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

Failed intentions? Meeting the water needs of people living with HIV in South Africa

Researchers, activists, practitioners and policy-makers have grappled with the challenge of providing people living with HIV (PLHIV) with an  adequate amount of safe water. Comprising 13% of the overall population of South Africa in 2018, 7.52 million PLHIV need water for drinking and  taking medication; preparing food; and personal hygiene and cleaning to minimise infections. This article examines the responses of the different stakeholders to this challenge and their impact on the water and health policy process. It finds that activists were able to emphasise the dimensions of the challenge; practitioners worked to implement provision more effectively within existing policy frameworks; and a range of stakeholders made a thoughtful and promising policy proposal for direct action, which the Department of Water and Sanitation ultimately failed to embrace. This article is based on an extensive review of academic research and publications by development agencies on HIV and water as well as engagement with policies and documents in the South African water sector related to water services delivery for PLHIV. While the widespread provision of antiretrovirals from 2004 has changed the context, the above findings are significant in understanding and reviewing the impact of various stakeholders on the water and health policy process. They raise questions regarding the effectiveness of NGO advocacy, the means of delivering improved services to specific populations, and the ability of a range of stakeholders to inform the policy approaches of government departments. Keywords: activist HIV multi-stakeholder targeting wate

Wildlife Conservation and the Role of the Indigenous Communities Living around Conservation Areas

The Indigenous Kenyan Maasai community has coexisted with the wildlife surrounding it for decades from Nairobi National Park, Maasai Mara and Amboseli. These parks border Maasai lands. Although the northern, eastern, and western perimeters of the Nairobi National Park are fenced, the southern part is not. It is at this point that the Maasai community’s land meets the park. This area also acts as a wildlife dispersal area where wildlife can freely migrate to other parks, including Maasai Mara and Amboseli. The park is only 117sq kms and its vitality depends on the plains to the south where the Maasai live so that the animals can migrate in and out. Without that open space, the park would be little more than a zoo. The fact is that approximately 60 to 80 percent of wildlife in Kenya is outside formally protected areas. For the Maasai community, wildlife poses an enormous threat. As herbivores migrate during the wet season, they are followed by predators such as lions. Livestock are an easy target for them. A lion attack can be devastating, ruining family lives and livelihoods. Lions have traditionally been the Maasai tribe’s greatest adversaries; they are a deadly threat to the cattle and other livestock that are both an integral part of the Maasai culture and the tribe’s greatest source of wealth. Despite the ongoing livestock predation, lions may be the tribe’s strongest hope of preserving their way of life. With these in mind, it becomes imperative that we think of systems for better correlation between all aspects of conservation and understand that the wildlife, livestock, and the surrounding pastoral communities play a key role in each other’s survival

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

Orbital operations study. Appendix B: Operational procedures

Operational procedures for each alternate approach for each interfacing activity of the orbital operations study are presented. The applicability of the procedures to interfacing element pairs is identified

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

Melting behavior of (Th,U)O2 and (Th,Pu)O2 mixed oxides

© 2016 Elsevier B.V.The melting behaviors of pure ThO2, UO2 and PuO2 as well as (Th,U)O2 and (Th,Pu)O2 mixed oxides (MOX) have been studied using molecular dynamics (MD) simulations. The MD calculated melting temperatures (MT) of ThO2, UO2 and PuO2 using two-phase simulations, lie between 3650-3675 K, 3050–3075 K and 2800–2825 K, respectively, which match well with experiments. Variation of enthalpy increments and density with temperature, for solid and liquid phases of ThO2, PuO2 as well as the ThO2 rich part of (Th,U)O2 and (Th,Pu)O2 MOX are also reported. The MD calculated MT of (Th,U)O2 and (Th,Pu)O2 MOX show good agreement with the ideal solidus line in the high thoria section of the phase diagram, and evidence for a minima is identified around 5 atom% of ThO2 in the phase diagram of (Th,Pu)O2 MOX

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