We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

In this paper, we generalize the point integral method to solve the general elliptic PDEs with variable coefficients and corresponding eigenvalue problems with Neumann, Robin and Dirichlet boundary conditions on point cloud. The main idea is using integral equations to approximate the original PDEs. The integral equations are easy to discretize on the point cloud. The truncation error of the integral approximation is analyzed. Numerical examples are presented to demonstrate that PIM is an effective method to solve the elliptic PDEs with smooth coefficients on point cloud.

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.

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.