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Here the authors are interested in the zero set of Sobolev functions and functions of bounded variation with negative power of integrability. The main result is a general Hausdorff dimension estimate on the size of zero set. The research is motivated by the model on van der waal force driven thin film, which is a singular elliptic equation. After obtaining some basic regularity result, the authors get an estimate on the size of singular set; such set corresponds to the thin film rupture set in the thin film model.

Few-shot named entity recognition (NER) systems aim at recognizing novel-class named entities based on only a few labeled examples. In this paper, we present a decomposed metalearning approach which addresses the problem of few-shot NER by sequentially tackling fewshot span detection and few-shot entity typing using meta-learning. In particular, we take the few-shot span detection as a sequence labeling problem and train the span detector by introducing the model-agnostic meta-learning (MAML) algorithm to find a good model parameter initialization that could fast adapt to new entity classes. For few-shot entity typing, we propose MAML-ProtoNet, i.e., MAML-enhanced prototypical networks to find a good embedding space that can better distinguish text span representations from different entity classes. Extensive experiments on various benchmarks show that our approach achieves superior performance over prior methods. 1 * Equal contributions.

Let Ω ⊂ R N , N ≥ 2 be a bounded smooth domain and α > 1. We are interested in the singular elliptic equationwith Neumann boundary conditions. In this paper, we gave a complete description of all continuous radially symmetric solutions. In particular, we constructed nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N = 2 and α = 3, the equation has been used to model steady states of van der Waals force driven thin films of viscous fluids. We also considered the physical problem when total volume of the fluid is prescribed.

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