Sur la stabilité du fibré de PicardOn the stability of the Picard bundle

We study the stability of the Picard bundle on the moduli space of stable vector bundles of fixed rank and determinant, over an irreducible, smooth, algebraic curve of genus g?2. To cite this article: V. Savin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 885–888.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Une axiomatisation de la substitution

We investigate the notion of substitution in an abstract way, without defining it explicitly. We single out the essential features of the operation of performing a substitution in order to define a concept of substitutive structure, called logos. We then prove a completeness theorem making precise and justifying the intuition that formulas true for the usual substitution can be proved from the logos axioms only. To cite this article: M. Crabbé, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Couplage pour la distance minimale

In this Note, we generalize a coupling result for real variables to the case of variables with values in some Polish space. This result follows from a conditional version of the Kantorovitch and Rubinstein theorem. To cite this article: J. Dedecker, C. Prieur, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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.     

The infinitesimal Lefschetz formulas: A heat equation proof

In this paper, the Lefschetz formulas of Atiyah and Bott (Ann. of Math.86 (1967), 374–407; 88 (1968), 451–491) and Atiyah and Singer (Ann. of Math.87 (1968), 546–604; 93 (1971), 119–138) are proved by heat equation methods in infinitesimal form, i.e., without using the usual localization on the fixed point set. The proof is obtained by a homotopic deformation of the Dirac operator and uses probabilistic methods. The main applications are potentially in noncompact situations where fixed point formulas do not always hold, while their infinitesimal analogues are still valid.     

Orthonormal Compactly Supported Wavelets with Optimal Sobolev Regularity: Numerical Results

Numerical optimization is used to construct new orthonormal compactly supported wavelets with a Sobolev regularity exponent as high as possible among those mother wavelets with a fixed support length and a fixed number of vanishing moments. The increased regularity is obtained by optimizing the locations of the roots the scaling filter has on the interval (π/2,π). The results improve those obtained by I. Daubechies (1988, Comm. Pure Appl. Math.41, 909–996), H. Volkmer (1995, SIAM J. Math. Anal.26, 1075–1087), and P. G. Lemarié-Rieusset and E. Zahrouni (1998, Appl. Comput. Harmon. Anal.5, 92–105).     

Estimation effective de la perte de positivité dans la régularisation des courants

Let (X,ω) be a compact complex Hermitian manifold, and let T?γ be a d-closed (1,1) almost positive current on X. A variant of Demailly's regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents T_m with analytic singularities of coefficient 1/m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C^∞, the regularizing currents T_m can be chosen such that $\text{T}{}_{\mathrm{m}}\text{?γ?}\text{C}\text{m}$ for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Obstruction à la transversalité de lagrangiens

In this Note, we give a necessary and sufficient condition for Lagrangians in a symplectic vector bundle to be deformed stably into transversal Lagrangians. In the case of three Lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group. To cite this article: M. Karoubi, M.L. Lapa de Souza, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Sur la complexité des nombres algébriques

Let b?2 be an integer. We prove that real numbers whose b-ary expansion satisfies some given, simple, combinatorial condition are transcendental. This implies that the b-ary expansion of any algebraic irrational number cannot be generated by a finite automaton. To cite this article: B. Adamczewski et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Sur une variante de la méthode des boites pour la détermination numérique de la dimension fractale d'un sous-ensemble du plan

In this work, we propose a particular discretization of the size of the grids in order to compute numerically, by the box-counting algorithm, the fractal dimension of a subset in two-dimensional space. The efficiency of the associated method is successfully tested on various examples of fractal sets which are derived from discret dynamical systems. To cite this article: N. Akroune, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

An analysis of (h,k, 1)-Shellsort

One classical sorting algorithm, whose performance in many cases remains unanalyzed, is Shellsort. Let h be a t-component vector of positive integers. An h-Shellsort will sort any given n elements in t passes, by means of comparisons and exchanges of elements. Let S>_j(h; n) denote the average number of element exchanges in the jth pass, assuming that all the n! initial orderings are equally likely. In this paper we derive asymptotic formulas of S_j(h; n) for any fixed h = (h, k, l), making use of a new combinatorial interpretation of S₃. For the special case h = (3, 2, 1), the analysis is further sharpened to yield exact expressions.     

Sur le théorème de la stabilité non asymptotique dans la méthode directe de Lyapunov

This paper provides a new theorem for nonasymptotic stability of autonomous dynamical systems by Lyapunov's direct method in the class of the indefinite functions. To cite this article: B. Kalitine, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.     

Sur la continuité de Hölder du semi-groupe de la chaleur sur les variétés coniques

In this Note, we study initially the heat kernel, p_t, on conic manifolds of dimension 2. Then, we improve the upper bound of p_t obtained in Li (Bull. Sci. Math. 124 (2000) 365–384) on conic manifolds of dimension ?3. Finally, we study the Hölder continuity of the heat semigroup on conic manifolds. Some new phenomenons are found on conic manifolds, in particular, on conic manifolds of dimension 2. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258??271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140??152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400??411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372??79 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336??46 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334??345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835??842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77??8 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766??76 (2007)].     

La conjecture de Karoubi pour la K-théorie lisse

Smooth K-functors are constructed on the category of locally convex real (resp. complex) algebras. The Karoubi's conjecture for the smooth K-theory on the isomorphism of algebraic and smooth K-functors is confirmed on the category of real (resp. complex) quasi-stable Fréchet algebras. To cite this article: H. Inassaridzé, T. Kandélaki, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Transcendance « à la Liouville » de certains nombres réels

Using a Liouville-like approach, we prove that real numbers like ∑_n=0^+∞(1/β^u_n), where β is a Pisot or a Salem number and where (u_n)_n?0 is a sufficiently lacunary sequence of positive integers, are transcendental. To cite this article: B. Adamczewski, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur la non-universalité des couples exacts

We define the notion of triangular (Leray–Koszul) spectral sequence and show that the Jordan spectral invariants of an endomorphism form a triangular spectral sequence. We introduce the concept of a universal coupling between a spectral sequence and a triangular spectral sequence and obtain a second spectral sequence associated to a Massey exact couple. Using these ideas, we construct an example of a spectral sequence which is not derived from an exact couple. To cite this article: B. Bendiffalah, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Conservation de la charge dans les codes PIC

This Note introduces a new integral form of the charge and current deposition scheme given in 1992 by Villasenor and Buneman, which permits the conservation of charge in PIC codes. This integral form extends directly to higher order spline shape factors, whose use produces less numerical noise. We thus get a discrete charge conserving method for high order shape factors. This method extends easily to nonuniform Cartesian grids for first order shape factors. To cite this article: R. Barthelmé, C. R. Acad. Sci. Paris, Ser. I 341 (2005).