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1. Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

   https://doi.org/10.1142/S021820251950057X  

   Borthagaray, Juan Pablo; Nochetto, Ricardo H.; Salgado, Abner J. (December 2019, Mathematical Models and Methods in Applied Sciences)

   We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian [Formula: see text] in a Lipschitz bounded domain [Formula: see text] satisfying the exterior ball condition. The weight is a power of the distance to the boundary [Formula: see text] of [Formula: see text] that accounts for the singular boundary behavior of the solution for any [Formula: see text]. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension [Formula: see text], which is optimal for [Formula: see text]. 

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2. 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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3. Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Directed Network Design Problems

   https://doi.org/10.1287/moor.2021.1181  

   Ghuge, Rohan; Nagarajan, Viswanath (May 2022, Mathematics of Operations Research)

   We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors). 

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4. Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

   https://doi.org/10.1142/S1793042120400126  

   Folsom, Amanda (March 2021, International Journal of Number Theory)

   null (Ed.)

   In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity. 

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5. 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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