查询词典 combinatorial manifold

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。

By the combinatorial proof or combinatorial interpretation, the identity is equipped with certain count meaning. The most general way in the combinatorial proof is to count two sides of the identity by two different methods. Generaly, through building a bijection from one set to another one, the number of the two sets respectively represents the two sides of the identity.

恒等式的组合证明或组合解释赋予了恒等式一定的计数意义，组合证明最常用的方法是分别用两种不同的方法对恒等式的两端进行计数，一般通过构造两个集合之间的双射，这两个集合的个数分别表示恒等式的两端，从而根据双射的一一对应性证明恒等式。

Moreover, by the combinatorial proof or combinatorial interpretation, the identity is equipped with certain count meaning. The most general way in the combinatorial proofs is to count two sides of the identity by two different methods. Generaly, through building a bijection from one set to another one, the number of the two sets respectively represents the two sides of the identity. Because of the 1-1 property of bijection, the identity is proved.

此外，恒等式的组合证明或组合解释赋予了恒等式一定的计数意义，组合证明最常用的方法是分别用两种不同的方法对恒等式的两端进行计数，一般通过构造两个集合之间的双射，这两个集合的个数分别表示恒等式的两端，从而根据双射的一一对应性证明恒等式。

In the thesis some vital identities with binomial coefficient are given with their combinatorial proof, and the q-analogues of some identities are offered with corresponding combinatorial proofs on the subspace-lattice. Especially, one remarkable result is that a new q-analogue of the cube-sums identity is obtained with its combinatorial proof over the vector space.

本文给出了一些重要的二项式系数恒等式的组合证明或组合解释，并从向量空间角度给出了一些恒等式的q-模拟及组合证明，其中突出的成果是给出了立方和恒等式的一个新的q-模拟及组合证明。

In the thesis , we give combinatorial proof of some q-identities using integer partition.In the thesis some vital identities with binomial coefficient are given with their combinatorial proof, and the q-analogues of some identities are offered with corresponding combinatorial proof using integer partition.

本文给出了一些重要的二项式系数恒等式的组合证明或组合解释，并从分拆的角度给出了一些q-恒等式的组合证明，其中突出的成果是给出了平方和恒等式的一个直接的组合证明并给出了偶数平方和与奇数平方和的q-模拟及组合证明。

If the power series ∑anxn an ∑anxn and ∑bnxn satisfy the condition r =∑bn xn, we obtain the twin combinatorial identity theorem between the sequences {an} and bn}.We also obtain some specific twin combinatorial identities by using the expansion on the binomial expression formula, including two expansions of combinatorial number(rsn and ...

如果幂级数∑anxn与∑bnxn满足条件r＝∑bnxn 时，获得数列｛an｝与｛bn｝之间孪生组合恒等式的定理，应用在二项式定理等展开式上得出具体的多组孪生组合恒等式，其中包含组合数的两种展开法，Bernoulli数直接表达式的新证等结果。

The chapter 3 investigates the constructive method which means that many combinatorial numbers, such as Stirling number, Bell number, harmonic number, Fibonacci number and the number of derangement, are represented to moments of some random variables. This method is used in the computation of combinatorial sums, discovery and proof of combinatorial identities.

第三章研究了构造性概率方法，即一些常见组合变量(以后统称组合数为组合变量)的概率表示，诸如Stirling数、Bell数、调和数、Fibonacci数、错排数都可以表示为一些随机变量的矩，这些概率表示可以用来研究组合和式的计算与恒等式的证明。

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。

Putt your way through 36 fun-filled holes of minigolf on 3D designed courses with elevated greens, bunkers, bridges and water hazards, among other crazy obstacles.

您的推杆方式，通过36个有趣的填孔迷你的三维设计的课程，以提升绿党，掩体，桥梁和水的危害，除其他疯狂的障碍。

Some participles can be used either as attributes or as predicatives.

有些分词既可当定语用，也可当表语用。