A nonconvex dissipative system and its applications (II)

This paper presents exact solutions in terms of implicit functions and hyperbolic functions to a nonconvex dissipative system, controlled by a Duffing–van der Pol nonlinear equation with a fifth-order nonlinearity. Applications to the complex Ginzburg–Landau equation are illustrated and several classes of uniformly translating solutions are obtained accordingly. Part of the work was announced at the International Conference on Complementarity, Duality, and Global Optimization in Science and Engineering, Virginia Tech. University, Blacksburg, Virginia, August 15–17, 2005. This work is supported by NSF Grant CCF–0514768.     

First passage time for a class of one-dimensional stochastic systems

We consider the time evolution of a class of stochastic systems of finite size with polynomial nearest neighbor transition rates. We obtain analytical expressions for the first passage time (FPT) and its moments. We show that the mean FPT, averaged over a uniform initial distribution, shows a simple asymptotoc behavior with the system size and the parameters of the transition rates.     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

High Power THz Undulator Radiation from Linear Accelerator

A 30-MeV femto-second electron linac is built at the Shanghai Institute of Applied Physics, which can produce high power, coherent THz undulator radiation. We report the experimental facility and measurement of the power, frequency spectrum. First experiments show the averaged power at THz to be about 20mW.     

Kramers-type elastic ratchet model for ATP gating during kinesin’s mechanochemical cycle

ATP binding, acting as a gate, plays an important role in kinesin stepping. To understand the physical mechanism of the ATP gate, we propose a Kramers-type elastic ratchet model in which the free head undergoes a biased diffusive search. By first passage time analysis, we investigate the dependence of the mean dwell time on the load force for forward steps of kinesin and find that the forward dwell time varies exponentially with the backward load force which is consistent with the data of Carter and Cross, Nature 435 (2005) 308. Our work suggests that the gating mechanism triggered by ATP binding involves both Kramers-type elastic ratchet mechanism and power stroke movement.     

The significant digit law in statistical physics

The occurrence of the nonzero leftmost digit, i.e., 1,2,…,9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic distribution, named Benford’s law. We investigate three kinds of widely used physical statistics, i.e., the Boltzmann-Gibbs (BG) distribution, the Fermi-Dirac (FD) distribution, and the Bose-Einstein (BE) distribution, and find that the BG and FD distributions both fluctuate slightly in a periodic manner around Benford’s distribution with respect to the temperature of the system, while the BE distribution conforms to it exactly whatever the temperature is. Thus Benford’s law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature. Furthermore, various elegant properties of Benford’s law, especially the mantissa distribution of data sets, are discussed.     

Transversely holomorphic structures with many invariant hypersurfaces

In this paper we show that a transversely holomorphic foliation in a compact manifold $M$ having an infinite number of invariant hypersurfaces admits a basic transversely meromorphic first integral $f:M?\overline{\mathbb{C}}$.     

Dyck paths with a first return decomposition constrained by height

We study the enumeration of Dyck paths having a first return decomposition with special properties based on a height constraint. We exhibit new restricted sets of Dyck paths counted by the Motzkin numbers, and we give a constructive bijection between these objects and Motzkin paths. As a byproduct, we provide a generating function for the number of Motzkin paths of height $k$ with a flat (resp. with no flats) at the maximal height.     

On solutions of stochastic differential equations with parameters modeled by random sets

We consider ordinary stochastic differential equations whose coefficients depend on parameters. After giving conditions under which the solution processes continuously depend on the parameters random compact sets are used to model the parameter uncertainty. This leads to continuous set-valued stochastic processes whose properties are investigated. Furthermore, we define analogues of first entrance times for set-valued processes called first entrance and inclusion times. The theoretical concept is applied to a simple example from mechanics.     

Strong （Weak） Exact Controllability and Strong （Weak） Exact Observability for Quasilinear Hyperbolic Systems

Numerical approximations of Cahn-Hilliard phase-field model for the two-phase incompressible flows are considered in this paper.Several efficient and energy stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase-field system for both the matched density case and the variable density case are constructed,and are shown to satisfy discrete energy laws which are analogous to the continuous energy laws.