零函数

We always reject zero as an eigenfunction on the ground of physics.

根据物理上的理由，我们总是剔除把零作为本征函数。

The influences of the shock thickness and Alfven waves on the particle acceleration by diffusive shock waves are numerically studied through solving one-dimensional diffusive equation including the second-order Fermi effect. It is shown that the spectral index of the energetic particles strongly depends on the shock thickness. For example, the spectral index increases from 2.1 to 3.7 in the low energy range of 3-10 MeV and from 2.5 to 5.0 in the high energy range of 20-60 MeV as the thickness increases. The spectral index decreases from 4.3 to 3.1 as the particle injection energy increases. The spectral index decreases from 4.0 to 1.8 at the quasi-steady stage with the enhancement of the compression ratio from 2 to 4. The results indicate that under the influence of Alfven waves, the energetic particle spectrum at lower energy becomes flat and the spectral index decreases from 2.5 to 0.6 in the low energy range of 3-10 MeV and from 11.6 to 5.0 in the high energy range of 20-60 MeV. At the same time, the rollover energy reaches 19.6 MeV. The spectral index decreases from 5.8 to 2.9 as the energy density of Alfven waves increase. All these results are basically consistent with the theoretical models, as well as the observations of typical energetic particle events.

通过数值求解包含二阶费米加速的一维扩散方程，探讨在准平行激波条件下激波厚度和级联阿尔芬波对粒子加速的影响，研究粒子分布函数的演化与激波厚度和阿尔芬波强度的内禀关系，计算结果表明：(1)考虑激波厚度时，谱指数明显依赖于激波厚度，随着厚度从0.32增大到2.56，低能端(3-10MeV)谱指数逐渐从2.1增加到3.7，高能端(20-60MeV)谱指数从2.4增大到5.0，能谱逐渐变软；当初始注入粒子动量增大1.3倍，质子能谱指数从4.3减小到3.1，且与零厚度激波加速的谱指数差值缩小；厚度不变时，随着压缩比从2增加到4，准稳态分布时低能端(3-10MeV)粒子能谱指数逐渐从4.0减小到1.8谱变硬；(2)在级联阿尔芬波的影响下，随着时间的增大，粒子在低能处(3-10MeV)的谱指数从2.5减小到0.6高能端(20-60MeV)谱指数从11.6减小到5.0，能谱变硬，拐点能量值从7.5MeV增大到为19.6MeV；随着波的能量密度增大，谱指数从5.8减小到2.9，这表明阿尔芬波强度越大，加速效率越高，通过与激波厚度解析结果和高能粒子事件的观测能谱比较发现两者是一致的，说明数值模拟结果是可靠的。

The expm1 function returns the value of Ex-1, where E is Euler's constant (approximately 2.7183) and x is the number passed to it.

expm1函数的作用是：计算 e（自然对数的底，其数值大约等于2.7183）的指数；它返回了exp 1的值，甚至当number的值接近零也能计算出准确结果。

The quasi-analytical expressions of the relative concentrations at the mobile region and the immobile region are deduced respectively by using hypergeometric equation, hypergeometric function and the method of Laplace transform under the initial concentration zero and the first or second boundary condition of a semi-infinite one-dimensional space.

并分别在初始浓度为零，半无限一维空间内第一类边界条件和第二类边界条件下，利用超几何方程、超几何函数和Laplace变换法推导出了可动区和不动区溶质相对浓度的准解析表达式。

By using the method for positive term resolution of equations of higher degree, all non-zero real roots of a real coefficient equation of higher degreewere obtained by determiningthe abscissas of intersection points of two monotonically increasing concave functions in the first quadrant of a planar rectangular coordinate system.

用高次方程正项分解方法，将求解实系数高次方程非零实数根的问题，转化成求解两单调上升凹函数在平面直角系第一象限内交点横坐标的等价问题；给出了基于共享存储多指令流多数据流并行计算模型求解任意实系数高次方程全部实数根的大范围收敛性异步并行迭代算法，并分析了算法计算的复杂程度。

Mastery: The Z-Transform, The Region of Convergence for the Z-Transform, The Inverse Z-Transform, Properties of the Z-Transform, System Function Algebra and Block Diagram Representations.

教学内容： Z 变换； Z 变换收敛域；逆 Z 变换；由零极点图对傅立叶变换进行几何求值； Z 变换性质；几个常用 Z 变换对；利用 Z 变换分析和表征线性时不变系统；系统函数的代数属性与方框图表示；单边 Z 变换。

The Laplace Transform; The Region of Convergence for Laplace Transforms; The Inverse Laplace Transform; Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot; Properties of the Laplace Transform; Analysis and Characterization of LTI Systems Using the Laplace Transform; System Function Algebra and Block Diagram Representations; The Unilateral Laplace Transform..

教学内容：拉普拉斯变换；拉普拉斯变换收敛域；拉普拉斯反变换；由零极点图对傅立叶变换进行几何求值；拉普拉斯变换性质；常用拉普拉斯变换对；用拉普拉斯变换分析和表征线性时不变系统；系统函数的代数属性与方框图表示；单边拉普拉斯变换。

Mastery: The Laplace Transform, The Region of Convergence for Laplace Transforms, The Inverse Laplace Transform, Properties of the Laplace Transform, Some Laplace Transform Pairs; Analysis and Characterization of LTI Systems Using the Laplace Transform, System Function Algebra and Block Diagram

基本要求：掌握拉普拉斯变换定义，拉普拉斯变换收敛域，拉普拉斯反变换，拉普拉斯变换性质，用拉普拉斯变换分析和表征线性时不变系统，系统函数的代数属性与方框图表示；熟悉常用拉普拉斯变换对，单边拉普拉斯变换；了解由零极点图对傅立叶变换进行几何求值。

Mastery: The Laplace Transform, The Region of Convergence for Laplace Transforms, The Inverse Laplace Transform, Properties of the Laplace Transform, Some Laplace Transform Pairs; Analysis and Characterization of LTI Systems Using the Laplace Transform, System Function Algebra and Block Diagram Representations;Roth Criterion;Mason Equation of the Signal Flow Graphs.

基本要求：掌握拉普拉斯变换定义，拉普拉斯变换收敛域，拉普拉斯反变换，拉普拉斯变换性质，用拉普拉斯变换分析和表征线性时不变系统，系统函数的代数属性与方框图表示，罗斯判别，信号流图的梅森公式；熟悉常用拉普拉斯变换对，单边拉普拉斯变换；了解由零极点图对傅立叶变换进行几何求值。

The Laplace Transform; The Region of Convergence for Laplace Transforms; The Inverse Laplace Transform; Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot; Properties of the Laplace Transform; Analysis and Characterization of LTI Systems Using the Laplace Transform; System Function Algebra and Block Diagram Representations; The Unilateral Laplace Transform.;Roth rule ;Signal Flow Graphs.

教学内容：拉普拉斯变换；拉普拉斯变换收敛域；拉普拉斯反变换；由零极点图对傅立叶变换进行几何求值；拉普拉斯变换性质；常用拉普拉斯变换对；用拉普拉斯变换分析和表征线性时不变系统；系统函数的代数属性与方框图表示；单边拉普拉斯变换；罗斯准则；信号流图。

If you are unfortunate enough to the lovelorn, please tell me, I will help you out, really, please contact me!

如果你不幸失恋了，请告诉我，我会帮助你摆脱困境，真的，请联系我啦！

China's plan to cut energy intensity by 20 percent and pollutant discharges by 10 percent between 2006 and 2010 is a case in point.

中国计划在2006年到2010间降低20%的能源强度和减少10%的主要污染物排放，就是一个这样的例子。