On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations

We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical study (Luo and Hou, 2013).Comment: 23 page

Relationship Between the Kinetic Power and Bolometric Luminosity of Jets: limitation from black hole X-ray binaries, active galactic nuclei, and gamma-ray bursts

The correlation between the kinetic power $P_{\rm jet}$ and intrinsic bolometric luminosity $L_{\rm jet}$ of jets may reveal the underlying jet physics in various black hole systems. Based on the recent work by Nemmen et al. (2012), we re-investigate this correlation with additional sources of black-hole X-ray binaries (BXBs) in hard/quiescent states and low-luminosity active galactic nuclei (LLAGNs). The new sample includes 29 sets of data from 7 BXBs and 20 LLAGNs, with $P_{\rm jet}$ and $L_{\rm jet}$ being derived from spectral modeling of the quasi-simultaneous multi-band spectra under the accretion-jet scenario. Compared to previous works, the range of luminosity is now enlarged to more than $20$ decades, i.e. from $10^{31}{\rm erg/s}$ to $10^{52}{\rm erg/s}$, which allows for better constraining of the correlation. One notable result is that the jets in BXBs and LLAGNs almost follow the same $P_{\rm jet} - L_{\rm jet}$ correlation that was obtained from blazars and gamma-ray bursts (GRBs). The slope indices we derived are $1.03\pm0.01$ for the whole sample, $0.85\pm0.06$ for BXB subsample, $0.71\pm0.11$ for LLAGN subsample, and $1.01\pm0.05$ for the LLAGN-blazar subsample, respectively. The correlation index around unit implies the independence of jet efficiency on the luminosity or kinetic power. Our results may further support the hypothesis that similar physical processes exist in the jets of various black hole systems.Comment: 5 pages, 3 figures, to appear in ApJL, updated versio

Achieving quantum precision limit in adaptive qubit state tomography

The precision limit in quantum state tomography is of great interest not only to practical applications but also to foundational studies. However, little is known about this subject in the multiparameter setting even theoretically due to the subtle information tradeoff among incompatible observables. In the case of a qubit, the theoretic precision limit was determined by Hayashi as well as Gill and Massar, but attaining the precision limit in experiments has remained a challenging task. Here we report the first experiment which achieves this precision limit in adaptive quantum state tomography on optical polarization qubits. The two-step adaptive strategy employed in our experiment is very easy to implement in practice. Yet it is surprisingly powerful in optimizing most figures of merit of practical interest. Our study may have significant implications for multiparameter quantum estimation problems, such as quantum metrology. Meanwhile, it may promote our understanding about the complementarity principle and uncertainty relations from the information theoretic perspective.Comment: 9 pages, 4 figures; titles changed and structure reorganise