On Diagonal Equations over Finite Fields

In this paper, we obtain a sufficient condition for the diagonal equation to have only the trivial solution over finite fields. This result improves a theorem of Sun (J. Sichuan Normal Univ. Nat. Sci. Ed.26 (1989), 55–59) greatly and proves that the conjecture posed by Powell (J. Number Theory18 (1984), 34–40) holds for general n∈N as well.     

A hyperbolic three-phase flow model

We introduce a hyperbolic entropy-consistent model to describe three-phase flows, which ensures that void fractions, mass fractions and pressures remain positive through single waves occurring in the one dimensional solution of the Riemann problem. To cite this article: J.-M. Hérard, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Vibration of a pre-constrained elastic thin shell II: Intrinsic exact modelVibration d'une coque élastique mince pré-contrainte II

We study the vibration of an elastic thin shell which is pre-constrained by a large displacement with a small deformation. In this second Note we come up with an exact model p(d,∞) in intrinsic geometry. We take advantage of the exactness of the model for the existence and regularity of its the solutions. To cite this article: J. Cagnol, J.-P. Zolésio, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 251–256.     

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.     

Coupling of a multilevel fast multipole method and a microlocal discretization for the 3-D integral equations of electromagnetism

The aim of this work is to propose an accurate and efficient numerical approximation for high frequency diffraction of electromagnetic waves. In the context of the boundary integral equations presented in F. Collino and B. Després, to be published in J. Comput. Appl. Math., the strategy we propose combines the microlocal discretization (T. Abboud et al., in: Third International Conference on Mathematical Aspects of Wave Propagation Phenomena, SIAM, 1995, pp. 178–187) and the multilevel fast multipole method (J.M. Song, W.C. Chew, Microw. Opt. Tech. Lett. 10 (1) (1995) 14–19). This leads to a numerical method with a reduced complexity, of order O(N^4/3ln(N)+N_iterN^2/3), instead of the complexity O(N_iterN²) for a classical numerical iterative solution of integral equations. Computations on an academic geometry show that the new method improves the efficiency, for a solution with a good level of accuracy. To cite this article: A. Bachelot et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Convergence systems and strong consistency of least squares estimates in regression models

A recent theorem of T. L. Hai, H. Robbins, and C. Z. Wei (J. Multivariate Anal.9 (1979), 343–362) is extended to a more general form which unifies previous results in the literature on the strong consistency of least squares estimates in multiple regression models with nonrandom regressors. In particular the issue of strong consistency of the least squares estimate in the Gauss-Markov model, in the i.i.d. model with infinite second moment, and in general time series models is examined. In this connection, some basic properties of convergence systems are also obtained and are applied to the strong consistency problem.     

Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation: An Improvement

In this paper, we consider a discrete delay problem with negative feedback x(t)=f(x(t), x(t−1)) along with a certain family of time discretizations with stepsize 1/n. In the original problem, the attractor admits a nice Morse decomposition. We proved in (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36-78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995, J. Dynam. Differential Equations7, 141-190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36-78) we proved certain continuity results about the individual Morse sets, including that if f(x, y)=f(y), then the individual Morse sets are upper semicontinuous at n=∞. In this paper we extend this result to the general case; that is, we prove for general f(x, y) with negative feedback that the Morse sets are upper semicontinuous.     

Une nouvelle méthode de raffinement de maillage spatio-temporel pour l'équation des ondes

In the present Note we introduce an extension of the conservative space–time mesh refinement method presented by Fouquet et al. We also propose a post-treatment of the solution that reduces the spurious phenomena due to the non-conformity between the time meshes. A reinterpretation of the equations in terms of new unknowns leads to a new scheme with second order consistent coupling equations. Numerical experiments in 2D and a plane wave analysis for the 1D model show that the method is second order accurate for an arbitrary refinement. To cite this article: J. Rodríguez, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Properties of the Principal Eigenvalues of a General Class of Non-classical Mixed Boundary Value Problems

In this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions. Our theory is a substantial extension of the classical theory by P. Hess and T. Kato (1980, Comm. Partial Differential Equations5, 999-1030). In obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself. These auxiliary results complement and in some sense complete the theory of D. Daners and E. N. Dancer (1997, J. Differential Equations138, 86-132). The main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to H. Amann and J. López-Gómez (1998, J. Differential Equations146, 336-374).     

