Éditeur(s) : HAL CCSD Cambridge University Press (CUP)
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: 0305-0041

hal-01530755
https://hal.univ-antilles.fr/hal-01530755

Éditeur(s) : HAL CCSD
Résumé : This work is devoted to the theoritical research and to the numerical calculus of weak solutions (in the sens of generalized functions) for the non linear transport equation((∂u)/(∂t))(x,y,t)+a(x,y,t)((∂f(u))/(∂x))(x,y,t)+b(x,y,t)((∂g(u))/(∂y))(x,y,t)=0 pour t>0with the initial condition u(x,y,0)=u₀(x,y) where the functions {(x,y,t)→a(x,y,t)} and {(x,y,t)→b(x,y,t)} belong to L^{∞}(R²×R⁺) (but can be discontinuous), the functions f and g are smooth and monotonous, the function {(x,y)→u₀(x,y)} belongs to L^{∞}(R²). We recall the necessary notions on nonlinear generalized functions for introducing their tensorial product. The main results (to determine the weak solutions) are sufficient conditions so that, when a sum of généralized functions (like Heaviside or Dirac products) is associated with zero, each terms of the sum is equal to zero. Thanks to these theoretical results, we can solve the Riemann problem with the help of a solver written like tensorial product of Heaviside functions (or like a sum of tensorial product of Heaviside functions) in order to obtain the weak solutions. These weak solutions allow to develop two dimensional numerical Godunov type schemes. Then, numerical tests are performed which give a comparison between the results obtained by these 2D schemes and the ones of the splitting method. These tests prove that the 2D numerical schemes are as reliable as the ones obtained by splitting. They are also more simple in their expression. Moreover, a more detailed comparative study of the two types of numerical schemes show that the 2D schemes are far less expensive in the linear case as well as in the non linear case. They are stable for the L^{∞} norm, unlike the splitting schemes. At least some perspectives as well as an existence theorem of a strong solution "almost eveywhere" for linear hyberbolic systems are studied.
Nous nous intéressons à la recherche et au calcul numérique de solutions faibles (au sens des fonctions généralisées) de l'équation de transport non linéaire((∂u)/(∂t))(x,y,t)+a(x,y,t)((∂f(u))/(∂x))(x,y,t)+b(x,y,t)((∂g(u))/(∂y))(x,y,t)=0 pour t>0complétée de la condition initiale : u(x,y,0)=u₀(x,y) où les fonctions {(x,y,t)→a(x,y,t)} et {(x,y,t)→b(x,y,t)} appartiennent à L^{∞}(R²×R⁺) (mais peuvent être discontinues), les fonctions f et g sont lisses et monotones, la fonction {(x,y)→u₀(x,y)} appartient à L^{∞}(R²). Des rappels sur les fonctions généralisées nous permettent d'introduire leur produit tensoriel. Un des résultats clés (pour déterminer ultérieurement les solutions faibles cherchées) donne des conditions suffisantes pour que, lorsqu'une somme de fonctions généralisées (de type produit d'Heaviside ou de Dirac) est associée à 0, chacun des termes de la somme est nul. Grâce à ces résultats théoriques, on résout le problème de Riemann 2D à l'aide d'un solveur s'écrivant comme produit tensoriel de fonctions type Heaviside (ou comme somme de produit tensoriel de fonctions type Heaviside) afin d'obtenir les solutions faibles. Ces solutions faibles permettent la construction des schémas numériques de type Godunov 2D. Nous les validons par des test numériques comparant les résultats obtenus par ces schémas 2D et ceux de la méthode du splitting. Ces tests montrent que les schémas numériques 2D sont aussi fiables que ceux par splitting, alors qu'ils sont plus simples dans leur écriture. Une étude comparative plus complète entre les deux types de schémas numériques montre de plus que les schémas 2D sont nettement moins coûteux à la fois dans les cas linéaire et non linéaire et qu'ils sont stables pour la norme L^{∞}, contrairement aux schémas par splitting. Enfin des perspectives ainsi qu'un théoréme d'existence de solutions fortes "presque partout" pour des systèmes hyperboliques linéaires sont abordés.
https://hal.archives-ouvertes.fr/tel-01487366

Droits : info:eu-repo/semantics/OpenAccess
tel-01487366
https://hal.archives-ouvertes.fr/tel-01487366
https://hal.archives-ouvertes.fr/tel-01487366/document
https://hal.archives-ouvertes.fr/tel-01487366/file/th%C3%A8se_olivier_jouannelle_HAL.pdf

