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A barrier certificate is an inductive invariant function which can be used for the safety verification of a hybrid system. Safety verification based on barrier certificate has the benefit of avoiding explicit computation of the exact reachable set which is usually intractable for nonlinear hybrid systems. In this paper, we propose a new barrier certificate condition, called Exponential Condition, for the safety verification of semi-algebraic hybrid systems. The most important benefit of Exponential Condition is that it has a lower conservativeness than the existing convex condition and meanwhile it possesses the property of convexity. On the one hand, a less conservative barrier certificate forms a tighter over-approximation for the reachable set and hence is able to verify critical safety properties. On the other hand, the property of convexity guarantees its solvability by semidefinite programming method. Some examples are presented to illustrate the effectiveness and practicality of our method., Comment: 18 pages, 7 figures

In this paper we present an improved dqds algorithm for computing all the singular values of a bidiagonal matrix to high relative accuracy. There are two key contributions: a novel deflation strategy that improves the convergence for badly scaled matrices, and some modifications to certain shift strategies that accelerate the convergence for most bidiagonal matrices. These techniques together ensure linear worst case complexity of the improved algorithm (denoted by V5). Our extensive numerical experiments indicate that V5 is typically 1.2x--4x faster than DLASQ (the LAPACK-3.4.0 implementation of dqds) without any degradation in accuracy. On matrices for which DLASQ shows very slow convergence, V5 can be 3x--10x faster. At the end of this paper, a hybrid algorithm (HDLASQ) is developed by combining our improvements with the aggressive early deflation strategy (AggDef2 in [SIAM J. Matrix Anal. Appl., 33(2012), 22-51]). Numerical results show that HDLASQ is the fastest among these different versions., Comment: submitted to SISC

We present the LU decomposition with panel rank revealing pivoting (LU_PRRP), an LU factorization algorithm based on strong rank revealing QR panel factorization. LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP). Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases and easily solves the Wilkinson matrix and the Foster matrix. We also present CALU_PRRP, a communication avoiding version of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail., Comment: No. RR-7867 (2012)

In this article, we propose an algorithm, NESTA-LASSO, for the LASSO problem, i.e., an underdetermined linear least-squares problem with a 1-norm constraint on the solution. We prove under the assumption of the restricted isometry property (RIP) and a sparsity condition on the solution, that NESTA-LASSO is guaranteed to be almost always locally linearly convergent. As in the case of the algorithm NESTA proposed by Becker, Bobin, and Candes, we rely on Nesterov's accelerated proximal gradient method, which takes O(e^{-1/2}) iterations to come within e > 0 of the optimal value. We introduce a modification to Nesterov's method that regularly updates the prox-center in a provably optimal manner, and the aforementioned linear convergence is in part due to this modification. In the second part of this article, we attempt to solve the basis pursuit denoising BPDN problem (i.e., approximating the minimum 1-norm solution to an underdetermined least squares problem) by using NESTA-LASSO in conjunction with the Pareto root-finding method employed by van den Berg and Friedlander in their SPGL1 solver. The resulting algorithm is called PARNES. We provide numerical evidence to show that it is comparable to currently available solvers., Comment: 22 pages, 1 figure

High resolution X-ray absorption spectroscopy is a powerful diagnostic tool for probing chemical and physical properties of the interstellar medium (ISM) at various phases. We present detections of K transition absorption lines from the low ionization ions of OI, OII, NeI, NeII, and NeIII, and the high ionization ones of OVI, OVII, OVIII, NeIX, and MgXI, as well as details of neutral absorption edges from Mg, Ne, and O in an unprecedented high quality spectrum of the low mass X-ray binary Cyg X-2. These absorption features trace the intervening interstellar medium which is indicated by the unshifted line centroids with respect to the rest frame wavelengths of the corresponding atomic transitions. We have measured the column densities of each ion. We complement these measurements with the radio HI and optical Halpha observations toward the same sight line and estimate the mean abundances of Ne, O, and Mg in the cool phase to Ne/H=0.84^{+0.13}_{-0.10}\times10^{-4}, O/H=3.83^{+0.48}_{-0.43}\times10^{-4}, and Mg/H=0.35^{+0.09}_{-0.11}\times10^{-4}, and O and Mg in the hot phase to O/H=5.81^{+1.30}_{-1.34}\times10^{-4} and Mg/H=0.33^{+0.09}_{-0.09}\times10^{-4}, respectively. These results indicate a mild depletion of oxygen into dust grains in the cool phase and little or no depletion of magnesium. We also find that absorption from highly ionized ions in the hot Galactic disk gas can account for most of the absorption observed toward the extragalactic sight lines like Mrk 421. The bulk of the observed OVI likely originates from the conductive interfaces between the cool and hot gases, from which a significant amount of NV and CIV emission is predicted., Comment: Accepted for publication in ApJ, 13 pages, 6 figures, and 7 tables

