Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media

In this paper,we establish the exponential convergence theory for the multipole and local expansions,shifting and translation operators for the Green's function of 3-dimensional Laplace equation in layered media.An immediate application of the theory is to ensure the exponential convergence of the FMM which has been shown by the numerical results reported in[27].As the Green's function in layered media consists of free space and reaction field components and the theory for the free space components is well known,this paper will focus on the analysis for the reaction components.We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex wave number plane.Then,by using the Cagniard-de Hoop transform and contour deformations,estimates for the remainder terms of the truncated expansions are given,and,as a result,the exponential convergence for the expansions and translation operators is proven.

The first localized back-to-back converter station started expansion

The first fully localized back-to-back DC intercon-nection project which connects Northwest and CentralChina power grids, 330-kV Lingbao Converter Stationhas now started construction for expansion.Lingbao back-to-back converter station situated inLingbao City of Henan Province is the first DC intercon-

Seed-Oriented Local Community Detection Based on Influence Spreading

In recent years,local community detection algorithms have developed rapidly because of their nearly linear computing time and the convenience of obtaining the local information of real-world networks.However,there are still some issues that need to be further studied.First,there is no local community detection algorithm dedicated to detecting a seed-oriented local community,that is,the local community with the seed as the core.The second and third issues are that the quality of local communities detected by the previous local community detection algorithms are largely dependent on the position of the seed and predefined parameters,respectively.To solve the existing problems,we propose a seed-oriented local community detection algorithm,named SOLCD,that is based on influence spreading.First,we propose a novel measure of node influence named k-core centrality that is based on the k-core value of adjacent nodes.Second,we obtain the seed-oriented local community,which is composed of the may-members and the must-member chain of the seed,by detecting the influence scope of the seed.The may-members and the must-members of the seed are determined by judging the influence relationship between the node and the seed.Five state-of-art algorithms are compared to SOLCD on six real-world networks and three groups of artificial networks.The experimental results show that SOLCD can achieve a high-quality seed-oriented local community for various real-world networks and artificial networks with different parameters.In addition,when taking nodes with different influence as seeds,SOLCD can stably obtain high-quality seed-oriented local communities.