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1. Linear instability of Z-pinch in plasma: Viscous case

   https://doi.org/10.1142/S0218202520500566  

   Bian, Dongfen; Guo, Yan; Tice, Ian (December 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   The [Formula: see text]-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the [Formula: see text]-direction. We develop a scaled variational framework to study its stability in the presence of viscosity effect, and demonstrate that any such [Formula: see text]-pinch is always unstable. We also establish the existence of a largest growing mode, which dominates the linear growth of the linear MHD system. 

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2. 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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3. Ordinal definability and combinatorics of equivalence relations

   https://doi.org/10.1142/S0219061319500090  

   Chan, William (December 2019, Journal of Mathematical Logic)

   null (Ed.)

   Assume [Formula: see text]. Let [Formula: see text] be a [Formula: see text] equivalence relation coded in [Formula: see text]. [Formula: see text] has an ordinal definable equivalence class without any ordinal definable elements if and only if [Formula: see text] is unpinned. [Formula: see text] proves [Formula: see text]-class section uniformization when [Formula: see text] is a [Formula: see text] equivalence relation on [Formula: see text] which is pinned in every transitive model of [Formula: see text] containing the real which codes [Formula: see text]: Suppose [Formula: see text] is a relation on [Formula: see text] such that each section [Formula: see text] is an [Formula: see text]-class, then there is a function [Formula: see text] such that for all [Formula: see text], [Formula: see text]. [Formula: see text] proves that [Formula: see text] is Jónsson whenever [Formula: see text] is an ordinal: For every function [Formula: see text], there is an [Formula: see text] with [Formula: see text] in bijection with [Formula: see text] and [Formula: see text]. 

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4. Smoothing and growth bound of periodic generalized Korteweg–De Vries equation

   https://doi.org/10.1142/S0219891621500260  

   Oh, Seungly; Stefanov, Atanas G. (December 2021, Journal of Hyperbolic Differential Equations)

   For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled. 

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5. 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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