Haros graphs: an exotic representation of real numbers

Author:

Calero-Sanz Jorge1ORCID,Luque Bartolo2,Lacasa Lucas3

Affiliation:

1. Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, ETSIAE, Universidad Politécnica de Madrid , 28040 Madrid, Spain and Signal and Communications Theory and Telematic Systems and Computing, Rey Juan Carlos University, 28942, Fuenlabrada, Madrid, Spain

2. Universidad Politécnica de Madrid Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, ETSIAE, , 28040 Madrid, Spain

3. Institute for Cross-Disciplinary Physics and Complex Systems IFISC (CSIC-UIB), E-07122, Palma de Mallorca, Spain

Abstract

Abstract This article introduces Haros graphs, a construction which provides a graph-theoretical representation of real numbers in the unit interval reached via paths in the Farey binary tree. We show how the topological structure of Haros graphs yields a natural classification of the real numbers into a hierarchy of families. To unveil such classification, we introduce an entropic functional on these graphs and show that it can be expressed, thanks to its fractal nature, in terms of a generalized de Rham curve. We show that this entropy reaches a global maximum at the reciprocal of the Golden number and otherwise displays a rich hierarchy of local maxima and minima that relate to specific families of irrationals (noble numbers) and rationals, overall providing an exotic classification and representation of the reals numbers according to entropic principles. We close the article with a number of conjectures and outline a research programme on Haros graphs.

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Mathematics,Control and Optimization,Management Science and Operations Research,Computer Networks and Communications

Reference34 articles.

1. Exact arithmetic on the Stern-Brocot tree;Niqui,;J. Discrete Algorithms,2007

2. Exact real computer arithmetic with continued fractions;Vuillemin,;IEEE Trans. Comput.,1999

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3