Random perturbation to the geodesic equation

We study random "perturbation" to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm $1$. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by ${\frac{8}{n(n-1)}}$ where $n$ is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.Comment: Published at http://dx.doi.org/10.1214/14-AOP981 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Low-Profile Fully-Printed Multifrequency Monopoles Loaded with Complementary Metamaterial Transmission Line

The design of a new class of multifrequency monopoles by loading a set of resonant-type complementary metamaterial transmission lines (CMTL) is firstly presented. Two types of CMTL elements are comprehensively explored: the former is the epsilon negative (ENG) one by loading complementary split ring resonators (CSRRs) with different configurations on the signal strip, whereas the latter is the double negative (DNG) one by incorporating the CSRRs and capacitive gaps. In both cases, the CMTLs are considered with different number of unit cells. By cautiously controlling the geometrical parameters of element structure, five antenna prototypes coving different communication standards (GSM, UMTS, DMB and WiMAX) are designed, fabricated and measured. Numerical and experimental results illustrate that the zeroth-order resonance frequencies of the ENG and DNG monopoles are in desirable consistency. Moreover, of all operating frequencies the antennas exhibit fairly good impedance matching performances better than -10dB and quasi-omnidirectional radiation patterns