{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/26109"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/26109","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Iterates of functions defined in terms of digital representations of the integers","abstract":"For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. In base 10, Conway discovered the curious sequence: 12333334444, 55666667777, 123333334444, 556666667777, .... Although the sequence is not periodic, it does display some periodic-like behavior which we refer to as “quasiperiodic.” Conway conjectured that all RATS sequences in base 10 are either eventually periodic, or they eventually lead to the previously mentioned quasiperiodic sequence. In this thesis, we study RATS sequences in various bases. In particular, we prove an Erd ̋os-Kac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverse-add process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome.","abstract_html":"For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. In base 10, Conway discovered the curious sequence: 12333334444, 55666667777, 123333334444, 556666667777, .... Although the sequence is not periodic, it does display some periodic-like behavior which we refer to as “quasiperiodic.” Conway conjectured that all RATS sequences in base 10 are either eventually periodic, or they eventually lead to the previously mentioned quasiperiodic sequence. In this thesis, we study RATS sequences in various bases. In particular, we prove an Erd ̋os-Kac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverse-add process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome.","abstract_has_math":false,"creators":["Thiel, Johann A."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hildebrand, A.J.","Berndt, Bruce C.","Reznick, Bruce A.","Stolarsky, Kenneth B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-08-25T22:14:28Z","date_published":"2011-08-25T22:14:28Z","updated_at":"2026-07-22T22:25:26Z","subjects":["Conway's RATS","iterative process","Lehmer","palindromes","problem 196","discrete dynamical systems","quasiperiodic","Erd ̋os-Kac","base","Lyndon words","Reverse-Add-Then-Sort (RATS)"],"languages":["en"],"rights":["Copyright 2011 Johann A. 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In this thesis, we study RATS sequences in various bases. In particular, we prove an Erd ̋os-Kac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverse-add process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-06-30T21:29:23Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Thiel_Johann.pdf: 605778 bytes, checksum: 5e4569f88ee1c946a1b4938f03663723 (MD5)","Made available in DSpace on 2011-08-25T22:14:28Z (GMT). No. of bitstreams: 2 Thiel_Johann.pdf: 605778 bytes, checksum: 5e4569f88ee1c946a1b4938f03663723 (MD5) license.txt: 4061 bytes, checksum: ed02214ef8e45bff6a871f116a3d6833 (MD5)"]},{"key":"dc:title","label":"Title","values":["Iterates of functions defined in terms of digital representations of the integers"]}]}],"canonical_facts":{"dc:contributor":["Hildebrand, A.J.","Berndt, Bruce C.","Reznick, Bruce A.","Stolarsky, Kenneth B."],"dc:creator":["Thiel, Johann A."],"dc:date":["2011-08-25T22:14:28Z","2011-08"],"dc:description":["For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. In base 10, Conway discovered the curious sequence: 12333334444, 55666667777, 123333334444, 556666667777, .... Although the sequence is not periodic, it does display some periodic-like behavior which we refer to as “quasiperiodic.” Conway conjectured that all RATS sequences in base 10 are either eventually periodic, or they eventually lead to the previously mentioned quasiperiodic sequence. In this thesis, we study RATS sequences in various bases. In particular, we prove an Erd ̋os-Kac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverse-add process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-06-30T21:29:23Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Thiel_Johann.pdf: 605778 bytes, checksum: 5e4569f88ee1c946a1b4938f03663723 (MD5)","Made available in DSpace on 2011-08-25T22:14:28Z (GMT). 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