奇异函数

Last, this paper introduced some concepts of multiscale geometrical analysis and curvelet transform theory. Because curvelet can showed this feature more 'sparsely' in the function included singular curve or singular side in two dimension space or multi-dimension space. This paper provided a robust digital watermarking technology algorithm based on curvelet transform. watermarking information was embedded into coarse scale coefficients after curvelet transforming, choose right embedding strength. Testing proved it has good invisibility and robustness.

最后，介绍了多尺度几何分析和曲波变换理论的相关概念，鉴于曲波能对二维乃至高维空间中含奇异曲线或者曲面的函数进行更"稀疏"地表示这一特性，本章提出了一种基于Curvelet变换的鲁棒数字图像水印算法，该算法将水印信息嵌入到经曲波变换后的图像的粗尺度系数中，合理选择的嵌入强度，试验表明该算法具有较好的隐蔽性和鲁棒性。

Firstly, we give out a reduction way for quasilinear generalized system A〓=G , and the concept of the index κ geometrically nonsingularity, where A is singular and of smooth kernal space. By using the canonical projection mappings onto the space of ker A , a class singular pointimpasse point only owned by nonlinear generalized system is also studied and we also prove that the regular index κ quasilinear generaized system A〓=G defined on the region W is equivalent with an ODEs 〓 defined on a submanifold W〓 of the W, so we reduce the problem of quasilinear generalized system to the problem of an ODEs.

首先利用投影函数，给出了拟线性广义系统A〓=G的一种归约方法，提出了指数κ几何非奇异的概念，同时对广义系统特有的一类奇点-寂点进行了研究，证明了正则指数κ的拟线性问题A〓=G等价于在其定义域W中某一个子流形W〓上的正常微分方程〓，从而将拟线性广义系统的问题归结为常微分方程问题。

This paper discusses the yield function, loading unloading criterion and constitutive equation of Lagrangian finite deformation elastic-plasticity theory. Finally, Lagrangian finite elastic plasticity theory is used on constitutive models of elastic-plasticity at infinitesimal strain, and the complete constitutive relation for singular yield surface theory is obtained.

本文讨论了Lagrange型有限变形弹塑性本构理论的屈服函数、加卸载准则、本构方程等，最终将Lagrange型有限变形弹塑性本构理论应用于小变形情况下的奇异屈服面弹塑性模型，得出了完整的Lagrange型有限变形弹塑性本构理论。

By means of singular model transformation technique, special Lyapunov-Krasovskii functional approach, similar Leibniz-Newton formula and linear matrix inequality concept, some new stability conditions are derived for above systems.

应用奇异模型转换技巧、特殊型Lyapunov-Krasovskii泛函数方法、相似型Leibniz-Newton公式、线性矩阵不等式观念针对上述类神经网路系统，提出新稳定测试准则。

According to static stability and avoiding the singularity induced bifurcation, saddle-node bifurcation, and Hopf bifurcation in power systems, this paper presents mathematical models for stability equilibrium solutions based on the semismooth property of spectral functions in mathematics.

针对电力系统中静态稳定和避免奇异诱导分岔、Saddle-Node分岔、Hopf分岔几类典型的系统稳定问题，基于数学上谱函数的半光滑特性，建立了系统稳定平衡解的数学模型。

Furthermore that the square measure error is not depending on the choice of generalized matrix inverses is proved. Finally the least square estimation and the maximum likelihood estimation for this model are proposed.

在此基础上对奇异度量误差下的函数关系模型的参数进行估计，得出了模型参数的最小二乘估计和极大似然估计。

By introducing the nonlinear complementarity problem function, the Karush-Kuhn-Tucker conditions of OPF model are transformed equivalently into a set of semismooth nonlinear algebraic equations. Then the set of semismooth equations can be solved by a semismooth L-M method based on the subdifferential. The method belongs to Newton-type method. It can ensure the positive defmitiveness of the iterative coefficient matrix by using the L-M parameter, which avoids the ill-conditioning of iterative equations. The method, requiring only the approximate solution of a linear system at each iteration, is quite applicable to the large-scale cases.

在利用非线性互补函数将OPF模型的KKT条件转化为半光滑非线性代数方程组后，采用基于次微分的半光滑L-M法求解，该方法属于牛顿法范畴，可通过对L-M参数的调整保证迭代系数矩阵的正定性，克服了系数矩阵的奇异引起的"病态"，且该方法在确定搜索方向时只需求解线性系统的加热器近似解，适用于大规模系统的求解。

The matrix =( xi, xjp having the e-th power of the greatest common P-divisorp of xi and xj as its-entry is called the e-th power GCD matrix on S. The matrix = having the e-th power of the least common P-multiple p of xi and xj as its-entry is called the e-th power LCM matrix on 5. We obtained the following results:(1) is nonsingular for any set S;(2) If S is an FC set, then the determined of has formula Det =Jpe(x1)...Jpe, where the function Jpe is the generalized Jordan totient function;(3) A formula of the inverse of is given when S is an FC set;(4) If S is an FC set, then |.

以_P的e次方为第i行j列元素的矩阵称为定义在S上的e次幂GCD矩阵，记为；以_P的e次方为第i行j列元素的矩阵称为S上的e次幂LCM矩阵，记为，我们得到了如下结果：①定义在集合S上的e次幂GCD矩阵是非奇异的；②若S是R上的FC集，则S上的e次幂GCD矩阵的行列式Det=J_p~e(x_1)J_P~e(x_2)…，J_p~e，其中J_p~e为R上的Jordan函数；③当S为FC集时，得到了的逆矩阵~-1的表达式；④证明了当S是FC集时，整除，即等于与R上另一个矩阵的乘积。

1B-4 is improper, not all functions are Laplace transformable

由于方程2-1B-4是奇异的，所以并非所有的函数的拉氏变换都存在。

A greedy feature selection algorithm has been employed, and a wieldy sample selection algorithm basing on the frequency histogram re-sampling has been proposed, which reduces the influence of redundant features and samples, and consequently improved the robustness of the resulting*recognition system.

从模式识别本身对训练样本集的要求出发，给出了基于频度统计函数重抽样的样本选择方法，一定程度上消除了冗余样本和奇异样本给分类器的学习、训练和分类识别带来的不良影响。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。