Number fields and function fields:Coalescences, contrasts and emerging applications

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained

The two-component giant radio halo in the galaxy cluster Abell 2142

We report on a spectral study at radio frequencies of the giant radio halo in A2142 (z=0.0909), which we performed to explore its nature and origin. A2142 is not a major merger and the presence of a giant radio halo is somewhat surprising. We performed deep radio observations with the GMRT at 608 MHz, 322 MHz, and 234 MHz and with the VLA in the 1-2 GHz band. We obtained high-quality images at all frequencies in a wide range of resolutions. The radio halo is well detected at all frequencies and extends out to the most distant cold front in A2142. We studied the spectral index in two regions: the central part of the halo and a second region in the direction of the most distant south-eastern cold front, selected to follow the bright part of the halo and X-ray emission. We complemented our observations with a preliminary LOFAR image at 118 MHz and with the re-analysis of archival VLA data at 1.4 GHz. The two components of the radio halo show different observational properties. The central brightest part has higher surface brightess and a spectrum whose steepness is similar to those of the known radio halos, i.e. $\alpha^{\rm 1.78~GHz}_{\rm 118~MHz}=1.33\pm 0.08$. The ridge, which fades into the larger scale emission, is broader in size and has considerably lower surface brightess and a moderately steeper spectrum, i.e. $\alpha^{\rm 1.78~GHz}_{\rm 118~MHz}\sim 1.5$. We propose that the brightest part of the radio halo is powered by the central sloshing in A2142, similar to what has been suggested for mini-halos, or by secondary electrons generated by hadronic collisions in the ICM. On the other hand, the steeper ridge may probe particle re-acceleration by turbulence generated either by stirring the gas and magnetic fields on a larger scale or by less energetic mechanisms, such as continuous infall of galaxy groups or an off-axis merger.Comment: 18 pages, 10 figures, 4 tables - A&A, accepte

The Three-Dimensional Structure of Interior Ejecta in Cassiopeia A at High Spectral Resolution

We used the Spitzer Space Telescope's Infrared Spectrograph to create a high resolution spectral map of the central region of the Cassiopeia A supernova remnant, allowing us to make a Doppler reconstruction of its 3D structure. The ejecta responsible for this emission have not yet encountered the remnant's reverse shock or the circumstellar medium, making it an ideal laboratory for exploring the dynamics of the supernova explosion itself. We observe that the O, Si, and S ejecta can form both sheet-like structures as well as filaments. Si and O, which come from different nucleosynthetic layers of the star, are observed to be coincident in velocity space in some regions, and separated by 500 km/s or more in others. Ejecta traveling toward us are, on average, ~900 km/s slower than the material traveling away from us. We compare our observations to recent supernova explosion models and find that no single model can simultaneously reproduce all the observed features. However, models of different supernova explosions can collectively produce the observed geometries and structures of the interior emission. We use the results from the models to address the conditions during the supernova explosion, concentrating on asymmetries in the shock structure. We also predict that the back surface of Cassiopeia A will begin brightening in ~30 years, and the front surface in ~100 years.Comment: 35 pages, 16 figures, accepted to Ap

The stripping of a galaxy group diving into the massive cluster A2142

Structure formation in the current Universe operates through the accretion of group-scale systems onto massive clusters. The detection and study of such accreting systems is crucial to understand the build-up of the most massive virialized structures we see today. We report the discovery with XMM-Newton of an irregular X-ray substructure in the outskirts of the massive galaxy cluster Abell 2142. The tip of the X-ray emission coincides with a concentration of galaxies. The bulk of the X-ray emission of this substructure appears to be lagging behind the galaxies and extends over a projected scale of at least 800 kpc. The temperature of the gas in this region is 1.4 keV, which is a factor of ~4 lower than the surrounding medium and is typical of the virialized plasma of a galaxy group with a mass of a few 10^13M_sun. For this reason, we interpret this structure as a galaxy group in the process of being accreted onto the main dark-matter halo. The X-ray structure trailing behind the group is due to gas stripped from its original dark-matter halo as it moves through the intracluster medium (ICM). This is the longest X-ray trail reported to date. For an infall velocity of ~1,200 km s-1 we estimate that the stripped gas has been surviving in the presence of the hot ICM for at least 600 Myr, which exceeds the Spitzer conduction timescale in the medium by a factor of >~400. Such a strong suppression of conductivity is likely related to a tangled magnetic field with small coherence length and to plasma microinstabilities. The long survival time of the low-entropy intragroup medium suggests that the infalling material can eventually settle within the core of the main cluster.Comment: 11 pages, 7 figures, accepted for publication in A&

The shape of invasion perclation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for both non-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two different anisotropy quantifiers, the anisotropy parameter and the asphericity are used for probing the degree of anisotropy of clusters. We observe that in spite of the difference in scaling properties of NTIP and TIP, there is no difference in the values of anisotropy quantifiers of these processes. Furthermore, we find that in completely random media, the invasion percolation clusters are on average slightly less isotropic than standard percolation clusters. Introducing isotropic long-range correlations into the media reduces the isotropy of the invasion percolation clusters. The effect is more pronounced for the case of persisting long-range correlations. The implication of boundary conditions on the shape of clusters is another subject of interest. Compared to the case of free boundary conditions, IP clusters of conventional rectangular geometry turn out to be more isotropic. Moreover, we see that in conventional rectangular geometry the NTIP clusters are more isotropic than TIP clusters

