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Avancées en Mathématiques Pures et Appliquées

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[FORTHCOMING] Forme des composantes connexes de la somme booléenne de deux digraphes (≤ 5)-hypomorphes à complémentaire près
Aymen Ben Amira, Jamel Dammak, Hamza Si Kaddour

Let $$$G=(V,E)$$$ and $$$G'=(V,E')$$$ be two digraphs, $$$(\leq 5)$$$-hypomorphic up to complementation, and $$$U:=G\dot{+} G'$$$ be the boolean sum of $$$G$$$ and $$$G'$$$. The case where $$$U$$$ and $$$\overline U$$$ are both connected was studied by the authors and B.Chaari giving the form of the pair$$$\{G, G'\}$$$. In this paper we study the case where $$$U$$$ is not connected and give the morphology of the pair $$$\{G_{\restriction {V({\mathcal C})}},G'_{\restriction {V({\mathcal C})}}\}$$$ whenever $$$C$$$ is a connected component of $$$U$$$.

[FORTHCOMING] Compatibilité d’une structure de Jacobi et une structure Riemannienne sur une algébroïde de Lie
Yacine Aït Amrane, Ahmed Zeglaoui

In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, (1/2)-Kenmotsu structures and locally conformally Kähler structures. In this paper we are generalizing this work to the framework of Lie algebroids.

[FORTHCOMING] D’Ingham à l’inégalité de Nazarov : un survey sur quelques inégalités trigonométriques
Philippe Jaming, Chadi Saba

The aim of this paper is to give an overview of some inequalities about $$$L^p$$$-norms ($$$p$$$ = 1 or $$$p$$$ = 2) of harmonic (periodic) and non-harmonic trigonometric polynomials. Among the material covered, we mention Ingham’s Inequality about $$$L^2$$$ norms of non-harmonic trigonometric polynmials, the proof of the Littlewood conjecture by McGehee, Pigno and Smith on the lower bound of the $$$L^1$$$ norm of harmonic trigonometric polynomials as well as its counterpart in the non-harmonic case due to Nazarov. For the later one, we give a quantitative estimate that completes our recent result with an estimate of $$$L^1$$$-norms over small intervals. We also give some stronger lower bounds when the frequencies satisfy some more restrictive conditions (lacunary Fourier series, “multi-step arithmetic sequences”). Most proofs are close to existing ones and some open questions are mentionned at the end.

[FORTHCOMING] Une nouvelle connection semi-symétrique et non-métrique sur les produits déformés
El Moctar El Id Mohamed, Abdoul Salam Diallo, Lessiad Ahmed Sid Ahmed

In this paper we study warped products endowed with a new semi-symmetric non-metric connection, which, we called Diallo-Massamba connection. We establish relationships between the Diallo-Massamba connection of the warped product to those of the base and the fiber. Also, we derive the curvature formulas for warped products with the Diallo-Massamba connection in terms of curvatures of its components. Examples of Diallo-Massamba connection are also given.

[FORTHCOMING] Une note sur le problème de courbure de Branson-Paneitz
Randa Ben Mahmoud

In this note we revise the perturbation result of [7] on the prescribed Branson-Paneitz curvature problem on the n-dimensional unit sphere, $$$n$$$ ≥ 6. We remove condition (A1) of ([7], Theorem 1.3) and we prove an entirely new perturbation theorem.