Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems

We suggest sufficient conditions for the existence of weak limits of solutions of the Liouville equation as time increases indefinitely. The presence of the weak limit of the probability distribution density leads to a new interpretation of the second law of thermodynamics for entropy increase.     

Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

The purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H , which stays unchanged during oscillation. This property is used to obtain approximate frequency–amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz–He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations. In contrast with the traditional methods, the proposed method does not require any small parameter in the equation. Copyright © 2011 John Wiley & Sons, Ltd.     

A new approach to young measure theory, relaxation and convergence in energy

The main idea of this paper is to reduce analysis of behavior of integral functionals along weakly convergent sequences to operations with Young measures generated by these sequences. We show that Young measures can be characterized as measurable functions with values in a special compact metric space and that these functions have a spectrum of properties sufficiently broad to realize this idea.These new observations allow us to give simplified proofs of the results of gradient Young measure theory and to use them for deriving the results on relaxation and convergence in energy under optimal assumptions on integrands.We think that this work helps to clarify role of Young measures.     

On the relation between energy conserving low-order models and Hamiltonian systems

In this paper we examine the relation between the energy conserving finite dimensional or low-order models obtained by the application of the Galerkin projection methods to infinite dimensional model, and the class of finite dimensional non-canonical Hamiltonian systems with quadratic Hamiltonian functions and structure matrix whose elements are linear function of the state. We also examine the conditions for the conservation of the phase volume.     

The rate of convergence of the Lax-Oleinik semigroup-degenerate fixed point case

For the standard Lagrangian in classical mechanics, which is defined as the kinetic energy of the system minus its potential energy, we study the rate of convergence of the corresponding Lax-Oleinik semigroup. Under the assumption that the unique global minimum point of the Lagrangian is a degenerate fixed point, we provide an upper bound estimate of the rate of convergence of the semigroup.     

Performance of the Gibbs,Hit-and-Run,and Metropolis Samplers

Abstract

We consider the performance of three Monte Carlo Markov-chain samplers—the Gibbs sampler, which cycles through coordinate directions; the Hit-and-Run (H&R) sampler, which randomly moves in any direction; and the Metropolis sampler, which moves with a probability that is a ratio of likelihoods. We obtain several analytical results. We provide a sufficient condition of the geometric convergence on a bounded region S for the H&R sampler. For a general region S, we review the Schervish and Carlin sufficient geometric convergence condition for the Gibbs sampler. We show that for a multivariate normal distribution this Gibbs sufficient condition holds and for a bivariate normal distribution the Gibbs marginal sample paths are each an AR(1) process, and we obtain the standard errors of sample means and sample variances, which we later use to verify empirical Monte Carlo results. We empirically compare the Gibbs and H&R samplers on bivariate normal examples. For zero correlation, the Gibbs sampler provides independent data, resulting in better performance than H&R. As the absolute value of the correlation increases, H&R performance improves, with H&R substantially better for correlations above .9. We also suggest and study methods for choosing the number of replications, for estimating the standard error of point estimators and for reducing point-estimator variance. We suggest using a single long run instead of using multiple iid separate runs. We suggest using overlapping batch statistics (obs) to get the standard errors of estimates; additional empirical results show that obs is accurate. Finally, we review the geometric convergence of the Metropolis algorithm and develop a Metropolisized H&R sampler. This sampler works well for high-dimensional and complicated integrands or Bayesian posterior densities.     

Weak convergence of the sequential empirical processes of residuals in TAR models

This paper studies the weak convergence of the sequential empirical process K n of the residuals in the threshold autoregressive(TAR)model of order p.Under some mild conditions,it is shown that K n converges weakly to a Kiefer process plus a random variable which converges to a multivariate normal.This differs from that given by Bai(1994)for a stationary autoregressive and moving average(ARMA)model.     

Extrapolation Algorithms for Filtering Series of Functions,and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the -algorithm) or by approximating the series by a rational function (in particular, a Padé approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.     

