Download Combinatorics, automata, and number theory by Valérie Berthé, Michel Rigo PDF

By Valérie Berthé, Michel Rigo

This collaborative quantity provides contemporary traits coming up from the fruitful interplay among the subjects of combinatorics on phrases, automata and formal language conception, and quantity concept. featuring a number of very important instruments and ideas, the authors additionally show a number of the fascinating and critical relationships that exist among those diversified fields. subject matters comprise numeration platforms, note complexity functionality, morphic phrases, Rauzy tilings and substitutive dynamical structures, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic homes of electronic capabilities, decidability concerns for D0L structures, matrix items and joint spectral radius. issues are provided in a fashion that hyperlinks them to the 3 major topics, but in addition extends them to dynamical structures and ergodic concept, fractals, tilings and spectral homes of matrices. Graduate scholars, learn mathematicians and machine scientists operating in combinatorics, conception of computation, quantity concept, symbolic dynamics, fractals, tilings and stringology will locate a lot of curiosity during this booklet.

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In particular, the number of words of length n ≥ 0 over {0, 1} not containing the factor “11” is Fn + Fn−1 = Fn+1 . Indeed, one has to count paths of length n from the initial state to one of the two terminal states. So we sum up the first two entries on the first row. The same kind of idea can be applied to morphisms. Let σ : A∗ → A∗ be an endomorphism. The matrix Mσ ∈ NA×A associated with σ is called the incidence matrix of σ and is defined by ∀a, b ∈ A, (Mσ )a,b = |σ(b)|a . Let us recall that P stands for the abelianisation map.

8 Let σ be a primitive substitution. 19. We denote it by (Xσ , S). Preliminaries 47 Let us quote an interesting property of substitutive dynamical systems associated with a primitive substitution. For more details, see (Queff´elec 1987) and (Pytheas Fogg 2002). 9 Let σ be a primitive substitution. The system (Xσ , S) is uniquely ergodic. 5. 1 Show that over a binary alphabet, any word of length ≥ 4 contains a square as factor. 2 Show that over a ternary alphabet A, it is possible to build an infinite word avoiding squares.

Fig. 5. Two redundancy transducers. † It has been given a renewed interest in computer arithmetic for the redundancy † It was known (at least) as early as Cauchy who advocated such system for p = 10 and h = 5 with the argument that it makes the learning of addition and multiplication easier: the size of the tables is rougly divided by 4 (see (Cauchy 1840)). 60 Ch. Frougny, J. Sakarovitch in the representations allows to improve the way operations are performed, as we shall see now. The following is to be found in (Avizienis 1961) for bases larger than 2, in (Chow and Robertson 1978) for the binary case — although the original statements and proofs are not formulated in terms of automata.

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