26. CMB 1999 (vol 42 pp. 221)
 Liu, Peide; Saksman, Eero; Tylli, HansOlav

Boundedness of the $q$MeanSquare Operator on VectorValued Analytic Martingales
We study boundedness properties of the $q$meansquare operator
$S^{(q)}$ on $E$valued analytic martingales, where $E$ is a
complex quasiBanach space and $2 \leq q < \infty$. We establish
that a.s. finiteness of $S^{(q)}$ for every bounded $E$valued
analytic martingale implies strong $(p,p)$type estimates for
$S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new
characterizations (in terms of analytic and stochastic properties
of the function $S^{(q)}$) of the complex spaces $E$ that admit an
equivalent $q$uniformly PLconvex quasinorm. We also obtain a
vectorvalued extension (and a characterization) of part of an
observation due to Bourgain and Davis concerning the
$L^p$boundedness of the usual squarefunction on scalarvalued
analytic martingales.
Categories:46B20, 60G46 

27. CMB 1998 (vol 41 pp. 166)
 Hof, A.

Percolation on Penrose tilings
In Bernoulli site percolation on Penrose tilings there are
two natural definitions of the critical probability.
This paper shows that they are equal on almost all Penrose tilings.
It also shows that for almost all Penrose tilings the number
of infinite clusters is almost surely~0 or~1.
The results generalize to percolation on a large class of aperiodic
tilings in arbitrary dimension, to percolation on ergodic subgraphs
of $\hbox{\Bbbvii Z}^d$, and to other percolation processes, including
Bernoulli bond percolation.
Categories:60K35, 82B43 
