关于凸曲面的几个定义的关系

本文讨论了凸曲面的几种定义及其关系，发现有的定义是局部凸的定义，有的是整体凸的定义，有的则对于局部凸和整体凸都适合，最后给出了各种定义之间互推的证明，对于局部凸和整体凸定义之间不能推证的，则说明了原因．     

正交变换在曲线,曲面积分计算中的应用

正交变换在曲线、曲面积分计算中的应用林元重（江西萍乡高等专科学校３３７０５５）对于三维空间的曲线积分与曲面积分，如果知道其积分曲线或积分曲面的参数形式，一般可按数学分析教材所介绍的公式计算．但是，对于某些曲线、曲面积分，要把积分曲线或曲面用适当的参数...     

一类具有任意次数的参数多项式极小曲面

极小曲面是在几何造型设计中有着重要应用的一类特殊曲面.本文从几何造型的视角提出一类次数任意的参数多项式极小曲面.所提出的极小曲面具有显式的参数表示,并具有一些重要的几何性质,如对称性、包含直线和自交性.根据几何性质,本文将该参数多项式极小曲面划分为4类:n=4k+1,n=4k+2,n=4k+3,n=4+4,其中n是极小曲面的次数,k是正整数.本文给出与之相对应的共轭极小曲面的显式参数形式,并实现其等距变形.     

高维旋转曲面

本文考虑欧氏空间中一种余一维的高维旋转曲面,通过发展出一种全新的复合映射、维数分解与分块矩阵递推法,我们系统性地研究了同它的面积和曲率有关的一系列问题.当母函数是多元函数时,这种高维旋转曲面的概念尚属首次提出.我们给出了这种高维旋转曲面的面积公式以及它的一些简单应用.我们发现:在任一直径方向上,单位球面的面积分布和低一维单位球体的体积分布完全相同,并且当维数趋于无穷时它们的密度函数的极限都是狄拉克函数.通过研究相应面积泛函的变分问题,我们得到了所谓的极小旋转曲面方程.我们证明了:满足极小旋转曲面方程的母函数对应的旋转曲面的平均曲率等于零.这种极小旋转曲面方程推广了传统的极小曲面方程,并且为非参数极小曲面理论提供了新的更一般的研究框架;通过计算径向对称解对应的常微分方程,我们研究了它的一些简单的特解.我们也简单讨论了相应的预定平均曲率和预定高斯曲率问题.     

第二型曲面积分的参数形式计算

给出"第二型曲面积分"的一种计算方法,即在曲面的参数形式下直接将曲面积分转化成参数区域上的一个二重积分,由此可使"第二型曲面积分"的计算问题得到简化.此法是对菲赫金哥尔茨《微积分学教程》所给"第二型曲面积分的参数形式计算"的一个改进.     

三维Minkowski空间中的一类直线汇

研究了由三维Minkowski空间$E^3_1$中一个类空曲面$S_1$上一个单参数测地曲线族的切线所构成的直线汇$T$,它以$S_1$为一个焦曲面.证明了$T$的两个可展曲面族沿着第二个焦曲面$S_2$的正交曲线网相交的充要条件是$S_1$是可展曲面.对于$T$的两个焦曲面$S_1$和$S_2$之间沿着同一光线的对应,还证明了其保持渐近曲线网的充要条件.最后,研究了$T$的正交曲面$S$,并且证明了如果$S$是$E^3_1$中的一个极大曲面,那么,$T$的焦曲面$S_1$和$S_2$之间沿着同一光线的对     

计算机辅助几何设计中极小曲面造型的研究进展

极小曲面在工程领域有着广泛应用,因此将其引入计算机辅助几何设计领域具有重要意义.详细概述了近年来计算机辅助几何设计领域中极小曲面造型的研究工作,按照造型方法的不同,可将现有工作分为精确造型方法和逼近造型方法两类.精确造型方法主要包括两个部分:某些特殊极小曲面的控制网格表示与构造;等温参数多项式极小曲面的挖掘与性质.逼近造型方法主要包括3个部分t基于数值计算的逼近方法;基于线性偏微分方程的逼近方法;基于能量函数最优化的逼近方法.最后对这些方法进行了分析比较,并讨论了极小曲面造型中有待进一步解决的问题.     

一般梯图的亏格分布

本文求出了-些曲面集的亏格分布的显式表达式.在联树的基础上,通过运用曲面分类法把一般梯图的亏格分布转化为这些曲面集的线性组合,从而可求出它们的显式表达式.     

