Counterexamples to C∞ C^{\infty} well posedness for some hyperbolic operators with triple characteristics

In this paper we prove that for a class of non-effectively hyperbolic operators with smooth triple characteristics the Cauchy problem is well posed in the Gevrey 2 class, beyond the generic Gevrey class $3/2$ (see e.g. \cite{Bro}). Moreover we show that this value is optimal

Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity

We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap \{0 \leq t \leq T\},\: \Omega \subset \R^{n+1}$ with triple characteristics on $t = 0$. We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t > 0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order terms $Q$

Higher secant varieties of Pn×Pm\mathbb{P}^n \times \mathbb{P}^m embedded in bi-degree (1,d)(1,d)

Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.Comment: 8 page

On an ODE Relevant for the General Theory of the Hyperbolic Cauchy Problem

2000 Mathematics Subject Classification: 34E20, 35L80, 35L15.In this paper we study an ODE in the complex plane. This is a key step in the search of new necessary conditions for the well posedness of the Cauchy Problem for hyperbolic operators with double characteristics