Deep Chandra observations of the stripped galaxy group falling into Abell 2142

In the local Universe, the growth of massive galaxy clusters mainly operates through the continuous accretion of group-scale systems. The infalling group in Abell 2142 is the poster child of such an accreting group, and as such, it is an ideal target to study the astrophysical processes induced by structure formation. We present the results of a deep (200 ks) observation of this structure with Chandra, which highlights the complexity of this system in exquisite detail. In the core of the group, the spatial resolution of Chandra reveals the presence of a leading edge and a complex AGN-induced activity. The morphology of the stripped gas tail appears straight in the innermost 250 kpc, suggesting that magnetic draping efficiently shields the gas from its surroundings. However, beyond $\sim300$ kpc from the core, the tail flares and the morphology becomes strongly irregular, which could be explained by a breaking of the drape, e.g. because of turbulent motions. The power spectrum of surface-brightness fluctuations is relatively flat ($P_{2D}\propto k^{-2.3}$), which indicates that thermal conduction is strongly inhibited even beyond the region where magnetic draping is effective. The amplitude of density fluctuations in the tail is consistent with a mild level of turbulence with a Mach number $M_{3D}\sim0.1-0.25$. Overall, our results show that the processes leading to the thermalization and mixing of the infalling gas are slow and relatively inefficient.Comment: Accepted for publication in A&

Freshly Formed Dust in the Cassiopeia A Supernova Remnant as Revealed by the Spitzer Space Telescope

We performed Spitzer Infrared Spectrograph mapping observations covering nearly the entire extent of the Cassiopeia A supernova remnant (SNR), producing mid-infrared (5.5-35 micron) spectra every 5-10". Gas lines of Ar, Ne, O, Si, S and Fe, and dust continua were strong for most positions. We identify three distinct ejecta dust populations based on their continuum shapes. The dominant dust continuum shape exhibits a strong peak at 21 micron. A line-free map of 21 micron-peak dust made from the 19-23 micron range closely resembles the [Ar II], [O IV], and [Ne II] ejecta-line maps implying that dust is freshly formed in the ejecta. Spectral fitting implies the presence of SiO2, Mg protosilicates, and FeO grains in these regions. The second dust type exhibits a rising continuum up to 21 micron and then flattens thereafter. This ``weak 21 micron'' dust is likely composed of Al2O3 and C grains. The third dust continuum shape is featureless with a gently rising spectrum and is likely composed of MgSiO3 and either Al2O3 or Fe grains. Using the least massive composition for each of the three dust classes yields a total mass of 0.02 Msun. Using the most-massive composition yields a total mass of 0.054 Msun. The primary uncertainty in the total dust mass stems from the selection of the dust composition necessary for fitting the featureless dust as well as 70 micron flux. The freshly formed dust mass derived from Cas A is sufficient from SNe to explain the lower limit on the dust masses in high redshift galaxies.Comment: 8 figures: Accepted for the publication in Ap

Theory of monolayers with boundaries: Exact results and Perturbative analysis

Domains and bubbles in tilted phases of Langmuir monolayers contain a class of textures knows as boojums. The boundaries of such domains and bubbles may display either cusp-like features or indentations. We derive analytic expressions for the textures within domains and surrounding bubbles, and for the shapes of the boundaries of these regions. The derivation is perturbative in the deviation of the bounding curve from a circle. This method is not expected to be accurate when the boundary suffers large distortions, but it does provide important clues with regard to the influence of various energetic terms on the order-parameter texture and the shape of the domain or bubble bounding curve. We also look into the effects of thermal fluctuations, which include a sample-size-dependent effective line tension.Comment: replaced with published version, 21 pages, 16 figures include

The Three-Dimensional Structure of Cassiopeia A

We used the Spitzer Space Telescope's Infrared Spectrograph to map nearly the entire extent of Cassiopeia A between 5-40 micron. Using infrared and Chandra X-ray Doppler velocity measurements, along with the locations of optical ejecta beyond the forward shock, we constructed a 3-D model of the remnant. The structure of Cas A can be characterized into a spherical component, a tilted thick disk, and multiple ejecta jets/pistons and optical fast-moving knots all populating the thick disk plane. The Bright Ring in Cas A identifies the intersection between the thick plane/pistons and a roughly spherical reverse shock. The ejecta pistons indicate a radial velocity gradient in the explosion. Some ejecta pistons are bipolar with oppositely-directed flows about the expansion center while some ejecta pistons show no such symmetry. Some ejecta pistons appear to maintain the integrity of the nuclear burning layers while others appear to have punched through the outer layers. The ejecta pistons indicate a radial velocity gradient in the explosion. In 3-D, the Fe jet in the southeast occupies a "hole" in the Si-group emission and does not represent "overturning", as previously thought. Although interaction with the circumstellar medium affects the detailed appearance of the remnant and may affect the visibility of the southeast Fe jet, the bulk of the symmetries and asymmetries in Cas A are intrinsic to the explosion.Comment: Accepted to ApJ. 54 pages, 21 figures. For high resolution figures and associated mpeg movie and 3D PDF files, see http://homepages.spa.umn.edu/~tdelaney/pape