近似积分

Intergral approximate calculation is widely used in many practical projects.

0引言积分的近似计算是实际工作中最常碰到的数学问题之一，例如在原子能科学技术领域和武器装备论证时会出现许多复杂的单重与多重积分。

The approximate calculation of integrals has a wide range of applications in many practical engineering, in fact a big problem of integrable system is the issue of high-dimensional points in the irregular region.

积分的近似计算在许多实际工程的应用十分广泛，可积系统的一大类问题事实上就是不规则区域上高维积分问题。

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。

Simple method based on contribution theory is proposed for congestion cost allocation.

与现有方法相比，该文方法的显著特点是：阻塞费用分摊可采用连续积分方法，而不是近似的离散数值积分方法。

In sphere approximate, GPS/Gravity-BVP is Neumann problem over exterior spherical domains. Then the natural boundary reduction, suggested by Feng and Yu, is applied to deal with the three dimensional problems. By expansion in spherical harmonics, the natural integral equations of harmonic problems over exterior spherical domains are obtained. Meanwhile, a numerical method for solving geodetic BVP is developed.

在球近似下将GPS-边值问题转化为Neumann外问题，用自然边界元法进行自然边界归化，得到自然边界积分方程，再求自然边界积分方程的数值解，与其他边界元方法相比，自然边界元法的计算量大大减小，并且具有很好的逼近性质，能有效地处理奇异积分，是求解GPS-边值问题有效方法。

The computing plots of the solutions from the two methods were inconsistent under the specified boundary-initial conditions, which stirred up the debate of the effectiveness of Maxwell's equations in lossy medium. It is found that the differences between the two numerical curves can be viewed as the Gibbs phenomenon caused by cutting off the high frequency components in computing the infinite integral of the numerical curve of Harmuth's solution.

通过比较两种解析解的数值曲线得出：两者的差异在于求解Harmuth解的数值曲线时，采取无穷积分的高频截断近似产生了吉布斯现象；在计算Harmuth解的数值曲线时不断增大积分上限，其数值曲线的振荡不断向不连续点处收缩，而且衰减加快，因此在求解Harmuth解的数值曲线时，若积分区间取为理想无限大，则可以预见相同计算精度下两种方法的电场解的数值曲线是一致的。

Results show that radial basis function and point interpolation methods possess Kronecker Delta function property, but the robustness is poor in some cases; the MLS approximation function does not possess Kronecker Delta function property, but it has good robustness. Differences among the three discretization schemes of meshless method are as follows:the collocation method requires no numerical integration and very little computational time while its robustness is poor; Galerkin method is not a truly meshless method due to the background meshes required for integration; the Petrov-Galerkin method is a truly meshless method and need numerical integration in each sub-domain, so it needs more computational time.

分析结果显示：径向基函数和点插值法均具有d 函数属性，但计算稳定性差；移动最小二乘近似函数不具有d 函数属性，但计算比较稳定；无网格方法中的3种离散方法不同之处在于：配点法不需要积分，计算量小，计算稳定性差；Galerkin方法需要借助背景网格进行积分，它不是真正的无网格方法；Petrov-Galerkin方法，是一种真正的无网格方法，它需要对每个子域进行积分，计算工作量较大。

LBIE, based on the local boundary equation, adopts the traditional moving least squares approximation which depends on only the values of the nodes in the domain of the problem or along its boundary. The whole process of integration is carried on over a local domain or its local boundary centered at the node in question. The local boundary equation can be rewritten to represent the values of the unknown function at the point of interest, and the essential boundary conditions can be directly and easily enforced by using the Green formula and the characters of the Dirac function.

它以局部边界积分方程为基础，采用移动最小二乘近似函数，从而只需要分布在问题域内及其边界上的节点的信息值，无需划分单元；整个积分是在以节点为中心的局部域及其边界上实现，所以不需要背景积分网格；借助于格林公式及Dirac函数的性质，将局部边界积分方程转化为所考虑点的未知函数的边界积分表达式，便于直接施加本质边界条件。

In this paper, firstly, the theory and the discretization schemes of LBIE and the concepts of MLS and GMLS are introduced; secondly, LBIE method is applied to solve the thin plate-bending problems and four typical examples are given to demonstrate the applicability and validity of the method. The results obtained from numerical examples agree well with the exact solutions and have an excellent rate of convergence.

本文首先介绍了无网格局部边界积分方程方法的理论基础和离散步骤，以及移动最小二乘近似法和广义移动最小二乘近似法等概念；其次将该方法初步应用于计算薄板的弯曲问题，建立了完整的积分公式，求解了四个典型算例，结果同精确解非常接近，且收敛快。

Since RKPM is a kind of Galerkin meshfree methods, in this thesis, penaltyfunction method is employed to enforce the essential boundary conditions,trapezoidal rule is used for the approximation of displacement field function, Gaussand Hammer quadratures are applied for the numerical integration of systemequation,self-adapting method is applied for confirming the support of the pointinterpolated.

针对 RKPM 是一种伽辽金型的无网格方法，本论文采用罚函数法引入位移边界条件，用梯形积分法则近似位移场函数，对系统方程进行积分时分别使用了 Gauss 积分和 Hammer 积分，采用自适应的方法确定计算点的定义域。

Objective To investigate the effects of interleukin-1 β converting enzyme gene on the biologic characteristics of ovarian cancer cells.

目的 探讨白细胞介素-1 β转换酶的表达，对卵巢癌细胞生物学特性的影响。

Campylobacter: This illness is the most commonly identified cause of diarrheal illness in the world.

弯曲：这种病是最常见的原因查明腹泻病，在世界上。