makar limanov s conjecture on free subalgebras abstract it is proved that over every countable field k there is a nil algebra r such that the algebra obtained from r by extending the field k contains noncommutative free subalgebras of arbitrarily high rank it is also shown that over every countable field k there is an algebra r without noncommutative free subalgebras of rank two such that the algebra obtained from r by extending the field k contains a noncommutative free subalgebra of rank two this answers a question of makar limanov lenny makar limanov private communication beijing june 2007

convolution generated motion and generalized huygens principles for interface motion a physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry accordingly there is great interest in algorithms that generate such geometric interface motion in this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion the classical huygens principle and diffusion generated motion we show that the resulting generalization can be viewed both geometrically as a type of huygens principle and algebraically as a convolution generated motion using the geometric algebraic duality from the unification we construct specific convolution generated motion algorithms for a common class of anisotropic curvature dependent motionlaws we validate these algorithms with numerical experiments and show that they can be implemented accurately and efficiently with adaptive resolution and fast fourier transform techniques

the universal relation between scaling exponents in first passage percolation it has been conjectured in numerous physics papers that in ordinary rst passage percolation on integer lattices the uctuation exponent and the wandering exponent are related through the universal relation 2 1 irrespective of the dimension this is sometimes called the kpz relation between the two exponents this article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense

the divergence of fluctuations for shape in first passage percolation we consider the first passage percolation model on z d for d 2 in this model we assign independently to each edge the value zero with probability p and the value one with probability 1 p we denote by t 0 the passage time from the origin to for r d and

efficient algorithms for diffusion generated motion by mean curvature the problem of simulating the motion of evolving surfaces with junctions according to some curvature dependent speed arises in a number of applications by alternately diffusing and sharpening characteristic functions for each region a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums however the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought even in two dimensions we propose a new spectral discretization of these diffusion generated methods which obtains greatly improved efficiency over the usual finite difference approach these efficiency gains are obtained in part through the use of a quadrature based refinement technique by integrating fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients indeed numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization our findings are demonstrated on several examples

lattices of jordan algebras commutative jordan algebras play a central part in orthogonal models the generations of these algebras is studied and applied in deriving lattices of such algebras these lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation

fluctuations of extensive functions of quenched random couplings an extensive quantity is a family of functions v of random parameters indexed by the finite regionsv subsets of d over which v are additive up to corrections satisfying the boundary estimate stated below it is shown that unless the randomness is nonessential in the sense that lim v v has a unique value in the absolute i e not just probabilistic sense the variance of such a quantity grows as the volume ofv of particular interest is the free energy of a system with random couplings for such v bounds are derived also for the generating functione et in a separate application variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment

the existence of asymptotically unbiased nonnegative quadratic estimates of variance components in anova models abstract conditions for the existence of asymptotically unbiased nonnegative quadratic estimates of individual variance components in general anova models are investigated a simple necessary condition for general models is obtained necessary and sufficient conditions are obtained in the case of random balanced nested classification models and the random unbalanced one way model the investigation demonstrates the need for exercising caution when employing nonnegative quadratic estimates of individual variance components in practice

directed polymers in a random medium 1 d expansion and the n tree approximation the authors develop a method which allows one to obtain 1 d expansions for the problem of directed polymers in a random medium using this method they expand thermal properties such as the free energy the specific heat the overlaps and the fluctuations of the transverse displacement their results are consistent with the existence of a finite upper critical dimension above which the low temperature phase is mean field like i e characterised by a broad distribution of overlaps broken replica symmetry and transverse fluctuations which scale as the length of the polymer

a sharper threshold for bootstrap percolation in two dimensions two dimensional bootstrap percolation is a cellular automaton in which sites become infected by contact with two or more already infected nearest neighbours we consider these dynamics which can be interpreted as a monotone version of the ising model on an n n square with sites initially infected independently with probability p the critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1 2 holroyd determined the sharp first order approximation pc 2 18 log n as n here we sharpen this result proving that the second term in the expansion is log n 3 2 o 1 and moreover determining it up to a poly log log n factor the exponent 3 2 corrects numerical predictions from the physics literature

one dimensional long range diffusion limited aggregation i we examine diffusion limited aggregation generated by a random walk on zz with long jumps we derive upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has under various regularity conditions on the tail of the step distribution we prove that the diameter grows as n o 1 n o 1 with an explicitly given the growth rate of the aggregate is shown to have three phase transitions when the walk steps have finite third moment finite variance and conjecturally finite half moment

a sharp transition for the two dimensional ising percolation we show that the percolation transition for the two dimensional ising model is sharp namely we show that for every reciprocal temperature 0 there exists a critical valueh c of external magnetic fieldh such that the following two statements hold we also shows that the percolation probability is continuous in h except on the half line 0 c

