单积分

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 arithmetic of the point charges integral is similar to that of nuclear attraction potential.

前者的计算类似于核吸引积分的单电子积分，后者可以通过点群对称性操作从原子簇内的积分矩阵元得到。

The final result indicates that in the expression of the completeness relation,the integral path is along the real axis from -∞ to ∞ but runs over near the origin,which is contrary to the Cauchy principal value appearing in previous works.

同时得到展开式中的积分是沿实轴从-∞到∞，但在原点附近将从上方绕过。这不同于过去所得的C auchy主值积分。为最明确显示这一差别，在单孤子情况下又用平方Jost函数的显式，直接作了验证。

By decomposing the problem system into an extended saturated soil system and a virtual pile system, through the strain compatibility condition between the extended saturated soil system and the virtual system at an arbitrary depth z along the pile, and combined the Laplace transform and its convolution, a Fredholm intergral equation of the second kind of the single pile in saturated layered transversely isotropic soils is established. The axial force and diaplacement of the pile are obtained by solving the Fredholm intergral equation.

将求解体系看作是饱和土体和虚拟桩的叠加，利用沿桩身任意一点z处饱和地基与虚拟桩的应变协调，并结合Laplace变换及其卷积性质，建立了饱和成层横观各向同性地基中竖向受荷单桩问题的第二类Fredhlom积分方程，对该积分方程进行求解可得桩的真实轴力、位移。

In order to solve the problem,We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold. The travel time are expressed as polynomials of the horizontal offset between the two points, and the single-square-root operator in frequency-wavenumber domain are expressed as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddle-point method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量(地面点到地下相点的水平距离)的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

In order to solve the problem, We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddlepoint method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量（地面点到地下相点的水平距离）的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

Secondly,the one-step and multistep methods of the precise integration algorithms arederived by means of the integration of functional matrix by parts and interpolation function,and some recursion schemes suitable for precise integration are obtained.

然后，以函数矩阵的分部积分法和插值函数为工具，推导出精细积分的单步法和多步法，给出了多种精细积分的递推格式。

In this paper, we generalized the reverse Shebyshev inequality and obtained the integral inequality of two vector function s, whose monotone of correspondence elements is reverse.

推广了反向Chebyshev不等式，得到了对应分量的单调性相反的连续向量函数的积分不等式和模的单调性相反的连续向量函数的积分不等式。

Furthermore, by using the corresponding inverse Laplace integral transform and numerical solution of integral equation, the displacement and axial force as well as the shear stress of the pile are obtained.

对变换域内的积分方程采用Schapery逆变换方法得到时域内单桩的近似积分方程。数值求解所得的积分方程可以得到桩的沉降、轴力、剪力和孔压。

Like the green jersey, the mountains jersey also requires consistency, but not in the same way. Most of the time, a rider wanting the dots will get in a breakaway on one of the first big mountain stages and win most of the points. The defense of it usually requires another long breakaway, but polka dot contenders don't have to concentrate on every single bonus like the sprinters do, because the big climbs are worth so much more than the small ones.

要想取得圆点衫，往往需要长距离的攻击，形成单飞之后逐一拿下爬坡赛点，与冲刺积分不同的是，爬坡积分会因爬坡赛点等级的不同而不同，一个S级的赛点会比4级的赛点积分高出很多，所以圆点衫的争夺要到山地赛段开始之后才会真正展开。

This one mode pays close attention to network credence foundation of the businessman very much.

这一模式非常关注商人的网络信用基础。

Cell morphology of bacterial ghost of Pasteurella multocida was observed by scanning electron microscopy and inactivation ratio was estimated by CFU analysi.

扫描电镜观察多杀性巴氏杆菌细菌幽灵和菌落形成单位评价遗传灭活率。