A Simple Recurrence for Covers of the Sphere With Branch Points of Arbitrary Ramification

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800’s. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-Mélou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called m-Eulerian trees. In this paper, we give a simple partial differential equation for Bousquet-Mélou and Schaeffer’s generating series, and for Goulden and Jackson’s generating series, as well as a new proof of the result by Bousquet-Mélou and Schaeffer. We apply algebraic methods based on Lagrange’s theorem, and combinatorial methods based on a new use of Bousquet-Mélou and Schaeffer’s m-Eulerian trees. Supported by a Discovery Grant from NSERC. Research supported by a Postgraduate Scholarship from NSERC. Received October 8, 2005     

广义A-proper映射的拓扑度

本文提出了广义A-proper映射,它更弱于A-proper映射。通过建立广义A-proper度,可用来研究(M)型映射与部分解答Browder问题(见[4];用类似[6]的方法)。本文推广了[3]、[6]和[2]的工作。 今往设X、y为实Banach空间,D X是开集。D表其边界,是之闭包。“→”和“”分别表强和弱收敛。N为正整数集合。     

Generalized Topological Index

This paper deals with an index which is different in general from the topological index defined by Atiyah and Singer because we loosen the normalization property. The intrinsic relation of this new index with operations in K-theory is explained. It is also shown that if we change the normalization axiom, the corresponding index is well-defined and may be expressed in terms of the topological index.     

Generalized Topological Representation for Complete Join-Semilattices

This paper describes a generalization of the Papert–Papert–Isbell adjunction between topological spaces and frames to a generalized one between the category of generalized topological spaces and the category of complete join-semilattices. Within these categories we aim to study certain separation axioms in a categorical point of view.     

拓扑群中广义度量性质的一个注记

主要讨论拓扑群中的一些广义度量性质.证明了对于拓扑群G和G的局部紧度量子群H,如果商群G/H是层空间(半层空间,κ半层空间,σ空间),则G也是层空间(半层空间, κ半层空间,σ空间),这肯定回答了Arhangel'skii A.V.和Uspenskij V.V.提出的一个问题.同时还讨论了弱拟第一可数的,不含Sω的闭拷贝的仿拓扑群.     

Connected Orbits in Topological Generalized Quadrangles

We study generalized quadrangles. After an investigation of the subgeometries that are generated by arbitrary sets of vertices, we consider orbits of connected subgroups of the automorphism group of topological generalized quadrangles. We deal with the problem of how a set of vertices has to be chosen in order that the union of the orbits generates a subquadrangle, or even the whole quadrangle.     

Generalized Principal Extensions in Topological Algebras

The ??generalized principal extension?? of a subset of a topological algebra is, by definition, a subset of another topological algebra. Hence, the generalized spectrum of an element, ??local spectrum??, of a topological algebra is thus a subset of a topological algebra, not necessarily of ${{\mathbb C}}$ . We also give a criterion for the continuity of the (generalized) Newburg map (Theorem 3.1).     

广义Kato分解与拓扑一致降指数

讨论了算子的广义Kato分解与拓扑一致降指数,给出算子既有广义Kato分解又有拓扑一致降指数的等价刻画,并由此对算子谱精细结构进行进一步的细化.     

广义近似空间的拓扑结构

在本文中,我们引入了论域X上的自反关系θ的传递化表示θs的概念,研究了拓扑空间(X,тθ)的基本性质,得到了一些有较大价值的结果,这不仅有助于理解粗糙集理论中的一些基本概念及其性质,而且对拓扑学本身都有理论和实际意义.     

Topological Multiple Recurrence of Weakly Mixing Minimal Systems for Generalized Polynomials

Acta Mathematica Sinica, English Series - Let (X, T) be a weakly mixing minimal system, p1, …, pd be integer-valued generalized polynomials and (p1, p2, …, pd) be non-degenerate. Then...     

A New Recursion Relation for the 6j-Symbol

The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This article presents a new recursion formula for the square of the 6j-symbol. In the asymptotic regime, the new recursion is shown to characterize the closure of the relevant tetrahedron. Since the 6j-symbol is the basic building block of the Ponzano–Regge model for pure three-dimensional quantum gravity, we also discuss how to generalize the method to derive more general recursion relations on the full amplitudes.     

Spline Surfaces over Arbitrary Topological Meshes: Theoretical Analysis and Application

Based on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to $C^2$-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate $C^1$-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational $C^1$-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The $C^1$−continuous connection schemes between two patches of the surfaces are presented.     

拓扑序空间中的广义向量值均衡问题

利用一个KKM型定理,在拓扑序空间中,建立了关于广义向量值均衡问题解的某些新的存在性定理.     

任意除环上矩阵的广义逆

该文使用投影算子方法研究任意除环上矩阵的广义逆, 建立了具有指定值域和零空间的{2} 逆的刻划和表示理论. 作为应用, 获得了带有对合函数的Moore Penrose逆, 群逆和Dra zin逆的一些新的表式.     

Generalized voronoi diagrams for moving a ladder. I: Topological analysis

Given a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line-segment B (a [ladder]) of length d inside Ω. FP is a three-dimensional manifold. A [Voronoi complex] in this manifold, a two-dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one-dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.     

关于局部凸拓扑向量空间中广义双拟变分不等式

In this paper we investigate generalized bi-quasi-variational inequalities in locally convex topological vector spaces. Motivated and inspired by the recent research work in this field,we establish several existence theorems of solutions for generalized bi-quasi-variational inequalities,which are the extension and improvements of the earlier and recent results obtained previously by many authors including Sun and Ding [18],Chang and Zhang [23] and Zhang [24].