Very deep spectroscopy of the bright Saturn Nebula NGC 7009 -- I. Observations and plasma diagnostics

We present very deep CCD spectrum of the bright, medium-excitation planetary nebula NGC 7009, with a wavelength coverage from 3040 to 11000 A. Traditional emission line identification is carried out to identify all the emission features in the spectra, based on the available laboratory atomic transition data. Since the spectra are of medium resolution, we use multi-Gaussian line profile fitting to deblend faint blended lines, most of which are optical recombination lines (ORLs) emitted by singly ionized ions of abundant second-row elements such as C, N, O and Ne. Computer-aided emission-line identification, using the code EMILI developed by Sharpee et al., is then employed to further identify all the emission lines thus obtained. In total about 1200 emission features are identified, with the faintest ones down to fluxes 10^{-4} of H_beta. The flux errors for all emission lines, estimated from multi-Gaussian fitting, are presented. Plots of the whole optical spectrum, identified emission lines labeled, are presented along with the results of multi-Gaussian fits. Plasma diagnostics using optical forbidden line ratios are carried out. Also derived are electron temperatures and densities from the H I, He I and He II recombination spectrum.Comment: 66 pages, 16 figures, 7 tables, paper accepted by MNRAS in Marc

A Bi-Hamiltonian Formulation for Triangular Systems by Perturbations

A bi-Hamiltonian formulation is proposed for triangular systems resulted by perturbations around solutions, from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed, provided that one operator of the Hamiltonian pair is invertible. Through our formulation, four examples of triangular systems are exhibited, which also show that bi-Hamiltonian systems in both lower dimensions and higher dimensions are many and varied. Two of four examples give local 2+1 dimensional bi-Hamiltonian systems and illustrate that multi-scale perturbations can lead to higher-dimensional bi-Hamiltonian systems.Comment: 16 pages, to appear in J. Math. Phy

Optical recombination lines as probes of conditions in planetary nebulae

Since the last IAU symposium on planetary nebulae (PNe), several deep spectroscopic surveys of the relatively faint optical recombination lines (ORLs) emitted by heavy element ions in PNe and H II regions have been completed. New diagnostic tools have been developed thanks to progress in the calculations of basic atomic data. Together, they have led to a better understanding of the physical conditions under which the various types of emission lines arise. The studies have strengthened the previous conjecture that nebulae contain another component of cold, high metallicity gas, which is too cool to excite any significant optical or UV CELs and is thus invisible via such lines. The existence of such a plasma component in PNe and possibly also in H II regions provides a natural solution to the long-standing problem in nebular astrophysics, i.e. the dichotomy of nebular plasma diagnostics and abundance determinations using ORLs and continua on the one hand and collisionally excited lines (CELs) on the other.Comment: 8 pages, 3 figures, review talk presented to the IAU Symposium #234, ``Planetary nebulae in our Galaxy and beyond'', held in Hawaii, USA, April 3-7 200

Towards efficient SimRank computation on large networks

SimRank has been a powerful model for assessing the similarity of pairs of vertices in a graph. It is based on the concept that two vertices are similar if they are referenced by similar vertices. Due to its self-referentiality, fast SimRank computation on large graphs poses significant challenges. The state-of-the-art work [17] exploits partial sums memorization for computing SimRank in O(Kmn) time on a graph with n vertices and m edges, where K is the number of iterations. Partial sums memorizing can reduce repeated calculations by caching part of similarity summations for later reuse. However, we observe that computations among different partial sums may have duplicate redundancy. Besides, for a desired accuracy ϵ, the existing SimRank model requires K = [logC ϵ] iterations [17], where C is a damping factor. Nevertheless, such a geometric rate of convergence is slow in practice if a high accuracy is desirable. In this paper, we address these gaps. (1) We propose an adaptive clustering strategy to eliminate partial sums redundancy (i.e., duplicate computations occurring in partial sums), and devise an efficient algorithm for speeding up the computation of SimRank to 0(Kdn2) time, where d is typically much smaller than the average in-degree of a graph. (2) We also present a new notion of SimRank that is based on a differential equation and can be represented as an exponential sum of transition matrices, as opposed to the geometric sum of the conventional counterpart. This leads to a further speedup in the convergence rate of SimRank iterations. (3) Using real and synthetic data, we empirically verify that our approach of partial sums sharing outperforms the best known algorithm by up to one order of magnitude, and that our revised notion of SimRank further achieves a 5X speedup on large graphs while also fairly preserving the relative order of original SimRank scores

Interlacing Log-concavity of the Boros-Moll Polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.Comment: 10 page

GRB 060206: hints of precession of the central engine?

Aims. The high-redshift (z=4.048) gamma-ray burst GRB 060206 showed unusual behavior, with a significant rebrightening by a factor of ~4 at about 3000 s after the burst. We argue that this rebrightening implies that the central engine became active again after the main burst produced by the first ejecta, then drove another more collimated jet-like ejecta with a larger viewing angle. The two ejecta both interacted with the ambient medium, giving rise to forward shocks that propagated into the ambient medium and reverse shocks that penetrated into the ejecta. The total emission was a combination of the emissions from the reverse- and forward- shocked regions. We discuss how this combined emission accounts for the observed rebrightening. Methods. We apply numerical models to calculate the light curves from the shocked regions, which include a forward shock originating in the first ejecta and a forward-reverse shock for the second ejecta. Results. We find evidence that the central engine became active again 2000 s after the main burst. The combined emission produced by interactions of these two ejecta with the ambient medium can describe the properties of the afterglow of this burst. We argue that the rapid rise in brightness at ~3000 s in the afterglow is due to the off-axis emission from the second ejecta. The precession of the torus or accretion disk of the central engine is a natural explanation for the departure of the second ejecta from the line of sight