The Cycle by Cycle Subtraction Process

The Cycle by Cycle Subtraction Process | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 3 December 2025 V1 Latest version Share on The Cycle by Cycle Subtraction Process Author : Anshveer Bindra 0009-0008-3593-1450 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176479451.12111592/v1 198 views 156 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We investigate the discrete dynamical system generated by the cycle-by-cycle subtraction map F_π (x) = |x − π(x)|, acting on k-digit base-B integers. The map induces a finite directed graph whose terminal components are finite cycles ("loops"). We develop a complete structural description of these loops by analysing the monogenic monoid C = ⟨F_π ⟩, and establish its fundamental properties, including the structure of its submonoids, the conjugacy-class invariance of loop counts, commutativity phenomena, and the nonexistence of units. We then classify the number and lengths of loops, derive explicit formulae for principal ideals, and give a full description of Green's relations in C, including the behaviour of exponents, the periodic subgroup, and the unique idempotent. This structural theory is complemented by asymptotic results describing the growth of loop counts for large bases and digit lengths, together with average-case estimates for random permutations, including the expected number of loops and probabilistic concentration bounds. We further identify permutations that extremise loop complexity and characterise the mechanisms by which their orbit structure differs from the typical case. A brief extension outlines how the main ideas adapt to semigroups generated by multiple digit-permutation subtraction maps. Overall, the paper provides a unified algebraic and asymptotic framework for understanding subtraction-permutation dynamics and the loops arising from a single map F_π . Supplementary Material File (cycle_by_cycle_subtraction_process.pdf) Download 553.00 KB Information & Authors Information Version history V1 Version 1 03 December 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords algebraic number theory dynamical systems number theory permutations semigroup theory Authors Affiliations Anshveer Bindra 0009-0008-3593-1450 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 198 views 156 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Anshveer Bindra. The Cycle by Cycle Subtraction Process. Authorea . 03 December 2025. DOI: https://doi.org/10.22541/au.176479451.12111592/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. 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