Universal Approximation with Deep Narrow Networks

The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural `dual' scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number of inputs neurons, $m$ be the number of output neurons, and let $\rho$ be any nonaffine continuous function, with a continuous nonzero derivative at some point. Then we show that the class of neural networks of arbitrary depth, width $n + m + 2$, and activation function $\rho$, is dense in $C(K; \mathbb{R}^m)$ for $K \subseteq \mathbb{R}^n$ with $K$ compact. This covers every activation function possible to use in practice, and also includes polynomial activation functions, which is unlike the classical version of the theorem, and provides a qualitative difference between deep narrow networks and shallow wide networks. We then consider several extensions of this result. In particular we consider nowhere differentiable activation functions, density in noncompact domains with respect to the $L^p$-norm, and how the width may be reduced to just $n + m + 1$ for `most' activation functions.Comment: Accepted at COLT 202

Photometric monitoring of the blazar 3C 345 for the period 1996 - 2006

We present the results of the blazar 3C 345 monitoring in Johnson-Cousins BVRI bands for the period 1996 - 2006. We have collected 29 V and 43 R data points for this period; the BI light curves contain a few measurements only. The accuracy of our photometry is not better than 0.03 mag in the VR bands. The total amplitude of the variability obtained from our data is 2.06 mag in the V band and 2.25 mag in the R one. 3C 345 showed periods of flaring activity during 1998/99 and 2001: a maximum of the blazar brightness was detected in 2001 February - 15.345 mag in the V band and 14.944 mag in the R one. We confirm that during brighter stages 3C 345 becomes redder; for higher fluxes the colour index seems to be less dependent on the magnitude. The intra-night monitoring of 3C 345 in three consecutive nights in 2001 August revealed no significant intra-night variability; 3C 345 did not show evident flux changes over timescales of weeks around the period of the intra-night monitoring. This result supports the existing facts that intra-night variability is correlated with rapid flux changes rather than with specific flux levels

Herschel PACS and SPIRE observations of blazar PKS 1510-089: a case for two blazar zones

We present the results of observations of blazar PKS 1510-089 with the Herschel Space Observatory PACS and SPIRE instruments, together with multiwavelength data from Fermi/LAT, Swift, SMARTS and SMA. The source was found in a quiet state, and its far-infrared spectrum is consistent with a power-law with a spectral index of alpha ~ 0.7. Our Herschel observations were preceded by two 'orphan' gamma-ray flares. The near-infrared data reveal the high-energy cut-off in the main synchrotron component, which cannot be associated with the main gamma-ray component in a one-zone leptonic model. This is because in such a model the luminosity ratio of the External-Compton and synchrotron components is tightly related to the frequency ratio of these components, and in this particular case an unrealistically high energy density of the external radiation would be implied. Therefore, we consider a well-constrained two-zone blazar model to interpret the entire dataset. In this framework, the observed infrared emission is associated with the synchrotron component produced in the hot-dust region at the supra-pc scale, while the gamma-ray emission is associated with the External-Compton component produced in the broad-line region at the sub-pc scale. In addition, the optical/UV emission is associated with the accretion disk thermal emission, with the accretion disk corona likely contributing to the X-ray emission.Comment: 13 pages, 8 figures, 7 tables; accepted for publication in the Astrophysical Journa

Optical Polarization of 52 Radio-Loud QSOs and BL Lac Objects

Polarization measurements are presented for 52 radio-loud QSOs and BL Lac objects. For 9 highly polarized (p >3%) AGN, these are the first published polarization measurements. Of these 9, 7 are highly-polarized QSOs (HPQs), one is a BL Lac object and another is a likely BL Lac object. Polarization variability is confirmed for some of these new and previously known highly-polarized AGN. While 6 of the HPQs have flat radio spectra are almost certainly blazars, PKS 1452-217 is probably a new member of the rare class of radio-loud QSOs that show high polarization by scattering, and is therefore important for testing orientation Unified Schemes. In competition for the highest redshift HPQ are the well-observed QSO PKS 0438-43 at z = 2.85, with maximum p = 4.7%, and PKS 0046-315 at z = 2.72, for which we find p = 13%.Comment: 6 pages. Astronomical Journal, in pres

Evolution of the Dust Coma in Comet 67P/Churyumov-Gerasimenko Before 2009 Perihelion

