By Edward B. Burger

ISBN-10: 1598034200

ISBN-13: 9781598034202

2 DVD set with 24 lectures half-hour every one for a complete of 720 minutes...

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Example text

240] first raised the problem of the existence of infinite abelian squarefree words. ) Evdokimov [1968] constructed such a sequence on 25 symbols. Pleasants [1970] improved this to 5 symbols. T. Brown [1971] gave a survey on constructing strongly nonrepetitive sequences. Entringer, Jackson, and Schatz [1974] proved that every infinite word over a 2-letter alphabet contains arbitrarily long abelian squares. Ker¨anen [1992] solved Erd˝os’s problem by exhibiting a strongly nonrepetitive sequence over a 4-letter alphabet.

Show that Jacobs’s “Mephisto waltz” infinite word 001001110001001110110110001 · · · , obtained by iterating the morphism which maps 0 → 001 and 1 → 110, is fourth-power-free. 17. (Berstel) A Langford string is a nonempty word over the infinite alphabet {2, 3, 4, . . } such that two consecutive occurrences of the letter a are separated by exactly a − 1 letters. For example, 24262425262 is a Langford string. Prove that every Langford string is squarefree. 26 Stringology 18. 12) of the game of chess, a player can claim a draw if “at least 50 consecutive moves have been made by each side without the capture of any piece and without the movement of any pawn”.

032. 0416. 109999. For more on enumeration of ternary squarefree words, see Baake, Elser, and Grimm [1997]. Berstel [1979] showed it was decidable that an infinite word on three letters obtained as the fixed point of a morphism is squarefree. De Luca [1984] investigated the question of when the product of squarefree words is squarefree. Although infinite binary words cannot be squarefree, it is possible to avoid arbitrarily long squares. Such a word is sometimes said to be of bounded repetition. This line was pursued by Entringer, Jackson, and Schatz [1974]; Dekking [1976]; Prodinger and Urbanek [1979]; Prodinger [1983]; Allouche [1984]; P.

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An Introduction to Number Theory 2 DVD Set with Guidebook by Edward B. Burger

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