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.     

Quotient probabilistic normed spaces and completeness results

We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.     

Sobolev-Type Classes of Functions with Values in a Metric Space. II

We prove equivalence of the definitions by the author and by Korevaar and Schoen of the Sobolev classes of mappings of a domain of an arithmetic n-dimensional space to a metric space.     

Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions

Let X be a Banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict Opial condition. Let C be a bounded closed convex subset of X, KC(X) the family of all compact convex subsets of X and T a nonexpansive mapping from C into KC(X) with bounded range. We prove that T has a fixed point. The nonstrict Opial condition can be removed if, in addition, T is an 1-χ-contractive mapping.     

On Covariant Phase Space and the Variational Bicomplex

The notion of a phase space in classical mechanics is well known. The extension of this concept to field theory however, is a challenging endeavor, and over the years numerous proposals for such a generalization have appeared in the literature. In this paper We review a Hamiltonian formulation of Lagrangian field theory based on an extension to infinite dimensions of J.-M. Souriau's symplectic approach to mechanics. Following G. Zuckerman, we state our results in terms of the modern geometric theory of differential equations and the variational bicomplex. As an elementary example, we construct a phase space for the Monge–Ampere equation.     

GENERALIZED CIRCLES AND THEIR CONFORMAL MAPPING IN A SUBSPACE OF A WEYL SPACE

The authors give the necessary and sufficient conditions for a generalized circle in a Weyl hypersurface to be generalized circle in the enveloping Weyl space. They then obtain the neccessary and sufficient conditions under which a generalized concircular transformation of one Weyl space onto another induces a generalized transformation oil its subspaces. Finally, it is shown that any totally geodesic or totally umbilical hypersurface of a generalized concircularly flat Weyl space is also generalized concircularly flat.     

一些随机重合点定理

本文引入了一种满足更一般的收缩不等式的多重函数类,并证明了属于该类的可测多重函数对的一些随机重合点定理。     

Complex Geometry of the Universal Teichmuller Space

We prove that all invariant distances on the universal Teichmuller space agree and are determined by the Grunsky coefficients of the naturally related conformal maps. This fact yields various important consequences; in particular, we obtain solutions of certain well-known geometric problems in complex analysis and related fields.     

K-强凸空间的一些性质

结合Banach空间的Drop性，利用K维体积给出了K—强凸空间的一个新的定义，同时也给出了K—强光滑空间定义的K维体积表示，然后利用单位圆的切片证明了K—强凸空间是自反空间，进而证明了K—强凸空间与K—强光滑空间是一对对偶空间．最后利用Drop性的切片描述证明了K—强凸空间具有Drop性．     

逼近严格伪压缩映象不动点的Ishikawa迭代程序

Let(X,‖·‖ ) be a Banach space.Let K be a nonempty closed,convex subset of Xand T∶K→K.Assume that T is Lipschitzian,i.e.there exists L>0 such that‖ T(x) -T(y)‖≤ L‖ x -y‖for all x,y∈K.Withoutloss of generality,assume that L≥ 1 .Assume also that T is strictly pseudocontractive.According to[1 ] this may be statedas:there exists k∈ (0 ,1 ) such that‖ x -y‖≤‖ x -y + r[(I -T -k I) x -(I -T -k I) y]‖for all r>0 and all x,y∈ K.Throughout,let N denote the set of positive in…