The spatial statistical properties of wave functions in a disordered finite one-dimensional sample

For a given wave function one can define a quantity $\mu_E$ having a meaning of its inverse spatial size. The Laplace transform of the distribution function $P(\mu_E)$ is calculated analytically for a 1D disordered sample with a finite length $L$.Comment: LaTEX, 7 pages, Preprint IFUM-456/FT, Milano, Jan.199

Co-ordinating retinal histogenesis: early cell cycle exit enhances early cell fate determination in the Xenopus retina

The laminar arrays of distinct cell types in the vertebrate retina are built by a histogenic process in which cell fate is correlated with birth order. To explore this co-ordination mechanistically, we altered the relative timing of cell cycle exit in the developing Xenopus retina and asked whether this affected the activity of neural determinants. We found that Xath5, a bHLH proneural gene that promotes retinal ganglion cell (RGC) fate, ( Kanekar, S., Perron, M., Dorsky, R., Harris, W. A., Jan, L. Y., Jan, Y. N. and Vetter, M. L. (1997) Neuron 19, 981-994), does not cause these cells to be born prematurely. To drive cells out of the cell cycle early, therefore, we misexpressed the cyclin kinase inhibitor, p27Xic1. We found that early cell cycle exit potentiates the ability of Xath5 to promote RGC fate. Conversely, the cell cycle activator, cyclin E1, which inhibits cell cycle exit, biases Xath5-expressing cells toward later neuronal fates. We found that Notch activation in this system caused cells to exit the cell cycle prematuely, and when it is misexpressed with Xath5, it also potentiates the induction of RGCs. The potentiation is counteracted by co-expression of cyclin E1. These results suggest a model of histogenesis in which the activity of factors that promote early cell cycle exit enhances the activity of factors that promote early cellular fates

Decomposition of Levy trees along their diameter

We study the diameter of L{\'e}vy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. L{\'e}vy trees have been introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of L{\'e}vy trees and we prove that it is realized by a unique pair of points. We prove that the law of L{\'e}vy trees conditioned to have a fixed diameter r $\in$ (0, $\infty$) is obtained by glueing at their respective roots two independent size-biased L{\'e}vy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of L{\'e}vy trees according to their diameter, we characterize the joint law of the height and the diameter of stable L{\'e}vy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (1983) in the Brownian case