华东师范大学学报(自然科学版) ›› 2008, Vol. 2008 ›› Issue (5): 78-83,1.

• 应用数学,统计学 • 上一篇    下一篇

Hausdorff测度和维数与所在空间度量的依赖性

陈海龙1, 桂咏新2   

  1. 1. 华东师范大学 数学系, 上海 200062;2. 咸宁学院 数学系, 咸宁 437005
  • 收稿日期:2007-10-01 修回日期:2008-01-10 出版日期:2008-09-25 发布日期:2008-09-25
  • 通讯作者: 陈海龙

Dependence of Hausdorff measure and dimension with the metric of the underlying space(Chinese)

CHEN Hai-long1, GUI Yong-xin2   

  1. 1. Department of Mathematics, East China Normal University, Shanghai 200062, China;2. Department of Mathematics, Xian Ning University, Xian Ning 437005, China
  • Received:2007-10-01 Revised:2008-01-10 Online:2008-09-25 Published:2008-09-25
  • Contact: CHEN Hai-long

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摘要： 讨论在Rn中具有嵌套结构的几何对象上构造新度量的问题. 主要结果是: K是Rn中 具有嵌套结构的几何对象, 那么对于任何一个的连续的纲函数h(t), 可以构造一个在K上的度量,在这个新的度量空间(K, p)中, 0<Hh (K)<+∞. 如果纲函数取h(t)=ts,那么 对于任意0K上的度量, 在 这个新的度量空间(K, p)中,Hh (K)=1并且有dimPK=dimBK=dimHK=s成立.

关键词: 嵌套结构, Hausdorff测度, 度量 , 嵌套结构, Hausdorff测度, 度量

Abstract: This paper discussed the construction of metric space on the nested geometrical object. Given a nested geometrical object K in Rn and a continuous gauge function h(t), a new metric p was constructed on K such that 0<Hh (K)<+∞ in the new metric space (K, p). Particularly, if the gauge function is h(t)=ts, then for any positive finite number s, it's also possible to construct a new metric p on K such that ~$\mathscrHh (K)=1 and
dimPK=dimBK=dimHK=s.

Key words: Hausdorff measure, metric, nested geometrical object, Hausdorff measure, metric

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