{"title":"大型水平集图像的尺寸","authors":"A. O’Farrell, G. Armstrong","doi":"10.7146/math.scand.a-129246","DOIUrl":null,"url":null,"abstract":"Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\\colon E\\to \\mathbb{R} $, where $E$ is some interval on the line having positive length. For $0<\\alpha <1$ let $I_\\alpha (f)$ denote the set of values $y\\in \\mathbb{R} $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\\alpha$. We consider how large can be the Hausdorff dimension of $I_\\alpha (f)$, as $f$ ranges over the set of all $k$-times continuously differentiable functions from $E$ into $\\mathbb{R} $. We show that the sharp upper bound on $\\dim I_\\alpha (f)$ is $(1-\\alpha )/k$.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dimension of images of large level sets\",\"authors\":\"A. O’Farrell, G. Armstrong\",\"doi\":\"10.7146/math.scand.a-129246\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\\\\colon E\\\\to \\\\mathbb{R} $, where $E$ is some interval on the line having positive length. For $0<\\\\alpha <1$ let $I_\\\\alpha (f)$ denote the set of values $y\\\\in \\\\mathbb{R} $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\\\\alpha$. We consider how large can be the Hausdorff dimension of $I_\\\\alpha (f)$, as $f$ ranges over the set of all $k$-times continuously differentiable functions from $E$ into $\\\\mathbb{R} $. We show that the sharp upper bound on $\\\\dim I_\\\\alpha (f)$ is $(1-\\\\alpha )/k$.\",\"PeriodicalId\":49873,\"journal\":{\"name\":\"Mathematica Scandinavica\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2020-10-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematica Scandinavica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7146/math.scand.a-129246\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Scandinavica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7146/math.scand.a-129246","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\colon E\to \mathbb{R} $, where $E$ is some interval on the line having positive length. For $0<\alpha <1$ let $I_\alpha (f)$ denote the set of values $y\in \mathbb{R} $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\alpha$. We consider how large can be the Hausdorff dimension of $I_\alpha (f)$, as $f$ ranges over the set of all $k$-times continuously differentiable functions from $E$ into $\mathbb{R} $. We show that the sharp upper bound on $\dim I_\alpha (f)$ is $(1-\alpha )/k$.
期刊介绍:
Mathematica Scandinavica is a peer-reviewed journal in mathematics that has been published regularly since 1953. Mathematica Scandinavica is run on a non-profit basis by the five mathematical societies in Scandinavia. It is the aim of the journal to publish high quality mathematical articles of moderate length.
Mathematica Scandinavica publishes about 640 pages per year. For 2020, these will be published as one volume consisting of 3 issues (of 160, 240 and 240 pages, respectively), enabling a slight increase in article pages compared to previous years. The journal aims to publish the first issue by the end of March. Subsequent issues will follow at intervals of approximately 4 months.
All back volumes are available in paper and online from 1953. There is free access to online articles more than five years old.