Multiple nodal solutions of nonlinear Choquard equations

In this paper, we consider the existence of multiple nodal solutions of the nonlinear Choquard equation \begin{equation*} \ \ \ \ (P)\ \ \ \ \begin{cases} -\Delta u+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \ \ \ \text{in}\ \mathbb{R}^3, \ \ \ \ \\ u\in H^1(\mathbb{R}^3),\\ \end{cases} \end{equation*} where $p\in (\frac{5}{2},5)$. We show that for any positive integer $k$, problem $(P)$ has at least a radially symmetrical solution changing sign exactly $k$-times

Local uniqueness of vortices for 2D steady Euler flow

We study the steady planar Euler flow in a bounded simply connected domain, where the vortex function is $f=t_+^p$ with $p>0$ and the vorticity strength is prescribed. By studying the location and local uniqueness of vortices, we prove that the vorticity method and the stream function method actually give the same solution. We also show that if the vorticity of flow is located near an isolated minimum point and non-degenerate critical point of the Kirchhoff-Routh function, it must be stable in the nonlinear sense.Comment: 47 pages. arXiv admin note: text overlap with arXiv:1703.0986

Reading Scene Text in Deep Convolutional Sequences

We develop a Deep-Text Recurrent Network (DTRN) that regards scene text reading as a sequence labelling problem. We leverage recent advances of deep convolutional neural networks to generate an ordered high-level sequence from a whole word image, avoiding the difficult character segmentation problem. Then a deep recurrent model, building on long short-term memory (LSTM), is developed to robustly recognize the generated CNN sequences, departing from most existing approaches recognising each character independently. Our model has a number of appealing properties in comparison to existing scene text recognition methods: (i) It can recognise highly ambiguous words by leveraging meaningful context information, allowing it to work reliably without either pre- or post-processing; (ii) the deep CNN feature is robust to various image distortions; (iii) it retains the explicit order information in word image, which is essential to discriminate word strings; (iv) the model does not depend on pre-defined dictionary, and it can process unknown words and arbitrary strings. Codes for the DTRN will be available.Comment: To appear in the 13th AAAI Conference on Artificial Intelligence (AAAI-16), 201