微分

In this work, we mainly study the numerical analysis of bifurcation theory, especiallythe numerical computation of Takens-Bogdanov point in delay differential equations.

本文中，我们主要考虑分支理论的数值方法，具体而言，时滞微分方程中Takens-Bogdanov点的数值计算方法。

The bifurcation theory of the plane polynomial vector fields is one important research domain of the theory of ordinary differential equation.

平面多项式向量场的分岔理论是常微分方程定性理论的重要研究领域之一，主要研究依赖于参数的向量场的全局轨线拓扑结构随参数变化的规律。

Instead of Gronwall\'s inequality,the non-linear Bihari inequality is crutial in dealing with the non-Lipschitz equations.Moreover,we emphasize that the dominating functions in non-Lipschitz conditions always satisfy concavity and some non-integrability near zero.The concavity is for using Jensen\'s inequality.Because of the non-integrability near zero,we can apply the comparison theorem of ordinary differential equations together with Bihari\'s inequality to yield many results.

3作为Gronwall不等式的推广，Bihari不等式在处理具有非Lipschitz系数的方程时不可或缺；同时应当注意，非Lipschitz条件中所涉及的控制函数总是满足凹性和某种零点处的不可积性：凹性是为了利用Jensen不等式；零点处的不可积性使得Bihari不等式与常微分方程中的比较定理结合在一起从而完成很多结论的证明。

This paper, firstly, after a deep study of image mathematical models, sets up two kinds of image models namely Laplace partial differential equation model and biharmonic equation model and then obtains the eigenvectors of image by the methods of eigenfunction expansion and boundary recurrence.

本文首先较深入地研究了图象的数学描述，提出图象的Laplace偏微分方程和双调和方程两种模型，继而采用特征函数展开法和边界递推法分别求得图象的特征矢量。

Now we study the properties of four multi-layer potentials and their high deriva-tives of biharmonic equation in the plane when they approach the boundary of adomain.

我们研究了重调和方程四种多层位势及其高阶微商在边界附近的性质，得到了这些微商新的表达式，从而将重调和方程的自由边值问题归化为新的边界积分-微分方程组。

Secondly, we discuss the boundary value problem of two dimensional biharmonic equation in a rectangular field and its variational problem,discretize it by using dual tensor product of and the direct product and lining up of matrices, we get some special matrices which are the presupposition in exploring the fast computation, then solve the system of linear equations.

其次本文讨论了矩形域上二维双调和方程边值问题及其相应的变分问题，利用二元张量积小波分析和矩阵的直积、拉直技巧将变分问题离散化，从而使求解偏微分方程问题变为求解线性方程组的问题。一维情况两组基的张量积下得到的系数矩阵分别为块状七对角阵和稀疏矩阵。这些特殊结构为以后快速算法的研究打下一个基础。

The grid generation technique for body_fitted coordinate system by means of numerical solution of biharmonic equations is studied, then the topical H_type grid and the flow field is generated and simulated numerically, respectively.

本文对利用双调和方程微分法生成贴体坐标网格的技术进行了探讨和尝试，生成了典型的H型网格，并对流场进行了数值模拟。

Chapter 3 introduces some methods seeking exact solutions for the nonlinear evolution equation, such as inverse scattering method, symmetry reduction method, Backlund transformation, Darboux transformation , Hirota bilinear method, Painleve analysis method, AC = BD model, and so on.

第三章简要介绍了非线性发展方程的若干求解方法，如反散射方法，对称与微分方程约化，Backlund变换和Darboux变换方法，Hirota双线性方法，Painleve奇性分析法，AC=BD框架下的精确求解等。

Its main tasks are to deal with binary images, gray scale processing, and the use of Sobel, Canny, Roberts, Laplacan operator of four digital image edge detection, the results show that: Roberts operator more precise positioning, but more sensitive to noise, Sobel operator son of gray-scale images with low noise gradient has a better detection of the effect of Laplacian can be detected in the vast majority of the edge did not appear at the same time the basic pseudo-edge, the edge can be precisely targeted, Canny operator is the traditional first-order differential operator in one of the best operator to detect the best edge detection, edges clear, make clear at a glance.

主要工作包括图像的二值处理，灰度处理，并运用Sobel、Canny、Roberts、Laplacan四个算子实现数字图像边缘检测，结果表明：Roberts 算子定位比较精确，但对噪声比较敏感，Sobel算子对灰度渐变低噪声的图像有较好的检测效果，拉普拉斯算子可以检测出绝大部分边缘，同时基本没有出现伪边缘，可以精确定位边缘，Canny算子是传统一阶微分算子中最好的算子之一，检测边缘检测的效果最好，边缘轮廓清晰，使人一目了然。问题补充：摘要英文翻译，不要在线翻译的，是在线等！急啊！！！！谢谢！！！！

Its main tasks are to deal with binary images, gray scale processing, and the use of Sobel, Canny, Roberts, Laplacan operator of four digital image edge detection, the results show that: Roberts operator more precise positioning, but more sensitive to noise, Sobel operator Gradient gray son of low-noise images have a better detection of the effect of Laplacian can be detected in the vast majority of the edge did not appear at the same time the basic pseudo-edge, the edge can be precisely targeted, Canny operator is the traditional first-order differential operator in one of the best operator to detect the best edge detection, edges clear, make clear at a glance

主要工作包括图像的二值处理，灰度处理，并运用Sobel、Canny、Roberts、Laplacan四个算子实现数字图像边缘检测，结果表明：Roberts 算子定位比较精确，但对噪声比较敏感，Sobel算子对灰度渐变低噪声的图像有较好的检测效果，拉普拉斯算子可以检测出绝大部分边缘，同时基本没有出现伪边缘，可以精确定位边缘，Canny算子是传统一阶微分算子中最好的算子之一，检测边缘检测的效果最好，边缘轮廓清晰，使人一目了然。不要在线翻译的那种！在线等！！急！！！谢谢！！！

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。