Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates

We give a new definition of the so-called overgenerated rings, which are the usual tool used to define the asymptotic structure of a (C,E,P)-algebra, written as a factor space M_{(A,E,P)}/N_{(I_{A},E,P)}. With this new definition and in the particular case of E=C^{∞}, we show that a moderate element i.e. in M_{(A,E,P)} is negligible if and only if it satisfies the C⁰-order estimate for the ideal N_{(I_{A},E,P)}.Comment: Note, 5 page

Microlocal Asymptotic Analysis in Algebras of Generalized Functions

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.Comment: 21 page

Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

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.Comment: Submitted to the Journal of Mathematical Analysis and Application

The partition bundle of type A_{N-1} (2, 0) theory

Six-dimensional (2, 0) theory can be defined on a large class of six-manifolds endowed with some additional topological and geometric data (i.e. an orientation, a spin structure, a conformal structure, and an R-symmetry bundle with connection). We discuss the nature of the object that generalizes the partition function of a more conventional quantum theory. This object takes its values in a certain complex vector space, which fits together into the total space of a complex vector bundle (the `partition bundle') as the data on the six-manifold is varied in its infinite-dimensional parameter space. In this context, an important role is played by the middle-dimensional intermediate Jacobian of the six-manifold endowed with some additional data (i.e. a symplectic structure, a quadratic form, and a complex structure). We define a certain hermitian vector bundle over this finite-dimensional parameter space. The partition bundle is then given by the pullback of the latter bundle by the map from the parameter space related to the six-manifold to the parameter space related to the intermediate Jacobian.Comment: 15 pages. Minor changes, added reference

Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces

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

La réforme de la formation des enseignants

La réforme de la formation des enseignants, voulue par le gouvernement actuel et appuyée sur une argumentation basée sur les recommandations européennes, nous semble pouvoir être analysée d'un triple point de vue : son inscription dans un processus d'évolution des dispositifs de formation des cadres de la fonction publique, marqué par une élévation du niveau de recrutement initial ; ses conséquences sur les institutions assurant la formation des enseignants ; les conséquences de son applicati..

Kernel Theorems in Spaces of Tempered Generalized Functions

In analogy to the classical isomorphism between $\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}) ,\mathcal{S}^{\prime}(\mathbb{R}^{m}) )$ and $\mathcal{S}^{\prime}(\mathbb{R}^{n+m})$, we show that a large class of moderate linear mappings acting between the space $\mathcal{G}\_{\mathcal{S}}(\mathbb{R}^{n})$ of Colombeau rapidly decreasing generalized functions and the space $\mathcal{G}\_{\tau}(\mathbb{R}^{n})$ of temperate ones admits generalized integral representations, with kernels belonging to $\mathcal{G}\_{\tau}(\mathbb{R}^{n+m})$. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.Comment: 15 page