A simple proof of exponential decay in the two dimensional percolation model

Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that $p_c=0.5$ for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten's theorem.Comment: 9 pages and one figur

Context-Aware Single-Shot Detector

SSD is one of the state-of-the-art object detection algorithms, and it combines high detection accuracy with real-time speed. However, it is widely recognized that SSD is less accurate in detecting small objects compared to large objects, because it ignores the context from outside the proposal boxes. In this paper, we present CSSD--a shorthand for context-aware single-shot multibox object detector. CSSD is built on top of SSD, with additional layers modeling multi-scale contexts. We describe two variants of CSSD, which differ in their context layers, using dilated convolution layers (DiCSSD) and deconvolution layers (DeCSSD) respectively. The experimental results show that the multi-scale context modeling significantly improves the detection accuracy. In addition, we study the relationship between effective receptive fields (ERFs) and the theoretical receptive fields (TRFs), particularly on a VGGNet. The empirical results further strengthen our conclusion that SSD coupled with context layers achieves better detection results especially for small objects ($+3.2\% {\rm AP}_{@0.5}$ on MS-COCO compared to the newest SSD), while maintaining comparable runtime performance

Observe matter falling into a black hole

It has been well known that in the point of view of a distant observer, all in-falling matter to a black hole (BH) will be eventually stalled and "frozen" just outside the event horizon of the BH, although an in-falling observer will see the matter falling straight through the event horizon. Thus in this "frozen star" scenario, as distant observers, we could never observe matter falling into a BH, neither could we see any "real" BH other than primordial ones, since all other BHs are believed to be formed by matter falling towards singularity. Here we first obtain the exact solution for a pressureless mass shell around a pre-existing BH. The metrics inside and interior to the shell are all different from the Schwarzschild metric of the enclosed mass. The metric interior to the shell can be transformed to the Schwarzschild metric for a slower clock which is dependent of the location and mass of the shell. Another result is that there does not exist a singularity nor event horizon in the shell. Therefore the "frozen star" scenario is incorrect. We also show that for all practical astrophysical settings the in-falling time recorded by an external observer is sufficiently short that future astrophysical instruments may be able to follow the whole process of matter falling into BHs. The distant observer could not distinguish between a "real" BH and a "frozen star", until two such objects merge together. It has been proposed that electromagnetic waves will be produced when two "frozen stars" merge together, but not true when two "real" bare BHs merge together. However gravitational waves will be produced in both cases. Thus our solution is testable by future high sensitivity astronomical observations.Comment: 7 pages, 2 figures. Proceeding of the conference "Astrophysics of Compact Objects", 1-7 July, Huangshan, China. Abridged abstrac