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 : 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
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 IRMA
Résumé : International audience
Colloque trajectorien à la mémoire de G. Reeb et J.L. Callot, Strasbourg/Obernai
Strasbourg, Obernai, France

hal-01529544
https://hal.univ-antilles.fr/hal-01529544

Éditeur(s) : HAL CCSD
Résumé : I propose a self contained research paper. I hope it adds some news ideas and results to the …xed point theory in the framework of generalized functions algebras, with application to the Cauchy-Lipschitz problem in a generalized formulation including strongly irregular cases. This leads to the transport equation with distributions as coe¢ cients we wish to treat later.
https://hal.univ-antilles.fr/hal-01231272

Droits : info:eu-repo/semantics/OpenAccess
hal-01231272
https://hal.univ-antilles.fr/hal-01231272
https://hal.univ-antilles.fr/hal-01231272/document
https://hal.univ-antilles.fr/hal-01231272/file/Fixed%20points%20in%20generalized%20algebras.pdf

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 : 18 pages
This article adresses questions on localization and microlocalization in the framework of the so called (C,E,P)-algebras of generalized functions. We give new results on nonlinear properties of the asymptotic singular spectrum in the multi-parameter case, a tool to describe the singularities of generalized solutions associated to problems which are classically ill-posed. Wa apply the theory to a transport equation with strongly irregular coefficients and initial data.
https://hal.univ-antilles.fr/hal-00761112

hal-00761112
https://hal.univ-antilles.fr/hal-00761112
https://hal.univ-antilles.fr/hal-00761112/document
https://hal.univ-antilles.fr/hal-00761112/file/DT2010-03_Devoue_al3.pdf

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 : 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 : 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 Elsevier
Résumé : International audience
In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz’ result is contained in our main theorem.
ISSN: 0022-247X

hal-01530748
https://hal.univ-antilles.fr/hal-01530748
https://hal.univ-antilles.fr/hal-01530748/document
https://hal.univ-antilles.fr/hal-01530748/file/delcroixgskttjan21.pdf
DOI : 10.1016/j.jmaa.2005.01.006

Éditeur(s) : HAL CCSD
Résumé : This thesis compiles two distinct studies that both document the control of microstructures on rock seismic properties. The first part deals with the development of crystallographic preferred orientations (CPO) in the uppermost mantle associated with melt/fluid-rock interactions, recorded in peridotites xenoliths from the Japan sea back-arc basin. The microstructural and geochemical characteristics of the studied samples reveal that active spreading is associated to uppermost mantle deformation similar to that observed in the Oman ophiolite. At the onset of back-arc spreading, there are no strong interactions between melt percolation and deformation in comparison to continental rift zones, probably due to the relatively small size and short duration of the spreading event. The second part presents a unique database of plagioclase CPO from variously deformed mafic rocks. CPO are grouped in three main types; their characteristics as a function of deformation regime (magmatic or crystal-plastic) are outlined and discussed. Calculated seismic properties of gabbroic rocks show that anisotropy tends to increase as a function of fabric strength, although it is generally weak, due to the competing effect of olivine/clinopyroxene and plagioclase.
Cette thèse regroupe deux études distinctes, qui documentent le contrôle des microstructures sur les propriétés sismiques des roches. La première partie traite du développement des orientations préférentielles cristallographiques (OPC) dans le manteau supérieur, associé aux interactions liquide/magma-roche, enregistré dans des xénolites de péridotites du bassin d'arrière-arc de la mer du Japon. Les caractéristiques microstucturales et géochimiques des échantillons étudiés montrent que l'ouverture arrière-arc active est associée à une déformation du manteau supérieure similaire à celle observée dans l'ophiolite d'Oman. L'initiation de l'extension d'arrière-arc n'est pas associée à de fortes interactions entre percolation magmatique et déformation, en comparaison avec les zones de rifting continentales, probablement en raison des taille et durée relativement petites de l'épisode d'ouverture. La seconde partie présente une base de données unique d'OPC de plagioclase de roches mafiques plus ou moins déformées. Les OPC sont classées en 3 types principaux; leurs caractéristiques en fonction du régime de déformation (magmatique ou plastique) sont présentées et discutées. Les propriétés sismiques calculées des roches gabbroiques montrent que l'anisotropie tend à croitre avec l'intensité des fabriques, bien qu'elle soit généralement faible, en raison des effets opposés des olivines/clinopyroxènes et du plagioclase.
https://tel.archives-ouvertes.fr/tel-00795505

tel-00795505
https://tel.archives-ouvertes.fr/tel-00795505
https://tel.archives-ouvertes.fr/tel-00795505/document
https://tel.archives-ouvertes.fr/tel-00795505/file/Satsukawa_thesis_UM2.pdf

