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Results on existence and uniqueness of the solution for the diffraction problem are obtained by variational method and integral equation method, respectively.

考虑一类具有周期系数的Helmholtz方程，它是一类衍射光栅问题的数学模型，研究这类问题的数值解法。

The integrand in the variational principle is rewritten in symmetrical form for duality variables.

变分原理被积函数可导向对于对偶变量为对称的形式。

The exchange energy for the electron transfer process at large internuclear distance may be calculated using either perturbation or variational method.

中文摘要与电子传递相关的远程交换能可用微扰法或变分法计算。

This thesis consists of five chapters. At first, We introduce the study history as well as present situation on KKM theory. Secondly, We discuss the invariability of transfer open valued multimap under the restriction of upper and lower semicontinuity on generalized convex spaces. Then, We obtain new KKM type theorems from the classical KKM theorem by applying upper and lower semicontinuity. We also obtain intersection theorems by applying the KKM type theorems for a multimap from a topological space to another topological space. Further, We obtain Ky-Fan type coincidence theorem by applying known variational results on the KKM type theorems. Finally, We give an application to new theorems.

本文共分五章，首先回顾了KKM理论的研究历史和现状，其次讨论了在一般化凸空间上转移开值映射在上、下半连续映射下的不变性，然后在上、下半连续映射下，以古典的KKM定理为基础得到新的KKM型定理，并利用从一个拓扑空间到另一个拓扑空间的集值映射及KKM型定理得到相交定理，进一步利用已知的KKM型定理的变形结论得到Ky-Fan型重叠定理，最后给出新定理的一个应用。

Especially in carbonate formation, owing to the intricate interstice, the variational size, shape of cave, and the different character of filling, it is almost impossible to get all-around cognition by a single way.

尤其对碳酸盐岩储层来说，由于孔隙空间非常复杂，缝洞的大小、形状、类型、充填物性质的变化很大，用单一的方法全面了解碳酸盐岩储层发育情况几乎是不可能的。

The main contents include: Some preliminary theory (introduction to Sobolev spaces and variational formulations for differential equations); finite element methods for one-dimensional elliptic problems; the construction methods for general finite elements; error estimates for interpolation operators and inverse inequalities for finite element spaces; a priori and a posteriori error estimates for the finite element method for high-dimensional elliptic problems; some typical spectral methods for partial differential equations; error analysis for the spectral approximation for some linear and nonlinear partial differential equations.

主要内容有：准备知识（Sobolev空间的基本概念和主要结果，微分方程的变分描述）；一维椭圆型方程有限元方法；一般有限元的构造；插值算子误差估计和逆不等式；高维椭圆型方程的先验、后验误差估计；求解偏微分方程的几类谱方法；线性与非线性问题谱逼近的误差分析等。

Inverse variational inequality, monotone mapping, proximal point algorithm

逆变分不等式，单调映射，临近点算法

First, UWPBL model(The University of Washington Planetary Boundary Layer Model) is used to retrieve typhoon's sea level pressure from sea surface winds data of SeaWinds which aboard on QuikSCAT and these strength results are analyzed. Then the variational method is applied to decompose the sea surface winds field into two parts: the vortex and irrotational environmental flow field, the UWPBL model is used again to retrieve ShanShan and YaGi typhoons' sea level pressure from the vortex flow field, the results of typhoon's central positions are analyzed by comparing the results with the results of direct retrieving, NCEP's data and typhoon annual reports. It turns out that the accuracy of the typhoon's sea level pressure central position has been obviously improved, This method offers a new idea for the study of typhoon from microwave scatterometer data.

本文首先利用QuikSCAT散射计获取的海面风场资料，利用华盛顿大学行星边界层模式对台风海平面气压场进行反演，并对其强度进行了分析；然后对获取的海面风场，采用变分方法对风场进行分解，分解出无旋和有旋风场，对分解出的有旋风场，再利用UWPBL模式对珊珊和魔羯台风的海平面气压场进行了反演，对所得结果与通常直接反演、NCEP海平面气压场及台风报文资料进行一一对比，结果表明：新的反演定位的方法对台风海平面气压场中心气压的定位精度有了大幅度的提高，此种定位的方法为散射计资料在台风中的实际应用开辟了一条新的思路。

The first section introduces some basic concepts[9].The second section introduces Schmidt"s method of proving isoperimetric inequality [2].The third section introduces Hurwitz"s method [9].The fourth section takes from my tutors lecture notes.The method is due to REILLY [10].Finally,I make use of variational method to prove the circle among the curves of length L encloses the biggest area on the plane.

首先介绍了平面上的一些基本概念[9]，其次介绍了文献[2]中Schmidt证明等周不等式的方法，再次介绍了文献[9]中Hurwitz证明等周不等式的方法，然后介绍了导师吴发恩整理的文献[8]中REILLY的方法证明平面上的等周不等式[10]，最后我利用变分的方法证明了平面上等长的曲线围成的面积最大时为圆。

In this paper, a novel method called the variational iteration method is proposed to solve perturbation problems.

本文提出了求解摄动问题的一种新方法——变分迭代算法。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。