Triviality of the prolongation algebra of the Kuramoto-Sivashinsky equation

The authors apply the well known Wahlquist-Estabrook prolongation technique to the Kuramoto-Sivashinsky equation. The prolongation algebra turns out to be trivial in the sense that it is commutative. This supports nonintegrability of the equation

Unique local determination of convex bodies

Barker and Larman asked the following. Let $K' \subset {\Bbb{R}}^d$ be a convex body, whose interior contains a given convex body $K \subset {\Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K' \cap H$ be given. Is $K'$ then uniquely determined? Yaskin and Zhang asked the analogous question when, for all supporting hyperplanes $H$ of $K$, the $d$-volumes of the "caps" cut off from $K'$ by $H$ are given. We give local positive answers to both of these questions, for small $C^2$-perturbations of $K$, provided the boundary of $K$ is $C^2_+$. In both cases, $(d-1)$-volumes or $d$-volumes can be replaced by $k$-dimensional quermassintegrals for $1 \le k \le d-1$ or for $1 \le k \le d$, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by $l$-dimensional affine planes, where $1 \le k \le l \le d-1$. In fact, here not all $l$-dimensional affine subspaces are needed, but only a small subset of them (actually, a $(d-1)$-manifold), for unique local determination of $K'$.Comment: 16 pdf-page

Quantum tomography of the GHZ state

We present a method of generation of the Greenberger-Horne-Zeilinger state involving type II and type I parametric downconversion, and triggering photodetectors. The state generated by the proposed experimental set-up can be reconstructed through multi-mode quantum homodyne tomography. The feasibility of the measurement is studied on the basis of Monte-Carlo simulations.Comment: Paper submitted to the proceedings of the III Adriatico Research Conference on Quantum Interferometry, ICTP, March 1-5,1999. 5 pages, 3 eps figure