Strong density for higher order Sobolev spaces into compact manifolds

Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $\pi_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is trivial. We also prove the density of maps that are smooth except for a set of dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy group of $N^n

Un altro sguardo al funzionale ortotropo nel piano

We address the C1&nbsp;regularity of the Lipschitz minimizers to the orthotropic functional in the plane.Studiamo la regolarit`a di classe C1 dei minimi lipschitziani del funzionale ortotropo nel piano

Distributed coloring in sparse graphs with fewer colors

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in \mbox{polylog}(n) rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.Comment: 16 pages, 4 figures - An extended abstract of this work was presented at PODC'18 (ACM Symposium on Principles of Distributed Computing

Enumeration of m-ary cacti

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table

Régler ses conflits dans un cadre spirituel : pouvoir, réparation et systèmes religieux chez les Anicinabek du Québec

Le présent article examine les conditions contemporaines d’assise de l’autorité dans les communautés algonquiennes du Québec, en analysant la régulation des conflits hors du contexte de la justice pénale. Qui départage les litiges, apaise les dissensions et rétablit l’ordre ? Comment une personne peut-elle acquérir ce pouvoir ? À la lumière de données recueillies lors de diverses enquêtes ethnographiques, replacées dans leur contexte historique, je montre l’importance de la mobilisation des religions, ou de ceux qui les dispensent, pour trouver des terrains d’entente. En particulier, la spiritualité panindienne joue un rôle croissant de restructuration des sociétés autochtones, que les gens y adhèrent ou non. Pour comprendre le phénomène, j’émets l’hypothèse que cette spiritualité, sans pour autant faire l’unanimité, offre une idéologie explicative sur l’origine des désordres qui fournit un cadre efficace de réconciliation. La religion n’est peut-être alors pas toujours l’opium du peuple, mais aussi, parfois, le moyen de s’en désintoxiquer…This article examines the contemporary foundations of authority in Algonquian communities in Quebec, by analyzing the regulation of conflicts outside the context of the criminal justice system. Who settles disputes, soothes dissension and restores order? How does a person acquire such power? Based on data collected during various ethnographic studies (situated in their historical contexts), I argue that the mobilization of religions or those who preach them – and in particular, pan-Indian spirituality – play an important role in finding common ground between disputing parties. To understand this phenomenon, I hypothesize that this spirituality, even if not unanimously accepted in communities, offers an explanatory ideology on the origin of conflicts, and as such provides an effective framework for reconciliation. Religion is perhaps not always the opium of the people, but sometimes it may be a way to detoxify…Este artículo analiza las condiciones contemporáneas de las bases de autoridad en las comunidades algonquinas de Quebec mediante el estudio de la regulación de conflictos fuera del contexto de la justicia penal. ¿Quién desempata los litigios, atenúa el disenso y restablece el orden ? ¿De qué forma una persona obtiene dicho poder ? Con base en datos recopilados en diversas investigaciones etnográficas puestas en su contexto histórico, mostramos la importancia de la utilización de las religiones, o de quienes las administran, para hallar terrenos de entendimiento. La espiritualidad pan-indígena, en particular, cumple una función creciente de reestructuración de las sociedades autóctonas, incluso si los involucrados no la suscriben. Para entender el fenómeno, adelantamos la hipótesis de que esta espiritualidad, sin llegar a la unanimidad, ofrece una ideología explicativa sobre el origen de los desórdenes y presenta un marco eficaz de reconciliación. La religión, quizá, no siempre es el opio del pueblo ; en ocasiones puede ser, también, una forma de desintoxicación