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.     

射影平面和环面上奇异地图的色和

一个地图的每条边,若在同一面的边界上,则称它为奇异地图．由于含环的地图是不可着色的,本文所有地图均不含环．本文研究射影平面和环面上带根奇异地图的色和．     

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。     

射影平面上的对偶无环不可分近三角剖分地图

本文研究了球面和射影平面上对偶无环不可分近三角剖分带根地图的以根面次和内面数为参数的计数问题,得到了这类地图在球面和射影平面上的计数函数满足的方程.还得到了射影平面上2连通地图一个参数的显示表达式和渐近估计式.     

高亏格曲面上地图的计数

本文给出了可定向曲面(亏格2,3)和不可定向曲面(亏格5)上根瓣丛以边数为参数时相应的计数显式.与此同时,考虑一类与瓣从拓扑等价的地图类: (无环,简单)近2-正则地图,通过一种组合方法,给出了多参数下平面近2一正则地图的计数显式,亦得到了任意亏格曲面上该类地图的具体个数.     

带根无环欧拉平面地图的计数(英文)

自20世纪60年代初Tutte的开创性工作以来,许多学者在带根地图的计数方面作了很多工作,但许多类无环地图的计数仍没有被处理.本文主要研究以根点次、非根点数和内面数为三个参数的带根无环欧拉平面地图的计数问题.     

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.     

关于简单平面地图的计数

关于简单平面地图的计数,首先是以递推的方式讨论的.它依赖一般有根平面地图的计数函数(Acta Math.Appl.Sinica,English Series 2(1985),101—111).继之,得到了一个计数显式(J.Math.Res.& Expos.4∶3(1984),37—46).近来,从面剖分计数的更一般情况导出了一个函数方程(已投应用数学学报).本文提供了便于依根节点的次和边数计数有根简单平面地图的一个新的函数方程.由此出发,更直接也更简单地导出了这个计数显式.     

双折数、自交数与环数——封闭折线复杂性的三项指标

给定ｎ边封闭折线Ｚｎ，怎样描述它的复杂程度呢？这自然要看它的边的折转情况、边相互间的交织情况和自相缠绕情况，我们分别以双折数、自交数和环数来描述．１　边的折性与双折数如果折线的一条边，其两邻边折向同侧，就叫做单折边，折向异侧的叫做双折边．关于边的折性，我们有如下定理［１］：定理１　封闭折线若有双折边，则有偶数条，左、右旋边各半且相间排列．如果把Ｚｎ双折边的条数ｓ（Ｚｎ）＝ｓ（ｎ）称为双折数，以ｓ０（ｎ）＝ｍａｘｓ（ｎ）表最大双折数，则我们有定理２　ｎ边闭折线的最大双折数ｓ０（ｎ）＝０，２，ｎ－１…     

球面和射影平面上不可分地图的色和

给出了球面和射影平面上带根不可分地图的色和方程,从色和方程导出了球面和射影平面上带根一般不可分地图、二部地图的计数函数方程. 利用色和理论,研究不同类地图的计数问题,得到了一种研究计数问题的新方法. 此外,还得到了一些计数显示表达式.     

Bisingular maps on the torus

A map is bisingular if each edge is either a loop or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps on the sphere and the torus, and we also present formula for such maps with four parameters: the root-valency,the number of isthmus, the number of planar loops and the number of essential loops.     

BISINGULAR MAPS ON SOME SURFACES

A map is bisingular if each edge is either a loop (This paper only considersplanar loop) or an isthmus (i.e., on the boundary of the same face). This paper studies thenumber of rooted bisingular maps on the sphere and the torus, and also presents formulaefor such maps with three parameters: the root-valency, the number of isthmus, and thenumber of planar loops.     

On the η‐function for bisingular pseudodifferential operators

In this work we consider the η‐invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the η‐function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the K‐theory of the algebra of 0‐order (global) bisingular operators. With these preparations we establish the regularity properties of the η‐function at the origin for global bisingular operators which are self‐adjoint, elliptic and of positive orders.     

Enumerating near-4-regular maps on the sphere and the torus

In this paper rooted near-4-regular maps on the plane and the torus are counted with formulae with respect to four parameters: the root valency, the number of edges, the inner faces, and nonroot-vertex loops. In particular, the number of rooted near-4-regular maps on those surfaces with exactly k nonroot-vertex loops is investigated.     

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.     

Generalization of the boundary function method for solving boundary-value problems for bisingularly perturbed second-order differential equations

A bisingular boundary-value problem for an ordinary differential equation is considered. The asymptotics of the solution as the sum of an outer expansion and an analog of a number of functions of the boundary layer is constructed.     

A simple reconnecting map

Generalized standard maps of the cylinder for which the rotation number is a rational function (a combination of the Fermi and Chirikov rotation functions) are considered. These symplectic maps often have degenerate resonant zones, and we establish two types resonance bifurcations: ??loops?? and ??vortex pairs??. Both the border of chaos and the existence of the chaotic web are discussed. Finally the transition to global chaos for a generalized map is considered.     

Intrinsic Covering Dimension for Nuclear C*-Algebras with Real Rank Zero

For operators belonging either to a class of global bisingular pseudodifferential operators on \({{\mathbb{R}^{m}} \times {\mathbb{R}^{n}}}\) or to a class of bisingular pseudodifferential operators on a product \({M \times N}\) of two closed smooth manifolds, we show the equivalence of their ellipticity (defined by the invertibility of certain operator-valued, homogeneous principal symbols) and their Fredholm mapping property in associated scales of Sobolev spaces. We also prove the spectral invariance of these operator classes and then extend these results to larger classes of Toeplitz type operators.     

Asymptotic Solution of Linear Bisingular Problems With Additional Boundary Layer

We study two bisingular Dirichlet problem with the additional boundary layer: 1) for the second order linear elliptic equation in a ring, 2) for linear ordinary differential equations of second order in a segment. We construct asymptotic solutions to the three-zone, bisingular Dirichlet problems by using the generalized method of boundary functions and obtain estimates for the residual functions.     

关于平面上双树梵和的一个注记

这篇文章得到了以根节点的次、割边的个数及环的个数为参数的双树梵和的色和方程，且导出了这类地图带以上三个参数的精确解及一些退化的情形。