Éditeur(s) : HAL CCSD
Résumé : International audience
Colloque en l'honneur de Robert Lutz.
Mulhouse, France

hal-01535440
https://hal.univ-antilles.fr/hal-01535440

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
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 Wiley
Résumé : Unveiling the genetic basis of local adaptation to environmental variation is a major goal in molecular ecology. In rugged landscapes characterized by environmental mosaics, living populations and communities can experience steep ecological gradients over very short geographical distances. In lowland tropical forests, interspecific divergence in edaphic specialization (for seasonally flooded bottomlands and seasonally dry terra firme soils) has been proven by ecological studies on adaptive traits. Some species are nevertheless capable of covering the entire span of the gradient; intraspecific variation for adaptation to contrasting conditions may explain the distribution of such ecological generalists. We investigated whether local divergence happens at small spatial scales in two stands of Eperua falcata (Fabaceae), a widespread tree species of the Guiana Shield. We investigated Single Nucleotide Polymorphisms (SNP) and sequence divergence as well as spatial genetic structure (SGS) at four genes putatively involved in stress response and three genes with unknown function. Significant genetic differentiation was observed among sub-populations within stands, and eight SNP loci showed patterns compatible with disruptive selection. SGS analysis showed genetic turnover along the gradients at three loci, and at least one haplotype was found to be in repulsion with one habitat. Taken together, these results suggest genetic differentiation at small spatial scale in spite of gene flow. We hypothesize that heterogeneous environments may cause molecular divergence, possibly associated to local adaptation in E. falcata.
ISSN: 1010-061X

hal-01032412
https://hal.archives-ouvertes.fr/hal-01032412
DOI : 10.1111/jeb.12069

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We investigated the response of the crystal structure, lattice parameters, and unit-cell volume of hydrous layered silicate phlogopite at conditions relevant to subduction zone settings. We have used first principles simulation based on density functional theory to calculate the equation of state and full elastic constant tensor. Based on the generalized gradient approximation, the full single crystal elastic constant tensor with monoclinic symmetry shows significant anisotropy with the compressional elastic constants: c11 = 181 GPa, c22 = 185 GPa, c33 = 62 GPa, the shear elastic constants c44 = 14 GPa, c55 = 20 GPa, c66 = 68 Ga, and c46 = −6 GPa; the off diagonal elastic constants c12 = 48 GPa, c13 = 12 GPa, c23 = 12 GPa, c15 = −16 GPa, c25 = −5 GPa and c35 = −1 GPa at zero pressure. The elastic anisotropy of phlogopite is larger than most of the layered hydrous phases relevant in the subduction zone conditions. The shear anisotropy, AVS for phlogopite is ∼77% at zero pressure condition and although it decreases upon compression it remains relatively high compared to other hydrous phases relevant in the subduction zone settings. We also note that the shear elastic constants for phlogopite are relatively low. Phlogopite also has a high isotropic bulk VP/VS ratio ∼2.0. However, the VP/VS ratio also exhibits significant anisotropy with values as low as 1.49. Thus, phlogopite bearing metasomatized mantle could readily explain unusual VP/VS ratio as observed from seismological studies from the mantle wedge regions of the subduction zone.
ISSN: 0031-9201

hal-01115670
https://hal.archives-ouvertes.fr/hal-01115670
DOI : 10.1016/j.pepi.2014.05.004

Éditeur(s) : HAL CCSD
Résumé : We unify the definitions of phylogenetic and functional entropy and diversity as a generalization of HCDT entropy when an ultrametric tree is considered. We derive the decomposition of phylodiversity and its estimation bias correction to allow its estimation from real, often undersampled data. Phyloentropy can be transformed into phylodiversity to provide a measure of true diversity, i.e. an effective number of species or communities.
https://hal-agroparistech.archives-ouvertes.fr/hal-00946177

hal-00946177
https://hal-agroparistech.archives-ouvertes.fr/hal-00946177
https://hal-agroparistech.archives-ouvertes.fr/hal-00946177/document
https://hal-agroparistech.archives-ouvertes.fr/hal-00946177/file/HAL_Decomposing_Phylodiversity.pdf