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Abstract We consider additive Schwarz methods for boundary value problems involving the $$p$$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. First, we present a new convergence theory for additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behaviour akin to the Bregman distance of the convex energy functional associated with the problem. Secondly, we provide a quasi-norm version of the Poincaré–Friedrichs inequality, which plays a crucial role in deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these key elements we establish the asymptotic linear convergence of additive Schwarz methods for the $$p$$-Laplacian. 

State-space models (SSMs), such as Mamba (Gu & Dao, 2023), have been proposed as alternatives to Transformer networks in language modeling, incorporating gating, convolutions, and input-dependent token selection to mitigate the quadratic cost of multi-head attention. Although SSMs exhibit competitive performance, their in-context learning (ICL) capabilities, a remarkable emergent property of modern language models that enables task execution without parameter optimization, remain less explored compared to Transformers. In this study, we evaluate the ICL performance of SSMs, focusing on Mamba, against Transformer models across various tasks. Our results show that SSMs perform comparably to Transformers in standard regression ICL tasks, while outperforming them in tasks like sparse parity learning. However, SSMs fall short in tasks involving non-standard retrieval functionality. To address these limitations, we introduce a hybrid model, MambaFormer, that combines Mamba with attention blocks, surpassing individual models in tasks where they struggle independently. Our findings suggest that hybrid architectures offer promising avenues for enhancing ICL in language models. 

Context. Because of its proximity and the large size of its black hole, M 87 is one of the best targets for studying the launching mechanism of active galactic nucleus jets. Currently, magnetic fields are considered to be an essential factor in the launching and accelerating of the jet. However, current observational estimates of the magnetic field strength of the M 87 jet are limited to the innermost part of the jet (≲100 r s ) or to HST-1 (∼10 5   r s ). No attempt has yet been made to measure the magnetic field strength in between. Aims. We aim to infer the magnetic field strength of the M 87 jet out to a distance of several thousand r s by tracking the distance-dependent changes in the synchrotron spectrum of the jet from high-resolution very long baseline interferometry observations. Methods. In order to obtain high-quality spectral index maps, quasi-simultaneous observations at 22 and 43 GHz were conducted using the KVN and VERA Array (KaVA) and the Very Long Baseline Array (VLBA). We compared the spectral index distributions obtained from the observations with a model and placed limits on the magnetic field strengths as a function of distance. Results. The overall spectral morphology is broadly consistent over the course of these observations. The observed synchrotron spectrum rapidly steepens from α 22 − 43 GHz  ∼ −0.7 at ∼2 mas to α 22 − 43 GHz  ∼ −2.5 at ∼6 mas. In the KaVA observations, the spectral index remains unchanged until ∼10 mas, but this trend is unclear in the VLBA observations. A spectral index model in which nonthermal electron injections inside the jet decrease with distance can adequately reproduce the observed trend. This suggests the magnetic field strength of the jet at a distance of 2−10 mas (∼900 r s  − ∼4500 r s in the deprojected distance) has a range of B  = (0.3−1.0 G)( z /2mas) −0.73 . Extrapolating to the Event Horizon Telescope scale yields consistent results, suggesting that the majority of the magnetic flux of the jet near the black hole is preserved out to ∼4500 r s without significant dissipation. 

We report three epochs of polarized images of M87* at 230 GHz using data from the Event Horizon Telescope (EHT) taken in 2017, 2018, and 2021. The baseline coverage of the 2021 observations is significantly improved through the addition of two new EHT stations: the 12 m Kitt Peak Telescope and the Northern Extended Millimetre Array (NOEMA). All observations result in images dominated by a bright, asymmetric ring with a persistent diameter of 43.9 ± 0.6 μas, consistent with expectations for lensed synchrotron emission encircling the apparent shadow of a supermassive black hole. We find that the total intensity and linear polarization of M87* vary significantly across the three epochs. Specifically, the azimuthal brightness distribution of the total intensity images varies from year to year, as expected for a stochastic accretion flow. However, despite a gamma-ray flare erupting in M87 quasi-contemporaneously to the 2018 observations, the 2018 and 2021 images look remarkably similar. The resolved linear polarization fractions in 2018 and 2021 peak at ∼5%, compared to ∼15% in 2017. The spiral polarization pattern on the ring also varies from year to year, including a change in the electric vector position angle helicity in 2021 that could reflect changes in the magnetized accretion flow or an external Faraday screen. The improved 2021 coverage also provides the first EHT constraints on jet emission outside the ring, on scales of ≲1 mas. Overall, these observations provide strong proof of the reliability of the EHT images and probe the dynamic properties of the horizon-scale accretion flow surrounding M87*.