Nombre chromatique total d'un hypergraphe

In this paper, the author defines a hypergraph total chromatic number and determines this number for the complete h-partite and for the complete h-uniform hypergraphs.     

Sur le joint d'une famille de graphes

Les graphes envisagés dans cet article sont non orientés, sans boucles ni arês multiples. On désigne par X(G) et E(G) les ensembles de sommets et d'arêtes d'un graphe G; ces ensembles seront toujours finis. Les notations usuelles sont ceiles de Berge [1].Dans [5], Zykov introduit la notion de graphe joint de deux graphes donnés. Une propriété du joint de deux graphes dont l'un est réduit à un sommet est donnée dans [3]; c'est une étude plus générate et systématique que nous présentons ici.     

Codes projectifs a deux ou trois poids associfs aux hyperquadriques d'une geometrie finie

The notion of closure structures of finite rank is introduced and several closure structures familiar from algebra and logic are shown to be of finite rank. The following theorem is established: Let k be any natural number and $\text{C}$ any closure structure of rank k + 2. If B is a finite base (generating set) of $\text{C}$and D is an irredundant (independent) base of $\text{C}$ such that |B| + k < |D|, then there is an irredundant base E of $\text{C}$ such that |B| < |E| < |B| + k.     

La reduction minimale d'un graphe a une reunion de cliques

On obtient la valeur maximale de l'indice de réduction minimale d'un graphe á une réunion de k cliques.     

Hypergraphes de chaines d'aretes d'un arbre

We introduce the ‘edges-paths hypergraph of a tree’ and study relations of this notion with graphic geometries, chordable graphs. As particular case, we give a simple characterization of intervals hypergraphs.     

Sur les extensions lineaires d'une famille particuliere d'ordres partiels

Corners are defined as ideals of an ordered integer half-dihedron; the paper develops a method of enumeration of the linear extensions of a given corner by means of an alternating sum of products of trinomials. The main result substantially generalizes previously known results and is by itself the starting point of generalizations to some further ordered sets.     

Produit croisé d'une algèbre de von Neumann par une algèbre de Kac

We introduce a notion of action of a Kac algebra (see [2, 9]) on a von Neumann algebra, and the cross-product of a von Neumann algebra by a Kac algebra with respect to an action α. The results of Takesaki [11, Chaps. 3 and 4] are then generalized, particularly the theorem of the double cross-product.     

Les 5-colorations d'un graphe planaire forment une classe de commutation unique

There is one class of interchanges for the 5-colorations of a planer graph. As a consequence it is always possible to reach a four coloration (if it exists) by a sequence of interchange from any 5-coloration. A theorem is given for surfaces of genus g.     

Petites valeurs de la fonction d'Euler

Let φ be the Euler's function. A question of Rosser and Schoenfeld is answered, showing that there exists infinitely many n such that $\text{n}\text{φ(n)}\text{> e}{}^{\mathrm{y}}\text{log log}\text{n}$, where γ is the Euler's constant. More precisely, if N_k is the product of the first k primes, it is proved that, under the Riemann's hypothesis, $\text{N}{}_{\mathrm{k}}\text{φ(N}{}_{\mathrm{k}}\text{)}\text{> e}{}^{\mathrm{y}}\text{log log}\text{N}{}_{\mathrm{k}}$ holds for any k ≥ 2, and, if the Riemann's hypothesis is false this inequality holds for infinitely many k, and is false for infinitely many k.     

La conjecture d'André–Oort pour le produit de deux courbes modulaires de DrinfeldThe André–Oort conjecture for the product of two Drinfeld modular curves

We prove an analogue of the André–Oort conjecture for the product of two Drinfeld modular curves, following S.J. Edixhoven's approach. To cite this article: F. Breuer, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 867–870.     

Estimation de la transition de probabilité d'une chaîne de Markov doëblin-récurrente. étude du cas du processus autorégressif général d'ordre 1

We consider an homogenous Markov chain {X_n}. We estimate its transition probability density with kernel estimators. We apply these methods to the estimation of the unknown function f of the process defined by X₁ and X_n+1 = f(X_n) + ε_n, where {ε_n} is a noise (sequence of independent identically distributed random variables) of unknown law. The mean quadratic integrated rates of convergence are identical to those of classical density estimations. These risks are used here because we want some global informations about our estimates. We also study the average of those risks when the variance changes; it is shown that they reach a minimal value for some optimal variance. We study uniform convergence of our estimators. We finally estimate the variance of the noise and its density.