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Let Δ be the closed unit disk in C, let Γ be the circle, let Π: Δ×C→Δ be projection, and let$A(Δ)$be the algebra of complex functions continuous on Δ and analytic in int Δ. Let$K$be a compact set in C^{2}such that Π($K$)=Γ, and let$K$_{λ}≠{w∈C|(λ,w)∈K}. Suppose further that (a) for every λ∈Γ,$K$_{λ}is the union of two nonempty disjoint connected compact sets with connected complement, (b) there exists a function Q(λ,w)≠(w-R(λ))^{2}-S(λ) quadratic in w with$R,S∈A(Δ)$such that for all λ∈Γ, {w∈C|Q(λ,w)=0}υ int$K$_{λ}, where$S$has only one zero in int Δ, counting multiplicity, and (c) for every λ∈Γ, the map ω→Q(λ,ω) is injective on each component of$K$_{λ}. Then we prove that К/K is the union of analytic disks 2-sheeted over int Δ, where К is the polynomial convex hull of$K$. Furthermore, we show that БК/K is the disjoint union of such disks.

▲With the rapid development of bicycle-sharing system since 2016, illegal parking has become one of the most significant social issues. While the existing management methods cannot meet the demand, object detection technology provides a possibility. The Video Surveillance System can be utilized as a rich source of information. However, there is technical challenge to handle complex scenarios. Faster R-CNN, chosen because of its excellent overall performance, cannot specify the proposals of shared bikes especially in distant view. Processing the consecutive images in videos is neither necessary nor efficient. The original results are limited by interference of moving bikes and blocking effects. ▲ In this paper, I introduce a Faster R-CNN over Attention (FoA) that combines the from-coarse-to-fine visual mechanism with deep neural networks. FoA realizes fine-grained detection of shared bikes in complex circumstances. It consists of an Attention Region Extraction (ARE) and an optimized Faster R-CNN. In ARE, I introduce the concept of Attention Region and define it as video background, then apply Gaussian Mixture Model. ARE processes the surveillance video into discrete regions, allowing concentration on certain areas and efficiency in detection. In Faster R-CNN, I put forward an anchor box optimization in regards of the remarkable region-based characteristic in R-CNN. The optimization applies k-means to refine the anchor boxes and generate more appropriate region proposals. ▲ I construct a categorized shared bike data set of 4,291 images and 12,697 labeled objects for FoA training and testing. FoA performs excellently with the detection and recognition ratios of 94.19% and 92.73% respectively. The optimization is proved to make a difference. ▲ The paper not only provides a new direction for solving illegal parking issue and managing bicycle-sharing system, but also generalizes FoA into a new detection framework of Attention Region Extraction plus Region-Based Convolutional Neural Network, which can be applied to certain object detection in various scenes. The mechanism that combines ARE and R-CNN organically allows the towards-real-time detection to possess both practicality and universality. FoA is one of its robust implements.

In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwells equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yees scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved

As a continuation of [28], the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on the velocity field and the perfect conducting boundary condition is given for the magnetic field. Under an assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a L8 estimate on the error by multi-scale analysis.