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1. A remark about the anomalies of cyclic holomorphic permutation orbifolds

   https://doi.org/10.1142/S0129167X20500809  

   Bischoff, M. (September 2020, International Journal of Mathematics)

   Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net [Formula: see text] with central charge [Formula: see text] depends on the “gravitational anomaly” [Formula: see text]. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed [Formula: see text]-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More generally, we show that cyclic permutations of order [Formula: see text] are non-anomalous if and only if [Formula: see text] or [Formula: see text]. We also show that all cyclic permutation gaugings of [Formula: see text] arise from conformal nets. 

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2. Breaking the r _max Barrier: Enhanced Approximation Algorithms for Partial Set Multicover Problem

   https://doi.org/10.1287/ijoc.2020.0975  

   Ran, Yingli; Zhang, Zhao; Tang, Shaojie; Du, Ding-Zhu (October 2020, INFORMS Journal on Computing)

   null (Ed.)

   Given an element set E of order n, a collection of subsets [Formula: see text], a cost c S on each set [Formula: see text], a covering requirement r e for each element [Formula: see text], and an integer k, the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection [Formula: see text] to fully cover at least k elements such that the cost of [Formula: see text] is as small as possible and element e is fully covered by [Formula: see text] if it belongs to at least r e sets of [Formula: see text]. This problem generalizes the minimum k-union problem (MinkU) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a [Formula: see text]-approximation algorithm for MinPSMC, in which [Formula: see text] is the maximum size of a set in S. And when [Formula: see text], we present a bicriteria algorithm fully covering at least [Formula: see text] elements with approximation ratio [Formula: see text], where [Formula: see text] is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself. 

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3. Reduction games, provability and compactness

   https://doi.org/10.1142/S021906132250009X  

   Dzhafarov, Damir D.; Hirschfeldt, Denis R.; Reitzes, Sarah (December 2022, Journal of Mathematical Logic)

   Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [Formula: see text] principles over [Formula: see text]-models of [Formula: see text]. They also introduced a version of this game that similarly captures provability over [Formula: see text]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [Formula: see text] between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [Formula: see text] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [Formula: see text], uncovering new differences between their logical strengths. 

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4. Good polynomials for locally recoverable codes: Classification results, asymptotics, and exact formulae

   https://doi.org/10.1142/S0219498825410178  

   Darwish, Mohamed O; Jaindungarwal, Anuvrat; Micheli, Giacomo; Vaghasiya, Divyesh; Zhao, Shujun (March 2025, Journal of Algebra and Its Applications)

   Let [Formula: see text] be a prime power and [Formula: see text]. In this paper we complete the classification of good polynomials of degree [Formula: see text] that achieve the best possible asymptotics (with an explicit error term) for the number of totally split places. Moreover, for degrees up to [Formula: see text], we provide an explicit lower bound and an asymptotic estimate for the number of totally split places of [Formula: see text]. Finally, we prove the general fact that the number [Formula: see text] of [Formula: see text] for which [Formula: see text] splits obeys a linear recurring sequence. For any [Formula: see text], this allows for the computation of [Formula: see text] for large [Formula: see text] by only computing a recurrence sequence over [Formula: see text]. 

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5. Inhomogeneous Diophantine approximation for generic homogeneous functions

   https://doi.org/10.1142/S1793042123500628  

   Kleinbock, Dmitry; Skenderi, Mishel (July 2023, International Journal of Number Theory)

   This paper is a sequel to [Monatsh. Math. 194 (2021) 523–554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers [Formula: see text] and [Formula: see text], any [Formula: see text], and any homogeneous function [Formula: see text] that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function [Formula: see text] for a generic element [Formula: see text] in the [Formula: see text]-orbit of [Formula: see text] to be (respectively, not to be) [Formula: see text]-approximable at [Formula: see text]: that is, for there to exist infinitely many (respectively, only finitely many) [Formula: see text] such that [Formula: see text] for each [Formula: see text]. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of [Formula: see text] that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace [Formula: see text] above by any closed subgroup of [Formula: see text] that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper. 

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