Results 1  10
of
31
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 ..."
Abstract

Cited by 68 (4 self)
 Add to MetaCart
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
The ubiquitous ProuhetThueMorse sequence
 Sequences and their applications, Proceedings of SETA’98
, 1999
"... We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be som ..."
Abstract

Cited by 59 (9 self)
 Add to MetaCart
We discuss a wellknown binary sequence called the ThueMorse sequence, or the ProuhetThueMorse sequence. This sequence was introduced by Thue in 1906 and rediscovered by Morse in 1921. However, it was already implicit in an 1851 paper of Prouhet. The ProuhetThueMorse sequence appears to be somewhat ubiquitous, and we describe many of its apparently unrelated occurrences.
Determination Of The Algebraic Relations Among Special Values In Positive Characteristic
 ANN. OF MATH
, 2003
"... We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual tmotives, we nd that all algebraic relations among special values of the geometricfunction over Fq [T ] are explained by the standard functional equations. ..."
Abstract

Cited by 36 (9 self)
 Add to MetaCart
We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual tmotives, we nd that all algebraic relations among special values of the geometricfunction over Fq [T ] are explained by the standard functional equations.
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 ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
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...
Real and padic Expansions Involving Symmetric Patterns
"... This paper is motivated by the nonArchimedean counterpart of a problem raised by Mahler and Mendès France, and by questions related to the expected normality of irrational algebraic numbers. We introduce a class of sequences enjoying a particular combinatorial property: the precocious occurrences o ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
This paper is motivated by the nonArchimedean counterpart of a problem raised by Mahler and Mendès France, and by questions related to the expected normality of irrational algebraic numbers. We introduce a class of sequences enjoying a particular combinatorial property: the precocious occurrences of infinitely many symmetric patterns. Then, we prove several transcendence statements involving both real and padic numbers associated with these palindromic sequences. 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 ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
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...
On Continued Fraction Expansions in Positive Characteristic: Equivalence Relations and Some Metric Properties
"... The aim of this paper is to survey some properties of analogues of continued fraction expansions for formal power series with coefficients in a finite field. We discuss in particular connections between equivalence relations for continued fractions and the action of SL(2; F q [X]). ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The aim of this paper is to survey some properties of analogues of continued fraction expansions for formal power series with coefficients in a finite field. We discuss in particular connections between equivalence relations for continued fractions and the action of SL(2; F q [X]).
WORDS AND TRANSCENDENCE
, 2009
"... Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should sati ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the bary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the gary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g � 3 is an integer, a a digit in {0,...,g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the gary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the gary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g � 2 be an integer. Any real number x has a unique expansion x = a−kg k + ···+ a−1g + a0 + a1g −1 + a2g −2 + ·· ·, where k � 0 is an integer and the ai for i � −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1,...,g − 1}. Uniqueness is subject to the condition that the sequence (ai)i�−k is not ultimately constant and equal to g − 1. We write this expansion x = a−k ···a−1a0.a1a2 ·· ·. in base 10 (decimal expansion), whereas
Computational Classification of Numbers and Algebraic Properties
"... In this paper, we propose a computational classification of finite characteristic numbers ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In this paper, we propose a computational classification of finite characteristic numbers