三角域上带两个形状参数的Bézier曲面的扩展

给出了三角域上带双参数λ1,λ2的类三次Bernstein基函数,它是三角域上三次Bernstein基函数的扩展.分析了该组基的性质并定义了三角域上带有两个形状参数λ1,λ2的类三次Bernstein-Bézier(B-B)参数曲面.该基函数及参数曲面分别具有与三次Bernstein基函数及三次B-B参数曲面类似的性质.当λ1,λ2取特殊的值时,可分别得到三次Bernstein基函数及三次B-B参数曲面以及参考文献中所定义的类三次Bernstein基函数及类三次B-B参数曲面.由实例可知,通过改变形状参数的取值,可以调整曲面的形状.     

NURBS曲面的形状修改的一种方法

NURBS曲面是计算机辅助几何设计和计算机图形中最常用的参数曲面。本文采用NURBS曲面的齐次坐标表示，给出了通过控制顶点和权因子同时改变来修改NURBS曲面形状的一种方法。     

参数曲面保凸的一个定理

本文讨论了参数曲面的凸性，给出了参数曲面为凸的充要条件．并给出了具体的应用实例．     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

微分几何观点下二元函数的极值

利用曲面的局部微分性质给出二元函数极值存在的必要条件和充分条件,并将之运用于具有明显几何特征的曲面对应的二元函数极值的判别问题中.     

On Shape Preserving C2 Hermite Interpolation

We propose a general parametric local approach for functional C ² Hermite shape preserving interpolation. The constructed interpolant is a parametric curve which interpolate values, first and second derivatives of a given function and reproduces the behavior of the data. The method is detailed for parametric curves with piecewise cubic components. For the selected space necessary and sufficient conditions are derived to ensure the convexity of the constructed interpolant. Monotonicity is also studied. The approximation order is investigated for both cases. The use of a parametric curves to interpolate data from a function can be considered a disadvantage of the scheme. However, the simple structure of the used curve greatly reduces such a disadvantage.     

Convexity conditions for parametric tensor‐product B‐spline surfaces

This paper provides four alternative sets of discrete conditions, which ensure that a patch of a parametric tensor‐product B‐spline surface is locally convex. These conditions are at most of degree six with respect to the control points of the surface. Their weakness is analytically investigated and graphically illustrated for a bicubic B‐spline surface of industrial interest. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bifurcations in parametric nonlinear programming

Bifurcation and continuation techniques are introduced as a class of methods for investigating the parametric nonlinear programming problem. Motivated by the Fritz John first-order necessary conditions, the parametric programming problem is first reformulated as a closed system of nonlinear equations which contains all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points. Since changes in the structure of the solution set and critical point type can occur only at singularities, necessary and sufficient conditions for the existence of a singularity are developed in terms of the loss of a complementarity condition, the linear dependence constraint qualification, and the singularity of the Hessian of the Lagrangian on a tangent space. After a brief introduction to elementary bifurcation theory, some simple singularities in this parametric problem are analyzed for both branching and persistence of local minima. Finally, a brief introduction to numerical continuation and bifurcation procedures is given to indicate how these facts can be used in a numerical investigation of the problem.This research was supported by the Air force Office of Scientific Research through grant number AFOSR-88-0059.     

Local asymptotic normality and asymptotical minimax efficiency of the MLE under random censorship

Here we study the problems of local asymptotic normality of the parametric family of distributions and asymptotic minimax efficient estimators when the observations are subject to right censoring. Local asymptotic normality will be established under some mild regularity conditions. A lower bound for local asymptotic minimax risk is given with respect to a bowl-shaped loss function, and furthermore a necessary and sufficient condition is given in order to achieve this lower bound. Finally, we show that this lower bound can be attained by the maximum likelihood estimator in the censored case and hence it is local asymptotic minimax efficient.     

On Minimizing Problems with a Volume Constraint in Hyperbolic 3-Manifolds

This paper investigates the existence of an area (or Dirichlet integral) minimizing parametric surface in a hyperbolic 3-manifold subject to a volume constraint. The existence of a minimizing surface is proved, assuming some conditions on the prescribed free homotopy class. This result implies a non-existence result of minimizing surfaces of prescribed mean curvature. A criterion for the existence of surfaces of prescribed mean curvature, which turns out to be optimal in view of the non-existence result, is also obtained.     

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

In this two-part study, we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite-dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e., whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can be performed with the use of optimality conditions and constraint qualifications. In the first paper, we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle, we recover existing simple, necessary, and sufficient conditions for the global exactness of linear penalty functions and for the existence of augmented Lagrange multipliers of Rockafellar–Wets’ augmented Lagrangian. We also present completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems as well.