Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.     

生长曲线模型中参数的Bayes线性无偏估计

本文在平方损失下导出了生长曲线模型中参数的Bayes线性无偏估计(LUE), 并在均方误差矩阵(MSEM)准则下研究了Bayes LUE相对于广义最小二乘估计(GLSE)的优良性. 对于非满秩情形,获得了可估函数的Bayes LUE并讨论了其优良性问题.     

一类四阶非线性抛物型方程的初边值问题

本文证明一类四阶非线性抛物型方程初边值问题整体广义解的存在性和唯一性,以及解的渐近性质,最后给出解爆破的充分条件.     

Center and isochronous center problems for quasi analytic systems

This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved. This research is partially supported by the National Natural Science Foundation of China (10671179 and 10771196) and the Natural Science Foundation of Yunnan Province (2005A0092M)     

一类广义指派问题的有效解法

提出一类广义指派问题,这类问题研究的是m个人执行n项任务,每个人执行的任务数、执行每项任务的人数以及总的指派人项数均有限制,要求最优指派.对这类广义指派问题建立了数学模型,并找到一种转换方法,将这类问题转换为平衡指派问题,从而用传统方法,如匈牙利法求解.最后用一个箅例来说明这种转换方法的简便和有效性.     

An application of Neustadt's abstract maximum principle to probability kinematics

An important class of problems in philosophy can be formulated as mathematical programming problems in an infinite-dimensional vector space. One such problem is that of probability kinematics: the study of how an individual ought to adjust his degree-of-belief function in response to new information. Much work has recently been done to establish maximum principles for these generalized programming problems (Refs. 3–4). Perhaps, the most general treatment of the problem presented to date is that by Neustadt (Ref. 1). In this paper, the problem of probability kinematics is formulated as a generalized mathematical programming problem and necessary conditions for the optimal revised degree-of-belief function are derived from an abstract maximum principle contained in Neustadt's paper.This work was supported by the National Research Council of Canada.The author is grateful to G. J. Lastman and J. A. Baker of the University of Waterloo for numerous suggestions made for improvement of this paper. The problem of probability kinematics was brought to the author's attention by W. L. Harper of the University of Western Ontario.     

Generalized Polak-Ribière algorithm

A new generalized Polak-Ribière conjugate gradient algorithm is proposed for unconstrained optimization, and its numerical and theoretical properties are discussed. The new method is, in fact, a particular type of two-dimensional Newton method and is based on a finite-difference approximation to the product of a Hessian and a vector.     

Comparison of a special-purpose algorithm with general-purpose algorithms for solving geometric programming problems

We study the performance of four general-purpose nonlinear programming algorithms and one special-purpose geometric programming algorithm when used to solve geometric programming problems. Experiments are reported which show that the special-purpose algorithm GGP often finds approximate solutions more quickly than the general-purpose algorithm GRG2, but is usually not significantly more efficient than GRG2 when greater accuracy is required. However, for some of the most difficult test problems attempted, GGP was dramatically superior to all of the other algorithms. The other algorithms are usually not as efficient as GGP or GRG2. The ellipsoid algorithm is most robust.This work was supported in part by the National Science Foundation, Grant No. MCS-81-02141.     

Common knowledge,communication, and convergence of beliefs

The aim of this paper is to examine the adequacy of the so-called expected utility theory. It is concluded that the theory is inadequate when the decision-maker is decreasingly risk averse and his initial wealth is risky.