解析函数

Asymmetrical dynamic propagation problems under the action of variable loads and (P^3t/x^2) located at the origin of the coordinates respectively, are resolved by the measures of self-similar functions, and the universal expressions of analytical solutions to stresses, displacements and stress intensity factors are gained.

采用自相似函数的方法对坐标原点分别受到变载荷Px/t、Pt^3/x^2作用下的非对称动态扩展问题进行求解，获得了应力、位移和动态应力强度因子解析解的一般表达式。

All equilibrium points of the original system are stable equilibrium points in the adjoint system.

对于一个非线性自治动力系统，存在一个伴随系统，该伴随系统是一个梯度系统，原始系统的所有平衡点都是伴随系统的渐近稳定平衡点，并且每一平衡点存在解析形式的Lyapunov函数。

For a nonlinear autonomous dynamic system, there exists an adjoint system, which is a gradient system. All equilibrium points of the original system are stable equilibrium points in the adjoint system.

对于一个非线性自治动力系统，存在一个伴随系统，该伴随系统是一个梯度系统，原始系统的所有平衡点都是伴随系统的渐近稳定平衡点，并且每一平衡点存在解析形式的Lyapunov函数。

Fall suppose in Swerling Ⅱ target, derivation detect below environment of ADT of threshold value of the MGF of quadrature generation function that this algorithm issues in even setting, average judgment, much target the maths of aiguille of the false alarm in probability Pd and clutter brim environment is analytic expression.

在SwerlingⅡ型目标假设下，推导了该算法在均匀背景下的矩产生函数MGF、平均判决阈值ADT、多目标环境下检测概率Pd和杂波边缘环境中虚警尖峰的数学解析表达式。

With the algebraic result in form of piecewise function expressions, the excitation and propagation processes of stress waves, including the longitudinal wave, the transverse wave, and Rayleigh-wave, are discussed in detail.

基于该分段函数形式的代数结果，纵波、横波、Rayleigh波等应力波成分在弹性表面的激发和传播过程得到详尽分析，其中很多结论是已有的解析积分结果或者数值计算结果不曾得到的。

However,in at early work of Dolan and Jackiw, after the analytic continuation ofthe imaginary time thermal propagator, the result was not the 2 × 2 matrix thermalpropagator in the real time formalisms, but only the 1-1 component of the matrix,which caused singularities like higher order δ function's.

然而在Dolan和Jackiw的工作中，由虚时热传播子解析延拓得到的结果不是实时方法中的二阶矩阵形式的热传播子，而只是其中的1-1矩阵元，从而会导致象高阶δ函数这样的奇异性，本文指出，这个结果是来源于一个隐含的假设，即认为实时方法的背景时空是一个明氏时空。

After all the angular integrations and spin summations completed, the needed parameters in the non-relativistic wave functions have been acquired using the variational method and so the functions themselves have been established in this way.

在第三章中，利用对角和法则，导出了类硼离子基态及第一激发态非相对论能量的解析表达式，完成涉及到的角向积分和自旋求和后，利用变分原理求出了非相对论性波函数中的待定参数，从而确定了非相对论性波函数，并获得类硼离子基态及第一激发态非相对论能量。

The potential energy contours describe the structure character of PdPb2 ground state molecule rightly. Furthermore, molecular static reaction pathway based on this potential energy function is investigated.

使用多体展式理论导出了势函数中的参数进而给出PdPb2分子基态势函数的解析表达式，其势能面准确地复现了PdPb2分子的两个稳定构型( C 2 V 和 C ∞ v )及其能量关系。

As same as the above-mentioned reason, because theconditions at the inside straight boundary of the binarycomposite material can not be easily satisfied, here, anelastic circular inclusion with large radius is used to replacethe upper half space of the binary composite material, so the〓mary binary composite mater ial problem can be changed〓the scattering problem of a circular elastic inclusion andto the scattering problem of a circular eastic inclusion and〓circular cavity in an infinite space to the steady incident〓wave.

与前面两个问题的理由一样，同样，由于问题中双质复合材料的内部直边界上的边界条件不易解析满足，为此，本文采用了一个半径很大的弹性实夹杂来代替问题中双质复合材料的上半空间部分，而将原来的双质复合材料问题转化成了无限大空间中一圆形弹性夹杂和一圆孔对稳态P波的共同散射问题，根据该问题的特点写出了弹性夹杂内外波场中波函数的一般形式解，利用该问题中夹杂和圆孔处的位移和应力的连续条件，并采用复数傅立叶级数展开技术将其化为一个仅包含问题波函数中未知系数的无穷线性代数方程组，在满足一定计算精度的前提下，通过有限项截断进行近似求解，进而讨论了双质复合材料中圆孔边界处的动应力集中系数随不同入射角、圆孔的不同位置、以及入射波的不同无量纲波数的变化和分布情况。

The propagation of Gaussian bea ms through a paraxial ABCD optical system with hard edge aperture is studied. In the treatment the rectangular function is expanded into a sum of complex Gaussia n functions, thus the approximate analytical propagation expression can be deriv ed.

对高斯光束通过有硬边光阑近轴 ABCD光学系统的传输进行了研究，采用将矩形函数展开为复高斯函数的技巧，推导出了近似的解析传输公式。

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