La conjecture de Casas Alvero pour les degrés 5pe5p^{e}

préprintAccording to Casas Alvero conjecture, if a one variable polynomial of degree $n$ over a field of characteristic 0 is not prime with each of the $n-1$ first derivees, then it is of the form $c (X-r)^{n}$. Let $p$ be a prime number and an integer $e$, the conjecture is showed to be true for polynomials of degree $p^{e}, 2p^{e}, 3p^{e} (p \ neq 2) \ and \ 4p^{e} \ (p \ neq 3,5, 7)$. In this work we show that the conjecture is true for polynomials of degree $5p^{e} \ (p \ neq 2,3,7,11,131,193,599,3541,8009)$ . It also corrects an error in \ cite {Dra-Jong } for the polynomials of degree $4p^{e}$Selon la conjecture de Casas Alvero, si un polyn\^{o}me \'{a} une variable de degr\'{e} $n$ sur un corps commutatif de caract\'{e}ristique 0 est non premier avec chacune de ses $n-1$ premi\'{e}res d\'{e}riv\'{e}s, alors il est de forme $c(X-r)^{n}$. Soient $p$ un nombre premier et $e$ un entier, la conjecture a \'{e}t\'{e} d\'{e}montr\'{e}e pour les polyn\^{o}mes de degr\'{e} $p^{e},2p^{e}, 3p^{e}\\ (p\neq 2) \ et \ 4p^{e}\ (p\neq 3,5,7)$. Dans ce travail on montre que la conjecture est vrai pour les polyn\^{o}mes de degr\'{e} $5p^{e}\ (p\neq 2,3,7,11,131,193,599,3541,8009)$. On corrige aussi une erreur dans \cite{Dra-Jong} pour les degr\'{e} $4p^{e}

A note on the independent roman domination in unicyclic graphs

A Roman dominating function (RDF) on a graph $G = (V;E)$ is a function $f : V \to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an RDF is the value $f(V(G)) = \sum _{u \in V (G)} f(u)$. An RDF $f$ in a graph $G$ is independent if no two vertices assigned positive values are adjacent. The Roman domination number $\gamma _R (G)$ (respectively, the independent Roman domination number $i_{R}(G)$) is the minimum weight of an RDF (respectively, independent RDF) on $G$. We say that $\gamma _R (G)$ strongly equals $i_R (G)$, denoted by $\gamma _R (G) \equiv i_R (G)$, if every RDF on $G$ of minimum weight is independent. In this note we characterize all unicyclic graphs $G$ with $\gamma _R (G) \equiv i_R (G)$

A note on a Vizing's generalized conjecture

In this note we give a generalized version of Vizing's conjecture concerning the distance domination number for the cartesian product of two graphs

WYFIWIF: A Haptic Communication Paradigm For Collaborative Motor Skills Learning

International audienceMotor skills transfer is a challenging issue for many applications such as surgery, design and industry. In order to design virtual environments that support motor skills learning, a deep understanding of humans' haptic interactions is required. To ensure skills transfer, experts and novices need to collaborate. This requires the construction of the common frame of reference between the teacher and the learner in order to understand each other. In this paper, human-human haptic collaboration is investigated in order to understand how haptic information is exchanged. Furthermore, WYFIWIF (What You Feel Is What I Feel), a haptic communication paradigm is introduced. This paradigm is based on a hand guidance metaphor. The paradigm helps operators to construct an efficient common frame of reference by allowing a direct haptic communication. A learning virtual environment is used to evaluate this haptic communication paradigm. Hence, 60 volunteer students performed a needle insertion learning task. The results of this experiment show that, compared to conventional methods, the learning method based on haptic communication improves the novices' performance in such a task. We conclude that the WYFIWIF paradigm facilitate expert-novice haptic collaboration to teach motor skills

Influence of Haptic Communication on a Shared Manual Task in a Collaborative Virtual Environment

