Éditeur(s) : HAL CCSD
Résumé : Submitted to the Journal of Mathematical Analysis and Applications
International audience
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{ infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.
J. Math. Anal. Appl.

Droits : info:eu-repo/semantics/OpenAccess
hal-00020280
https://hal.archives-ouvertes.fr/hal-00020280
https://hal.archives-ouvertes.fr/hal-00020280/document
https://hal.archives-ouvertes.fr/hal-00020280/file/ADelcroixGRegularJan27.pdf
ARXIV : math/0603183

Éditeur(s) : Paris : Michel Sonnius
Résumé : 5e édition

Provenance : Ville de Bordeaux. Bibliothèque municipale
Permalien : http://www.manioc.org/patrimon/BBX19009
BBX19009

Éditeur(s) : Cayenne : Imprimerie du gouvernement
Résumé : Année 1916
Siècle(s) traité(s) : 20
Droits : Domaine public
Provenance : Collectivité territoriale de Guyane. Bibliothèque Alexandre-Franconie
Permalien : http://www.manioc.org/patrimon/FRA11243
FRA11243

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

hal-01535462
https://hal.univ-antilles.fr/hal-01535462

Éditeur(s) : HAL CCSD
Résumé : International audience
In this paper we investigate solutions to the semi-linear wave equation in canonical form with non Lipschitz non-linearity and distributions or other generalized functions as data. To give a meaning to the Goursat problem with irregular data, we replace it by a biparametric family of problems. The first parameter turns the problem into a family of Lipschitz problems, the second one regularizes the data. Finally, the problem is solved in an appropriate algebra. We show that the solution is equal to the non-regularized one. In the examples, we take advantage of our results to give a new approach of the blow-up problem.
Differential Equations & Applications

Droits : info:eu-repo/semantics/OpenAccess
hal-00345090
https://hal.archives-ouvertes.fr/hal-00345090
https://hal.archives-ouvertes.fr/hal-00345090v2/document
https://hal.archives-ouvertes.fr/hal-00345090/file/Devoue-Goursat-wellposed-12-08.pdf

Éditeur(s) : HAL CCSD
Résumé : 21 pages Publié sous le titre: Spectral asymptotic analysis in algebras of generalized functions dans Asymptot. Anal. 59/1-2, 83-107, 2008.
International audience
We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.
Asymptot. Anal.

Droits : info:eu-repo/semantics/OpenAccess
hal-00140656
https://hal.archives-ouvertes.fr/hal-00140656
https://hal.archives-ouvertes.fr/hal-00140656/document
https://hal.archives-ouvertes.fr/hal-00140656/file/MicrolocalDelcroixMartiOberguggenbergerArkiv.pdf
ARXIV : 0704.1077

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We discuss various methods of regularization of singular differential problems. Their common point is that we use the flexibility of the theories of nonlinear generalized functions for adapting the regularization to the singularity of the problem. We particularly underline the relationship between the generalized solutions and those classical or distribution, when they exist, giving a general result for the case of the regularization of data.
Journal of Mathematical Analysis and applications

hal-01529550
https://hal.univ-antilles.fr/hal-01529550
DOI : 10.1016/j.jmaa.2008.11.077

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 : 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 Taylor & Francis
Résumé : International audience
ISSN: 0003-6811

hal-01529548
https://hal.univ-antilles.fr/hal-01529548
DOI : 10.1080/00036811.2010.511474

Éditeur(s) : London : H. Colburn and R. Bentley
Siècle(s) traité(s) : 19
Droits : Domaine public
Provenance : Collectivité territoriale de Martinique. Bibliothèque Schoelcher
Permalien : http://www.manioc.org/patrimon/SCH13081
SCH13081

Éditeur(s) : HAL CCSD American Physiological Society
Résumé : International audience
The IL-33/ST2 axis plays a protective role in T-cell-mediated hepatitis, but little is known about the functional impact of endogenous IL-33 on liver immunopathology. We used IL-33-deficient mice to investigate the functional effect of endogenous IL-33 in concanavalin A (Con A)-hepatitis. IL-33(-/-) mice displayed more severe Con A liver injury than wild-type (WT) mice, consistent with a hepatoprotective effect of IL-33. The more severe hepatic injury in IL-33(-/-) mice was associated with significantly higher levels of TNF-alpha and IL-1 beta and a larger number of NK cells infiltrating the liver. The expression of Th2 cytokines (IL-4, IL-10) and IL-17 was not significantly varied between WT and IL-33(-/-) mice following Con A-hepatitis. The percentage of CD25(+) NK cells was significantly higher in the livers of IL-33(-/-) mice than in WT mice in association with upregulated expression of CXCR3 in the liver. Regulatory T cells (Treg cells) strongly infiltrated the liver in both WT and IL-33(-/-) mice, but Con A treatment increased their membrane expression of ST2 and CD25 only in WT mice. In vitro, IL-33 had a significant survival effect, increasing the total number of splenocytes, including B cells, CD4(+) and CD8(+) T cells, and the frequency of ST2(+) Treg cells. In conclusion, IL-33 acts as a potent immune modulator protecting the liver through activation of ST2(+) Treg cells and control of NK cells.
ISSN: 0193-1857

hal-01366372
https://hal-univ-rennes1.archives-ouvertes.fr/hal-01366372
https://hal-univ-rennes1.archives-ouvertes.fr/hal-01366372/document
https://hal-univ-rennes1.archives-ouvertes.fr/hal-01366372/file/Ablation%20of%20interaction%20between%20IL-33%20and%201%20ST2.pdf
DOI : 10.1152/ajpgi.00097.2016

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) : Cayenne : Imprimerie du Roi
Siècle(s) traité(s) : 19
Droits : Domaine public
Provenance : Collectivité territoriale de Guyane. Bibliothèque Alexandre-Franconie
Permalien : http://www.manioc.org/patrimon/FRA11124
FRA11124

Éditeur(s) : Cayenne : Imprimerie du gouvernement
Résumé : Année 1868
Siècle(s) traité(s) : 19
Droits : Domaine public
Provenance : Collectivité territoriale de Guyane. Bibliothèque Alexandre-Franconie
Permalien : http://www.manioc.org/patrimon/FRA11256
FRA11256

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
By means of convenient regularization for an ill posed Cauchy problem, we define an associated generalized problem and discuss the conditions for the solvability of it. To illustrate this, starting from the semilinear unidirectional wave equation with data given on a characteristic curve, we show existence and uniqueness of the solution
ISSN: 0973-5348

hal-01529519
https://hal.univ-antilles.fr/hal-01529519
DOI : 10.1051/mmnp/201611207

Éditeur(s) : London : H. Colburn and R. Bentley
Siècle(s) traité(s) : 19
Droits : Domaine public
Provenance : Collectivité territoriale de Martinique. Bibliothèque Schoelcher
Permalien : http://www.manioc.org/patrimon/SCH13082
SCH13082

Éditeur(s) : [S.l.] : L.G. Stahl
Siècle(s) traité(s) : 20
Droits : Domaine public
Provenance : Archives départementales de Guadeloupe
Permalien : http://www.manioc.org/patrimon/ADG18107
ADG18107

Éditeur(s) : Londres : Sampson Law, Marston, Searle & Rivington
Siècle(s) traité(s) : 19
Provenance : Archives départementales de Guadeloupe
Permalien : http://www.manioc.org/patrimon/ADG18114
ADG18114

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