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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. 

Galaxy cluster mergers are representative of a wide range of physics, making them an excellent probe of the properties of dark matter and the ionized plasma of the intracluster medium. To date, most studies have focused on mergers occurring in the plane of the sky, where morphological features can be readily identified. To allow study of mergers with arbitrary orientation, we have assembled multi-probe data for the eight-cluster ICM-SHOX sample sensitive to both morphology and line of sight velocity. The first ICM-SHOX paper [1] provided an overview of our methodology applied to one member of the sample, MACS J0018.5+1626, in order to constrain its merger geometry. That work resulted in an exciting new discovery of a velocity space decoupling of its gas and dark matter distributions. In this work, we describe the availability and quality of multi-probe data for the full ICM-SHOX galaxy cluster sample. These datasets will form the observational basis of an upcoming full ICM-SHOX galaxy cluster sample analysis. 

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. 

Context.The nearby elliptical galaxy M87 contains one of only two supermassive black holes whose emission surrounding the event horizon has been imaged by the Event Horizon Telescope (EHT). In 2018, more than two dozen multi-wavelength (MWL) facilities (from radio toγ-ray energies) took part in the second M87 EHT campaign. Aims.The goal of this extensive MWL campaign was to better understand the physics of the accreting black hole M87*, the relationship between the inflow and inner jets, and the high-energy particle acceleration. Understanding the complex astrophysics is also a necessary first step towards performing further tests of general relativity. Methods.The MWL campaign took place in April 2018, overlapping with the EHT M87* observations. We present a new, contemporaneous spectral energy distribution (SED) ranging from radio to very high-energy (VHE)γ-rays as well as details of the individual observations and light curves. We also conducted phenomenological modelling to investigate the basic source properties. Results.We present the first VHEγ-ray flare from M87 detected since 2010. The flux above 350 GeV more than doubled within a period of ≈36 hours. We find that the X-ray flux is enhanced by about a factor of two compared to 2017, while the radio and millimetre core fluxes are consistent between 2017 and 2018. We detect evidence for a monotonically increasing jet position angle that corresponds to variations in the bright spot of the EHT image. Conclusions.Our results show the value of continued MWL monitoring together with precision imaging for addressing the origins of high-energy particle acceleration. While we cannot currently pinpoint the precise location where such acceleration takes place, the new VHEγ-ray flare already presents a challenge to simple one-zone leptonic emission model approaches, and it emphasises the need for combined image and spectral modelling. 

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.