In this paper, we present an efficient \GQR\ algorithm for solving the linear response eigenvalue problem $\scrH\bx ={\lambda}\bx $, where $\scrH$ is $\bPi^-$-symmetric with respect to $\Gamma_0 = \diag(I_n,-I_n)$. Based on newly introduced $\Gamma$-orthogonal transformations, the \GQR\ algorithm preserves the $\bPi^-$-symmetric structure of $\scrH$ throughout the whole process, which guarantees the computed eigenvalues to appear pairwise $(\lambda,-\lambda)$ as they should. With the help of a newly established implicit $\Gamma$-orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the \GQR\ algorithm. Numerical experiments are given to show the effectiveness of the algorithm.

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With the rapid development of infrastructure including power grids, managers in power industry these days are faced with an increasingly severe problem of theft of electricity. Theft of electricity has negative effects on many socioeconomic aspects, including impacting stable growth of economic for power enterprises and social development. The traditional anti-theft means require officers checking the integrity of kilowatt-hour meter and the correctness of wiring house by house, which requires enormous manpower and material resources. With the advancement of information collecting technology, power enterprises now possess relatively complete database of power consumption. As a result, performing data mining on existing database and identifying abnormal users has become a hot topic in the field of information technology. The purpose of this paper is to identify abnormal users with machine learning algorithms, providing power enterprises with means to detect theft of electricity at lower cost. The contributions of this paper are listed as follows: 1) Propose various features, providing means to describe users’ behaviors. First, monistic features (mean, difference, coefficient of variation, range, standard deviation), binary features (cosine similarity, Pearson product-moment correlation coefficient) and multivariate features are calculated based on user power consumption (monthly, seasonally, yearly, per holiday, per workday). Then principal component analysis is used to perform dimensionality reduction on the dataset. The dataset is finally scaled and oversampled to form the feature dataset. 2) Study various classification algorithms, optimize hyperparameters, providing models to detect theft of electricity. In this paper, the mechanism behind support vector machine, back-propagation neural network, random forest and XGBoost is studied and the hyperparameters are optimized. 3) Compare and evaluate different classifiers, and provide suggestions for real-world application. In this paper, different classifiers are evaluated with different metrics and the best classifier is recommended based on real-world dataset and different application scenarios.

The Cauchy problem for the barotropic compressible Navier--Stokes equations on the whole two-dimensional space with vacuum as far field density is considered. When the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds, the global existence and uniqueness of strong and classical solutions is established. It should be remarked that there are no restrictions on the size of the data.

In this paper, we establish an analytic foundation for a fully non-linear equation 2 1 = f on manifolds with metrics of positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f > 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a manifold with metrics of positive scalar curvature admits a conformal metric of positive scalar curvature and positive 2-scalar curvature if and only if Y2,1 > 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with 2(g) = 1(g), for some constant . Using these analytic results, we give a rough classification of the space of manifolds with metrics of positive scalar curvature.