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) : 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 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 : 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) : CNRS Programme Environnement : Colloque Recherche et Environnement Strasbourg, 24-25 Septembre 1990
Résumé : As the third millennium draws near, industrial societies are becoming aware of the limits of what they can do to constrain the biosphere, ending up where "primitive" societies left off. If the perverse effects of a century of technical revolution and economic expansion are above all perceptible in the area of multiple contaminations that put in danger the well-being of people, environments and in a general way the biosphere's functioning, they also concern the standardisation of products and landscapes and overinvestment in the means of exploiting resources (inputs increasing more rapidly than outputs, when they are not bringing about their decrease). The search for solutions to the growing imbalance of the relationships Man – Nature – Resources cannot reside in a partial approach or a technical resolution of the problems posed by using renewable resources.
(OCR non controlé) A l'approche du troisième millénaire, les sociétés industrielles prennent la mesure des limites de leurs possibilités de contraindre la biosphère, aboutissant là d'où partaient les sociétés "primitives". Si les effets pervers d'un siècle de révolution technique et d'expansion économique sont surtout perceptibles dans le domaine des pollutions multiples qui mettent en péril le bien-être des personnes, les milieux et d'une manière générale le fonctionnement de la biosphère, ils concernent également l'uniformisation des produits et des paysages et le surinvestissement dans les moyens d'exploitation des ressources (les intrants croissant plus vite que les rendements, quand ils n'entraînent pas leur baisse). La recherche de solutions au déséquilibre croissant des relations Hommes - Nature - Ressources ne peut résider dans une approche partielle ou une résolution technique des problèmes posés par l'utilisation des ressources renouvelables.

Droits : info:eu-repo/semantics/openAccess
http://archimer.ifremer.fr/doc/1990/acte-2416.pdf
http://archimer.ifremer.fr/doc/00000/2416/

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