17,950 research outputs found
Negative association, ordering and convergence of resampling methods
We study convergence and convergence rates for resampling schemes. Our first
main result is a general consistency theorem based on the notion of negative
association, which is applied to establish the almost-sure weak convergence of
measures output from Kitagawa's (1996) stratified resampling method. Carpenter
et al's (1999) systematic resampling method is similar in structure but can
fail to converge depending on the order of the input samples. We introduce a
new resampling algorithm based on a stochastic rounding technique of Srinivasan
(2001), which shares some attractive properties of systematic resampling, but
which exhibits negative association and therefore converges irrespective of the
order of the input samples. We confirm a conjecture made by Kitagawa (1996)
that ordering input samples by their states in yields a faster
rate of convergence; we establish that when particles are ordered using the
Hilbert curve in , the variance of the resampling error is
under mild conditions, where
is the number of particles. We use these results to establish asymptotic
properties of particle algorithms based on resampling schemes that differ from
multinomial resampling.Comment: 54 pages, including 30 pages of supplementary materials (a typo in
Algorithm 1 has been corrected
Planetesimal and Protoplanet Dynamics in a Turbulent Protoplanetary Disk: Ideal Stratified Disks
Due to the gravitational influence of density fluctuations driven by
magneto-rotational instability in the gas disk, planetesimals and protoplanets
undergo diffusive radial migration as well as changes in other orbital
properties. The magnitude of the effect on particle orbits can have important
consequences for planet formation scenarios. We use the local-shearing-box
approximation to simulate an ideal, isothermal, magnetized gas disk with
vertical density stratification and simultaneously evolve numerous massless
particles moving under the gravitational field of the gas and the host star. We
measure the evolution of the particle orbital properties, including mean
radius, eccentricity, inclination, and velocity dispersion, and its dependence
on the disk properties and the particle initial conditions. Although the
results converge with resolution for fixed box dimensions, we find the response
of the particles to the gravity of the turbulent gas correlates with the
horizontal box size, up to 16 disk scale heights. This correlation indicates
that caution should be exercised when interpreting local-shearing-box models
involving gravitational physics of magneto-rotational turbulence. Based on
heuristic arguments, nevertheless, the criterion L_h / R ~ O(1), where L_h is
the horizontal box size and R is the distance to the host star, is proposed to
possibly circumvent this conundrum. If this criterion holds, we can still
conclude that magneto-rotational turbulence seems likely to be ineffective at
driving either diffusive migration or collisional erosion under most
circumstances.Comment: Accepted to ApJ. Major expansion in Secs. 2.1 & 2.2 and new Sec. 4.
The family Floer functor is faithful
Family Floer theory yields a functor from the Fukaya category of a symplectic
manifold admitting a Lagrangian torus fibration to a (twisted) category of
perfect complexes on the mirror rigid analytic space. This functor is shown to
be faithful by a degeneration argument involving moduli spaces of annuli.Comment: 70 pages, 24 figures. Final version, with substantially enhanced
exposition, accepted for publication at JEM
On Hydrodynamic Motions in Dead Zones
We investigate fluid motions near the midplane of vertically stratified
accretion disks with highly resistive midplanes. In such disks, the
magnetorotational instability drives turbulence in thin layers surrounding a
resistive, stable dead zone. The turbulent layers in turn drive motions in the
dead zone. We examine the properties of these motions using three-dimensional,
stratified, local, shearing-box, non-ideal, magnetohydrodynamical simulations.
Although the turbulence in the active zones provides a source of vorticity to
the midplane, no evidence for coherent vortices is found in our simulations. It
appears that this is because of strong vertical oscillations in the dead zone.
By analyzing time series of azimuthally-averaged flow quantities, we identify
an axisymmetric wave mode particular to models with dead zones. This mode is
reduced in amplitude, but not suppressed entirely, by changing the equation of
state from isothermal to ideal. These waves are too low-frequency to affect
sedimentation of dust to the midplane, but may have significance for the
gravitational stability of the resulting midplane dust layers.Comment: 36 pages, 19 figures. ApJ accepte
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