定理

1Because some conditions can not be fulfilled,one can not applythe known results in stochastic approximation directly.

1由于有些条件不能满足，现有的随机逼近结果无法直接利用，我们巧妙地克服了这一困难，在较弱的条件(文中H.4.1—H.4.3)下，给出了这类算法的严格收敛性分析(定理4.3.2，定理4.3.3)。

By using the concept of the stop-loss order, we have discussed the total compensations S of random risks while individual risk model approximates collective risk model and generalized some known results.

从停止损失序的角度，探讨了用集体风险模型来近似个体风险模型时，随机风险理赔总额S的一般情况，我们推广现有文献的关于单因模型的结果，得到了各模型间S的序的关系(定理1)，并给出了各模型S之间的误差公式(定理3)。

Using the method of analysis to study strong limit theorem proposed by professor Liu Wen, and a plying Legbesuge theoremon differentiability of monotone function, a new proof of the theorem of strong law of large numbers for the sequence of random variables of the product distribution of Poisson is forwarded.

利用文献[1]刘文教授提出的研究强极限的纯分析方法，通过构造适当的辅助函数，然后利用单调函数异数存在定理，给出具有乘积泊松分布的随机变量序列的一个强大数定理的新证明。

We would like to point out that our proof of the singularity层次structure theorem (Theorem 7.1.1) is different from that of Perelman in two aspects:(1) we avoid using his crucial estimate in Claim 2 in Section 12.1 of [103];(2) we give a new approach to extend the limit backward in time to an ancient solution.

这个特解大致上走的正是小佩避免使用定理7.4的那条路，不过曹朱强调：在这个"完整证明"中所用的定理7.1.1证明与小佩所采用的方式有两点不同

Study on proof of sum of products equality and its application

给出了积分上限函数在证明等式和不等式、计算累次积分、证明微分中值定理和积分中值定理中的应用。

In this paper, the inertial theorem of real symmetric matrix has been proved by three methods in three aspects: the relationship between real symmetric matrix and real quadratic form, the relationship between real symmetric matrix and symmetric bilinear function of real linear space.

从实对称矩阵与实二次型的联系、实对称矩阵与实线性空间的对称双线性函数的联系以及将实对称矩阵作为研究主体这三个角度，介绍实对称矩阵的惯性定理的三种证明，以期加深对实对称矩阵的惯性定理的理解。

In the third chapter, we study complete minimal submanifolds of locallysymmetric space, and we obtain a pinching theorem about the Ricci curvature of the minimal submanifolds, which generalizes the result of Norio Ejiri\'s from sphere space to locally symmetric space.

第三章节研究局部对称空间中具有正Ricci曲率的完备极小子流形，得到了关于子流形Ricci曲率的一个pinching定理，该定理把Norio Ejiri的结论从外围空间为球空间推广到局部对称空间中。

In addition, by utilizing Jacobi two square numbers theorem and Lagrange four square numbers theorem and some theta function identities, we also prove the known results of number theory: two triangular number theorem, four triangular number theorem, and the number of representations of a positive integer by various quadratic forms in terms of divisor functions

包括Jacobi二平方数定理，Lagrange四平方数定理等，然后利用这些结果结合几个theta函数恒等式，我们获得了把任意一个正整数表示成两个三角数或四个三角数的和以及其他的二次形式的方法数，这些方法数都是用因子函数来表示的。

In spite of the second-best theory,it isagain enforced by the third-best theory.

第一定理说明完全竞争是帕累托最优的，虽然存在次优问题，但是三优理论重新肯定了第一定理的作用和意义。

In this thesis, we mainly study problems on global geometry and geometricanalysis of Riemannian submanifolds, including vanishing theorem for homologygroups, topological sphere theorem, L~2 harmonic 1-forms, finiteness of end andthe spectrum of the Laplacian. In 1973, by using the Federer-Fleming existence theorem and the techniquesfrom the calculus of variations in the geometric measure theory, H.

本文着重研究黎曼子流形上整体几何与几何分析的若干问题，主要内容包括子流形的同调群消没定理、拓扑球面定理、L~2调和1-形式、端的有限性和Laplace算子谱等问题。

The labia have now been sutured together almost completely.The drains and the Foley catheter come out at the top.

此刻阴唇已经几乎完全的缝在一起了，排除多余淤血体液的管子和Foley导管从顶端冒出来。

To get the business done, I suggest we split the difference in price.

为了做成这笔生意，我建议我们在价格上大家各让一半。