13,191 research outputs found
Angled decompositions of arborescent link complements
This paper describes a way to subdivide a 3-manifold into angled blocks,
namely polyhedral pieces that need not be simply connected. When the individual
blocks carry dihedral angles that fit together in a consistent fashion, we
prove that a manifold constructed from these blocks must be hyperbolic. The
main application is a new proof of a classical, unpublished theorem of Bonahon
and Siebenmann: that all arborescent links, except for three simple families of
exceptions, have hyperbolic complements.Comment: 42 pages, 23 figures. Slightly expanded exposition and reference
Kepler-10 c: a 2.2 Earth Radius Transiting Planet in a Multiple System
The Kepler mission has recently announced the discovery of Kepler-10 b, the smallest exoplanet discovered to date and the first rocky planet found by the spacecraft. A second, 45 day period transit-like signal present in the photometry from the first eight months of data could not be confirmed as being caused by a planet at the time of that announcement. Here we apply the light curve modeling technique known as BLENDER to explore the possibility that the signal might be due to an astrophysical false positive (blend). To aid in this analysis we report the observation of two transits with the Spitzer Space Telescope at 4.5 ÎŒm. When combined, they yield a transit depth of 344 ± 85 ppm that is consistent with the depth in the Kepler passband (376 ± 9 ppm, ignoring limb darkening), which rules out blends with an eclipsing binary of a significantly different color than the target. Using these observations along with other constraints from high-resolution imaging and spectroscopy, we are able to exclude the vast majority of possible false positives. We assess the likelihood of the remaining blends, and arrive conservatively at a false alarm rate of 1.6 Ă 10^(â5) that is small enough to validate the candidate as a planet (designated Kepler-10 c) with a very high level of confidence. The radius of this object is measured to be R_p = 2.227^(+0.052)_(â0.057) R_â (in which the error includes the uncertainty in the stellar properties), but currently available radial-velocity measurements only place an upper limit on its mass of about 20 M_â. Kepler-10 c represents another example (with Kepler-9 d and Kepler-11 g) of statistical "validation" of a transiting exoplanet, as opposed to the usual "confirmation" that can take place when the Doppler signal is detected or transit timing variations are measured. It is anticipated that many of Kepler's smaller candidates will receive a similar treatment since dynamical confirmation may be difficult or impractical with the sensitivity of current instrumentation
Compact convex sets of the plane and probability theory
The Gauss-Minkowski correspondence in states the existence of
a homeomorphism between the probability measures on such that
and the compact convex sets (CCS) of the plane
with perimeter~1. In this article, we bring out explicit formulas relating the
border of a CCS to its probability measure. As a consequence, we show that some
natural operations on CCS -- for example, the Minkowski sum -- have natural
translations in terms of probability measure operations, and reciprocally, the
convolution of measures translates into a new notion of convolution of CCS.
Additionally, we give a proof that a polygonal curve associated with a sample
of random variables (satisfying ) converges
to a CCS associated with at speed , a result much similar to
the convergence of the empirical process in statistics. Finally, we employ this
correspondence to present models of smooth random CCS and simulations
Fast quantum control in dissipative systems using dissipationless solutions
We report on a systematic geometric procedure, built up on solutions designed
in the absence of dissipation, to mitigate the effects of dissipation in the
control of open quantum systems. Our method addresses a standard class of open
quantum systems modeled by non-Hermitian Hamiltonians. It provides the
analytical expression of the extra magnetic field to be superimposed to the
driving field in order to compensate the geometric distortion induced by
dissipation, and produces an exact geometric optimization of fast population
transfer. Interestingly, it also preserves the robustness properties of
protocols originally optimized against noise. Its extension to two interacting
spins restores a fidelity close to unity for the fast generation of Bell state
in the presence of dissipation
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