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1. A stability inequality for planar lens partition

   https://doi.org/10.1017/prm.2025.2  

   Bonacini, Marco; Cristoferi, Riccardo; Topaloglu, Ihsan (January 2025, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Recently it has been shown that the unique local perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens, hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem. 

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2. Statistical Efficiency of Score Matching: The View from Isoperimetry

   Koehler, F; Heckett, A; Risteski, A. (May 2023, International Conference on Learning Representations (ICLR), 2024)

   Deep generative models parametrized up to a normalizing constant (e.g. energy-based models) are difficult to train by maximizing the likelihood of the data because the likelihood and/or gradients thereof cannot be explicitly or efficiently written down. Score matching is a training method, whereby instead of fitting the likelihood for the training data, we instead fit the score function, obviating the need to evaluate the partition function. Though this estimator is known to be consistent, it's unclear whether (and when) its statistical efficiency is comparable to that of maximum likelihood---which is known to be (asymptotically) optimal. We initiate this line of inquiry in this paper and show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated---i.e. the Poincar\'e, log-Sobolev and isoperimetric constant---quantities which govern the mixing time of Markov processes like Langevin dynamics. Roughly, we show that the score matching estimator is statistically comparable to the maximum likelihood when the distribution has a small isoperimetric constant. Conversely, if the distribution has a large isoperimetric constant---even for simple families of distributions like exponential families with rich enough sufficient statistics---score matching will be substantially less efficient than maximum likelihood. We suitably formalize these results both in the finite sample regime, and in the asymptotic regime. Finally, we identify a direct parallel in the discrete setting, where we connect the statistical properties of pseudolikelihood estimation with approximate tensorization of entropy and the Glauber dynamics. 

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3. Nonexistence of isoperimetric sets in spaces of positive curvature

   https://doi.org/10.1515/crelle-2024-0032  

   Antonelli, Gioacchino; Glaudo, Federico (May 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract For every $d\ge 3$d\geq 3, we construct a noncompact smooth 𝑑-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below 1.We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in ${\mathbb{R}}^d$\mathbb{R}^{d}.The examples we construct have nondegenerate asymptotic cone.The dimensional constraint $d\ge 3$d\geq 3is sharp.Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed, in nonnegatively curved spaces with nondegenerate asymptotic cones, isoperimetric sets with large volumes always exist.This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets. 

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4. Supernova remnant candidates discovered by the SARAO MeerKAT Galactic Plane Survey

   https://doi.org/10.1051/0004-6361/202451038  

   Anderson, L D; Camilo, F; Faerber, T; Bietenholz, M; Bordiu, C; Bufano, F; Chibueze, J O; Cotton, W D; Ingallinera, A; Loru, S; et al (January 2025, Astronomy & Astrophysics)

   Context. Sensitive radio continuum data could bring the number of known supernova remnants (SNRs) in the Galaxy more in line with what is expected. Due to confusion in the Galactic plane, however, faint SNRs can be challenging to distinguish from brighter HIIregions and filamentary radio emission. Aims. We exploited new 1.3 GHz SARAO MeerKAT Galactic Plane Survey (SMGPS) radio continuum data, which cover 251° ≤ℓ≤ 358° and 2° ≤ℓ≤ 61° at |b| ≤ 1.5°, to search for SNR candidates in the Milky Way disk. Methods. We also used mid-infrared data from theSpitzerGLIMPSE,SpitzerMIPSGAL, and WISE surveys to help identify SNR candidates. These candidates are sources of extended radio continuum emission that lack mid-infrared counterparts, are not known as HIIregions in the WISE Catalog of Galactic HIIRegions, and have not been previously identified as SNRs. Results. We locate 237 new Galactic SNR candidates in the SMGPS data. We also identify and confirm the expected radio morphology for 201 objects classified in the literature as SNRs and 130 previously identified SNR candidates. The known and candidate SNRs have similar spatial distributions and angular sizes. Conclusions. The SMGPS data allowed us to identify a large population of SNR candidates that can be confirmed as true SNRs using radio polarization measurements or by deriving radio spectral indices. If the 237 candidates are confirmed as true SNRs, it would approximately double the number of known Galactic SNRs in the survey area, alleviating much of the discrepancy between the known and expected populations. 

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5. An atlas of gas motions in the TNG-Cluster simulation: From cluster cores to the outskirts

   https://doi.org/10.1051/0004-6361/202348612  

   Ayromlou, Mohammadreza; Nelson, Dylan; Pillepich, Annalisa; Rohr, Eric; Truong, Nhut; Li, Yuan; Simionescu, Aurora; Lehle, Katrin; Lee, Wonki (October 2024, Astronomy & Astrophysics)

   Galaxy clusters are unique laboratories for studying astrophysical processes and their impact on halo gas kinematics. Despite their importance, the full complexity of gas motion within and around these clusters remains poorly known. This paper is part of a series presenting the first results from the new TNG-Cluster simulation, a suite comprising 352 high-mass galaxy clusters including the full cosmological context, mergers and accretion, baryonic processes and feedback, and magnetic fields. Studying the dynamics and coherence of gas flows, we find that gas motions in galaxy cluster cores and intermediate regions are largely balanced between inflows and outflows, exhibiting a Gaussian distribution centered at zero velocity. In the outskirts, even the net velocity distribution becomes asymmetric, featuring a double peak where the second peak reflects cosmic accretion. Across all cluster regions, the resulting net flow distribution reveals complex gas dynamics. These are strongly correlated with halo properties: at a given total cluster mass, unrelaxed, late-forming halos with fewer massive black holes and lower accretion rates exhibit a more dynamic behavior. Our analysis shows no clear relationship between line-of-sight and radial gas velocities, suggesting that line-of-sight velocity alone is insufficient to distinguish between inflowing and outflowing gas. Additional properties, such as temperature, can help break this degeneracy. A velocity structure function (VSF) analysis indicates more coherent gas motion in the outskirts and more disturbed kinematics toward halo centers. In all cluster regions, the VSF shows a slope close to the theoretical models of Kolmogorov (∼1/3), except within 50 kpc of the cluster centers, where the slope is significantly steeper. The outcome of TNG-Cluster broadly aligns with observations of the VSF of multiphase gas across different scales in galaxy clusters, ranging from ∼1 kpc to megaparsec scales. 

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