传递辛矩阵群收敛于辛Lie群

通过作用量变分原理,给出了Hamilton正则方程离散积分的传递辛矩阵表示,利用Hamilton正则方程给出了其对应的Lie代数．说明了当时间区段长度趋近于0时,离散系统积分的传递辛矩阵群收敛于连续时间Hamilton系统微分方程分析积分得到的辛Lie群．     

四阶杆振动方程的两类高精度辛格式

In this paper, we present two classes of symplectic schemes with high order accuracy for solving four-order rod vibration equation utt uxxxx=0 via the third type generating function method. First, the equation of four order rod vibration is written into the canonical Hamilton system; second, overcoming successfully the essential difficult on the calculus of high order variations derivative, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs by the third type generating function method. Furthermore the discretization of the related modified equation of original equation is obtained. Finally, arbitrary order accuracy symplectic schemes are obtained. Numerical results are also presented to show the effectiveness of the scheme, high order accuracy and properties of excellent long-time numerical behavior.     

The Steinberg formula for orbit groups

The aim of this note is to give a direct proof of the fact that the Steinberg formula holds for the Vaserstein symbol and the universal weak Mennicke symbol. We also give an alternate proof of the fact that the Steinberg formula holds for the Vaserstein symbol.     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept. The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.     

Convergence of the Lavrent’ev method

The convergence of the Lavrent’ev method, which is a well-known regularization method for integral equations of the first kind, is analyzed as applied to equations with arbitrary linear bounded operators. A theorem concerning necessary and sufficient conditions for this convergence is proved. It is shown that these conditions are satisfied for two classes of integral equations that do not possess the properties required by the classical Lavrent’ev method.     

Solving the quasi-variational Signorini inequality by the method of successive approximations

The method of successive approximations is examined as a tool for solving the semicoercive quasi-variational Signorini inequality. The auxiliary problems with given friction arising at each step of this method are solved using the Uzawa method with an iterative proximal regularization of the modified Lagrangian functional.     

Equivalence of curves with respect to the action of the symplectic group

We establish an equivalence criterion for finite systems of curves with respect to the action of the symplectic group.     

Regularization in the Mosolov and Myasnikov problem with boundary friction

We propose an iterative algorithm for solving a semicoercive nonsmooth variational inequality. The algorithm is based on the stepwise partial smoothing of the minimized functional and an iterative proximal regularization method.We obtain a solution to the variational Mosolov and Myasnikov problem with boundary friction as a limit point of a sequence of solutions to stable auxiliary problems.