Molecular gas associated with the IRAS-Vela shell

We present a survey of molecular gas in the J = 1 -> 0 transition of 12CO towards the IRAS Vela shell. The shell, previously identified from IRAS maps, is a ring-like structure seen in the region of the Gum Nebula. We confirm the presence of molecular gas associated with some of the infrared point sources seen along the Shell. We have studied the morphology and kinematics of the gas and conclude that the shell is expanding at the rate of ~ 13 km/s from a common center. We go on to include in this study the Southern Dark Clouds seen in the region. The distribution and motion of these objects firmly identify them as being part of the shell of molecular gas. Estimates of the mass of gas involved in this expansion reveal that the shell is a massive object comparable to a GMC. From the expansion and various other signatures like the presence of bright-rimmed clouds with head-tail morphology, clumpy distribution of the gas etc., we conjecture that the molecular gas we have detected is the remnant of a GMC in the process of being disrupted and swept outwards through the influence of a central OB association, itself born of the parent cloud.Comment: 21 pages, 9 figures. Figure 1 is a separate jpeg file. To appear in Journal of Astrophysics and Astronom

Improved bounds and algorithms for graph cuts and network reliability

Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial approximation scheme to estimate the probability that a graph $G$ becomes disconnected, given that its edges are removed independently with probability $p$. This algorithm runs in $n^{5+o(1)} \epsilon^{-3}$ time to obtain an estimate within relative error $\epsilon$. We improve this run-time through algorithmic and graph-theoretic advances. First, there is a certain key sub-problem encountered by Karger, for which a generic estimation procedure is employed, we show that this has a special structure for which a much more efficient algorithm can be used. Second, we show better bounds on the number of edge cuts which are likely to fail. Here, Karger's analysis uses a variety of bounds for various graph parameters, we show that these bounds cannot be simultaneously tight. We describe a new graph parameter, which simultaneously influences all the bounds used by Karger, and obtain much tighter estimates of the cut structure of $G$. These techniques allow us to improve the runtime to $n^{3+o(1)} \epsilon^{-2}$, our results also rigorously prove certain experimental observations of Karger & Tai (Proc. ACM-SIAM Symposium on Discrete Algorithms, 1997). Our rigorous proofs are motivated by certain non-rigorous differential-equation approximations which, however, provably track the worst-case trajectories of the relevant parameters. A key driver of Karger's approach (and other cut-related results) is a bound on the number of small cuts: we improve these estimates when the min-cut size is "small" and odd, augmenting, in part, a result of Bixby (Bulletin of the AMS, 1974)

The Moser-Tardos Framework with Partial Resampling

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994)

Algorithmic and enumerative aspects of the Moser-Tardos distribution

Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables which arise during intermediate stages of MT. We show that these configurations have a more or less "random" form, building further on the "MT-distribution" concept of Haeupler et al. in understanding the (intermediate and) output distribution of MT. This has a variety of algorithmic applications; the most important is that bad events can be found relatively quickly, improving upon MT across the complexity spectrum: it makes some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which are of basic combinatorial interest), gives lower-degree polynomial run-times in some settings, transforms certain super-polynomial-time algorithms into polynomial-time ones, and leads to Las Vegas algorithms for some coloring problems for which only Monte Carlo algorithms were known. We show that in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution which avoids most of the bad events. We show in some cases this MT variant can run faster than the original MT algorithm itself, and develop the first-known criterion for the case of the asymmetric LLL. This can be used to find partial Latin transversals -- improving upon earlier bounds of Stein (1975) -- among other applications. We furthermore give applications in enumeration, showing that most applications (where we aim for all or most of the bad events to be avoided) have many more solutions than known before by proving that the MT-distribution has "large" min-entropy and hence that its support-size is large

A High Galactic Latitude HI 21cm-line Absorption Survey using the GMRT: II. Results and Interpretation

We have carried out a sensitive high-latitude (|b| > 15deg.) HI 21cm-line absorption survey towards 102 sources using the GMRT. With a 3-sigma detection limit in optical depth of ~0.01, this is the most sensitive HI absorption survey. We detected 126 absorption features most of which also have corresponding HI emission features in the Leiden Dwingeloo Survey of Galactic neutral Hydrogen. The histogram of random velocities of the absorption features is well-fit by two Gaussians centered at V(lsr) ~ 0 km/s with velocity dispersions of 7.6 +/- 0.3 km/s and 21 +/- 4 km/s respectively. About 20% of the HI absorption features form the larger velocity dispersion component. The HI absorption features forming the narrow Gaussian have a mean optical depth of 0.20 +/- 0.19, a mean HI column density of (1.46 +/- 1.03) X 10^{20} cm^{-2}, and a mean spin temperature of 121 +/- 69 K. These HI concentrations can be identified with the standard HI clouds in the cold neutral medium of the Galaxy. The HI absorption features forming the wider Gaussian have a mean optical depth of 0.04 +/- 0.02, a mean HI column density of (4.3 +/- 3.4) X 10^{19} cm^{-2}, and a mean spin temperature of 125 +/- 82 K. The HI column densities of these fast clouds decrease with their increasing random velocities. These fast clouds can be identified with a population of clouds detected so far only in optical absorption and in HI emission lines with a similar velocity dispersion. This population of fast clouds is likely to be in the lower Galactic Halo.Comment: 19 pages, 19 figures. Accepted for publication in Journal of Astrophysics & Astronom

Adaptive multigrid domain decomposition solutions for viscous interacting flows

Several viscous incompressible flows with strong pressure interaction and/or axial flow reversal are considered with an adaptive multigrid domain decomposition procedure. Specific examples include the triple deck structure surrounding the trailing edge of a flat plate, the flow recirculation in a trough geometry, and the flow in a rearward facing step channel. For the latter case, there are multiple recirculation zones, of different character, for laminar and turbulent flow conditions. A pressure-based form of flux-vector splitting is applied to the Navier-Stokes equations, which are represented by an implicit lowest-order reduced Navier-Stokes (RNS) system and a purely diffusive, higher-order, deferred-corrector. A trapezoidal or box-like form of discretization insures that all mass conservation properties are satisfied at interfacial and outflow boundaries, even for this primitive-variable, non-staggered grid computation

Partial resampling to approximate covering integer programs

We consider column-sparse covering integer programs, a generalization of set cover, which have a long line of research of (randomized) approximation algorithms. We develop a new rounding scheme based on the Partial Resampling variant of the Lov\'{a}sz Local Lemma developed by Harris & Srinivasan (2019). This achieves an approximation ratio of $1 + \frac{\ln (\Delta_1+1)}{a_{\min}} + O\Big( \log(1 + \sqrt{ \frac{\log (\Delta_1+1)}{a_{\min}}} \Big)$, where $a_{\min}$ is the minimum covering constraint and $\Delta_1$ is the maximum $\ell_1$-norm of any column of the covering matrix (whose entries are scaled to lie in $[0,1]$). When there are additional constraints on the variable sizes, we show an approximation ratio of $\ln \Delta_0 + O(\log \log \Delta_0)$ (where $\Delta_0$ is the maximum number of non-zero entries in any column of the covering matrix). These results improve asymptotically, in several different ways, over results of Srinivasan (2006) and Kolliopoulos & Young (2005). We show nearly-matching inapproximability and integrality-gap lower bounds. We also show that the rounding process leads to negative correlation among the variables, which allows us to handle multi-criteria programs