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In this article, we show explicitly the Wedderburn decomposition of the metacyclic group algebra $\mathbb F_qG$, where $G$ has a cyclic subgroup of index 2 and $\gcd(|G|,q)=1$. We also construct the complete set of central and left idempotents of these group algebras.

In this data release from the LOFAR Two-metre Sky Survey (LoTSS) we present 120-168MHz images covering 27% of the northern sky. Our coverage is split into two regions centred at approximately 12h45m +44$^\circ$30' and 1h00m +28$^\circ$00' and spanning 4178 and 1457 square degrees respectively. The images were derived from 3,451hrs (7.6PB) of LOFAR High Band Antenna data which were corrected for the direction-independent instrumental properties as well as direction-dependent ionospheric distortions during extensive, but fully automated, data processing. A catalogue of 4,396,228 radio sources is derived from our total intensity (Stokes I) maps, where the majority of these have never been detected at radio wavelengths before. At 6" resolution, our full bandwidth Stokes I continuum maps with a central frequency of 144MHz have: a median rms sensitivity of 83$\mu$Jy/beam; a flux density scale accuracy of approximately 10%; an astrometric accuracy of 0.2"; and we estimate the point-source completeness to be 90% at a peak brightness of 0.8mJy/beam. By creating three 16MHz bandwidth images across the band we are able to measure the in-band spectral index of many sources, albeit with an error on the derived spectral index of +/-0.2 which is a consequence of our flux-density scale accuracy and small fractional bandwidth. Our circular polarisation (Stokes V) 20" resolution 120-168MHz continuum images have a median rms sensitivity of 95$\mu$Jy/beam, and we estimate a Stokes I to Stokes V leakage of 0.056%. Our linear polarisation (Stokes Q and Stokes U) image cubes consist of 480 x 97.6 kHz wide planes and have a median rms sensitivity per plane of 10.8mJy/beam at 4' and 2.2mJy/beam at 20"; we estimate the Stokes I to Stokes Q/U leakage to be approximately 0.2%. Here we characterise and publicly release our Stokes I, Q, U and V images in addition to the calibrated uv-data., Comment: 23 figures, 1 table and 29 pages. The catalogues, images and uv-data associated with this data release are publicly available via https://lofar-surveys.org/

In this paper we present the X-ray analysis of SDSS DR8 redMaPPer (SDSSRM) clusters using data products from the $XMM$ Cluster Survey (XCS). In total, 1189 SDSSRM clusters fall within the $XMM$-Newton footprint. This has yielded 456 confirmed detections accompanied by X-ray luminosity ($L_{X}$) measurements. Of the detected clusters, 382 have an associated X-ray temperature measurement ($T_{X}$). This represents one of the largest samples of coherently derived cluster $T_{X}$ values to date. Our analysis of the X-ray observable to richness ($\lambda$) scaling relations has demonstrated that scatter in the $T_{X}-\lambda$ relation is roughly a third of that in the $L_{X}-\lambda$ relation, and that the $L_{X}-\lambda$ scatter is intrinsic, i.e. will not be significantly reduced with larger sample sizes. Our analysis of the scaling relation between $L_{X}$ and $T_{X}$ has shown that the fits are sensitive to the selection method of the sample, i.e. whether the sample is made up of clusters detected "serendipitously" compared to those deliberately targeted by $XMM$. These differences are also seen in the $L_{X}-\lambda$ relation and, to a lesser extent, in the $T_{X}-\lambda$ relation. Exclusion of the emission from the cluster core does not make a significant impact to the findings. A combination of selection biases is a likely, but as yet unproven, reason for these differences. Finally, we have also used our data to probe recent claims of anisotropy in the $L_{X}-T_{X}$ relation across the sky. We find no evidence of anistropy, but stress that this may be masked in our analysis by the incomplete declination coverage of the SDSS DR8 sample., Comment: 23 pages, 20 figures, Accepted for publication to MNRAS

Cool gas (T$\sim$10$^{4}$~K) traced by hydrogen Ly$\alpha$ emission is now routinely detected around $z\sim3$ quasars, but little is known about their molecular gas reservoirs. Here, we present an APEX spectroscopic survey of the CO(6-5), CO(7-6) and [CI](2-1) emission lines for 9 quasars from the QSO MUSEUM survey which have similar UV luminosities, but very diverse Ly$\alpha$ nebulae. These observations ($\langle~\rm rms~\rangle=2.6$~mJy in 300~km~s$^{-1}$) detected three CO(6-5) lines with 3.4$\leq I_{\rm CO(6-5)} \leq$5.1~Jy~km~s$^{-1}$, 620$\leq$FWHM$\leq$707~km~s$^{-1}$, and three [CI](2-1) lines with 2.3$\leq I_{\rm [CI](2-1)} \leq$15.7~Jy~km~s$^{-1}$, 329$\leq$FWHM$\leq$943~km~s$^{-1}$. For the CO and [CI] detected sources, we constrain the molecular gas reservoirs to be $\rm M_{H_{2}} = (0.4-6.9) \times 10^{11} M_{\odot}$, while the non-detections imply $\rm M_{H_{2}} < 1.1\times 10^{11} M_{\odot}$. We compare our observations with the extended Ly$\alpha$ properties to understand the link between the cool and the molecular gas phases. We find large velocity shifts between the bulk of Ly$\alpha$ and the molecular gas systemic redshift in five sources (from $\sim$-400 to $\sim+$1200~km~s$^{-1}$). The sources with the largest shifts have the largest Ly$\alpha$ line widths in the sample, suggesting more turbulent gas conditions and/or large-scale inflows/outflows around these quasars. We also find that the brightest ($I_{\rm[CI](2-1)}=15.7\pm3.7~\rm Jy~km~s^{-1}$) and the widest (FWHM$\sim$900~km~s$^{-1}$) lines are detected for the smallest and dimmest Ly$\alpha$ nebulae. From this, we speculate that host galaxy obscuration can play an important role in reducing the ionizing and Ly$\alpha$ photons able to escape to halo scales, and/or that these systems are hosted by more massive halos., Comment: Accepted on January 3, MNRAS. 25 pages, 17 figures

