Edge-Fault Tolerance of Hypercube-like Networks

This paper considers a kind of generalized measure $\lambda_s^{(h)}$ of fault tolerance in a hypercube-like graph $G_n$ which contain several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes and M\"obius cubes, and proves $\lambda_s^{(h)}(G_n)= 2^h(n-h)$ for any $h$ with $0\leqslant h\leqslant n-1$ by the induction on $n$ and a new technique. This result shows that at least $2^h(n-h)$ edges of $G_n$ have to be removed to get a disconnected graph that contains no vertices of degree less than $h$. Compared with previous results, this result enhances fault-tolerant ability of the above-mentioned networks theoretically

GRB/GW association: Long-short GRB candidates, time-lag, measuring gravitational wave velocity and testing Einstein's equivalence principle

Short-duration gamma-ray bursts (SGRBs) are widely believed to be powered by the mergers of compact binaries, such as binary neutron stars or possibly neutron star-black hole binaries. Though the prospect of detecting SGRBs with gravitational wave (GW) signals by the advanced Laser Interferometer Gravitational-Wave Observatory (LIGO)/VIRGO network is promising, no known SGRB has been found within the expected advanced LIGO/VIRGO sensitivity range for binary neutron star systems. We find, however, that the two long-short GRBs (GRB 060505 and GRB 060614) may be within the horizon of advanced GW detectors. In the upcoming era of GW astronomy, the merger origin of some long-short GRBs, as favored by the macronova signature displayed in GRB 060614, can be unambiguously tested. The model-dependent time lags between the merger and the onset of the prompt emission of the GRB are estimated. The comparison of such time lags between model predictions and the real data expected in the era of the GW astronomy would be helpful in revealing the physical processes taking place at the central engine (including the launch of the relativistic outflow, the emergence of the outflow from the dense material ejected during the merger, and the radiation of gamma rays). We also show that the speed of GWs, with or without a simultaneous test of Einstein's equivalence principle, can be directly measured to an accuracy of $\sim 3\times 10^{-8}~{\rm cm~s^{-1}}$ or even better in the advanced LIGO/VIRGO era. The Astrophysical Journal, VolumeComment: 12 pages, 3 figures, published in The Astrophysical Journa

Calculation of the Branching Ratio of B−→hc+K−B^{-}\to h_{c}+K^{-} in PQCD

The branching ratio of $B^-\to h_c+K^-$ is re-evaluated in the PQCD approach. In this theoretical framework all the phenomenological parameters in the wavefunctions and Sudakov factor are priori fixed by fitting other experimental data, and in the whole numerical computations we do not introduce any new parameter. Our results are consistent with the upper bounds set by the Babar and Belle measurements.Comment: 12 pages, 1 figure, version to appear in Phys. Rev.