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编号:____________ 审定成绩:____________ 重庆邮电学院毕业设计(论文) 设计(论文)题目: 关于函数凸性定义的等价性及其判定的讨论单位( 二级学院): 计算机科学与技术学院学生姓名:桑松专业: 信息与计算科学班级: 430101 学号: 01430121 指导教师:李玲答辩组负责人: 填表时间: 2005 年6月重庆邮电学院教务处制重庆邮电学院本科毕业设计( 论文) II 关于函数凸性定义的等价性及其判定的讨论摘要凸分析是数学中相对年轻的一个分支。凸函数作为凸分析的主要研究对象, 在凸分析中占有重要地位,其定义、性质经常作为解决数学规划论、对策论、数理经济学、逼近论、变分学、最优控制理论这些方面的问题的工具被加以使用。本文深入地讨论了凸函数的几种不同定义的等价性, 判定定理及凸函数的应用。首先给出了凸函数的六个等价定义,包括三个常见的定义和另外三个等价定义。我们知道,在前三个定义中下(上)凸函数的本质是连接函数图形上任意两点的线段,处处都不在函数图形的下方(或上方) 。后三个定义中下(上)凸函数的本质是左差商不大于(不小于)右差商,左右差商当自变量差分减小时是不减(不增) 的。然后给出四个判定凸函数的定理及其证明, 并列出了凸函数的一些运算。最后举例说明凸函数的相关结论在不等式的证明、验证级数的收敛性等方面的应用。关键词:凸函数;等价定义;判定定理。重庆邮电学院本科毕业设计( 论文) On the discussion of the equivalent differences of convex function and their judgment ABSTRACT Convex analysis isa young branch in mathematics . As the main research object of convex analysis, convex function is very important . Its definitions, natures are ofen regarded as tools to solve the problems like the plan of mathematics , game theory , mathematics and physics economics, theory of approaching, optimum control theory respect and so on. This paper discusses the equivalent of several kinds of different definitions of convex functions in deeply, judging theorems and its applications. At first ,we get six equivalent definitions of convex function, including mon definitions and other three equivalent definitions . As we know, in the first three definitions , the essence of convex (concave) function is that everywhere at function below (or above) the figure when connect function wanton line segment for two acmes at the figure. The last three definitions , their essence of convex (concave) is left trader is not more above ( below ) than right trader, different traders is increasing (reducing) when the independent variable reduces. And then, we give four judging theorems of convex function and its proving. At last, illustrates some operstions of convex function in such aspects like the identifications of the inequality, convergence property of verification progre