Average equidistribution and statistical independence properties of digital inversive pseudorandom numbers over parts of the period

This article deals with the digital inversive method for generating uniform pseudorandom numbers. Equidistribution and statistical independence properties of the generated pseudorandom number sequences over parts of the period are studied based on the distribution of tuples of successive terms in the sequence. The main result is an upper bound for the average value of the star discrepancy of the corresponding point sets. Additionally, lower bounds for the star discrepancy are established. The method of proof relies on bounds for exponential sums.

     

Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems

We establish the existence of weak limits of solutions (in the class L _p, p%thinsp;1) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.     

On asymmetry of the future and the past for limit self-joinings

Let be an off-diagonal joining of a transformation . We construct a non-typical transformation having asymmetry between limit sets of for positive and negative powers of . It follows from a correspondence between subpolymorphisms and positive operators, and from the structure of limit polynomial operators. We apply this technique to find all polynomial operators of degree in the weak closure (in the space of positive operators on ) of powers of Chacon's automorphism and its generalizations.

     

Extreme and high-level sojourns of the single server queue in heavy traffic

This study characterizes the behavior of large queue lengths in heavy traffic. We show that the distribution of the maximum queue length in a random time interval for a queueing systems in heavy traffic converges to a novel extreme value distribution. We also study the processes that record the times that the queue length exceeds a high level and the cumulative time the queue is above the level. We show that these processes converge in distribution to compound Poisson processes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weak convergence and weak compactness in the space of almost periodic functions on the real line

We give necessary and sufficient conditions for sequences in the space AP(R) of continuous almost periodic functions on the real line to converge in the weak topology. The abstract results are illustrated by a number of examples which show that weak convergence seems to be a rare phenomenon. We also characterize the weakly compact subsets in AP(R). In particular, earlier statements made in the monograph by Dunford and Schwartz are refined and completed. We close with some open problems.     

On the empirical process of multivariate, dependent random variables

A convergence theorem of Billingsley for the empirical process of stationary, real valued radom variables under a mixing condition is generalized to the k-dimensional and nonstationary case. Further a more general empirical process is treated, including the upper summation boundary as argument. Some applications are given to a Kolmogorov-Smirnov and a Cramér-von Mises type statistic.     

超布朗运动:质量过程的极限行为及值空间的扩张

本文给出了超布朗运动的质量过程在不灭绝条件下的极限分布的拉普拉斯变换的具体形式,并得出了在较一般的分支特征下使其值空间从M_F（R^d）扩张到M_P（R^d）的一个充分条件．     

On the cusum of squares test for variance change in nonstationary and nonparametric time series models

In this paper we consider the problem of testing for a variance change in nonstationary and nonparametric time series models. The models under consideration are the unstable AR(q) model and the fixed design nonparametric regression model with a strong mixing error process. In order to perform a test, we employ the cusum of squares test introduced by Inclán and Tiao (1994,J. Amer. Statist. Assoc.,89, 913–923). It is shown that the limiting distribution of the test statistic is the sup of a standard Brownian bridge as seen in iid random samples. Simulation results are provided for illustration.     

Necessary and sufficient conditions for the convergence of convolution products of non-identical distributions on finite abelian semigroups

Given a sequence of probability measures ( _n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products _{k,n k+1}*···* _n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( _n ).     

一个Lie代数的子代数及其相关的两类Loop代数

本文构造了Lie代数A2的一个子代数A2,通过选取恰当的基元阶数得到相应的一个loop代数A2,由此设计一个等谱问题,利用屠格式得到了一个新的Liouville可积的Hamilton方程族.作为其约化情形,得到了一个非线性有理分式型演化方程.再由一个矩阵变换,得到了换位运算与A2等价的Lie代数A1的一个子代数A1,将A1再扩展成一个新的高维loop代数G,利用G获得了所得方程族的一类扩展可积系统.