Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We show that a new definition of the asymptotic structure of the (C,E,P)-algebras (which are a refinement of the Colombeau Generalized Algebras) allows to define a natural topology on them. Furthermore, this topology can be mixed with functorial aspects of the construction in order to give extension principles, which give powerful tools to solve singular differential problems in this framework.
ISSN: 0022-247X

hal-01529551
https://hal.univ-antilles.fr/hal-01529551
DOI : 10.1016/j.jmaa.2009.05.046

Éditeur(s) : HAL CCSD
Résumé : We extend classical results from the Colombeau algebra, concerning point-value characterizations of generalized functions, to the more general case of multi-parameter (C,E,P)-algebras. Our investigations include considerations about different definitions of subspaces related to tempered generalized functions.
https://hal.univ-antilles.fr/hal-00761114

hal-00761114
https://hal.univ-antilles.fr/hal-00761114
https://hal.univ-antilles.fr/hal-00761114/document
https://hal.univ-antilles.fr/hal-00761114/file/DT2010-05_Hasler_Marti.pdf

Éditeur(s) : HAL CCSD
Résumé : Prelimary version - 13 pages.
International audience
A new approach to the algebra G_{τ} of temperate nonlinear generalized functions is proposed, in which G_{τ} is based on the space O_{M} endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of G_{τ}. This is illustrated by the natural introduction of a regularity theory in G_{τ}, of the Fourier transform, with the definition of G_{O_{C}′}, the space of rapidly generalized distributions which is the Fourier image of G_{τ}.
Publ. Inst.Math. (Beograd) (N.S.)

hal-00201458
https://hal.archives-ouvertes.fr/hal-00201458
https://hal.archives-ouvertes.fr/hal-00201458/document
https://hal.archives-ouvertes.fr/hal-00201458/file/NewGOMdec29WithAdditions.pdf
ARXIV : 0801.0074

Éditeur(s) : HAL CCSD
Résumé : Un IREM est un Institut de Recherche sur l'Enseignement des Mathématiques
International audience
L'apprentissage de l'algèbre et de l'arithmétique en lycée et dans les premières années d'université peut être considéré comme inutile par de nombreux étudiants. Et pourtant les congruences, le Théorème de Bezout, les espaces vectoriels, les matrices et les polynômes sont devenues inséparables de nos techniques d'information. Nous les utilisons quotidiennement lorsque nous écoutons un CD Audio ou dès que nous tapons le code de notre carte bancaire sur le clavier d'un distributeur de billets. Le but de cet exposé est de raconter deux applications récentes de l'algèbre : les codes linéaires (et à travers eux les codes BCH) et le désormais célèbre système de chi¤rement à clés publiques RSA.
Actes du 5e Colloque de l'IREM des Antilles-Guyane

Droits : info:eu-repo/semantics/OpenAccess
hal-00767442
https://hal.univ-antilles.fr/hal-00767442
https://hal.univ-antilles.fr/hal-00767442/document
https://hal.univ-antilles.fr/hal-00767442/file/ccod0013.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Nous étudions dans l'anneau Fq[X0, X1] des polynômes à m + 1 variables et à coefficients dans le corps fini à q éléments, l'idéal homogène J engendré par les polynômes homogènes qui s'annulent sur tout l'espace. Cet idéal s'introduit naturellement lors de l'étude des codes de Reed-Muller projectifs ([7], [8]). Nous donnons une résolution libre du quotient Fq[X0, ..., Xm]/J en utilisant le complexe de Eagon et Northcott [4] qui généralise le complexe de Koszul [5]. Ceci permet en particulier de calculer directement les dimensions des composantes homogènes de l'idéal.
ISSN: 0022-4049

hal-00767441
https://hal.univ-antilles.fr/hal-00767441
https://hal.univ-antilles.fr/hal-00767441/document
https://hal.univ-antilles.fr/hal-00767441/file/ccod0010.pdf

Éditeur(s) : HAL CCSD
Résumé : We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a "frequential microlocal analysis " and into a "microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations.
https://hal.archives-ouvertes.fr/hal-00190006

Droits : info:eu-repo/semantics/OpenAccess
hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006
https://hal.archives-ouvertes.fr/hal-00190006/document
https://hal.archives-ouvertes.fr/hal-00190006/file/Regularity-Local-Microlocal-Analysis5.pdf
ARXIV : 0711.3688

Éditeur(s) : HAL CCSD APMEP
Résumé : L'APMEP est l'association des professeurs de mathématiques de l'enseignement public.
National audience
Cet article montre à quel point l’algèbre linéaire, les polynômes et les corps fi…nis sont présents dans de nombreuses applications concrètes touchant aux domaines de l’information et de la communication. Il in- forme sur l’enjeu actuel du transfert des informations et sur les moyens algébriques mis en oeuvre pour réaliser une ”communication parfaite”.
Bulletin de l'APMED

Droits : info:eu-repo/semantics/OpenAccess
hal-00764247
https://hal.univ-antilles.fr/hal-00764247
https://hal.univ-antilles.fr/hal-00764247/document
https://hal.univ-antilles.fr/hal-00764247/file/ccod0002.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove functorial properties of those algebras and show how weak equalities, in the sense of various associations, can be described in this setting.
International Journal of Mathematics and Mathematical Sciencess

hal-00761083
https://hal.univ-antilles.fr/hal-00761083
https://hal.univ-antilles.fr/hal-00761083/document
https://hal.univ-antilles.fr/hal-00761083/file/0210249v1.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The aim of this paper is to prove that the framework of generalized functions of Sobolev type is more suitable to pose and solve some PDEs problems with very irregular data, than the one introduced by J.-F. Colombeau, when ${mathcal{C}}^{infty}$ estimates are not available or out of reach. In such type of algebras, one shows the existence and some qualitative properties of solutions for problems appearing in the mathematical modelisation of oil activities.
Journal of Mathematical Analysis and applications

hal-00685042
https://hal.archives-ouvertes.fr/hal-00685042
DOI : 10.1016/j.jmaa.2011.07.063

Éditeur(s) : HAL CCSD
Résumé : International audience
In this paper, via the algebra of generalized functions, we investigate the generalized Riemann's problem associated to conservation laws with analytical coefficients. This allows us to transform the problem into a system of ordinary differential equations. In some particular cases, such that Burgers' and conservative Richard's equation, approximated solutions are obtained by the truncation of the so called Hugoniot-Maslov's chain and numerical simulations are also presented in the case of equations with polynomial coefficients.
Novi Sad Journal of Mathematics

hal-00771405
https://hal.archives-ouvertes.fr/hal-00771405

Éditeur(s) : HAL CCSD
Résumé : International audience
Given a topological R-module or algebra E and an asymptotic scale MR Λ, we define a natural M-extended topology on the sequence space E Λ, and the M-extension of E as the Hausdorff space associated with the subspace of sequences for which multiplication is continuous with respect to this topology. Colombeau's generalized functions and similar constructions are obtained as special cases, but this new approach also allows the iteration of the construction, which was not possible with previously existing theories. We use only the topology, i.e. neigbourhoods of zero, but not its explicit definition in terms of seminorms, inductive or projective limits etc., which is particularly useful in non-metrizable spaces. Many ideas commonly used in generalized function spaces (functoriality, association, sheaf structure, algebraic analysis, ...) can be applied to a large extent; but reasoning on a category-theoretic level allows the establishment of several results so far only known for particular cases, to the whole class of such spaces.
Integral Transforms and Special Functions
Bedlewo, Poland

hal-00761666
https://hal.univ-antilles.fr/hal-00761666
DOI : 10.1080/10652460802565040

Éditeur(s) : HAL CCSD Chapman & Hall/CRC
Résumé : International audience
Nonlinear Theory of Generalized Functions

ISBN : 9780849306495
hal-01529543
https://hal.univ-antilles.fr/hal-01529543

Éditeur(s) : HAL CCSD
Résumé : In this paper we compute the first vertex of a generic Newton polygon in some special cases, and the corresponding Hasse polynomial. This allows us to show the nonexistence of $p$-cyclic coverings of the projective line in characteristic $p$ with supersingular jacobian for some (infinite families of) genera.
https://hal.archives-ouvertes.fr/hal-00551456

hal-00551456
https://hal.archives-ouvertes.fr/hal-00551456
ARXIV : 0912.2051

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretisation of differential systems. In this article the authors propose generating numerical methods in numerical linear algebra by modelling the linear system to be solved as a given state of a dynamical system; the solution can be reached asymptotically, as a (asymptotically stable) steady state, but also as a finite time (shooting methods). In that way, any (stable) numerical scheme for the integration of such a problem can be presented as a method for solving linear systems. The authors discuss aspects of this approach, which allows them to recover some known methods but also to introduce new ones. Finally, some convergence results and numerical illustrations are presented.
ISSN: 1017-1398

hal-00730194
https://hal.archives-ouvertes.fr/hal-00730194

Éditeur(s) : HAL CCSD
Résumé : International audience
8th Congress of the International Society for Analysis, its Applications, and Computation
Moscou, Russia

hal-01529545
https://hal.univ-antilles.fr/hal-01529545

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
This article is intended as a preliminary report on the implementation of a finite volume multilevel scheme for the discretization of the incompressible Navier-Stokes equations. As is well known, the use of staggered grids (e.g., MAC grids [M. Perić, R. Kessler and G. Scheuerer, Comput. & Fluids 16 (1988), no. 4, 389-403; Zbl 0672.76018]) is a serious impediment for the implementation of multilevel schemes in the context of finite differences. This difficulty is circumvented here by the use of a colocated finite volume discretization [S. Faure, J Laminie and R. Temam, "Colocated finite volume schemes for fluid flows'', submitted; M. Perić, R. Kessler and G. Scheuerer, op. cit.], for which the algebra of multilevel methods is much simpler than in the context of MAC type finite differences. The general ideas and the numerical simulations are presented in this article in the simplified context of a two-dimensional Burgers equation; the two- and three-dimensional Navier-Stokes equations, which introduce new difficulties related to the incompressibility condition and the time discretization, will be considered elsewhere [see S. Faure, J Laminie and R. Temam, op. cit.; "Finite volume discretization and multilevel methods for the Navier-Stokes equations'', in preparation].
ISSN: 0885-7474

hal-00730193
https://hal.archives-ouvertes.fr/hal-00730193

Éditeur(s) : HAL CCSD Juliusz Schauder University Centre for Nonlinear Studies
Résumé : International audience
ISSN: 1230-3429

hal-01545552
https://hal.archives-ouvertes.fr/hal-01545552
DOI : 10.12775/TMNA.1999.016

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We show that, for any finite field Fq , there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.
ISSN: 0022-314X

hal-00978911
https://hal.archives-ouvertes.fr/hal-00978911
https://hal.archives-ouvertes.fr/hal-00978911/document
https://hal.archives-ouvertes.fr/hal-00978911/file/AubryBlache5_10.pdf
DOI : 10.1016/j.jnt.2007.10.014

Éditeur(s) : HAL CCSD
Résumé : accepté pour publication par l'International Journal of Number Theory
In this paper, we precise the asymptotic behaviour of Newton polygons of $L$-functions associated to character sums, coming from certain $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $Delta=Delta_1oplus Delta_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.
https://hal.archives-ouvertes.fr/hal-00551461

Droits : info:eu-repo/semantics/OpenAccess
hal-00551461
https://hal.archives-ouvertes.fr/hal-00551461
https://hal.archives-ouvertes.fr/hal-00551461/document
https://hal.archives-ouvertes.fr/hal-00551461/file/prodpolytopespubli.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Nous étudions dans l'anneau Fq[Xo, X1, ..., Xm] des polynômes à m + 1 variables et à coefficients dans le corps fini à q éléments, l'idéal homogène J engendré par les polynômes homogènes qui s'annulent sur tout l'espace. Cet idéal s'introduit naturellement lors de l'étude des codes de Reed-Muller projectifs ([7], [8]). Nous donnons une résolution libre du quotient Fq [Xo, ..., Xm]/J en utilisant le complexe de Eagon et Northcott [4] qui généralise le complexe de Koszul [5]. Ceci permet en particulier de calculer directement les dimensions des composantes homogènes de l'idéal.
ISSN: 0022-4049

hal-00767463
https://hal.univ-antilles.fr/hal-00767463
https://hal.univ-antilles.fr/hal-00767463/document
https://hal.univ-antilles.fr/hal-00767463/file/ccod0010.pdf