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Let Aand Bbe Banach algebras with σ(B)≠∅. Let θ,ϕ,γ∈σ(B)and Der(A×θϕ,γB)be the set of all linear mappings d:A×B→A×Bsatisfying d((a,b)·θ(x,y))=d(a,b)·ϕ(x,y)+(a,b)·γd(x,y)for all a,x∈Aand b,y∈B. In this paper, we characterize elements of Der(A×θϕ,γB)in the case where Ahas a right identity. We then investigate the concept of centralizing for elements of Der(A×θϕ,γB)and determine dependent elements of Der(A×θϕ,γB). We also apply some results to group algebras.

Let {Mk}k∈Zbe a sequence of closed subspaces of Hilbert space H, and let {Θk}k∈Zbe a sequence of linear operators from Hinto Mk, k∈Z. In the case where, Θkis self-adjoint and Θk(Mk)=Mkfor all k∈Z, we show that if a g-frame {(Mk,Θk)}k∈Zis represented via a linear operator Ton span{Mk}k∈Z, then Tis bounded; moreover, if {(Mk,Θk)}k∈Zis a tight g-frame, then Tis not invertible. We also study the perturbation and the stability of these g-frames. Finally, we give some examples to show the validity of the results. A preliminary version of this manuscript was submitted to https://arxiv.org/abs/2305.08182This version is a reedited copy of it.

The effects of lamellar aluminum (Al) and micaceous iron oxide (MIO) pigments on the anticorrosion properties of an epoxy zinc-rich coating were studied. To this end, the epoxy zinc-rich coatings containing 70% w/w spherical Zn particles, 60% w/w Zn + 10% w/w MIO, and 60% w/w Zn + 10% w/w Al were prepared. The electrochemical noise (ECN), potentiostatic polarization technique, and salt spray test were employed in order to investigate the anticorrosion performances of the zinc-rich coatings. The zinc-rich coatings morphologies were studied by scanning electron microscope (SEM) before and after the salt spray test. The open-circuit potential values were also measured at different immersion times. Results showed that MIO particles could enhance the cathodic protection duration of the zinc-rich coating by enhancing its barrier properties and reducing the zinc particles oxidation rate. It was also shown that Al particles reduced zinc-rich coating sacrificial behavior at short immersion times and increased it at long immersion times. Unlike MIO particles, Al particles behaved both as barrier and sacrificial pigment.

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Context.The study of abundances in the nucleus of active galaxies allows us to investigate the evolution of the abundance by comparing local and higher redshift galaxies. However, the methods used so far have substantial drawbacks or rather large uncertainties. Some of the measurements are at odds with the initial mass function derived from the older stellar population of local elliptical galaxies.

We present the results of our 600 ks RGS observation as part of the multiwavelength campaign on Mrk 509. The very high quality of the spectrum allows us to investigate the ionized outflow with an unprecedented accuracy due to the long exposure and the use of the RGS multipointing mode. We detect multiple absorption lines from the interstellar medium and from the ionized absorber in Mrk 509. A number of emission components are also detected, including broad emission lines consistent with an origin in the broad line region, the narrow O vii forbidden emission line and also (narrow) radiative recombination continua. The ionized absorber consists of two velocity components (v= −13 ± 11 km s-1and v= −319 ± 14 km s-1), which both are consistent with earlier results, including UV data. There is another tentative component outflowing at high velocity, −770 ± 109 km s-1, which is only seen in a few highly ionized absorption lines. The outflow shows discrete ionization components, spanning four orders of magnitude in ionization parameter. Due to the excellent statistics of our spectrum, we demonstrate for the first time that the outflow in Mrk 509 in the important range of log ξbetween 1−3 cannot be described by a smooth, continuous absorption measure distribution, but instead shows two strong, discrete peaks. At the highest and lowest ionization parameters we cannot differentiate smooth and discrete components.

Context.We present here the results of a 180 ks Chandra-LETGS observation as part of a large multi-wavelength campaign on Mrk 509.

