引理

And some results on the isometries and equivalences of quantum codes are presented, one of which generalizes one of results of Bogart et al. By these results, we construct an isometry of quantum codes which is not an equivalent mapping but preserves the symplectic inner product, i.e.

研究了量子纠错码的等价性和保距同构，推广了Bogart等人的一些概念，并给出若干基本引理和定理，这些结论对进一步研究量子码的等价性和保距同构是非常有用的。

By changing differential equations into nonlinear integral equation,we study the existence of positive solution of a class of second order differential equations problems by the fixed point theorem of cone expansion and compression,and the fixed point index theorem,then we obtain two multiple positive .

通过将常微分方程转化为非线性积分方程，利用锥拉伸和锥压缩不动点定理和不动点指数讨论了一类二阶常微分方程的正解存在性问题，在一定条件下，得到了几个多重正解定理，同时证明了与此相关的主要引理

Although the localic Urysohn lemma was proved by Papert in 1958 early, and then Dowker and Papert improved this proof, but the localic Tietze extension theorem, which is stronger in form and is more convenient to be used, has not been proved for more than thirty years.

虽然Locale形式的Urysohn引理早在1958年即已由Papert证明，9年后Dowker和Papert又对此证明加以改进，但形式上更强、更便于应用的Locale形式的Tietze扩张定理却三十余年未获证明。

The contents are the following:In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.

在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题，为应用非光滑的山路引理证明解的存在性，在证明方程所对应的能量泛函满足非光滑的PS条件时，需利用Sobolev空间的一致凸性，但是对于具有更一般形式的算子的p-Laplacian型方程，不具备上述性质，在文中为克服这一困难，本人对位势泛函做了一致凸的假设，从而证明了解的存在性，并应用推广的Ricceri定理，证明了方程三个解的存在性。

We now prove the inequality in the other direction (which also follows from Fatou's lemma), that is

我们现在来证明另一个方向的不等式（它也可以通过法图引理证明），即

For the calculus of set valued functions,some convergence theorems about thePettis-Aumann integral,such as Fatou's Lemma,are mainly investigated;theproperties of the parametric Pettis-Aumann integral are gotten.

在集值映射的微积分方面，本文着重讨论了Pettis-Aumann积分的一些收敛定理，如Fatou引理等，并研究了含参变量Pettis-Aumann积分的一些性质。

Lemma 2 Let x be an fixed vector in V = Vn(F2) and let z be an indeterminate.

引理2 令x是V = Vn中的一个固定矢量，并令z是一个不确定值。

Using Bruck抯 Lemma [2], Passty [7] extended to the results of [4,5] to a uniformly convex Banach space with a Frechet differentiable norm.

随后，Passty［7］又利用Bruck引理[2]将[4][5]的结果推广到具有范数Frechet可微的一致凸Banach空间中。

In the condition weaker than the condition, the corresponding functional of the equation is first proved satisfying the condition.

在比条件更弱的条件下，先证明方程相应的泛函满足条件，再应用山路引理得到了该问题无穷多解的存在性。

When it was checked this year and confirmed to be correct, mathematicians around the globe breathed a sigh of relief. Mathematicians' work in this area in the last three decades was predicated on the principle that the fundamental lemma was indeed accurate and would one day be proved.

这让全世界的数学家终于松了一口气，因为在过去30年中，数学家在这一领域的工作都是在"基本引理"是正确的并且终将有一天得到证实的基础上进行的。

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