查询词典 existence theorem of implicit function

We consider Theorem 1 to be highly significant and apply it to obtaining the rank of a certain augmentation quotient group by proposing Theorem 3 and giving its complete proof.

应用具有Np-序列有限p-群的特殊性质和重量函数，基本序列等概念以及已有的一些结果，分别研究了类为1的pk(k 2)阶A bel基本p-群和类为2的p4阶基本p-群之增广商群Qn的结构，得到了当n足够大时Qn作为A bel基本p-群的秩。

The use of the Chinese remainder theorem transforms the operation of producing the multiplicative inverses into two corresponding operations with shorter numbers and a faster combination step so that, compared with methods not using the Chinese remainder theorem, an acceleration by factor 4 can be achieved.

采用中国余数定理，形成将倍数逆数的运算转换为两个具有较短数字和一个较快结合的运算的步骤，使得相较于没有中国余数定理的方法可获得因子4的加速。

Recently , the classical Riesz representation theorem on Hilbert spaces has been generalized onto complete random inner product modules ,in this paper the classical Friedrichs theorem on Hilbert spaces is generalized onto complete random inner product modules.

最近，经典的Riesz 表示定理已经被推广到完备随机内积模上，在此基础上本文将Hilbert 空间上经典的Friedrichs 定理推广到完备随机内积模上。首先，证明完备随机内积模上任一正Her2 mite 型惟一地对应一个正自共轭算子。

Finally, it investigate QF-ring with right gpm-injective, and obtain the following theorem:Theorem The following statements are equivalent:(1) R is quasi-Frobeniusean ring:(2) R is right noetherian, left pm-injective ring, right pm-injective ring;(3) R is right noetherian, left pm-injective ring, right GP-injective ring;(4) R is right noetherian, left gpm-injective ring, right gpm-injective ring;(5) R is right noetherian, right gpm-injective ring and each right minimal ideal is right annihilator.

最后用右gpm-内射对QF环进行研究，得到了如下定理：定理下列条件等价：(1) R是quasi-Frobeniusean环；(2) R是右Noether，左pm-内射环，右pm-内射环：(3) R是右Noether，左pm-内射环，右GP-内射环；(4) R是右Noether，左gpm-内射环，右gpm-内射环；(5) R是右Noether，右gpm-内射环且每个极小右理想是右零化子。

In Chapter 3 Firstly, the exponential stability in mean square and almost sure exponential stability for neutral stochstic neural networks with time-varying delay can be discussed by using the linear matrix inequality and the semimartingale convergence theorem; And our sufficient conditions are less conserative; Secondly, by utilizing the fixed point theorem, some sufficient and necessary conditions ensuring the asymp- totic stability in mean square for neutral stochastic neural networks with time-varying delays can be given.

第三章首先，利用线性矩阵不等式方法和半鞅收敛定理，考虑了一类具变时滞中立型随机神经网络的二阶矩指数稳定性和几乎必然指数稳定性，所得到的条件保守性要好；其次，采用不动点方法，给出了具变时滞中立型随机神经网络的二阶矩渐近稳定性的充要条件。

Then relative interior is introduced and an alternative theorem of generalized convex set-valued maps is established by using the separation theorem.

引进了相对内部，应用凸集分离定理建立了一个广义凸集值映射的择一性定理。

In the paper the separate theorem of convex set with weak topology σ(L ∞,L 1) is proved, and the topological characterization about No Free Lunch with Vanishing Risk Condition is given, from it, the equivalency between exist of equivalent martingale measures and NFLVR is proved, this is the first fundamental theorem of asset pricing.

资产定价的第一基本定理是数量金融学中核心的定理之一，本文证明了在 L∞的弱*拓扑σ(L∞，L1)中的凸集分离定理，并在此定理的基础上给出了没有无风险免费午餐的拓扑描述，证明了市场公平性与没有无风险免费午餐条件的等价性，从而重新证明了资产定价的第一基本定理。

In chapter 1, similarity and perturbations of nest subalgebras of factor von Neumann algebras are studied. A similarity theorem is given (see Theorem 1. 2. 3), and it is shown that similar nests in a factor von Neumann algebra are close.

第一章研究因子von Neumann代数中套的相似性和套子代数的摄动，获到一个相似定理（定理1.2.3），其几何意义是：因子von Neumann代数中相似的套聚拢在一起并可向外扩展。

Since Bloch's Theorem applies only to periodic potentials, and since unceasing random movements of atoms in a crystal disrupt periodicity, Bloch's Theorem is only an approximation, but it has proven to be a tremendously valuable approximation, without which most solid-state physics analysis would be intractable.

由于布洛赫定理仅对周期性势场的情况严格成立，而由于实际的晶体中不可避免的有各种缺陷或者原子相对于平衡位置的位移，所以布洛赫定理对于实际晶体来说只是一个近似。但是布洛赫定理对于固体物理的研究不可或缺。

Using two new methods, this paper prores one pertubation theorem on the finite Hilbert space of 2×2 operator matrices, and gereralizes this theorem to the infinite Hilbert space of selfadjoint operators.

给出了有限维Hilbert空间中2×2算子矩阵的数值域扰动定理的两种证明方法，并且将该定理推广到无限维Hilbert空间上的自伴算子。

A strange and mysterious deep sea creature.

一种奇异、神秘的深海生物。

It's a yellow rectangle .

这是黄色的长方形。