Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some "asymmetric systems" where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems

Du Discours sur l’Inégalité à l’Émile : le parcours anthropologique de Rousseau

Le point de départ de cette étude est un passage de la préface du Discours sur l'Inégalité, passage qui, quoique souligné par Rousseau, n'a pas longtemps retenu l'attention des commentateurs. Il s'agit du texte suivant : « ... une bonne solution du problème suivant ne me paraîtrait pas indigne des Aristotes et des Plines de notre siècle : Quelles expériences seraient nécessaires pour parvenir à connaître l'homme naturel? et quels sont les moyens de faire ces expériences au sein de la société? ». La lecture du Manuscrit Favre, où l'on retrouve les mêmes interrogations, nous autorise à voir dans l'Emile la solution de Rousseau au problème posé et à mieux saisir les limites du Second Discours en tant que discours anthropologique.This study is a commentary on a passage of the preface of Rousseau's Discourse on Inequality : " It appears to me that a good solution of the following problem would be not unworthy of the Aristotles and Plinys of the present age : What experiments would have to be made, to discover the natural man? And how are those experiments to be made in a state of society ? The "Manuscrit Favre", in which Rousseau formulates the same questions, allows us to state that the author sets forth his solution in his Emile, and to perceive the limits of the Second Discourse as an anthropological work

Optimal Online Selection of a Monotone Subsequence: a Central Limit Theorem

Consider a sequence of $n$ independent random variables with a common continuous distribution $F$, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy $\pi_n^*$ that is optimal in the sense that it maximizes the expected value of $L_n(\pi_n^*)$, the number of selected observations. We investigate the distribution of $L_n(\pi_n^*)$; in particular, we obtain a central limit theorem for $L_n(\pi_n^*)$ and a detailed understanding of its mean and variance for large $n$. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality $N$ where $N$ is a Poisson random variable that is independent of the sequence.Comment: 26 page

Classical solution of the Cauchy problem for biwave equation: Application of Fourier transform

In this paper, we use some Fourier analysis techniques to find an exact solution to the Cauchy problem for the $n$-dimensional biwave equation in the upper half-space $\mathbb{R}^n\times [0,+\infty)$.Comment: 11 page