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

free group algebras in division rings with valuation i abstract let r be an algebra over a commutative ring k suppose that r is endowed with a descending filtration indexed on an ordered group g such that the restriction to k is positive we show that the existence of free algebras on a certain set of generators in the induced graded ring grad r implies the existence of free group algebras in r our best results are obtained for division rings endowed with a valuation

busemann functions and infinite geodesics in two dimensional first passage percolation we study first passage percolation on mathbb z 2 where the edge weights are given by a translation ergodic distribution addressing questions related to existence and coalescence of infinite geodesics some of these were studied in the late 1990s by c newman and collaborators under strong assumptions on the limiting shape and weight distribution in this paper we develop a framework for working with distributional limits of busemann functions and use it to prove forms of newman s results under minimal assumptions for instance we show a form of coalescence of long finite geodesics in any deterministic direction we also introduce a purely directional condition which replaces newman s global curvature condition and whose assumption we show implies the existence of directional geodesics without this condition we prove existence of infinite geodesics which are directed in sectors last we analyze distributional limits of geodesic graphs proving almost sure coalescence and nonexistence of infinite backward paths this result relates to the conjecture of nonexistence of bigeodesics

covariance hypotheses which are linear in both the covariance and the inverse covariance both the covariance and the inverse covariance are products of models each of which consists of either i independent identically distributed random vectors which have a covariance with a real complex or quaternion structure or ii independent identically distributed random vectors with a parametrization of the covariance which is given by means of the clifford algebra the models i are well known for models ii we have found under the assumption that the distribution is nornal the exact distributions of the maximum likelihood estimates and the likelihood ratio test statistics 1 introduction the purpose of the present paper is to describe for the family of normal distributions the structure of those statistical hypotheses models which are linear in both the covariance and the inverse covariance in section 2 we show that under such a hypothesis the problem of maximum likelihood estimation of the covariance has an explicit solution and that these hypotheses are parametrized by jordan algebras quadratic subspaces based on the theory of jordan algebras it is then possible to give a canonical form for such hypotheses in section 3 we show that the jordan algebras are products of so called simple jordan algebras and in section 4 we prove that the hypotheses can be decomposed into products of hypotheses parametrized by simple jordan algebras the simple jordan algebras are classified according to their so called degree the only simple jordan algebras of degree 1 is the set of real numbers and corresponding to that we have the hypothesis under which the covariance is proportional to a known covariance in section 5 we give a canonical form for the hypotheses which are parametrized by simple jordan algebras of degree greater than 3 it tums out that these hypotheses are equivalent to those consisting of independent identically distributed random vectors which have a covariance with a real complex or quatemion structure and the distributions of the maximum likelihood estimates are given by wishart 1928 goodman 1963 and andersson 1975 respectively the statistical hypotheses which are parametrized by simple jordan algebras of degree 2 are new and in section 6 we give a complete solution to the problems of maximum likelihood inference it turns out that the distribution problems are not more complicated than those for the two dimensional normal distribution the mathematical structure of the hypotheses is closely related to the theory of spinors in quantum mechanics based on that theory we give a canonical form for such hypotheses in section 7

free algebras and free groups in ore extensions and free group algebras in division rings abstract let k be a field of characteristic zero let be an automorphism of k and let be a derivation of k we show that the division ring d k x either has the property that every finitely generated subring satisfies a polynomial identity or d contains a free algebra on two generators over its center in the case when k is finitely generated over a subfield k we then see that for a k algebra automorphism of k and a k linear derivation of k k x having a free subalgebra on two generators is equivalent to having infinite order and k x having a free subalgebra is equivalent to being nonzero as an application we show that if d is a division ring with center k of characteristic zero and d contains a solvable subgroup that is not locally abelian by finite then d contains a free k algebra on two generators moreover if we assume that k is uncountable without any restrictions on the characteristic of k then d contains the k group algebra of the free group of rank two

