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Soave N.,Justus Liebig University | Zilio A.,Center Danalyse Et Of Mathematique Sociales
Archive for Rational Mechanics and Analysis | Year: 2015

For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one. © 2015, Springer-Verlag Berlin Heidelberg.

Nesetril J.,Charles University | Ossona de Mendez P.,Center Danalyse Et Of Mathematique Sociales
European Journal of Combinatorics | Year: 2011

We prove that the asymptotic logarithmic density of copies of a graph F in the graphs of a nowhere dense class C is integral and we determine the range of its possible values. This leads to a generalization of the trichotomy theorem of Nešetřil and Ossona de Mendez (2011) [18] and to a notion of the degree of freedom of a graph F in a class C. This provides yet another formulation of the somewhere dense-nowhere dense classification. We obtain a structural result concerning the asymptotic shape of graphs with given degree of freedom. © 2011 Elsevier Ltd.

Nesetril J.,Charles University | Ossona de Mendez P.,Center Danalyse Et Of Mathematique Sociales | Wood D.R.,University of Melbourne
European Journal of Combinatorics | Year: 2012

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties.In this paper, we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters and the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with the Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d>0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class.We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with 'linear' crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class. © 2011 Elsevier Ltd.

Pitcher A.B.,Center Danalyse Et Of Mathematique Sociales
Mathematical and Computer Modelling | Year: 2013

A model of rule-breaking is proposed. The violation rate is assumed to respond to the expected payoff of violating, which is composed of the probability and the severity of punishment as well as the gain associated with violating. The probability of punishment is itself a function of the number of violators: for a given enforcement expenditure, the probability of punishment will decrease as the number of violators increases, simply because there would be a smaller expenditure allocated to ensuring punishment per violation. The problem of determining the optimal enforcement expenditure as a function of time is treated as a constrained optimal control problem. The results show that a crackdown (very high enforcement expenditure in the beginning) is optimal and can shift the system to a low violation state requiring a smaller enforcement expenditure to maintain a high probability of punishment. A punishment severity increase is also explored. In all cases considered, a punishment severity increase coincides with a jump up in optimal expenditure when the harsher punishment is implemented which can subsequently be reduced as the violation rate is driven down. Only when the cost of imposing the punishment is not too high can the optimal enforcement expenditure be reduced down the line to a level lower than the optimal level in the case of no punishment severity increase. This study highlights the importance of punishment costs when considering harsher penalties for violations. © 2013 Elsevier Ltd.

Leclerc B.,Center Danalyse Et Of Mathematique Sociales | Monjardet B.,University of Paris 13 | Monjardet B.,French School for Advanced Studies in the Social Sciences
Order | Year: 2013

In this note we give a characterization of meet-projections in simple atomistic lattices which generalizes previous results on the aggregation of partitions obtained in a cluster analysis framework. © 2011 Springer Science+Business Media B.V.

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