基于可变模糊集的质变与量变定理——兼论集对分析

基于可变模糊集理论中的相对比例函数定义,第一次提出唯物辩证法三大规律之一"质量互变规律"的严密数学定理:质变定理与量变定理.据此提出博弈定理、投票定理.根据提出的这些定理与集对分析的基础概念进行了比较分析.     

问题驱动下的微分与全微分教学探索

微分概念是高等数学中的核心概念,通过问题教学法和几何图形演示,将一元函数的微分教学和二元函数的全微分教学统一起来,使学生更深刻理解微分这一概念的核心本质,并认识到从一元函数到多元函数许多结论要发生变化,量变引发质变的变化过程.     

多元函数微分学的几个基本问题

<正> 从一元函数到多元函数是一个从特殊到一般的过程.一元函数的许多概念和方法可以很自然地推广到多元函数,这种推广有着极大的理论意义和实际意义.但是特殊毕竟不能代替一般.我们既要注意到一元函数和多元函数的共性,又要注意到它们的差别,确切地说,一元函数的许多性质是多元函数所不具备的.     

微积分中若干思政元素的探讨

文中探讨了微积分中若干思政元素.首先，从微积分的简史中让学生树立正确的科学观，启发学生们要具备勇于探索，不断创新的精神.微积分的发展历程体现了唯物辩证法的量变引起质变的规律.进一步，利用极限理论对唯物辩证法的质量互变规律进行量化表示，其中也蕴涵了动态变化和相对静止的辩证统一.最后，通过连续函数建立自然哲理的数学模型，探讨连续所蕴涵的人生启发，以及对芝诺悖论的新解释.     

多元微分学中探究式的教学案例

在学习多元函数性质时,一定要和一元函数性质加以比较,找出它们的异同点,因此有很多探究性问题可供学生思考.文中给出了多元微分学的五个探究式教学案例,供同行参考.     

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.     

漫谈多元函数与一元函数的差别

<正> 研究多元函数的主要方法是将其视为一元函数的推广,把多元函数看作点的函数,在形式上与一元函数相类似,与一元函数相对照很容易建立多元函数的有关概念、理论和方法。我们知道多元函数保留着一元函     

多元函数最值的求解策略

在近几年的高考及各种测试试题中,多元函数的最值及其衍生问题频频出现,因为变量多、解析式复杂、方法技巧性强、题目灵活多变而具有较强的挑战性,成为最值问题中的一个难点,也是考查学生的数学素养和能力的一个热点.根据课程标准的要求,求多元函数的最值,总的策略是转化为一元函数或二元函数最值问题,转化的具体策略多种多样,本文对此进行了归纳和梳理.     

非光滑函数的凸性

本文借助于一元函数左、右导数的定义及其性质 ,将多元函数的方向导数转化为一元函数的左、右导数 ,并利用一元函数的凸性判别准则给出并证明了判别多元函数凸性的充分必要条件 .     

利用偏反函数求解多元隐函数的偏导数

本文利用类比联想给出了一种类似于一元函数利用反函数求导法则的新方法来求解多元隐函数的偏导数,并在此基础上利用多元函数的一阶微分形式不变性得出了以二元函数为例6个偏导数中的某三个满足特殊的链式法则,并将此法则推广到任意n元函数.     

On the rigidity of Hardy submodules

In contrast with the one-variable case, there is a large number of distinct submodules of the Hardy module over the polydisk algebra in the multi-variable use. We show that under hypotheses on the zero sets, two submodules which are equivalent in any reasonable sense must be equal. This is the rigidity referred to in the title.The research of the first author was partially supported by a grant from the National Science Foundation.     

含边界在内的一般极值的必要条件与拉格朗日乘数法

讨论包括定义域边界点在内的极值,称为一般极值.对可导的一元和多元函数给出了一般极值点的必要条件,这些必要条件与经典极值的必要条件是相容的.还利用一般极值的必要条件导出了条件极值的拉格朗日乘数法.     

关于二重变上限积分的等价无穷小

将二重变上限积分看作是一类特殊的一元诱导函数,本文给出了两种二重变上限积分的定义方式,分别对积分限和被积函数做相应的等价无穷小量替换.在一定的条件下,替换后的二重变上限积分与替换前的二重变上限积分是等价无穷小,从而得到一类求极限的方法,并给出了应用实例.     

基于Matlab的数学分析极限概念教学实践

极限概念是数学分析理论的基础,贯穿数学分析教学的始终,在数学分析的理论体系中占有十分重要的地位.由于极限概念的严谨性和抽象性,在教学实践当中发现学生对极限概念难以理解.本文借助Matlab软件的图形处理功能,将数列极限,一元函数极限以及多元函数极限的形成过程展示出来,从而强化学生对极限概念的理解.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

二元Thiele型向量有理插值

本文对二元Thiele型连分式的渐近分式施行Samelson逆变换,建立了平面矩形域上的二元向量值有理插值,所得结果是一元向量值有理插值的推广和改进.     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.     

Control optimization in a hyperbolic linear system

An optimal control problem is considered for a distributed-parameter Goursat-Darboux system with controlled boundary conditions. For the numerical solution of the problem, an algorithm based on separability and minimax theorems is constructed, which reduces the problem to finding the maximum of a concave functional defined in the class of one-variable functions.     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

Optimality conditions of the maximum principle type in bilinear control problems

The optimization of a bilinear functional related to a linear state system with a modular control constraint is considered. Exact formulas for the functional increment are used to obtain sufficient conditions for the optimality of extremal controls that supplement the maximum principle. These conditions are represented in the form of inequalities and equalities for one-variable functions on a time interval. The optimization of a quadratic functional with the help of a matrix conjugate function is reduced to the bilinear case.