MAGIC observations of MWC 656, the only known Be/BH system

Context: MWC 656 has recently been established as the first observationally detected high-mass X-ray binary system containing a Be star and a black hole (BH). The system has been associated with a gamma-ray flaring event detected by the AGILE satellite in July 2010. Aims: Our aim is to evaluate if the MWC 656 gamma-ray emission extends to very high energy (VHE > 100 GeV) gamma rays. Methods. We have observed MWC 656 with the MAGIC telescopes for $\sim$23 hours during two observation periods: between May and June 2012 and June 2013. During the last period, observations were performed contemporaneously with X-ray (XMM-Newton) and optical (STELLA) instruments. Results: We have not detected the MWC 656 binary system at TeV energies with the MAGIC Telescopes in either of the two campaigns carried out. Upper limits (ULs) to the integral flux above 300 GeV have been set, as well as differential ULs at a level of $\sim$5% of the Crab Nebula flux. The results obtained from the MAGIC observations do not support persistent emission of very high energy gamma rays from this system at a level of 2.4% the Crab flux.Comment: Accepted for publication in A&A. 5 pages, 2 figures, 2 table

Planck intermediate results. XLI. A map of lensing-induced B-modes

The secondary cosmic microwave background (CMB) $B$-modes stem from the post-decoupling distortion of the polarization $E$-modes due to the gravitational lensing effect of large-scale structures. These lensing-induced $B$-modes constitute both a valuable probe of the dark matter distribution and an important contaminant for the extraction of the primary CMB $B$-modes from inflation. Planck provides accurate nearly all-sky measurements of both the polarization $E$-modes and the integrated mass distribution via the reconstruction of the CMB lensing potential. By combining these two data products, we have produced an all-sky template map of the lensing-induced $B$-modes using a real-space algorithm that minimizes the impact of sky masks. The cross-correlation of this template with an observed (primordial and secondary) $B$-mode map can be used to measure the lensing $B$-mode power spectrum at multipoles up to $2000$. In particular, when cross-correlating with the $B$-mode contribution directly derived from the Planck polarization maps, we obtain lensing-induced $B$-mode power spectrum measurement at a significance level of $12\,\sigma$, which agrees with the theoretical expectation derived from the Planck best-fit $\Lambda$CDM model. This unique nearly all-sky secondary $B$-mode template, which includes the lensing-induced information from intermediate to small ($10\lesssim \ell\lesssim 1000$) angular scales, is delivered as part of the Planck 2015 public data release. It will be particularly useful for experiments searching for primordial $B$-modes, such as BICEP2/Keck Array or LiteBIRD, since it will enable an estimate to be made of the lensing-induced contribution to the measured total CMB $B$-modes.Comment: 20 pages, 12 figures; Accepted for publication in A&A; The B-mode map is part of the PR2-2015 Cosmology Products; available as Lensing Products in the Planck Legacy Archive http://pla.esac.esa.int/pla/#cosmology; and described in the 'Explanatory Supplement' https://wiki.cosmos.esa.int/planckpla2015/index.php/Specially_processed_maps#2015_Lensing-induced_B-mode_ma

Planck intermediate results: XLIV. Structure of the Galactic magnetic field from dust polarization maps of the southern Galactic cap

Using data from the Planck satellite, we study the statistical properties of interstellar dust polarization at high Galactic latitudes around the south pole (b < −60°). Our aim is to advance the understanding of the magnetized interstellar medium (ISM), and to provide a modelling framework of the polarized dust foreground for use in cosmic microwave background (CMB) component-separation procedures. We examine the Stokes I, Q, and U maps at 353 GHz, and particularly the statistical distribution of the polarization fraction (p) and angle (ψ), in order to characterize the ordered and turbulent components of the Galactic magnetic field (GMF) in the solar neighbourhood. The Q and U maps show patterns at large angular scales, which we relate to the mean orientation of the GMF towards Galactic coordinates (l0,b0) = (70° ± 5°,24° ± 5°). The histogram of the observed p values shows a wide dispersion up to 25%. The histogram of ψ has a standard deviation of 12° about the regular pattern expected from the ordered GMF. We build a phenomenological model that connects the distributions of p and ψ to a statistical description of the turbulent component of the GMF, assuming a uniform effective polarization fraction (p0) of dust emission. To compute the Stokes parameters, we approximate the integration along the line of sight (LOS) as a sum over a set of N independent polarization layers, in each of which the turbulent component of the GMF is obtained from Gaussian realizations of a power-law power spectrum. We are able to reproduce the observed p and ψ distributions using a p0 value of 26%, a ratio of 0.9 between the strengths of the turbulent and mean components of the GMF, and a small value of N. The mean value of p (inferred from the fit of the large-scale patterns in the Stokes maps) is 12 ± 1%. We relate the polarization layers to the density structure and to the correlation length of the GMF along the LOS. We emphasize the simplicity of our model (involving only a few parameters), which can be easily computed on the celestial sphere to produce simulated maps of dust polarization. Our work is an important step towards a model that can be used to assess the accuracy of component-separation methods in present and future CMB experiments designed to search the B mode CMB polarization from primordial gravity waves

