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Vol 14 - Numéro 2 (Spécial CSMT 2022)

Avancées en Mathématiques Pures et Appliquées

Liste des articles

Extension des operations semistar
Gmiza Wafa, Hizem Sana

Let RT be an extension of integral domains and ∗ be a semistar operation stable of finite type on R. We define a semistar operation ∗1 on T in the following way : for each nonzero T-submodule E of the quotient field K1 of T, let E∗1 = ∪ {E :K1 JT | J ∈ $$$\mathcal{F}$$$∗}, where K1 denotes the quotient field of T and $$$\mathcal{F}$$$∗ the localizing system associated to ∗. In this paper we investigate the basic properties of ∗1. Moreover, we show that the map $$$\varphi$$$ which associates to a semistar operation ∗ stable and of finite type on R, the semistar operation ∗1 is continuous. Furthermore, we give sufficient conditions for $$$\varphi$$$ to be a homeomorphism.

Bifurcation au-delà des valeurs propres principales pour les problèmes de Neumann avec des poids indéfinis
Marta Calanchi, Bernhard Ruf

This paper is devoted to the study of the effects of indefinite weights on the following nonlinear Neumann problems
$$$ {(P^\pm)} \begin{cases} -\Delta u &= \lambda \, a(x) u \pm |u|^{p-1}u\ &\quad \hbox{in } \Omega \subset ℝ^N \\ \frac{\partial u}{\partial \nu} &=\ 0 \ & \hbox{on } \partial \Omega \end{cases}$$$
The function $$$ a = a(x)$$$ is assumed to be continuous and sign-changing. Then the linear part has two sequences of eigenvalues. Our results establish a relation between the position of the parameter $$$\lambda$$$ and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem.

Invariants topologiques pour le problème de la courbure scalaire sur les variétés
Khadijah Abdullah Sharaf, Hichem Chtioui

In [7], A.Bahri introduced two topological invariants μ and τ to study the prescribed scalar curvature problem on standard spheres of high dimensions. In this paper we first extend μ and τ to the problem on general riemannian manifolds. Second we analyze, as suggested in [7], the relation between these two quantities and we prove under topological conditions that μ = τ.

Les foncteurs de type Gysin-(ℤ/2ℤ)d
Dorra Bourguiba, Said Zarati

Let d ≥ 1 be an integer and Kd be a contravariant functor from the category of subgroups of (ℤ/2ℤ)d to the category of graded and finite 𝔽2-algebras. In this paper, we generalize the conjecture of G. Carlsson [C3], concerning free actions of (ℤ/2ℤ)d on finite CW-complexes, by suggesting, that if Kd is a Gysin-(ℤ/2ℤ)d-functor (that is to say, the functor Kd satisfies some properties, see 2.2), then we have :
$$$\big(C_{d} \big): \; \underset{i \geq 0}{\sum}dim_{\mathbb{F}_{2}} \big(\mathcal{K}_{d}(0)\big)^{i} \geq 2^{d}$$$
We prove this conjecture for 1 ≤ d ≤ 3 and we show that, in certain cases, we get an independent proof of the following
results (for d = 3 see [C4]) :
If the group (ℤ/2ℤ)d, 1 ≤ d ≤ 3, acts freely and cellularly on a finite CW-complex X, then $$${\underset{i \geq 0}{\sum}}dim_{\mathbb{F}_{2}}H^{i}(X;\; \mathbb{F}_{2}) \geq 2^{d}$$$