Interprétation bijective d'une récurrence des nombres de Motzkin

The purpose of this note is to give a combinatorial proof of the three-term linear recurrence for Motzkin numbers. The present work is inspired by Rémy's combinatorial proof of the linear recurrence for Catalan numbers (RAIRO Inform. Theor. 19(2) (1985) 179) and the more recent proof given by Foata and Zeilberger (J. Combin. Theory Ser. A 80(2) (1997) 380) for Schröder numbers (Z. Math. Phys. 15 (1870) 361).     

Nouvelles statistiques de partitions pour les q-nombres de Stirling de seconde espèce

Steingr?́msson (Preprint, 1999) has recently introduced a partition analogue of Foata-Zeilberger's mak statistic for permutations and conjectured that its generating function is equal to the classical q-Stirling numbers of second kind. In this paper, we prove a generalization of Steingr?́msson's Conjecture 12.     

Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique

We give a combinatorial interpretation of the negative moments of the values at the edge of the critical strip of the L functions of modular forms of GL(2) and GL(3). We deduce some results about the size of these numbers.     

Le lemme de Schur pour les représentations orthogonales

Let σ be an orthogonal representation of a group G on a real Hilbert space. We show that σ is irreducible if and only if its commutant σ(G)' is isomorphic to , or . This result is an analogue of the classical Schur lemma for unitary representations. In both cases (orthogonal and unitary), a representation is irreducible if and only if its commutant is a field. If σ is irreducible, we show that there exists a unitary irreducible representation π of G such that the complexification σ is unitarily equivalent to π if σ(G)' , to π π̄ if σ(G)' , and to π π if σ(G)' (here π̄ denotes the contragredient representation of π). These results are classical for a finite-dimensional σ, but seem to be new in the general case.     

Analyse de complexité pour un théorème de Hall sur les fractions continues

We give a polynomial time controlled version of a theorem of M. Hall: every real number can be written as the sum of two irrational numbers whose developments into a continued fraction contain only 1, 2, 3 or 4. Mathematics Subject Classification: 03D15, 03F60, 11A55, 68Q25.     

Approximation simultanée d'un nombre et de son carré

In 1969, H. Davenport and W.M. Schmidt established a measure of the simultaneous approximation for a real number ξ and its square by rational numbers with the same denominator, assuming only that ξ is not rational nor quadratic over $\text{Q}$. Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three. To cite this article: D. Roy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur l'indépendance l-adique de nombres algébriques

Let l a prime number and K a Galois extension over the field of rational numbers, with Galois group G. A conjecture is put forward on l-adic independence of algebraic numbers, which generalizes the classical ones of Leopoldt and Gross, and asserts that the l-adic rank of a G submodule of K^x depends only on the character of its Galois representation. When G is abelian and in some other cases, a proof is given of this conjecture by using l-adic transcendence results.     

Sur une question d'équirépartition de nombres algébriques

We give an elementary counter-example for a conjecture of Pineiro, Szpiro and Tucker about the uniform distribution of algebraic numbers with small height. To cite this article: P. Autissier, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Congruences dyadiques entre nombres de classes de corps quadratiques

We prove here a general mod 16 congruence between the class numbers of the two fields and , for any square free positive integer d. Similarly we obtain congruences mod 64 relating the class numbers of the quadratic extensions corresponding to the integers d, 2d, -d, -2d. We use the p-adic analytic class number formula, which leads to study the difference of the values at 0 and 1 of the KUBOTA-LEOPOLDT's p-adic L-function (for p=2, and for the quadratic character associated to the real quadratic field). We use then IWASAWA's series, conveniently described for our purpose by Barsky. This gives a new proof for mod 16 congruences formally or simultaneously obtained by various authors. Moreover we get new results mod 16 and mod 64.Part A) presents the method and results, Part B) the proofs. All the details can be found in [4].     

Une application des nombres de Pisot à l'algorithme de Jacobi-Perron

SinceO. Perron introduced in 1907 Jacobi-Perron algorithm, which is the simplest generalization of continued fractions to finite sets of real numbers, the main question of characterising the periodicity is still open. The usual conjecture is that the development of any basis of a real number field by this algorithm is periodic. But we only know some infinite families for which this is true. In this paper we prove that for any real number field there exists a basis for which we have periodicity.     

Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár

The Kalmár function K(n) counts the factorizations n=x₁x₂…xr with xi?2(1?i?r). Its Dirichlet series is where ζ(s) denotes the Riemann ζ function. Let ρ=1.728… be the root greater than 1 of the equation ζ(s)=2. Improving on preceding results of Kalmár, Hille, Erd?s, Evans, and Klazar and Luca, we show that there exist two constants C₅ and C₆ such that, for all n, holds, while, for infinitely many n's, .An integer N is called a K-champion number if M<N⇒K(M)<K(N). Several properties of K-champion numbers are given, mainly about the size of the exponents and the number of prime factors in the standard factorization into primes of a large enough K-champion number.The proof of these results is based on the asymptotic formula of K(n) given by Evans, and on the solution of a problem of optimization.     

