多函数

Though comparing Canny operator and center B spline dyadic wavelet, the following conclusion is proven in this dissertation: a Center B spline function has tight support and Canny operator hasn't. b Center B spline function asymptotic convergence to Gaussian function and the derivative of Center B spline function asymptotic convergence to Canny operator. c The derivative of fourth order center spline B function is more suitable as a optimal edge detector than Canny operator. d Center B spline function can balance the smoothing and approximation of original data, and the fourth center B spline function is the only optimal solution of two order smoothing problem. e The error between the valve of time-frequency uncertainty of the fourth center B spline function and the lower bound of time-frequency uncertainty does not exceed 0.143% of the lower bound. f The derivative of center spline B function can construct a stability dyadic wavelet and can give a fast algorithm for multiscale edge detection, but Canny operator can do neither.

作者给出了Canny算子与中心B样条二进小波严格的比较证明，得出如下结论：a中心B样条函数具有紧支集，Canny算子不具有紧支集。b中心B样条函数的极限收敛于高斯函数，中心B样条函数的导数收敛于Canny算子。c四阶中心B样条函数的导数比Canny算子更接近最佳边缘检测滤波器。d中心B样条函数比高斯函数更能兼顾对原函数平滑和逼近的折中要求，并且四阶中心B样条函数是二阶逼近问题的唯一最优解。e四阶中心B样条函数的时频测不准关系值与时频测不准关系下界的逼近误差不超过0.143％。f中心B样条函数的导数可以构成稳定的二进小波，存在快速的多尺度算法；而Canny算子不构成稳定的二进小波，无法给出快速的多尺度算法。

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。

Based on analyzing the problems,we propose an evolutionary algorithm by using niches technology and multi-parent-crossover technology to experiment on complex multimodal function optimization with samples.

在分析了简单遗传算法求解带约束的多峰值函数优化中存在的问题的基础上，提出了利用小生境技术和多父体杂交的演化算法，对复杂的多峰值函数优化问题进行了测试，证实该算法能够快速求出多峰函数的所有全局最优解，而且搜索效率高，达到了令人满意的效果。

Another part of our work begins with definition of multioutput generalized bent functions. We show their spectral characteristics and give equivalence definition using generalized first order Chrestenson cyclic spectrum. Also we calculate rate of accordance between multioutput generalized bent functions and multioutput m-valued affine functions. Then we define multioutput m-valued perfect nonlinear functions and show relationship between them and multioutput generalized bent functions. Finally we give two ways of constructions of multioutput generalized bent functions.

第三章中首先给出了剩余类环上的多输出广义Bent函数的定义，得到了多输出广义Bent函数的广义一阶Chrestenson谱特征，然后计算了多输出广义Bent函数与多输出m值仿射函数的符合率；接着定义了多输出m值完全非线性函数，讨论了多输出m值完全非线性函数与多输出广义Bent函数之间的关系；最后给出了两种构造多输出广义Bent函数的方法。

The monotone class theorem of multiple dimension functional format.This is the generalization from the one dimension monotone class theorem.

给出了多维函数形式的单调类定理，它是一维函数形式单调类定理的推广，也是处理多维可测函数、多维随机过程的有力工具。

The research on integral function and meromorphic function oriented to three branches and profoundly guided to different function fields from the end of 19~ to beginning of 20~ century.

魏尔斯特拉斯对整函数和亚纯函数的研究开启了三个方向的系统研究，对19世纪末至20世纪诸多函数论分支的发展产生深刻的启发与导向。

The problem of multi-objective usually meet many conflict each other and can not use the objective of the same standard unit, then to employ membership function of fuzzy theory,at first each objective function to change fuzzy set and express with membership function, membership function include the maximum constraint, minimum constraint, the decrease function of monotonic property, to become optimal of the multi-objective function, but still to scanty of weighting value with regard to adjustment of objective function, thus this thesis proposes the method of combine orthogonal arrays and particle swarm optimization to solve the problem of multi-objective optimal power flow, each objective function separately add weighting value, to setting each weighting value of objective function in order to the result of anticipating.

多目标最佳化的问题通常会面临到许多互相冲突且不能用同ㄧ标准单位的目标，於是利用模糊理论中的归属函数，首先将各目标函数转换成模糊集合并以归属函数表示，归属函数包含了最大限制值、最小限制值、单调的递减函数所组成，将各目标函数利用归属函数表示，形成单一目标函数的最佳化，但是仍缺乏权重值对於目标函数的调整，於是本篇论文提出ㄧ种权重值设定与粒子群优演算法的方法去解决多目标最佳电力潮流的问题，将各目标函上分别加上权重值设定各目标函数的权重值，得到预期的效果。

Firstly, this paper definited pseudometric d for n -foldproduct of σ-algebra A of subsets of a given set X , and definited indicator function ; secondly, gave the concept of differentiable of set function at S° and concept of partial derivative at S° with respect to the ith argument Si through indicator function, and defound the weak efficient solution of and optimality necessary condition; Finally, obtained three weak duality results and strong duality results under generalized convexity conditions.

首先，对已知集X的子集的σ-代数A的n-折积A^n，定义了伪度量d，给出了相应的特征函数〈h，Is〉；其次，通过特征函数给出了集函数在S 可微的定义及集函数在S 关于第i个变量Si的偏导数定义；给出了多目标规划问题的弱有效解概念及的最优性必要条件；最后，分别在目标函数和约束函数的3种较弱凸性条件下，研究n-集函数多目标规划问题的对偶问题，获得了3个弱对偶结果和强对偶结果。

The multi-wavelets functions are supported on [-1, 1], and one wavelet function is symmetric, the other is anti-symmetric.

所以，可以由这两个尺度函数构造一类多小波函数，这类多小波函数不但在[-1，1]上具有紧支撑性，而且一个小波函数具有对称性，另一个小波函数具有反对称性，因此这种紧支撑多小波函数适用于区间[0，1]。

TheMRA with more then one-wavelet function is presented and the multi-wavelet basis isproduced by unitary matrix.

我们提出了多函数小波概念，并按照单函数的多分辨分析方法，讨论了多函数小波的一些性质。

I will endeavour to find you some assistance.

我尽力帮你找人帮忙。

At first I only know bruck is the idol of American younglings, afterwards I returned back to Taiwan ,even in Beijing last year ,I saw her poster everywhere, I was so surprised at her charm.

起初我只晓得布鲁克雷德丝是美国少男少女崇拜的偶像，后来回台湾，甚至去年在北京，居然也四处看见她的海报，才惊讶她的魅力之大。