从一道运动学题目的求解谈坐标系选取的重要性

通过比较一道典型运动学题目的近似求解方法与精确求解方法，探讨坐标系选取对研究曲线运动的重要性．     

The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.     

Internal Rotation of the Methyl Group in Cations of Toluene Derivatives

The potential of the internal rotation of the methyl group was determined for o-, m-, and p-fluorotoluene cations by pulsed field ionization spectroscopy. The potential of the internal rotational motion was also surveyed for other toluene derivative cations. It was found that the barrier height generally increases by ionization. The increase in the barrier height has been discussed in connection with the reduction of the internal rotational constant B by ionization. The geometrical distortion of the methyl group during the internal rotation has been suggested.     

Model-dependent nature of blowtorch effect on the escape and equilibration rates

We analyze the relaxation behavior of a bistable system when the background temperature profile is inhomogeneous due to the presence of a localized hot region (blowtorch) on one side of the potential barrier. Since the diffusion equation for inhomogeneous medium is model-dependent, we consider two physical models to study the kinetics of such system. Using a conventional stochastic method, we obtain the escape and equilibration rates of the system for the two physical models. For both models, we find that the hot region enhances the escape rate from the well where it is placed while it retards the escape rate from the other well. However, the value of the escape rate from the well where the hot region is placed differs for the two models while that of the escape rate from the other well is identical for both. This work, for the first time, gives a detailed report of the similarities and differences of the escape rates and, hence, exposes the common and distinct features of the two known physical models in determining the way the bistable system relaxes. Received 25 September 2001     

角域上的调和函数与条件Brown运动

设中的角域．该文绘出了A上正调和函数的Martin表示，讨论了极小调和函数与条件Brown运动的一个0—1律之间的关系，并给出了A上极小调和函数的表现形式，     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

用参考圆法演示简谐振动

对受迫振动仪进行了改进，通过参考圆法，演示了作镁速圆周运动的质点在直径方向上的投影所作的运动为简谐振动。     

Nonlinear control of systems with multiple equilibria and unknown sinusoidal disturbance

In this paper, nonlinear systems having multiple equilibrium points and low order dynamics are investigated. Roll motions of ships are studied by means of modern nonlinear techniques to exemplify the behavior of such nonlinear systems in the case when they are under the influence of external sinusoidal disturbances with unknown amplitudes. The main objective is to analyze the performance of this system at different operating conditions, including those giving rise to chaos, and to design a controller with an overparameterized structure to stabilize the system at the origin. A nonlinear recursive backstepping controller is proposed and the transient performance is investigated. Lyapunov-based techniques are used to force systematic following of a reference model while introducing a nonlinear parameter estimator to guarantee adaptivity. Robustness problems as well as ways to tune the controller parameters are examined. Simulation results are submitted for the uncontrolled and controlled cases, verifying the effectiveness of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.     

Propagation of Light in an Ensemble of Tripod Level Atoms

We study the propagation of a quantum probe light in an ensemble of tripod level atoms when the atoms are coupled to two other classical control fields. First we calculate the dispersion properties, such as susceptibility and group velocity, of the probe light within such an atomic medium under the case of three-photon resonance via the dynamical algebra method of collective atomic excitations. Then we calculate the dispersion of the probe light in the case that two classical control fields have the different detunings to the relative atomic transitions. Our results show that in both cases the phenomenon of electromagnetically induced transparency can occur. Especially under the second case, we can find two transparency windows for the probe light.