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.     

一个周期函数不等式及其应用

本文研究一类特殊的周期函数,利用Fourier级数的方法,获得了关于这类周期函数的一个积分不等式.此函数积分不等式等价于著名的关于平面两凸集混合面积的Minkowski不等式.     

探讨积分上限函数的应用

积分上限函数是一种特殊但很重要的函数,它有着很广泛的应用.本文主要讨论并总结了积分上限函数的几大应用.     

一类延迟积分方程的渐近概周期解

积分方程解的性态是积分理论中一个重要而又基本的问题.其中关于方程的周期解、概周期型解的存在性问题更是具有重要的理论意义和应用价值.已有文献中,有的延迟是常数,而有的延迟虽为函数,但对延迟做了是概周期函数的假设.本文在某些延迟积分方程的概周期解的存在性的基础上,并利用渐近概周期函数的定义和不动点理论,讨论了一类延迟是渐近概周期函数的积分方程的渐近概周期解的存在性.     

变上限积分的特性及应用

变上限积分是一个很重要的函数,在我们的微积分教材中,用它来证明了微积分基本公式.许多其它问题用它处理也是既方便又简单.这些主要用到它的两条特性:①变上限积分在积分区间上是可导的,且其导函数就是被积函数;②变上限积分和被积函数比起来,其可导的阶数大1.下面举几例说明其特性及应用.     

积分上限函数相关问题的探讨

探讨积分上限函数的七个相关问题,给出对应例题和解析过程,旨在使学生对积分上限函数有更深的理解和掌握.     

|sinx|原函数的直接计算法

利用变限积分函数和周期函数的性质,给出了求|sinx|原函数的一种直接计算法.     

浅谈一元变上限积分的等价无穷小

本文探讨了当一元变上限积分上限和被积函数均为无穷小量时,分别的用另一等价无穷小替换.在一定条件下,替换前的变上限积分与替换后的变上限积分等价无穷小.     

关于一元含参量变上限积分的等价无穷小

本文应用等价无穷小量理论,对一类一元含参量变上限积分,在满足一定条件下,分别对积分限和被积函数做相应的等价无穷小量替换.,使替换后的一元含参量变上限积分与替换前的一元含参量变上限积分是等价无穷小.     

微积分学中辅助函数的应用

利用积分的方法以及积分上限函数,适当作辅助函数,能较方便地证明某些等式和不等式.     

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。     

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.     

On the solvability of a complete two-dimensional hypersingular integral equation

We study the solvability of a complete two-dimensional linear hypersingular integral equation that contains a hypersingular integral operator in which the integral is understood in the sense of Hadamard finite value as well as an integral operator in which the integral is understood in the sense of principal value, an integral operator with a weakly singular kernel, and an integral-free term. We consider smooth solutions in the class of functions that have Hölder continuous derivatives outside a neighborhood of the boundary. We prove the Fredholm alternative and estimate the norm of the solution in a special metric.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.     

The behavior of a singular integral with the Hilbert kernel near a point where the density of the integral is weakly continuous

We study the behavior of a singular integral with the Hilbert kernel near a point where the continuous density of the integral does not satisfy the H?lder condition and, as a result, the integral, possibly, diverges.     

A new boundary integral equation for molecular electrostatics with charges over whole space

In this paper, a new integral equation of electrostatics is proposed as an integral form of a basic dielectric continuum model, which is traditionally represented in a form of Poisson differential equation. As an application in protein simulations, the new integral equation is reduced to a second kind Fredholm boundary integral equation on the interface between the solute and solvent regions for a piecewise constant permittivity function, together with two new integral expressions for the electrostatics within the solute and solvent regions. The new integral equation and expressions work for any charge problem over the whole space (including the one with charges on the interface). This valuable feature is verified numerically for a dielectric sphere model with a point charge inside, outside, or on the sphere in this paper.     

On Set-Valued Differentiation and Integration

Both the notion of differential of a set-valued mapping and the concept of a set-valued integral are introduced in such a way that, on the one hand, the class of differentiable mappings is rather extensive, while on the other hand, the concept of integral includes as special cases Dinghas' approach and Pontryagin's alternated integral. At the same time, the differential of an integral with respect to the upper limit of integration equals in some sense the integrant. Thus the differentiation of a set-valued mapping can be interpreted as an operation inverse to set-valued integration. Both the differential and the integral are defined via approximation by quasi-affine mappings introduced below. The class of quasi-affine mappings is compared with the classes of affine, semi-affine, eclipsing and multi-affine mappings.     

关于重积分与线面积分内容体系的探讨

本文将二重积分、三重积分、第一类曲线积分及第一类曲面积分统一为多元数量值函数的积分，并且用第一类曲线、曲面积分定义第二类曲线、曲面积分。     

Estimates on level set integral operators in dimension two

Both oscillatory integral operators and level set operators appear naturally in the study of properties of degenerate Fourier integral operators (such as generalized Randon transforms). The properties of oscillatory integral operators have a longer history and are better understood. On the other hand, level set operators, while sharing many common characteristics with oscillatory integral operators, are easier to handle. We study L²-estimates on level set operators in dimension two and compare them with what is known about oscillatory integral operators. The cases include operators with non-degenerate phase functions and the level set version of Melrose-Taylor transform (as an example of a degenerate phase function). The estimates are formulated in terms of the Newton polyhedra and type conditions.     

积分问题中分段估值的解题方法

细致分析和研究积分不等式、积分值序列的极限、含参变量积分的极限等若干方面诸多问题的解决过程,提炼出其中蕴含的一类解题方法--分段估值法.