A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl., Christiana, Nr. 1 (1912) 1--67.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....say that w contains a factor of the form xuxux with x 2 A and u 2 A . A word w is called overlap free if it contains no overlap. In the language of patterns an overlap free word is a word which simultaneously avoids the patterns fffifffiff and ff . 1. 2 Previous Results In 1906, Axel Thue [10, 11] proved that there are infinitely many overlap free binary words, or, equivalently: there are arbitrarily long overlap free binary words) To do this, he constructed an infinite overlap free word, a) where is the morphism: A Gamma A a 7 Gamma ab b 7 Gamma ba and (a) ....

A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr., I. Mat. Nat. Kl., Christiana 1 (1912), 1--67. 9

.... such that the order of every k generated subsemigroup of S is smaller than or equal to f(k) The general Burnside problem is the following: Is every periodic semigroup locally finite Morse and Hedlund [33] observed that the existence of an infinite squarefree word over a three letter alphabet [50, 51, 28] shows that the quotient of [f0g by the relations x = 0 is infinite if jAj 3. This semigroup satisfies the identity x and thus the answer is negative for semigroups. Actually, as shown in [6] the monoid presented by hA j x i is infinite even if jAj = 2. Note that, however, the ....

A. Thue (1912), Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr. I. Math. Nat. Kl., Christiana 1, 1--67.

....word is not ultimately periodic, both proven di erently in the literature. Section 4 contains the constructions of in nite sequences of words in four letters with density one for every word, in nite sequences of ternary words which has a limit of their densities at one, using square free words [12, 1, 2], and in nite sequences of binary words which has a limit of their densities at two, using Thue Morse words [11, 10, 1, 2] We also show that these limits are optimal. 2 Preliminaries In this section we x the notations for this paper. We refer to [8, 4] for more basic and general de nitions. ....

....a k ) 1 that is, every position is critical. Moreover, there exists a sequence of words of index one in the alphabet A = fa; b; cg such that the limit of their densities is one. 7 Example 12 (Square free words) Let us consider the endomorphism # : A a 7 abc b 7 ac c 7 b by Thue [12], cf. 1, 2] and let T 2k 1 = a # 2k 1 (a) c and T 2k = a # 2k (a) b; for all k 0, then (T n ) 1 because every word T n has a square pre x and sux and # (a) is square free, so, T n ) jT n j 3 and (T n ) 1 2= jT n j 1) Of course, any square free word with suitable borders ....

Axel Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Det Kongelige Norske Videnskabersselskabs Skrifter, I Mat.-nat.

....if all the symbols in j(0) are distinct (i.e. if t i 6= t j for i 6= j) then the symmetric D0L word w is overlap free, i.e. contains no factor of the form axaxa for any x 2 S and a 2 S. Keywords: overlap free word, D0L word, symmetric morphism 1 Introduction In his classical 1912 paper [15] (see also [3] A. Thue gave the first example of an overlap free infinite word, i.e. of a word which contains no subword of the form axaxa for any symbol a and word x. Thue s example is known now as the Thue Morse word w TM = 01101001100101101001011001101001 : It was rediscovered several ....

A. Thue. Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (

....The algorithm can be extended to report all squares and maximal repetitions in a string. TUCS Research Group Mathematical Structures of Computer Science 1 Introduction Studying regularities in strings has a long tradition in combinatorics on words. Starting from the fundamental work of Thue [23, 24], who considered the existence of infinite repetition free words, many authors investigated such words in various contexts, see for a review [7] In the case of infinite words which are defined by iterated morphisms the problems of repetition freeness become difficult even for a restricted class ....

Thue A., Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreichen, Norske Videnskabers Selskabe Skrifter Mat-Nat. Kl.(Kristania), 1, 1-67, 1912.

....recognition, coding, automata and formal language theory. A widely studied regularity in strings are consecutive occurrences of the same substring. Two consecutive occurrences of the same substring is called an occurrence of a square or a tandem repeat. In the beginning of the last century, Thue [25, 26] showed how to construct arbitrary long strings over any alphabet of more than two characters that contain no squares. Since then a lot of work have focused on developing ecient methods for counting or detecting squares in strings. Several methods that determine if a string of length n contains a ....

A. Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Skrifter udgivet af Videnskabsselskabet i Christiania, Mathematiskog Naturvidenskabeligklasse, 1:1-67, 1912.

....do Lago In these terms the conjecture of Brzozowski we deal with here can be formulated as follows: 8w 2 A ; w] is recognizable. That is, for all words w there is a finite semigroup S and a morphism f : A Gamma S such that f Gamma1 f(w) w] From an application of Thue Morse words [11] and from a piece of work by Brzozowski et al. published in 1971 [2] we know that these Burnside semigroups are infinite. Even though we consider only the cases in which n 2 and m 1 in this work. It can be noted, however, that the idempotent semigroup the Burnside semigroup for which n = 1 ....

A. Thue. Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I Mat. Nat. Kl., 1:1--67, 1912.

.... in di erent areas such as algebra, number theory, game theory (see [15, 21] The oldest results of this kind, dating back to the beginning of the century, are Thue s famous constructions of in nite square free and (strongly) cube free words over alphabets of three and two letters respectively [22, 23] (see also [4] A word is square free (respectively cube free, strongly cube free) if it does not contain a subword uu (respectively uuu, uua) where u is a non empty word and a is the rst letter of u. During the last two decades, di erent generalizations of Thue s results have been studied. A ....

