are the so-called monsters, which are
problematic to the Anti-de Sitter/Conformal Field Theory (AdS/CFT)
correspondence paradigm since there is far insufficient degrees of freedom on
the field theory side to account for the enormous entropy of monsters in AdS
bulk. The physics of the bulk however may be considerably modified at
semi-classical level due to the presence of branes. We show that this is
especially so since monster spacetimes are unstable due to brane nucleation. As
a consequence, it is not clear what the final fate of monsters is. We argue
that in some cases there is no real threat from monsters since although they
are solutions to Einstein's Field Equations, they are very likely to be
completely unstable when embedded in string theory, and thus probably are not
solutions to the full quantum theory of gravity. Our analysis, while suggestive
and supportive of the claim that such pathological objects are not allowed in
the final theory, by itself does not rule out all monsters. We comment on
various kin of monsters such as the bag-of-gold spacetime, and also discuss
briefly the implications of our work to some puzzles related to black hole
entropy.Comment: Version accepted by JHE
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