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In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov., Comment: 33 pages. References added

We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution., Comment: Contribution to the Proceedings of the ICCM 2018, Taipei, Taiwan, 29 pages

Although neural machine translation(NMT) yields promising translation performance, it unfortunately suffers from over- and under-translation is- sues [Tu et al., 2016], of which studies have become research hotspots in NMT. At present, these studies mainly apply the dominant automatic evaluation metrics, such as BLEU, to evaluate the overall translation quality with respect to both adequacy and uency. However, they are unable to accurately measure the ability of NMT systems in dealing with the above-mentioned issues. In this paper, we propose two quantitative metrics, the Otem and Utem, to automatically evaluate the system perfor- mance in terms of over- and under-translation respectively. Both metrics are based on the proportion of mismatched n-grams between gold ref- erence and system translation. We evaluate both metrics by comparing their scores with human evaluations, where the values of Pearson Cor- relation Coefficient reveal their strong correlation. Moreover, in-depth analyses on various translation systems indicate some inconsistency be- tween BLEU and our proposed metrics, highlighting the necessity and significance of our metrics.

This is a survey of some of the recent developments on the geometric and analytic aspects of the Anomaly flow. It is a flow of $(2,2)$-forms on a $3$-fold which was originally motivated by string theory and the need to preserve the conformally balanced property of a Hermitian metric in the absence of a $\partial\bar\partial$-Lemma. It has revealed itself since to be a remarkable higher order extension of the Ricci flow. It has also led to several other curvature flows which may be interesting from the point of view of both non-K\"ahler geometry and the theory of non-linear partial differential equations., Comment: References and a short section added

We show a second order a priori estimate for solutions to the complex $k$-Hessian equation on a compact K\"ahler manifold provided the $(k$-$1)$-st root of the right hand side is $\mathcal C^{1,1}$. This improves an estimate of Hou-Ma-Wu. An example is provided to show that the exponent is sharp., Comment: 25 pages

While the Anomaly flow was originally motivated by string theory, its zero slope case is potentially of considerable interest in non-Kahler geometry, as it is a flow of conformally balanced metrics whose stationary points are precisely Kahler metrics. We establish its convergence on Kahler manifolds for suitable initial data. We also discuss its relation to some current problems in complex geometry., Comment: 34 pages; references added

In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass., Comment: 8 pages

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale., Comment: 36 pages

The dominant neural machine translation (NMT) models apply unified attentional encoder-decoder neural networks for translation. Traditionally, the NMT decoders adopt recurrent neural networks (RNNs) to perform translation in a left-toright manner, leaving the target-side contexts generated from right to left unexploited during translation. In this paper, we equip the conventional attentional encoder-decoder NMT framework with a backward decoder, in order to explore bidirectional decoding for NMT. Attending to the hidden state sequence produced by the encoder, our backward decoder first learns to generate the target-side hidden state sequence from right to left. Then, the forward decoder performs translation in the forward direction, while in each translation prediction timestep, it simultaneously applies two attention models to consider the source-side and reverse target-side hidden states, respectively. With this new architecture, our model is able to fully exploit source- and target-side contexts to improve translation quality altogether. Experimental results on NIST Chinese-English and WMT English-German translation tasks demonstrate that our model achieves substantial improvements over the conventional NMT by 3.14 and 1.38 BLEU points, respectively. The source code of this work can be obtained from https://github.com/DeepLearnXMU/ABDNMT., Comment: accepted by AAAI 18