Éditeur(s) : HAL CCSD
Résumé : To appear in the Proceeding of GF2004 in ITSF (8 pages).
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential of a subclass of such operators.
https://hal.archives-ouvertes.fr/hal-00004897

hal-00004897
https://hal.archives-ouvertes.fr/hal-00004897
https://hal.archives-ouvertes.fr/hal-00004897/document
https://hal.archives-ouvertes.fr/hal-00004897/file/GF04BernardDelcroixColombeauNov01.pdf
ARXIV : math.GM/0505179

Éditeur(s) : HAL CCSD
Résumé : To appear in TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS (16 pages).
In the last two decades, many algebras of generalized functions have been constructed, particularly the so-called generalized Sobolev algebras. Our goal is to study the latter and some of their main properties. In this framework, we pose and solve a nonlinear degenerated Dirichlet problem with irregular data such as Dirac generalized functions.
https://hal.archives-ouvertes.fr/hal-00004903

Droits : info:eu-repo/semantics/OpenAccess
hal-00004903
https://hal.archives-ouvertes.fr/hal-00004903
https://hal.archives-ouvertes.fr/hal-00004903/document
https://hal.archives-ouvertes.fr/hal-00004903/file/sb_psn_pjm.pdf
ARXIV : math.GM/0505228

Éditeur(s) : HAL CCSD
Résumé : International audience
Generalized functions 2004
Novi Sad, Serbia

hal-01535463
https://hal.univ-antilles.fr/hal-01535463

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
Modelling environmental problems leads to mathematical systems with missing data. For instance, weather problems have generally missing initial conditions. The paper is concerned with identifying pollution terms arising in the state equation of some dissipative system with incomplete initial condition. To this aim the so-called sentinel method is used. Here, the problem of determining a sentinel is equivalent to a null-controllability problem for which Carleman inequalities are revisited.
ISSN: 1270-900X

hal-00720818
https://hal.univ-antilles.fr/hal-00720818
https://hal.univ-antilles.fr/hal-00720818/document
https://hal.univ-antilles.fr/hal-00720818/file/esaimpr2.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
Investigacion Operacionnal

hal-01185666
https://hal.univ-antilles.fr/hal-01185666

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
Résumé : 15 pages
International audience
In analogy to the classical isomorphism between $mathcal{L}left( mathcal{S}left( mathbb{R}^{n}right) ,mathcal{S}^{prime}left( mathbb{R}^{m}right) right) $ and $mathcal{S}^{prime}left( mathbb{R}^{n+m}right) $, we show that a large class of moderate linear mappings acting between the space $mathcal{G}_{mathcal{S}}left( mathbb{R}^{n}right) $ of Colombeau rapidly decreasing generalized functions and the space $mathcal{G}_{ au}left( mathbb{R}^{n}right) $ of temperate ones admits generalized integral representations, with kernels belonging to $mathcal{G}_{ au}left( mathbb{R}^{n+m}right) $. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.
Math. Proc. Camb. Philos. Soc.

Droits : info:eu-repo/semantics/OpenAccess
hal-00019916
https://hal.archives-ouvertes.fr/hal-00019916
https://hal.archives-ouvertes.fr/hal-00019916/document
https://hal.archives-ouvertes.fr/hal-00019916/file/KernelSADelcroixfeb7th.pdf
ARXIV : math.FA/0603035

Éditeur(s) : HAL CCSD Taylor & Francis
Résumé : International audience
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.
ISSN: 1065-2469

hal-01530753
https://hal.univ-antilles.fr/hal-01530753

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
Résumé : International audience
Rencontres européennes d'Analyse Non Standard
Luminy - Marseille, France

hal-01535424
https://hal.univ-antilles.fr/hal-01535424

Éditeur(s) : HAL CCSD Institute of Mathematics Polish Academy of Sciences
Résumé : International audience
In analogy to the classical isomorphism between L(D(Rn);D0(Rm)) and D0(Rm+n) (resp. L(S(Rn); S0(Rm)) and S0(Rm+n)), we show that a large class of moderate linear mappings acting between the space GC(Rn) of compactly supported generalized functions and G(Rn) of generalized functions (resp. the space GS(Rn) of Colombeau rapidly decreasing generalized functions and the space G (Rn) of temperate ones) admits generalized integral representations, with kernels belonging to specic regular subspaces of G(Rm+n) (resp. G (Rm+n)). The main novelty is to use accelerated -nets, which are unit elements for the convolution product in these regular subspaces, to construct the kernels. Finally, we establish a strong relationship between these results and the classical ones.
ISSN: 0137-6934

hal-01529541
https://hal.univ-antilles.fr/hal-01529541
https://hal.univ-antilles.fr/hal-01529541/document
https://hal.univ-antilles.fr/hal-01529541/file/delcroix.pdf
DOI : 10.4064/bc88-0-7

Éditeur(s) : HAL CCSD
Résumé : L'introduction des restitutions organisées de connaissances (ROC) dans les épreuves du baccalauréat, à partir de 1995, est une réponse de l'institution à la volonté de rendre plus efficace l'enseignement en cycle terminal. Cet article s'appuie sur l'analyse de la ROC du sujet Antilles Guyane session 2006 par une approche anthropologique en proposant la construction de son « site mathématique ». Au travers de cet exemple sont interrogées certaines caractéristiques du concept ROC comme sa cohérence (notamment institutionnelle) et sa transparence de cette évaluation.
https://hal.archives-ouvertes.fr/hal-00319519

Droits : info:eu-repo/semantics/OpenAccess
hal-00319519
https://hal.archives-ouvertes.fr/hal-00319519
https://hal.archives-ouvertes.fr/hal-00319519/document
https://hal.archives-ouvertes.fr/hal-00319519/file/PropositionArticleVersionHAL1.pdf

Éditeur(s) : HAL CCSD
Résumé : We study various proofs of the caracterization of constant functions, more precisely of the theorem: a derivable function, defined on a real interval, is constant if, and only if, its derivative is null. Our aim is to study the relationships of these proofs with the mathematical curriculum of secondary schools and the begining of undergraduate studies in France, from various point of views (epistemological, historical, didactical).
https://hal.archives-ouvertes.fr/hal-00318449

hal-00318449
https://hal.archives-ouvertes.fr/hal-00318449
https://hal.archives-ouvertes.fr/hal-00318449v2/document
https://hal.archives-ouvertes.fr/hal-00318449/file/CroissanceSansTAF10HALV2.pdf
ARXIV : 0809.0965

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We introduce the Convex Circuit-Free coloration and Convex Circuit-Free chromatic numberView the MathML source of an oriented graph View the MathML source and establish various basic results. We show that the problem of deciding if an oriented graph verifies View the MathML source is NP-complete if k≥3, and polynomial if k≤2. We introduce an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are vertex-critical for the convex circuit-free coloration.
ISSN: 0195-6698

hal-00716574
https://hal.archives-ouvertes.fr/hal-00716574
DOI : 10.1016/j.ejc.2008.02.012

Éditeur(s) : HAL CCSD Académie Serbe des Sciences et des Arts
Résumé : International audience
ISSN: 0561-7332

hal-01530760
https://hal.univ-antilles.fr/hal-01530760

Éditeur(s) : HAL CCSD
Résumé : International audience
This paper proposes Chebyshev-dependent inhomogeneous second order differential equation for the m-Boubaker polynomials (or Boubaker-Turki polynomials). This differential equation is also presented as a guide to applied physics studies. A concrete example is given through an attempt to solve the Bloch NMR flow equation inside blood vessels.
Applications and Applied Mathematics: An International Journal

hal-00766227
https://hal.univ-antilles.fr/hal-00766227

Éditeur(s) : HAL CCSD
Résumé : In this thesis, we propose a method to solve some Cauchy problems with irregular or characteristic data by using the recent theories of generalized functions. We study a regular Cauchy problem and a regular Goursat problem in the first part with data on a monotonic curve. The second part is devoted to the setting up of an algebra adapted to the generalized Cauchy problem. In the third part, we give a meaning to a generalized Cauchy problem and we show that the problem admits a unique solution. We study a generalized Goursat problem in the same way. In the fourth part, we approach a characteristic Cauchy problem by a family of non-characteristic ones. The family of solutions is a representative of a generalized function we consider as the generalized solution of the problem in an appropriate algebra. We give a meaning to the characteristic Cauchy problem with irregular data by replacing it by a family of non-characteristic problems in an appropriate biparametric algebra. The first parameter permits to replace the given problem by a non-characteristic one, whereas the second parameter makes it regular. The family of solutions is a representative of a generalized function considered as the solution of the problem.
Dans cette thèse nous proposons une méthode pour résoudre certains problèmes de Cauchy à données irrégulières ou caractéristiques en utilisant les récentes théories des fonctions généralisées. Nous étudions dans la première partie un problème de Cauchy et un problème de Goursat réguliers avec des données sur une courbe monotone. La deuxième partie est consacrée à la mise en place d'une algèbre adaptée à la résolution du problème de Cauchy généralisé. Dans la troisième partie nous donnons un sens à un problème de Cauchy généralisé et nous montrons qu'il admet une unique solution. Nous étudions de même un problème de Goursat généralisé. Dans la quatrième partie nous approchons un problème de Cauchy caractéristique par une famille de problèmes non caractéristiques. La famille de solutions est un représentant d'une fonction généralisée que nous considérons comme la solution généralisée du problème dans une algèbre appropriée. Nous donnons un sens au problème de Cauchy caractéristique dans le cas de données irrégulières en le remplaçant par une famille de problèmes non caractéristiques dans une algèbre convenable dépendant de deux paramètres. Le premier paramètre permet de se ramener à un problème non caractéristique que le second rend régulier. La famille de solutions est un représentant d'une fonction généralisée que nous considérons comme la solution du problème.
https://tel.archives-ouvertes.fr/tel-00012098

Droits : info:eu-repo/semantics/OpenAccess
tel-00012098
https://tel.archives-ouvertes.fr/tel-00012098
https://tel.archives-ouvertes.fr/tel-00012098v2/document
https://tel.archives-ouvertes.fr/tel-00012098/file/These-Devoue-29-12-2006.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Let C be a smooth curve over a Galois ring R. Let f be a function over C, and Ψ be an additive character of order p^l over R; in this paper we study the exponential sums associated to f and Ψ over C, and their L-functions. We show the rationality of the L-functions in a more general setting, then in the case of curves we express them as products of L-functions associated to polynomials over the affine line, each factor coming from a singularity of f. Finally we show that in the case of Morse functions (i.e. having only simple singularities), the degree of the L-functions are, up to sign, the same as in the case of finite fields, yielding very similar bounds for exponential sums.
ISSN: 1071-5797

hal-00551470
https://hal.archives-ouvertes.fr/hal-00551470
https://hal.archives-ouvertes.fr/hal-00551470/document
https://hal.archives-ouvertes.fr/hal-00551470/file/Charlefinal.pdf
DOI : 10.1016/j.ffa.2009.01.001