Nonlinear Differential Equations and Related Topics

在我们的研究计划下，我们学习了各种类型的微分方程，线性和非线性，常微分方程和偏微分方程，以及相关的主题。我们将我们的研究结果总结如下：研究了系数在无限阶点退化的半椭圆算子，得到了这种算子是次椭圆的充分条件。我们找到了一些非微次椭圆算子的例子。研究了围绕不动点的解析变换的局部结构，得到了与不动点相连的稳定流形的几个性质。我们研究了二维复流形中带节点的国家曲线邻域的几何结构。5 .为了研究阿诺德的舌头和燕子的尾巴在一维复杂动力系统中的出现，我们进行了一个漫长而麻烦的数值实验，得到了各种有趣的图形和图片。在研究一个非常方差高斯过程时，我们得到了分布尾部的一个明显的上界。我们发现了玻尔兹曼方程中出现的线性和非线性碰撞积分的精细结构。但是这篇论文还没有完成。总之，我们认为我们在数学中几种非线性现象的研究上迈出了重要的一步。这项研究应继续进行，以获得进一步的结果。

Under our research program we studied various types of differential equations, linear and nonlinear, ordinary and partial, and related topics. We state the summary of our results below :1. We studied semi-elliptic operators with coefficients degenerating at a point with infinite order and found sufficient conditions for such operators to be hypoelliptic.2. We found some examples of hypoelliptic operators wich are not micro hypoelliptic3. We studied the local structure of analytic transformations around its fixed point, and obtained several properties of the stable manifold connected with the fixed point4. We studied the geometric structure of a neighborhood of a national curve with a node in a two dimensional complex manifold5. We pursued a long, troublesome numerical experiment to investigate Arnold's tongues and Swallow's tails appearing in 1-dimensional complex dynamical systems, and obtained various interesting figures and pictures.6. In our study of a Gaussian process with nonconstant variance, we obtained a sharp upper bound of the distribution tail.7. We found fine structures of the linear and nonlinear collision integrals appearing in the Boltzmann equation. But the paper is not completed yet. In conclusion, we belive we made important steps in the study of several nonlinear phenomena in mathematics. This research should be continued to obtain further results.