定积分不等式几种典型证法

通过几道常见的定积分不等式证明例题,从不同角度分析、研究定积分不等式的特点,归纳总结出构造辅助函数,利用重要积分公式、性质、定积分中值定理及重要不等式等证明定积分不等式的七种典型方法．     

一个积分不等式的九种证明方法

积分不等式是微积分学中一类常见而又重要的不等式,其证明方法多种多样．分别用定积分的定义、积分变限函数、积分第一、第二中值定理、微分中值定理等九种方法证明积分不等式∫0^1xf（x）dx≥1/2∫0^1f（x）dx（其中f（x）在[0,1]上连续而且单调递增）,借此介绍证明积分不等式的几种常用的方法．     

积分不等式证明技巧解析

基于定积分不等式的证明是高等数学教学中的一个难点的认识,重点解析定积分不等式证明过程中所涉及的知识点,并对不等式证明技巧进行分析与归纳,阐述定积分不等式证明的基本思路和解题技巧．     

两类积分不等式的一个应用

针对北京市大学生数学竞赛试题中一道积分不等式证明题，在重述原参考解答后，重点介绍Steffesen不等式和Chebyshev不等式，并分别运用它们证明该积分不等式．此外，利用积分第二中值定理也可证明该积分不等式。且方法更为简洁．     

从一个积分不等式证明到一般性结论的推广

采用积分的区间可加性和保序性对一个积分不等式进行了证明,该证明虽然比经典证明复杂,但是从其证明过程中可探索出更加一般的积分不等式.     

两个积分不等式的推广

应用交换积分次序方法和微分中值定理等原理,对两个积分不等式进行拓展,使得这两个积分不等式的适用范围更广,证明过程更为简洁.     

用积分法证明一类不等式

由于不等式的形式是多种多样的,所以证明不等式的方法可以因题而异来选择,关于不等式的证明,中学课本主要介绍了比较法、分析法、综合法与数学归纳法．本文主要讨论用积分的方法证明一类不等式．     

抽象与具体函数积分不等式的证明

在区间I上连续、单调的抽象函数的积分不等式证明的基本思路是适当进行积分变换、分拆积分区间，使不等式恒等变形等手段，使能应用函数的单调性质。具体函数的积分不等式的一般证明方法是把被积函数适当缩放、求出最值(或上下确界)等。     

关于Pan—Yang不等式的新证明

曲率积分不等式是研究平面曲线的演化问题的重要组成部分,Pan—Yang在研究一类缩短流时得到一个关于曲率积分的不等式．本文主要利用傅里叶分析的方法给出了此不等式的一种简单的证明方法．     

对一道积分不等式题的再思考

利用提升维度的方法并结合几何图形直观分析,给出一道一元函数积分均值不等式的新证明,并将原不等式推广至形式较为对称的不等式,使得原不等式成为新不等式的特例.最后证明新不等式与函数单调递减的定义等价.     

关于 C-Pettis 积分的一个控制收敛定理

In this paper, we give the Riemann-type extensions of Dunford integral and Pettis integral, C-Dunford integral and C-Pettis integral. We discuss the relationship between the C-Pettis integral and Pettis integral, and prove a controlled convergence theorem for the C-Pettis integral.     

Integral Equations in Measure Spaces

In this paper, we consider a class of integral equations in measure spaces, and the corresponding integral inequalities. Special cases are Volterra type integral equations and Gronwall type integral inequalities. We give different necessary and su.cient, and only su.cient conditions which together with the Lipschitz condition imply the existence and the uniqueness of solutions of the considered integral equations. We study the successive approximations for the considered integral equations. We derive estimates for the solutions of the studied integral equations and integral inequalities. Submitted: June 20, 2000?Revised: July 10, 2001     

Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

非线性非完整系统相对于非惯性系动力学的积分理论

本建立非线性非完整系统相对于非惯性系动力学的积分理论。首先，由这种相对运动的Ｒｏｕｔｈ方程给出系统的第一积分；其次，分别利用系统的循环积分、能量积分降阶运动方程，得到推广的Ｒｏｕｔｈ方程和推广的Ｗｈｉｔｔａｋｅｒ方程；再次，建立这类系统运动的正则方程和主分方程，并由第一积分构造系统的积分不变量；然后，给出系统的Ｐｏｉｎｃａｒｅ－Ｃａｒｔａｎ型积分变量关系和积分不变量。最后，给出一系列推论。     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results

For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.     

共轭A-调和张量的双权Hardy-Littlewood不等式

本文证明了共轭A-调和张量的局部双权积分不等式,此结果类似于共轭调和函数的古典Hardy-Littlewood不等式.作为局部结果的应用,还证明了John域上的共轭A-调和张量的全局双权积分不等式.     

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems

In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.     

多元整系数多项式因式分解的一种理论和算法

本文将ｔ（ｔ是大于２的整数）元整系数多项式看成为系数为ｔ－２元整系数多项式的二元多项式，建立了多元整系数多项式因式分解的一种新理论，进而得到了分解多元整系数多项式的一个有力的算法。