General multifractal dimensions of measures

IF 3.2 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Fuzzy Sets and Systems Pub Date : 2024-11-05 DOI:10.1016/j.fss.2024.109177

Bilel Selmi

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.

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测量的一般多分形维度

本文的主要目的是全面探讨Borel概率度量的一般多分形维度，并特别强调了估算或计算这些维度的各种方法。我们证明，这些一般维度使我们能够比较概率度量ϑ与一般多分形Hausdorff或打包度量。此外，我们还旨在测量ϑ的规则性或奇异性程度，并以这些度量的基本支撑的分形分析为基础，研究了定义博尔概率度量的一般多分形精确维度的各种方法之间的关联。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Fuzzy Sets and Systems 数学-计算机：理论方法

CiteScore

6.50

自引率

17.90%

发文量

321

审稿时长

6.1 months

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.