The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kahn. A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq 5$ then $${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and recently proved for all $d$ by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most $3$ the search could be done by hand, but for the case of maximum degree $4$ or $5$, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics

20 cm VLA Radio-Continuum Study of M31 - Images and Point Source Catalogues

We present a series of new high-sensitivity and high-resolution radio-continuum images of M31 at \lambda=20 cm (\nu=1.4 GHz). These new images were produced by merging archived 20 cm radio-continuum observations from the Very Large Array (VLA) telescope. Images presented here are sensitive to rms=60 \mu Jy and feature high angular resolution (<10"). A complete sample of discrete radio sources have been catalogued and analysed across 17 individual VLA projects. We identified a total of 864 unique discrete radio sources across the field of M31. One of the most prominent regions in M31 is the ring feature for which we estimated total integrated flux of 706 mJy at \lambda=20 cm. We compare here, detected sources to those listed in Gelfand et al. (2004) at \lambda=92 cm and find 118 sources in common to both surveys. The majority (61%) of these sources exhibit a spectral index of \alpha <-0.6 indicating that their emission is predominantly non-thermal in nature. That is more typical for background objects.Comment: 28 pages, 25 figures, accepted for publication in the Serbian Astronomical Journa

The ATLAS 9.0 GHz Survey of the Extended Chandra Deep Field South: The Faint 9.0 GHz Radio Population

We present a new image of the 9.0 GHz radio emission from the extended Chandra Deep Field South. A total of 181 hours of integration with the Australia Telescope Compact Array has resulted in a 0.276 square degree image with a median sensitivity of $\sim$20 $\mu$Jy/beam rms, for a synthesised beam of 4.0 $\times$ 1.3 arcsec. We present a catalogue of the 9.0 GHz radio sources, identifying 70 source components and 55 individual radio galaxies. Source counts derived from this sample are consistent with those reported in the literature. The observed source counts are also generally consistent with the source counts from simulations of the faint radio population. Using the wealth of multiwavelength data available for this region, we classify the faint 9 GHz population and find that 91% are radio loud AGN, 7% are radio quiet AGN and 2% are star forming galaxies. The 9.0 GHz radio sources were matched to 5.5 and 1.4 GHz sources in the literature and we find a significant fraction of flat or inverted spectrum sources, with 36% of the 9 GHz sources having $\alpha_{5.5GHz}^{9.0GHz}$ $>$ -0.3 (for $S \propto \nu^\alpha$). This flat or inverted population is not well reproduced by current simulations of radio source populations.Comment: 17 pages, 12 figures. Accepted for publication in MNRA

Discovery of magnetic fields along stacked cosmic filaments as revealed by radio and X-ray emission

Diffuse filaments connect galaxy clusters to form the cosmic web. Detecting these filaments could yield information on the magnetic field strength, cosmic ray population and temperature of intercluster gas, yet, the faint and large-scale nature of these bridges makes direct detections very challenging. Using multiple independent all-sky radio and X-ray maps we stack pairs of luminous red galaxies as tracers for cluster pairs. For the first time, we detect an average surface brightness between the clusters from synchrotron (radio) and thermal (X-ray) emission with 73 5\u3c3 significance, on physical scales larger than observed to date ( 653 Mpc). We obtain a synchrotron spectral index of \u3b1 43 -1.0 and estimates of the average magnetic field strength of 30 64 B 64 60 nG, derived from both equipartition and Inverse Compton arguments, implying a 5 to 15 per cent degree of field regularity when compared with Faraday rotation measure estimates. While the X-ray detection is inline with predictions, the average radio signal comes out higher than predicted by cosmological simulations and dark matter annihilation and decay models. This discovery demonstrates that there are connective structures between mass concentrations that are significantly magnetised, and the presence of sufficient cosmic rays to produce detectable synchrotron radiation

Forecasting Periods of Strong Southward Magnetic Field Following Interplanetary Shocks

Long periods of strong southward magnetic fields are known to be the primary cause of intense geomagnetic storms. The majority of such events are caused by the passage over Earth of a magnetic ejecta. Irrespective of the interplanetary cause, fast-forward shocks often precede such strong southward B$_{z}$ periods. Here, we first look at all long periods of strong southward magnetic fields as well as fast-forward shocks measured by the \textit{Wind} spacecraft in a 22.4-year span. We find that 76{\%} of strong southward B$_{z}$ periods are preceded within 48 hours by at least a fast-forward shock but only about 23{\%} of all shocks are followed within 48 hours by strong southward B$_{z}$ periods. Then, we devise a threshold-based probabilistic forecasting method based on the shock properties and the pre-shock near-Earth solar wind plasma and interplanetary magnetic field characteristics adopting a `superposed epoch analysis'-like approach. Our analysis shows that the solar wind conditions in the 30 minutes interval around the arrival of fast-forward shocks have a significant contribution to the prediction of long-duration southward B$_{z}$ periods. This probabilistic model may provide on average a 14-hour warning time for an intense and long-duration southward B$_{z}$ period. Evaluating the forecast capability of the model through a statistical and skill score-based approach reveals that it outperforms a coin-flipping forecast. By using the information provided by the arrival of a fast-forward shock at L1, this model represents a marked improvement over similar forecasting methods. We outline a number of future potential improvements.Comment: published in Space Weather, 22 Nov 201

Towards the probabilistic analysis of small bowel capsule endoscopy features to predict severity of duodenal histology in patients with villous atrophy

