R3中双弦幂积分

本文研究R3中凸体的双弦幂积分,利用线偶的运动密度,得到了一些性质及球体上的部分双弦幂积分值,并由此解决了R3中一几何概率问题.     

Hardy-Hilbert积分不等式的一个新推广及其应用

通过引入一个形如x1 x(x∈[0, ∞))的幂指函数建立了带权的Hardy-Hilbert积分不等式的新推广.并证明了系数(2)(sinπp)是最佳值.作为应用,给出了Hardy-Littlewood积分不等式的一个推广.     

对一类定积分问题的再研究

文献[1-3]中对一类积分进行了讨论。本文给出一种递推方法并推广到二次幂(Fejer积分)至五次幂的情况,最后给出了六次幂猜想的结果.     

凸体的弦幂积分公式

本文研究了高维欧氏空间中凸体的弦幂积分问题.利用积分的方法,获得了凸体弦幂积分的几个积分公式.     

三大著名不等式的拓广与深化

利用一个分式型的双向积分不等式 ,将 H lder不等式、H.Minkowski不等式、Schl milch不等式(幂平均不等式 )三大世界著名不等式进行拓广与深化 ,使对此问题的研究更具深刻性、系统性 .     

区间上似凸函数的幂平均不等式

定义了区间上似凸函数的概念.利用定积分的性质把凸函数的幂平均不等式Mα(f ) 相似文献   

对Fejer积分的推广

利用对积分∫π0sinnx/sinx dx的递推解法,方便地得到Fejer积分的结果,并且对三次幂的情况进行推广.     

利用重积分证明定积分不等式

利用重积分与定积分的关系,举例说明利用重积分证明定积分不等式。     

若干有关欧阳不等式的非线性积分不等式和离散不等式

获得几个非线性的积分不等式及离散不等式．它们和近期Ｐａｃｈｐａｔｅ［１］中欧阳亮不等式［２］的推广有关．作为特殊情形还导出了一些具有幂非线性的新不等式．为说明结果的有用性，讨论了某个非线性差分方程解的有界性．     

利用二重积分证明积分不等式

有时将一元函数的积分问题转化为二元函数的二重积分问题 ,会给解题带来方便 .本文通过几个范例说明利用二重积分证明积分不等式的方法 .例 1　设函数 f (x)与 g(x)在 [a,b]上连续 ,证明 Cauchy-Schwarz积分不等式(∫baf (x) g(x) dx) 2≤∫baf 2 (x) dx∫bag2 (x) dx　　证明　记积分区域 D =[a,b]× [a,b],利用定积分与积分变量符号无关的性质等 ,有(∫baf (x) g(x) dx) 2 =∫baf (x) g(x) dx∫baf (y) g(y) dy = Df (x) g(x) f (y) g(y) dxdy≤ D12 [f2 (x) g2 (y) f2 (y) g2 (x) ]dxdy=12 ∫baf 2 (x) dx∫bag2 (y) dy 12 ∫baf …     

等腰梯形域内两点间平均距离

本文研究了凸域内两点间平均距离的问题.利用广义支持函数和凸域的弦幂积分的方法,获得了计算凸域内两点间平均距离的一般公式,并将公式推广到等腰梯形域,得出了等腰梯形域内两点间平均距离的计算结果.     

二重积分变量代换公式证明的探讨

在一般教材中二重积分变量代换公式的证明通常采用几何的方法,也有部分数学分析教材给与了严格的分析证明,但证明不便直观的几何说明.本文对二重积分变量代换公式进行了一些探讨,给出了一个较简洁直观的分析证明方法.     

The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities

This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.     

一类双中心平面二次系统的Abel积分零点个数

利用Abel积分与第一、第二型完全椭圆积分,本文研究一类具有两个中心奇点的平面二次系统在n次小扰动下的Abel积分零点个数上界问题,得到了较小的上界估计．     

Critical perspectives of pedagogical approaches to reversing the order of integration in double integrals

This paper presents some critical perspectives regarding pedagogical approaches to the method of reversing the order of integration in double integrals from prevailing educational literature on multivariable calculus. First, we question the message found in popular textbooks that the traditional process of reversing the order of integration is necessary when solving well-known problems. Second, we illustrate that the method of integration by parts can be directly applied to many of the classic pedagogical problems in the literature concerning double integrals, without taking the well-worn steps associated with reversing the order of integration. Third, we examine the benefits and limitations of such a method. In our conclusion, we advocate for integration by parts to be a part of the pedagogical conversation in the learning and teaching of double integral methods; and call for more debate around its use in the learning and teaching of other areas of mathematics. Finally, we emphasize the need for critical approaches in the pedagogy of mathematics more broadly.     

Numerical evaluation of the two-dimensional partition of unity boundary integrals for Helmholtz problems

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.     

函数粗集生成的粗积分(英文)

The concepts of the lower approximation integral,the upper approximation integral and rough integrals are given on the basis of function rough sets.Based on these concepts,the relation of the lower approximation integrals,the relation of the upper approximation integrals,the relation of rough integrals,and the double median theorem of rough integrals are discussed.Rough integrals have finite contraction characteristic and finite extension characteristic.     

重积分的对称性质

在重积分的计算中,经常要用到对称性质.而限于时间和篇幅,现行大多数教科书上都不加讨论的引用对称性质.这就使初学者感到重积分的对称性质不好掌握.本文通过重积分的变量替换公式,给出重积分对称性质的一个一般定理和两个推论,最后给出一个具体实例.