....not contain a subword uua, where u is a non empty word and a is the rst letter of u, then w is called strongly cube free. An equivalent property (see [20] is overlap freeness w is overlap free if it does not contain two overlapping occurrences of a non empty word u. Well known Thue s results [22, 23] state that there exist square free words of unbounded length on the 3 letter alphabet, and strongly cube free words of unbounded length on the 2 letter alphabet. An in nite sequence of strongly cube free words can be constructed by iterating the morphism h(0) 01; h(1) 10, known as Thue Morse ....

A. Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 10:1-67, 1912. 14

....coding, automata and formal language theory. A widely studied regularity in strings are consecutive occurrences of the same substring. Two consecutive occurrences of the same substring is called an occurrence of a square or a tandem repeat. This type of regularity was rst studied by Thue [27, 28] at the beginning of this century. Thue showed that it is possible to construct arbitrary long strings over any alphabet of more than two characters that contain no squares. Since then a lot of work has been done to develop ecient methods to detect or count squares in strings. Several methods [12, ....

A. Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Skrifter udgivne af Videnskabs-Selskabet i Christiania, MathematiskNaturvidenskabelig Klasse, 1:1-67, 1912.

....recognition, coding, automata and formal language theory. A widely studied regularity in strings are consecutive occurrences of the same substring. Two consecutive occurrences of the same substring is called an occurrence of a square or a tandem repeat. In the beginning of the last century, Thue [25, 26] showed how to construct arbitrary long strings over any alphabet of more than two characters that contain no squares. Since then a lot of work have focused on developing ecient methods to count or detect squares in strings. Several methods [12, 18, 23] can determine if a string of length n ....

A. Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Skrifter udgivet af Videnskabsselskabet i Christiania, Mathematisk- og Naturvidenskabeligklasse, 1:1-67, 1912.

....semigroups (these papers contain the definition of what is now called a Thue system ) He was able to solve the word problem in special cases. It was only in 1947 that the general case was shown to be unsolvable independently by E. L. Post [28] and A. A. Markov [24] The other two papers [43, 45] deal with repetitions in finite and infinite words. Perhaps because these papers were published in a journal with restricted availability (this is guessed by G. A. Hedlund [20] this work of Thue was widely ignored during a long time, and consequently some of his results have been rediscovered ....

....a result that was discovered later also in [42] T. Harju [19] gives a result which is similar, but different. Theorem (Satz 9) For every twosided infinite overlap free word x, there exists a unique infinite overlap free word y such that x = y) 1 The mention Satz n refers to theorem n in [45] 5 This gives, in some sense, a complete description of the set of overlap free twosided infinite words; indeed, it means that this set is a minimal set. More precisely, recall that a dynamical system is a set X of infinite words that is closed for the shift operator, defined by T (x) n) x(n ....

A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., Christiana 1912, Nr. 10.

....2 (0) T 3 (0) is a prefix of some (possibly infinite) limit word #,calledtheThue Morse sequence for (#,T,0) This is the least word in lexicographic order such that T (#) #.TospecifyaThue Morse word, it su#ces to list the rewrite rules for T,sincewetakew =0. The famous sequence of Thue [25, 24] and Morse [19, 20] is generated by the rewrite rules 0 # 01 and 1 # 10. It begins 01101001100101101001100101100110 and has many interesting combinatorial properties [2] The sequence at the beginning of this section is generated by the rewrite rules 0 # 001 and 1 # 101. This sequence ....

A. Thue. Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. In T. Nagell, A. Selberg, S. Selberg, and K. Thalberg, editors, Selected Mathematical Papers of Axel Thue, pages 413--477. Universitetsforlaget, Oslo, 1977.

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A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl., Christiana, Nr. 1 (1912) 1--67.

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A. Thue, "U ber die Gegenseitige Lage Gleicher Teile Gewissser Zeichenreinen," Videnskapsselskapets Skrifter. I. Mat.-naturv., Klasse, Kristiania, 1912, pp. 1-- 67.

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A. Thue,  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiana 1 (1912) 1-67; reprinted in Selected Mathematical Papers of Axel Thue, eds. T. Nagell, A. Selberg, S. Selberg and K. Thalberg, Universitetsforlaget, Oslo (1977), pp. 413-477.

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A. Thue. Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. I. Mat. Nat. Kl., 10:1--67, 1912. Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, pp. 413--478.

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A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiana 1 (1912) 1--67; reprinted in Selected Mathematical Papers of Axel Thue, eds. T. Nagell, A. Selberg, S. Selberg and K. Thalberg, Universitetsforlaget, Oslo (1977), pp. 413--477.

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A. Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 10:1-67, 1912.

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A. Thue. Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl. 1 (

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A. Thue. Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 10:1--67, 1912.

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A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr., I. Mat. Nat. Kl., Christiana 1 (1912), 1--67.

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A. Thue.  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912), 1-67. Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, pp. 413-478.

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A. Thue [1912], Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr., I. Mat. Nat. Kl. Christiania I, 1--67.

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A. Thue [1912], Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr., I. Mat. Nat. Kl. Christiania I, 1--67.

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Thue, A. (1912).  Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Kra. Vidensk. Selsk. Skrifter, I. Mat. Nat. Kl., 1912 (1), 1-67.

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