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

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Paperfolding Take a rectangular sheet of paper and fold it in half, right half over left. Repeat the operation, and again, and again, and . . . . On unfolding the sheet of paper we see the creases as a sequence of valleys← 1 and ridges← 0. Then, reading from left to right, after five folds we obtain 1 1 0 1 0 1 1 0 0 We now recall that after one, two, three, four, five folds we saw 1 110 1101100 110110011100100 1101100111001001110110001100100 so we obtain an infinite paperfolding sequence by adding a new middle fold 1 and noting that the second half of the sequence is the reverse mirror image of the first half. This justifies my adding . . . after those thirty-one creases. 5
Paperfolding Take a rectangular sheet of paper and fold it in half, right half over left. Repeat the operation, and again, and again, and . . . . On unfolding the sheet of paper we see the creases as a sequence of valleys← 1 and ridges← 0. Then, reading from left to right, after five folds we obtain 1 1 0 1 1 0 1 1 0 0 We now recall that after one, two, three, four, five folds we saw 1 110 1101100 110110011100100 1101100111001001110110001100100 so we obtain an infinite paperfolding sequence by adding a new middle fold 1 and noting that the second half of the sequence is the reverse mirror image of the first half. This justifies my adding . . . after those thirty-one creases. 5
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Page 7 and 8: Paperfolding Take a rectangular she
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Page 31 and 32: A Mahler Functional Equation Aside.
Page 47 and 48: Next, if we pair the sequence Regul
Page 49 and 50: Next, if we pair the sequence Regul
Page 51 and 52: The uniform, or regular, 2-substitu
Page 61 and 62: Characteristic Functions I found th
Page 63 and 64: Characteristic Functions I found th
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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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But for expansions over the complex
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Power Series in Several Variables R
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Diagonals and Hadamard Products The
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Diagonals and Hadamard Products The
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A Theorem of Furstenberg It follows
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As said, in characteristic zero, ne
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A Beautiful Transcendence Argument
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A Beautiful Transcendence Argument
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Breaking Up in Characteristic p The
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Algebraic power series in character
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Hence, since Fp is finite, there ar
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Hence, since Fp is finite, there ar
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Suppose the series P aνx ν is gen
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So we have: Theorem. P aνx ν ∈
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The Lifting Theorem Looking careful
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Diagonals of Rational Functions Now
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Whilst (i) and (ii) are reasonably
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References MICHEL DEKKING, MICHEL M
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Paperfolding, Automata, and Rational Functions - Diagonals and ...

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