Coloration des graphes de reines

Until 2003 no chromatic numbers (${\chi }_n$) for the queen graphs were available for $n>9$ except where n is not a multiple of 2 or 3. In this research announcement we present an exact algorithm which provides coloring solutions for $n=12,14,15,16,18,20,21,22,24,26,28$ and 32 such as ${\chi }_n=n$. Then we prove that there exists an infinite number of values for n such that $n=2p$ or $n=3p$, and ${\chi }_n=n$. To cite this article: M. Vasquez, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Inverse spectral problem for singular AKNS and Schrödinger operators on [0,1]

We consider an inverse spectral problem for singular Sturm–Liouville equations on the unit interval with explicit singularity $a(a+1)/x^2$, $a\in \mathbb{N}$. This problem arises by splitting of the Schrödinger operator with radial potential acting on the unit ball of ${\mathbb{R}}^3$. Our goal is the global parametrization of potentials by spectral data noted by ${\lambda }^a$, and some norming constants noted by ${\kappa }^a$. For $a=0$ and 1, ${\lambda }^a\times {\kappa }^a$ was already known to be a global coordinate system on $L_{\mathbb{R}}^2(0,1)$. We extend this to any non-negative integer a. Similar result is obtained for singular AKNS operator. To cite this article: F. Serier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity

We study positive solutions of the equation ${\mathrm{?}}^2\mathrm{\Delta }u?u+u^{\frac{n+2}{n?2}}=0$, where $n=3,4,5$ and $\mathrm{?}>0$ is small, with Neumann boundary condition in a unit ball B. We prove the existence of solutions with an interior bubble at the center and a boundary layer at the boundary ?B. To cite this article: J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Asymptotics of instability zones of Hill operators with a two term potential

We give a sharp asymptotics of the instability zones of the Hill operator $Ly=?y^{″}+(acos2x+bcos4x)y$ for arbitrary real $a\text{,}b\ne 0$. To cite this article: P. Djakov, B. Mityagin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Une propriété de composition dans l'espace Hs

Let us assume that $1<p?q?+\infty$, $1/p<s?[s]<1$, and $[s]?1$. If f and g are functions in the Besov space $B_p^{s\text{,}q}(\mathbb{R})$, such that g is real valued and such that $f(0)=0$, then the composed function $f○g$ belongs to $B_p^{s\text{,}q}(\mathbb{R})$. To cite this article: G. Bourdaud, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains

We study the qualitative properties of sign changing solutions of the Dirichlet problem $\mathrm{\Delta }u+f(u)=0$ in Ω, $u=0$ on ?Ω, where Ω is a ball or an annulus and f is a $C^1$ function with $f(0)?0$. We prove that any radial sign changing solution has a Morse index bigger or equal to $N+1$ and give sufficient conditions for the nodal surface of a solution to intersect the boundary. In particular, we prove that any least energy nodal solution is non radial and its nodal surface touches the boundary. To cite this article: A. Aftalion, F. Pacella, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Duality of the space of germs of harmonic vector fields on a compact

A vector field $\overline{u}$ in ${\mathbb{R}}^3$ is said to be harmonic in the open set U if $\mathrm{rot}\overline{u}=\overline{0}$, $\mathrm{div}\overline{u}=0$ in U. Harmonic vector fields are a natural extension to ${\mathbb{R}}^3$ of the concept of analytic function of complex variable. We characterize continuous linear functionals acting on the space $h(K)$ of germs of harmonic vector fields on a compact set K. This result provides an ${\mathbb{R}}^3$-analog of a theorem by G. Köthe on the dual of the space of germs of analytic functions of complex variable on a compact. To cite this article: R. Dáger, A. Presa, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Version explicite du théorème de Beilinson pour la courbe modulaire X1(N)

We state an explicit version of Beilinson's theorem for the modular curve $X_1(N)$. We deduce from it, for any elliptic curve E of prime conductor N, a formula giving $L(E,2)$ in terms of the twisted values $L(E,\chi ,1)$, where χ is a character modulo N. We illustrate this result and its consequences in the case of the elliptic curve $E=X_1(11)$. To cite this article: F. Brunault, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Favorite sites of randomly biased walks on a supercritical Galton–Watson tree

Erd?s and Révész (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We prove that there is some parameter $\kappa \in (1,\infty ]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $\kappa \in (2,\infty ]$, tight in the case $\kappa =2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $\kappa \in (1,2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $\kappa \in (2,\infty ]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton–Watson trees.     

Existence et unicité pour un fluide inhomogène

Recently R. Danchin showed the existence and uniqueness for an inhomogenous fluid in the homogeneous Besov space ${\dot{B}}_{21}^{\frac{N}{2}}({\mathbb{R}}^N)\times {\dot{B}}_{21}^{?1+\frac{N}{2}}({\mathbb{R}}^N)$, under the condition that ${\rho }_0?1$ is small in ${\dot{B}}_{2\infty }^{\frac{N}{2}}\cap L^{\infty }$ if $2<N,$ in ${\dot{B}}_{21}^{\frac{N}{2}}$ if $N=2.$ In this Note, one shows that the condition $6{\rho }_0?16_{L^{\infty }}?1$ is sufficient to have the existence and uniqueness. To cite this article: H. Abidi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Three examples of three-dimensional continued fractions in the sense of Klein

The problem of the investigation of the simplest n-dimensional continued fraction in the sense of Klein for $n?2$ was posed by V. Arnold. The answer for the case $n=2$ can be found in the works of E. Korkina (1995) and G. Lachaud (1995). In present Note we study the case $n=3$. To cite this article: O. Karpenkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Contacts et auto-contacts sans frottement

Let M be a submanifold of ${\mathbb{R}}^n$ ($n=2,3$) considered as the reference configuration of a hyperelastic solid. A topological constraint is imposed on the admissible deformations $\psi :M\to {\mathbb{R}}^n$ of the solid in order to satisfy a non penetration condition. We show that the associated minimization problem has at least one solution and, in the case $\dim (M)\ne 2$ or $n\ne 3$, provides a mathematical model of body that allows frictionless self-contact. A numerical application is presented. To cite this article: O. Pantz, C. R. Acad. Sci. Paris, Ser. I 341 (2005).