a shape theorem for the spread of an infection in ksb we studied the following model for the spread of a rumor or infection there is a gas of so called a particles each of which performs a continuous time simple random walk on z d with jump rate da we assume that just before the start the number of a particles at x na x 0 has a mean a poisson distribution and that the na x 0 x z d are independent in addition there are b particles which perform continuous time simple random walks with jump rate db we start with a finite number of b particles in the system at time 0 the positions of these initial b particles are arbitrary but they are nonrandom the b particles move independently of each other the only interaction occurs when a b particle and an a particle coincide the latter instantaneously turns into a b particle ksb gave some basic estimates for the growth of the set b t x z d a b particle visits x during 0 t

growth and roughness of the interface for ballistic deposition in ballistic deposition bd d 1 dimensional particles fall sequentially at random towards an initially flat large but bounded d dimensional surface and each particle sticks to the first point of contact for both lattice and continuum bd a law of large numbers in the thermodynamic limit establishes convergence of the mean height and surface width sample standard deviation of the height of the interface to constants h t and w t respectively depending on time t we show that h t is asymptotically linear in t while w t 2 grows at least logarithmically in t when d 1 we use duality results showing that w t can be interpreted as the standard deviation of the height for deposition onto a surface growing from a single point

convolution generated motion as a link between cellular automata and continuum pattern dynamics cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical biological and ecological systems however for patterns in which sharp interfaces form and propagate automata simulations can exhibit undesirable properties including spurious anisotropy and poor representation of interface curvature effects these simulations are also prohibitively slow when high accuracy is required even in two dimensions also the highly discrete nature of automata models makes theoretical analysis difficult in this paper we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata but based on continuous convolutions rather than discrete sums these convolution generated motions naturally achieve the fine grid limit of the corresponding automata and they are also well suited to numerical and theoretical analysis because of this the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast fourier transform techniques and for a large class of convolutions the limiting interface motion laws can be derived analytically thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics this is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics in this latter role it also has a number of benefits over the traditional reaction diffusion ginzburg landau continuum pde models of pattern formation which yield true moving interfaces only as singular limits we illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media

on tests of the overall treatment effect in meta analysis with normally distributed responses for the meta analysis of controlled clinical trials or epidemiological studies in which the responses are at least approximately normally distributed a refined test for the hypothesis of no overall treatment effect is proposed the test statistic is based on a direct estimation function for the variance of the overall treatment effect estimator as outcome measures the absolute and the standardized difference between means are considered in simulation studies it is shown that the proposed test keeps the prescribed significance level very well in contrast to the commonly used tests in the fixed effects and random effects model respectively which can become very liberal furthermore just for using the proposed test it is not necessary to choose between the fixed effects and the random effects approach in advance

convergence of nonlocal threshold dynamics approximations to front propagation in this note we prove that appropriately scaled threshold dynamics type algorithms corresponding to the fractional laplacian of order 0 2 converge to moving fronts when 1 the resulting interface moves by weighted mean curvature while for 1 the normal velocity is nonlocal of fractional type the results easily extend to general nonlocal anisotropic threshold dynamics schemes

threshold growth dynamics we study the asymptotic shape of the occupied region for monotone deterministic dynamics in d dimensional euclidean space parametrized by a threshold 0 and a borel set n r d with positive and finite lebesgue measure if a n denotes the oocupied set of the dynamics at integer time n then a n 1 is obtained by adjoining any point x for which the volume of overlap between x n and a n exceeds except in some degenerate cases we prove that n 1 a n converges to a unique limiting shape l starting from any bounded initial region a 0 that is suitably large moreover l is computed as the polar transform for 1 w where w is an explicit width function that depends on n and

on the number of infinite geodesics and ground states in disordered systems we study first passage percolation models and their higher dimensional analogs models of surfaces with random weights we prove that under very general conditions the number of lines or in the second case hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to as corollaries we show that in any dimensiond 2 the number of ground states of an ising ferromagnet with random coupling constants equals with probability one 2 or proofs employ simple large deviation estimates and ergodic arguments

