On se donne un intervalle ouvert non vide $\omega $ de $\mathbb{R}$, un ouvert connexe non vide $\Omega $ de ${{\mathbb{R}}_{5}}$ et unpolynôme unitaire

$${{P}_{m}}\left( z,\text{ }\!\!\lambda\!\!\text{ } \right)={{z}^{m}}+{{a}_{1}}\left( \text{ }\!\!\lambda\!\!\text{ } \right){{z}^{m-1}}=+\cdot \cdot \cdot +{{a}_{m-1}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)z+{{a}_{m}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)$$

de degré $m>0$, dépendant du paramètre $\lambda \in \Omega $. Un tel polynôme est dit $\omega $-hyperbolique si, pour tout $\lambda \in \Omega $, ses racines sont réelles et appartiennent à $\omega $.

On suppose que les fonctions ${{a}_{k}},k=1,\cdot \cdot \cdot ,m$, appartiennent à une classe ultradifférentiable ${{C}_{M}}\left( \Omega \right)$. On s‘intéresse au problème suivant. Soit $f$ appartient à ${{C}_{M}}\left( \Omega \right)$, existe-t-il des fonctions ${{Q}_{f}}$ et ${{R}_{f,k}},k=0,\cdot \cdot \cdot ,m-1$, appartenant respectivement à ${{C}_{M}}\left( \omega \,\times \,\Omega \right)$ et à ${{C}_{M}}\left( \Omega \right)$, telles que l’on ait, pour $\left( x,\,\lambda \right)\,\in \,\omega \,\times \,\Omega$,

$$f\left( x \right)={{P}_{m}}\left( x,\text{ }\!\!\lambda\!\!\text{ } \right){{Q}_{f}}\left( x,\text{ }\!\!\lambda\!\!\text{ } \right)+\sum\limits_{k=0}^{m-1}{{{x}^{k}}{{R}_{f,k}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)?}$$

On donne ici une réponse positive dès que le polynôme est $\omega$-hyperbolique, que la class untradifférentiable soit quasi-analytique ou non ; on obtient alors, des exemples d’idéaux fermés dans ${{C}_{M}}\left( {{\mathbb{R}}^{n}} \right)$ . On complète ce travail par une généralisation d’un résultat de C. L. Childress dans le cadre quasi-analytique et quelques remarques.

We classify normal affine surfaces with trivial Makar-Limanov invariant and finite Picard group of the smooth locus, realizing them as open subsets of weighted projective planes. We also show that such a surface admits, up to conjugacy, one or two ${{G}_{a}}$-actions.

In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of ${{f}^{\left( j \right)}}$ for $j=1,2,3,4$.

We study the connectedness of the moduli space of gauge equivalence classes of flat $G$-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups.

We are concerned with a unital separable nuclear purely infinite simple ${{C}^{*}}$-algebra $A$ satisfying UCT with a Rohlin flow, as a continuation of [12]. Our first result (which is independent of the Rohlin flow) is to characterize when two central projections in $A$ are equivalent by a central partial isometry. Our second result shows that the $\text{K}$-theory of the central sequence algebra ${A}'\cap {{A}^{\omega }}$ (for an $\omega \in \beta \text{N }\!\!\backslash\!\!\text{ N}$) and its fixed point algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in [12].

We introduce and investigate using Hilbert modules the properties of the Fourier algebra$A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use ${{S}^{1}}$ -equivariant minimal models to give a number of examples of ${{S}^{1}}$-spaces illustrating the properties of the various definitions.

It is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight $K$ when $K=2$ and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.