Combinatorics, Automata and Number Theory
Valrie Berth,Michel Rigo +1 more
- 30 Sep 2010
TL;DR: In this article, the authors present recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory and reveal some of the exciting and important relationships that exist between these different fields.
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Abstract: This collaborative volume presents recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Topics include numeration systems, word complexity function, morphic words, Rauzy tilings and substitutive dynamical systems, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic properties of digital functions, decidability issues for D0L systems, matrix products and joint spectral radius. Topics are presented in a way that links them to the three main themes, but also extends them to dynamical systems and ergodic theory, fractals, tilings and spectral properties of matrices. Graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, fractals, tilings and stringology will find much of interest in this book.
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Citations
•Posted Content
Beyond substitutive dynamical systems: S-adic expansions
Valérie Berthé,Vincent Delecroix +1 more
TL;DR: This article studied S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.
104
AF inverse monoids and the structure of countable MV-algebras
Mark V. Lawson,Philip J. Scott +1 more
TL;DR: In this paper, a special class of Boolean inverse monoids having the property that their lattices of principal ideals naturally form an MV-algebra is defined, and it is shown how these two generalizations are connected.
36
•Posted Content
Iterative algebras
Jason P. Bell,Blake W. Madill +1 more
TL;DR: In this article, it was shown that one can construct an iterative algebra in a natural way given a finitely generated free monoid $X$ and a morphism $\phi : X\to X to X'' and that many ring theoretic properties of iterative algebras can be easily characterized in terms of linear algebra and combinatorial data from the morphism.
Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences
TL;DR: For each radix-rational sequence with complex values, the precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first-order differences as discussed by the authors.
26
Counting the number of non-zero coefficients in rows of generalized Pascal triangles
TL;DR: This paper considers the generalization of the Pascal triangle to binomial coefficients of words and the sequence ( S ( n ) n ≥ 0 counting the number of positive entries on each row, and introduces a convenient tree structure that leads to a connection with the 2 -regular Stern–Brocot sequence and the sequences of denominators occurring in the Farey tree.
24
References
AUTOMATIC CONVERSION FROM FIBONACCI REPRESENTATION TO REPRESENTATION IN BASE φ, AND A GENERALIZATION
TL;DR: It is shown that there exists a letter-to-letter finite two-tape automaton that maps the Fibonacci representation of any positive integer onto its φ-expansion, provided the latter is folded around the radix point.
16
The generalized spectral radius is strictly increasing
TL;DR: In this article, it was shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense, and sufficient conditions for Lipschitz continuity were derived.
16
Construction of regular languages and recognizability of polynomials
TL;DR: A generalization of numeration system in which N is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language by showing that if P ∈ Q[x] is a polynomial such that P(N) ⊂ N then it can be constructed in which the set of representations of P( N) is regular.
15
Substitutions, abstract number systems and the space filling property
Clemens Fuchs,Robert Tijdeman +1 more
TL;DR: In this paper, a set of mots multidimensionnels engendres par des points fixes de substitutions, and obtenus en projetant les points entiers sur la demi-droite brisee correspondante.