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1. Small fractional parts of binary forms

   https://doi.org/10.1093/qmath/haad024  

   Yeon, Kiseok (June 2023, The Quarterly Journal of Mathematics)

   Abstract We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with $$\alpha_k,\alpha_l,\ldots,\alpha_0\in{\mathbb R}$$ and $$l\leq k-2.$$ By exploiting recent progress on Vinogradov’s mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ, depending on k and $l,$ such that $$ \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}. $$ 

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2. A Systematic Analysis of the X-Ray Emission in Optically Selected Tidal Disruption Events: Observational Evidence for the Unification of the Optically and X-Ray-selected Populations

   https://doi.org/10.3847/1538-4357/ad2f9f  

   Guolo, Muryel; Gezari, Suvi; Yao, Yuhan; van_Velzen, Sjoert; Hammerstein, Erica; Cenko, S Bradley; Tokayer, Yarone M (May 2024, The Astrophysical Journal)

   Abstract We present a systematic analysis of the X-ray emission of a sample of 17 optically selected, X-ray-detected tidal disruption events (TDEs) discovered between 2014 and 2021. The X-ray light curves show a diverse range of temporal behaviors, with most sources not following the expected power-law decline. The X-ray spectra are mostly extremely soft and consistent with thermal emission from the innermost region of an accretion disk, which cools as the accretion rate decreases. Three sources show formation of a hard X-ray corona at late times. The spectral energy distribution shape, probed by the ratio (L_BB/L_X) between the UV/optical and X-ray, shows a wide range ofL_BB/L_X∈ (0.5, 3000) at early times and converges to disklike values ofL_BB/L_X∈ (0.5, 10) at late times. We estimate the fraction of optically discovered TDEs withL_X≥ 10⁴²erg s⁻¹to be at least 40% and show that X-ray loudness is independent of black hole mass. We argue that distinct disk formation timescales are unlikely to be able to explain the diverse range of X-ray evolution. We combine our sample with X-ray-discovered ones to construct an X-ray luminosity function, best fit by a broken power law, with a break atL_X≈ 10⁴⁴erg s⁻¹. We show that there is no dichotomy between optically and X-ray-selected TDEs; instead, there is a continuum of early-timeL_BB/L_X, at least as wide asL_BB/L_X∈ (0.1, 3000), with optical/X-ray surveys selecting preferentially, but not exclusively, from the higher/lower end of the distribution. Our findings are consistent with unification models for the overall TDE population. 

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3. Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

   https://doi.org/10.1090/bproc/164  

   Chirvasitu, Alexandru (July 2024, Proceedings of the American Mathematical Society, Series B)

   An extended derivation (endomorphism) of a (restricted) Lie algebra $L$is an assignment of a derivation (respectively) of $L^{\prime }$for any (restricted) Lie morphism $f:L\to <\#comment/>L^{\prime }$, functorial in $f$in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of $L^{\prime }$to every $f$; and (b) if $L$is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then $L$is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. 

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4. Intersection complexes and unramified 𝐿-factors

   https://doi.org/10.1090/jams/990  

   Sakellaridis, Yiannis; Wang, Jonathan (July 2022, Journal of the American Mathematical Society)

   Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G ˇ \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G ˇ \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties. 

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5. Radio Scrutiny of the X-Ray-weak Tail of Low-mass Active Galactic Nuclei: A Novel Signature of High-Eddington Accretion?

   https://doi.org/10.3847/1538-4357/ad67d1  

   Paul, Jeremiah_D; Plotkin, Richard_M; Brandt, W_N; Ellis, Christopher_H; Gallo, Elena; Greene, Jenny_E; Ho, Luis_C; Kimball, Amy_E; Haggard, Daryl (October 2024, The Astrophysical Journal)

   Abstract The supermassive black holes (M_BH∼ 10⁶–10¹⁰M_⊙) that power luminous active galactic nuclei (AGNs), i.e., quasars, generally show a correlation between thermal disk emission in the ultraviolet (UV) and coronal emission in hard X-rays. In contrast, some “massive” black holes (mBHs;M_BH∼ 10⁵–10⁶M_⊙) in low-mass galaxies present curious X-ray properties with coronal radiative output up to 100× weaker than expected. To examine this issue, we present a pilot study incorporating Very Large Array radio observations of a sample of 18 high-accretion-rate (Eddington ratiosL_bol/L_Edd> 0.1), mBH-powered AGNs (M_BH∼ 10⁶M_⊙) with Chandra X-ray coverage. Empirical correlations previously revealed in samples of radio-quiet, high-Eddington AGNs indicate that the radio–X-ray luminosity ratio,L_R/L_X, is approximately constant. Through multiwavelength analysis, we instead find that the X-ray-weaker mBHs in our sample tend toward larger values ofL_R/L_Xeven though they remain radio-quiet per their optical–UV properties. This trend results in a tentative but highly intriguing correlation betweenL_R/L_Xand X-ray weakness, which we argue is consistent with a scenario in which X-rays may be preferentially obscured from our line of sight by a “slim” accretion disk. We compare this observation to weak emission-line quasars (AGNs with exceptionally weak broad-line emission and a significant X-ray-weak fraction) and conclude by suggesting that our results may offer a new observational signature for finding high-accretion-rate AGNs. 

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