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1. Volume-preserving mappings between Hermitian symmetric spaces of compact type

   https://doi.org/10.1016/j.aim.2019.106885  

   Fang, Hanlong; Huang, Xiaojun; Xiao, Ming (January 2020, Advances in mathematics)

   In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian symmetric space of compact type into its Cartesian products. 

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2. An uncountable Moore–Schmidt theorem

   https://doi.org/10.1017/etds.2022.36  

   JAMNESHAN, ASGAR; TAO, TERENCE (July 2023, Ergodic Theory and Dynamical Systems)

   Abstract We prove an extension of the Moore–Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a ‘conditional’ Pontryagin duality for spaces of abstract measurable maps. 

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3. Compactness results for a Dirichlet energy of nonlocal gradient with applications

   https://doi.org/10.1002/num.23149  

   Han, Zhaolong; Mengesha, Tadele; Tian, Xiaochuan (November 2024, Numerical Methods for Partial Differential Equations)

   Abstract We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem. 

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4. PHANGS-HST Catalogs for ∼100,000 Star Clusters and Compact Associations in 38 Galaxies. I. Observed Properties

   https://doi.org/10.3847/1538-4365/ad3cd3  

   Maschmann, Daniel; Lee, Janice C; Thilker, David A; Whitmore, Bradley C; Deger, Sinan; Boquien, Médéric; Chandar, Rupali; Dale, Daniel A; Wofford, Aida; Hannon, Stephen; et al (July 2024, The Astrophysical Journal Supplement Series)

   Abstract We present the largest catalog to date of star clusters and compact associations in nearby galaxies. We have performed aV-band-selected census of clusters across the 38 spiral galaxies of the PHANGS–Hubble Space Telescope (HST) Treasury Survey, and measured integrated, aperture-corrected near-ultraviolet-U-B-V-Iphotometry. This work has resulted in uniform catalogs that contain ∼20,000 clusters and compact associations, which have passed human inspection and morphological classification, and a larger sample of ∼100,000 classified by neural network models. Here, we report on the observed properties of these samples, and demonstrate that tremendous insight can be gained from just the observed properties of clusters, even in the absence of their transformation into physical quantities. In particular, we show the utility of the UBVI color–color diagram, and the three principal features revealed by the PHANGS-HST cluster sample: the young cluster locus, the middle-age plume, and the old globular cluster clump. We present an atlas of maps of the 2D spatial distribution of clusters and compact associations in the context of the molecular clouds from PHANGS–Atacama Large Millimeter/submillimeter Array. We explore new ways of understanding this large data set in a multiscale context by bringing together once-separate techniques for the characterization of clusters (color–color diagrams and spatial distributions) and their parent galaxies (galaxy morphology and location relative to the galaxy main sequence). A companion paper presents the physical properties: ages, masses, and dust reddenings derived using improved spectral energy distribution fitting techniques. 

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5. (In)equality distance patterns and embeddability into Hilbert spaces

   Alexandru Chirvasitu (October 2022, Palestine journal of mathematics)

   We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddability: if every finite subspace thereof is loosely embeddable into a common R^N , then the metric space as a whole is loosely embeddable into R^N in a weakened sense. 

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