关于rq-凸性相关问题的研究-应用数学专业论文.docxVIP

AbstractA Abstract A set M C Rd is called rq-convex if, for any two distinct points z，Y∈M，there Z，W∈M such that conv{x，Y，z，叫)is non-degenerate rectangle． In this thesis we first discribe some non-discrete rq-convex sets according to the definition，and then probe into the rq-convexity about discrete point sets in the plane and the vertex sets of Platonic solids．The main results follows：(1)Discuss the rq-convexity of closed unbounded sectors in the plane，squares and the complement of any bounded set in Rd(d≥2)；give necessary and sufficient condition on the difference set of triangle being rq-convex；and provide some examples of non-connected sets which not rqoconvex．(2)Study the rq-convexity of eleven Archimedean tilings finfinite sets)，and prove that the vertex sets of Archimedean tilings(38)，(44)，(63)，(3．6．3．6)， (34．6)，(32．4．3．4)，(4．82)are rq-convex sets，while the vertex sets of Archimedean tilings (33．42)，(3．4．6．4)，(4．6．12)，(3．122)are not．(3)Investigate the rq-convexity about finite point sets in the plane，and obtain an upper bound of the rq-convex completion number of any finite point set；characterize the configurations of 6-point rq-convex sets and 8- point rq-convex sets，and prove that the smallest odd cardinality of an rq-convex set i11 R2 is 9．(4)Investigate the rq-convexity about the vertex sets of Platonic solids in R3， and prove that the vertex set of the regular tetrahedron is not rq-convex and find its rq-convex completion number is 3． Keywords：rq-convexity；Archimedean tiling；finite point set；rq-convex completion number V 万方数据 目 目 录 中文摘要 ．．III 英文摘要 ．．V 引言 ． 1 第一章基本概念和一些rg-凸集 ．． 5 第二章阿基米德铺砌顶点集的7．g-凸性 ． 9 第三章有限点集的rg．凸性 ．35 3．1 R2中有限点集的rg．凸性 ··35 3．2柏拉图体顶点集的rg-凸性 ·51 结论 ．53 参考文献 ．．55 后记 ．57 攻读学位期问取得的科研成果清单 ．59 ⅥI 万方数据 引 引 言 凸性理论是离散与组合几何学的基础研究内容之一，在美国数学会(AMS)的学科 分类中，将其列为专门的一个子类(52AXX)．凸性理论的研究可以追溯到阿基米德时代， 近代凸性理论则是二十世纪初由Minkowski等人创始的．二十世纪中叶，凸性在优化理 论中的应用引发了国际数学界对凸性理论的广泛关注，促进了凸性理论研究的发展．二 十世纪七十年代以后，凸性理论不仅成为线性规划，非线性规划，最优化理论，对策论， 数理经济学，计算数学等学科的理论基础，也是现代概率论，Banach