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In particular, when thereis an alternating group A_5 action on a Spin 4-manifold X, we obtain: If X is smooth Spin4-manifold with non-positive signature and b_1=0, denote k=-σ/16 and m=b_2~+,then 2k+3≤m if b_2~++b_2~+≠2b_2~+(Z/A_5) and b_2~+≠0,where and are subgroups of A_5 generated by the elements s=∈A_5and t=∈A_5 respectively.

特别地，在Spin 4-流形具有非交换群A_5作用时，我们得到了：若X为光滑的具有非正符号差的Spin 4-流形，b_1=0，令k=-σ／16，m=b_2~+，如果X上具有Spin交错群A_5作用，且满足b_2~++b_2~+≠2b_2~+(X／A_5)，b_2~+≠0，则2k+3≤m，其中和分别为s=∈A_5和t=∈A_5生成的A_5的子群。

Based on the theory, the differential geometry of statistical model proposed by Amari is generalized from probability space to Sugeno measure space, and a new notion of fuzzy manifold based on Sugeno measure is proposed, including the fundamental differential-geometrical structures of statistical models, the tangent space, and the Riemannian metric in a fuzzy manifold.

第四章介绍了Sugeno测度空间上随机变量的数字特征，在此基础上将学者Amari提出的统计流形从概率测度空间推广到Sugeno测度空间，提出了一种基于Sugeno测度的模糊流形分析方法，包括Sugeno模糊模型的基本微分几何结构，模糊流形中的切空间、黎曼度量。

The linear manifold-learning algorithms represented by PCA, with their substantial mathematical foundation and simple implementation, but they are not able to show complex nonlinear manifold structure with their linear essence, which gives rise to nonlinear manifold-learning algorithms such as Isomap, SIE etc.

以PCA为代表的线性流形学习算法有较强的数学基础，实现简单，但是其线性本质并不能揭示复杂的非线性流形结构；于是又出现了以Isomap、SIE等为代表的非线性流形学习算法。

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。

As a new technique, the manifold method, proposed by Genhua, Shi in 1990s is based on the concept of topology manifold and differential manifold in mathematics.

其中尤以数值流形方法最为突出，以其新颖的数值思想、先进的数值技术得到了学术界的广泛关注。

As mathematicians say,"Every simply connected closed 3-manifold is homeomorphic to a 3-sphere."

用数学语言表达就是"每个单连通3重（3-manifold）封闭体与一个三维球体同胚。"

In this paper, the existence of isometric mapping between the manifold in the high-dimensional data space and the parameter space is proved. By distinguishing the intrinsic dimensionality of high-dimensional data space from the manifold dimensionality, and it is proved that the intrinsic dimensionality is the upper bound of the manifold dimensionality in the high-dimensional space in which there is a toroidal manifold.

首先给出了高维数据的连续流形和低维参数空间之间的等距映射存在性证明，然后区分了嵌入空间维数、高维数据空间的固有维数和流形维数，并证明存在环状流形高维数据空间的参数空间维数小于嵌入空间维数。

We get some results about the scalar curvature of 2 and n(n≥3)dimensions Einstein manifold :ρ Maybe is constant and ρ is constant, the relation between Einstein Manifold and Riemannian Manifold with constant sectional curvature.

本文对其有关性质进行了讨论，得到了2维和n(n≥3)维爱因斯坦流形的数曲率的一些结果：ρ可能为常数和ρ为常数，以及爱因斯坦流形与常曲率黎曼流形之间的关系

Secondly, method of Taylor series is used to approximate the slow manifold and the O approximation of the slow manifold is given; slow dynamics such as saddle node bifurcation which has been proven to be the same as that for the full system is analyzed; at the same time, fast dynamics including bifurcation conditions and domain of attractor of the stable equilibrium are also analyzed; the attracting region which consists of the stable manifold of the unstable equilibrium and stable equilibrium is given; we found that when the inner flux decay increases, the domain of attractor is enlarged.

首先，给出了电力系统SMIB模型的时间尺度分解，得到标准的奇异摄动模型；其次，利用幂级数方法，给出了慢流形的O近似M〓，分析了在慢流形M〓上的慢动态，得到了慢子系统的鞍结分支值，并证明此分支值与原系统的鞍结分支值相同；分析了快子系统的平衡点和全局分支的存在条件，给出由不稳定平衡点的稳定流形与稳定平衡点形成的吸引域；并研究内部电压对稳定域的影响，发现稳定域随内部电压增加而扩大；最后给出整个系统的稳定平衡点的近似吸引域。

The simulation results of inlet air manifold show that the uniformity of the velocity at outlet of the manifold are improved by using the appropriate bell-mouthed inlet and clipper-built baffle, and inlet pressure needed of the manifold decrease.

对进气箱的模拟结果表明，采用扩压型进口加流线型挡块的方案可以提高出口速度的均匀性，同时降低了进气箱入口所需的气体压力。

The absorption and distribution of chromium were studied in ryeusing nutrient culture technique and pot experiment.

采用不同浓度K2CrO4(0，0.4，0.8和1.2 mmol/L)的Hoagland营养液处理黑麦幼苗，测定铬在黑麦体内的亚细胞分布、铬化学形态及不同部位的积累。

By analyzing theory foundation of mathematical morphology in the digital image processing, researching morphology arithmetic of the binary Image, discussing two basic forms for the least structure element: dilation and erosion.

通过分析数学形态学在图像中的理论基础，研究二值图像的形态分析算法，探讨最小结构元素的两种基本形态：膨胀和腐蚀；分析了数学形态学复杂算法的基本原理，把数学形态学的部分并行处理理念引入到家实际应用中。