等价函数

Limit the type of function is more extensive and complex, involving a bounded function, infinitely small amount, equivalent to infinitesimal, the continuity of function, and many other content.

函数极限的类型较为广泛、复杂，涉及到有界函数，无穷小量，等价无穷小，函数的连续性等多方面的内容。

In presented method, the correlated variables are transformed into uncorrelated variables equivalently, and the performance functions in the correlated variables space are transformed into those in the equivalent uncorrelated variables space, in which the reliability sensitivity of the system failure probability with respect to the distribution parameters of the uncorrelated variables can be solved by the moment method for the uncorrelated variables. Finally the equivalent relationship between the correlated variables and the uncorrelated variables, the chain rule for derivative are employed to obtain the reliability sensitivity of the system failure probability with respect to the means of the correlated variables.

在所提的方法中，首先将相关变量空间等效转化为不相关变量空间，并将系统多个失效模式的功能函数转化为不相关变量空间的功能函数，进而利用矩方法得到多模式系统失效概率对不相关变量空间基本分布参数的灵敏度，最后利用相关变量空间与不相关变量空间的等价关系及复合函数求导法则，求得多模式系统失效概率对相关变量空间中变量均值的灵敏度。

By level function,ε-subdifferential, normal cone and dual function, global optimal solutions for a convex function in locally convex topological vector space and a few of equivalent forms on it are obtained. Maximum points are characterized.

讨论局部凸拓扑向量空间中凸函数的全局最优解，用水平函数、ε-次微分及法向锥和共轭函数等工具刻画其最大值点，得到最优解的几个等价形式。

Secondly,the deterministic equivalence problem of theprobabilistic inequality is discussed,and the expression formula of thedeterministic equivalences for several commonly used probabilitydistributions,such as normal,exponent,uniform,β and Γ distributions etc.,are given for the linear function with respect to random variables,which solvesthe problem of the deterministic equivalence for the linear-function-class.

其次，本章讨论了概率不等式的确定性等价问题，就随机变量的线性函数类，给出了与几种常用的概率分布，如正态分布、指数分布、均匀分布、β分布、Γ分布等等，相关的确定性等价问题。

Based on a kind of orthogonal complete functions in L2[0,1], another one (called V-system) which has equivalence relation with U-system is created. It is a kind of new and effective mathematical tool on signal multi-resolution analysis. Besides that V-system keeps the favorable properties of U-system and can accurately reconstruct geometry information which expressed in piecewise polynomials by finite sum of functions in V-system, it has more special properties, such as simple construction, abundant levels, convenience of application, quick calculation, local support, and so on. V-system is the generalization of Haar orthogonal functions, and to be a class of wavelet base. It has a good future for some applications in the fields of wavelet analysis and signal processing.

从L2[0，1]空间的一类正交完备函数系U系统出发，构造了另一类与之等价的正交完备函数系，称之为V系统，它是信号多分辨分析方面的一种新型的、有效的数学工具。V系统不仅保持了U系统的优良特性，对多项式表达的几何信息能够做到有限项精确重构，并且较之U系统，它更有结构简单、层次分明、计算快捷、局部支集等特点，应用起来将更加灵活方便。V系统可以看作是Haar函数系的推广，是一类小波基，在某些数字信号处理及小波分析问题中有良好的应用前景。

Robust estimation, RLS Estimation, M estimation, Independent observations, Correlated observations, Function model, Random model, Outliers analysis, Outliers Confirmation, Outliers Disposal, Infectant Distribution, Equivalent weights, Equivalent variance-covariance

稳健估计，稳健最小二乘估计， M 估计，独立观测，相关观测，函数模型，随机模型，粗差分析，粗差定位，粗差处理，污染分布，等价权，等价方差-协方差

In chapter 2, first of all,we characterize subspace V_0 of multiresultion analysis {V_j}_~ based on invariant subspace. Subsequently, we consider the equivalent conditions among orthogonal multiwavelet Ψ=(Ψ_1,Ψ_2,...Ψ_r)~T, subspace, basis and dimension of subspace, properties of filter function matrix P are dealt with.

第二章讨论了多尺度分析{V_j}_~中子空间V_0的性质，进而讨论了多尺度分析生成的多重正交小波Ψ=(ψ_1，ψ_2，…，ψ_r)~T和子空间V_0以及基和维数之间的等价关系，分析了滤子函数矩阵P的性质，最后给出了r阶矩阵函数P生成尺度函数Φ的充分条件。

In this paper the Karnaugh map for simplifying logic functions is used in computing canonical covers based on the equivalence between the theory of functional dependencies and logic algebra.

本文依据函数依赖理论与逻辑代数对应部分的等价性，引入化简逻辑函数的卡诺图法求函数依赖集的规范覆盖。

The forth algorithm is the same as the third one with an addition of finite termination criteria. Although the latter two algorithms have only linear rate of convergence, they are especially suitable for large-scale and sparse problems, with features of simple formula, small storage, sparsity preservation and easy implementation.

前者借助NCP函数把互补问题转化为等价的非光滑方程组，再用带参数的光滑方程组近似这些非光滑方程组，最后用牛顿型方法求解所得到的光滑方程组，希望通过光滑参数趋于零得到原来互补问题的解；后者基于等价不动点格式，构造了一个光滑迭代算法和一个具有有限终止性质的算法，虽然这种迭代算法仅有线性收敛速度，但由于其格式简单、存储量小、保稀疏性、非常易于计算机实现等特点，故较适用于求解大规模稀疏问题。

Using the properties of the harmonic function and the theories of the expansion of the spherical harmonics,the equivalence between Moritz analytical continuation and Bjerhammar s virtual spherical solution is proved according to the power series expanded form of the real number field function,the differences in meaning between the two solutions are analyzed.

利用调和函数的性质以及球谐函数展开理论，并根据实数域上函数的幂级数展开式证明了Moritz解析延拓解与Bjerhammar虚拟球面解的等价性，同时分析了两种解的内在区别。

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