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  1. Existence and uniqueness results for stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of a strong solution is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solution of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to the degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential and the resulting solutions to the dequantized Schrödinger equation. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold (up to codimension one). The second consists of singularities that may occur on a lower-dimensional manifold (up to codimension two). 
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  2. Air pollution is a major global risk to human health and environment. Particle matter (PM) with diameters less than 2.5 micrometers (PM2.5) is more harmful to human health than other air pollutants because it can penetrate deeply into lungs and damage human respiratory system. A new image-based deep feature analysis method is presented in this paper for PM2.5 concentration estimation. Firstly, low level and high level features are extracted from images and their spectrums by a deep learning neural network, and then regression models are created using the extracted deep features to estimate the PM2.5 concentrations, which are future refined by the collected weather information. The proposed method was evaluated using a PM2.5 dataset with 1460 photos and the experimental results demonstrated that our method outperformed other state-of-the-art methods. 
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