Koch曲线的Hausdorff测度上界的研究  

Research on the upper bound of Hausdorff measure of Koch curve

作　　者：郭东亮 李聪端 秦家银 Guo Dongliang;Li Congduan;Qin Jiayin(School of electronics and communication engineering,Sun Yat-sen University,Shenzhen 518107,China;School of electronics and information technology,Sun Yat-sen University,Guangzhou 510006,China)

机构地区：[1]中山大学电子与通信工程学院,广东深圳518107 [2]中山大学电子与信息工程学院,广东广州510006

出　　处：《纯粹数学与应用数学》2021年第1期71-80,共10页Pure and Applied Mathematics

基　　金：国家自然科学基金(61901534);广东省自然科学基金(2018B0303110016);广东省基础与应用基础研究重点项目(2019B1515120032);深圳市科创委面上项目JCYJ20190807155617099。

摘　　要：研究了Koch曲线的Hausdorff测度的上界估计,通过在Koch曲线上构造分形级更高的新覆盖,得到新覆盖与Koch曲线的交集对应的连通弧,并利用相关定理计算出了Koch曲线的Hausdorff测度,得到了更好的上界估计值H^(s)(K)≤0.58764947.这是迄今所知的Koch曲线的Hausdorff测度的最好上界.进一步的分析得出:Koch曲线的Hausdorff测度的精确上界在0.58764946至0.58764947之间.The upper bound estimation of Hausdorff measure of Koch curve is studied.By constructing a new cover with higher fractal level on Koch curve,the connected arc corresponding to the intersection of the new cover and Koch curve is obtained.The Hausdorff measure of Koch curve is calculated by using the relevant theorem,and a better upper bound estimation H^(s)(K)≤0.58764947 is obtained.This is the best upper bound of the Hausdorff measure of Koch curve so far known.Further analysis shows that the exact upper bound of Hausdorff measure of Koch curve is between 0.58764946 and0.58764947.

关 键 词：分形 KOCH曲线 HAUSDORFF测度 上界估计 

分 类 号：O174.12[理学—数学]