Séries formelles et algèbres syntactiques

AbstractThe notion of the syntactic monoid is well known to be very important for formal languages, and in particular for rational languages; examples of that importance are Kleene's theorem, Schützenberger's theorem about aperiodic monoid and Eilenberg's theorem about varieties. We introduce here, for formal power series, a similar object: to each formal power series we associate its syntactic algebra. The Kleene-Schützenberger theorem can then be stated in the following way: a series is rational if and only if its syntactic algebra has finite dimension. A rational central series (this means that the coefficient of a word depends only on its conjugacy class) is a linear combination of characters if and only if its syntactic algebra is semisimple. Fatou properties of rational series in one variable are extended to series in several variables and a special case of the rationality of the Hadamard quotient of two series is positively answered. The correspondence between pseudovarieties of finite monoids and varieties of rational languages, as studied by Eilenberg, is extended between pseudovarieties of finite dimensional algebras and varieties of rational series. We study different kinds of varieties that are defined by closure properties and prove a theorem similar to Schützenberger's theorem on aperiodic monoids

Opérations polynomiales et hiérarchies de concaténation

RésuméSoit C une classe de langages. Notons Pol(C) la fermeture polynomiale de C. Pol(C) est la plus petite classe de langages contenant C et fermée par union finie et produit marqué LaL' où a est une lettre. Nous déterminons les clôtures polynomiales de diverses classes de langages rationnels puis nous étudions les propriétés des fermetures polynomiales. Par exemple, si C est fermée par quotients (resp. quotients et morphisme inverse), alors il en est de même de Pol(C). Notre résultat principal montre que si C est une algèbre de Boole fermée par résiduels alors Pol(C) est fermée par intersection. Comme application, nous affinons la hiérarchie de concaténation introduite par Straubing et nous prouvons la décidabilité des niveaux 12 et 32 de cette hiérarchie.AbstractGiven a class C of languages, let Pol(C) be the polynomial closure of C, that is, the smallest class of languages containing C and closed under the operations union and marked product LaL', where a is a letter. We determine the polynomial closure of various classes of rational languages and we study the properties of polynomial closures. For instance, if C is closed under quotients (resp. quotients and inverse morphism) then Pol(C) has the same property. Our main result shows that if C is a boolean algebra closed under quotients then Pol(C) is closed under intersection. As an application, we refine the concatenation hierarchy introduced by Straubing and we show that the levels 12 and 32 of this hierarcy are decidable

Quelques remarques sur les vari{\'e}t{\'e}s, fonctions de Green et formule de Stokes

We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces of Sobolev spaces, the distance function, the notion of degree and a duality theorem, the variational formulation and conformal map in dimension 2, the metric on the boundary of a Lipschitz domain and polar geodesic coordinates and the Gauss-Bonnet formula and the positive mass theorem in dimension 3 and in the locally conformally flat case

Théorèmes de finitude pour les variétés pfaffiennes

Guérin Pierre, Lofficial Louis-Prosper, Treilhard Jean-Baptiste. Discussion sur l'acte d'accusation de Carrier, lors de la séance du 5 frimaire an III (25 novembre 1794). In: Archives Parlementaires de 1787 à 1860 - Première série (1787-1799) Tome CII - Du 1er au 12 frimaire An III (21 novembre au 2 décembre 1794) Paris : CNRS éditions, 2012. pp. 176-177