二元函数条件极值的充分条件

[1]文从一元函数极值的第二种充分条件推出二元函数极值的一个充分条件.本文从一元函数极值的第一种充分条件推出二元函数条件极值的另一个充分条件.不难看出,一元函数极值的第一种充分条件可有以下等价表达形式:     

二元函数极值充分条件的评注

本对二元函数极值的充分条件作了进一步的讨论，得到了AC－B^2=0时，二元函数极值判定的充分条件。     

解二元函数无条件极值问题的一个有效方法

针对高等数学教学内容中二元函数无条件极值存在的第二充分条件的缺点,在一定假设下给出了二元函数无条件极值存在的一个充分且必要条件,并举例说明在一定条件下该方法比原有方法更好.     

二元函数极值充分条件的评注

本文对二元函数极值的充分条件作了进一步的讨论 ,得到了 AC-B2 =0时 ,二元函数极值判定的充分条件     

多元函数极值的求法及其应用

在实际问题中,往往会遇到多元函数的最大值、最小值问题.多元函数的最大值、最小值问题与极大值、极小值有密切联系.求多元函数极值,一般可以利用偏导数来解决.与一元函数相类似,可以利用函数的极值来求函数的最大值和最小值.但是由于自变量个数的增加,应特别注意概念中的一些变化和计算复杂性.这里主要讨论二元函数,对于二元以上的函数极值可以类似加以解决.     

二元函数条件极值的初等解法

二元函数的极值问题,在中学课本中出现较少,部分同学往往碰到这类问题感到束手无策,难以作出正确的解答。为此,本文试图通过一些典型的例题。来谈谈二元一次函数和二元二次函数的极值求法。一图象法图象法就是把二元函数的图象及其条件所表示的范围在平面上表现出来,然后,据图象进行分析,确定出所求函数取得极值     

多元函数极值充分条件证明的一元方法

从多元函数极值的定义出发,用一元函数的方法给出了二元和三元函数极值判定的充分条件的证明,其中只涉及了偏导数的求法.相对于多元函数极值充分条件证明的多元泰勒公式方法,本文所用的方法更为直接而且简明.     

用向量函数证明积分换元公式

运用二元向量函数和三元向量函数证明二重积分和三重积分的换元公式,推导曲面的表面积公式,并计算第二类曲面积分.     

多元函数条件极值的必要条件

推导证明了一般n元函数及常用的二元、三元函数在等式约束条件下用行列式表示的极值的必要条件,并从几何上对二元、三元函数在等式约束条件下取极值的必要条件予以了直观解释.利用这些必要条件求解条件极值,因除去了Lagrange乘数法带来的Lagrange乘子对解方程组的困扰,而使得最终方程组的求解变得明快简洁.     

利用二阶方向导数证明极值的充分条件

介绍二元函数二阶方向导数的概念与计算方法.利用线性代数中的二次型知识,对二元函数在驻点处是否取得极值的充分性定理给出有几何意义的证明.     

Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.     

鞍点定理在Lagrange乘数法上的应用

将鞍点的概念运用在Lagrange乘数法上,给出了多元函数的条件极值问题存在的一个充要条件.     

Vector Moment Problems for Rapidly Decreasing Smooth Functions of Several Variables

The existence of rapidly decreasing smooth solutions of moment problems for functions of several variables with values in a Fréchet space is obtained. It is shown that the corresponding results for functions with values in a general topological vector space do not hold.

     

New computer-based search strategies for extreme functions of the Gomory–Johnson infinite group problem

We describe new computer-based search strategies for extreme functions for the Gomory–Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions.     

Martingales with sufficient statistics

The property of countable convexity of the set which contains all dominated martingale laws, for a family of measurable functions with values in a separable Borel space, is proved to be equivalent to the existence of a sufficient statistic. The result is then used to derive a representation of such laws in terms of their extreme points, which are laws of uniformly integrable martingales. Finally, we show that martingales with sufficient statistics converge with probability 1.     

Extreme problems on classes of polynomials and splines

By the methods of the theory of extremum problems, existence theorems for functions with given sequence of extreme values are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 494–502, April, 1998.The author wishes to express his sincere gratitude to Professor V. M. Tikhomirov for setting the problem and for his permanent attention and help and also to the referee for the remarks, which aided in the amelioration of the present paper.     

正定矩阵在函数极值问题中的应用

通过正定矩阵来处理多元函数的极值问题,通过一个简单的方法,证明了关于多元函数极值存在的一个充分条件.     

A tail empirical process approach to some nonstandard laws of the iterated logarithm

We show that a number of nonstandard laws of the iterated logarithm have limiting constants which may be expressed as the extreme values of functionals off, wheref varies over suitable compact sets of functions. By solving the corresponding extremal problems, we show how these constants are generated. In addition, several new laws are presented.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

Topological structure of candidates for positive curvature

In light of recent advances in the study of manifolds admitting Riemannian metrics of positive sectional curvature, the study of certain infinite families of seven dimensional manifolds has become a matter of interest. We determine the cohomology ring structures of manifolds belonging to these families. This particular ring structure indicates the existence of topological invariants distinguishing the corresponding homeomorphism and diffeomorphism type. We show that all families contain representatives of infinitely many homotopy types.