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1. A Holocene record of Pacific Decadal Oscillation (PDO)-related hydrologic variability in Southern California (Lake Elsinore, CA)

   https://doi.org/10.1007/s10933-010-9454-0  

   Kirby, M. E.; Lund, S. P.; Patterson, W. P.; Anderson, M. A.; Bird, B. W.; Ivanovici, L.; Monarrez, P.; Nielsen, S. (October 2010, Journal of Paleolimnology)

   High-resolution terrestrial records of Holocene climate from Southern California are scarce. Moreover, there are no records of Pacific Decadal Oscillation (PDO) variability, a major driver of decadal to multi-decadal climate variability for the region, older than 1,000 years. Recent research on Lake Elsinore, however, has shown that the lake’s sediments hold excellent potential for paleoenvironmental analysis and reconstruction. New 1-cm contiguous grain size data reveal a more complex Holocene climate history for Southern California than previously recognized at the site. A modern comparison between the twentieth century PDO index, lake level change, San Jacinto River discharge, and percent sand suggests that sand content is a reasonable, qualitative proxy for PDO-related, hydrologic variability at both multi-decadal-to-centennial as well as event (i.e. storm) timescales. A depositional model is proposed to explain the sand-hydrologic proxy. The sand-hydrologic proxy data reveal nine centennial-scale intervals of wet and dry climate throughout the Holocene. Percent total sand values >1.5 standard deviation above the 150–9,700 cal year BP average are frequent between 9,700 and 3,200 cal year BP (n = 41), but they are rare from 3,200 to 150 cal year BP (n = 6). This disparity is interpreted as a change in the frequency of exceptionally wet (high discharge) years and/or changes in large storm activity. A comparison to other regional hydrologic proxies (10 sites) shows more then occasional similarities across the region (i.e. 6 of 9 Elsinore wet intervals are present at >50% of the comparison sites). Only the early Holocene and the Little Ice Age intervals, however, are interpreted consistently across the region as uniformly wet (≥80% of the comparison sites). A comparison to two ENSO reconstructions indicates little, if any, correlation to the Elsinore data, suggesting that ENSO variability is not the predominant forcing of Holocene climate in Southern California. 

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2. Submodular Clustering in Low Dimensions

   https://doi.org/10.4230/LIPIcs.SWAT.2020.8  

   Backurs, Arturs; Har-Peled, Sariel (January 2020, 17th Scandinavian Symposium and Workshops on Algorithm Theory)

   We study a clustering problem where the goal is to maximize the coverage of the input points by k chosen centers. Specifically, given a set of n points P ⊆ ℝ^d, the goal is to pick k centers C ⊆ ℝ^d that maximize the service ∑_{p∈P}φ(𝖽(p,C)) to the points P, where 𝖽(p,C) is the distance of p to its nearest center in C, and φ is a non-increasing service function φ: ℝ+ → ℝ+. This includes problems of placing k base stations as to maximize the total bandwidth to the clients - indeed, the closer the client is to its nearest base station, the more data it can send/receive, and the target is to place k base stations so that the total bandwidth is maximized. We provide an n^{ε^-O(d)} time algorithm for this problem that achieves a (1-ε)-approximation. Notably, the runtime does not depend on the parameter k and it works for an arbitrary non-increasing service function φ: ℝ+ → ℝ+. 

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3. From quantum groups to Liouville and dilaton quantum gravity

   https://doi.org/10.1007/JHEP05(2022)092  

   Fan, Yale; Mertens, Thomas G. (May 2022, Journal of High Energy Physics)

   A bstract We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with $$ \mathcal{N} $$ N = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of U q ( $$ \mathfrak{sl} $$ sl (2 , ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for U q ( $$ \mathfrak{osp} $$ osp (1 | 2 , ℝ)) and apply it to explain structural features of $$ \mathcal{N} $$ N = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp + (1 | 2 , ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in $$ \mathcal{N} $$ N = 1 Liouville supergravity, and (3) it leads to 3 j -symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for $$ \mathcal{N} $$ N = 1 Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group U q ( $$ \mathfrak{sl} $$ sl (2 , ℝ)) or the quantum supergroup U q ( $$ \mathfrak{osp} $$ osp (1 | 2 , ℝ)). 

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4. Active Learning a Convex Body in Low Dimensions

   Har-Peled, Sariel; Jones, Mitchell; Saladi, Rahul (July 2020, Leibniz international proceedings in informatics)

   Czumaj, Artur; Dawar, Anuj; Merelli, Emanuela (Ed.)

   Consider a set P ⊆ ℝ^d of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O(⬡_P log n) queries, where ⬡_P is the largest subset of points of P in convex position. In 2D, we provide an algorithm which efficiently generates these adaptive queries. Furthermore, we show that in two dimensions one can solve this problem using O(⊚(P,C) log² n) oracle queries, where ⊚(P,C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires. Finally, we consider other variations on the problem, such as using the fewest number of queries to decide if C contains all points of P. As an application of the above, we show that the discrete geometric median of a point set P in ℝ² can be computed in O(n log² n (log n log log n + ⬡(P))) expected time. 

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5. Affine actions with Hitchin linear part

   https://doi.org/10.1007/s00039-019-00511-6  

   Danciger, Jeffrey; Zhang, Tengren (July 2019, Geometric and Functional Analysis)

   Properly discontinuous actions of a surface group by affine automorphisms of ℝ^d were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in 𝖲𝖮(n,n−1), so that the affine action is by isometries of a flat pseudo-Riemannian metric on ℝ^d of signature (n,n−1). Here, the translational part determines a deformation of the linear part into 𝖯𝖲𝖮(n,n)-Hitchin representations and the crucial step is to show that such representations are not Anosov in 𝖯𝖲𝖫(2n,ℝ) with respect to the stabilizer of an n-plane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudo-Riemannian hyperbolic space of signature (n,n−1) by a 𝖯𝖲𝖮(n,n)-Hitchin representation fails to be properly discontinuous. 

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