P~3中3次曲面与平面4次曲线间的关系

本文通过二重覆盖建立了P3中仅有二重的孤立奇点的3次曲面与平面带孤立奇点的4次曲线的对应关系，这种关系使我们能用平面4次曲线的奇点分类及几何性质研究P3中3次曲面的奇点类型及几何性质     

一类三次系统的细奇点

本文研究了具有一个高阶奇点的系统（１,２）,通过在实系统中引入奇点量的概念使焦点量和鞍点量得到统一,并且计算了系统（１．２）的各阶奇点量,证明了这样的系统存在珐多三阶细奇点,而且同时存在细奇点的最大个数是四。当系统是四个细奇点时,它们可能是四个一阶细奇点或四个三阶细奇点。     

线性奇点的对称性质

本文研究奇点集为光滑流形的非孤立奇点的对称性质，证明了这类奇点具有与孤立奇点相类似的对称性质．     

一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量，然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

从R^n到R^3的光滑映射的generic奇点

本文的目的在于用微分拓扑的方法讨论从Ｒ＾ｎ到Ｒ＾３的光滑映射的ｇｅｎｅｒｉｃ奇点的分类问题,我们证明这样的映射只有三类奇点：折点,尖点和燕尾点,并给出了它们的标准型。它推广了Ｊ．Ｍａｒｔｉｎｅｔ（１）的结果。     

一类二阶方程的解的收敛性

本文研究一类二阶常微分方程解的渐近性态，假设方程具有正阻尼及适当保证所有解有界的条件成立，我们证明了每个解收敛于奇点，该结果能够用于系统的奇点集是不可数的情形。     

一类平面奇次系统的全局结构

通过研究一类平面奇次系统的有限远奇点和无穷远奇点的类型,对其进行定性分析,得到系统的全局结构及其相图.     

论复自治微分系统的奇点量

本文研究复自治微分系统(E),得到如下的主要结果:(ⅰ)引入奇点量概念,实现了实系统中焦点量和鞍点量的统一;(ⅱ)定义Lie不变量概念,得到奇点量结构定理和广义对称原理;(ⅲ)计算了(E₃)的全部120个基本Lie不变量,并应用奇点量结构定理得到(E₂)和(E₃⁽³⁾)的奇点量公式及可积性条件。     

微分方程奇点指数的几何计算法

本文提供一种简单的计算奇点指数的方法，文中的结论和例子说明了新方法在计算微分方程指数方面的简洁性     

多项式微分代数系统的奇点性质

本讨论多项式微分代数方程的奇点性质，证明了经用吴方法整序后的系统的奇点与原系统的相应奇点有相同的鞍点性质。     

On the Milnor Number of an Equivariant Singularity

Mathematical Notes -     

Gorenstein Isolated Quotient Singularities of Odd Prime Dimension are Cyclic

In this article, we shall prove that Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated noncyclic quotient singularities.     

On the splash singularity for the free-surface of a Navier–Stokes fluid

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for d-dimensional flows, $d=2$ or 3, the free-surface of a viscous water wave, modeled by the incompressible Navier–Stokes equations with moving free-boundary, has a finite-time splash singularity for a large class of specially prepared initial data. In particular, we prove that given a sufficiently smooth initial boundary (which is close to self-intersection) and a divergence-free velocity field designed to push the boundary towards self-intersection, the interface will indeed self-intersect in finite time.     

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in Rn, n=2,3, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in Rn. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.     

Pure Log Terminal Blow-Ups

The existence of a pure log terminal blow-up for a log terminal singularity and a criterion for a singularity to be weakly exceptional are proved.     

Universal abelian covers of superisolated singularities

We give explicit examples of Gorenstein surface singularities with integral homology sphere link, which are not complete intersections. Their existence was shown by Luengo–Velasco, Melle–Hernández and Némethi, thereby providing counterexamples to the universal abelian covering conjecture of Neumann and Wahl (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Saddlepoint Approximation for the Distribution Function Near the Mean

We repair numerical difficulties in applying saddlepoint tail probability approximations when the ordinate at which the approximation is evaluated is near the mean of the distribution approximated. These modifications apply to double saddlepoint approximations to conditional distributions as well.     

Deformations of canonical singularities

We prove that small deformations of canonical singularities are canonical.