variance component estimation with model based diagnostics the problem of estimating variance components in mixed linear models has no satisfactory solution for the unbalanced data case further there has been little work on diagnostic methods for assessing the data and the model in this article we propose a new class of unbiased estimators these estimators have simple closed form expressions allowing for smallsample analysis and easy computations the structure of the estimators reveals natural diagnostics for examining the data and the model assumptions the source of negative estimates of variance components is revealed limited efficiency studies indicate that these estimators are comparable to existing estimators the estimators are illustrated by two numerical examples that reveal the importance of the diagnostic analysis

jordan algebras and bayesian quadratic estimation of variance components the authors consider a linear model ey x cov y v where v is a quadratic subspace a jordan algebra using this property it is shown how multiplication computation of g inverses determination of a neutral element can be done more efficiently in v finally this technique is applied to obtain formulae for locally optimal estimators and bayesian estimators

free subalgebras of division algebras over uncountable fields we study the existence of free subalgebras in division algebras and prove the following general result if a is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0 then either the quotient division algebra of a contains a free algebra on two generators or it is left algebraic over every maximal subfield as an application we prove that if k is an uncountable algebraically closed field and a is a finitely generated k algebra that is a domain of gk dimension strictly less than 3 then either a satisfies a polynomial identity or the quotient division algebra of a contains a free k algebra on two generators

geodesics and spanning trees for euclidean first passage percolation the metric d q q on the set q of particle locations of a homogeneous poisson process on r d defined as the infimum of i q i q i 1 1 over sequences in q starting with q and ending with q where denotes euclidean distance has nontrivial geodesics when a 1 the cases 1 are the euclidean first passage percolation fpp models introduced earlier by the authors while the geodesics in the case are exactly the paths from the euclidean minimal spanning trees forests of aldous and steele we compare and contrast results and conjectures for these two situations new results for 1 and any d include inequalities on the fluctuation exponents for the metric x 1 2 and for the geodesies 3 4 in strong enough versions to yield conclusions not yet obtained for lattice fpp almost surely every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic from every q for d 2 and 2 further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

random growth models with polygonal shapes we consider discrete time random perturbations of monotone cellular automata ca in two dimensions under general conditions we prove the existence of half space velocities and then establish the validity of the wulff construction for asymptotic shapes arising from finite initial seeds such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases in many cases exact stability is observed that is for small perturbations the shapes of the deterministic and random processes agree exactly we give a complete characterization of such cases and show that they are prevalent among threshold growth ca with box neighborhood we also design a nontrivial family of ca in which the shape is exactly computable for all values of its probability parameter

lower bounds for boundary roughness for droplets in bernoulli percolation we consider boundary roughness for the droplet created when supercritical two dimensional bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l 2 for large l the maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull we show that for large l this maximum is at least of order l 1 3 logl 2 3 this complements the upper bound of order l 1 3 logl 2 3 proved in al3 for the average local roughness the exponent 1 3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions

coexistence of infinite clusters ii ising percolation in two dimensions we show a strong type of conditionally mixing property for the gibbs states ofd dimensional ising model when the temperature is above the critical one by using this property we show that there is always coexistence of infinite and clusters when is smaller than c andh 0 in two dimensions it is also possible to show that this coexistence region extends to some non zero external field case i e for every c there exists someh c 0 such that h h c implies the coexistence of infinite clusters with respect to the gibbs state for h

directed polymers in a random environment some results on fluctuations we consider a polymer model on where to each edgee is associated a random variable v e a polymer configuration is represented by a directed pathr and has a weight exp e r e with 1 t the inverse temperature we extend some rigorous results that have been obtained for the ground state of this model to finite temperatures in particular we obtain some upper and lower bounds on sample to sample free energy fluctuations and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations

free groups in normal subgroups of the multiplicative group of a division ring abstract let d be a division ring with center z and multiplicative group d 0 d and let n be a normal subgroup of d we investigate various conditions under which n must contain a free noncyclic subgroup in one instance assuming that the transcendence degree of z over its prime field is infinite and that n contains a nonabelian solvable subgroup we use a construction method due to chiba to exhibit free generators of the free subgroup

