Singular sets of Sobolev functionsEnsembles singuliers des fonctions de Sobolev

We are interested in finding Sobolev functions with “large” singular sets. Given $\text{N,k∈}\text{N}$, 1<p<∞, kp<N, for any compact subset A of $\text{R}{}^{\mathrm{N}}$, such that its upper box dimension is less than N?kp, we construct a Sobolev function $\text{u∈}\text{W}{}^{\mathrm{k,p}}\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N?kp. To cite this article: D. ?ubrini?, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.     

On some infinite sums of integer valued Dirac's massesSur certaines sommes infinies de masses de Dirac entières

We give a simple proof of a result obtained by Bourgain, Brezis and Mironescu [2] concerning special distributions arising as singular Jacobian determinants. The strong relation of the problem with boundary rectifiability theorems is discussed, and an interesting question remains open. To cite this article: D. Smets, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374.     

Holonomic systems with solutions ramified along a cuspSystèmes holonomes avec solutions ramifiées le long d'un cusp

We classify the holonomic systems of (micro) differential equations of multiplicity one along a singular Lagrangian irreducible variety contained in an involutive submanifold of maximal codimension. We show that their solutions are related to _kF_k?1 hypergeometric functions on the Riemann sphere. To cite this article: O. Neto, P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 171–176.     

Classification et mélanges de processusClustering and mixtures of processes

We propose a clustering method based on the estimation of mixtures of probability distributions, the new point being that the statistical units are described by probability distributions. The components of the mixtures are Dirichlet processes, normalized weighted Gamma processes, and Kraft processes. Mixtures obtained by applying some algorithms to the finite dimensional distributions of the components converge to the desired mixture as the dimension increases, since the components are mutually singular due to a theorem of Kakutani. The desired clusters are then the support of these components. To cite this article: R. Emilion, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 189–193.     

On the projectors FF and FF

A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF^† and F^†F, determined by a complex square matrix F and its Moore-Penrose inverse F^†. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rako?evi? [J. Benítez, V. Rako?evi?, Matrices A such that AA^† − A^†A are nonsingular, Appl. Math. Comput. 217 (2010) 3493-3503]. Further characteristics of FF^† and F^†F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF^† and F^†F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.     

Structures de Poisson sur une intersection complète à singularités isoléesPoisson structures over a complete intersection with isolated singularities

We study Poisson structures over singular varieties. For this purpose, we consider the Koszul complex associated to the equations of a complete intersection. This complex forms a differential graded algebra which is equivalent to the algebra of the variety. We show that a Poisson structure is equivalent to a sequence of multiderivations over the Koszul complex. If the variety has isolated singularities, then we can construct a sequence of multiderivations of reduced form. To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 5–10.     

Discrete-to-continuum limit of magnetic forcesLarge limites de forces magnétiques du discret au continu

We derive a formula for the forces within a magnetized body, starting from a discrete configuration of magnetic dipoles on a Bravais lattice. The resulting force consists of the usual (nonlocal) volume term and an additional local surface term, whose coefficients involve a singular sum over the lattice. The force thus obtained is different from the usual continuum expression, reflecting the different character of the lattice regularization of the underlying hypersingular integral. To cite this article: S. Müller, A. Schlömerkemper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 393–398.     

Justification de la théorie non linéaire de Kirchhoff–Love,comme application d'une nouvelle méthode d'inversion singulièreJustification of the nonlinear Kirchhoff–Love theory,as the application of a new singular inverse method

In the framework of nonlinear elasticity, we consider a three-dimensional plate made of a St Venant–Kirchhoff isotropic and homogeneous material of thickness 2ε and periodic in the two other directions. By a change of scales, the problem can be mapped on a fixed open set, and seen as a nonlinear singular perturbation problem. We introduce a new singular inverse method. Applying this method, we prove that for fixed and small enough exterior forces, the three-dimensional displacement converges to the solution of the nonlinear Kirchhoff–Love theory of plate as the thickness 2ε tends to zero. The limit plate model contains in particular that of von Kármán. We also give a quantitative estimate of the convergence. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615–620.     

Lp-bounds on curvature,elliptic estimates and rectifiability of singular setsBornes Lp sur la courbure,estimées elliptiques et rectifiabilité d'ensembles singuliers

We announce results on rectifiability of singular sets of pointed metric spaces which are pointed Gromov–Hausdorff limits on sequences of Riemannian manifolds, satisfying uniform lower bounds on Ricci curvature and volume, and uniform L_p-bounds on curvature. The rectifiability theorems depend on estimates for |Hess_h|_L2p, (|?Hess_h·|Hess_h|^p?2)_L2, where Δh=c, for some constant c. We also observe that (absent any integral bound on curvature) in the Kähler case, given a uniform 2-sided bound on Ricci curvature, the singular set has complex codimension 2. To cite this article: J. Cheeger, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 195–198.     

The 3-XORSAT thresholdLe seuil de 3-XORSAT

We prove the existence of a threshold phenomenon regarding the random 3-XORSAT problem (or more generally k-XORSAT). We provide the value of the threshold as the solution of two transcendental equations. These results confirm rigorously those obtained by physicists using the one-step replica symmetry breaking method and thus give for the first time the proof of the validity of this method for a problem of the class of satisfiability problems. To cite this article: O. Dubois, J. Mandler, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 963–966.     

Stable Stationary Harmonic Maps to Spheres

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.     

Travelling waves and dispersion relation in the spatial unfolding of a periodic orbitOndes progressives et relation de dispersion dans le déploiement spatial d'une orbite périodique

For a partial differential equation in spatial dimension one, admitting a spatially homogeneous time periodic solution, we show the generic existence, close to this solution, of a one-parameter family of travelling waves parametrized by their wave number k (k=0 corresponding to the spatially homogeneous initial solution). The argument is elementary and relies on a direct application of singular perturbation theory (Fenichel's global center manifold theorem). To cite this article: E. Risler, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 833–838.     

Extended quasi-homogeneous polynomial system in {\mathbb{R}^{3}}

In this paper we define an extended quasi-homogeneous polynomial system d x/dt = Q = Q ₁ + Q ₂ + ... + Q _δ , where Q _i are some 3-dimensional quasi-homogeneous vectors with weight α and degree i, i = 1, . . . ,δ. Firstly we investigate the limit set of trajectory of this system. Secondly let Q _T be the projective vector field of Q. We show that if δ ≤ 3 and the number of closed orbits of Q _T is known, then an upper bound for the number of isolated closed orbits of the system is obtained. Moreover this upper bound is sharp for δ = 3. As an application, we show that a 3-dimensional polynomial system of degree 3 (resp. 5) admits 26 (resp. 112) isolated closed orbits. Finally, we prove that a 3-dimensional Lotka-Volterra system has no isolated closed orbits in the first octant if it is extended quasi-homogeneous.     

Strong approximation by Fourier--Laplace series on the unit sphere S n-1

We study the strong approximation properties of the Cesáro means of order δ of the Fourier--Laplace expansion of functions integrable on the unit sphere S ^n-1, where δ ≥λ? (n-2)/2, the latter being the critical index for Cesáro summability of Fourier--Laplace series on S ^n-1. The main purpose of this paper is to extend known results from the unit circle S ¹to the general sphere S ^n-1 with n≥3. We prove six theorems. To prove Theorems 1-3, our machinery is based on the equiconvergent operator E ^δ _N (f) of the Cesáro means σ^δ _N (f) on S ^n-1 introduced by Wang Kunyang for δ>-1. We prove in Theorem 6 that E ^δ _N (f) is also equiconvergent with σ^δ _N (f) for δ>0 in the case of strong approximation. To prove Theorems 4 and 5, we rely on known equivalence relations between K-functionals and moduli of continuity.     

Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structureAnalyse de perturbation singulière pour la réduction de la chimie complexe des mélanges gazeux avec structure entropique

In this Note, we investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions, the equilibrium constant of which are compatible with thermodynamics, thus providing an entropic structure. We assume that a subset of the reactions is consituted of fast reactions and define a constant and linear projection onto the partial equilibrium manifold compatible with the entropy production. This reduction step is used for the study of a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. To cite this article: M. Massot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 93–98.     

Invertibility of functional Galois connectionsInversibilité des correspondances de Galois fonctionnelles

We consider equations of the form Bf=g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann's covering theorem for max-plus linear equations. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 883–888.     

δ-Uniform BSS Machines

Aδ-uniform BSS machine is a standard BSS machine which does not rely on exact equality tests. We prove that, for any real closed archimedean fieldR, a set isδ-uniformly semi-decidable iff it is open and semi-decidable by a BSS machine which is locally time bounded; we also prove that the local time bound condition is nontrivial. This entails a number of results about BSS machines, in particular the existence of decidable sets whose interior (closure) is not even semi-decidable without adding constants. Finally, we show that the sets semi-decidable by Turing machines are the sets semi-decidable byδ-uniform machines with coefficients inQorT, the field of Turing computable numbers.     

Pinceau intégral enlacéLinked integral pencils

Let γ₀ be a integral curve of an analytic vector field on a manifold of dimension 3. We suppose that γ₀ has oriented, iterated tangents. The integral pencil PI(γ₀) is the set of integral curves γ which have the same oriented, iterated tangent as γ₀. The curves of PI(γ₀) are either subanalitically separated or asymptotically linked. In this case PI(γ₀) has a formal axis which is divergent if and only if the curves of PI(γ₀) are not oscillating. To cite this article: F. Cano et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 855–858.     

S-arrangements avec répétitionsS-arrangements with repetitions

We study natural correspondances between the set of arrangements of subsets with repetitions and families of objects met in various areas of mathematics (walks in lattices, pre-order) and computer science (formal languages, Petri nets, Artificial Intelligence), using the framework of formal languages. For n=2, they are enumerated by Delannoy numbers. To cite this article: S.R. Schwer, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 261–266.     

Nombres de Liouville et nombres normauxLiouville numbers and normal numbers

We prove that there exist Liouville numbers which are normal, as well as Liouville numbers which are non-normal to any base. To cite this article: Y. Bugeaud, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 117–120.