Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

Hierarchical progressive surveys. Multi-resolution HEALPix data structures for astronomical images, catalogues, and 3-dimensional data cubes

Scientific exploitation of the ever increasing volumes of astronomical data requires efficient and practical methods for data access, visualisation, and analysis. Hierarchical sky tessellation techniques enable a multi-resolution approach to organising data on angular scales from the full sky down to the individual image pixels. Aims. We aim to show that the Hierarchical progressive survey (HiPS) scheme for describing astronomical images, source catalogues, and three-dimensional data cubes is a practical solution to managing large volumes of heterogeneous data and that it enables a new level of scientific interoperability across large collections of data of these different data types. Methods. HiPS uses the HEALPix tessellation of the sphere to define a hierarchical tile and pixel structure to describe and organise astronomical data. HiPS is designed to conserve the scientific properties of the data alongside both visualisation considerations and emphasis on the ease of implementation. We describe the development of HiPS to manage a large number of diverse image surveys, as well as the extension of hierarchical image systems to cube and catalogue data. We demonstrate the interoperability of HiPS and Multi-Order Coverage (MOC) maps and highlight the HiPS mechanism to provide links to the original data. Results. Hierarchical progressive surveys have been generated by various data centres and groups for ~200 data collections including many wide area sky surveys, and archives of pointed observations. These can be accessed and visualised in Aladin, Aladin Lite, and other applications. HiPS provides a basis for further innovations in the use of hierarchical data structures to facilitate the description and statistical analysis of large astronomical data sets.Comment: 21 pages, 6 figures. Accepted for publication in Astronomy & Astrophysic

The CDS information hub

The Centre de Donnees astronomiques de Strasbourg (CDS) provides homogeneousaccess to heterogeneous information of various origins: information aboutastronomical objects in Simbad; catalogs and observation logs in VizieR and inthe catalogue service; reference images and overlays in Aladin; nomenclature inthe Dictionary of Nomenclature; Yellow Page services; the AstroGLU resourcediscovery tool; mirror copies of other reference services; and documentation.With the implementation of links between the CDS services, and with otheron--line reference information, CDS has become a major hub in the rapidlyevolving world of information retrieval in astronomy, developing efficienttools to help astronomers to navigate in the world-wide `Virtual Observatory'under construction, from data in the observatory archives to results publishedin journals. The WWW interface to the CDS services is available at:http://cdsweb.u-strasbg.fr

Weak local rules for planar octagonal tilings

We provide an effective characterization of the planar octagonal tilings which admit weak local rules. As a corollary, we show that they are all based on quadratic irrationalities, as conjectured by Thang Le in the 90s.Comment: 23 pages, 6 figure

Bounds on the heat kernel of the Schroedinger operator in a random electromagnetic field

We obtain lower and upper bounds on the heat kernel and Green functions of the Schroedinger operator in a random Gaussian magnetic field and a fixed scalar potential. We apply stochastic Feynman-Kac representation, diamagnetic upper bounds and the Jensen inequality for the lower bound. We show that if the covariance of the electromagnetic (vector) potential is increasing at large distances then the lower bound is decreasing exponentially fast for large distances and a large time.Comment: some technical improvements, new references, to appear in Journ.Phys.

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.Comment: To appear in Czech. Math.

<i>Gaia</i> Data Release 1. Summary of the astrometric, photometric, and survey properties

Context. At about 1000 days after the launch of Gaia we present the first Gaia data release, Gaia DR1, consisting of astrometry and photometry for over 1 billion sources brighter than magnitude 20.7. Aims. A summary of Gaia DR1 is presented along with illustrations of the scientific quality of the data, followed by a discussion of the limitations due to the preliminary nature of this release. Methods. The raw data collected by Gaia during the first 14 months of the mission have been processed by the Gaia Data Processing and Analysis Consortium (DPAC) and turned into an astrometric and photometric catalogue. Results. Gaia DR1 consists of three components: a primary astrometric data set which contains the positions, parallaxes, and mean proper motions for about 2 million of the brightest stars in common with the HIPPARCOS and Tycho-2 catalogues – a realisation of the Tycho-Gaia Astrometric Solution (TGAS) – and a secondary astrometric data set containing the positions for an additional 1.1 billion sources. The second component is the photometric data set, consisting of mean G-band magnitudes for all sources. The G-band light curves and the characteristics of ∼3000 Cepheid and RR-Lyrae stars, observed at high cadence around the south ecliptic pole, form the third component. For the primary astrometric data set the typical uncertainty is about 0.3 mas for the positions and parallaxes, and about 1 mas yr−1 for the proper motions. A systematic component of ∼0.3 mas should be added to the parallax uncertainties. For the subset of ∼94 000 HIPPARCOS stars in the primary data set, the proper motions are much more precise at about 0.06 mas yr−1. For the secondary astrometric data set, the typical uncertainty of the positions is ∼10 mas. The median uncertainties on the mean G-band magnitudes range from the mmag level to ∼0.03 mag over the magnitude range 5 to 20.7. Conclusions. Gaia DR1 is an important milestone ahead of the next Gaia data release, which will feature five-parameter astrometry for all sources. Extensive validation shows that Gaia DR1 represents a major advance in the mapping of the heavens and the availability of basic stellar data that underpin observational astrophysics. Nevertheless, the very preliminary nature of this first Gaia data release does lead to a number of important limitations to the data quality which should be carefully considered before drawing conclusions from the data

The Herschel–ATLAS data release 2, Paper I. Submillimeter and far-infrared images of the South and North Galactic Poles: the largest Herschel survey of the extragalactic sky

We present the largest submillimeter images that have been made of the extragalactic sky. The Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS) is a survey of 660 deg2 with the PACS and SPIRE cameras in five photometric bands: 100, 160, 250, 350, and 500 μm. In this paper we present the images from our two largest fields, which account for ~75% of the survey. The first field is 180.1 deg2 in size, centered on the north Galactic pole (NGP), and the second is 317.6 deg2 in size, centered on the south Galactic pole. The NGP field serendipitously contains the Coma cluster. Over most (~80%) of the images, the pixel noise, including both instrumental noise and confusion noise, is approximately 3.6, and 3.5 mJy pix−1 at 100 and 160 μm, and 11.0, 11.1 and 12.3 mJy beam−1 at 250, 350 and 500 μm, respectively, but reaches lower values in some parts of the images. If a matched filter is applied to optimize point-source detection, our total 1σ map sensitivity is 5.7, 6.0, and 7.3 mJy at 250, 350, and 500 μm, respectively. We describe the results of an investigation of the noise properties of the images. We make the most precise estimate of confusion in SPIRE maps to date, finding values of 3.12 ± 0.07, 4.13 ± 0.02, and 4.45 ± 0.04 mJy beam−1 at 250, 350, and 500 μm in our un-convolved maps. For PACS we find an estimate of the confusion noise in our fast-parallel observations of 4.23 and 4.62 mJy beam−1 at 100 and 160 μm. Finally, we give recipes for using these images to carry out photometry, both for unresolved and extended sources