We present medium-resolution (λ/Δλ~ 20   000) ultraviolet spectra covering the 1155−1760 Å spectral range of the Seyfert 1 galaxy Mrk 509obtained using the Cosmic Origins Spectrograph (COS) on the Hubble Space Telescope (HST). Our observations were obtained simultaneously with a Low Energy Transmission Grating Spectrometer observation using the Chandra X-ray Observatory, and they are part of a multiwavelength campaign in September through December 2009 which also included observations with XMM-Newton, Swift, and INTEGRAL. Our spectra are the highest signal-to-noise observations to date of the intrinsic absorption components seen in numerous prior ultraviolet observations. To take advantage of the high S/N, we describe special calibrations for wavelength, flat-field and line-spread function corrections that we applied to the COS data. We detect additional complexity in the absorption troughs compared to prior observations made with the Space Telescope Imaging Spectrograph (STIS) on HST. We attribute the UV absorption to a variety of sources in Mrk 509, including an outflow from the active nucleus, the interstellar medium and halo of the host galaxy, and possible infalling clouds or stripped gaseous material from a merger that are illuminated by the ionizing radiation of the active nucleus. Variability between the STIS and COS observation of the most blue-shifted component (#1) allows us to set an upper limit on its distance of  <250 pc. Similarly, variability of component 6 between FUSE observations limits its distance to  <1.5 kpc. The absorption lines in all components only partially cover the emission from the active nucleus with covering fractions that are lower than those seen in the prior STIS observations and are comparable to those seen in spectra from the Far Ultraviolet Spectroscopic Explorer (FUSE). Given the larger apertures of COS and FUSE compared to STIS, we favor scattered light from an extended region near the active nucleus as the explanation for the partial covering. As observed in prior X-ray and UV spectra, the UV absorption has velocities comparable to the X-ray absorption,but the bulk of the ultraviolet absorption is in a lower ionization state with lower total column density than the gas responsible for the X-ray absorption. We conclude that the outflow from the active nucleus is a multiphase wind.

Aims.We study the bright Seyfert 1 galaxy Mrk 509 with the Reflection Grating Spectrometers (RGS) of XMM-Newton, using for the first time the RGS multi-pointing mode of XMM-Newtonto constrain the properties of the outflow in this object. We obtain very accurate spectral properties from a 600 ks spectrogram of Mrk 509 with excellent quality.

We present the analysis of XMM-Newtonand Swiftoptical-UV and X-ray observations of the Seyfert-1/QSO Mrk 509, part of an unprecedented multi-wavelength campaign, investigating the nuclear environment of this AGN. The XMM-Newtondata are from a series of 10 observations of about 60 ks each, spaced from each other by about 4 days, taken in Oct.-Nov. 2009. During our campaign, Mrk 509 was also observed with Swiftfor a period of about 100 days, monitoring the behaviour of the source before and after the XMM-Newtonobservations. With these data we have established the continuum spectrum in the optical-UV and X-ray bands and investigated its variability on the timescale of our campaign with a resolution time of a few days. In order to measure and model the continuum as far as possible into the UV, we also made use of Hubble Space Telescope (HST) cosmic origin spectrograph (COS) observations of Mrk 509 (part of our coordinated campaign) and of an archival Far Ultraviolet Spectroscopic Explorer (FUSE) observation. We have found that in addition to an X-ray power-law, the spectrum displays soft X-ray excess emission below 2 keV, which interestingly varies in association with the thermal optical-UV emission from the accretion disc. The change in the X-ray power-law component flux (albeit smaller than that of the soft excess), on the other hand, is uncorrelated to the flux variability of the soft X-ray excess and the disc component on the probed timescale. The results of our simultaneous broad-band spectral and timing analysis suggest that, on a resolution time of a few days, the soft X-ray excess of Mrk 509 is produced by the Comptonisation of the thermal optical-UV photons from the accretion disc by a warm (0.2 keV) optically thick (τ~ 17) corona surrounding the inner regions of the disc. This makes Mrk 509, with a black hole mass of about 1–3 × 108M⊙, the highest mass known system to display such behaviour and origin for the soft X-ray excess.

Context.Active galactic nuclei (AGN) show a wealth of interesting physical processes, some of which are poorly understood. In a broader context, they play an important role in processes that are far beyond their immediate surroundings, owing to the high emitted power.

Aims. We present a new analysis of the soft and medium energy X-ray spectrum of the Seyfert 1 galaxy NGC 3516 taken with the Reflection Grating Spectrometer (RGS) and European Photon Imaging Camera (EPIC) on board the XMM-Newtonobservatory. We examine four observations made in October 2006. We investigate whether the observed variability is due to absorption by the warm absorber and/or is intrinsic to the source emission.Methods. We analyse in detail the EPIC-pn and RGS spectra of each observation separately.Results. The warm absorber in NGC 3516 is found to consist of three phases of ionisation, two of which have outflow velocities of more than 1000 km s-1. The third phase (the least ionised one) is much slower at 100 km s-1. One of the high ionisation phases, with log $\xi $of 2.4, is found to have a partial covering fraction of about 60%. It has previously been suggested that the passage of a cloud, part of a disc wind, in front of the source (producing a change in the covering fraction) was the cause of a significant dip in the lightcurve during one of the observations. From our modelling of the EPIC-pn and RGS spectra, we find that variation in the covering fraction cannot be solely responsible for this. We show that intrinsic change in the source continuum plays a much more significant role in explaining the observed flux and spectral variability than originally thought.

Aims. We present a new analysis of the soft and medium energy X-ray spectrum of the Seyfert 1 galaxy NGC 3516 taken with the Reflection Grating Spectrometer (RGS) and European Photon Imaging Camera (EPIC) on board the XMM-Newtonobservatory. We examine four observations made in October 2006. We investigate whether the observed variability is due to absorption by the warm absorber and/or is intrinsic to the source emission.