A Hamilton-Jacobi approach for front propagation in kinetic equations

In this paper we use the theory of viscosity solutions for Hamilton-Jacobi equations to study propagation phenomena in kinetic equations. We perform the hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our models describe particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The scattering operator is supposed to satisfy a maximum principle. When the velocity space is bounded, we show, under suitable hypotheses, that the phase converges towards the viscosity solution of some constrained Hamilton-Jacobi equation which effective Hamiltonian is obtained solving a suitable eigenvalue problem in the velocity space. In the case of unbounded velocities, the non-solvability of the spectral problem can lead to different behavior. In particular, a front acceleration phenomena can occur. Nevertheless, we expect that when the spectral problem is solvable one can extend the convergence result

Runner-up patents: is monopoly inevitable?.

Exclusive patents sacrifice product competition to provide firms incentives to innovate. We characterize an alternative mechanism whereby later inventors are allowed to share the patent if they discover within a certain time period of the first inventor. These runner-up patents increase social welfare under very general conditions. Furthermore, we show that the time window during which later inventors can share the patent should become a new policy tool at the disposal of the designer. This instrument will be used in a socially optimal mix with the breadth and length of the patent and could allow sorting between more or less efficient firms.

Runner-up patents: is monopoly inevitable?.

Exclusive patents sacrifice product competition to provide firms incentives to innovate. We characterize an alternative mechanism whereby later inventors are allowed to share the patent if they discover within a certain time period of the first inventor. These runner-up patents increase social welfare under very general conditions. Furthermore, we show that the time window during which later inventors can share the patent should become a new policy tool at the disposal of the designer. This instrument will be used in a socially optimal mix with the breadth and length of the patent and could allow sorting between more or less efficient firms.

Disclosure of research results: the cost of proving your honesty.

In situations where a biased sender provides verifiable information to a receiver, I study how strategic reporting affects the incentives to search for information. Research provides series of signals that can be used selectively in reporting. I show that the sender is strictly worse off when his research effort is not observed by the receiver: he has to conduct more research than in the observable case and in equilibrium, discloses all the information he obtained. However this extra research can be socially beneficial and mandatory disclosure of results can thus be welfare reducing. Finally I identify cases where the sender withholds evidence and for which mandatory disclosure rules become more attractive.persuasion games, search for information, mandatory disclosure, clinical trials;

The largest and the smallest fixed points of permutations

We give a new interpretation of the derangement numbers d_n as the sum of the values of the largest fixed points of all non-derangements of length n-1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length n with at least two fixed points. We provide analytic and bijective proofs of both results, as well as a new recurrence for the derangement numbers.Comment: 7 page

Field strength scaling in quasi-phase-matching of high-order harmonic generation by low-intensity assisting fields

High-order harmonic generation in gas targets is a widespread scheme used to produce extreme ultraviolet radiation, however, it has a limited microscopic efficiency. Macroscopic enhancement of the produced radiation relies on phase-matching, often only achievable in quasi-phase-matching arrangements. In the present work we numerically study quasi-phase-matching induced by low-intensity assisting fields. We investigate the required assisting field strength dependence on the wavelength and intensity of the driving field, harmonic order, trajectory class and period of the assisting field. We comment on the optimal spatial beam profile of the assisting field

A simple and unusual bijection for Dyck paths and its consequences

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. We also present a generalization of the bijection, as well as several applications of it to enumeration problems of statistics in restricted permutations.Comment: 13 pages, 8 figures, submitted to Annals of Combinatoric

A kinetic eikonal equation

We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit (t,x)\to (t/\eps,x/\eps). We derive a new type of limiting Hamilton-Jacobi equation, which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity. We prove well-posedness of the phase problem and convergence towards the viscosity solution of the Hamilton-Jacobi equation. This is a preliminary work before analysing the propagation of reaction fronts in kinetic equations