必要条件

The necessary and sufficient conditions for the semidirect product of S and T to be Clifford eventually semigroup are given.

给出了两个左正则拟半群S和T的半直积S×αT和圈积SωXT是左正则拟半群的充分必要条件。

Using operator-theoretic-methods, we give some characterizations for a dual generalized frame of a generalized frame in a separable Hilbert space H. We also prove a result concerning two strongly disjiont generalized frames.

本文研究了可分的Hilbert空间H中的广义框架，运用算子理论方法，研究了可分的Hilbert空间H中广义框架的性质，给出了广义框架的对偶广义框架的一些刻画，并且证明了两个广义框架是强非交的一个充分必要条件。

Under generalized cone-subconvexlikeness for vector valued mappings in locally-convex Hausdorff topological vector spaces, by using separation theorem for convex sets, optimality necessary conditions are derived.

在局部凸Hausdorff拓扑线性空间中，在近似锥－次类凸假设下，利用凸集分离定理得到了最优性必要条件。

This paper discusses convergence of Ishikawa iteration sequence and existence of fixed points for set-valued nonexpansive mapping in uniformly covex Banach space, and the conditions are shown which guarantee the convergence of the iteration sequence to a fixed point.

讨论了δ集值非扩张映象在一致凸Banach空间中不动点非空的充分必要条件与Ishikawa迭代序列的收敛性及确保迭代程序收敛到不动点的条件，所得结果是单值非扩张映象的推广和发展。

Negative strain rate sensitivity is proved to be a necessary condition for the occurrence of the serrated flow.

负的应变速率敏感性只是锯齿流变产生的必要条件。

Chapter four studies the chaotic responses in a system consisting of simple pendulum and harmonic oscillator under bounded noise excitation. Firstly, the Melnikov function of the two-degree-of-freedom system under Hamiltonian perturbation is derived. The essential condition of the autonomous system for the probable onset of chaos is obtained, the Poincare maps of the system under small Hamiltonian perturbation and the effect of increasing perturbation on the Poincare maps are studies. Then for the non-autonomous system under damping and harmonic or bounded noise excitation, the largest Lyapunov exponent and Poincare maps are calculated. From the analysis of the largest Lyapunov exponent, the critical criterion for the onset of chaos, and the conclusion that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity parameter of Wiener process increases are obtained. The result from the analysis of Poincare maps is in agreement with that obtained from the Largest Lyapunov exponent. The effect of varying damping coefficient and intensity parameter of Wiener process is also analyzed.

第四章研究了有界噪声激励下的两自由度单摆—谐振子系统的混沌运动，首先推导了该两自由度系统仅在Hamilton扰动下的Melnikov函数，得到该自治系统可能产生混沌的必要条件；研究了该系统在小的Hamilton扰动和增大摄动情形下的Poincare截面；然后对有阻尼、谐和或有界噪声激励下的非自治系统数值计算了其最大Lyapunov指数和Poincare截面；从Lyapunov指数分析得到了这个两自由度系统产生混沌运动的临界条件及产生混沌的临界激励幅值随Wiener过程强度参数值的增大而增大的结论，Poincare截面分析的结果亦符合Lyapunov指数分析的结论；研究了Wiener过程强度参数、阻尼系数变化对Poincare截面的影响。

Reproduction is the sine qua non for the propagation of species and continuation of life.

再生产是必要条件为种类的生活的传播和继续。

There are others, of course, but better roads and airports are the sine qua non for catching up with China.

当然还有其它因素，但改善公路和机场则是赶上中国的必要条件。

A necessary and sufficient coindition for the existence of positive solutions of a class of singular boundary value problems of higher order differential equations is given by means of the method of lower and upper solutions with the maximum principle.

摘要利用上下解方法和极大植原理给出了一类高阶微分方程的奇异边值问题正解存在的充分必要条件。

We obtain some neccessary and sufficient conditions of reverse order laws for g-inverse and reflexive g-inverse of matrix products on an arbitrary skew field.

获得了任意体上矩阵乘积的g一逆和自反g-逆的反序律的充分必要条件。

Singer Leona Lewis and former Led Zeppelin guitarist Jimmy Page emerged as the bus transformed into a grass-covered carnival float, and the pair combined for a rendition of "Whole Lotta Love".

歌手leona刘易斯和前率领的飞艇的吉他手吉米页出现巴士转化为基层所涵盖的嘉年华花车，和一双合并为一移交&整个lotta爱&。

This is Kate, and that's Erin.

这是凯特，那个是爱朗。