Residual error estimators for the time-dependent Stokes equations

We present a posteriori residual error estimators for the approximate time-dependent Stokes model Chorin–Temam (Chorin, Math. Comp. 23 (1969) 341–353) projection scheme using a conforming finite element discretization. We prove a global upper bound and local lower bounds for the error on the velocity field only. To cite this article: N. Kharrat, Z. Mghazli, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The entropy condition and the admissibility of shocks

The author proposed (Trans. Amer. Math. Soc.199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author (J. Differential Equations18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand [Uspehi Mat. Nauk14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller [Comm. Pure Appl. Math.23 (1970), 867–884] proved that a shock satisfies Lax's shock inequalities [cf. Comm. Pure Appl. Math.14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.     

Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels

This work is concerned with stabilization of a wave equation stabilized by a boundary feedback. When the feedback is both frictional and with memory, we prove exponential stability of the solutions. In case of a boundary feedback which is only of memory type, uniform stability is not expected. We prove in this latter case, that the solutions decay polynomially. The method is new and uses the method of higher order energies (see [F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation; F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems, J. Evolution Equations 2 (2002) 127–150; F. Alabau, Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim. 41 (2002) 511–541]), the multiplier method and the properties of a large class of singular kernels. Moreover, our method can be extended to include cases of nonsingular kernels (see [V. Vergara, R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008) 287–309; R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint.]). To cite this article: F. Alabau-Boussouira et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Réalisation de Hodge du polylogarithme d'un schéma abélien

The Hodge realization of the polylogarithm of an Abelian scheme is an extension of Hodge modules. In [A. Levin, Polylogarithmic currents on abelian varieties, in: A. Reznikov, N. Schappacher (Eds.), Regulators in Analysis, Geometry and Number Theory, in: Progr. Math., vol. 171, Birkhäuser, 2000, pp. 207–229], Levin constructs some currents (generalized Eisenstein series) and conjectures that they describe the polylogarithmic extension. Our main result (Thm 3.1 and Cor 3.2) is a proof of this conjecture. This provides a tool to study the Eisenstein classes (see Section 4), which have a motivic origin (see [G. Kings, K-theory elements for the polylogarithm of abelian schemes, J. Reine Angew. Math. 517 (1999) 103–116]); an example of its application will be given in a second Note. To cite this article: D. Blottière, C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

Cellular covers of totally ordered abelian groups

Cellular covers of groups, and in particular, those of divisible abelian groups, were studied in [FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of groups, J. Pure Appl. Algebra 208, (2007), 61–76], [CHA-CHÓLSKI, W.—FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of divisible abelian groups. In: Contemp. Math. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 77–97], and continued in [FUCHS, L.—GÖBEL, R.: Cellular covers of abelian groups, Results Math. 53, (2009), 59–76] for abelian groups in general. In this note we are investigating cellular covers in the category of totally ordered abelian groups (called o-cellular covers; for definition see Section 2). Some results are similar to those on torsion-free abelian groups (unordered), while others are completely different. For instance, though kernels of o-cellular covers can not be non-zero divisible groups (Lemma 3.1), they may contain non-zero divisible subgroups (Example 3.2); however, the divisible part can not be much larger than the reduced part (Theorem 3.4). There are o-groups, even among the additive subgroups of the rationals, whose o-cellular covers form a proper class (Theorem 4.3).     

Asymptotic behaviour of certain second order integro-differential equations

The asymptotic behaviour of certain second order integro-differential equations which are more general than those equations studied in [R. P. Agarwal, J. Math. Anal. Appl.86 (1982), 471–475] and [S. R. Grace and B. S. Lalli, J. Math. Anal. Appl.76 (1980), 84–90] are discussed. It is pointed out that a defect appeared in the basic Assumption 1 made in both papers, and we avoid this defect in our discussion by using more natural conditions.     

Équations aux différences dans les vecteurs de Witt

We prove an approximation property for Frobenius difference equations in the Witt vectors, analog to a theorem of Greenberg [Publ. Math. IHES 31 (1966) 59–64]. To cite this article: L. Bélair, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On sum rules of special form for Jacobi matrices

We use sum rules of a special form to study spectral properties of Jacobi matrices. As a consequence of the main theorem, we obtain a discrete counterpart of a result by Molchanov, Novitskii and Vainberg (Comm. Math. Phys. 216 (2001) 195–213). To cite this article: S. Kupin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Spaces of lower semicontinuous set-valued maps

We give a negative answer to Problem 2 posed by R. A. McCoy in his paper [McCoy R. A.: Spaces of lower semicontinuous set-valued maps II, Math. Slovaca 60 (2010), 541–570]. Some topological properties of the space L ^?(X) introduced in [McCOY R. A.: Spaces of lower semicontinuous set-valued maps I, Math. Slovaca 60 (2010), 521–540] equipped with the Vietoris topology are also investigated.     

