Digital lines, Sturmian words, and continued fractions
In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based descri...
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Uppsala universitet, Matematiska institutionen
2009
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ndltd-UPSALLA1-oai-DiVA.org-uu-1072742013-01-08T13:05:25ZDigital lines, Sturmian words, and continued fractionsengUscka-Wehlou, HannaUppsala universitet, Matematiska institutionenUppsala : Matematiska institutionen2009digital geometrydigital linehierarchy of runscombinatorics on wordsSturmian wordupper mechanical wordcharacteristic wordirrational slopecontinued fractionGauss mapfixed pointDiscrete mathematicsDiskret matematikIn this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator. Doctoral thesis, comprehensive summaryinfo:eu-repo/semantics/doctoralThesistexthttp://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107274urn:isbn:978-91-506-2090-0Uppsala Dissertations in Mathematics, 1401-2049 ; 65application/pdfinfo:eu-repo/semantics/openAccess |
collection |
NDLTD |
language |
English |
format |
Doctoral Thesis |
sources |
NDLTD |
topic |
digital geometry digital line hierarchy of runs combinatorics on words Sturmian word upper mechanical word characteristic word irrational slope continued fraction Gauss map fixed point Discrete mathematics Diskret matematik |
spellingShingle |
digital geometry digital line hierarchy of runs combinatorics on words Sturmian word upper mechanical word characteristic word irrational slope continued fraction Gauss map fixed point Discrete mathematics Diskret matematik Uscka-Wehlou, Hanna Digital lines, Sturmian words, and continued fractions |
description |
In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator. |
author |
Uscka-Wehlou, Hanna |
author_facet |
Uscka-Wehlou, Hanna |
author_sort |
Uscka-Wehlou, Hanna |
title |
Digital lines, Sturmian words, and continued fractions |
title_short |
Digital lines, Sturmian words, and continued fractions |
title_full |
Digital lines, Sturmian words, and continued fractions |
title_fullStr |
Digital lines, Sturmian words, and continued fractions |
title_full_unstemmed |
Digital lines, Sturmian words, and continued fractions |
title_sort |
digital lines, sturmian words, and continued fractions |
publisher |
Uppsala universitet, Matematiska institutionen |
publishDate |
2009 |
url |
http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107274 http://nbn-resolving.de/urn:isbn:978-91-506-2090-0 |
work_keys_str_mv |
AT usckawehlouhanna digitallinessturmianwordsandcontinuedfractions |
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1716508539235598336 |