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The Frobenius-Perron theory of an endofunctor of a k \Bbbk -linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras. 

We present an observational study of the luminous red nova (LRN) AT 2021biy in the nearby galaxy NGC 4631. The field of the object was routinely imaged during the pre-eruptive stage by synoptic surveys, but the transient was detected only at a few epochs from ∼231 days before maximum brightness. The LRN outburst was monitored with unprecedented cadence both photometrically and spectroscopically. AT 2021biy shows a short-duration blue peak, with a bolometric luminosity of ∼1.6 × 10 41 erg s −1 , followed by the longest plateau among LRNe to date, with a duration of 210 days. A late-time hump in the light curve was also observed, possibly produced by a shell-shell collision. AT 2021biy exhibits the typical spectral evolution of LRNe. Early-time spectra are characterised by a blue continuum and prominent H emission lines. Then, the continuum becomes redder, resembling that of a K-type star with a forest of metal absorption lines during the plateau phase. Finally, late-time spectra show a very red continuum ( T BB  ≈ 2050 K) with molecular features (e.g., TiO) resembling those of M-type stars. Spectropolarimetric analysis indicates that AT 2021biy has local dust properties similar to those of V838 Mon in the Milky Way Galaxy. Inspection of archival Hubble Space Telescope data taken on 2003 August 3 reveals a ∼20 M ⊙ progenitor candidate with log ( L / L ⊙ ) = 5.0 dex and T eff  = 5900 K at solar metallicity. The above luminosity and colour match those of a luminous yellow supergiant. Most likely, this source is a close binary, with a 17–24 M ⊙ primary component. 

ABSTRACT GRANDMA (Global Rapid Advanced Network Devoted to the Multi-messenger Addicts) is a network of 25 telescopes of different sizes, including both photometric and spectroscopic facilities. The network aims to coordinate follow-up observations of gravitational-wave (GW) candidate alerts, especially those with large localization uncertainties, to reduce the delay between the initial detection and the optical confirmation. In this paper, we detail GRANDMA’s observational performance during Advanced LIGO/Advanced Virgo Observing Run 3 (O3), focusing on the second part of O3; this includes summary statistics pertaining to coverage and possible astrophysical origin of the candidates. To do so, we quantify our observation efficiency in terms of delay between GW candidate trigger time, observations, and the total coverage. Using an optimized and robust coordination system, GRANDMA followed-up about 90 per cent of the GW candidate alerts, that is 49 out of 56 candidates. This led to coverage of over 9000 deg2 during O3. The delay between the GW candidate trigger and the first observation was below 1.5 h for 50 per cent of the alerts. We did not detect any electromagnetic counterparts to the GW candidates during O3, likely due to the very large localization areas (on average thousands of degrees squares) and relatively large distance of the candidates (above 200 Mpc for 60 per cent of binary neutron star, BNS candidates). We derive constraints on potential kilonova properties for two potential BNS coalescences (GW190425 and S200213t), assuming that the events’ locations were imaged.