Small bowel capsule endoscopy (SBCE) can be complementary to histological assessment of celiac disease (CD) and serology negative villous atrophy (SNVA). Determining the severity of disease on SBCE using statistical machine learning methods can be useful in the follow up of patients. SBCE can play an additional role in differentiating between CD and SNVA. De-identified SBCEs of patients with CD and SNVA were included. Probabilistic analysis of features on SBCE were used to predict severity of duodenal histology and to distinguish between CD and SNVA. Patients with higher Marsh scores were more likely to have a positive SBCE and a continuous distribution of macroscopic features of disease than those with lower Marsh scores. The same pattern was also true for patients with CD when compared to patients with SNVA. The validation accuracy when predicting the severity of Marsh scores and when distinguishing between CD and SNVA was 69.1% in both cases. When the proportions of each SBCE class group within the dataset were included in the classification model, to distinguish between the two pathologies, the validation accuracy increased to 75.3%. The findings of this work suggest that by using features of CD and SNVA on SBCE, predictions can be made of the type of pathology and the severity of disease

Pulmonary tuberculosis in intensive care setting, with a focus on the use of severity scores, a multinational collaborative systematic review.

BACKGROUND AND AIM: Tuberculosis (TB) is associated with a high mortality in the intensive care unit (ICU), especially in subjects with Acute Respiratory Distress Syndrome (ARDS) requiring mechanical ventilation. Despite its global burden on morbidity and mortality, TB is an uncommon cause of ICU admission, however mortality is disproportionate to the advances in diagnosis and treatment made. Herein we report a systematic review of published studies. METHODS: Our Literature search was conducted to identify studies on outcomes of individuals with TB admitted to ICU. We report and review in-hospital mortality, predictors of poorer outcomes, usefulness of severity scoring systems and potential benefits of intravenous antibiotics. Searches from Pubmed, Embase, Cochrane and Medline were conducted from inception to March 2020. Only literature in English was included. RESULTS: Out of 529 potentially relevant articles, 17 were included. Mortality across all studies ranged from 29-95% with an average of 52.9%. All severity scores underestimated average mortality. The most common indication for ICU admission was acute respiratory failure (36.3%). Negative predictors of outcome included hospital acquired infections, need of mechanical ventilation and vasopressors, delay in initiation of anti-TB treatment, more than one organ failure and a higher severity score. Low income, high incidence countries showed a 23.4% higher mortality rate compared to high income, low TB incidence countries. CONCLUSION: Mortality in individuals with TB admitted to ICU is high. Earlier detection and treatment initiation is needed

The Magnetic Field Geometry of Small Solar Wind Flux Ropes Inferred from their Twist Distribution

This work extends recent efforts on the force-free modeling of large flux rope-type structures (magnetic clouds, MCs) to much smaller spatial scales. We first select small flux ropes (SFRs) by eye whose duration is unambiguous and which were observed by the Solar Terrestrial Relations Observatory (STEREO) or Wind spacecraft during solar maximum years. We inquire into which analytical technique is physically most appropriate. We consider three models: (i) linear force-free field ($\bigtriangledown\times$ B = $\alpha (r)$ B) with a specific, prescribed constant $\alpha$ (Lundquist solution), and (ii) with $\alpha$ as a free constant parameter (Lundquist-alpha solution), (iii) uniform twist field (Gold-Hoyle solution). We retain only those cases where the impact parameter is less than one-half the FR radius, $R$, so the results should be robust (29 cases). The SFR radii lie in the range [$\sim$ 0.003, 0.059] AU. Comparing results, we find that the Lundquist-alpha and uniform twist solutions yielded comparable and small normalized $\chi^2$ values in most cases. We then use Grad-Shafranov (GS) reconstruction to analyze these events further. We then considered the twist per unit length, $\tau$, both its profile through the FR and its absolute value. We find $\tau$ to lie in the range [5.6, 34] turns/AU. The GH model-derived $\tau$ values are comparable to those obtained from GS reconstruction. We find that twist unit length ($L$) is inversely proportional to $R$, as $\tau \sim 0.17/R$. We combine MC and SFR results on $\tau (R)$ and give a relation which is approximately valid for both sets. The axial and azimuthal fluxes, $F_z$ and $F_\phi$, vary as $\approx 2.1 B_0 R^2 \times10^{21}$ Mx and $F_{\phi}/L \approx 0.36 B_0 R \times10^{21}$Mx/AU. The relative helicity per unit length, $H/L \approx 0.75 B_0^2 R^3$$\times 10^{42}$ Mx$^2$/AU.Comment: abstract shortened for arxiv, 31 pages, 15 Figures, in press at Solar Physic

Investigating the Magnetic Structure of Interplanetary Coronal Mass Ejections using Simultaneous Multi-Spacecraft In situ Measurements

In situ measurements from spacecraft typically provide a time series at a single location through coronal mass ejections (CMEs) and they have been one of the main methods to investigate CMEs. CME properties derived from these in situ measurements are affected by temporal changes that occur as the CME passes over the spacecraft, such as radial expansion and ageing, as well as spatial variations within a CME. This study uses multi-spacecraft measurements of the same CME at close separations to investigate both the spatial variability (how different a CME profile is when probed by two spacecraft close to each other) and the so-called ageing effect (the effect of the time evolution on in situ properties). We compile a database of 19 events from the past four decades measured by two spacecraft with a radial separation <0.2 au and an angular separation <10{\deg}. We find that the average magnetic field strength measured by the two spacecraft differs by 18% of the typical average value, which highlights non-negligible spatial or temporal variations. For one particular event, measurements taken by the two spacecraft allow us to quantify and significantly reduce the ageing effect to estimate the asymmetry of the magnetic field strength profile. This study reveals that single-spacecraft time series near 1 au can be strongly affected by ageing and that correcting for self-similar expansion does not capture the whole ageing effect.Comment: Accepted for publication at ApJ. 18 pages, 6 figures, 2 table