Paperfolding, Automata, and Rational Functions - Diagonals and ...

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Hence, since Fp is finite, there are only a finite number of distinct yα 1...αe and we have a result of Christol, Kamae, Mendès France, and Rauzy, here proved by the elegant independent methods of Jan Denef and Leonard Lipshitz. Theorem. If y = P aνx ν is algebraic then (F) there is an e such that for every (α1, . . . , αe) ∈ Se there is an e ′ < e and a (β1, . . . , βe ′) ∈ Se′ such that yα 1...αe = yβ 1...β e ′ ; (A) equivalently, there is an e such that for all j = (j1, . . . , jn) with the ji < pe there is an e ′ < e and a j ′ = (j ′ 1 , . . . , j′ n) with the j ′ i < pe′ such that ap e ν+j = a p e ′ ν+j ′ for all ν . Conversely, if y satisfies (F) then taking yβ 1...β e ′ and breaking it up e − e ′ times, we see that the yα 1...αe satisfy a system of the form yα 1...αe = X x γ y pe−e′ β 1...βe . By an easy lemma it follows that the yα 1...αe , and hence y , must be algebraic. 33
Suppose the series P aνx ν is generated by a finite automaton M . Choose e so large that every state that M enters in the course of any computation is entered in a computation of length less than e . Then the aν satisfy version (A) of the Theorem. Conversely, if the aν satisfy (A) one can construct a finite automaton M that generates P aνx ν . M will be equipped with a table detailing the identifications (A) and an output list of the values of the aj for j = (j1, . . . , jn) with the ji and iterates. At each stage it outputs the appropriate value from its output list. 34
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Paperfolding, Automata, and Rationa
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Paperfolding, Automata, and Rationa
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Paperfolding Take a rectangular she
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A Mahler Functional Equation Aside.
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Next, if we pair the sequence Regul
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Next, if we pair the sequence Regul
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The uniform, or regular, 2-substitu
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Characteristic Functions I found th
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An Algebraic Equation in Characteri
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These remarks show that the paperfo
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These remarks show that the paperfo
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The Thue-Morse Sequence 0 1 10 11 1
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The Thue-Morse Sequence 0 1 10 11 1
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Euler’s Identity and a Functional
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An Algebraic Equation The function
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A Counter-example in Analysis I cla
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The Shapiro Sequence Consider, the
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A Remark on the Shapiro Function Se
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Transcendence of Automatic Numbers
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Complexity of a Sequence Given an i
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Complexity of a Sequence Given an i
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Algebraicity and Automaticity The p
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Comments on the Proof The best proo
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Page 141 and 142: But for expansions over the complex
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References MICHEL DEKKING, MICHEL M
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References MICHEL DEKKING, MICHEL M
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