Dans cet article nous considérons des germes de feuilletages holomorphes dans le plan complexe à singularité isolée à l'origine dont la désingularisation possède des diviseurs avec des groupes d'holonomie projective non résolubles possédant des séparatrices au sens de Nakai. Nous étudions les ensembles invariants fermés construits, au voisinage de chaque diviseur, par saturation des séparatrices et nous établissons des conditions pour recoller et préserver ces ensembles après passage des coins.

In this paper we consider germs of holomorphic foliations in the complex plain with an isolated singularity at the origin whose desingularization has divisors with non solvable projective holonomy groups with separatrices. We study closed invariant sets obtained, in neighborhoods of each divisor, by saturation of the separatrices and we give conditions in order to preserve and glue those sets after go through the corners.

The purpose of this paper is to study an analogue of Euler's constant for the Selberg zeta functions of a compact Riemann surface and the Dedekind zeta function of an algebraic number field. Especially, we establish similar expressions of such Euler's constants as de la Vall\'ee-Poussin obtained in 1896 for the Riemann zeta function. We also discuss, so to speak, higher Euler's constants and establish certain formulas concerning the power sums of essential zeroes of these zeta functions similar to Riemann's explicit formula.

For any collection of spaces ${\cal A}$, we investigate two non-negative integer homotopy invariants of maps: $L_{\cal A}(f)$, the ${\cal A}$-cone length of $f$, and ${\cal L}_{\cal A}(f)$, the ${\cal A}$-category of $f$. When ${\cal A}$ is the collection of all spaces, these are the cone length and category of $f$, respectively, both of which have been studied previously. The following results are obtained: (1) For a map of one homotopy pushout diagram into another, we derive an upper bound for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the induced map of homotopy pushouts in terms of $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the other maps. This has many applications, including an inequality for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the maps in a mapping of one mapping cone sequence into another. (2) We establish an upper bound for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the product of two maps in terms of $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the given maps and the ${\cal A}$-cone length of their domains. (3) We study our invariants in a pullback square and obtain as a consequence an upper bound for the ${\cal A}$-cone length and ${\cal A}$-category of the total space of a fibration in terms of the ${\cal A}$-cone length and ${\cal A}$-category of the base and fiber. We conclude with several remarks, examples and open questions.

In this paper we prove the global existence and study decay property of the solutions to the initial boundary value problem for the solutions to the quasilinear wave equation of Kirchhoff-Carrier type with a general weakly nonlinear dissipative term by constructing a stable set in $H^{2}\bigcap H_{0}^{1}$. }

We say $x \in \{ 0,1,2 \}^{\mathbb{N}}$ is a word with Sturmian erasures if for any $a\in \{ 0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word. A large family of such words is given coding trajectories of balls in the game of billiards in the cube. We prove that the monoid of morphisms mapping all words with Sturmian erasures to words with Sturmian erasures is not finitely generated.

We prove that in a certain statistical sense the Cayley graph of almost every finitely presented group with $m\ge 2$ generators contains a subdivision of the complete graph on $l\le 2m+1$ vertices. In particular, this Cayley graph is non planar. We also show that some group constructions preserve the planarity.

This article provides new and elementary proofs for some of the crucial theorems in the theory of central simple algebras with involution of the first kind. In the first place Albert's criterion for the existence of an involution of the first kind and Kneser's extension theorem for such involutions are presented in a unified way. These two results are retrieved as corollaries of a new theorem which gives a criterion to decide whether an antiautomorphism of a central simple algebra is an involution of the first kind. Two examples are given to indicate that the analogous approach cannot be applied to involutions of the second kind. Quaternion algebras give the easiest nontrivial examples of central simple algebras which carry an involution of the first kind. Albert has shown that any central simple algebra of dimension $16$ with involution of the first kind is a tensor product of two quaternion algebras. This theorem is presented here with a new proof essentially using basic linear algebra.

Le présent travail a pour but de fournir des preuves nouvelles et élémentaires de quelques résultats clés en théorie des algèbres simples centrales à involution. Il vise à permettre au lecteur familier des notions de base sur les algèbres simples centrales et sur les involutions (rappelées en Section~2) d'aborder directement la question de la structure des algèbres admettant une involution de première espèce. On traite en premier lieu le critère d'existence d'involutions de première espèce, dû à A. A. Albert, ainsi qu'un théorème d'extension d'une involution de M. Kneser, ici pour une involution de première espèce. Ces deux résultats sont ob\-tenus comme corollaires du théorème~1 qui donne un critère simple pour qu'un anti-automorphisme d'une algèbre simple centrale soit une involution. Deux exemples indiquent que cette approche ne s'applique pas aux involutions de deuxième espèce. On démontre ensuite, à partir des propriétés usuelles des algèbres de quaternions, deux résultats, dûs à \mbox{Albert} et à J.-P. Tignol et L.~H.~Rowen, sur la structure des algèbres simples centrales à involution de première espèce et de degré~$4$.

In this paper, using an idea from the direct method of Hyers and Ulam, we investigate the situations so that the generalized Hyers-Ulam stability problem for Appolonius' equation \begin{eqnarray} \nonumber f(x-z)+f(y-z)={1 \over 2}f(x-y)+2f\Big(z-{x+y \over 2}\Big) \end{eqnarray} is satisfied.

Let $A$ be a p-Banach algebra and $\alpha$ a two-sided ideal of $A$ with a complete p-norm stronger than the p-norm inherited from $A$. By integral methods, we give here a holomorphic functional calculus relatively to $\alpha$ which coïncides with the holomorphic functional calculus defined in $A\vert\alpha$\, considered as a quotient quasi-Banach algebra. As application, we get a version of Šilov's decomposition theorem. We also give the spectral mapping theorem and an integral formula for the image of the k-th derivative of a holomorphic function.