Klein瓶上的双奇异地图

一个地图的每条边如果不是环就是割边(即该边的两边是同一个面的边界)，则称之为双奇异地图，本文研究Klein瓶上带根双奇异地图的计数问题，得到了此类地图以边数、平面环数、手柄上本质环数和又帽上本质环数为参数的计数公式，并得到了部分计数显式。     

关于双奇异平面地图的计数

本文讨论了带根双奇异平面地图的计数问题,提供了以根面次、度和内面数为参数及以根面次、奇异边数和自环数为参数的计数函数所满足的计数方程,并且导出了所有的计数显式.     

左非奇异左CS-环和左遗传左CS-环

本文首先证明了：环为左非奇异左CS-环当且仅当它为左余非奇异Baer环,然后给出左遗传左CS-环,正则左遗传左CS-环和左遗传左连续环的某些刻划.     

平面上一般地图的色和与双色和

本文研究了平面上一般带根地图的色和与双色和,得到了这类地图的色和与双色和函数方程。从这类地图的色和函数方程,导出了平面上一般无环地图、平面上二部地图和平面上欧拉地图的计数函数方程。还得到了一些计数函数的计数显式。     

一类奇摄动系统奇异极限环的不变环面分支

对一类奇异摄动系统中由奇异极限环产生的不变环面分支进行了研究并利用不变环面的分支理论,讨论了由快系统的二重极限环和三重环分支出的不变环面的存在性．     

环面上有根欧拉地图的计数

本文探讨了环面上有限欧拉迹的计算并且提供了一个解析表达式．在此基础上，我们给出了环面上有根欧地图的计算公式．     

柱面上有根近三角剖分的计数

柱面上的三角剖分是一类与环面上的地图紧密相关的地图．本文提供了一个计算柱面上有根近三角剖分的具有三个变量的精确公式．     

广义幂级数环的本质理想和非奇异性

本文研究了广义幂级数环与其系数环在本质理想和非奇异性上的关系.利用本质理想的定义和性质,得到了广义幂级数环的左理想为本质左理想的菪干充分必要条件.在此基础上,给出了广义幂级数环为左非奇异环的充分必要条件.     

关于右FP~2F环的一些结果

本文证明了满足~⊥dcc的右FP~2F环半素与非奇异等价,对于满足~⊥dcc的非奇异右FP~2F环,它的左右极大商环一致,并且极大商环是半单Artin环和右FPF环,最后给出了它的一个特征。     

模的弱消去问题与qu－正则环

本文考虑模的弱消去问题与正则环的Ｓｔａｂｌｅｒａｎｇｅ条件，引进了ｑｕ－正则环的概念，并用它刻划了一类满足弱消去条件的模，刻划了ｑｕ－正则环，指出一切满足比较公理的正则环均为ｑｕ－正则环．文中还考虑这些结果的应用．例如，刻划了满足弱消去律的拟内射模；证明了素的正则右自内射环上的任意非奇异内射右Ｒ－模可以从直和项中弱消去．     

克莱茵瓶上奇异地图的色数

A map is singular if each edge is on the same face on a surface (i.e., it has only one face on a surface). In this paper we present the chromatic enumeration for rooted singular maps on the Klein bottle.     

Chromatic sums of singular maps on some surfaces

A map is singular if each edge is on the same face on a sruface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper povides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enum erating functions of such maps are also derived.     

Klein瓶上的奇异地图

It is well known that singular maps (i. e. ,those have only one face on a surface)play a key role in the theory of up-embeddability of graphs. In this paper the number of rooted singular maps on the Klein bottle is studied. An explicit form of the enumerating function according to the root-valency and the size of the map is determined. Further ,an expression of the vertex partition function is also found.     

可定向曲面上两个地图的交叉数

In this paper, we discuss the crossing numbers of two one-vertex maps on orientable surfaces. By using a reductive method, we give the crossing number of two one-vertex maps with one face on an orientable surface and the crossing number of a one-vertex map with one face and a one-vertex map with two faces on an orientable surface. This provides a lower bound for the crossing number of two general maps on an orientable surface.     

Partial regularity of energy minimizing harmonic maps into a complete manifold

In this paper, we study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at mostn–2, wheren is the dimension of the domain. We will also give an example of an energy minimizing map from surface to surface that has a singular point. Thus then–2 dimension estimate is optimal, in contrast to then–3 dimension estimate of Schoen-Uhlenbeck [SU] for compact targets.     

On the Regularity for Stationary Harmonic Maps

We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.     

Counting unrooted maps on the plane

A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside (“infinite”) face. An unrooted map is an equivalence class of maps under orientation-preserving homeomorphism, and a rooted map is a map with a distinguished oriented edge. Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps. In this article, using the same technique we obtain closed formulae for counting unrooted plane maps of all these classes and their duals. The corresponding formulae for rooted maps are known to be all sum-free; the formulae that we obtain for unrooted maps contain only a sum over the divisors of n. We count also unrooted two-vertex plane maps.     

Oriented bordism of codimension one immersions of 7-manifolds and relative Thom polynomials

We give a formula to detect the oriented bordism class of a codimension one immersion of an oriented 7-manifold in terms of singularities of its singular Seifert surface, that is, a generic map from a compact 8-manifold which extends the given immersion. Our argument involves a study of a relative version of the Thom polynomials for certain singularities of generic maps from manifolds with non-empty boundaries.     

FRACTAL DIMENSION ESTIMATES FOR INVARIANT SETS OF NON-INJECTIVE MAPS

For a special class of non-injective maps on Riemannian manifolds an upper bound for the fractal dimension of invariant set in terms of singular values of the tangent map and degree of non-injectivity is given     

Integrable Riemannian Submersion with Singularities

A map of a Riemannian manifold into an euclidian space is said to be transnormal if its restrictions to neighbourhoods of regular level sets are integrable Riemannian submersions. Analytic transnormal maps can be used to describe isoparametric submanifolds in spaces of constant curvature and equifocal submanifolds with flat sections in simply connected symmetric spaces. These submanifolds are also regular leaves of singular Riemannian foliations with sections. We prove that regular level sets of an analytic transnormal map on a real analytic complete Riemannian manifold are equifocal submanifolds and leaves of a singular Riemannian foliation with sections.