on a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain soit un domaine non borne dans un espace euclidien a n dimensions r n a frontiere compacte et c on considere le probleme mixte u u f t x d 1 d 1 u f t x dans 0 u 0 sur 0 u 0 x u 0 x t u 0 x u 1 x dans f est une fonction non lineaire a valeur reelle et d 1 d 1 u t j x u 1 j 2

the contact process on a finite set iii the critical case we show that in n is the time that the contact process on 1 n first hits the empty set then for c the critical value for the contact process on z n n and n n 4 0 in probability as n as a consequence of the result we get bounds on some critical exponents we also study the analogous problem for bond percolation in 1 n z and investigate the limit distribution of n e n

smooth threefolds in bbb p 5 without apparent triple or quadruple points and a quadruple point formula for a projective variety x of codimension 2 in bbb p n 2 defined over the complex number field bbb c it is traditionally said that x has no apparent k 1 ple points if the k 1 secant lines of x do not fill up the ambient projective space bbb p n 2 equivalently the locus of k 1 ple points of a generic projection of x to bbb p n 1 is empty we show that a smooth threefold in bbb p 5 has no apparent triple points if and only if it is contained in a quadric hypersurface we also obtain an enumerative formula counting the quadrisecant lines of x passing through a general point of bbb p 5 and give necessary cohomological conditions for smooth threefolds in bbb p 5 without apparent quadruple points this work is intended to generalize the work of f severi fse and a aure au where it was shown that a smooth surface in bbb p 4 has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric furthermore we give open questions along these lines

simulation of three dimensional bootstrap percolation on a simple cubic lattie which is initially occupied randomly with concentration p a site is emptied if it does not have at least four of its six neighbors occupied the threshold concentration below which all sites are emptied after sufficiently many iterations is found to vary logarithmically with l for l l l lattices and l between 32 and 704 it is extrapolated to about 0 936 for infinite lattices two other cases are roughly compatible with the theoretical prediction of unity as a threshold concentration

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

some extremal contractions between smooth varieties arising from projective geometry we construct explicit examples of elementary extremal contractions both birational and of fiber type from smooth projective n dimensional varieties with n geq 4 onto smooth projective varieties arising from classical projective geometry and defined over sufficiently small fields not necessarily algebraically closed the examples considered come from particular special homaloidal and subhomaloidal linear systems which are usually degenerations of general phenomena classically investigated by bordiga severi todd room fano semple and tyrrell and more recently by ein and shepherd barron the first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of mathbf p m with 3 leq m leq 5 we get another series of examples by considering special cubic hypersurfaces through some surfaces in mathbf p 5 or some 3 folds in mathbf p 7 having one apparent double point the last examples come from an intriguing birational elementary extremal contraction in dimension 6 studied by semple and tyrrell and fully described in the last section of the paper

proof of straley s argument for bootstrap percolation we prove thatpc 1 for bootstrap percolation with large void instabilities in particular ifm 3 on the square lattice

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

remarks on global existence for nonlinear parabolic equations on considere le probleme suivant u t u f u d x u d x 2 u u t 0 x avec l hypothese que f 0 p pres de 0 si p 2 et f ne contient pas u alors ce probleme pour tout n 1 admet une solution unique globale lisse pourvu que la donnee initiale soit suffisamment petite

correlation lengths for oriented percolation oriented percolation has two correlation lengths one in the space and one in the time direction in this paper we define these quantities for the two dimensional model in terms of the exponential decay of suitably chosen quantities and study the relationship between the various definitions the definitions are used in a companion paper to prove inequalities between critical exponents

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives