特征函数

However, we can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and zero everywhere else.

然而，我们可以仅单独测量一个运动的自由粒子的位置用多数情况下特别位置X和零其它地方的一波函数来产生一个特征态。

After we perform the measurement, obtaining some result x, the wavefunction collapses into a position eigenstate centered at x.

在我们完成测量后，获得某些结果x，波函数坍塌为以x为圆心的位置特征态。

Conditions of integral chain can be closed.

基于函数S粗集和F粗积分，本文研究粗积分的动态特征，定义粗积分链和粗积分环的概念，给出上积分链，下积分链定理，粗积分链定理，以及积分链的存在性定理，讨论了积分链的可闭条件，得到积分环定理，粗积分环定理。

This paper reviews and summarizes the progress in this field as follows: the mathematical model and the difficulty of the analytical solutions is introduced; discusses the latest progress of the numerical solutions is discussed; the finite difference method, the Godunov method, the Boltzmann method, and the KFVS method of one-dimensional dam-break problem are introduced; the TVD method, the discontinuous finite element method, the finite volume method and characteristic method of two-dimensional dam-break problem are discussed; the applicability, the advantage and the disadvantage of each method are analyzed; and the progress in the numerical calculations of flow of fluid with free surface are presented.

回顾和总结了国内外对溃坝水流演进问题的研究进展：介绍了溃坝水流的数学模型及解析解法存在的困难，进而讨论了数值解法的最新进展；论述了求解溃坝水流一维问题的有限差分法、近似黎曼解的Godunov格式法、Boltzmann法、KFVS法和二维问题的TVD格式法、间断有限元法、有限体积法、特征线法，并分析了各种方法的适用范围和优缺点，及讨论了限制函数的使用；介绍了利用自由水面追踪方法计算溃坝水流的研究进展，并根据目前存在的不足和实际工程的需要，提出了进一步研究的方向和发展趋势。

In order to help for evaluating properties of soil in interaction more precisely, the paper applies elastic theory solution on different modulus in tension and compression for vertical load into semi-infinite elastic body. Through quoting Heaviside function, the solution of elasticity theory with different modulus in tension and compression is established founded on the solution of elasticity theory for the material incapable in tension and Mindlinsolution for equal modulus.

为使地基土在共同作用中的行为特征评价更为合理，本文应用了半空间问题竖向荷载下地基土的拉压模量不同弹性理论解，该理论解是通过引入Heaviside函数，利用不能承受拉应力材料的弹性理论解并结合拉压模量相等的经典Mindlin解而获得。

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