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Recommender systems build user profiles using concept analysis of usage matrices. The concepts are mined as spectra and form Galois connections. Descent is a general method for spectral decomposition in algebraic geometry and topology which also leads to generalized Galois connections. Both recommender systems and descent theory are vast research areas, separated by a technical gap so large that trying to establish a link would seem foolish. Yet a formal link emerged, all on its own, bottom-up, against authors’ intentions and better judgment. Familiar problems of data analysis led to a novel solution in category theory. The present paper arose from a series of earlier efforts to provide a top-down account of these developments. 

The Planck All-Sky Survey to Analyze Gravitationally-lensed Extreme Starbursts project aims to identify a population of extremely luminous galaxies using the Planck all-sky survey and to explore the nature of their gas fuelling, induced starburst, and the resulting feedback that shape their evolution. Here, we report the identification of 22 high-redshift luminous dusty star-forming galaxies (DSFGs) at z = 1.1–3.3 drawn from a candidate list constructed using the Planck Catalogue of Compact Sources and Wide-field Infrared Survey Explorer all-sky survey. They are confirmed through follow-up dust continuum imaging and CO spectroscopy using AzTEC and the Redshift Search Receiver on the Large Millimeter Telescope Alfonso Serrano. Their apparent infrared luminosities span (0.1–3.1) × 1014 L⊙ (median of 1.2 × 1014 L⊙), making them some of the most luminous galaxies found so far. They are also some of the rarest objects in the sky with a source density of ≲0.01 deg−2. Our Atacama Large Millimeter/submillimeter Array 1.1 mm continuum observations with θ ≈ 0.4 arcsec resolution show clear ring or arc morphologies characteristic of strong lensing. Their lensing-corrected luminosity of LIR ≳ 1013 L⊙ (star-formation rate ≳ 103 M⊙ yr−1) indicates that they are the magnified versions of the most intrinsically luminous DSFGs found at these redshifts. Our spectral energy distribution analysis finds little detectable active galactic nucleus (AGN) activity despite their enormous luminosity, and any AGN activity present must be extremely heavily obscured.

 

Context.3C 84 is a nearby radio source with a complex total intensity structure, showing linear polarisation and spectral patterns. A detailed investigation of the central engine region necessitates the use of very-long-baseline interferometry (VLBI) above the hitherto available maximum frequency of 86 GHz.

Aims.Using ultrahigh resolution VLBI observations at the currently highest available frequency of 228 GHz, we aim to perform a direct detection of compact structures and understand the physical conditions in the compact region of 3C 84.

Methods.We used Event Horizon Telescope (EHT) 228 GHz observations and, given the limited (u, v)-coverage, applied geometric model fitting to the data. Furthermore, we employed quasi-simultaneously observed, ancillary multi-frequency VLBI data for the source in order to carry out a comprehensive analysis of the core structure.

Results.We report the detection of a highly ordered, strong magnetic field around the central, supermassive black hole of 3C 84. The brightness temperature analysis suggests that the system is in equipartition. We also determined a turnover frequency ofν_m = (113 ± 4) GHz, a corresponding synchrotron self-absorbed magnetic field ofB_SSA = (2.9 ± 1.6) G, and an equipartition magnetic field ofB_eq = (5.2 ± 0.6) G. Three components are resolved with the highest fractional polarisation detected for this object (m_net = (17.0 ± 3.9)%). The positions of the components are compatible with those seen in low-frequency VLBI observations since 2017–2018. We report a steeply negative slope of the spectrum at 228 GHz. We used these findings to test existing models of jet formation, propagation, and Faraday rotation in 3C 84.

Conclusions.The findings of our investigation into different flow geometries and black hole spins support an advection-dominated accretion flow in a magnetically arrested state around a rapidly rotating supermassive black hole as a model of the jet-launching system in the core of 3C 84. However, systematic uncertainties due to the limited (u, v)-coverage, however, cannot be ignored. Our upcoming work using new EHT data, which offer full imaging capabilities, will shed more light on the compact region of 3C 84.

 

