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Abstract: We extend Saito's filtration to the Chow groups of a representable morphism of presheaves on , the category of schemes over . Moreover, for a presheaf on the category of smooth and projective schemes over , we define its higher Chow groups along with their filtrations. [Copyright &y& Elsevier]

Abstract: The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ of a singular surface as there are essential components of a desingularisation of . It is known that this problem can be reduced to the study of quasi-rational singularities. In this Note we give a positive answer to the Nash problem for a family of non-rational quasi-rational hypersurfaces. This same method applies to give a positive answer in some other cases, for instance, the and type singularities, and gives simple proofs of known cases. [Copyright &y& Elsevier]

Abstract: We consider the Springer fiber corresponding to a nilpotent endomorphism u of nilpotent order 2. We establish a necessary and sufficient condition of singularity for the components of . To cite this article: L. Fresse, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: We give a variant of the proof of Y.-T. Siu''s theorem concerning the non-existence of Levi-flat real hypersurface of Sobolev class , , in , . To cite this article: A. Iordan, F. Matthey, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: The aim of this Note is to present a very simple original, purely formal, proof of Quillen''s adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for ‘absolute’ derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. The ‘absolute derived functors’ are a non-additive generalization of Deligne''s ‘everywhere defined derived functors’. To cite this article: G. Maltsiniotis, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: We give an effective version of a result of Cohen, Shiga and Wolfart, which is a generalisation to the case of Siegel spaces of arbitrary degree, of the classical theorem of Schneider on the modular invariant . Given a point τ of the Siegel space parameterizing a principally polarised Abelian variety defined over , we obtain a lower bound for the distance between τ and algebraic points β of the Siegel space, in terms of the geometrical data of the problem. To achieve this, we establish a simultaneous measure of linear independence for periods of Abelian integrals, using Baker''s method. To cite this article: E. Villani, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: We sketch a proof of the conjecture due to Goncharov and Manin which states that the relative periods of the moduli space of Riemann spheres with n ordered marked points are multiple zeta values. To cite this article: F.C.S. Brown, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: For all , we define a p-adic regulator given by an explicit formula and we show that is non-trivial for . The main ideas come from a Note published in 2000 for the transcendental case. To cite this article: N. Hamida, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: Let Dumont has conjectured the identity , which generalizes, in particular, the classical results of Lagrange, Gauß, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont''s conjecture. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Abstract: In this Note, we give a geometric proof of Kolchin''s Theorem, using resolution of singularities. We give a counter-example in positive characteristic. To cite this article: J. Nicaise, J. Sebag, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Abstract: In this Note, we present the construction of a Neumann–Neumann domain decomposition preconditioner with scalability properties to solve nonlinear elastostatic and elastodynamic problems. By using the results given in Alart et al. [C. R. Acad. Sci. Paris, Ser. I 331 (2000) 399–404] and according to the work realized in Farhat et al. [Int. J. Numer. Methods Engrg. 38 (1995) 3831–3853] and Fragakis and Papadrakakis [Comput. Methods Appl. Mech. Engrg. 193 (2004) 4611–4662], we give the construction steps of the two level preconditioner which are based on the introduction of a coarse space especially adapted to finite deformations problems with static and dynamic process. To cite this article: M. Barboteu, C. R. Acad. Sci. Paris, Ser. I 340 (2005). [Copyright &y& Elsevier]

We give an explicit example of an Enriques surface without a rational point over C((t)). To cite this article: G. Lafon, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

Thanks to results of A. Beauville (Acta Math. 164 (1990) 211–235), we give an explicit description of translation-invariant vector fields on affine Jacobians of spectral curves, which gives a mathematical support for the work of F.A. Smirnov and V. Zeitlin (preprint math-ph/0203037). In the hyperelliptic case, this description is due to D. Mumford (Tata Lectures on Theta, Vol. II, Birkha¨user, Boston, 1983). To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]