In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.
The two-sided rank-one (TR1) update method was introduced by Griewank and Walther (2002) for solving nonlinear equations. It generates dense approximations of the Jacobian and thus is not applicable to large-scale sparse problems. To overcome this difficulty, we propose sparse extensions of the TR1 update and give some convergence analysis. The numerical experiments show that some of our extensions are superior to the TR1 update method. Some convergence analysis is also presented.
CHENG MingHou & DAI YuHong State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
