3D shape classification plays a fundamental role in geometric big data analysis. This work proposes a novel method for shape classification based on optimal mass transportation theory. The Riemann surfaces are mapped onto canonical domains conformally based on uniformization theorem. The conformal factor function is treated as probability distribution on the canonical domain. For each pair of probability distributions, the optimal mass transportation map is computed by solving Monge-Amper´e equation. The transportation cost is theWasserstein distance between two distributions. By using this distance, geometric classification based on clustering can be performed. The method is applied to 3D human facial expression recognition, which demonstrates the efficiency and efficacy of the method.

This paper is concerned with an optimal strategy for simultaneously trading of a pair of stocks. The idea of pairs trading is to monitor their price movements and compare their relative strength over time. A pairs trade is triggered by their prices divergence and consists of a pair of positions to short the strong stock and to long the weak one. Such a strategy bets on the reversal of their price strengths. From the viewpoint of technical tractability, typical pairs-trading models usually assume a difference of the stock prices satisfies a mean-reversion equation. In this paper, we consider the optimal pairs-trading problem by allowing the stock prices to follow general geometric Brownian motions. The objective is to trade the pairs over time to maximize an overall return with a fixed commission cost for each transaction. The optimal policy is characterized by threshold curves obtained by solving the associated HJB equations. Numerical examples are included to demonstrate the dependence of our trading rules on various parameters and to illustrate how to implement the results in practice.

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Traditional point cloud registration methods require large overlap between scans, which imposes strict constraints on data acquisition. To facilitate registration, users have to carefully position scanners to ensure sufficient overlap. In this work, we propose to use high-level structural information (i.e., plane/line features and their inter-relationship) for registration, which is capable of registering point clouds with small overlap, allowing more freedom in data acquisition. We design a novel plane/linebased descriptor dedicated to establishing structure level correspondences between point clouds. Based on this descriptor, we propose a simple but effective registration algorithm. We also provide a dataset of real-world scenes containing a larger number of scans with a wide range of overlap. Experiments and comparisons with state-of-the-art methods on various datasets reveal that our method is superior to existing techniques. Though the proposed algorithm outperforms state-of-the-art methods on the most challenging dataset, the point cloud registration problem is still far from being solved, leaving significant room for improvement and future work.