必要条件

Several criteria are given for judging whether ideal class number of a realquadratic function field to be one.

研究了实二次函数域的理想类数为1的一些必要条件以及等价条件。

The trick is to grow the middle class and keep reducing the number of poor people.

在全球经济的背景下，相当数量的富裕人口是一个社会成功的必要条件之一。

Then it proves a necessary and sufficient condition for anamalgamation of two 3-manifolds with toms boundary with no new incompressible surfaces.Lastly, with the help of the condition and the theorem about the classification of allautomorphisms of a torus, it is proveed that there exists infinitely many 3-manifolds each ofwhich contains no incompressible surfaces besides a torus.

最后，文章给出了以环面作为边界的两个3维流形在作融合积时不产生新的不可压缩曲面的一个充分必要条件，并将这个结果与关于环面到自身的自同胚类的分类的结果相结合，证明了本文的一个核心结论，即存在无穷多个互不同胚的3维流形满足除了一个环面以外不含有其他的不可压缩曲面。

It sounds like a cliche, but that's the real necessity

这听起来像一句老生常谈，但它却是真正的必要条件。

In this paper, a sufficient and necessary condition such that the Ricci form of continuous loop group s is closed is given.

本文给出了连续环群的Ricci形式是闭的充分必要条件。

Patience is a very necessary for coastguard skippers the commanders appear to have patience of job .

耐心是海岸警卫队船长们的必要条件，指挥官们似乎都有极大的耐心。

Patience is a very necessary qualification for coastguard skippers;the commanders appear to have patience of job.

耐心是海岸警卫队船长们的必要条件，指挥官们似乎都有极大的耐心。

If is cyclic, then for every codeword the word is also a codeword.

必要条件：如果是循环码，那么仍然是码字，故线性码是理想。a是多项式，码字也看成是多项式。

As a second year graduate textbook, Cohomology of Groups introduces students to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites.

作为第2本年毕业生教科书，组的Cohomology有最少的必要条件向学生介绍cohomology 理论(与在代数和拓扑之间的丰富的相关作用有关)。

In particular, we compute Hochschild homology and cohomology groups of infinitedimensional path algebras and some of their quotient algebras, and we prove that for a general monomial algebra (not necessary finite-dimensional), all Hochschild cohomology groups of positive degrees vanish if and only if its Gabriel quiver is a finite tree.

特别地，我们计算了无限维路代数以及某些商代数的Hochschild同调群和上同调群，而且给出了一般单项代数的各正次Hochschild上同调群为零的充分必要条件，即它的Gabriel箭图是有限树。

Singer Leona Lewis and former Led Zeppelin guitarist Jimmy Page emerged as the bus transformed into a grass-covered carnival float, and the pair combined for a rendition of "Whole Lotta Love".

歌手leona刘易斯和前率领的飞艇的吉他手吉米页出现巴士转化为基层所涵盖的嘉年华花车，和一双合并为一移交&整个lotta爱&。

This is Kate, and that's Erin.

这是凯特，那个是爱朗。