随机逼近

The random spline plays very important role in the approximation and fitting of random function.

随机样条在对随机函数的逼近与拟合中都有重要作用。

Comparing with deterministic system, the point of bifurcation in stochastic Duffing-van der Pol system is moved as increasing the intensity of random parameter.

进一步的推广证实正交多项式逼近法在实际问题中有广阔的应用前景，它可以解决含有多个随机参数的非线性随机动力学系统的分岔问题。

In the fourth chapter, firstly it introduces the Boolean functions algebraic expressions of the 2-value clock-controlled stop-and-go generator and Gunther generator. It reveals the balanced property of the two kinds of Boolean functions, and studies the Walsh cycle spectrum and the autocorrelation function. It also obtains the coincidence rate of their output sequences with affine sum of some bits of input sequences, and analyzes their ability of resisting the best affine approximation cryptanalysis and differential cryptanalysis. Secondly, we properly present a new definition of the Best Affine Approximation, namely BAA on the Boolean vector functions, followed by the spectral characteristic of such defined BAA attacks through using the decomposition formula of the union distribution for random variables. A lower bound of such BAA attacks is proposed. Finally, we also study the spectral characteristic of the second kind of nonlinearity of Boolean vector functions, followed by a higher bound of such nonlinearity. Furthermore, the limited relationship between the second kind of nonlinearity of Boolean vector functions and the linear structure of the linear combination of every component is analyzed.

在第四章中，首先给出了2值密钥流"停走生成器"和"衮特生成器"中实际存在的布尔函数的代数表示，揭示了这两类布尔函数的平衡性，随后研究了它们的Walsh循环谱和自相关函数等，得到了它们的输出序列与输入序列中的某些bit的仿射项的符合率，分析了它们抵抗最佳仿射逼近攻击和差分攻击的能力；其次，我们合理地给出了布尔向量函数最佳仿射逼近的新定义，利用布尔随机变量联合分布的分解式考察了相应的谱特征，并给出了布尔向量函数与其最佳仿射逼近的符合率的一个下界；最后，我们还考察了布尔向量函数第二类非线性度的谱特征，给出了布尔向量函数第二类非线性度的一个上界，并揭示了布尔向量函数第二类非线性度与其各个分量的线性和的线性结构之间存在的制约关系。

The Chebyshev polynomial approximation is used to investigate the stochastic bifurcation of van der Pol system with two-sided barriers.

利用随机光滑动力系统的Chebyshev正交多项式逼近方法，研究了双边约束条件下随机van der Pol系统的分岔现象。

The random system is reduced to its equivalent deterministic one by Chebyshev polynomial approximation, and the response of the stochastic system can be obtained by the deterministic methods.

首先用Chebyshev多项式逼近法将随机系统化成与其等价的确定性系统，然后通过等价确定性系统来探索随机Duffing-Van der pol系统的对称破裂分岔现象。

The stochastic Bonhoeffer-Van der Pol system is first transformed into an equivalent deterministic system using the Chebyshev polynomial approximation, so that the problem of controlling stochastic chaos is reduced to the problem of controlling deterministic chaos in the equivalent system.

首先运用Chebyshev多项式逼近的方法，将随机Bonhoeffer-Van der Pol系统转化为等价的确定性系统，使原系统的随机混沌控制问题转换为等价的确定性系统的确定性混沌控制问题，继而可用Lyapunov指数指标来研究等价确定性系统的确定性混沌现象和控制问题。数值结果表明，随机Bonhoeffer-Van der Pol系统的随机混沌现象与相应的确定性Bonhoeffer-Van der Pol系统极为相似。

In the following part,we define a couple of new approximate operators and a couple of corresponding new approximate measure operators,in this way,inner measure belief function and lower probability all can be considered as the particular cases of this lower approximate measure .

通过定义一对新的逼近算子以及相应的一对逼近测度，使内测度、信任函数、随机集上的下概率均可表示为此下逼近测度结构下的一种特例。

In chapter 10, we propose the random weighting approximation method for distribution of sample mean under simple random sampling in one-dimensional finite population.

第十章 将随机加权法首次应用到简单随机抽样方案下的样本均值分布逼近问题中，从数字特征角度说明了随机加权方法在抽样理论中的可用性，从而拓广了随机加权法的应用范围。

Based on Bellman stochastic nonlinear dynamic programming, we develop an optimal control algorithm for the discrete stochastic system with conditional Markov structure.

基于Bellman随机非线性动态规划法，提出了具有条件马尔科夫跳变结构的离散随机系统的最优控制方法，应用随机变结构系统的性质对最优控制算法进行了简化处理，并将后验概率密度函数用条件高斯函数来逼近，针对一类具有条件马尔科夫跳变结构的线性离散随机系统，给出了其逼近最优控制算法。

Firstly, an empirical approximation model of stochastic programming is obtained by replacing the probability measure of original program with empirical probability measure. Sequentially, the constrained stochastic programming is transformed into an equivalent unconstrained stochastic programming. Finally, using the epi-convergence theory, the almost everywhere upper semiconvergence of optimal solution set of empirical approximations for stochastic programming is obtained.

首先通过经验概率测度替代初始规划的概率测度得到随机规划的经验逼近模型，然后将带有约束的随机规划问题转化成与其等价的无约束的随机规划问题，最后利用上图收敛性理论，给出了随机规划经验逼近最优解集的几乎处处上半收敛性。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了