New PDF release: AdobePhotoshopCS3: официальный учебный курс

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The proof is similar to the previous one. Thanks to the factor (01) of f , we verify that 0 = (0) is a return word of w in f . Thanks to the factor (10l 1) of f , we verify that 0k 1 = (1) is the other return word of w in f . The case (0) = 1 and (1) = 1k 0 is similar. Finally, if | (0)| > | (1)| the proof is analogous. Remark 22. In Theorem 21, statement (c) may be substituted by “f is a morphic word”. Indeed, a characteristic Sturmian word is morphic if and only if it is the fixed point of a (non-erasing, non-trivial) morphism.

Lemma 17. Let f be the characteristic Sturmian word of slope , where has continued fraction expansion [0, a1 + 1, a2 , . ]. M. Araújo, V. Bruyère / Theoretical Computer Science 340 (2005) 204 – 219 (a) E(f ) = f1− and 1 − has continued fraction expansion • [0, 1, a1 , a2 , . ] if a1 > 0 and • [0, a2 + 1, a3 , . ] if a1 = 0. (b) If (tn )n and (sn )n are the standard sequences associated to f and f1− , respectively, then • if a1 > 0 then E(tn ) = sn+1 for all n 0 and • if a1 = 0 then E(tn ) = sn−1 for all n 1.

The next proposition is Proposition 9 in the case n = 1. Proposition 25. Let n = 1 and i ∈ {1, . . , a2 }. Let w be a prefix of f of length in the interval ]iq1 + q0 − 2, (i + 1)q1 + q0 − 2] = [iq1 , (i + 1)q1 − 1]. The shortest return word of w is t1 , and its first occurrence is 0 if i < a2 , a3 q2 if i = a2 . The longest return word of w is t1i t0 , and its first occurrence is (a2 − i)q1 . M. Araújo, V. Bruyère / Theoretical Computer Science 340 (2005) 204 – 219 Fig. 5. Illustration of an occurrence of t1i in ]a3 q2 , q3 [.

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AdobePhotoshopCS3: официальный учебный курс


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