函数的积分

Secondly, by applying inverse Fourier integral transform to the displacement, and uniting the constitutive and geometrical equations, the analytical expression of stress in Laplace domain were derived. Thirdly, by defining dislocation density functions, the Cauchy singular integral equations were obtained according to the boundary condition and interface connection conditions, and the problem was reduced to algebraic equations by Chebyshev orthogonal polynomial. Based the these, the unknown coefficient of the algebraic equations can be solved by Schmidt method. Finally, the time response of dynamic stress intensity factor and energy release rate are obtained by inverse Laplace transform.

首先，利用积分变换方法，推导出粘弹性层的控制方程组；其次，引入位错密度函数，并结合边界条件和界面连接条件，导出反映裂纹尖端奇异性的Cauchy型奇异积分方程组，然后，应用Chebyshev正交多项式化奇异积分方程组为代数方程组，并采用Schmidt方法对其数值求解，最后，经过Laplace逆变换，求得动态应力强度因子和能量释放率的时间响应。

Monotone functions; functions of bounded variation and Borel measures; absolute continuity; differentiation and integration; convex functions; semicontinuity; Borel sets; properties of the Cantor set.

单调函数；囿变函数；绝对连续性；微分和积分；凸函数；半连续性；博雷尔集类；康托集的性质

This text will talk about some new methods about the limit of disjunction function in the boundary and the operation of lead number and definite integral.

分段函数是函数问题中难点，本文就分段函数在分界点的极限，导数、定积分的运算问题探讨一些新方法。

The research achievement are acquired as follows: 1From mathematical models for inverse scattering in two dimensional homogenous media including Dirichlet, Neumann, Robin,all kinds of probable mixed boundaries and cracks, direct and inverse scattering are discussed, and ill-posed integral equation and indicator function method are formulated for the diverse of boundary identification. It is shown that the kernel of the integral equation characters the boundary of scatterer, which is determined by solving it by virtual of regularity method, meanwhile, some numerical tests are given.

本研究取得了如下研究成果： 1）在二维均匀介质逆散射各种边界识别的数学模型（包括Dirichlet，Neumann，Robin，各种可能的混合边界问题，裂纹问题）下，分别考虑了正散射问题和逆散射问题，推导了上述各种边界识别的不适定积分方程以及指示函数方法，由于积分方程的核充分表征了散射物的边界，由此说明只要利用正则化方法求解该积分方程，就可以确定散射物的边界，并给出了一些数值实验。

For the joint density function of 2-D continuous random variables, a new function can be obtained by upper limit-variable integral, and then, by normalization of the new function according to the property of density functions, a general marginal density function was constructed.

对二维连续型随机变量的联合密度函数，在一定要求下变上限积分得到新函数，依据密度函数性质进行规范化，从而构造出一个广义的边缘密度函数。

From our results we know that the average error of the Lagrange interpolation sequence and the Hermite interpolation sequence based on the Chebyshev nodes in the 1-fold integrated Wiener space equal weakly to the average error of their corresponding optimal approximation polynomial in the 1-fold integrated Wiener space,and as a kind of information-based operation,they have simple form and their recover functions are polynomials,in the 1-fold integrated wiener space,their average error equal weakly to the corresponding minimal information radius whose permissible information operators class is function values.

通过我们的结果可以知道，基于第一类Chebyshev多项式零点的Lagrange插值算子列和Hermite插值算子列在1-重积分Wiener空间下的平均误差弱等价于相应的最佳逼近多项式在1-重积分Wiener空间下的平均误差，并且作为形式简单且恢复函数为多项式的一种信息基算法，其在1-重积分Wiener空间下的平均误差弱等价于相应的以函数值计算为可允许信息算子的最小平均信息半径。

This paper discusses the integrability of Riemann integral systematically: By analyzing the common characters of a lot of integral calculus, it abstracts the concept of Riemann integral and discusses its integrability of Riemann integral and then gets integrable functions.

摘要本文较为系统地讨论了积分的可积性：通过分析诸多积分概念的共性，抽象定义了积分并详细讨论了其可积性，得出了可积函数类。

Stability of theoretical and numerical solution to stiff integro-differential equa-tionsand to stiff integro-differential equations of integral limit which depends on state function yare discussed in this paper.

中文摘要：本文研究刚性积分微分方程初值问题及积分限依赖于状态函数y的刚性积分微分方程初值问题的理论解及数值解的稳定性和收敛性。

In the fourth paragraph, how the theory of the damping to forging hammer foundation advances at home and abroad, the research and application of the implement for damping to forging hammer, the structure of damping to forging hammer and the calculation methods for forging hammer foundation vibration system, which are used at home nowadays, are particularly introduced.In the fifth paragraph, aiming at direct damping under anvil block of forging hammer, the author bases a mechanics model of two-degree with damping, and optimizes the vibration parameters both in frequency field and in time field, then finds out the new simple method of optimizing the vibration parameters of the damping device under anvil block of forging hammer.

第五章针对锻锤砧下直接支承式减振基础，建立了锻锤减振系统的二自由度有阻尼振动力学模型，然后用傅立叶变换法求出振动传递率的频率关系，用直接积分法求出砧座和基础位移的时域响应，并分别对频域函数和时域函数进行参数优化，最终建立起频率函数—时域函数—优化参数间的相互关系，从而找到了锻锤砧下隔振器参数优化的简便新方法。

The equation of time constant of an integrating sphere is derived.

用分析方法推导了积分球时间常数的一般表达式、积分球对δ函数形式的功率波形的变换表达式、积分球对任意光功率波形的变换表达式及其反变换表达式。

The concept of equivalent rotationally rigidity is offered and the formula of rotationally rigidity is obtained.

主要做了如下几个方面的工作：对伸臂位于顶部的单层框架—筒体模型进行分析，提出了等效转动约束的概念和转动约束刚度的表达式。

Male cats normally do not need aftercare with the exception of the night after the anesthetic.

男猫通常不需要善后除了晚上的麻醉。