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We then introduce a conformal invariant quantity which is formulated in terms of the square of the length of the trace free part of the second fundamental form and the mean curvature, and prove that if this quantity is bounded above by that value of the Clifford torus then x is either a totally umbilical sphereor a conformal Clifford torus.

接著我们介绍一个由第二基本型式迹为零部份的张量长度平方与平均曲率形成的保角不变量，并证明此保角不变量的上界若为Clifford环面之值，则x不是全脐球面就是Clifford环面的保角类。

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。

In this design, the dimension M of the memory vector field may be not necessarily equal to the dimension N of the network vector field. This may be the usually case in applications. Designing a network with either static or oscillating behavior, after analysing the dynamical. behavior of its four-dimension subspace, we get 2-torus in the subspace. If we vary the ratio of the imagine parts of the complex conjugate eigenvalues, the periodic motion of all kinds of period ratio on 2-torus are obtained.

设计一个既具有静态平衡点又具有动态振荡的网络，其静态平衡点由多余维叉形分叉得到，动态振荡形式由多余维Hopf分叉得到，通过对其中一个具有振荡行为的四维子空间的进一步分析，可以得到网络在四维子空间中的二环面运动，适当调整约当形中共轭复特征根虚部的比值，可以得到各种周期比的二环面上的周期运动。

Every connected, compact Lie groups G contains a maximal torus T. This maximal torus is unique up to conjugation, and every element of G is conjugate to the element of the maximal torus.

每个连通紧李群具有极大环面子群，任二极大环面子群彼此共轭，且连通紧李群中每个元素共轭于极大环面的一个元素。

According to the helix theories of cylinder,the essential factors of helix with torus are analyzed,and the parametric equation of helix on torus is given based on analytic geometry.

根据圆柱螺旋线理论，分析了形成圆环螺旋线的要素，并运用解析几何方法建立了圆环螺旋线的参数方程。

Next in section 2.3,usingthe finite isolated regular representations in SO(3)character varity of 〓,weobtained a computation formula of SO(3)Donaldson invariants with degree 1 of4—manifold 〓(Theorem 2.3.1).Finally in section 2.4 wediscussed the Donaldson invariants of 4—manifolds with smoothly embedded 2—torus 〓 whose selfintersection is +1,with the help of the single irreducibleSU(2)representation of the principal 〓—bundle over a 2—torus with Eulernumber—1,we got a gluing formula of the Donaldson invariant on the 2—homology represented by 2—torus 〓(Theorem 2.4.1),it is a supplement of P.

最后在§2.4我们讨论了具有自交数为+1的嵌入环面〓的4—流形的Donladson不变量，运用环面上欧拉数为-1的圆丛的单个不可约SU（2）表示给出了Donaldson不变量在环面表示的同调类上的一个粘合公式（定理2.4.1），这一结果补充了P。

We also define two families of functions, named a torus height functionand an AdS-distance-squared function respectively, and use them to investigate thesingularities of the AdS-null surfaces and the torus Gauss images as applications ofsingularity theory of functions.

作为函数奇点理论的应用，利用这两个函数族来研究AdS-null曲面和环面高斯像的奇点。

Of, relating to, or shaped like a torus or part of a torus.

环形圆纹曲面环形圆纹曲面的或其某一部分的；形状象环形圆纹曲面的或其某一部分的；与之有关的

Caption:: The Jovian magnetosphere, including the Io flux tube, plasma torus, Jovian aurorae, the sodium cloud, and sulfur torus is shown in this illustration.

在木星附近，它的卫星活在这个庞大的磁层内，如木卫一艾奥就形成了等离子圆环，而且也跟木星通电。

The paper discusses the position relationship between the litter circles of torus and the section circle of sphere figured out the intersection of torus and sphere when the central axis of torus and sphere center in XOZ coordinate plane through coordinate transformation.

利用坐标变换使圆环中心轴与球心同在XOZ平面内，讨论了在该平面内圆环截圆与球圆的位置不相交、相切和相交时圆环面和球面的相交情况，并给出了判别圆环面、球面相交情况的流程图以及圆环面和球面相交实例。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。