幂向量,复合向量数及其函数理论

本文提出向量为其幂向量和向量幂级数．向量幂级数由一实数和某一向量联合组成的“复合向量数”及其函数有重要涵义．这数也有运算法则．从复合向量数的函数理论分析知其函数有导数和解析函数的必要和充分条件．这些条件构成了“双曲型”方程的特征以及函数的积分性质等．     

平面向量基本定理的相关性质及应用

平面向量的基本定理是平面向量坐标表示的基础，说明了同一平面内任一向量都可以表示为两个不共线向量的线性组合．即：如果e1，e2是同一平面内的两个不共线向量，那么对于这一平面内的任一向量a.     

基底的选择

在平面向量中,有了共线向量定理和平面向量基本定理,平面内任一向量都可以通过选择一组基底向量来表示,这样,若所选择的两基向量的夹角及其模长都可知,则称这组基底为“已知基底”,那么涉及有关向量运算问题可以将所求向量转化到这组“已知基底”向量上解决,这样就形成向量问题解决的一个通法,在空间向量中这一方法同样是奏效的．     

伪单调型向量F-互补问题可行集的最小元问题

本文引入了几类向量F-互补问题并给出了向量F-互补问题与广义向量变分不等式之间的关系.通过定义向量F-互补问题的可行集,研究了伪单调型向量F-互补问题的可行集的最小问题,推广了已有的一些结果.     

向量共线和不共线

预备知识 :方向相同或相反的非零向量叫做平行向量 .规定 0 → 与任一向量平行 .任一组平行向量都可移到同一条直线上 ,因此平行向量也叫做共线向量 .由预备知识易证定理 1.定理 1　一组平行向量共线 ,0→ 与任一向量共线 .定理 2　向量ｂ→ 与非零向量ａ→ 共线的充要条件是有且只有一个实数λ ,使得ｂ→ =λａ→ .(参见新教材高一《数学》第一册下第 10 4页 )定理 3　ａ→ ,ｂ→ 具备下列情况中的任何一种情况 ,都可以说ａ→ ,ｂ→ 共线 .1)ａ→ ,ｂ→ 中至少有一个为 0 → ;2 )ａ→ ,ｂ→ 都不为 0 → ,存在一个实数λ ,使得ｂ→=λａ→ …     

向量OP=xOA＋yOB中x,y的性质

一、问题由平面向量基本定理知道：以平面内两个不共线的向量OA、OB为基底,对于平面内任一向量0P都存在唯一的有序实数对（z,y）使得0P=xOA＋yOB成立．     

0被“双规”以后的话题

高中数学新教材 [1 ]第一册 (下 )第五章“平面向量”在给出定义“长度为 0的向量叫做零向量 ,记作 0”后 ,先后对它作了下面两个“规定”(本文简称“双规”) :“0与任一向量平行”(第 95页 )以及“零向量与任一向量的数量积为 0”(第 1 1 6页 ) ,并且接着在给出“a⊥b a·b=0”的前面明确指出“a、b都是非零向量”(第 1 1 7页 ) .这也就是说 ,新教材把平面向量集内的零元——— 0与其他元素的关系只归入共线而回避垂直 :当a·b=0时当且仅当a、b都不是零向量时 ,有a⊥b ;若a或b中有零向量时 ,未说a⊥b是否成立 .两向量的共线和垂直是两向量…     

向量互补问题和极小元问题

本文不仅引入了非空集的有效下界的概念,而且引入了新的严格C-伪单调映射,在C-伪单调性的假设下,文内证明了向量互补问题（VCP）的可行集及其相关集合的有效下界和下有效点的存在性。     