On the number of rooted c-nets

W. T. Tutte (Canad. J. Math.15 (1963), 249–271) was the first person to find the number of rooted c-nets. Then, R. C. Mullin and P. J. Schellenberg found it again in another way (J. Combin. Theory4 (1968), 259–276). This note presents two simpler recursive formulae and an explicit one as a summation with all the terms positive except for at most one negative term.     

Division des distributions et applications à l'étude d'idéaux de fonctions holomorphes

In this Note, we construct residue currents without Hironaka's theorem. As an application, we obtain a representation formula for holomorphic function as in Passare (Math. Scand. 62 (1988) 75–152). To cite this article: E. Mazzilli, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On orthogonal groups over the integers of a non-dyadic local field

Let L be a lattice in a quadratic space over a non-dyadic local field. We shall answer the question: What are the lattices whose unit groups coincide with that of L? If the residue class field has more than three elements the question is easy. In this case such a lattice must be $\text{a}$L or $\text{a}$L^# with a fractional ideal $\text{a}$ and the dual lattice L^# by Satz 2 of A. Kallmann, M. Kneser, and U. Stuhler (J. Reine Angew. Math.258 (1978), 51–54) or Theorem 5.2 of C. R. Riehm (Amer. J. Math.89 (1967), 549–577). But it is not easy in the case of the residue class field of three elements.     

Alternating sign matrices and descending plane partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1, ?1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math.53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math.66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.     

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.     

Sur la compacité des multimesures (II)

This paper completes the article by Siggini (C. R. Acad. Sci. Paris, Sér. I 334 (2002) 949–952). We prove another criterion of compactness in the space of positive $\text{K}$-regular set-valued measures endowed with the s-topology. We deduce from this Topsøe's criterion for real nonnegative measures (Stud. Math. XXXVI (1970) 208). To cite this article: K.K. Siggini, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur un théorème de Kronecker concernant les variétés algébriques

A classical result of Kronecker, stated at the end of the Section 10 of Kronecker (J. Reine Angew. Math. 92 (1882) 1–123), is that any radical of a finitely generated ideal in a polynomial ring of n variables is the radical of an ideal generated by n+1 elements. We give a constructive and elementary proof of a generalisation presented in (Michigan Math. J. 31 (1984) (2) 167–180): in a ring of Krull dimension $\text{?}\text{n}$ a radical of a finitely generated ideal is the radical of an ideal generated by n+1 elements. To cite this article: T. Coquand, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Some non-existence results for critical equations on step-two stratified groups

We discuss a non-existence result for a semilinear sub-elliptic Dirichlet problem with critical growth on half-spaces of stratified groups of step two. Our result improves a recent theorem (N. Garofalo, D. Vassilev, Math. Ann. 318 (3) (2000) 453–516). To cite this article: A. Bonfiglioli, F. Uguzzoni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Unobstructed Lagrangian deformations

We prove that deformations of a Lagrangian singularity are unobstructed if the usual (flat) deformations are unobstructed and if a cohomological vanishing condition is satisfied. This gives another application to deformation theory of the Lagrangian de Rham complex introduced in Sevenheck and van Straten (Math. Ann. 327 (1) (2003) 79–102). To prove our theorem, we use the T¹-lifting criterion due to Ran, Kawamata and others. To cite this article: C. Sevenheck, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On n-skein isomorphisms of graphs

H. Whitney [Amer. J. Math.54 (1932), 150–168] proved that edge isomorphisms between connected graphs with at least five vertices are induced by isomorphisms and that circuit isomorphisms between 3-connected graphs are induced by isomorphisms. R. Halin and H. A. Jung [J. London Math. Soc.42 (1967), 254–256] generalized these results by showing that for n ≥ 2, n-skein isomorphisms between (n+1)-connected graphs are induced by isomorphisms. In this paper we show that for n ≥ 2, n-skein isomorphisms between 3-connected graphs having (n+1)-skeins are induced by isomorphisms.     

Riemann–Poisson manifolds and Kähler–Riemann foliations

Riemann–Poisson manifolds were introduced by the author in C. R. Acad. Sci. Paris, Ser. I 333 (2001) 763–768, and studied in detail in preprint math.DG/0206102. Kähler–Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see Ann. Global Anal. Geom. 21 (2002) 377–399). In this Note we will show that to give a regular Riemann–Poisson structure on a manifold P is equivalent to give a Kähler–Riemann foliation on P such that the leafwise symplectic form is invariant with respect to all local foliation-preserving perpendicular vector fields. Finally, we give some examples of such manifolds. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 336 (2003).