Exploiting the sensitivity of the IRAM NOrthern Extended Millimeter Array (NOEMA) and its ability to process large instantaneous bandwidths, we have studied the morphology and other properties of the molecular gas and dust in the star forming galaxy, H-ATLAS J131611.5+281219 (HerBS-89a), at z = 2.95. High angular resolution (0 . ″3) images reveal a partial 1 . ″0 diameter Einstein ring in the dust continuum emission and the molecular emission lines of 12 CO(9−8) and H 2 O(2 02  − 1 11 ). Together with lower angular resolution (0 . ″6) images, we report the detection of a series of molecular lines including the three fundamental transitions of the molecular ion OH + , namely (1 1  − 0 1 ), (1 2  − 0 1 ), and (1 0  − 0 1 ), seen in absorption; the molecular ion CH + (1 − 0) seen in absorption, and tentatively in emission; two transitions of amidogen (NH 2 ), namely (2 02  − 1 11 ) and (2 20  − 2 11 ) seen in emission; and HCN(11 − 10) and/or NH(1 2  − 0 1 ) seen in absorption. The NOEMA data are complemented with Very Large Array data tracing the 12 CO(1 − 0) emission line, which provides a measurement of the total mass of molecular gas and an anchor for a CO excitation analysis. In addition, we present Hubble Space Telescope imaging that reveals the foreground lensing galaxy in the near-infrared (1.15  μ m). Together with photometric data from the Gran Telescopio Canarias, we derive a photometric redshift of z phot = 0.9 −0.5 +0.3 for the foreground lensing galaxy. Modeling the lensing of HerBS-89a, we reconstruct the dust continuum (magnified by a factor μ  ≃ 5.0) and molecular emission lines (magnified by μ  ∼ 4 − 5) in the source plane, which probe scales of ∼0 . ″1 (or 800 pc). The 12 CO(9 − 8) and H 2 O(2 02  − 1 11 ) emission lines have comparable spatial and kinematic distributions; the source-plane reconstructions do not clearly distinguish between a one-component and a two-component scenario, but the latter, which reveals two compact rotating components with sizes of ≈1 kpc that are likely merging, more naturally accounts for the broad line widths observed in HerBS-89a. In the core of HerBS-89a, very dense gas with n H 2  ∼ 10 7 − 9 cm −3 is revealed by the NH 2 emission lines and the possible HCN(11 − 10) absorption line. HerBS-89a is a powerful star forming galaxy with a molecular gas mass of M mol  = (2.1 ± 0.4) × 10 11   M ⊙ , an infrared luminosity of L IR  = (4.6 ± 0.4) × 10 12   L ⊙ , and a dust mass of M dust  = (2.6 ± 0.2) × 10 9   M ⊙ , yielding a dust-to-gas ratio δ GDR  ≈ 80. We derive a star formation rate SFR = 614 ± 59  M ⊙ yr −1 and a depletion timescale τ depl  = (3.4 ± 1.0) × 10 8 years. The OH + and CH + absorption lines, which trace low (∼100 cm −3 ) density molecular gas, all have their main velocity component red-shifted by Δ V  ∼ 100 km s −1 relative to the global CO reservoir. We argue that these absorption lines trace a rare example of gas inflow toward the center of a galaxy, indicating that HerBS-89a is accreting gas from its surroundings. 

Using the IRAM NOrthern Extended Millimeter Array (NOEMA), we conducted a program to measure redshifts for 13 bright galaxies detected in the Herschel Astrophysical Large Area Survey with S 500  μ m  ≥ 80 mJy. We report reliable spectroscopic redshifts for 12 individual sources, which are derived from scans of the 3 and 2 mm bands, covering up to 31 GHz in each band, and are based on the detection of at least two emission lines. The spectroscopic redshifts are in the range 2.08 <   z  <  4.05 with a median value of z  = 2.9 ± 0.6. The sources are unresolved or barely resolved on scales of 10 kpc. In one field, two galaxies with different redshifts were detected. In two cases the sources are found to be binary galaxies with projected distances of ∼140 kpc. The linewidths of the sources are large, with a mean value for the full width at half maximum of 700 ± 300 km s −1 and a median of 800 km s −1 . We analyze the nature of the sources with currently available ancillary data to determine if they are lensed or hyper-luminous ( L FIR  >  10 13   L ⊙ ) galaxies. We also present a reanalysis of the spectral energy distributions including the continuum flux densities measured at 3 and 2 mm to derive the overall properties of the sources. Future prospects based on these efficient measurements of redshifts of high- z galaxies using NOEMA are outlined, including a comprehensive survey of all the brightest Herschel galaxies. 

We present spectroscopic measurements for 71 galaxies associated with 62 of the brightest high-redshift submillimetre sources from the Southern fields of the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS), while targeting 85 sources which resolved into 142. We have obtained robust redshift measurements for all sources using the 12-m Array and an efficient tuning of ALMA to optimize its use as a redshift hunter, with 73 per cent of the sources having a robust redshift identification. Nine of these redshift identifications also rely on observations from the Atacama Compact Array. The spectroscopic redshifts span a range 1.41 < z < 4.53 with a mean value of 2.75, and the CO emission line full-width at half-maxima range between $\rm 110\, km\, s^{-1} \lt FWHM \lt 1290\, km\, s^{-1}$ with a mean value of ∼500 km s−1, in line with other high-z samples. The derived CO(1-0) luminosity is significantly elevated relative to line-width to CO(1-0) luminosity scaling relation, which is suggestive of lensing magnification across our sources. In fact, the distribution of magnification factors inferred from the CO equivalent widths is consistent with expectations from galaxy–galaxy lensing models, though there is a hint of an excess at large magnifications that may be attributable to the additional lensing optical depth from galaxy groups or clusters.