引理

But he was convinced that one stepping stone that needed confirmation — dubbed the "fundamental lemma"— would be reasonably straightforward.

他认为，证明所谓的"基本引理"将是完成这项任务的一个合理跳板。

But he was convinced that one stepping stone that needed confirmation — dubbed the "fundamental lemma"— would be reasonably straightforward. He, his collaborators and his students were able to prove special cases of this fundamental theorem.

朗兰兹知道证明支撑自己构想的假设需要几代人的努力，但证明所有这些首先需要一个垫脚石，它被称作"基本引理"。

Over the past few years, Ngo Bao Chau, a Vietnamese mathematician working at Université Paris-Sud and the Institute for Advanced Study in Princeton, formulated an ingenious proof of the fundamental lemma.

过去5年来，就职于巴黎第十一大学和普林斯顿高等研究院的越南数学家吴宝珠试图用公式表述一项有关基本引理的精巧证法。2009年进行的验证证明了这一证法的正确性，全世界的数学家终于可以松一口气。

Under some moderate conditions, the prior estimate of the solutions of the system is obtained. Making use of constructing Picard iterative sequence, Doob martingale inequality, Gronwall inequality, Borel-Cantelli lemma and some fundamental inequalities, together with the uniform Lipschitz conditions, the existence and uniqueness of the solution for stochastic functional differential equations with infinite delay is derived on the interval t0,∞.

在适当的条件下，得到了随机泛函微分方程的解的先验估计；再结合一致Lipschitz条件，通过构造Picard迭代序列，利用Doob较不等式、Gronwall不等式、Borel-Cantelli引理及一些基本不等式，得到该方程的解在区间[t0，∞]上是存在且唯一的。

In this paper, we just consider the initial-boundary value problems of aclass of two-dimensional nonlinear integro-differential equation, simultaneouslyconsidering the damped term and memory termThe details will go as follows. Firstly, the c...

本文同时考虑耗散项和记忆项等因素，讨论了一类具有记忆项的二维非线性积分微分方程的如下初边值问题具体研究内容如下：首先，文章简单介绍了国内外当前对非线性积分微分方程的研究现状；其次，文章给出了一些重要概念和引理，并对部分符号做了说明；第三，利用Galerkin方法证明了问题(1)-(3)的变形问题弱解的存在唯一性；第四，证明了问题(1)-(3)强解的存在性和能量的指数衰减性；第五，进一步证明了问题(1)-(3)经典解的存在性。

In this paper, we just consider the initial-boundary value problems of aclass of two-dimensional nonlinear integro-differential equation, simultaneouslyconsidering the damped term and memory termThe details will go as follows. Firstly, the current st...

本文同时考虑耗散项和记忆项等因素，讨论了一类具有记忆项的二维非线性积分微分方程的如下初边值问题具体研究内容如下：首先，文章简单介绍了国内外当前对非线性积分微分方程的研究现状；其次，文章给出了一些重要概念和引理，并对部分符号做了说明；第三，利用Galerkin方法证明了问题(1)-(3)的变形问题弱解的存在唯一性；第四，证明了问题(1)-(3)强解的存在性和能量的指数衰减性；第五，进一步证明了问题(1)-(3)经典解的存在性。

In the second chapter,we introduce some lemmas and concepts about iterative roots of piecewise homeomorphic maps of y-stars.

在第二章，我们介绍了与Y星上逐片同胚映射的迭代根有关的概念和引理。

On the whole, we do as follows: Firstly, we list some conceptions and lemmas for later use. Secondly, we define δ-fine partitions for infinite interval and integral of vector-valued functions on infinite interval, and discuss the properties of integral, and characterize its primitives...

主要包括以下五部分内容：在第一部分中，我们介绍了本文所用到的基本概念和引理；在第二部分中，通过定义无穷区间上δ-精细的分法，我们给出了无穷区间上向量值函数的积分的定义，并讨论其性质，还给出了原函数的刻划；在本文的第三部分中，我们着重讨论了无穷区间上向量值函数积分的收敛定理；在本文的在第四部分中，我们首先应用无穷区间上向量值函数积分的收敛定理给出了常微分方程整体广义解的存在性定理，其次应用强积分对Banach空间常微分方程广义解进行了讨论；最后，在第五部分中，我们将模糊积分推广到无穷区间上并给出了其数值计算方法。

This is an intuitionistic methods, and using it can prove Mountain Pass lemma.

这是一种较为直观的方法，用它可以证明山路引理。

Secondly, the concept of column space, cyclic invariant subspace and generalized cyclic invariant subspace were introduced.And some basic properties, such as separation lemma, were presented.

其次，引入列空间、循环不变子空间和广义循环不变子空间等基本几何概念，给出一些有关概念的基本性质，特别是分离引理。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新