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 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

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

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 $\gamma _R (G) \equiv i_R (G)$

Haptic communication to support biopsy procedures learning in virtual environments

International audienceIn interventional radiology, physicians require high haptic sensitivity and fine motor skills development because of the limited real-time visual feedback of the surgical site. The transfer of this type of surgical skill to novices is a challenging issue. This paper presents a study on the design of a biopsy procedure learning system. Our methodology, based on a task-centered design approach, aims to bring out new design rules for virtual learning environments. A new collaborative haptic training paradigm is introduced to support human-haptic interaction in a virtual environment. The interaction paradigm supports haptic communication between two distant users to teach a surgical skill. In order to evaluate this paradigm, a user experiment was conducted. Sixty volunteer medical students participated in the study to assess the influence of the teaching method on their performance in a biopsy procedure task. The results show that to transfer the skills, the combination of haptic communication with verbal and visual communications improves the novices' performance compared to conventional teaching methods. Furthermore, the results show that, depending on the teaching method, participants developed different needle insertion profiles. We conclude that our interaction paradigm facilitates expert-novice haptic communication and improves skills transfer; and new skills acquisition depends on the availability of different communication channels between experts and novices. Our findings indicate that the traditional fellowship methods in surgery should evolve to an off-patient collaborative environment that will continue to support visual and verbal communication, but also haptic communication, in order to achieve a better and more complete skills training

Combination of Annealing Particle Filter and Belief Propagation for 3D Upper Body Tracking

3D upper body pose estimation is a topic greatly studied by the computer vision society because it is useful in a great number of applications, mainly for human robots interactions including communications with companion robots. However there is a challenging problem: the complexity of classical algorithms that increases exponentially with the dimension of the vectors’ state becomes too difficult to handle. To tackle this problem, we propose a new approach that combines several annealing particle filters defined independently for each limb and belief propagation method to add geometrical constraints between individual filters. Experimental results on a real human gestures sequence will show that this combined approach leads to reliable results

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