多元函数微分学中的反例构造技巧

文中给出了多元函数微分学的四个定义"连续、可偏导、可微、偏导数连续"之间关系的反例及其构造思路.     

谈多元微分学中几个概念间的关系

学习多元函数微分学,一定要弄清连续、偏导数、全微分、方向导数之间的关系,并与一元函数中连续、可导、可微之间的关系比较,看看有何类似.有何区别,才能更好地掌握和使用这些基本概念.从教材中我们知道这几个基本概念间的关系(以二元函数为例)由下面定理给出:     

二元函数极限、连续、微分之间的关系与反例

<正> 多元函数连续、可导、可微是极其重要的概念,初学者常常将其与一元函数相应概念混淆.“多元函数微分学中的几个重要问题”一文已从理论上予以分析,为加深理解,本文以例题进一步揭示它们之间的关系。     

多元函数微分学总结

各位同学:今天我们把多元函数的微分学总结一下,着重总结二元函数的微分学.因为二元函数是多元函数的一个代表,把二元函数的微分学掌握好了,多元函数的微分学也就可以类似地掌握住了.     

二元函数的可微性研究

研究二元函数的可微性,得到两个充分条件,它们与已有的判别二元函数可微性的充分条件相比更为简单,因而使得应用更为简便.     

谈谈多元函数中可导与可微的关系

大家熟悉,在一元函数中,可导必可微,可微必可导.但对于多元函数,偏导数存在与可微却是两个不等价的概念.本文以二元函数为例,谈谈二者之间的关系.     

多元函数微分学导引

掌握了一元函数微分学之后,便可以进一步学习多元函数微分学。由于函数的自变量个数增多,引起了一系列的变化,使多元函数与一元函数在若干方面存在本貭的差异。虽然如此,处理多元函数問題时,在相当程度上,于一定条件下可借用一元函数的有关概念与方法。所以,随时注意这种区别和联系,对掌握多元函数微分学会有一定的帮助。上述的区别和联系,当从一元函数过渡到二元函数的研究吋,便充分得到显示;至于从二元推广到多元,則仅需在技巧方面下工夫,而沒有原則上的困难。因此,本文重点討論二元函数,其結果不难推广到多元函数。由于篇幅有限,仅討論最基本的概念:极限,連續,微商与微分,并涉及一些初步应用。学过一元函数微分学的讀者都可以看懂。进一步的材料可参考[1],[2],[3],[4],[5]各书。     

一个关于多元函数可微的定理

大家知道,多元函数的可微与可导是两个概念,但如果一个多元函数的所有偏导数在某一点都存在并连续,则它一定在该点可微.本文给出了一个在较弱条件下关于多元函数可微的定理.     

关于多元函数可微性的一个注记

研究了多元函数的可微性,给出了多元函数连续、可微的一些条件.     

多元函数微分学中几个基本概念之间的关系

针对多元函数微分学中用以刻画函数局部性态的基本概念,给出连续、偏导数、可微、方向导数之间的关系图,采用证明和举反例的方式．深入分析这些概念之间的关系．     

Subvexormal Functions and Subvex Functions

Subvexormal functions and subinvexormal functions are proposed, whose properties are shared commonly by most generalized convex functions and most generalized invex functions, respectively. A necessary and sufficient condition for a subvexormal function to be subinvexormal is given in the locally Lipschitz and regular case. Furthermore, subvex functions and subinvex functions are introduced. It is proved that the class of strictly subvex functions is equivalent to that of functions whose local minima are global and that, in the locally Lipschitz and regular case, both strongly subvex functions and strongly subinvex functions can be characterized as functions whose relatively stationary points (slight extension of stationary points) are global minima.     

回收函数与函数的无界性

主要利用回收锥和回收函数来研究函数的下无界性。首先, 针对凸函数在非可微条件下,利用中值定理和回收锥刻画了凸函数次微分的性质, 并在此基础上给出了基于次可微条件下回收向量的充要条件。其次,将凸性推广到E-凸, 在一定条件下,利用回收函数研究了E-凸函数的下无界性。最后,通过举例说明这些结果不能推广到拟凸条件.     

Cauchy and Poisson formulas for polyanalytic functions and applications

We obtain new Cauchy and Poisson integral formulas for polyanalytic functions. As an application, we establish mean value theorems for polyanalytic and real polyharmonic functions in a disk. We also give applications to sharp estimates of generalized maximum modulus principle type for associated functions, and, in particular, to estimates for rational functions (components) in the problem of singularity separation for polyrational functions.     

半严格-G-E-半预不变凸型函数及其在数学规划中的应用研究

提出了一类新的广义凸函数——半严格-G-E-半预不变凸函数,它是一类非常重要的广义凸函数,为半严格-G-半预不变凸函数与半严格-E-预不变凸函数的推广.首先给出例子,以说明半严格-G-E-半预不变凸函数的存在性及其与其他相关广义凸函数间的关系.然后讨论了半严格-G-E-半预不变凸函数的一些基本性质.最后,探究了半严格-G-E-半预不变凸型函数分别在无约束和有约束非线性规划问题中的重要应用,获得一系列最优性结论,并举例验证了所得结果的正确性.     

Characterizations and Applications of Prequasi-Invex Functions

In this paper, two new types of generalized convex functions are introduced. They are called strictly prequasi-invex functions and semistrictly prequasi-invex functions. Note that prequasi-invexity does not imply semistrict prequasi-invexity. The characterization of prequasi-invex functions is established under the condition of lower semicontinuity, upper semicontinuity, and semistrict prequasi-invexity, respectively. Furthermore, the characterization of semistrictly prequasi-invex functions is also obtained under the condition of prequasi-invexity and lower semicontinuity, respectively. A similar result is also obtained for strictly prequasi-invex functions. It is worth noting that these characterizations reveal various interesting relationships among prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions. Finally, prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions are used in the study of optimization problems.     

Properties and construction of NCP functions

The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions.     

Staircase and fractional part functions

The staircase and fractional part functions are basic examples of real functions. They can be applied in several parts of mathematics, such as analysis, number theory, formulas for primes, and so on; in computer programming, the floor and ceiling functions are provided by a significant number of programming languages – they have some basic uses in various programming tasks. In this paper, we view the staircase and fractional part functions as a classical example of non-continuous real functions. We introduce some of their basic properties, present some interesting constructions concerning them, and explore some intriguing interpretations of such functions. Throughout the paper, we use these functions in order to explain basic concepts in a first calculus course, such as domain of definition, discontinuity, and oddness of functions. We also explain in detail how, after researching the properties of such functions, one can draw their graph; this is a crucial part in the process of understanding their nature. In the paper, we present some subjects that the first-year student in the exact sciences may not encounter. We try to clarify those subjects and show that such ideas are important in the understanding of non-continuous functions, as a part of studying analysis in general.     

Optimality and duality with generalized convexity

Hanson and Mond have given sets of necessary and sufficient conditions for optimality and duality in constrained optimization by introducing classes of generalized convex functions, called type I and type II functions. Recently, Bector defined univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, optimality and duality results for several mathematical programs are obtained combining the concepts of type I and univex functions. Examples of functions satisfying these conditions are given.     

p-Adic Multiresolution Analysis and Wavelet Frames

We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.     

P-凸函数及其性质

给出P-凸函数的定义及其性质,将一个有关凸函数的命题推广到P-凸函数情形,并指出赵海清等人关于该命题的证明过程是错误的.