广义de Sitter空间中的类时超曲面的切触性质

从切触几何及Legendrian奇点理论的角度研究了广义de sitter空间中的类时超曲面的切触性质及gdS-高斯像的奇点的分类和几何意义.     

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     

Heisenberg群中曲线的球面指标线之间的勒让德对偶

3维Heisenberg群H_3是二阶幂零李群,是Thurston几何化猜想中几何结构的8种模型结构之一.主要从勒让德对偶的视角考察3维Heisenberg群上正则曲线的切球面指标线和副法球面指标线之间的对偶关系,发现了刻画副法球面指标线奇点的几何不变量.     

关于平面四次Bézier曲线的拐点与奇点

在计算几何中,已给出了三次Bezier曲线的保凸性的充要条件,并进行了几何解释。本文则是导出形式简洁的拐点和奇点方程并对四次Bezier曲线的拐点和奇点的分布进行讨论。按Bezier曲线的拐点个数进行分类,还得到了四次Bezier曲线有奇点的充分必要条件,并给出几个数值实例,实例说明,不但非凸的单纯特征多角形可以有凸的Bezier曲线段,而且非单纯特征多角形也可以有凸的Bezier曲线段。四次Bezier曲线的奇点和拐点是可以共存的。     

平面三次H-Bézier曲线的形状分析

本文对平面三次H-Bézier曲线的形状进行分析,讨论其诸如奇点、拐点、局部凸和全局凸的几何特征,得出曲线上含有奇点、拐点和曲线为局部凸或全局凸的用控制多边形边向量相对位置表示的充分必要条件.     

平面三次H-Bézier曲线的形状分析

本文对平面三次H-Bézier曲线的形状进行分析,讨论其诸如奇点、拐点、局部凸和全局凸的几何特征,得出曲线上含有奇点、拐点和曲线为局部凸或全局凸的用控制多边形边向量相对位置表示的充分必要条件.     

异点与非异奇点

异点与非异奇点郭时光（四川轻化工学院）为了讨论三维空间中曲线的奇点处切线问题，本文给出了切线存在性定理和切向量计算公式，由此把奇点分为异点和非异奇点两类。１问题我们知道，设点Ｐ０是曲线Γ上一定点，Ｐ是Γ上一动点，如果Ｐ趋近于Ｐ０时，，割线Ｐ０Ｐ有极限...     

计算平面代数曲线亏格上界的符号-数值算法

亏格是代数曲线的重要不变量.文章给出计算一类平面代数曲线亏格上界的符号-数值混合算法.首先通过数值稳定的符号-数值混合算法把代数曲线的定义多项式系统约化到几何对合形式,然后考察奇点的性质.如果曲线的奇点是寻常的,那么由奇点的重数可以计算出代数曲线的亏格;否则算法仅给出亏格的一个上界.     

三次参数曲线段和三次Bézier曲线形状控制

文献[1—5]研究了代数参数曲线的仿射不变量及代数参数曲线段上实奇点和实拐点的分布问题,运用经典的代数几何方法,在计算几何中成功地讨论了对代数参数曲线段的控制问题。本文是这一工作的继续。     

有理Bzier曲线

§1.前言 计算几何中曲线造型的主要工具是代数参数曲线。其中,按照Bernstein基和B样条基表示的参数曲线,称为Bezier曲线和B样条曲线,尤其应用广泛。 苏步青教授最早把代数曲线论的仿射不变量理论导进计算几何领域,用以研究仿射平面参数曲线的几何性质,特别是关于那些以实拐点和实奇点个数为特征的仿射分类,从而获得一系列具有重要应用价值的结果,推动了计算几何的理论发展。近来,这些结果被应用到CAGD的工程技术课题中去,收到了成效。     

Variation homological diagrams,¶and corner singularities

We describe the general homological framework (the variation arrays and variation homological diagrams) in which can be studied hypersurface isolated singularities as well as boundary singularities and corner singularities from the point of view of duality. We then show that any corner singularity is extension, in a sense which is defined, of the corner singularities of less dimension on which it is built. This framework is also used to rewrite Thom–Sebastiani type properties for isolated singularities and to establish them for boundary singularities. Received: 27 June 2000 / Revised version: 18 October 2000     

Resolution except for minimal singularities II. The case of four variables

In this sequel to Bierstone and Milman [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philosophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.     

Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.     

A note on irregularity of two dimensional simple hyperbolic singularities

Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singularities. In this note we give an optimal upper bound for a class of surface singularities by the computation of cohomology. Moreover a sufficient condition is given for the positivity of irregularity of some simple hyperbolic surface singularities. Therefore a class of surface singularities with non-rigid deformation is constructed.     

The heat equation and reflected Brownian motion in time-dependent domains.: II. Singularities of solutions

We study the space-time Brownian motion and the heat equation in non-cylindrical domains. The paper is mostly devoted to singularities of the heat equation near rough points of the boundary. Two types of singularities are identified—heat atoms and heat singularities. A number of explicit geometric conditions are given for the existence of singularities. Other properties of the heat equation solutions are analyzed as well.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

Singularities on the boundary of the hyperbolicity region

The singularities of hyperbolic polynomials (hypersurfaces) and the singularities of the boundary of the hyperbolicity region are investigated. Theorems on stabilization of these singularities in families with a fixed number of parameters and on their relationship with elliptic singularities are proved. The problems considered in this study are part of a research program focusing on singularities of boundaries of spaces of differential equations, proposed by V. I. Arnol'd.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 193–214, 1988.     

Deformation of isolated hypersurface singularities and their moduli algebras

By the using of determinantal varieties from moduli algebras of hypersurface singularites the relation of the deformation of hypersurface singularities and the deformation of their moduli algebras is studied. For a type of hypersurface singularities a weak Torelli type result is proved. This weak Torelli type result showes that for families of hypersurface singularities the moduli algebras can be used to distinguish the complex structures of singularities at least in some weak sence. Research supported by NNSF     

Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems

In this article we give a list of 10 rank zero and 6 rank one singularities of 2-degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system.      

Resolution of Corank 1 Singularities of a Generic Front

We construct a resolution of singularities for wave fronts having only stable singularities of corank 1. It is based on a transformation that takes a given front to a new front with singularities of the same type in a space of smaller dimension. This transformation is defined by the class A_µ of Legendre singularities. The front and the ambient space obtained by the A_µ-transformation inherit topological information on the closure of the manifold of singularities A_µ of the original front. The resolution of every (reducible) singularity of a front is determined by a suitable iteration of A_µ-transformations. As a corollary, we obtain new conditions for the coexistence of singularities of generic fronts.