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1. Adic tropicalizations and cofinality of Gubler models

   https://doi.org/10.1112/blms.70001  

   Foster, Tyler; Payne, Sam (January 2025, Bulletin of the London Mathematical Society)

   Abstract We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a ‐topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi. 

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2. Smart quantum statistical imaging beyond the Abbe-Rayleigh criterion

   https://doi.org/10.1038/s41534-022-00593-5  

   Bhusal, Narayan; Hong, Mingyuan; Miller, Ashe; Quiroz-Juárez, Mario A.; León-Montiel, Roberto de; You, Chenglong; Magaña-Loaiza, Omar S. (December 2022, npj Quantum Information)

   Mary Francina (Ed.)

   Abstract The wave nature of light imposes limits on the resolution of optical imaging systems. For over a century, the Abbe-Rayleigh criterion has been utilized to assess the spatial resolution limits of imaging instruments. Recently, there has been interest in using spatial projective measurements to enhance the resolution of imaging systems. Unfortunately, these schemes require a priori information regarding the coherence properties of “unknown” light beams and impose stringent alignment conditions. Here, we introduce a smart quantum camera for superresolving imaging that exploits the self-learning features of artificial intelligence to identify the statistical fluctuations of unknown mixtures of light sources at each pixel. This is achieved through a universal quantum model that enables the design of artificial neural networks for the identification of photon fluctuations. Our protocol overcomes limitations of existing superresolution schemes based on spatial mode projections, and consequently provides alternative methods for microscopy, remote sensing, and astronomy. 

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3. Geometric generalizations of the square sieve, with an application to cyclic covers

   https://doi.org/10.1112/mtk.12180  

   Bucur, Alina; Cojocaru, Alina Carmen; Lalín, Matilde N.; Pierce, Lillian B. (December 2022, Mathematika)

   Abstract We formulate a general problem: Given projective schemes and over a global fieldKand aK‐morphism η from to of finite degree, how many points in of height at mostBhave a pre‐image under η in ? This problem is inspired by a well‐known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when and is a prime degree cyclic cover of . Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields. 

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4. Perfect Conditional ε ‐Equilibria of Multi‐Stage Games With Infinite Sets of Signals and Actions

   https://doi.org/10.3982/ECTA13426  

   Myerson, Roger B.; Reny, Philip J. (January 2020, Econometrica)

   We extend Kreps and Wilson's concept of sequential equilibrium to games with infinite sets of signals and actions. A strategy profile is a conditional ε ‐equilibrium if, for any of a player's positive probability signal events, his conditional expected utility is within ε of the best that he can achieve by deviating. With topologies on action sets, a conditional ε ‐equilibrium is full if strategies give every open set of actions positive probability. Such full conditional ε ‐equilibria need not be subgame perfect, so we consider a non‐topological approach. Perfect conditional ε ‐equilibria are defined by testing conditional ε ‐rationality along nets of small perturbations of the players' strategies and of nature's probability function that, for any action and for almost any state, make this action and state eventually (in the net) always have positive probability. Every perfect conditional ε ‐equilibrium is a subgame perfect ε ‐equilibrium, and, in finite games, limits of perfect conditional ε ‐equilibria as ε  → 0 are sequential equilibrium strategy profiles. But limit strategies need not exist in infinite games so we consider instead the limit distributions over outcomes. We call such outcome distributions perfect conditional equilibrium distributions and establish their existence for a large class of regular projective games. Nature's perturbations can produce equilibria that seem unintuitive and so we augment the game with a net of permissible perturbations. 

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5. OPTIMAL LINE PACKINGS FROM FINITE GROUP ACTIONS

   https://doi.org/10.1017/fms.2019.48  

   IVERSON, JOSEPH W.; JASPER, JOHN; MIXON, DUSTIN G. (January 2020, Forum of Mathematics, Sigma)

   We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry. 

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