微分拓扑

For example, we study the properties of the"centroid curve" and the"auxiliary curve", and identify two planes in which these two curves are defined respectively; we also use the idea of homotopy method to prove a fact that the orbit of a differential system has no triple contact point with the line we considered. Another key technique, which we use several times, is deformation argument.

例如研究由Abel积分之比定义的"质心曲线"和"辅助曲线"的几何性质，并把这两条在不同平面上定义的曲线放在同一坐标系中考虑；利用拓扑中同伦的思想来证明微分系统的轨线不可能与某条直线三重相切；利用几何方法与分析计算相结合证明了某一曲线是正则的；以及连续变动参数的技巧。

InSection 2,we give several sufficient conditions for the existence of one solution,multiplesolutions and infinitely many solutions of the Sturm-Liouville bvps via the generalizedpolar coordinates.Then in Section 3,the existence of positive solutions are proved undersuperlinearity,sublinearity and many other conditions by a fixed point theorem in cones.Our results have generalized those in many articles.A detailed discussion of periodic solu-tions of a kind of functional differential equations with high-order Laplacian-like operatorcan be found in Section 4 and this subject has not been studied before.

在第二节，我们定义了一种新的坐标变换-广义极坐标，并利用它讨论了p-Laplacian算子和Laplacian-型算子的Sturm-Liouville边值问题，分别得到了存在一个解、多个解、无穷多个解的多个充分条件；第三节研究p-Laplacian算子的Sturm-Liouville边值问题正解的存在性与多重性，采用的是锥上的不动点定理，全面推广了这一方面已有的结果；对目前研究较少的高维Laplacian-型算子及带有Laplacian-型算子的泛函微分方程的周期解问题，我们在第四节做了一些研究，这也是拓扑方法的一个应用。

And Satan entered into Judas who was called Iscariot and was of the number of the twelve.

22：3 这时，撒但进了那称为加略人的犹大里面，他本是十二数中的一个。

Teachers to consciously reflect on the study of theory and practice, constantly adjust and perfect their knowledge structure, in understanding the future of education, and social life on the basis of a correct conception of modern education bioethics.

教师在自觉进行理论学习和实践反思，不断调整和完美自己的知识结构，在认识生物教育的未来性，生命性和社会性的基础上形成正确的现代教育生命伦理观。