On the Colin de Verdière number of graphs

Let μ(G) and ω(G) be the Colin de Verdière and clique numbers of a graph G, respectively. It is well-known that μ(G)?ω(G)-1 for all graphs. Our main results include μ(G)?ω(G) for all chordal graphs; μ(G)?tw(G)+1 for all graphs (where tw is the tree-width), and a characterization of those split (⊆ chordal) graphs for which μ(G)=ω(G). The bound μ(G)?tw(G)+1 improves a result of Colin de Verdière by a factor of 2.     

Asymptotique des nombres de Betti,invariants $l^2$ et laminations

Let K be a finite simplicial complex. We are interested in the asymptotic behavior of the Betti numbers of a sequence of finite sheeted covers of $K$, when normalized by the index of the covers. W. Lück, has proved that for regular coverings, these sequences of numbers converge to the $l^2$ Betti numbers of the associated (in general infinite) limit regular cover of K. In this article we investigate the non regular case. We show that the sequences of normalized Betti numbers still converge. But this time the good limit object is no longer the associated limit cover of K, but a lamination by simplicial complexes. We prove that the limits of sequences of normalized Betti numbers are equal to the $l^2$ Betti numbers of this lamination. Even if the associated limit cover of K is contractible, its $l^2$ Betti numbers are in general different from those of the lamination. We construct such examples. We also give a dynamical condition for these numbers to be equal. It turns out that this condition is equivalent to a former criterion due to M. Farber. We hope that our results clarify its meaning and show to which extent it is optimal. In a second part of this paper we study non free measure-preserving ergodic actions of a countable group $\Gamma$ on a standard Borel probability space. Extending group-theoretic similar results of the second author, we obtain relations between the $l^{2}$ Betti numbers of $\Gamma$ and those of the generic stabilizers. For example, if $b_1^{(2)} (\Gamma ) \neq 0$, then either almost each stabilizer is finite or almost each stabilizer has an infinite first $l^2$ Betti number.

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Asymptotique des nombres de Betti, invariants $l^2$ et laminations

     

Nombres de Tamagawa des groupes réductifs quasi-connexes

We define Tamagawa numbers for quasi-connected linear algebraic groups. This category of non-connected linear algebraic groups occurs naturally in the stabilization of the Arthur-Selberg trace formula. An expression for their Tamagawa numbers, needed for the stabilization process, is given in term of cohomological invariants generalizing results of Ono, Sansuc, Kottwitz and Kottwitz & Shestad. Received: 7 March 2000     

Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes

In the first part of the paper, we prove the existence of a unique global solution to the axisymmetric Navier-Stokes system with initial data and external force with . This improves the result obtained by S. Leonardi, J. Málek, J. Necǎs and M. Pokorný [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649], where H²(R³) regularity was required. In the second part, we state global existence and uniqueness for the axisymmetric Navier-Stokes system with initial data in W2,p(R³) and external force in with 1<p<2. This also improves [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649] because less integrability is required on v₀ and on f.     

Sur l'ensemble exceptionnel de Weyl

It is well known since H. Weyl's work that for any given real number $r>1$, the set $W_r$ consisting of positive real numbers λ for which the sequence ${(\lambda r^n)}_{n\in \mathbb{N}}$ is not uniformly distributed modulo 1, has Lebesgue measure zero. In this Note, without use the concept of Hausdorff dimension, one shows among other things that these sets $W_r$ are uncountable. To cite this article: B. Farhi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Construction de quasimodes de Rayleigh à grande durée de vie

For the Neumann problem in linear elasticity at the exterior of a convex body with analytic boundary in two dimension, we know that exists exactly two sequences of resonances (called Rayleigh resonances), such that . If the boundary is close enough to a circle and analytic in a wide enough complex band, we construct quasimodes associated to these resonances as far as possible from the obstacle. We study the exponential decay of these quasimodes according to the geometry of the obstacle.     

Mesures d'indépendance linéaire de carrés de périodes et quasi-périodes de courbes elliptiques

A linear independence measure over Q is obtained for values of some generalized hypergeometric functions at rational points and for the squares of real periods and quasi-periods of elliptic curves in Legendre form y²=x(x−1)(x−1/q) for almost every q∈Z.     

The Legendre–Stirling numbers

The Legendre–Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre–Stirling numbers. In this paper, we establish several properties of the Legendre–Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.     

Zeta functions of<Emphasis Type="Italic">q</Emphasis>-perfect numbers

We introduce a notion ofq-analogue of the perfect numbers. We also define a new zeta function which we call a zeta function ofq-perfect numbers. In this paper, the properties of theq-perfect numbers and the zeta functions are studied. Especially, we determine theq-perfect numbers whenq is a root of unity.