一般方程

In the method,thematrix exponential is calculated precisely by using 2N algorithm and the integration offunctional matrix by parts is applied.

首先依据常微分方程的经典理论，运用指数矩阵的2N类精细算法及函数矩阵的分部积分法建立数值求解常微分方程的一般方法。

Subsequently, using the mass-energy relation, the general expression of the solution of the energy-curvature equation on the medium shell curve method is discussed, and the general expression of Binet equation about orbit of the celestial bodies motion is given.

随后应用质能关系探讨介质层壳弯曲方法中能量方程解的一般形式，给出天体运行轨道的一般性 Binet 方程，给出了行星近日点进动、光线弯曲的解析分析。

For the application of Mathematica in the differential geometry,this paper introduces how to processe the curve curvature and the torsion questions,the graphic expression of the curvature and the torsions,and draw curves by the curvature and the torsions. Curvature ; torsion ; solution curve ; osculating plane ; natural parameter ; natural equation

利用Mathematica，只要输入有关系统命令就可以得到期望的结果。1曲率和挠率及其图形意义假设R3上的曲线一般参数表示为r=r它的曲率和挠率计算公式是：κ=‖r′×r"‖‖r′‖3τ=‖r′×r‖我们知道，空间曲线r=r在一点(s=s0)邻近的近似方程为ξ=sη=120κs2ζ=160κ0τs3通过近似方程可以确定曲线在其基本三棱形的三个平面(即密切平面，法平面，从切平面)上的投

This thesis is divided into six parts. The first chapter is preface, the current status of research in the inverse problems for parabolic partial differential equations is reported; the second chapter is "regularization methods for numerical differentiation and their applications ", in this chapter we investigate many regularization methods from a viewpoint of regularization theory and algorithm, some applications in the inverse problems for parabolic partial differential equations are given; the third chapter is "spectral regularization methods". Based on Fourier analysis, within the framework of regularization theory, we apply the spectral methods to some ill-posed problems. Many numerical experiments are done in order to show the validity of the methods; the fourth chapter is devoted to wavelet dual least squares method and a revised wavelet method; in the fifth chapter,we combine finite difference method with method of lines and apply it to the backward heat conduction problem in time; in the sixth chapter "identification problems for unknown source ", the essence and the degree of two problems related to source identification are pointed out, at the same time, some numerical methods are reported.

本文分为六个部分，第一章前言简要分析了国内外抛物型偏微分方程反问题的研究现状；第二章数值微分的正则化及其应用从正则化理论和算法的角度出发，考察了许多正则化方法，还给出了数值微分在抛物型偏微分方程反问题的一些应用；第三章谱正则化方法是在Fourier分析的基础上，在一般正则化理论的框架下，给出了这种方法在各种不适定问题中的应用，数值实验表明谱方法是有效的；第四章研究了小波对偶最小二乘方法和改进的小波方法；第五章主要研究了有限差分方法结合线方法在时间反向热传导问题中的应用；第六章是未知源识别问题，主要指出了两类未知源问题的不适定程度和不适定本质，同时报告了一些数值方法。

To apply the method of Chapter Ⅱ to ring dynamics, we develop the streamline formalism to a canonical form in Chapter Ⅲ. The perturbation equations which describe the perturbed motion of streamlines are derived, with the general form of a near-conservative system.

为将上述映射方法应用到环动力学的研究，在第三章"流线表示法的正则描述"，我们研究该表示法的正则形式，导出描述流线运动的受振运动方程，该方程具有近保守系统的一般形式。

The second part deals with renormalization group equations in general gauge field theories. In this part, the complete set of two-loop renormalization group equations in general gauge field theories is presented.

第二个部分主要讲述了规范场理论中的两圈重整化群方程，我们给出了一组完整的一般规范场的两圈重整化群方程。

Based on them, we calculated precisely some enegy levels of P-wave and D-wave of a helium atom, presented and proved the general interdimensional degeneracies for a quantum N-body system, derived the general spherical spinor functions and the exact solution to the Dirac equation for the hydrogen atom, and proved uniformly the Levinson theorem for the Dirac equation in D-dimensions.

在此基础上，我们精确计算了氦原子P波和D波若干较低能级，建立和证明了量子N体系统不同维数状态间最普遍的简并，推导出任意 D 维空间广义球旋函数和Dirac方程氢原子的精确解，统一地提出和证明了任意 D维空间Dirac方程Levinson定理的一般形式。

It is proved that equilibrium solutions of this differential system are solutions to the complementarity problem and a numerical algorithm is given based on the numerical integration of the system of ordinary differential equations.

在一定条件下，证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法。

However, we can not get the film parameters directly because the ellipsometer equation is a transcendental equation. We can hardly get the analytical solution for film parameters from the measured ellipsometric parameters Ψ and Δ. Therefore, to find an algorithm to inverse the measured data Ψ and Δ becomes a primary but important problem.

但是通过椭圆偏振测量只能得到椭偏参数Ψ和Δ，由椭偏参数求解薄膜结构参数的椭偏方程是一个超越方程，很难得到精确的解析解，因此一般采用数值反演迭代不断逼近测量数据，将最优解作为测量结果。

Thus this paper first introduces a general approach to continuous element methods for one-dimensional structures,in which the continuous element methods is computed numerically and directly from the transfer matrix from the state vector equation of motion of a structure.

因此，介绍一维结构的连续单元法的一般方法：由结构运动方程建立状态向量方程和利用单元传递矩阵得到单元的刚度矩阵。

In the negative and interrogative forms, of course, this is identical to the non-emphatic forms.

。但是，在否定句或疑问句里，这种带有"do"的方法表达的效果却没有什么强调的意思。

Go down on one's knees;kneel down

屈膝跪下。。。下跪祈祷