39,525 research outputs found
The Detectability of Transit Depth Variations due to Exoplanetary Oblateness and Spin Precession
Knowledge of an exoplanet's oblateness and obliquity would give clues about
its formation and internal structure. In principle, a light curve of a
transiting planet bears information about the planet's shape, but previous work
has shown that the oblateness-induced signal will be extremely difficult to
detect. Here we investigate the potentially larger signals due to planetary
spin precession. The most readily detectable effects are transit depth
variations (TV) in a sequence of light curves. For a planet as oblate
as Jupiter or Saturn, the transit depth will undergo fractional variations of
order 1%. The most promising systems are those with orbital periods of
approximately 15--30 days, which is short enough for the precession period to
be less than about 40 years, and long enough to avoid spin-down due to tidal
friction. The detectability of the TV signal would be enhanced by moons
(which would decrease the precession period) or planetary rings (which would
increase the amplitude). The Kepler mission should find several planets for
which precession-induced TV signals will be detectable. Due to modeling
degeneracies, Kepler photometry would yield only a lower bound on oblateness.
The degeneracy could be lifted by observing the oblateness-induced asymmetry in
at least one transit light curve, or by making assumptions about the planetary
interior.Comment: Accepted for publication in The Astrophysical Journa
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
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Includes bibliographical references
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