Éditeur(s) : HAL CCSD Springer Verlag
Résumé : 44 pages, 14 figures
International audience
With any non necessarily orientable unpunctured marked surface (S,M) we associate a commutative algebra, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the set of arcs and one-sided simple closed curves in (S,M). Quasi-cluster variables are naturally gathered into possibly overlapping sets of fixed cardinality, called quasi-clusters, corresponding to maximal non-intersecting families of arcs and one-sided simple closed curves in (S,M). If the surface S is orientable, then the quasi-cluster algebra is the cluster algebra associated with the marked surface (S,M) in the sense of Fomin, Shapiro and Thurston. We classify quasi-cluster algebras with finitely many quasi-cluster variables and prove that for these quasi-cluster algebras, quasi-cluster monomials form a linear basis. Finally, we attach to (S,M) a family of discrete integrable systems satisfied by quasi-cluster variables associated to arcs in the quasi-cluster algebra and we prove that solutions of these systems can be expressed in terms of cluster variables of type A.
ISSN: 0925-9899

hal-00821878
https://hal.archives-ouvertes.fr/hal-00821878
ARXIV : 1105.1560
DOI : 10.1007/s10801-015-0586-1

Éditeur(s) : HAL CCSD Cambridge University Press (CUP)
Résumé : International audience
In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz spaces into the Colombeau algebra G are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable or quasianalytic), embedded into corresponding algebras, is the ordinary multiplication.
ISSN: 0305-0041

hal-00758568
https://hal.univ-antilles.fr/hal-00758568
https://hal.univ-antilles.fr/hal-00758568/document
https://hal.univ-antilles.fr/hal-00758568/file/DHPV2-9sep03-singlespaced.pdf
DOI : 10.1017/S0305004104007923

Éditeur(s) : HAL CCSD American Mathematical Society
Résumé : International audience
A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra( pseudo)metrics defined by sequences of exponential weights. Such an algebra with embedded Dirac's delta distribution induces discrete topology on the basic space. This result is in analogy to Schwartz' impossibility result concerning multiplication of distributions.
ISSN: 0002-9939

hal-00758571
https://hal.univ-antilles.fr/hal-00758571
https://hal.univ-antilles.fr/hal-00758571/document
https://hal.univ-antilles.fr/hal-00758571/file/0206039v1.pdf

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 : International audience
Based on a previously established framework of asymptotic extensions of topological modules and algebras, we use results concerning sheaf theoretic properties and functoriality of the construction, to develop some tools for microlocal algebraic analysis in this setting. In particular, we introduce the notion of singular spectrum of sections, and give results concerning its linear and non-linear properties.
Progress in Analysis and its Applications
London, United Kingdom

hal-00761660
https://hal.univ-antilles.fr/hal-00761660
DOI : 10.1142/9789814313179_0079

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 : 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 : 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
We review our recent formulation of Colombeau-type algebras as Hausdorff sequence spaces with ultranorms, defined by sequences of exponential weights. We extend previous results and give new perspectives related to echelon-type spaces, possible generalizations, asymptotic algebras, concepts of association, and applications thereof.
ISSN: 1065-2469

hal-00758570
https://hal.univ-antilles.fr/hal-00758570
https://hal.univ-antilles.fr/hal-00758570/document
https://hal.univ-antilles.fr/hal-00758570/file/0705.2739v1.pdf
DOI : 10.1080/10652460500437807

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