Papers by Michel Mendès-France
Uniform distribution one, A geometrical viewpoint
On lacunary formal power series and their continued fraction expansion
Abstract. We note that the continued fraction expansion of a lacunary formal power series is a fo... more Abstract. We note that the continued fraction expansion of a lacunary formal power series is a folded continued fraction with monomial partial quotients, and with the property that its convergents have denominators that are the sums of distinct monomials, that is, they are polynomials with coefficients 0, 1, and��� 1 only. Our results generalise, simplify and refine remarks of a previous note 'Convergents of folded continued fractions'
Folds! II: Symmetry disturbed
The Mathematical Intelligencer, 1982
Entropy, degrees of freedom, and free climbing: A thermodynamic study of a complex behavior based on trajectory analysis
International journal of sport psychology
... thermodynamic study of a complex behavior based on trajectory analysis. Cordier, Patrick; Fra... more ... thermodynamic study of a complex behavior based on trajectory analysis. Cordier, Patrick; France, Michel Mendès; Bolon, Philippe; Pailhous, Jean. International Journal of Sport Psychology, Vol 24(4), Oct-Dec 1993, 370-378. Abstract. ...
Dimensions des spirales
Bulletin de la Société mathématique de France
Le Journal de Physique Colloques
Un automate fini est une machine qui engendre des suites dbterministes (p6riodiques ou non pbriod... more Un automate fini est une machine qui engendre des suites dbterministes (p6riodiques ou non pbriodiques). Nous 6tudions de telles suites et nous considhrons des chafnes d'Ising dont la suite des coefficients de couplage est "quasicrystalline", c'est-L-dire engendrke par automate fini.
De nouveaux curieux produits infinis. (On new curious infinite products)
Acta Arithmetica
Paper folding, theta functions and the circle method
Acta Mathematica
J. J. Sylvester’s two convex sets theorem and G.-L. Lesage’s theory of gravity
Chaotic curves
Lecture Notes in Biomathematics, 1983
The Planck constant of a curve
Fractal Geometry and Analysis, 1991
On lacunary formal power series and their continued fraction expansion
Proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the 60th Birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997, 1999
Abstract. We note that the continued fraction expansion of a lacunary formal power series is a fo... more Abstract. We note that the continued fraction expansion of a lacunary formal power series is a folded continued fraction with monomial partial quotients, and with the property that its convergents have denominators that are the sums of distinct monomials, that is, they are polynomials with coefficients 0, 1, and��� 1 only. Our results generalise, simplify and refine remarks of a previous note 'Convergents of folded continued fractions'
Vibrational Modes in a One Dimensional "Quasi-Alloy" : The Morse Case
Le Journal de Physique Colloques, 1986
... Having derived 2" frequencies w2 one numerically calculates the corres-3 ponding modes. ... more ... Having derived 2" frequencies w2 one numerically calculates the corres-3 ponding modes. Figure 2 shows an example of such a mode, localized/l5A that has a non trivial decrease from the center peak. ... l/ G. Christol, T. Kamae. M. Mendès-France, G. Rauzy, Bull Soc. Math. Pr. ...
An Exercise on the Average Number of Real Zeros of Random Real Polynomials
Bolyai Society Mathematical Studies, 2006
The average number of real zeros of random n degree real polynomials is well known since M. Kac’s... more The average number of real zeros of random n degree real polynomials is well known since M. Kac’s seminal article of 1943 [12] which states that it is logn + O(1). Some fifty years later, A. Edelman and E. Kostlan found a beautiful geometrical proof which allowed them to give many other related results [10]. Using their method we discuss the average number of real zeros of random real polynomials å</font >j = 0n Aj Xj \sum\limits_{j = 0}^n {A_j X^j } where the A j ’s are independent Gaussian variables with mean 0 and with variance s</font >2 ( Aj ) = ( *20c n j )n -</font > b</font >j \sigma ^2 \left( {A_j } \right) = \left( {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right)n^{ - \beta j} where β ∈ ℝ is a given parameter. The average number of real zeros in the interval (a, b) is shown to be E( n;a,b ) = \frac1p</font >Ö</font >n ( Arctan\fracbnb</font >/2 -</font > Arctan\fracanb</font >/2 ). E\left( {n;a,b} \right) = \frac{1} {\pi }\sqrt n \left( {Arc\tan \frac{b} {{n^{\beta /2} }} - Arc\tan \frac{a} {{n^{\beta /2} }}} \right). While discussing special polynomials we are led to show that under general conditions, polynomials of the type å</font >i = 1k Ai ( X )( ai X + bi )n \sum\limits_{i = 1}^k {A_i \left( X \right)\left( {a_i X + b_i } \right)^n } have at most O(1) real zeros as n increases to infinity.
Genesis: From Euclid to Chebyshev
The Student Mathematical Library, 2000
Stochastic distribution of prime numbers
The Student Mathematical Library, 2000
In this paper we study the three-dimensional curves generated by the iterated bending of a piece ... more In this paper we study the three-dimensional curves generated by the iterated bending of a piece of wire, which are generalizations of the so-called "dragon curves" or "paper-folding sequences" previously studied by Davis and Knuth, Mend&s France, and other writers. These "wire-bending sequences" have several surprising properties. We characterize the nth term of a wire-bending sequence in terms of the binary expansion of n. We prove that the curves traced out in R3 by many wirebending sequences are actually bounded, although they are all aperiodic. Finally, we illustrate the close connection between wire-bending and the continued fractions for the transcendental numbers C,, ~ a a, g -*", where E, = k 1 and g 3 3 is an integer.
AN EXERCISE ON THE AVERAGE NUMBER or REAL
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Papers by Michel Mendès-France