International audienceWith the advent of new haptic feedback devices, researchers are giving serious consideration to the incorporation of haptic communication in collaborative virtual environments. For instance, haptic interactions based tools can be used for medical and related education whereby students can train in minimal invasive surgery using virtual reality before approaching human subjects. To design virtual environments that support haptic communication, a deeper understanding of humans' haptic interactions is required. In this paper, human's haptic collaboration is investigated. A collaborative virtual environment was designed to support performing a shared manual task. To evaluate this system, 60 medical students participated to an experimental study. Participants were asked to perform in dyads a needle insertion task after a training period. Results show that compared to conventional training methods, a visual-haptic training improves user's collaborative performance. In addition, we found that haptic interaction influences the partners' verbal communication when sharing haptic information. This indicates that the haptic communication training changes the nature of the users' mental representations. Finally, we found that haptic interactions increased the sense of copresence in the virtual environment: haptic communication facilitates users' collaboration in a shared manual task within a shared virtual environment. Design implications for including haptic communication in virtual environments are outlined

Elaboration of a common frame of reference in Collaborative Virtual Environments

International audienceMotivation – To design virtual environments that support collaborative activities. Research approach – An experimental approach in which 44 students were asked to work in pairs to reconstruct five 3D figures. Findings/Design – The results show that including a contextual clue in virtual environments improves collaboration between operators. Research limitations – Further investigative work must be carried out to extract accurate female collaboration profiles. Originality/Value – The results enable three collaboration profiles to be identified. They also allow the extraction of some characteristics of a contextual clue which can be added to a virtual environment to improve collaboration. Take away message – The contents of a collaborative virtual environment influences the way that users collaborate

A note on global alliances in trees

For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive (respectively, defensive) alliance if for each vertex in $V-S$ (respectively, in $S$) at least half the vertices from the closed neighborhood of $v$ are in $S$. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set of $G$, and the global offensive alliance number $\gamma_{o}(G)$ (respectively, global defensive alliance number $\gamma_{a}(G)$) is the minimum cardinality of a global offensive alliance (respectively, global deffensive alliance) of $G$. We show that if $T$ is a tree of order $n$, then $\gamma_{o}(T)\leq 2\gamma(T)-1$ and if $n\geq 3$, then $\gamma_{o}(T)\leq \frac{3}{2}\gamma_{a}(T)-1$. Moreover, all extremal trees attaining the first bound are characterized

Influence of Contextual Objects on Spatial Interactions and viewpoints sharing in Virtual Environments

International audienceCollaborative virtual environments (CVEs) are 3D spaces in which users share virtual objects, communicate, and work together. To collaborate efficiently, users must develop a common representation of their shared virtual space. In this work, we investigated spatial communication in virtual environments. In order to perform an object co-manipulation task, the users must be able to communicate and exchange spatial information, such as object position, in a virtual environment. We conducted an experiment in which we manipulated the contents of the shared virtual space to understand how users verbally construct a common spatial representation of their environment. Forty-four students participated in the experiment to assess the influence of contextual objects on spatial communication and sharing of viewpoints. The participants were asked to perform in dyads an object co-manipulation task. The results show that the presence of a contextual object such as fixed and lateralized visual landmarks in the virtual environment positively influences the way male operators collaborate to perform this task. These results allow us to provide some design recommendations for CVEs for object manipulation tasks

Influence des systèmes de référence sur les profils de collaboration dans un environnement virtuel

International audienceLes Environnements Virtuels Collaboratifs sont des espaces 3D partagés dans lesquels les personnes travaillent ensemble. Pour que la collaboration soit efficace, les utilisateurs doivent avoir des compréhensions compatibles du monde virtuel. Dans cette étude exploratoire nous avons manipulé le contenu de l'environnement virtuel partagé pour comprendre comment les utilisateurs construisent verbalement des représentations communes de cet environnement. Les résultats montrent que la présence d'un indice visuel fixe dans l'environnement virtuel influence la façon dont les opérateurs collaborent