微分同胚

The definitions of generalized directional derivative and generalized gradient of Lipschitz functions defined on Riemannian manifold are presented. Some properties of the directional derivative and gradient are proved by using tangent and cotangent mapping. The minimization necessary condition of nonsmooth Lipschitz functions is given. Moreover, Fritz John necessary optimality condition in mathematical programming is provided on Riemannian manifold.

在黎曼流形上给出了Lipschitz函数的广义方向导数和广义梯度的概念，利用黎曼流形局部上与欧氏空间开集微分同胚的性质以及切映射和余切映射导出了广义梯度的性质和运算法则，证明了定义在黎曼流形上的函数取得极小值的必要条件是广义梯度包含零元素，并利用这些性质给出了黎曼流形上数学规划问题的Fritz John型最优性条件。

But diffeomorphism invariant theory leaves some problems to be explained, and the meanings of spacetime and physical theory are still in question.

但微分同胚不变理论存在需要解释的问题，空时及物理理论的意义仍不明确。

Further, when the coefficients are smooth, the solutions form a stochastic diffeomorphism flow.

而且，若系数是光滑的，则方程的解形成一随机微分同胚流。

A diffeomorphism is applied to transform the nonlinear system into a new coordinate system. The stability and robustness of the system are analysized in detail.

根据微分同胚，将含有建模误差的非线性系统变换为易于分析的规范形式，并在此基础上分析了故障诊断系统的稳定性和鲁棒性。

Results show that the normal forms of diffeomorphism theory can reflect the nonlinear characteristic of power grid better, and the location to implement load control and place to allocate SVC can be effectively decided.

结果表明，微分同胚正规形方法能更好地反映电力系统的非线性特性，有效地确定实施负荷控制措施的地点以及SVC的安装地点。

Analysis shows that the robot model containing uncertainty can be transformed into a quasi—linearized form by using nonlinear feedback and diffeomorphism. Based on this result, a simple design method is proposed for robot tracking control.

第一种算法将谈自忠等提出的机器人反馈线性化控制推广到存在不确定性的情形，指出通过非线性反馈和微分同胚变换可将含不确定性的机器人模型变换成准线性形式，基于此给出一种简单的跟踪器设计方案。

As the special characteristic of nonlinearity, several notions are introduced and used in this paper, such as differential manifold, vector field, diffeomorphism, tangent space, topological equivalence. Stable manifold and unstable manifold are also introduced.

针对非线性的特殊性，本文专门介绍了非线性中几个比较特殊的概念，如微分流形、向量场、微分同胚、切空间以及等价关系等，对稳定和不稳定流形也作了介绍。

The basic idea of the method is: take a diffeomorphism mapping and coordinate transformation to the nonlinear model, to get exact linearization model of the power system, with the state feedback. At last, optimal control method is adopted to design the chaos controller of the power system.

该方法基本路线为：对系统非线性模型通过微分同胚进行坐标变换，再采用状态反馈，完成对电力系统非线性模型的精确线性化处理，之后，采用最优控制方法，设计控制器。

In differentiable manifolds, one studies for instance differentiable structure, definition of differential manifolds, diffeomorphism, tangent space, Embedding theorem, partitions of unity etc.

微分流形部分主要涉及微分结构，微分流形的定义及例子，可微映射，微分同胚，切空间与余切空间，流形的嵌入，单位分解定理等。

In section 3, the theorem is applied to interval analysis.

本文将Banach空间之间的同胚问题归之为一类动力系统非负解的存在性问题，证明了一个全局微分同胚定理，给出了一些推论，并应用于区间分析，推广了一些已有的结果。

Lugalbanda was a god and shepherd king of Uruk where he was worshipped for over a thousand years.

Lugalbanda 是神和被崇拜了一千年多 Uruk古埃及喜克索王朝国王。

I am coming just now,' and went on perfuming himself with Hunut, then he came and sat.

我来只是现在，'歼灭战perfuming自己与胡努特，那麼，他来到和SAT 。