曲面的亏格

Hence the associated surface of minimum genus can be got by doing a sequence of exchangers on some associated surface.

通过对任一关联曲面作一系列调位，从而可得最小亏格的关联曲面，此亏格即为所求图类的最小亏格。

In this paper, firstly, the embeddability of near-triangular graphs on the orientable surface is studied. By using Petersen's Theorem about 1-factor, we first prove that for a near-triangulation on the orientable surface, its geometry dual graph has a 1-factor; then after a procedure of operation leaded by the 1-factor, we show that if a graph G triangulates some orientable surface S_g, then G has a near-triangular embedding in S_h for h = g,g + 1,...,β(G/2」, where β is the Betti number of G. Hence we obtain a conclusion: A near-triangulation of the orientable surface is upper-embeddable. As a generalization, a class of near-quadrangulation is studied, and similar results are obtained.

本文首先研究了近三角剖分图的可定向曲面嵌入性质，通过运用Petersen关于1-因子的定理，首先证明了对于可定向曲面上的三角剖分图，其几何对偶图具有1-因子；然后在1-因子的导向下，通过做一系列增加亏格的手术，证明了如果一个图G三角剖分可定向曲面S_g，那么G可以近三角剖分可定向曲面S_h，这里h=g，g+1，…，「β(G／2」，β是指图G的Betti数，从而得出推论：可定向曲面上的三角剖分图是上可嵌入的，作为推广，又研究了一类近四角剖分图的可定向曲面嵌入性质，并得到类似的结论。

For genus zero closed surfaces,the parameter domain is sphere,so the cutting of the surface meshes in the parametrization process is avoided.

对于零亏格封闭网格曲面，以球面为参数域的参数化方法可以有效避免对网格的切割，是近年来的研究热点。

Chapter 7,discusses some unresolved problems about minimum genus and genus distribution of orientable embeddings for a graph,such as based on the joint tree model,calculation of minimum genus of complete graphs,determination of minimum genera for further types of graphs or even arbitrary graphs,etc.

第七章介绍了图的最小亏格及亏格分布领域中一些需要更进一步研究的问题，如如何利用联树模型计算完全图的最小亏格，以及如何通过刻画最小亏格关联曲面的特性来确定更广泛图类，乃至任意图类的最小亏格问题等等。

This paper addresses the problem of spherical parameterization, i.e., mapping a given polygonal surface of genus-zero onto a unit sphere.

考虑零亏格网格的球面参数化问题，即将给定的零亏格多边形曲面一一映射到单位球面上。

Based on the properties of associated surfaces for complete bipartite graphs,a new type of graph J with weak symmetry is constructed.At the same time,the minimum genus of it is derived.Finally,the number of genus embeddings of complete bipartite graphs is easily estimated.

然后基于完全二部图关联曲面的特点进行推广，构造了对称性比较弱的新图类J且得其最小亏格，最后对完全二部图不同最小亏格嵌入的数目作了估算。

I didn't watch TV last night, because it .

昨晚我没有看电视，因为电视机坏了。

Since this year, in a lot of villages of Beijing, TV of elevator liquid crystal was removed.

今年以来，在北京的很多小区里，电梯液晶电视被撤了下来。