Variétés complexes dont l'éclatée en un point est de FanoComplex manifolds whose blow-up at a point is Fano

We classify complex projective manifolds X for which there exists a point a such that the blow-up of X at a is Fano. As a consequence, we get that, in dimension greater or equal than three, the quadric is the only complex manifold X for which there exists two distinct points a and b such that the blow-up of X with center {a,b} is Fano. To cite this article: L. Bonavero et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 463–468.     

Intersecting random half spaces with a cubeVolume de l'intersection d'un cube et de sous-espaces aleatoires

We compute the typical number of points of the discrete cube {?1,1}^N that belong to the intersection of M random half-spaces, when M is a small proportion of N. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 807–809.     

Hypersurfaces d'un fibré vectoriel Riemannien à courbure de Gauss prescriteHypersurfaces of a Riemannian vector bundle with prescribed Gaussian curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed Gaussian curvatures of various types. To cite this article: A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 927–930.     

The effect of perturbations on the first eigenvalue of the p-LaplacianL'éffet des pérturbations sur la première valeur propre du p-Laplacien

Let $\text{Ω}$ be a domain with Lipschitzian boundary of a compact Riemannian manifold (M,g) and p>1. We prove that we can make the volume of M arbitrarily close to the volume of $\text{(Ω,g)}$ while the first eigenvalue of the p-Laplacian on M remains uniformly bounded from below in terms of the the first eigenvalue of the Neumann problem for the p-Laplacian on $\text{(Ω,g)}$. To cite this article: A.-M. Matei, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 255–258.     

On a canonical placement of knots in irreducible 3-manifoldsSur un placement canonique des nœuds dans les 3-variétés irréductibles

If M is a compact connected orientable irreducible 3-manifold and T is a minimal Jaco–Shalen–Johannson system of tori inside M, we define the pieces of M to be regular neighborhoods of incompressible tori in T∪?M, the components of their complement or regular neighborhoods of Seifert fibres in those components that admit Seifert fibrations. For a given isotopy class $\text{K}$ of knots inside M we describe, with some restrictions on M, the set of pieces which contain representatives of $\text{K}$. If the knots of $\text{K}$ are not contained in balls, we show that the isotopy class of a representative of $\text{K}$ inside a piece P is independent of the chosen representative. To cite this article: P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 677–682.     

Hypersurfaces compactes d'un fibré vectoriel riemannien à courbure moyenne prescriteCompact hypersurfaces of a Riemannian vector bundle with prescribed mean curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed mean curvatures of various types. To cite this article: P. Cherrier, A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 525–528.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

Particules collantes signées et lois de conservation scalaires 1DSigned sticky particles and 1D scalar conservation laws

We present an approximation of the entropy solution of a 1D scalar conservation law based on signed sticky particles when the variation of the initial condition is bounded. This method is a generalization of the one studied by Brenier and Grenier [2] in case the initial condition is monotone. When they collide, particles with the same sign stick together with conservation of the momentum whereas particles with opposite sign are destroyed. We prove the convergence of the approximate solution to the entropy solution when the initial number of particles goes to +∞. To cite this article: B. Jourdain, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 233–238.     

On Linear Preservers of Two-Sided Gut-Majorization On <Emphasis Type="Bold">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis></Subscript>

For X, Y ∈ M_n,m it is said that X is gut-majorized by Y, and we write X ?_gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~_gut as follows. X ~_gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ?_gut on ?ⁿ and strong linear preservers of this relation on M_n,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~_gut on ?ⁿ and M_n,m.     

Extensive Lyapounov functionals for moment-preserving evolution equationsFonctions de Lyapounov extensives pour des problèmes d'évolution preservant certains moments

We consider a certain class of moment-preserving equations from the point of view of their stationary solutions. Starting from a given stationary distribution, we construct a convex entropy functional which is (in a class of functions with prescribed moments) minimal precisely at this point. Under general assumptions, we show that the entropy which is canonically associated to a stationary distribution is, up to a polynomial change of variables, its Legendre–Fenchel transform. We then show that, if this entropy is extensive, necessarily the stationary distribution is a Gibbs state. Such a state being given by the exponential of the energy density, this clarifies the duality relationship between energy and entropy. To cite this article: J.F. Collet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 429–434.     