The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier–Stokes equations

In this Note, we formulate sufficient conditions for regularity of a so called suitable weak solution (v;p) in a sub-domain D of the time–space cylinder Q_T by means of requirements on one of the eigenvalues or on the eigenvectors of the symmetrized gradient of velocity. To cite this article: J. Neustupa, P. Penel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Étude d'un modèle de fragmentation secondaire pour les brouillards de gouttelettes

In this Note, we first present a model for droplet secondary breakup, in liquid sprays. This model is based on the kinetic theory formalism and uses experimental correlations found in the literature (L.-P. Hsiang, G.M. Faeth, Int. J. Multiphase Flow 19 (5) (1993) 721–735; R. Maxey, J. Riley, Phys. Fluids 26 (4) (1983) 883–889; M. Pilch, C.A. Erdman, Int. J. Multiphase Flow 13 (6) (1987) 741–757) to determine the fragmentation rate and its outcome. We then conduct a mathematical study of the resulting kinetic equation. We prove, under some physically reasonable assumptions, an existence and uniqueness theorem and characterize the long-time behaviour of the solution. To cite this article: G. Dufour et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A class of stochastic differential equations with non-Lipschitzian coefficients: pathwise uniqueness and no explosion

A new result for the pathwise uniqueness of solutions of stochastic differential equations with non-Lipschitzian coefficients is established. Furthermore, we prove that the solution has no explosion under the growth ξlogξ. To cite this article: S. Fang, T. Zhang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of E. J. McShane (SIAM J. Control5 (1967), 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of L. W. Neustadt (J. Math. Anal. Appl.7 (1963), 110–117), C. Olech (J. Differential Equations2 (1966), 74–101), M. Q. Jacobs (“Mathematical Theory of Control,” pp. 46–53, Academic Press, 1967), L. Cesari (SIAM J. Control12 (1974), 319–331) and T. S. Angell (J. Optim. Theory Appl.19 (1976), 63–79).     

A counterexample concerning iteratively generated sequences

We present a counterexample concerning the cluster set of a sequence (x_n) iteratively generated by a relation of sort x_{n + 1} = f(x_n). This counterexample answers a question posed in 1970 by F. T. Metcalf and T. D. Rogers (J. Math. Anal. Appl.31 (1970), 206–212).     

Extending Greene's theorem to directed graphs

The purpose of this note is to prove a theorem on acyclic digraphs, conjectured by Linial (J. Combin. Theory Ser. A, 30 (1981), 331–334), which includes Greene's theorem (J. Combin. Theory Ser. A, 20 (1976), 69–79) on decompositions of a partially ordered set into antichains.     

Applications of autoreproducing kernel grammian moduli to S (U,H)-valued stationary random functions

It is shown that the analytical characterizations of q-variate interpolable and minimal stationary processes obtained by H. Salehi (Ark. Mat., 7 (1967), 305–311; Ark. Mat., 8 (1968), 1–6; J. Math. Anal. Appl., 25 (1969), 653–662), and later by A. Weron (Studia Math., 49 (1974), 165–183), can be easily extended to Hilbert space valued stationary processes when using the two grammian moduli that respectively autoreproduce their correlation kernel and their spectral measure. Furthermore, for these processes, a Wold-Cramér concordance theorem is obtained that generalizes an earlier result established by H. Salehi and J. K. Scheidt (J. Multivar. Anal., 2 (1972), 307–331) and by A. Makagon and A. Weron (J. Multivar. Anal., 6 (1976), 123–137).     

On the uniqueness of the graphs G(n, k) of the Johnson schemes

T. A. Dowling (J. Combin. Theory6 (1969), 251–263) proved the uniqueness of the graphs G(n, k) of the Johnson schemes for n > 2k(k ? 1) + 4. We improve this result by showing the uniqueness of G(n, k) for n > 4k.     

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).     

Derivation of the k–ε model for locally homogeneous turbulence by homogenization techniques

We derive the incompressible and compressible k–ε model for locally homogeneous turbulence. The model is rigorously derived on formal mathematical grounds using the MPP modelling technique. This lets us calculate by either analytical or numerical means the closure constants of the model. To cite this article: T. Chacón Rebollo, D. Franco Coronil, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Contrôlabilité exacte frontière pour les équations des ondes quasi linéaires unidimensionnelles

By means of the existence and uniqueness of semi-global C¹ solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues, we present a unified method to establish the exact boundary controllability for 1-D quasilinear wave equations with boundary conditions of different types. To cite this article: T.T. Li, L.X. Yu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).