Distributed Estimation of Graph Spectrum

In this paper, we develop a two-stage distributed algorithm that enables nodes in a graph to cooperatively estimate the spectrum of a matrix $W$ associated with the graph, which includes the adjacency and Laplacian matrices as special cases. In the first stage, the algorithm uses a discrete-time linear iteration and the Cayley-Hamilton theorem to convert the problem into one of solving a set of linear equations, where each equation is known to a node. In the second stage, if the nodes happen to know that $W$ is cyclic, the algorithm uses a Lyapunov approach to asymptotically solve the equations with an exponential rate of convergence. If they do not know whether $W$ is cyclic, the algorithm uses a random perturbation approach and a structural controllability result to approximately solve the equations with an error that can be made small. Finally, we provide simulation results that illustrate the algorithm.Comment: 15 pages, 2 figure

A pathway-based mean-field model for E. coli chemotaxis: Mathematical derivation and Keller-Segel limit

A pathway-based mean-field theory (PBMFT) was recently proposed for E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we derived a new moment system of PBMFT by using the moment closure technique in kinetic theory under the assumption that the methylation level is locally concentrated. The new system is hyperbolic with linear convection terms. Under certain assumptions, the new system can recover the original model. Especially the assumption on the methylation difference made there can be understood explicitly in this new moment system. We obtain the Keller-Segel limit by taking into account the different physical time scales of tumbling, adaptation and the experimental observations. We also present numerical evidence to show the quantitative agreement of the moment system with the individual based E. coli chemotaxis simulator.Comment: 21 pages, 3 figure

First-principles calculations of current-induced spin-transfer torques in magnetic domain walls

Current-induced spin-transfer torques (STTs) have been studied in Fe, Co and Ni domain walls (DWs) by the method based on the first-principles noncollinear calculations of scattering wave functions expanded in the tight-binding linearized muffin-tin orbital (TB-LMTO) basis. The results show that the out-of-plane component of nonadiabatic STT in Fe DW has localized form, which is in contrast to the typical nonlocal oscillating nonadiabatic torques obtained in Co and Ni DWs. Meanwhile, the degree of nonadiabaticity in STT is also much greater for Fe DW. Further, our results demonstrate that compared to the well-known first-order nonadiabatic STT, the torque in the third-order spatial derivative of local spin can better describe the distribution of localized nonadiabatic STT in Fe DW. The dynamics of local spin driven by this third-order torques in Fe DW have been investigated by the Landau-Lifshitz-Gilbert (LLG) equation. The calculated results show that with the same amplitude of STTs the DW velocity induced by this third-order term is about half of the wall speed for the case of the first-order nonadiabatic STT.Comment: 8 pages, 8 figure