Sur l'analyticité des applications CR lisses à valeurs dans un ensemble algébrique réelOn the analyticity of smooth CR maps into a real algebraic set

Let f:M→M′ be a $\text{C}{}^{\mathrm{\infty }}$-smooth CR mapping between a generic real analytic submanifold $\text{M?}\text{C}{}^{\mathrm{n}}$ and a real algebraic subset $\text{M′?}\text{C}{}^{\mathrm{n\prime }}$. We prove that if M is minimal at a point p and if M′ does not contain complex curves, then f is real-analytic at p. To cite this article: B. Coupet et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 953–956.     

Sur la compacité des multimesures (I)On compactness of set-valued measures (I)

Let $\text{M}{}_{\mathrm{+}}\text{(T,}\text{K}\text{,}\text{ck}\text{(E))}$ be the space of positive $\text{K}$-regular set-valued measures defined on a σ-algebra $\text{B}$ with values in the space of all compact non empty convex subsets of a Banach space E. We characterize the compact subsets of $\text{M}{}_{\mathrm{+}}\text{(T,}\text{K}\text{,}\text{ck}\text{(E))}$ endowed with the weakest topology for which all mappings M→M(A), $\text{A∈}\text{B}$ are continuous. The case of real nonnegative measures has been investigated by Topsøe [6], Grothendieck [3] and others. To cite this article: K.K. Siggini, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 949–952.     

Types d'orbites et dynamique minimale pour les applications continues de graphesOrbit types and minimal dynamics for graph maps

We define the type of a periodic orbit of a graph map. We consider the class of ‘train-track’ representatives, that is, those graph maps which minimise the topological entropy of the topological representatives of a given free group endomorphism. We prove that each type of periodic orbit realised by an efficient representative is also realised by any representative of the same free group endomorphism. Moreover, the number of periodic orbits of a given type is minimised by the efficient representatives. To cite this article: Ll. Alsedà et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 479–482.     

Finite extensions and unipotent shadows of affine crystallographic groupsExtensions finies et ombres unipotentes des groupes affines cristallographiques

Let Γ be a virtually polycyclic group so that the Fitting subgroup is torsion-free and contains its centralizer. We prove that an effective extension of Γ by a finite group μ is isomorphic to an affine crystallographic group if and only if there exists a fixed point for the action of μ on the deformation space of affine crystallographic actions of Γ. We associate to Γ a finitely generated torsion-free nilpotent group Θ which is called the unipotent shadow of Γ, and we relate the deformation space of Γ to the deformation space of Θ. As an application, we show that Γ is isomorphic to an affine crystallographic group if, e.g., Θ has nilpotency class ?3, or if the polycylic rank of Γ is ?5, and also in some other cases. To cite this article: O. Baues, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 785–788.     

Un lemme de Mañé bilatéralA bilateral version of Mañé's lemma

We prove that, assuming some hyperbolicity on the dynamical system T:X→X and some regularity on $\text{f:X→}\text{R}$, there exists $\text{θ:X→}\text{R}$ in the same regularity class and such that α(f)?f?θ+θ°T?β(f), where α(f), β(f) are the infimum and the supremum of the averages of f along periodic orbits. To cite this article: T. Bousch, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 533–536.     

Solutions entropiques des problèmes non locauxEntropy solutions of nonlocal problems

We introduce a notion of entropy solution for the nonlocal problem $\text{C}\text{f+f=ψ}$ on $\text{?Ω}$, where $\text{ψ∈}\text{L}{}^1\text{(?Ω)}$ and $\text{C}$ is a nonlinear capacity operator. We prove its existence and uniqueness. This notion of solution allows also to solve a general elliptic problem with nonlinear boundary conditions. To cite this article: K. Ammar, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 751–756.     

Holomorphic vector bundles on open subsets of Stein manifolds

LetM be a connected two-dimensional Stein manifold withH ²(M,Z)=0 andS∈M a discrete subset withS≠ Ø. SetX:=M/S. Fix an integerr≥2. Then there exists a rankr vector bundleE onX such that there is no line bundleL onX with a non-zero mapL →E.     

Rigidité d'Einstein du plan hyperbolique complexeEinstein rigidity of the complex hyperbolic plane

We prove that every Einstein metric on $\text{B}{}^4\text{?}\text{C}{}^2$ asymptotic to the Bergmann metric is equal to it up to a diffeomorphism. The proof relies on the construction of a solution of Seiberg–Witten equations in this infinite volume setting. Therefore, and more generally, if M⁴ is a manifold with a CR-boundary at infinity, an adapted spin^c-structure which has a nonzero Kronheimer–Mrowka invariant and an asymptotically complex hyperbolic Einstein metric, we produce a solution of Seiberg–Witten equations with an strong exponential decay property. To cite this article: Y. Rollin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 671–676.     

Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.