解析函数

In part one,because there is no long-range order in one-dimensional(1D)system,generalized second-order Green's function theory isused to 1D spin-1 antiferromagnetic spin chain.The Haldane gapin the excitation spectrum appears naturally in the analytical re-sult;When k is zero,there is a gap 2Δ.

第一部分中，在长程序为零的条件下，我们应用推广的二级格林函数理论，讨论了自旋为1的反铁磁一维链，Haldane能隙在解析结果中自然地得出，在k＝0时，能谱中存在一个2Δ能隙，Δ＝0.5J。

Using Density Function Theory we receive the ground states of GaN and GaN2 are X 、X2B2 respectively.Murrell-Sorbie analytical potential energy functions of GaN have been derived by using ab initio data and the least-square fitting method, and harmonic frequency, force constant and spectroscopic data also have been calculated.

基于量子化学计算得出GaN和GaN2 基电子态及其离解极限之后，利用计算数据和最小二乘法得到了GaN 的M-S解析势能函数，并计算出各态的谐振频率、力常数和光谱数据。

Analytical solutions for displacements of the viscoelatic half-space were obtained using Green's function method for various moving loads,including constant and harmonic point,line and area load.All velocity cases including the subsonic,transonic and supersonic velocities were considered.

首先采用Green函数法求解黏弹性半空间体在各种移动荷载模式作用下的动力响应的解析解，包括恒常和简谐移动点源、线源和面源荷载。

First of all,we have given some of the basic concepts of differential equations, described the constant coefficient linear ordinary differential equation solution, for a class of second-order variable coefficient linear ordinary differential equation initial value problem, an approximate solution, the method is first unknown function of a definition for N sub-interval, and then in between each district within a constant coefficient ordinary differential equations similar to the replacement, the solution has been the problem as similar to the original analytical solution, and then gives a detailed second-order change order coefficient of linear homogeneous ordinary differential equation solution examples, the examples of the approximate method proposed in this paper is valid.

首先给出了微分方程的一些基本概念，讲述了常系数线性常微分方程的解法，针对一类二阶变系数线性常微分方程初值问题，提出了一个近似解法，本方法是先对未知函数的一个定义区间作N等分，然后在每一个小区间内用一个常系数常微分方程近似替换，所得到的解作为原问题的近似解析解，随后详细给出了一个求二阶变系数齐次线性常微分方程的解的实例，该实例说明本文提出的近似方法是有效的。

A general class of solutions to nonlinear scalar equations with static cylindrical symmetry is obtained in the form of a hyperbolic function series.

利用双曲函数级数的技术，研究了静态轴对称非线性标量方程的解析解。

However, the solutions of the Cauchy problem of the nonlinear hyperbolic system generally develop singularities in finite time even if the initial data are small and smooth.For this reason, solutions must be found in the space of discontinuous functions. Therefore, one can not directly use the classical analytic techniques that predominate in the theory of partial differential equations of other types. Instead, we construct approximate solutions via the singular perturbation methods. By using the compactness of the approximate solutions, we can obtain the existence of the solutions of original system.

然而一般而言，即使是在初值很小且光滑的情况下，非线性双曲型守恒律的Cauchy问题的解在有限时间内也会出现奇异性，为此，我们必须在不连续函数空间中寻找上述问题的解，因此对这类问题的研究，我们也不可能直接利用在其它类型的偏微分方程中占主导地位的解析方法来解决问题，而是通过奇异扰动法构造近似解，由近似解的紧性得到原问题解的存在性。

This thermal structure is approximate to a hyperbolic tangent profile, which allows theoretically that a mode-2 ISW propagates along the thermocline with permanent form (Benjamin, 1967 and Davis and Acrivos, 1967, etc).

这种密度垂直结构近似於双曲正切函数，理论上可以求得一个在斜温层传递且形状不变的第二斜压模之内孤立波的解析解(Benjamin， 1967 和 Davis and Acrivos， 1967，等人)。

The large deformation for incompressibility rubber cylinder under inner pressure is considered by a kind of new rubber materials strain energy function.

针对一类新的橡胶材料应变能函数，推导了受内压作用下不可压缩橡胶圆柱的大变形公式，给出了位移、应力的解析表达式。

Because the workpiece is in dynamic balance, the resultant moment acting on it should be zero. Introducing the moment from lapping tool and press head a moving differential equation is built. In this equation only workpiece rotating speed is unknown. It means that workpiece rotating speed can be got by this equation. Since there is a transcendental function in integral function of the equation an analytic result can not be got, only digital result can be got by a computer. Thus a change rule of workpiece rotating speed with lapping parameters is known, e.g. the radius of press disc ball socket, radius of press head ball, the distance between the workpiece rotating axis and lapping tool rotating axis, workpiece radius, modulus of elasticity of the press head and press disc, friction coefficients between the lapping tool and workpiece, and between the head and disc, pressure between the lapping tool and workpiece and pressure between the press head and press disc.

因工件处于动态平衡状态，其上所受到的来自于磨具和压头的力矩应相等，将二力矩表达式联立，建立工件运动微分方程，该方程中只有工件的旋转角速度是未知量，因此可以求出工件旋转角速度，但该方程积分式中含有超越函数，得不出解析解，只能通过计算机求出数值解，得出工件旋转角速度随各研磨参数的变化规律，如压盖球座半径、压头的球头半径、工件回转中心相对磨具回转中心的偏心距、工件半径、压头和压盖材料的弹性模量、磨具与工件间的摩擦系数、压头压盖间的摩擦系数、磨具与工件间的压强、压头压盖间的压力等。

We have considered the analytic property of the integrand in the complex plane for the calculation of the Casimir force when the metamaterials are concerned, and based on the theory of the Casimir effect for real media situation, we obtained the formula of the Casimir force for the planar structures containing metamaterials.

通过考察分析复频率空间内的含特异材料的介质平面板间Casimir力被积函数的解析性质，根据实际的介质平板间Casimir作用力的计算理论，推导得到包含特异材料的平面板间Casimir作塑料托盘用力的计算公式。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失