Abstract
The Rudin-Shapiro sequence appears both in Fourier Analysis and Number Theory. The Ising chain is a crude model for magnetic substance and plays a fundamental rôle in Statistical Mechanics. The study of patterns of folds on a sheet of paper is linked to many domains including Number Theory and Dynamical Systems. These three concepts are different facets of one and the same object.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.P. Allouche, M. Mendès France, Suite de Rudin-Shapiro et modèle d’Ising, Bull. Soc. Math. France 113 (1985), 273–283.
J.P. Allouche, M. Mendès France, On an extremal property of the Rudin-Shapiro sequence, Mathematika 32 (1985), 33–38.
R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982.
J. Brillhart, L. Carlitz, Note on the Shapiro polynomials, Proc. Amer. Math. Soc. 25 (1970), 114–118.
B.A. Cipra, An introduction to the Ising Model, Amer. Math. Monthly 94 (1987), 937–959.
Ch. Davis, D. Knuth, Number representations and dragon curves, J. Recreational Math. 3 (1970), 61–81; 133–149.
M.Dekking, M. Mendès France, A.J.van der Poorten, Folds !, Math. Intelligencer 4 (1982), 130–138; 173–181; 190–195.
A.O.Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. 13 (1968), 259–265.
H.A. Kramers, G.H. Wannier, Statistics of the two-dimensional ferromagnet, Phys. Rev. 60 (1941), 252–262; 263–276.
J. Loxton, A.J. van der Poorten, Arithmetic properties of the solutions of a class of functional equations, V. reine angew. Math. 330 (1982), 159–172.
J. Loxton, A.J. van der Poorten, Arithmetic properties of automata: regular sequences, V. reine angew. Math. 392 (1988), 57–69.
S.-K. Ma, Statistical Mechanics, World Scientific, 1985.
B.M.McCoy, T.T.Wu, The Two-Dimensional Ising Model, Harvard University Press, 1973.
M.Mendès France, The inhomogeneous Ising chain and paperfolding, in Physics and Number TheorySpringer-Verlag (to appear).
M.Mendès France, G. Tenenbaum, Dimension des courbes planes, pa- piers plies et suites de Rudin-Shapiro, Bull. Soc. Math. France 109 (1981), 207–215.
M.Mendès France, A.van der Poorten, Arithmetic and analytic properties of paperfolding sequences (dedicated to K. Mahler), Bull. Austral. Math. Soc. 24 (1981), 123–131.
L.Onsager, Crystal statistics. A Two-Dimensional Model with an Order-Disorder transition, Phys. Rev 65 (1944), 117–149.
W. Rudin, Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), 855–859.
B.Saffari, Une fonction extrémale liée à la suite de Rudin-Shapiro, C.R.Acad.Sc.Paris 303 (1986), 97–100.
H.S.Shapiro, Extremal problems for polynomials and power series, Thesis MIT, 1951.
C.Thompson, Mathematical Statistical Mechanics, Princeton University Press, 1972.
C.Thompson, Classical Equilibrium Statistical Mechanics, Oxford University Publications, 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
To Paul Bateman, his wife Felice, and his daughter Sally
Rights and permissions
Copyright information
© 1990 Birkhäuser Boston
About this chapter
Cite this chapter
France, M.M. (1990). The Rudin-Shapiro Sequence, Ising Chain, and Paperfolding. In: Berndt, B.C., Diamond, H.G., Halberstam, H., Hildebrand, A. (eds) Analytic Number Theory. Progress in Mathematics, vol 85. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3464-7_23
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3464-7_23
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3481-0
Online ISBN: 978-1-4612-3464-7
eBook Packages: Springer Book Archive