Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.Comment: 8 page

Entre les foot-ball lillois, quelle bataille du milieu urbain ?

Avec Lille pour terrain d’enquête, ce travail évalue la pertinence des secteurs privilégiés du football, du futsal et du football indoor. Mises en comparaison, ces trois pratiques font ressortir une cartographie locale des foot-ball à la fois hiérarchisée mais surtout différenciée. Appuyée par des surfaces de jeu spécifiques, une empreinte spatiale singulière se dessine pour chaque culture sportive « balle au pied » selon le nombre et la localisation de leurs prestataires de services respectifs. L’hypothèse d’un basculement culturel au profit des pratiques alternatives se heurte alors à la réalité du terrain : le futsal et le football indoor ne remplacent pas le football. L’analyse effectuée montre que l’un et l’autre viennent en complément du football, par leur faculté à se greffer avec succès au sein de zones urbaines ciblées mais limitées, ne parvenant pas à s’exporter au-delà de ces quelques espaces conquis.Focused on metropolitan Lille (France), this study evaluates the relevance of favourite targets for football, futsal and five-a-side football. A comparison between these three sporting cultures highlights a ranked and differentiated local geographical space. Sustained by specific fields, a singular spatial print is shaped by each foot-ball as stated by the number and location of their service providers. The hypothesis of a cultural change for the benefit of alternative physical activities is denied by facts: futsal and five-a-side football don’t replace football. Both come as complementary options alongside football, by their abilities to settle themselves into specific zones of the city. Even though successful, there seems to be no development of new implantations nor reproduction of implementation schemes for alternatives football activities

Le sport, un outil géopolitique au service des puissances contemporaines

De la guerre froide et ses boycotts olympiques, au lobbying pour obtenir l’attribution des méga-événements, il arrive régulièrement que les agendas politiques et sportifs se confondent. De ce point de vue, les contributions de Pascal Boniface ont permis de mettre en évidence cette collusion d’intérêts et d’envisager le sport comme un contributeur de puissance, façonneur d’un ordre mondial particulier. En dépit de cette filiation théorique directe, per­ceptible jusqu’à son titre, l’ouvrage de ..

Marquer pour égaliser : quelle place pour les femmes dans l’espace du football ?

L’année 2019 donnera l’occasion à la Fédération française de football de célébrer son centenaire. Pour le fêter, la FFF s’est offert un cadeau sans précédent, en organisant pour la première fois la Coupe du monde féminine de football. Cette désignation en tant que pays hôte permet de mesurer le chemin parcouru en matière de reconnaissance, de soutien et d’égalité d’accès à la pratique du football. Car 2019 marque également un autre centenaire, celui du premier Championnat de France de footbal..

The set of realizations of a max-plus linear sequence is semi-polyhedral

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coefficients of a commutative rational expression in one letter that yield a given max-plus linear sequence is a semi-algebraic set in the max-plus sense. In particular, it is a finite union of polyhedral sets

Le football : une culture mondialisée et territorialisée

Le football est marqué par la géographie. Comme tout sport, il est la délimi­tation d’un espace de jeu normé, standardisé (Gay, 1997). Ce rapport entre­tenu avec l’espace situe nécessairement telle ou telle activité physique dans une logique culturelle spécifique, entre sport et jeu, entre système sportif et système ludo-sportif (Darbon, 2014), entre le football et d’autres jeux « balle au pied ». De par la simplicité de ses règles, y compris donc celles de son cadre d’expression, le football..

Le dimanche à Paris en 2030: Enquête sur les rythmes urbains

Economic, social and technical development have transformed the urban rhythms during the last 30 years and determining a new horizon for the cities. Among those rhythms, Sundays'one occupy a special symbolic place. This research program aims at questioning the transformation of Parisian rhythms. Specifically: How does socio-economical evolution change Sunday activities? What makes it still different from the other days of the week? Is it becoming normal to offer a city without interruption, constantly opened, on a 24/7? How will our Sundays look like in 2030? Confronting territorial changes with the customs and representations of the populations and with placed-bases projects will help to understand urban rhythms, as well as Sunday evolution trends on the 2030 horizon.Les évolutions économiques, sociales, religieuses et techniques bouleversent l’ensemble des rythmes urbains depuis une trentaine d'années et interrogent le devenir des villes. Parmi ceux-ci le dimanche occupe une place singulière. Cette recherche vise à interroger les transformations des rythmes urbains parisiens. De quelles manières les évolutions socio-économiques en cours transforment-elles le dimanche ? En quoi celui-ci reste-il différent des autres jours de la semaine ? Tend-t-il à se banaliser pour offrir une ville sans interruption, vivant 24 heures sur 24, 7 jours sur 7 ? Quel(s) dimanche(s) se profilent à l’horizon 2030

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update

3D individual based model for simultaneous growth and interaction of L. Monocytogenes and lactic acid bacteria

International audienceBy interacting with pathogens, lactic acid bacteria (LAB) are able to contribute to food safety. By means of their lactic acid production which induces pH decrease, LAB influence the growth of pathogens. The aim of this study is to model and simulate lactic acid production, pH evolution, according to carbohydrate concentration in media, temperature, water activity and ratio of both population

Analysis and Multiple Steady States of an Industrial Heterogeneous Azeotropic Distillation

A study of an industrial multicomponent heterogeneous azeotropic distillation is presented. The process concerns an organic acid dehydration using an immiscible entrainer. First, a validation of the MESH and thermodynamic models through a comparison between pilot-plant experimental reconciled data and simulation results is conducted. A four-component mixture is considered for the simulation. Case studies of the boiler heat duty are automatically generated by an operating path tool. An ¥/¥ analysis is performed for the heterogeneous azeotropic pilot column and an industrial column with a decanter. Because of practical constraints, the pilot and the industrial columns do not have the same reflux policies. This leads the ¥/¥ analysis to predict multiple steady states for the industrial unit but not for the pilot column. However, multiple steady states are found by simulation both for the pilot and for the industrial unit. Multiple steady states are confirmed by simulation and experimental data for the industrial unit. Because of the positive ¥/¥ analysis, they are attributed to the phase equilibrium properties of the quaternary system. For the pilot column, multiple steady states are found by the simulation and linked to experimental observations. The multiplicity is not caused by the phase equilibrium properties; rather, it is attributed to interactions between the material and energy balances. An analysis of the simulation results helps explain the behavior of the industrial unit: the temperature of the sensitive tray gives rise to a peak in heat. This peak is located very close to the industrial temperature set point and is correlated with an impurity content minimum in the main product stream. An impurity minimum is also evidenced by the simulation for the pilot column. This complex behavior can explain observed difficulties in controlling the process at the industrial set point