{"title":"General multifractal dimensions of measures","authors":"Bilel Selmi","doi":"10.1016/j.fss.2024.109177","DOIUrl":null,"url":null,"abstract":"<div><div>The main aim of this paper is to present a comprehensive exploration of general multifractal dimensions of Borel probability measures, with a specific emphasis on the diverse methodologies available for their estimation or computation. We prove that these general dimensions enable us to compare a probability measure <em>ϑ</em> with the general multifractal Hausdorff or packing measure. Also, we aim to measure the degree of regularity or singularity of <em>ϑ</em>, and we examine the correlations between various approaches to defining the general multifractal exact dimension of Borel probability measures, using fractal analysis of the essential supports of these measures as the foundation.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424003233","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
The main aim of this paper is to present a comprehensive exploration of general multifractal dimensions of Borel probability measures, with a specific emphasis on the diverse methodologies available for their estimation or computation. We prove that these general dimensions enable us to compare a probability measure ϑ with the general multifractal Hausdorff or packing measure. Also, we aim to measure the degree of regularity or singularity of ϑ, and we examine the correlations between various approaches to defining the general multifractal exact dimension of Borel probability measures, using fractal analysis of the essential supports of these measures as the foundation.
期刊介绍：
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.
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