Newton's method for sections on Riemannian manifolds: Generalized covariant -theory

One kind of the L-average Lipschitz condition is introduced to covariant derivatives of sections on Riemannian manifolds. A convergence criterion of Newton's method and the radii of the uniqueness balls of the singular points for sections on Riemannian manifolds, which is independent of the curvatures, are established under the assumption that the covariant derivatives of the sections satisfy this kind of the L-average Lipschitz condition. Some applications to special cases including Kantorovich's condition and the γ-condition as well as Smale's α-theory are provided. In particular, the result due to Ferreira and Svaiter [Kantorovich's Theorem on Newton's method in Riemannian manifolds, J. Complexity 18 (2002) 304–329] is extended while the results due to Dedieu Priouret, Malajovich [Newton's method on Riemannian manifolds: covariant alpha theory, IMA J. Numer. Anal. 23 (2003) 395–419] are improved significantly. Moreover, the corresponding results due to Alvarez, Bolter, Munier [A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. to appear] for vector fields and mappings on Riemannian manifolds are also extended.     

Discrete inertial manifolds

This work is devoted to attractive invariant manifolds for nonautonomous difference equations, occurring in the discretization theory for evolution equations. Such invariant sets provide a discrete counterpart to inertial manifolds of dissipative FDEs and evolutionary PDEs. We discuss their essential properties, like smoothness, the existence of an asymptotic phase, normal hyperbolicity and attractivity in a nonautonomous framework of pullback attraction. As application we show that inertial manifolds of the Allen–Cahn and complex Ginzburg–Landau equation persist under discretization. For the Ginzburg–Landau equation we can also estimate the dimension of the inertial manifold. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flight dynamics over irregular asteroids

小行星探测是当前深空探测的主要方向之一,具有重要的科学意义.绝大多数小行星引力场极不规则,探测器在小行星附近运动形态复杂多样.由于同时受到中心引力、快速自旋的不规则形状摄动力、以及光压摄动等作用,探测器容易与小行星发生碰撞或逃逸.概述小行星研究现状和不规则引力场建模方法.重点介绍不规则引力场内动力学特性,包括引力平衡点、局部流形、自然周期轨道和悬停探测轨道等,尝试提出新的研究方向.     

Continuous dependence of the controllability set of a nonlinear control system on a parameter

In this note, we consider the mapping which associates a controllability set to each value of the parameter of a system. Sufficient conditions for the existence of breaks of this mapping are obtained. They show that nearly all such mappings have a finite number of breaks.This note was written during the author's stay at the University of Hamburg. He expresses his gratitude to the Institut für Angewandte Mathematik for its warm hospitality.     

最小熵产生定理的推广和定态的稳定性

本文将线性非平衡热力学中的唯象系数从常数推广成描述系统的宗量的函数,在局域平衡条件下,考虑到热力学内部空间坐标变换时熵产生的不变性,证明了线性非平衡热力学系统的定态是稳定的.     

Some Examples of Metrics with Positive Curvature and Negative Curvature on R￣n

ＳｏｍｅＥｘａｍｐｌｅｓｏｆＭｅｔｒｉｃｓｗｉｔｈＰｏｓｉｔｉｖｅＣｕｒｖａｔｕｒｅａｎｄＮｅｇａｔｉｖｅＣｕｒｖａｔｕｒｅｏｎＲ￣ｎ￥ＺｈｕＤａｘｉｎ；ＨｏｕＷｅｎｈｕａ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＴｉａｎｊｉｎＵｎｉｖｅｒｓ...     

HARMONIC MAPS AND FUNDAMENTAL GROUPS OF NONPOSITIVELY CURVEDRIEMANNIAN MANIFOLDS

Ｔｈｅｔｈｅｏｒｙｏｆｈａｒｍｏｎｉｃｍａｐｓｉｎｔｏａｃｏｍｐａｃｔｎｏｎｐｏｓｉｔｉｖｅｌｙｃｕｒｖｅｄｔａｒｇｅｔｉｓｗｅｌｌ-ｄｅｖｅｌｏｐｅｄ.ＴｈｅｅｘｉｓｔｅｎｃｅｏｆｓｕｃｈａｈａｒｍｏｎｉｃｍａｐｐｉｎｇｉｎａｎｙｇｉｖｅｎｈｏｍｏｔｏｐｙｃｌａｓｓｉｓｅｓｔａｂｌｉｓｈｅｄｂｙＥｅｌｌｓａｎｄＳａｍｐｓｏｎ[1]ｉｎ1964,ａｎｄｌａｔｅｒ,ｉｎ1967,Ｈａｒｔｍａｎ[2]ｇａｖｅａｐｒｏｏｆｏｆｔｈｅｕｎｉｑｕｅｎｅｓｓ.Ｔｈｉｓｐａｐｅｒｉｓｄｅｖｏｔｅｄｔｏａｎａｐｐｌｉｃａｔｉｏｎｏ…     

Strictly pseudoconvex domains and algebraic varieties

In the paper, we consider applications of strictly pseudoconvex domains to the problems of algebraicity and rationality. We give a new proof of the Kodaira theorem on the algebraicity of a surface and we also prove a multidimensional version of this theorem. Theorems analogous to the Hodge index theorem and the Lefschetz theorem about (1, 1)-classes are obtained for strictly pseudoconvex domains. Conjectures on the geometry of strictly pseudoconvex domains on algebraic surfaces are formulated. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 414–422, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the International Science Foundation under grant No. 508.