Comet 67P/Churyumov-Gerasimenko is the main target of ESA's Rosetta mission and will be encountered in May 2014. As the spacecraft shall be in orbit the comet nucleus before and after release of the lander {\it Philae}, it is necessary necessary to know the conditions in the coma. Study the dust environment, including the dust production rate and its variations along its preperihelion orbit. The comet was observed during its approach to the Sun on four epochs between early-June 2008 and mid-January 2009, over a large range of heliocentric distances that will be covered by the mission in 2014. An anomalous enhancement of the coma dust density was measured towards the comet nucleus. The scalelength of this enhancement increased with decreasing heliocentric distance of the comet. This is interpreted as a result of an unusually slow expansion of the dust coma. Assuming a spherical symmetric coma, the average amount of dust as well as its ejection velocity have been derived. The latter increases exponentially with decreasing heliocentric distance (\rh), ranging from about 1 m/s at 3 AU to about 25-35 m/s at 1.4 AU. Based on these results we describe the dust environment at those nucleocentric distances at which the spacecraft will presumably be in orbit. Astronomy and Astrophysics, in pressComment: 5 pages, 4 figure

Optical Observations of PSR J0205+6449

PSR J0205+6449 is a X-ray and radio pulsar in supernova remnant 3C 58. We report on observations of the central region of 3C 58 using the 4.2-m William Herschel Telescope with the intention of identifying the optical counterpart of PSR J0205+6449 and characterising its pulsar wind nebula. Around the pulsar position we identified extended emission with a magnitude of B = 23 \fm 97 \pm 0.10, V = 22 \fm 95 \pm 0.05 and R = 22 \fm 15 \pm 0.03 consistent with a pulsar wind nebula. From the R-band image we identified three knots with $m_R$ = 24 \fm 08 \pm 0.07 (o1), 24 \fm 15 \pm 0.07 (o2) and 24 \fm 24 \pm 0.08 (o3). We confirm the presence of an optical pulsar wind nebula around PSR J0205+6449 and give an upper limit of $m_R \approx$ 24 for the optical magnitude of the pulsar. Furthermore we make the tentative suggestion that our object o1, with an $m_R \approx$ 24.08 is the optical counterpart. If confirmed the pulsar would have an $L_R/L_x\approx 0.004$ and an optical efficiency of about 5% of the Crab pulsar. Such a low efficiency is more consistent with the characteristic age of the pulsar rather than that of SN 1181.Comment: The paper contains 3 figures and 3 tables, accepted for publication in MNRA

Optical and Radio monitoring of S5 1803+74

The optical (BVRI) and radio (8.4 GHz) light curves of S5 1803+784 on a time span of nearly 6 years are presented and discussed. The optical light curve showed an overall variation greater than 3 mag, and the largest changes occured in three strong flares. No periodicity was found in the light curve on time scales up to a year. The variability in the radio band is very different, and shows moderate oscillations around an average constant flux density rather than relevant flares, with a maximum amplitude of $\sim$30%, without a simultaneous correspondence between optical and radio luminosity. The optical spectral energy distribution was always well fitted by a power law. The spectral index shows small variations and there is indication of a positive correlation with the source luminosity. Possible explanations of the source behaviour are discussed in the framework of current models.Comment: 25 pages, 12 figure

Polarisation Observations of H2_{2}O JK−1K1=532−441J_{K_{-1}K_{1}} = 5_{32} - 4_{41} 620.701 GHz Maser Emission with Herschel/HIFI in Orion KL

Context. The high intensities and narrow bandwidths exhibited by some astronomical masers make them ideal tools for studying star-forming giant molecular clouds. The water maser transition $J_{K_{-1}K_{1}}=5_{32}-4_{41}$ at 620.701 GHz can only be observed from above Earth's strongly absorbing atmosphere; its emission has recently been detected from space. Aims. We sought to further characterize the star-forming environment of Orion KL by investigating the linear polarisation of a source emitting a narrow 620.701 GHz maser feature with the heterodyne spectrometer HIFI on board the Herschel Space Observatory. Methods. High-resolution spectral datasets were collected over a thirteen month period beginning in 2011 March, to establish not only the linear polarisation but also the temporal variability of the source. Results. Within a $3\sigma$ uncertainty, no polarisation was detected to an upper limit of approximately 2%. These results are compared with coeval linear polarisation measurements of the 22.235 GHz $J_{K_{-1}K_{1}}=6_{16}-5_{23}$ maser line from the Effelsberg 100-m radio telescope, typically a much stronger maser transition. Although strongly polarised emission is observed for one component of the 22.235 GHz maser at 7.2 km s$^{-1}$, a weaker component at the same velocity as the 620.701 GHz maser at 11.7 km s$^{-1}$ is much less polarised.Comment: Accepted for publication in A&