自然映射

For a monotone twist map the well-known Birkhoff fixed point theoremsays that the circles with rational rotation number will break up whence a finitenumber of pairs of periodic points arise,half of which are elliptic and the othershyperbolic.Moreover,around each elliptic periodic point a few of new invariantcurves will appear and make a picture of phase space with a self-similar structure.Again one may ask is there a similar picture for higher dimensional Hamiltoniansystems?

对于单调扭转映射，著名的Birkhoff不动点定理指出，在通有情况下旋转数为有理数的不变曲线一般会破裂，在原来的不变曲线附近会产生有限多对周期点，其中一半是双曲的，一半是椭圆的，而在椭圆型周期点的周围又会产生一圈圈的不变曲线，使得椭圆周期点附近的动力学行为和整个相空间的动力学行为十分相象，换句话说，整个相空间具有无限嵌套的自相似结构，自然又产生了一个问题：对于自由度数目大于2的高维哈密顿系统，其相空间是否有类似的图象？

On the product of unit spheres,we give a kind of natural embeddings from the product unit spheres to the unit sphere in which the product of unit spheres can be viewed as a hypersurface.

第一个是在乘积单位球面上，我们给出自然的由乘积单位球面到高一维球面的嵌入映射，然后考虑了在乘积单位球面上预给定Gauss-Kronecker曲率后，我们所考虑的嵌入的存在性。

To build the models for immune computation, the natural computing model of immune computation is proposed and the mapping from the natural immune system to the artificial immune system is established.

在免疫计算的建模方面，提出了人工免疫系统的自然计算模型，建立了从自然免疫系统到人工免疫系统的映射。

Note: Methods or procedures may be specified for mapping other natural linear orders onto the natural numbers.

注：为把其它自然线性顺序映射成自然数，可以采用各种方法或过程，例如，排序可以按字母或时间。

Simulation shows that the proposed algorithm can achieve similar performance to that of the ITS using natural binary bit mapping and has about 1dB gap with the ITS using Gray bit mapping. In the high SNR region, the complexity of the proposed bit-level ITS is about 6％~40％ of the symbol-level ITS.

仿真结果表明：在误码率性能上，该算法略次于使用自然二进制映射的基于符号的ITS算法，与使用格雷映射的基于符号的ITS算法只有1dB左右的差距；而该算法的计算量在高信噪比时只相当于基于符号的ITS算法的6％~40％。

Order.Note: Methods or procedures may be specified for mapping other natural linear orders onto the natural numbers.

注：为把其它自然线性顺序映射成自然数，可以采用各种方法或过程，例如，排序可以按字母或时间。

Thirdly,we define a equivalence relation Rin the topology space X by the fibrewise function f ,and then we get the fibrewise quotient space X / R,we call it the fibrewise decomposition space of X .And last we discuss the relation between f and the natural mapping.

其三是，通过一个纤维映射f ： X→Y，在B上的纤维拓扑空间X中定义了一个等价关系R，得到X的纤维商空间X / R，称其为X的分解空间，最后也讨论了f与自然投射的关系。

The unnaturalness of the mapping compels users to figure this relatship out anew every time they use the stove .

用户必须通过试验或者参考旋钮下的图标来确定，这种不自然的映射迫使用户在每次使用气炉时都不得不重新确定映射关系。

Users must find out by trial and error or by referring to the tiny icons next to the knobs. The unnaturalness of the mapping compels users to figure this relationship out anew every time they use the stove.

用户必须通过试验或者参考旋钮下的图标来确定，这种不自然的映射迫使用户在每次使用气炉时都不得不重新确定映射关系。

First of all, Chinese unique geographical surroundings result in cultivation civilization, in which people's living and production were utterly influenced or even controlled by sun and such disasters as floods and droughts caused by sun made our ancestors much more dread and hate the moon. On the contrary, in the view of our ancestors, moon not only is beneficial to the growth of crops, but brings about brightness, coolness and enjoyment, which in turn makes moon much more popular than sun. As a carrier of national emotion, mythology also contributes to the development of "valuing moon above sun"in that the collective unconscious of hating sun but loving moon reflected in mythologies such as "Hou Yi Shot the suns"and "Chang E's fly to moon"easily finds its expression in the course of selecting poetic images. Philosophically,"masculine"and "feminine"are realized in many aspects of social life, but the trinity of sun, monarchy and father, which is identical with "masculine"dominates the political and ethical field while the trinity of moon, mother and daughter, which represents "feminine", plays a leading part in culture and art. So far as poetry is concerned, the image of moon has an overwhelming superiority. Aesthetically speaking, the moon spirit has been embodied in several ways. Moon is both the origin and the summit of such aesthetic standards as softy, sweetness and clearness, so it is reasonable that the image of moon has been regarded as aesthetic object in poetry much more frequently than the image of sun.

中华民族所处的独特的地理环境孕育出了农耕文明，在农业社会中，太阳对人们的生产生活具有绝对的控制与影响，因日而起的水旱灾害更加深了先民对太阳的畏惧与仇恨；而月亮在先民心中，不仅能促进农作物的生长，而且带来了与太阳相反的皎洁、清凉、闲适，所以月亮比太阳受到更多的欢迎；神话是是民族情感的载体，&后羿射日&与&嫦娥奔月&所映射的恨日爱月的集体无意识，在诗歌意象的选取中自然地流露出来，有助于&重月轻日&现象的形成；哲学上的&阴&&阳&范畴在人们社会生活的多个方面均有体现，但与&阳&直接同一的&日—君—父&三位一体控制了政治、伦理领域；而代表&阴&的&月—母—女&则在文化、艺术领域占据了突出地位，具体到诗歌中，则是月亮意象完全压倒日意象；审美上的月神精神，在月亮意象上得到了多方面的反映，月亮既是阴柔美、恬静美、澄澈美等艺术审美准则的来源，也是其最高体现，月亮意象自然比日意象更多地进入诗歌审美领域，成为审美对象。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了