7,464 research outputs found
A Spatial Structural Derivative Model for Ultraslow Diffusion
This study investigates the ultraslow diffusion by a spatial structural
derivative, in which the exponential function exp(x)is selected as the
structural function to construct the local structural derivative diffusion
equation model. The analytical solution of the diffusion equation is a form of
Biexponential distribution. Its corresponding mean squared displacement is
numerically calculated, and increases more slowly than the logarithmic function
of time. The local structural derivative diffusion equation with the structural
function exp(x)in space is an alternative physical and mathematical modeling
model to characterize a kind of ultraslow diffusion.Comment: 13 pages, 3 figure
Statistical Properties of Multiple Optical Emission Components in Gamma-Ray Bursts and Implications
Well-sampled optical lightcurves of 146 gamma-ray bursts (GRBs) are complied
from the literature. Multiple optical emission components are extracted with
power-law function fits to these lightcurves. We present a systematical
analysis for statistical properties and their relations to prompt gamma-ray
emission and X-ray afterglow for each component. We show that peak luminosity
in the prompt and late flares are correlated and the evolution of the peak
luminosity may signal the evolution of the accretion rate. No tight correlation
between the shallow decay phase/plateau and prompt gamma-ray emission is found.
Assuming that they are due to a long-lasting wind injected by a compact object,
we show that the injected behavior favors the scenarios of a long-lasting wind
after the main burst episode. The peak luminosity of the afterglow onset is
tightly correlated with Eiso, and it is dimmer as peaking later. Assuming that
the onset bump is due to the fireball deceleration by the external medium, we
examine the Gamma_0-Eiso relation and find that it is confirmed with the
current sample. Optical re-brightening is observed in 30 GRBs in our sample. It
shares the same relation between the width and the peak time as found in the
onset bump, but no clear correlation between the peak luminosity and Eiso as
observed in the onset bumps is found. Although its peak luminosity also decays
with time, the slope is much shallower than that of the onset peak. We get L
t^{-1}_{p}$, being consistent with off-axis observations to an expanding
external fireball in a wind-like circum medium. The late re-brightening may
signal another jet component. Mixing of different emission components may be
the reason for the observed chromatic breaks in different energy bands.Comment: 10 pages, 5 figures, to be published by IJMPD (Proceedings of "The
Third Galileo - Xu Guangqi meeting", Beijing, October 11-15, 2011
Nearly Optimal Stochastic Approximation for Online Principal Subspace Estimation
Processing streaming data as they arrive is often necessary for high
dimensional data analysis. In this paper, we analyse the convergence of a
subspace online PCA iteration, as a followup of the recent work of Li, Wang,
Liu, and Zhang [Math. Program., Ser. B, DOI 10.1007/s10107-017-1182-z] who
considered the case for the most significant principal component only, i.e., a
single vector. Under the sub-Gaussian assumption, we obtain a finite-sample
error bound that closely matches the minimax information lower bound of Vu and
Lei [Ann. Statist. 41:6 (2013), 2905-2947].Comment: 37 page
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