A new type of bubble solutions for a critical fractional Schr\"{o}dinger equation

We consider the following critical fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^s u+V(|y'|,y'')u=u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^3\times\mathbb{R}^{N-3}$. If $r^{2s}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, by using a modified finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y'')$. We choose cleverly one of the reduced parameters $\bar{h}$ which depends on the scaling parameter $\lambda$ and avoid to compute the first partial derivative of the reduced functional with respect to $\bar{h}$ directly. Also we have to overcome some difficulties caused by the fractional Laplacian.Comment: 47 page

RocketQAv2: A Joint Training Method for Dense Passage Retrieval and Passage Re-ranking

In various natural language processing tasks, passage retrieval and passage re-ranking are two key procedures in finding and ranking relevant information. Since both the two procedures contribute to the final performance, it is important to jointly optimize them in order to achieve mutual improvement. In this paper, we propose a novel joint training approach for dense passage retrieval and passage re-ranking. A major contribution is that we introduce the dynamic listwise distillation, where we design a unified listwise training approach for both the retriever and the re-ranker. During the dynamic distillation, the retriever and the re-ranker can be adaptively improved according to each other's relevance information. We also propose a hybrid data augmentation strategy to construct diverse training instances for listwise training approach. Extensive experiments show the effectiveness of our approach on both MSMARCO and Natural Questions datasets. Our code is available at https://github.com/PaddlePaddle/RocketQA.Comment: EMNLP 202

Solutions with multiple peaks for nonlinear Kirchhoff equations on R^3

In this paper, we mainly investigate the following nonlinear Kirchhoff equation $-\left(\epsilon^2 a+\epsilon b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u +u =Q(x)u^{q-1}$, u>0, $x\in\mathbb{R}^{3}$, $u\to 0$, as $|x|\to +\infty$, &nbsp; where a,b>0 are constants, 2<q<6, and \epsilon>0 is a parameter. Under some suitable assumptions on the function $Q(x)$, we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of $Q(x)$ for \epsilon>0 sufficiently small, by using the finite dimensional reduction method. Different from the local Schrödinger problem, here the corresponding limit problem is a system. Moreover, the nonlocal term brings some new difficulties which involve some technical and complicated estimates

Microstructure, Wear Resistance and Oxidation Behavior of Ni-Ti-Si Coatings Fabricated on Ti6Al4V by Laser Cladding

The Ni-Ti-Si composite coatings were successfully fabricated on Ti6Al4V by laser cladding. The microstructure were studied by SEM (scanning electron microscopy) and EDS (energy dispersive spectrometer). It has been found that Ti2Ni and Ti5Si3 phases exist in all coatings, and some samples have TiSi2 phases. Moreover, due to the existence of these phases, coatings presented relatively higher microhardness than that of the substrate (826 HV (Vickers hardness)) and the microhardness value of coating 3 is about twice larger than that of the substrate. During the dry sliding friction and wear test, due to the distribution of the relatively ductile phase of Ti2Ni and reinforcement phases of Ti5Si3 and TiSi2, the coatings performed good wear resistance. The oxidation process contains two stages: the rapid oxidation and slow oxidation by high temperature oxidation test at 800 °C for 50 h. Meanwhile, the value of the oxidation weight gain of the substrate is approximately three times larger than that of the coating 4. During the oxidation process, the oxidation film formed on the coating is mainly consisted of TiO2, Al2O3 and SiO2. Phases Ti2Ni, Ti5Si3, TiSi2 and TiSi were still found and it could be responsible for the improvement in oxidation resistance of the coatings by laser cladding

DuReader_retrieval: A Large-scale Chinese Benchmark for Passage Retrieval from Web Search Engine

In this paper, we present DuReader-retrieval, a large-scale Chinese dataset for passage retrieval. DuReader-retrieval contains more than 90K queries and over 8M unique passages from Baidu search. To ensure the quality of our benchmark and address the shortcomings in other existing datasets, we (1) reduce the false negatives in development and testing sets by pooling the results from multiple retrievers with human annotations, (2) and remove the training queries that are semantically similar to the development and testing queries. Additionally, we provide two out-of-domain testing sets for cross-domain evaluation, as well as a cross-lingual set that has been manually translated for cross-lingual retrieval. The experiments demonstrate that DuReader-retrieval is challenging and there is still plenty of room for improvement, e.g. salient phrase and syntax mismatch between query and paragraph. These experimental results show that the dense retriever does not generalize well across domains, and cross-lingual retrieval is essentially challenging. DuReader-retrieval will be publicly available at https://github.com/baidu/DuReader/tree/master/DuReader-Retrieval