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