积分论

Secondly,we firstly study the properties of functions with values in a uni-versal Clifford algebra 〓,and we obtain the following very important basictheorems in universal Clifford analysis:Cauchy's integral formula,Cauchy's inte-gral theorem,the mean value theorem,the three versions of the maximum mod-ulus theorem,the Taylor's expansion,the Laurent's expansion and the residuetheorem etc..All of these results generalized the classical results.

第二，本文所讨论的各种函数性质以及所得的结果都在泛Clifford代数〓上所做的工作，它一方面包含了从前在泛Clifford代数〓上所做的工作，所得到的结果更广泛、更漂亮、更自然，另一方面，本文也是迄今为止第一次建立起来了在泛Clifford分析中与经典函数论相对照处基础地位的LR正则函数在特异边界上的Cauchy积分公式、Cauchy积分定理、平均值定理、极大模原理的三种表达形式、Taylor展式、Laurent展式留数定理等深刻的结果。

For the Riemann boundary value problems for the first order elliptic systems , we translates them to equivalent singular integral equations and proves the existence of the solution to the discussed problems under some assumptions by means of generalized analytic function theory , singular integral equation theory , contract principle or generaliezed contract principle ; For the Riemann-Hilbert boundary value problems for the first order elliptic systems , we proves the problems solvable under some assumptions by means of generalized analytic function theory , Cauchy integral formula , function theoretic approaches and fixed point theorem ; the boundary element method for the Riemann-Hilbert boundary value problems for the generalized analytic function , we obtains the boundary integral equations by means of the generalized Cauchy integral formula of the generalized analytic function , introducing Cauchy principal value integration , dispersing the boundary of the area , and we obtains the solution to the problems using the boundary conditions .

对于一阶椭圆型方程组的Riemann边值问题，是通过把它们转化为与原问题等价的奇异积分方程，利用广义解析函数理论、奇异积分方程理论、压缩原理或广义压缩原理，证明在某些假设条件下所讨论问题的解的存在性；对于一阶椭圆型方程组的Riemann-Hilbert边值问题，利用广义解析函数理论、Cauchy积分公式、函数论方法和不动点原理，证明在某些假设条件下所讨论问题的可解性；广义解析函数的Riemann-Hilbert边值问题的边界元方法是利用广义解析函数的广义Cauchy积分公式，引入Cauchy主值积分，通过对区域边界的离散化，得到边界积分方程，再利用边界条件得到问题的解。

The main subjects of the volume include:- spectral analysis of periodic differential operators and delay equations, stabilizing controllers, Fourier multipliers;- multivariable operator theory, model theory, commutant lifting theorems, coisometric realizations;- Hankel operators and forms;- operator algebras;- the Bellman function approach in singular integrals and harmonic analysis, singular integral operators and integral representations;- approximation in holomorphic spaces.

卷包括：主要课程-定期微分算子和延迟方程谱分析，稳定控制器，傅立叶乘数；-多变量算子理论，模型论，换位解除定理，coisometric变现；- Hankel算子和形式；-算子代数；-在奇异积分和谐波分析，奇异积分算子和积分表示贝尔曼函数方法；-在全纯空间近似。

Using difference method, the first-order and the second-order differential equations of fractional derivative can become the first-order ordinary differential equation, and the high precision direct integration can be used for the solution.

所论方法首先引入差分格式，将含有分数阶导数的一阶和二阶微分方程变为一阶的常微分方程，然后再用精细积分方法逐步积分进行求解。

The theorem of Lebesgue integral is one of the most important parts in real variable function. About Lebesgue integral,there are different definition modes.

O引言Lebesgue积分是实变函数论的中心论题，L积分取代Riemann积分是大学数学教育改革发展的必然。

By applying the law of energy transformation and conservation in electromagnetism to a sinusoidal steady state linear exchange network,the aauthors obtain a new expression of the law in which the merits of such energy forms as resistance,capacitance,self induction and mutual induction of four kinds of elements are embodied.

利用式中统一体现的电阻、电容、自感和互感4种元件能量形式的优点，以及网络现代场论关于元件复阻抗率和积分形式的复阻抗等概念，分别给出4种元件具体的积分形式复阻抗。

In recent years, Feng Kang has advanced a more natural and direct redu-ction, i. e. the reduction via Green's formula and Green's function.

近年来冯康又提出一种更自然而直接的归化，即从Green公式及Green函数出发将微分方程边值问题化为边界上的含有广义函数意义下发散积分有限部分的奇异积分方程，这种归化在各种边界归化中占有特殊地位，被称为正则边界归化，本文将这一理论应用于重调和椭圆边值问题，研究了其正则归化的性质，并通过利用Green函数、Fourier分析及复变函数论方法等不同途径求出了在上半平面、单位圆内部、单位圆外部三种区域的Poisson积分公式及正则积分方程，其离散化可用于实际计算。

Its theoretic foundation is integration geometry, random theory of set, topology, modern probability, algebra, theory of set, graph theory, and so on.

其最基本的数学理论基础是积分几何和随机集论，同时还涉及到拓扑学、现代概率论、近代代数、集合论、图论等一系列数学分支。

New properties and improvements on VSI. Combining the density processof local absolute continuous measure, properties of the martingale space H1 and semi-martingale space, and the closed image theorem of functional analysis (see the proofs ofTheorem 2.4.5 and Theorem 2.4.10), we obtain the general form of Girsanov Theorem forsemi-martingale vector stochastic integral .

特别是利用局部绝对连续测度的密度过程、鞅空间H1与半鞅空间的性质以及泛函分析中的闭图像定理(见定理2.4.5与定理2.4.10的证明)获得了一般形式的半鞅向(来源：Ab6BC论a1文网www.abclunwen.com)量随机积分的Girsa-nov定理，它对于随机分析的理论与随机积分的应用都具有重要价值。

The biggest advantage of this mean is that don"t need to computer the perturbation matrix element by using zero level wave function, even don"t need to give out high rank amendatory result using infinite series to sum like the ordinary perturbation theory. Thus, it not only avoids many complicated integral operation, but also obtains high rank amendatory result that it is very difficult to get or cant get.

这种方法最大的优点就是不需要用零级波函数来计算微扰矩阵元，更不需要像普通微扰论那样用无穷级数求和给出高阶修正的结果，这样一来，不仅避免了许多繁复的积分运算，而且求得了普通微扰论很难得到或不可能得到的高阶修正的结果。

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。