Equisingular Deformations of Sandwiched Singularities

We relate the equisingular deformation theory of plane curve singularities and sandwiched surface singularities. We show the existence of a smooth map between the two corresponding deformation functors and study the kernel of this map. In particular we show that the map is an isomorphism when a certain invariant is large enough.     

关于平面曲线的分形维数

关于平面曲线的分形维数邓冠铁（武汉华中理工大学教学系，武汉４３００７４）关键词分形维数，平面曲线，周期函数．分类号ＡＭＳ（１９９１）２８Ａ８０／ＣＣＬＯ１７４．１２为方便起见，采用如下定义和符号：设为平面上有界非空集，对ε＞０，用Ｇ（ε）＝｛（ｘ，ｙ...     

Limits of Puiseux Expansions and Parametric Deformations of Plane Curve Singularities

We consider a family of parametrizations of unibranched plane curve singularities. Each member of the family can be expressed via the Puiseux expansion. Studying the limits of suitably renormalized coefficients of the Puiseux expansions, we obtain in some cases obstructions for a possible adjacency of unibranched plane curve singularities.

Furthermore, we show that the coefficients of the Puiseux expansion are not generic (in a precisely defined sense) as functions of the coefficients of parametrization of the singularity.     

An Algorithm for Computing the Singularity of the Generic Germ of a Pencil of Plane Curves

Abstract

We give an algorithm that describes the singularity of all but finitely-many germs in a pencil generated by two arbitrary germs of plane curve.     

Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

Let γ be a smooth generic curve in ?P ₃. Denote by C the number of its flattening points, and by T the number of planes tangent to γ at three distinct points. Consider the osculating planes to γ at the flattening points. Let N denote the total number of points where γ intersects these osculating plane transversally. Then T ≡ [N + θ(γ)C]/2 (mod 2), where θ(γ) is the number of noncontractible components of γ. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in ?³ has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.     

平面C-B样条的奇拐点分析

平面C-B样条曲线是三次均匀B样条的推广.通过移动C-B样条曲线段的一个控制点而固定其余三个控制点的方法,讨论了在曲线上形成零曲率点的移动控制点的轨迹,得到了C-B样条曲线段的尖点判别曲线、拐点判别区域,同时也给出了在曲线段上生成重结点的移动控制点的轨迹区域.     

On the Diameter of Plane Curves

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini--Study metric from₂ () and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.     

带有加权Hardy-Sobolev临界指数的非齐次Neumann边界奇异的多解问题

主要研究了非齐次Neumann边界奇异的问题,利用Ekeland变分原理、山路引理和一些分析技巧,证明了正解的存在性.     

Approximation of the Singularities of a Bounded Function by the Partial Sums of Its Differentiated Fourier Series

In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas.     

On the Number of Galois Points for a Plane Curve in Positive Characteristic

We study Galois points for a plane smooth curve C ? P ² of degree d ≥ 4 in characteristic p > 2. We generalize Yoshihara's result on the number of inner (resp., outer) Galois points to positive characteristic under the assumption that d ? 1 (resp., d ? 0) modulo p. As an application, we also find the number of Galois points in the case that d = p.     

Convexity and the Average Curvature of Plane Curves

The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve D which bounds the convex set.     

一类具有强弱奇性的Duffing方程周期正解的存在性

通过Manasevich-Mawhin连续定理和一些分析技巧,本文证明一类具有强弱奇性的Duffing方程周期正解的存在性.     

Classification of critical sets and their images for quadratic maps of the plane

We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose critical sets are either ellipses or hyperbolas, was published by Delgado et al. in 2013. This work completes the classification by including all the nongeneric cases: the empty set, a single point, a single line, a parabola, two parallel lines, two intersecting lines, or the whole plane. We describe all possible images for each critical set case and illustrate the geometry of representative maps for each case.     

关于弹性力学平面问题中的主轴应力坐标

本文中讨论了弹性力学平面问题中由主轴应力曲线构成的正交曲线坐标系上的平衡方程,以及解的特性.同时,认为在弹性力学中还存在另一种构造解的方式,即可通过直接构造主轴应力正交网络获得主轴应力的解.     

具有五对特殊方向的一类平面齐五次系统的全局拓扑结构

研究了一类具有五对特殊方向的平面齐五次系统的全局拓扑分类及系数条件.     

具有六对特殊方向的一类平面齐五次系统的全局拓扑结构

In this paper,a class of plane homogeneous fifth system with six special direction is studied,and the global topological classification and coefficient conditions is discussed.     

Nontrivial Critical Networks. Singularities of Lagrangians and a Criterion for Criticality

We single out the class of so-called quasiregular Lagrangians, which have singularities on the zero section of the cotangent bundle to the manifold on which extremal networks are considered. A criterion for a network to be extremal is proved for such Lagrangians: the Euler--Lagrange equations must be satisfied on each edge, and some matching conditions must be valid at the vertices.     

Primitive Illumination Systems for Families of Convex Bodies in the Plane

Let F= {C₁,C₂,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let p_max(F) denote the maximum number of points forming a primitive illumination system for F, and letp_max(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities p_max(F) and p_max(n).     

Notes on pedal and contrapedal curves of fronts in the Euclidean plane

A pedal curve (a contrapedal curve) of a regular plane curve is the locus of the feet of the perpendiculars from a point to the tangents (normals) to the curve. These curves can be parametrized by using the Frenet frame of the curve. Yet provided that the curve has some singular points, the Frenet frame at these singular points is not well‐defined. Thus, we cannot use the Frenet frame to examine pedal or contrapedal curves. In this paper, pedal and contrapedal curves of plane curves, which have singular points, are investigated. By using the Legendrian Frenet frame along a front, the pedal and contrapedal curves of a front are introduced and properties of these curves are given. Then, the condition for a pedal (and a contrapedal) curve of a front to be a frontal is obtained. Furthermore, by considering the definitions of the evolute, the involute, and the offset of a front, some relationships are given. Finally, some illustrated examples are presented.