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

on the existence of unbiased nonnegative estimates of variance convariance components the existence of unbiased nonnegative definite quadratic estimates for linear combinations of variance covariance components is characterized by means of the natural parameter set in a residual model in the presence of a quadratic subspace condition the following disjunction for nonnegative estimability is derived either standard methods suffice or the concepts of unbiasedness and nonnegative definiteness are incompatible for the case of a single variance component it is shown that unbiasedness and nonnegative definiteness always entail a reduction to a trivial model in which the variance component under investigation is the sole remaining parameter several examples illustrate these results

the wetting transition in a random surface model we continue our analysis of the phase diagram of a discrete random surface with no downward fingers lying above a flat two dimensional substrate the surface is closely related to the 2d ising model and its free energy is exactly solvable in much but not all of the phase diagram there is a transition at temperaturet w from a high t infinite height or wet phase to a low t finite height or partially wet phase previously it was shown that when a parameterb related to the contact interaction is positive t w is independent ofb and there is a logarithmic specific heat divergence ast w is approached fromeither side here we show that forb 0 t w does depend onb and there isno thermodynamic singularity from the wet phase the partially wet phases forb 0 andb 0 differ in the absence or presence of a monolayer covering the entire substrate this results in a first order transition across the lineb 0 t t w

scaling identity for crossing brownian motion in a poissonian potential we consider d dimensional brownian motion in a truncated poissonian potential d 2 if brownian motion starts at the origin and ends in the closed ball with center y and radius 1 then the transverse fluctuation of the path is expected to be of order y whereas the distance fluctuation is of order y physics literature tells us that and should satisfy a scaling identity 2 1 we give here rigorous results for this conjecture

linear toeplitz covariance structure models with optimal estimators of variance components this paper deals with the problem of optimal quadratic unbiased estimation for statistical models with linear toeplitz covariance structure the main result is a necessary and sufficient condition for these models to have an optimal unbiased estimator for any linear combination of variance components this result is obtained by means of special jordan algebras which are a powerful tool to characterize optimality in quadratic unbiased estimation

euclidean models of first passage percolation we introduce a new family of first passage percolation fpp models in the context of poisson voronoi tesselations of d compared to standard fpp on d these models have some technical complications but also have the advantage of statistical isotropy we prove two almost sure results a shape theorem where isotropy implies an exact euclidean ball for the asymptotic shape and nonexistence of certain doubly infinite geodesics where isotropy yields a stronger result than in standard fpp

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

first passage percolation on mathbb z 2 a simulation study first passage percolation on mathbb z 2 is a model for describing the spread of an infection on the sites of the square lattice the infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges in this paper the speed of the growth and the shape of the infected set is studied by aid of large scale computer simulations with focus on continuous passage time distributions it is found that the most important quantity for determining the value of the time constant which indicates the inverse asymptotic speed of the growth is mathbf e min tau 1 ldots tau 4 where tau 1 ldots tau 4 are i i d passage time variables the relation is linear for a large class of passage time distributions furthermore the directional time constants are seen to be increasing when moving from the axis towards the diagonal so that the limiting shape is contained in a circle with radius defined by the speed along the axes the shape comes closer to the circle for distributions with larger variability

differentiability at the edge of the percolation cone and related results in first passage percolation we study first passage percolation in two dimensions using measures on passage times with b inf supp 0 and mu b p geq vec p c the threshold for oriented percolation we first show that for each such the boundary of the limit shape for is differentiable at the endpoints of flat edges in the so called percolation cone we then conclude that the limit shape must be non polygonal for all of these measures furthermore the associated richardson type growth model admits infinite coexistence and if is not purely atomic the graph of infection has infinitely many ends we go on to show that lower bounds for fluctuations of the passage time given by newman piza extend to these measures we establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for this result confirms a prediction of newman and piza ann probab 23 977 1005 1995 and zhang ann probab 36 331 362 2008 under the assumption of finite radius of curvature for the limit shape in these directions we obtain a power law lower bound for the variance and an inequality between the exponents and

approximation schemes for propagation of fronts with nonlocal velocities and neumann boundary conditions the convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied the velocities considered depend on the principal curvatures the normal direction the location as well as some nonlocal properties of the front most of the schemes considered are in essence threshold dynamics type approximation schemes modified for neumann boundary conditions and nonlocal terms the existence and uniqueness of appropriately defined viscosity solutions of the level set equations describing the nonlocal motions is also shown

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