THE NUMBER OF ROOTED NEARLY CUBIC C-NETS

1. IntroductionW.T. Tutte's original papers[1--3) on the enumerative theory of rooted planar maps havebrought forth a series of papers on enumerating triangulations. The enumeration of generalrooted planar maps has then also been investigated and a number of elegant results havebeen obtained, although relatively fewer than that of triangulations. As the dual case oftriangulations, the enumerative theory of cubic maps has also been developed, though thereare a lot of problems waiting for solut…     

数有根近2-正则平面地图

The number of rooted nearly 2-regular maps with the valency of root-vertex, the number of non-rooted vertices and the valency of root-face as three parameters is obtained. Furthermore, the explicit expressions of the special cases including loopless nearly 2-regular maps and simple nearly 2-regular maps in terms of the above three parameters are derived.     

带根单行平面地图的计数

提供了根点为一个奇点的带根单行平面地图以其边数、根点次和非根奇点次为参数的生成函数所满足的一些函数方程,并且导出了这些函数的显式,它们有两个是无和式.     

Chromatic sums of general maps on the sphere and the projective plane

In this paper, we study the chromatic sum functions of rooted general maps on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of rooted loopless maps, bipartite maps and Eulerian maps are also derived. Moreover, some explicit expressions of enumerating functions are also derived.     

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。     

Couningg-essential maps on surfaces with small genera

This paper provides some functional equations and parametric expressions ofg-essential maps on the projective plane, on the torus and on the Klein bottle with the size as a parameter and gives their explicit formulae for exact enumeration further.     

The number of rooted circuit boundary maps

In this article, the authors discuss two kinds of new planar maps： pan-fan maps and circuit boundary maps, and provide explicit expressions about their enumerating functions with different parameters. Meanwhile, two explicit counting formulas for circuit cubic boundary maps with two parameters; the size and the valency of the root-face, are also extracted.     

Weak monotonicity inequality and partial regularity for harmonic maps

The notion of locally weak monotonicity inequality for weakly harmonic maps is introduced and various results on this class of maps are obtained. For example, the locally weak monotonicity inequality is nearly equivalent to theε-regularity. Project supported by the National Natural Science Foundation of China (Grant No. 19571028) and the Guangdong Provincial Natural Science Foundation of China.     

Conformal,geometric and invariant measures for transcendental expanding functions

We describe the fractal structure of expanding meromorphic maps of the form , where H and Q are rational functions whose most transparent examples are among the functions of the form with . In particular we show that depending upon whether the Hausdorff dimension of the Julia set is greater or less than 1, the finite non-zero geometric measure is provided by the Hausdorff or packing measure. In order to describe this fractal structure we introduce and explore in detail Walters expanding conformal maps and jump-like conformal maps. Received: 3 May 2001 / Published online: 5 September 2002     

Gibbs state for some class of meromorphic functions

This paper is a continuation of our earlier works [1,2] on the fractal structure of expanding and subexpanding meromorphic functions of the form F = H o exp o Q, where H and Q are non-constant rational maps. Under some assumptions on the forward trajectories of asymptotic values ofF we define a class of summable potentials for the maps f of the punctured cylinder induced by F. We prove the existence and uniqueness of Gibbs states for these potentials.     

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains. This work is supported by the project No: SB 2005-0199, Spain.     

Quadratic maps and smooth vector-valued functions: Euler characteristics of level sets

Quadratic maps of R^N into R^K are studied. Explicit expressions are obtained for the Euler characteristics of level sets of such maps. The Euler characteristics of level sets of smooth vector-valued functions are also evaluated in terms of their values at critical points.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 35, pp. 179–239, 1989.     

A New Uniform Convergence for Partial Functions

Suppose X, Y are topological spaces. In this paper maps are not necessarily continuous. A map f from a non-empty subset of X to Y is called a partial map. Partial maps occur as inverse functions in elementary analysis, as solution of ordinary differential equations, as utility functions in mathematical economics, etc. In many applications, X and Y are metric spaces and there is a need to have a uniform convergence on a family of partial functions. Since partial maps do not have a common domain, the usual uniform convergence (u.c.) is not available. Noting that in many situations, all maps of a family under consideration, have a common range, we define a new uniform convergence (n.u.c.) that is complementary to the usual one. This n.u.c. does not preserve continuity but preserves (uniform) openness. Its usefulness stems from the fact that it can be used when u.c. cannot be defined. Moreover, in some situations where both u.c. and n.u.c. are available, the latter satisfies our intuition but not the former. We give applications to ODE's and throw some light on earlier literature.     

Robot navigation functions on manifolds with boundary

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are constructed on an arbitrary sphere world—a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n − l)-spheres. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider.     

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.     

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.     

A systematization of convexity and quasiconvexity concepts for set-valued maps,defined by l-type and u-type preorder relations

Abstract

In this paper, we study different classes of generalized convex/quasiconvex set-valued maps, defined by means of the l-type and u-type preorder relations, currently used in set-valued optimization. In particular, we identify those classes of set-valued maps for which it is possible to extend the classical characterization of convex real-valued functions by quasiconvexity of their affine perturbations.     

Generalized isotone optimization with applications to starshaped functions

This article considers a curve-fitting problem involving the minimization of the distance from a functionf to a convex cone of functions. A weighted uniform norm is considered as a measure of the distance. The domain of the functions is a partially ordered set, and the convex cone is defined by the isotonicity and nonnegativity conditions on functions. The problem has a linear programming formulation; however, explicit expressions for the optimal solutions have been obtained directly, thereby eliminating the necessity of using linear programming techniques. The results are applied to approximation by starshaped functions.     

Stability of generalized monotone maps with respect to their characterizations

We show that the known types of generalized monotone maps are not stable with respect to their characterizations (i.e. the characterizations are not maintained if an arbitrary map of this type is disturbed by an element with sufficiently small norm) and introduce s-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, s-quasimonotonicity is related to s-quasiconvexity (introduced by Phu in Optimization, 38, 1996) of the underlying function. A necessary and sufficient condition for a univariate polynomial to be s-quasimonotone is given. Furthermore, some stability properties of s-quasiconvex functions are presented.     

The degree of maps of free G-manifolds and the average

Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying map for M, then the degree is one. The condition that the map be equivariant can be relaxed: it is enough to require that it be “nearly equivariant”, up to a positive constant. We will also discuss the G-average construction and show that the requirement that the map be equivariant can be replaced by a somewhat weaker condition involving the average of the map. These results are applied to maps into real, complex and quaternionic Stiefel manifolds. In particular, we show that a nearly equivariant map of a complex or quaternionic Stiefel manifold into itself has degree one. Dedicated to the memory of Jean Leray