This paper has been withdrawn by the authors due to its publication, Comment: This paper has been withdrawn

We provide improved atomic calculation of wavelengths, oscillator strengths, and autoionization rates relevant to the $2\to 3$ inner-shell transitions of Fe~VI--XVI, the so-called Fe~M-shell unresolved transition array (UTA). A second order many-body perturbation theory is employed to obtain accurate transition wavelengths, which are systematically larger than previous theoretical results by 15--45~m{\AA}. For a few transitions of Fe~XVI and Fe~XV where laboratory measurements exist, our new wavelengths are accurate to within a few m{\AA}. Using these new calculations, the apparent discrepancy in the velocities between the Fe~M-shell UTA and other highly ionized absorption lines in the outflow of NGC 3783 disappears. The oscillator strengths in our new calculation agree well with the previous theoretical data, while the new autoionization rates are significantly larger, especially for lower charge states. We attribute this discrepancy to the missing autoionization channels in the previous calculation. The increased autoionization rates may slightly affect the column density analysis of the Fe~M-shell UTA for sources with high column density and very low turbulent broadening. The complete set of atomic data is provided as an electronic table., Comment: Accepted for publication in ApJ

We present high resolution laboratory spectra of K-shell X-ray lines from inner-shell excited and ionized ions of oxygen, obtained with a reflection grating spectrometer on the electron beam ion trap (EBIT-I) at the Lawrence Livermore National Laboratory. Only with a multi-ion model including all major atomic collisional and radiative processes, are we able to identify the observed K-shell transitions of oxygen ions from \ion{O}{3} to \ion{O}{6}. The wavelengths and associated errors for some of the strongest transitions are given, taking into account both the experimental and modeling uncertainties. The present data should be useful in identifying the absorption features present in astrophysical sources, such as active galactic nuclei and X-ray binaries. They are also useful in providing benchmarks for the testing of theoretical atomic structure calculations., Comment: 17 pages, 2 figures, to appear in ApJ

We report on the observation of OVII Doppler line broadening in a compact knot at the edge of SN 1006 detected with the Reflective Grating Spectrometer on board XMM-Newton. The observed line width of sigma = 3.4+/-0.5 eV at a line energy of 574 eV indicates an oxygen temperature of kT = 528+/-150 keV. Combined with the observed electron temperature of 1.5 keV the observed broadening is direct evidence for temperature non-equilibration in high Mach number shocks, and slow subsequent equilibration. The OVII line emission allows an accurate determination of the ionization state of the plasma, which is characterized by a relatively high forbidden line contribution, indicating log(n_e t) ~ 9.2., Comment: Accepted by Astrophysical Journal Letters, 4 pages, 4 figures. One of the original figures has been replaced by a color version (Fig. 1a)

High resolution spectroscopic observations of Seyfert galaxies with Chandra and XMM-Newton allows us to study the detailed ionization and thermal structures of the X-ray absorbing/emitting material in the circumnuclear environment. The vast improvement in the spectral resolving power by more than an order of magnitude enables us, for the first time, to unambiguously distinguish the dominant line emission mechanisms and to measure its dynamical properties as well. The X-ray band harbors spectral transitions from a wide range of ionization states, including valence-shell transitions in K-shell and L-shell ions from most cosmically abundant elements, as well as inner-shell transitions of iron and other mid-Z elements, which can be probed through absorption measurements. The X-ray spectrum, therefore, provides simultaneous velocity and column density constraints of highly ionized to only slightly ionized gas harbored in many of these systems. We summarize recent results that have emerged from observations of Seyfert 2 galaxies with the grating spectrometers onboard Chandra and XMM-Newton. We give particular emphasis to an empirical physical model that we have developed based on the observed spectra, and how it can be used for comparative studies with Seyfert 1 galaxies to test the AGN unification scenarios., Comment: 8 pages, 7 figures. Invited review. To appear in the proceedings of the workshop "X-ray spectroscopy of AGN with Chandra and XMM-Newton", eds., Th. Boller, S. Komossa, S. Kahn, and H. Kunieda

We derive a semi-analytic solution for the structure of conduction-mediated transition layers above an X-ray illuminated accretion disk, and calculate explicitly the X-ray line radiation resulting from both resonance line scattering and radiative recombination in these layers. The vertical thermal structure of the illuminated disk is found to depend on the illuminating continuum: for a hard power law continuum, there are two stable phases connected by a single transition layer; while for a softer continuum, there may exist three stable phases connected by two separate transition layers, with an intermediate stable layer in between. We show that the structure can be written as a function of the electron scattering optical depth through these layers, which leads to unique predictions of the equivalent width of the resulting line radiation from both recombination cascades and resonance line scattering. We find that resonance line scattering plays an important role, especially for the case where there is no intermediate stable layer., Comment: 27 pages, 6 figures, submitted to ApJ