On the Jacobi Last Multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé–Gambier classification

We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé–Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.     

A remark on the Beale-Kato-Majda criterion for the 3D MHD equations with zero kinematic viscosity

In this paper,we study the blow-up criterion of smooth solutions to the 3D magneto-hydrodynamic system in ˙ B 0 ∞,∞.We show that a smooth solution of the 3D MHD equations with zero kinematic viscosity in the whole space R 3 breaks down if and only if certain norm of the vorticity blows up at the same time.     

On Location and Approximation of Clusters of Zeros of Analytic Functions

At the beginning of the 1980s, M. Shub and S. Smale developed a quantitative analysis of Newton's method for multivariate analytic maps. In particular, their α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this paper we focus on one complex variable function. We study general criteria for detecting clusters and analyze the convergence of Schroder's iteration to a cluster. In the case of a multiple root, it is well known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which is of the order of its diameter.     

Stopping criteria, forward and backward errors for perturbed asynchronous linear fixed point methods in finite precision

^** Email: pierre.spiteri{at}enseeiht.fr This paper deals with perturbed linear fixed point methods inthe presence of round-off errors. Successive approximationsas well as the more general asynchronous iterations are treated.Forward and backward error estimates are presented, and theseare used to propose theoretical stopping criteria for thesemethods. In the case of asynchronous iterations, macro-iterationsare used as a tool in order to obtain estimates.     

Geometric existence theory for the control-affine nonlinear optimal regulator

For infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space.     

增广Lagrangian对偶分解方法求解变分不等式系统

本文考虑带约束的变分不等式系统.提出一个基于增广Lagrangian对偶的分解算法,本文给出了算法的收敛性分析.     

Bayes and empirical Bayes iteration estimators in two seemingly unrelated regression equations

For a system of two seemingly unrelated regression equations given by (?)(y_1 is an m×1 vector and y_2 is an n×1 vector,m≠n),employ- ing the covariance adjusted technique,we propose the parametric Bayes and empirical Bayes iteration estimator sequences for regression coefficients.We prove that both the covariance matrices converge monotonically and the Bayes iteration estimator squence is consistent as well.Based on the mean square error (MSE) criterion,we elaborate the su- periority of empirical Bayes iteration estimator over the Bayes estimator of single equation when the covariance matrix of errors is unknown.The results obtained in this paper further show the power of the covariance adiusted approach.     

多重均衡选择中的风险占优与混合策略

均衡选择理论是博弈理论的重要组成部分.风险占优均衡是人们经济决策或行为的一个主要结果.利用混合策略及其性质和“抵制”的概念,“支持”了海萨尼和泽尔腾用公理定义的风险占优概念,且具体给出了识别风险占优均衡的标准和方法,并把它们推广到对称博弈中去.最后对均衡占优的直觉概念和风险占优相冲突的一些博弈进行了类似的讨论.     

On an inequality of Oprea for Lagrangian submanifolds

We show that a Lagrangian submanifold of a complex space form attaining equality in the inequality obtained by Oprea in [8], must be totally geodesic.      

A Dual Approach for Solving Nonlinear Infinity-Norm Minimization Problems with Applications in Separable Cases

In this paper,we consider nonlinear infinity-norm minimization problems.We device a reliable Lagrangian dual approach for solving this kind of problems and based on this method we propose an algorithm for the mixed linear and nonlinear infinity- norm minimization problems.Numerical results are presented.