On the independence ratio of distance graphs

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.Comment: 39 pages, 12 figures, 6 table

Supermassive black hole mass in the massive elliptical galaxy M87 from integral-field stellar dynamics using OASIS and MUSE with adaptive optics: assessing systematic uncertainties

The massive elliptical galaxy M87 has been the subject of several supermassive black hole mass measurements from stellar dynamics, gas dynamics, and recently the black hole shadow by the Event Horizon Telescope (EHT). This uniquely positions M87 as a benchmark for alternative black hole mass determination methods. Here we use stellar kinematics extracted from integral-field spectroscopy observations with Adaptive Optics (AO) using MUSE and OASIS. We exploit our high-resolution integral field spectroscopy to spectrally decompose the central AGN from the stars. We derive an accurate inner stellar-density profile and find it is flatter than previously assumed. We also use the spectrally-extracted AGN as a reference to accurately determine the observed MUSE and OASIS AO PSF. We then perform Jeans Anisotropic Modelling (JAM), with a new flexible spatially-variable anisotropy, and measure the anisotropy profile, stellar mass-to-light variations, inner dark matter fraction, and black hole mass. Our preferred black hole mass is $M_{\rm BH}=(8.7\pm1.2) \times 10^9 \ M_\odot$. However, using the inner stellar density from previous studies, we find a preferred black hole mass of $M_{\rm BH} = (5.5^{+0.5}_{-0.3}) \times 10^9 \ M_\odot$, consistent with previous work. We conduct numerous systematic tests of the kinematics and model assumptions and conclude that uncertainties in the black hole mass of M87 from previous determinations may have been underestimated and further analyses are needed.Comment: 19 pages, 16 figures, 4 tables, Submitted to MNRA

Structural Transitions and Global Minima of Sodium Chloride Clusters

In recent experiments on sodium chloride clusters structural transitions between nanocrystals with different cuboidal shapes were detected. Here we determine reaction pathways between the low energy isomers of one of these clusters, (NaCl)35Cl-. The key process in these structural transitions is a highly cooperative rearrangement in which two parts of the nanocrystal slip past one another on a {110} plane in a direction. In this way the nanocrystals can plastically deform, in contrast to the brittle behaviour of bulk sodium chloride crystals at the same temperatures; the nanocrystals have mechanical properties which are a unique feature of their finite size. We also report and compare the global potential energy minima for (NaCl)NCl- using two empirical potentials, and comment on the effect of polarization.Comment: extended version, 13 pages, 8 figures, revte

Uncovering treatment burden as a key concept for stroke care: a systematic review of qualitative research

<b>Background</b> Patients with chronic disease may experience complicated management plans requiring significant personal investment. This has been termed ‘treatment burden’ and has been associated with unfavourable outcomes. The aim of this systematic review is to examine the qualitative literature on treatment burden in stroke from the patient perspective.<p></p> <b>Methods and findings</b> The search strategy centred on: stroke, treatment burden, patient experience, and qualitative methods. We searched: Scopus, CINAHL, Embase, Medline, and PsycINFO. We tracked references, footnotes, and citations. Restrictions included: English language, date of publication January 2000 until February 2013. Two reviewers independently carried out the following: paper screening, data extraction, and data analysis. Data were analysed using framework synthesis, as informed by Normalization Process Theory. Sixty-nine papers were included. Treatment burden includes: (1) making sense of stroke management and planning care, (2) interacting with others, (3) enacting management strategies, and (4) reflecting on management. Health care is fragmented, with poor communication between patient and health care providers. Patients report inadequate information provision. Inpatient care is unsatisfactory, with a perceived lack of empathy from professionals and a shortage of stimulating activities on the ward. Discharge services are poorly coordinated, and accessing health and social care in the community is difficult. The study has potential limitations because it was restricted to studies published in English only and data from low-income countries were scarce.<p></p> <b>Conclusions</b> Stroke management is extremely demanding for patients, and treatment burden is influenced by micro and macro organisation of health services. Knowledge deficits mean patients are ill equipped to organise their care and develop coping strategies, making adherence less likely. There is a need to transform the approach to care provision so that services are configured to prioritise patient needs rather than those of health care systems

A general notion of visibility graphs

Abstract. We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results giving necessary properties of visibility graphs, and giving some examples of classes of graphs that can be so represented. Finally, we give some applications of the concept, and we provide potential avenues for future research in the area

Inferring the Galactic gravitational potential with Gaia and friends

In the coming decade the Gaia satellite will measure the positions and velocities of an unprecedented number of stars in our Galaxy, with unprecedented precision. Among many firsts, this revolutionary new data set will include full six-dimensional phase space information for millions of stars in the Galactic halo, including stars in many tidal streams. These streams, the products of hierarchical accretion, can be used to infer the Galactic gravitational potential thanks to the common origin of the stars in each one. We present a method for doing so by maximizing the information content (i.e. clumpiness) of the action space of the stream stars. This statistical approach eliminates the need to assign stars to particular streams. Using a toy model of the stellar halo in a known potential, and including updated error models for Gaia, we show that ground-based spectroscopic follow-up of faint halo stars is essential to complete the six-dimensional Gaia catalog and properly constrain the scale radius of the potential. By fitting a spherical NFW potential to streams in a cosmologically simulated halo, we show how oversimplification of the potential model affects fit results. This material is based upon work supported by the National Science Foundation under Award No. AST-1400989