A nonsmooth version of Newton's method

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC ²-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.This author's work is supported in part by the Australian Research Council.This author's work is supported in part by the National Science Foundation under grant DDM-8721709.  

一类二阶非线性阻尼微分方程的振动性

研究了一类二阶非线性阻尼微分方程解的振动性,建立了三个新的振动性定理,推广了Cecchi M和Marini M(Rocky Mount J Math,1992,22:1259-1276)的结果.  

脉冲时滞抛物型方程解的振动性

本文研究了两类边界条件下有振动系数的脉冲时滞抛物型方程解的振动性，得到了若干解振动的判别准则．其结果即使用于有固定符号系数的方程，也显著地改进了已有的结果．  

The approximation power of moving least-squares

A general method for near-best approximations to functionals on , using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives' approximations. For the interpolation problem this approach gives Mclain's method. The method is near-best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation in is shown to be a function, and an approximation order result is proven for quasi-uniform sets of data points.

  

图的距离不大于β的任意两点可区别的边染色

本文提出了图的距离不大于β的任意两点可区别的边染色,即D（β）-点可区别的边染色（简记为D（β）-VDPEC）.并得到了一些特殊图类,如圈、完全图、完全二部图、扇、轮、树以及一些联图的D（β）-点可区别的边色数,文后提出了相关的猜想。  

Flow past a stretching plate

Zusammenfassung Eine Platte aus plastischem Material fliesst aus einem Spalt mit einer Geschwindigkeit, die proportional zum Abstand vom Spalt ist. Eine exakte Lösung der Grenzschichtgleichungen für die von der Platte erzeugte Luftbewegung wird gegeben. Oberflächenreibung und Wärmeleitungskoeffizient werden berechnet.  

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995  

Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

The smoothing Newton method for solving a system of nonsmooth equations , which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the th step, the nonsmooth function is approximated by a smooth function , and the derivative of at is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if is semismooth at the solution and satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is -uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

  

Another note on Weyl's theorem

``Weyl's theorem holds" for an operator on a Banach space when the complement in the spectrum of the ``Weyl spectrum" coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the ``Browder spectrum", which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in . In this note we try to explore these distinctions.

  

模糊粗糙集理论研究进展

介绍模糊粗糙集的概念及发展进程．提出了理论建立过程中，分别以推广到模糊集、引入模糊逻辑算子、拓展到两个论域为特点的三个发展阶段；分析、比较了各阶段代表性理论的特点，并对模糊粗糙集的未来发展作出了预期。