Sharp well-posedness for Kadomtsev-Petviashvili-Burgers (KPBII) equation in R2R^2

We prove global well-posedness for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers equation (KPBII) in $\mathbb R^2$ when the initial value belongs to the anisotropic Sobolev space $H^{s_1,s_2}(\mathbb R^2)$ for all $s_1>-\frac12$ and $s_2\geq0$. On the other hand, we prove in some sense that our result is sharp.Comment: 31 page

On the ergodicity of the Weyl sums cocycle

For $\theta \in [0,1]$, we consider the map T_\a: \T^2 \to \T^2 given by $T_\theta(x,y)=(x+\theta,y+2x+\theta)$. The skew product f_\a: \T^2 \times \C \to \T^2 \times \C given by $f_\theta(x,y,z)=(T_\theta(x,y),z+e^{2 \pi i y})$ generates the so called Weyl sums cocycle a_\a(x,n) = \sum_{k=0}^{n-1} e^{2\pi i(k^2\theta+kx)} since the $n^{{\rm th}}$ iterate of f_\a writes as f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2\pi iy} a_\a(2x,n)). In this note, we improve the study developed by Forrest in \cite{forrest2,forrest} around the density for x \in \T of the complex sequence {\{a_\a(x,n)\}}_{n\in \N}, by proving the ergodicity of $f_\theta$ for a class of numbers \a that contains a residual set of positive Hausdorff dimension in $[0,1]$. The ergodicity of f_\a implies the existence of a residual set of full Haar measure of x \in \T for which the sequence {\{a_\a(x,n) \}}_{n \in \N} is dense

A note on a relationship between the inverse eigenvalue problems for nonnegative and doubly stochastic matrices and some applications

In this note, we establish some connection between the nonnegative inverse eigenvalue problem and that of doubly stochastic one. More precisely, we prove that if $(r; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$ nonnegative matrix A with Perron eigenvalue r, then there exists a least real number $k_A\geq -r$ such that $(r+\epsilon; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$ nonnegative generalized doubly stochastic matrix for all $\epsilon\geq k_A.$ As a consequence, any solutions for the nonnegative inverse eigenvalue problem will yield solutions to the doubly stochastic inverse eigenvalue problem. In addition, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Some related results are also discussed.Comment: 8 page

Non uniform hyperbolicity and elliptic dynamics

We present some constructions that are merely the fruit of combining recent results from two areas of smooth dynamics: nonuniformly hyperbolic systems and elliptic constructions.Comment: 6 pages, 0 figur