Projection theorems for intermediate dimensions

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2019-07-17 DOI:10.4171/JFG/99

Stuart A. Burrell, K. Falconer, J. Fraser

引用次数: 17

Abstract

Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set $E \subset \R^n$ onto almost all $m$-dimensional subspaces depend only on $m$ and $E$, that is, they are almost surely independent of the choice of subspace. Our approach is based on `intermediate dimension profiles' which are expressed in terms of capacities.

查看原文本刊更多论文

中间维的投影定理

中间维最近被引入到分形的豪斯多夫维和盒计数维之间进行插值。首先，我们证明了这些中间维度可以根据某些核的容量来定义。然后，在此基础上，我们证明了一个集合$E \子集$ R^n$在几乎所有$m$维子空间上的投影的中间维数只依赖于$m$和$E$，也就是说，它们几乎肯定与子空间的选择无关。我们的方法是基于“中间维度轮廓”，用能力来表达。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.