We investigate the radio properties of a sample of 53 sources selected at 850 $\mu$m from the SCUBA-2 Cosmology Legacy Survey using new deep, low-frequency radio imaging of the Lockman Hole field from the Low Frequency Array. Combining these data with additional radio observations from the GMRT and the JVLA, we find a variety of radio spectral shapes and luminosities within our sample despite their similarly bright submillimetre flux densities. We characterise their spectral shapes in terms of multi-band radio spectral indices. Finding strong spectral flattening at low frequencies in ~20% of sources, we investigate the differences between sources with extremely flat low-frequency spectra and those with `normal' radio spectral indices. As there are no other statistically significant differences between the two subgroups of our sample as split by the radio spectral index, we suggest that any differences are undetectable in galaxy-averaged properties that we can observe with our unresolved images, and likely relate to galaxy properties that we cannot resolve, on scales $\lesssim$ 1 kpc. We attribute the observed spectral flattening in the radio to free-free absorption, proposing that those sources with significant low-frequency spectral flattening have a clumpy distribution of star-forming gas. We estimate an average spatial extent of absorbing material of at most several hundred parsecs to produce the levels of absorption observed in the radio spectra. This estimate is consistent with the highest-resolution observations of submillimetre galaxies in the literature, which find examples of non-uniform dust distributions on scales of ~100 pc, with evidence for clumps and knots in the interstellar medium. Additionally, we find two bright (> 6 mJy) submm sources undetected at all other wavelengths. We speculate that these objects may be very high redshift sources, likely residing at z > 4., Comment: 15 pages, 10 figures, accepted by A&A

There have been several reports of a detection of an unexplained excess of X-ray emission at $\simeq$ 3.5 keV in astrophysical systems. One interpretation of this excess is the decay of sterile neutrino dark matter. The most influential study to date analysed 73 clusters observed by the XMM-Newton satellite. We explore evidence for a $\simeq$ 3.5 keV excess in the XMM-PN spectra of 117 redMaPPer galaxy clusters ($0.1 < z < 0.6$). In our analysis of individual spectra, we identify three systems with an excess of flux at $\simeq$ 3.5 keV. In one case (XCS J0003.3+0204) this excess may result from a discrete emission line. None of these systems are the most dark matter dominated in our sample. We group the remaining 114 clusters into four temperature ($T_{\rm X}$) bins to search for an increase in $\simeq$ 3.5 keV flux excess with $T_{\rm X}$ - a reliable tracer of halo mass. However, we do not find evidence of a significant excess in flux at $\simeq$ 3.5 keV in any $T_{\rm X}$ bins. To maximise sensitivity to a potentially weak dark matter decay feature at $\simeq$ 3.5 keV, we jointly fit 114 clusters. Again, no significant excess is found at $\simeq$ 3.5 keV. We estimate the upper limit of an undetected emission line at $\simeq$ 3.5 keV to be $2.41 \times 10^{-6}$ photons cm$^{-2}$ s$^{-1}$, corresponding to a mixing angle of $\sin^2(2\theta)=4.4 \times 10^{-11}$, lower than previous estimates from cluster studies. We conclude that a flux excess at $\simeq$ 3.5 keV is not a ubiquitous feature in clusters and therefore unlikely to originate from sterile neutrino dark matter decay., Comment: 17 pages, 14 figures. Accepted for publication in MNRAS

Using infrared spectra of the z = 1.693 quadruply lensed quasar HE0435-1223 acquired in 2009 with the spectrograph SINFONI at the ESO Very Large Telescope, we have detected a clear microlensing effect in images A and D. While microlensing affects the blue and red wings of the H{\alpha} line profile in image D very differently, it de-magnifies the line core in image A. The combination of these different effects sets constraints on the line-emitting region; these constraints suggest that a rotating ring is at the origin of the H{\alpha} line. Visible spectra obtained in 2004 and 2012 indicate that the MgII line profile is microlensed in the same way as the H{\alpha} line. Our results therefore favour flattened geometries for the low-ionization line-emitting region, for example, a Keplerian disk. Biconical models cannot be ruled out but require more fine-tuning. Flux ratios between the different images are also derived and confirm flux anomalies with respect to estimates from lens models with smooth mass distributions., Comment: 6 pages, 4 figures, 3 tables, accepted by A&A on 10 April 2014

In this article, we show explicitly all possible weight enumerators for every irreducible cyclic code of length $n$ over a finite field $\mathbb F_q$, in the case which each prime divisor of $n$ is also a divisor of $q-1$., Comment: Submitted to Designs, Codes and Cryptography, 8 pages

Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into irreducible factors is given. Finally, weakening one of our hypothesis, we also obtain factors of the form $x^{2t}-ax^t+b$ and explicit splitting of $x^n-1$ into irreducible factors is given., Comment: Submitted to Designs, Codes and Cryptography. 9 pages

Let $\mathbb F_q$ be a finite field with $q$ elements, $G$ a finite cyclic group of order $p^k$ and $p$ is an odd prime with ${\rm gcd}(q,p)=1$. In this article, we determine an explicit expression for the primitive idempotents of $\mathbb F_qG$. This result extends the result in Arora-Pruthi [1] and Sharma-Bakshi-Dumir-Raka [8]., Comment: 9 pages