Thèse de Doctorat
DOI
https://doi.org/10.11606/T.45.2020.tde-28012020-182713
Document
Auteur
Nom complet
Unité de l'USP
Domain de Connaissance
Date de Soutenance
Editeur
São Paulo, 2019
Directeur
Jury
Juriaans, Orlando Stanley (Président)
Cortes, Wagner de Oliveira
Ferraz, Raul Antonio
Miranda, Manuel Antolino Milla
Titre en portugais
As álgebras plena de Colombeau e de Aragona
Mots-clés en portugais
Álgebra de Aragona
Conjunto interno
Resumé en portugais
Titre en anglais
The full algebras of Colombeau and Aragona
Mots-clés en anglais
Aragona algebra
Generalized holomorphic functions
Internal set
Resumé en anglais
Colombeau generalized functions are natural environment in which linear and nonlinear relationship can be treated. This PhD thesis aims to continue the developing the Colombeau theory of generalized functions, as well as building new differential algebras in order to use a monomorphism with Schwartz distributions can be dipped. The first chapter of this thesis brings an introduction to the Colombeau theory and describes the main results that have been obtained in the literature, which are important for the development, we describe the main results - which are important for the development of this thesis. In order to help the reader and to make the text more accessible, mathematical evidence of these results are also presented in this chapter. In the second chapter, it was studied the maximal ideal of full Colombeau algebra of generalized functions and completely classified these ideals. We intro- duced the Aragona algebras and used a compactification process similar to Stone - Check, which was introduced by Khelif - Scarpalezos to obtain this classification. In particular, we completed a classification of Khelif - Scarpalezos that is applied for simplified algebras. This chapter also serves as the basis for this thesis. In chapter 3, we generalize the internal sets and membranes, introduced by Aragona - Fernandes -Juriaans -Oberguggenberger - Ver- naev, for the context of full algebra . Several results are then generalized to the context of full algebra. If f H() is a classic holomorphic function, Rf C the zero set of f and Gf Rf K the set of generalized zeros from f , we then related these two sets using the latest theories developed in this field and this work is presented in the Chapter 4.

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