Planck 2015 results: XV. gravitational lensing

We present the most significant measurement of the cosmic microwave background (CMB) lensing potential to date (at a level of 40 sigma), using temperature and polarization data from the Planck 2015 full-mission release. Using a polarization-only estimator we detect lensing at a significance of 5 sigma. We cross-check the accuracy of our measurement using the wide frequency coverage and complementarity of the temperature and polarization measurements. Public products based on this measurement include an estimate of the lensing potential over approximately 70% of the sky, an estimate of the lensing potential power spectrum in bandpowers for the multipole range 40<L<400 and an associated likelihood for cosmological parameter constraints. We find good agreement between our measurement of the lensing potential power spectrum and that found in the best-fitting LCDM model based on the Planck temperature and polarization power spectra. Using the lensing likelihood alone we obtain a percent-level measurement of the parameter combination σ 8 Ω 0.25 m =0.591±0.021 . We combine our determination of the lensing potential with the E-mode polarization also measured by Planck to generate an estimate of the lensing B-mode. We show that this lensing B-mode estimate is correlated with the B-modes observed directly by Planck at the expected level and with a statistical significance of 10 sigma, confirming Planck's sensitivity to this known sky signal. We also correlate our lensing potential estimate with the large-scale temperature anisotropies, detecting a cross-correlation at the 3 sigma level, as expected due to dark energy in the concordance LCDM model

Planck 2015 results XV. Gravitational lensing

We present the most significant measurement of the cosmic microwave background (CMB) lensing potential to date (at a level of 40σ), using temperature and polarization data from the Planck 2015 full-mission release. Using a polarization-only estimator, we detect lensing at a significance of 5σ. We cross-check the accuracy of our measurement using the wide frequency coverage and complementarity of the temperature and polarization measurements. Public products based on this measurement include an estimate of the lensing potential over approximately 70% of the sky, an estimate of the lensing potential power spectrum in bandpowers for the multipole range 40 ≤ L ≤ 400, and an associated likelihood for cosmological parameter constraints. We find good agreement between our measurement of the lensing potential power spectrum and that found in the ΛCDM model that best fits the Planck temperature and polarization power spectra. Using the lensing likelihood alone we obtain a percent-level measurement of the parameter combination σ8Ω0.25m = 0.591 ± 0.021. We combine our determination of the lensing potential with the E-mode polarization, also measured by Planck, to generate an estimate of the lensing B-mode. We show that this lensing B-mode estimate is correlated with the B-modes observed directly by Planck at the expected level and with a statistical significance of 10σ, confirming Planck’s sensitivity to this known sky signal. We also correlate our lensing potential estimate with the large-scale temperature anisotropies, detecting a cross-correlation at the 3σ level, as expected because of dark energy in the concordance ΛCDM model

Waring's problem: A survey

tribus, &amp;c. usque ad novemdecim compositus, &amp; sic deinceps.” Waring [150, pp. 204-5]. “Every integer is a cube or the sum of two, three,... nine cubes; every integer is also the square of a square, or the sum of up to nineteen such; and so forth.” Waring [152, p. 336]. It is presumed that by this, in modern notation, Waring meant that for every k ≥ 3 there are numbers s such that every natural number is the sum of at most s k-th powers of natural numbers and that the smallest such number g(k) satisfies g(3) = 9, g(4) = 19. By the end of the nineteenth century, the existence of g(k) was known for only a finite number of values of k. There is an account of this work in Dickson [48], and as far as we have been able to ascertain, by 1909 its existence was known for k = 3, 4, 5, 6, 7, 8, 10, but not for any larger k (of course, with the natural extension of the definition of g(k), Lagrange proved in 1770 that g(2) = 4). However, starting with Hilbert [69], who showed that g(k) does indeed exist for every k, the twentieth century has seen an almost complete solution of this problem. Let [x] denote the greatest integer not exceeding x and write {x} for x − [x]. As the result of the work of many mathematicians we now know that g(k) = 2k + [(3/2)k] − 2, provided tha