Number Theory

Automatic Sequences: Theory, Applications, Generalizations by Jean-Paul Allouche

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By Jean-Paul Allouche

Combining innovations of arithmetic and desktop technological know-how, this e-book is set the sequences of symbols that may be generated through basic types of computation known as ''finite automata''. compatible for graduate scholars or complex undergraduates, it begins from straightforward rules and develops the fundamental conception. The research then progresses to teach how those rules could be utilized to unravel difficulties in quantity thought and physics.

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Extra resources for Automatic Sequences: Theory, Applications, Generalizations

Example text

Show that this factorization into palindromes is in fact unique. 9 Exercises 29 51. Let x, y ∈ + . We say x is a conjugate of y, and we write x ∼ y, if there exist u, v ∈ ∗ such that x = uv and y = vu. In other words, x is a cyclic shift of the symbols of y. (a) Show that ∼ is an equivalence relation. (b) Suppose x ∈ + , j ≥ 1, is an integer, and there exist u, v ∈ ∗ such that x j = uv. Then there exist r, s ∈ ∗ such that x = r s, and vu = y j for y = sr . (c) Suppose w is primitive and w ∼ x. Then x is primitive.

3 above, this implies that cv is the image by µ of a binary word. Let r, s, t be words in 2∗ such that µ(r ) = x, µ(s) = cv, and µ(t) = cy. Then µ(w) = µ(r )µ(s)µ(s)µ(t) = µ(r sst). Hence w = r sst. But µ(s) and µ(t) both begin with c; hence s and t both begin with c. This implies that sst (and hence w) contains an overlap. Case 2: |x| is odd. Hence |y| is even. Then µ(w) = (xc)(vcvc)y, so by the same reasoning as in case 1, there exist r, s, t ∈ 2∗ such that µ(r ) = xc, µ(s) = vc, and µ(t) = y.

The word z is overlap-free. Since |z| < |x|, the induction hypothesis shows that z = uµ(y)v, where u, v ∈ { , 0, 1, 00, 11}. If u = or if u = a, then x = (au)µ(y)v gives the desired factorization. If u = a, then x = (aa)µ(y)v and we get the factorization x = µ(ay)v. If u = a a, then x begins with aaa, which is impossible. If u = a a, then x = a a a µ(y)v. If |y| = 0, then x = a a a v. Hence v ∈ { , a, aa}. The three corresponding desired factorizations for x are respectively x = µ(a)a, x = µ(aa), and x = µ(aa)a.

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