Note on the optimal transportation of distributions

We extend our work on the optimal mapping of distributions in three directions: (a) we consider any set-valued mapping, not just diffeomorphisms; (b) we distinguish between weak and strong optimality, and identify strong optimality with cyclic monotonicity; and (c) we prove that, under some restrictions, a weakly optimal mapping has the subgradient property.     

一类调和映射的边界正则性

讨论一类映入球面的满足拟单调不等式的弱调和映射的边界正则性。利用函数的延拓技巧以及Hardy空间和BMO空间的对偶性，对这类弱调和映射的边界正则性给出一个简明的证明。     

Reducibility and contractivity of Runge–Kutta methods revisited

The exact relation between a Cooper-like reducibility concept and the reducibilities introduced by Hundsdorfer, Spijker and by Dahlquist and Jeltsch is given. A shifted Runge–Kutta scheme and a transplanted differential equation is introduced in such a fashion that the input/output relation remains unchanged under these transformations. This gives a technique to prove stability and contractivity results. This is demonstrated on the example of contractivity disks. AMS subject classification (2000) 65L07     

导数在函数研究上的应用

本文利用函数某处的各阶导数的特点，以确定函数在该处的单调、极大值、极小值、拐点及浙近线等。     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

Tensor-product monotonicity preservation

This paper studies systems of tensor-product functions for which the functions they span are monotonic in any direction when their control nets are monotonic in that direction. It is shown that Bernstein polynomials and B-splines have this property but that totally positive systems in general, such as certain trigonometric and rational bases, do not. This revised version was published online in August 2006 with corrections to the Cover Date.     

An analysis of single-index model with monotonic link function

The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h（.） can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h（.）. Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.     

Generalized Invex Monotonicity and Its Role in Solving Variational-Like Inequalities

The paper stresses the role of new classes of generalized invex monotonicity in the convergence of iterative schemes for solving a variational-like inequality problem on a closed convex set. This work was supported by Grant NSFC 70432001.     

Paths to constrained Nash equilibria

We propose and analyze a primal-dual, infinitesimal method for locating Nash equilibria of constrained, non-cooperative games. The main object is a family of nonstandard Lagrangian functions, one for each player. With respect to these functions the algorithm yields separately, in differential form, directions of steepest-descent in all decision variables and steepest-ascent in all multipliers. For convergence we need marginal costs to be monotone and constraints to be convex inequalities. The method is largely decomposed and amenable for parallel computing. Other noteworthy features are: non-smooth data can be accommodated; no projection or optimization is needed as subroutines; multipliers converge monotonically upward; and, finally, the implementation amounts, in essence, only to numerical integration.