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

   https://doi.org/10.1142/S0218202521500093  

   Bian, Dongfen; Guo, Yan; Tice, Ian (February 2021, 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 variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that such a [Formula: see text]-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system. 

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2. 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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3. Minimum norm interpolation in the ℓ1(ℕ) space

   https://doi.org/10.1142/S0219530520400059  

   Cheng, Raymond; Xu, Yuesheng (January 2021, Analysis and Applications)

   null (Ed.)

   We consider the minimum norm interpolation problem in the [Formula: see text] space, aiming at constructing a sparse interpolation solution. The original problem is reformulated in the pre-dual space, thereby inducing a norm in a related finite-dimensional Euclidean space. The dual problem is then transformed into a linear programming problem, which can be solved by existing methods. With that done, the original interpolation problem is reduced by solving an elementary finite-dimensional linear algebra equation. A specific example is presented to illustrate the proposed method, in which a sparse solution in the [Formula: see text] space is compared to the dense solution in the [Formula: see text] space. This example shows that a solution of the minimum norm interpolation problem in the [Formula: see text] space is indeed sparse, while that of the minimum norm interpolation problem in the [Formula: see text] space is not. 

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4. Weakly minimal groups with a new predicate

   https://doi.org/10.1142/S0219061320500117  

   Conant, Gabriel; Laskowski, Michael C. (August 2020, Journal of Mathematical Logic)

   Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text]. 

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5. Fast Fréchet Distance Between Curves with Long Edges

   https://doi.org/10.1142/S0218195919500043  

   Gudmundsson, Joachim; Mirzanezhad, Majid; Mohades, Ali; Wenk, Carola (June 2019, International Journal of Computational Geometry & Applications)

   Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve. 

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