与中学老师谈微积分（4）——强可导的函数和导数概念

3 强可导的函数和导数概念 3．1两个函数的差商——中值关系 定义3．1设F（x）和f（x）都是I上的函数．如果对I中任意的u和v，都有[u，v]上的p和q，使得     

(h,)-Lipschitz函数及其广义方向导数和广义梯度

借助于Ben—Tal广义代数运算引进了一种新的函数-(h,)-Lipschitz函数．讨论了它与Lipschitz函数之间的关系,给出了它的广义方向导数和广义梯度,得到了它们的若干性质．作为应用,给出了广义方向导数与切锥之间的关系。     

近似已知函数的求导方法

1 引言 在实际问题中会遇到求近似已知函数的微商，这是一个典型的不适定问题[1-2]，即函数的一个微小的扰动会使得导数值有巨大的变化，因此求导数是相当不稳定的，对这类问题需要用特殊的方法。     

广义函数的集值导数

本文应用广义函数的调和表示,引进了一维广义函数的集值导数,并给出了连续函数的集值导数的几种等价定义.局部Lipschitz函数的集值导数同Clarke定义的广义梯度一致;广义函数在一点附近是Lipschitz 函数之充要条件是它在该点的集值导数是有限的.当广义函数在某点的集值导数不同时包含+∞和-∞时,它的广义导函数在该点的某邻域上是Radon测度.利用一阶集值导数,给出了连续函数的逆函数存在定理;应用高阶集值导数,得到了广义函数取极值的两种非常一般的充分条件.广义函数在一个开区间上成为凸函数的充要条件是它在该区间内每点处的二阶集值导数都包含在[0,+∞]之中.于是,本文建立起一元非可微函数的一套令人满意的微分理论.     

n维广义函数的集值导数

本文将文献［9］中给出的一维广义函数的集值导数的某些结果，推广到n维情形，并给出一些与优化有关的其他结果．     

区域函数的广义导数及其应用

本文以[7]的基本概念为基础,并根据Clarke的广义导数[1],以及Lasotra和Strauss[6]的多值函数f(x)的广义微分D_f(x)的定义.从而建立了区域函数F(x)的广义导数D_F=∪∩{G(x)?B(R),?x∈B(R);G(x)=F_x=F(x)}讨论了区域函数F(x)的广义导数的存在性;建立了区域函数的广义Fréchet导数存在的必要充分条件.     

参数函数和复合函数高阶导数的计算

用函数的链式法则和乘积公式给出了参数函数和复合函数的高阶导数的计算公式．     

从2006年高考看用导数探讨函数图象的交点问题

2006年高考数学导数命题在方向基本没变的基础上，又有所创新．导数命题创新的两个方面：一是研究对象的多元化，由研究单一函数转向研究两个函数或多个函数，二是研究内容的多元化，由用导数研究函数性质（单调性、最值、极值）转向运用导数综合研究函数的性质、函数图象的交点和方程根的分布等，实际上就是运用导数考查函数图象的交点个数问题．     

一类广义差商的Leibniz公式与Green函数的递推关系

本文研究具有幂基解组的微分算子所定义的广义差商的Leibniz公式及其格林(Green)函数的递推关系.     

Multiquadric 拟插值对高阶导数逼近的稳定性分析

基于样本数据来数值模拟函数的高阶导数是数值逼近中遇到的一类重要而且基本的问题, 差商方法是数值微分的传统方法. 但是在实际问题的求解中, 它表现出强烈的不稳定性. 在实际应用中, 由于差商计算的不稳定性, 它仅能用来模拟函数的低阶导数. 为了更好地模拟函数的高阶导数, 本文利用multiquadric 拟插值提出了一种新的方法. 并将multiquadric 拟插值方法模拟函数导数的稳定性与传统差商方法所得结果进行了对比. 数值例子很好地验证了本文的理论. 从理论论证和数值例子比较来看, multiquadric 拟插值方法比差商方法更为稳定. 这个性质也表明, 基于散乱甚至有干扰的数据, 在逼近函数的高阶导数时, multiquadric 拟插值方法是一个有效的工具.     

某些新广义函数类(英文)

给出由非标准离散函数及其差商所定义的新广义函数的某些类,它们密切联系于通常的直至某阶为连续可微的函数．周的一个深刻的定理被用来建立这些类与非标准Ｓｏｂｏｌｅｖ空间之间的关系．     

Generalized Functions Valued in a Smooth Manifold

 Based on Colombeau’s theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalized functions and indicate applications of the resulting theory in general relativity. Received February 13, 2002     

Generalized Functions in Signal Theory

Within a self-contained signal theory, generalized functions have to be taken into account, because without them notions like impulse response or transmission function cannot be defined. Starting from the requirements that have to be taken for a function space, if it should be suitable for a signal theory, generalized functions are introduced. Moreover, the connections between such a signal theory and the theory of white noise are discussed.     

广义函数法边界积分方程的建立

从广义函数论出发,本文引入一特殊广义函数δθ_P,通过它以及它的各阶导数建立了任一足够光滑函数的各阶导数的边界积分方程。对于由线性偏微分算子定义的问题,只要存在着相应的基本解,问题的偏微分方程总可转换成边界积分方程。     

Microlocal Analysis of Generalized Pullbacks of?Colombeau Functions

In distribution theory the pullback of a general distribution by a C ^∞-function is well-defined whenever the normal bundle of the C ^∞-function does not intersect the wave front set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C ^∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wave front sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.      

广义迹函数

本文引进两类新的广义迹函数,用矩阵理论和有限群的表示论研究它们的性质,给出它们在分块厄米特矩阵上的应用.     

Sobolev Embedding Theorems and Generalizations for Functions on a Metric Measure Space

Considering the metric case, we define an analog of the Sobolev space of functions with generalized derivatives of order greater than 1. The space of functions with fractional generalized derivatives is also treated. We prove generalizations of the Sobolev embedding theorems and Gagliardo–Nirenberg interpolation inequalities to the metric case.     

Higher order fundamental forms of a surface and their applications to wave propagation and generalized derivatives

We present a general theory of moving and deforming wave fronts by first defining and discussing the higher order fundamental forms for such a surface. These fundamental forms are then used to find the general formula for the jump of the Nth order differential of generalized functions which are discontinuous across this surface. Subsequently, we consider the wave fronts which carry multilayers and develop the theory of the generalized derivatives of the multilayers. In the process we find many new results in the theories of differential geometry and generalized functions.     

On Some Concepts of Generalized Differentials

In this paper, we study some concepts of generalized differentials for set-valued maps and introduce some new ones. In particular we first focus on the concept of Generalized Differential Quotients, briefly GDQs. It is shown that minimal GDQs are unique for scalar single-valued functions, then GDQs are compared with contingent and Dini derivatives, finally some other results characterizing GDQs are given. A new definition of generalized differentiation theory is presented, namely weak GDQs that are a modification of GDQs. We clarify the relationships with other concepts of generalized differentiability: Clarke generalized Jacobians, path-integral generalized differentials and Warga derivate containers. Finally, some applications of GDQs end the paper.