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   Li, Lei; Liu, Jian-Guo (April 2018, Journal of Differential Equations)
2. Singular Abreu equations and linearized Monge–Ampère equations with drifts

   https://doi.org/10.4171/JEMS/1548  

   Kim, Young Ho; Le, Nam Q; Wang, Ling; Zhou, Bin (October 2024, Journal of the European Mathematical Society)

   We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher-dimensional case by transforming singular Abreu equations into linearized Monge–Ampère equations with drifts. We establish global Hölder estimates for linearized Monge–Ampère equations with drifts under suitable hypotheses, and then apply them to prove the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side are also discussed. 

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3. Derived KZ Equations

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4. 2D Voigt Boussinesq Equations

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5. Adaptive Linear Estimating Equations

   Ying, Mufang; Khamaru, Koulik; Zhang, Cun-Hui (May 2024, 2023 Conference on Neural Information Processing Systems)

   Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least squares estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality. 

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