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Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 68 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
"... Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asym ..."
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Cited by 34 (14 self)
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Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morsetheoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have
Transcendence of Formal Power Series With Rational Coefficients
, 1999
"... We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal po ..."
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Cited by 9 (2 self)
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We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, pLucas sequences, ChomskySchutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the ChomskySchutzenberger theorem [16] asserts that the generating seri...
A new method for computing asymptotics of diagonal coefficients of multivariate generating functions
, 2007
"... Let P n∈N d fnx n be a multivariate generating function that converges in a neighborhood of the origin of C d. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients fa1n,...,a dn and show its superiority over the standard, univariate diagonal method. ..."
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Cited by 8 (3 self)
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Let P n∈N d fnx n be a multivariate generating function that converges in a neighborhood of the origin of C d. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients fa1n,...,a dn and show its superiority over the standard, univariate diagonal method.
Cauchy type integrals of algebraic functions
 Israel J. Math
"... Abstract We consider Cauchy type integrals I(t) = 1 2πi γ g(z)dz z−t with g(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions for I(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by ..."
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Cited by 6 (5 self)
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Abstract We consider Cauchy type integrals I(t) = 1 2πi γ g(z)dz z−t with g(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions for I(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the Monodromy group of the algebraic function g, the geometry of the integration curve γ, and the analytic properties of the Cauchy type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincaré CenterFocus problem and the second part of the Hilbert 16th problem. 1.
Sequences Of Low Complexity: Automatic And Sturmian Sequences
"... The complexity function is a classical measure of disorder for sequences with values in a finite alphabet: this function counts the number of factors of given length. We introduce here two characteristic families of sequences of low complexity function: automatic sequences and Sturmian sequences ..."
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Cited by 6 (0 self)
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The complexity function is a classical measure of disorder for sequences with values in a finite alphabet: this function counts the number of factors of given length. We introduce here two characteristic families of sequences of low complexity function: automatic sequences and Sturmian sequences. We discuss their topological and measuretheoretic properties, by introducing some classical tools in combinatorics on words and in the study of symbolic dynamical systems. 1 Introduction The aim of this course is to introduce two characteristic families of sequences of low "complexity": automatic sequences and Sturmian sequences (complexity is defined here as the combinatorial function which counts the number of factors of given length of a sequence over a finite alphabet). These sequences not only occur in many mathematical fields but also in various domains as theoretical computer science, biology, physics, cristallography... We first define some classical tools in combinatorics on...
Computational Classification of Numbers and Algebraic Properties
"... In this paper, we propose a computational classification of finite characteristic numbers ..."
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Cited by 4 (2 self)
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In this paper, we propose a computational classification of finite characteristic numbers
Additive Cellular Automata and Algebraic Series
, 1993
"... Introduction A cellular automaton is a grid of elementary automata, each communicating only with a finite number of its neighbours. In the simplest models each elementary device, a cell or site, can take only two values and is updated at intervals according to a rule which expresses the actual valu ..."
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Cited by 2 (0 self)
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Introduction A cellular automaton is a grid of elementary automata, each communicating only with a finite number of its neighbours. In the simplest models each elementary device, a cell or site, can take only two values and is updated at intervals according to a rule which expresses the actual value from its preceding value and that of its neighbours. Here the values are elements of a field K, finite or infinite. As O. Martin, A. Odlyzko and S. Wolfram [7] emphasized, the behaviour of a cellular automaton with a finite number of cells on a finite field is ultimately periodic. It is natural to consider also automata with cells in a line, which we call one dimensional automata. So a cell is indexed by an integer n 2 Z. At each time all the cells but a finite number are in state 0. Throughout the paper we make use of generating series and, from this point of view, a configuration is a Lauren