空间向量在立体几何中的初步应用

《全日制普通高级中学教科书》数学 (第二册下Ｂ)课本中 ,对第九章“直线、平面、简单几何体”(简称“9Ｂ”)的内容 ,引进了较新的数学内容———空间向量 ,在进行“9Ｂ”内容的教学实践中 ,我们引导学生将“平面向量”知识引申拓宽到“空间向量” ,较好地完善了向量的知识体系 ,并通过“空间向量”的知识性和工具性这两大特性的教学 ,增强了学生分析问题的能力 ,开阔了学生解决立体几何问题的视野 .现就“空间向量”在立体几何中的初步应用 ,谈谈我们的具体做法 .1　实现由“平面向量”到“空间向量”的自然转化 ,调动学生学习“空间向量”…     

平面向量基本定理可以这样用

1．定理的呈现如果a,b是同一平面内的两个不共线向量,那么对于这一平面内的任一向量p,存在唯一一对实数入,弘,使得P=λa＋μb     

具有控制结构的向量均衡问题与解集的性质

提出一种具有控制结构的向量均衡问题与向量映射的新的伪单调性概念,得到具有控制结构的向量均衡问题解的存在性及其解集的紧凸性．作为应用,得到具有控制结构的向量变分不等式与互补问题的解．     

Stampacchia向量均衡问题与向量相补问题

引进Stampacchia向量均衡问题与一种新的向量相补问题．用数值方法,得到它们的存在定理,并讨论Stampacchia广义向量变分不等式,向量隐相补问题与极小元问题的关系．     

Vector complementarity and minimal element problems

In this paper, vector complementarity problems are introduced as weak versions of vector variational inequalities in ordered Banach spaces. New dual cones are introduced and proved to be closed. In the sense of efficient point, we prove that the minimal element problem is solvable if a vector variational inequality is solvable; we also prove that any solution of a strong vector variational inequality or positive vector complementarity problem is a solution of the minimal element problem.This work was done while the author was with the Chongqing Institute of Architecture and Engineering, Chongqing, P. R. China.     

集值映射的对称向量拟均衡问题

研究向量集值映射的拟均衡问题的有效解,利用数值化方法与不动点定理,得到解的存在性定理.作为应用,得到广义向量鞍点,向量变分不等式与向量互补问题的存在性定理.     

向量变分和向量拟变分不等式解的存在性定理及应用

本文利用纯量化方法及ＫＫＭ技巧,研究向量变分和向量拟变分不等式解的存在性问题,作为应用本文还研究了向量互补问题及最优化问题解的存在性及其等价性刻划。本文结果改进和推广了已有的结果。     

一类集值映射的向量均衡问题

在拓扑向量空间内,研究了一类新的集值映射的广义向量均衡问题,利用KKM定理,证明了解的几个存在性定理,并讨论了解的性状.所得结果推广了近期一些作者的研究成果.     

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

Vector optimization problems with nonconvex preferences

In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. While this paper was being revised in September 2006, Professor Alex Rubinov (the second author of the paper) left us due to the illness. This is a very sad news to us. We dedicate this paper to the memory of Professor Rubinov as a mathematician and truly friend.     

On implicit vector equilibrium problems

In this paper, we introduce and study a class of implicit vector equilibrium problems, which includes a number of (scalar) implicit equilibrium problems, implicit variational inequalities, and implicit complementarity problems as special cases. By using KKM-Fan theorem, we prove some new existence theorems of solutions for this kind of implicit vector equilibrium problems in Hausdorff topological vector spaces. Our results extend and unify some corresponding results of several authors.     

Feasibility and Solvability for Vector Complementarity Problems1

The purpose of this paper is to discuss the feasibility and solvability of vector complementarity problems. We prove that, under suitable conditions, the vector complementarity problem with a pseudomonotonicity assumption is solvable whenever it is strictly feasible. By strengthening the generalized monotonicity condition, we show also that the homogeneous vector complementarity problem is solvable whenever it is feasible. At last, we study the solvability of the vector complementarity problem on product spaces.The authors thank Professor Franco Giannessi for valuable comments and suggestions leading to improvements of this paper.