equilibria of set valued maps on nonconvex domains we present new theorems on the existence of equilibria or zeros of convex as well as nonconvex set valued maps defined on compact neighborhood retracts of normed spaces the maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets among other things we show that if k is a compact neighborhood retract with nontrivial euler characteristic in a banach space e and 1 k 2e is an upper hemicontinuous set valued map with nonempty closed convex values satisfying the tangency condition x n tk x 0 for all x c k then there exists xo c k such that 0 c 4 xo here tk x denotes a new concept of retraction tangent cone to k at x suited for compact neighborhood retracts when k is locally convex at x tk x coincides with the usual tangent cone of convex analysis special attention is given to neighborhood retracts having lipschitzian behavior called l retracts below this class of sets is very broad it contains compact homeomorphically convex subsets of banach spaces epi lipschitz subsets of banach spaces as well as proximate retracts our results thus generalize classical theorems for convex domains as well as recent results for nonconvex sets

multiple spectral kernel learning and a gaussian complexity computation multiple kernel learning mkl partially solves the kernel selection problem in support vector machines and similar classifiers by minimizing the empirical risk over a subset of the linear combination of given kernel matrices for large sample sets the size of the kernel matrices becomes a numerical issue in many cases the kernel matrix is of low efficient rank however the low rank property is not efficiently utilized in mkl algorithms here we suggest multiple spectral kernel learning that efficiently uses the low rank property by finding a kernel matrix from a set of gram matrices of a few eigenvectors from all given kernel matrices called a spectral kernel set we provide a new bound for the gaussian complexity of the proposed kernel set which depends on both the geometry of the kernel set and the number of gram matrices this characterization of the complexity implies that in an mkl setting adding more kernels may not monotonically increase the complexity while previous bounds show otherwise

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

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

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

representations of epi lipschitzian sets abstract a closed subset m of a banach space e is epi lipschitzian i e can be represented locally as the epigraph of a lipschitz function if and only if it is the level set of some locally lipschitz function f e r for which clarke s generalized gradient does not contain 0 at points in the boundary of m i e such that m x f x 0 0 f x if x bd m this extends the characterization previously known in finite dimension and answers to a standing question

dual truncated toeplitz operators abstract let u be a nonconstant inner function in this paper we study the dual truncated toeplitz operators on the orthogonal complement of the model space k u 2 this is a new class of toeplitz operator we show the product of two dual truncated toeplitz operators d f d g to be zero if and only if either f or g is zero we give a necessary and sufficient condition for the product of two dual truncated toeplitz operators to be a finite rank operator furthermore a necessary and sufficient condition is found for the product of two dual truncated toeplitz operators to be a dual truncated toeplitz operator the last two results are different from the classical toeplitz operator theory

a sharp threshold for a modified bootstrap percolation with recovery bootstrap percolation is a type of cellular automaton on graphs introduced as a simple model of the dynamics of ferromagnetism vertices in a graph can be in one of two states healthy or infected and from an initial configuration of states healthy vertices become infected by local rules while the usual bootstrap processes are monotone in the sets of infected vertices in this paper a modification is examined in which infected vertices can return to a healthy state vertices are initially infected independently at random and the central question is whether all vertices eventually become infected the model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected

a d dimensional nucleation and growth model we analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice the model is characterized by d parameters which represent the activation energies of a site depending on the number of occupied nearest neighbours this model is a natural generalisation of the model studied by dehghanpour and schonmann probab theory relat fields 107 1 123 135 1997 where the activation energy of a site with more than two occupied neighbours is zero

metastable behavior for bootstrap percolation on regular trees we examine bootstrap percolation on a regular b 1 ary tree with initial law given by bernoulli p the sites are updated according to the usual rule a vacant site becomes occupied if it has at least occupied neighbors occupied sites remain occupied forever it is known that when b 2 the limiting density q q p of occupied sites exhibits a jump at some p t p t b 0 1 from q t q p t p t we investigate the metastable behavior associated with this transition explicitly we pick p p t h with h 0 and show that as h 0 the system lingers around the critical state for time order h 1 2 and then passes to fully occupied state in time o 1 the law of the entire configuration observed when the occupation density is q q t 1 converges as h 0 to a well defined measure

bootstrap percolation on complex networks we consider bootstrap percolation on uncorrelated complex networks we obtain the phase diagram for this process with respect to two parameters f the fraction of vertices initially activated and p the fraction of undamaged vertices in the graph we observe two transitions the giant active component appears continuously at a first threshold there may also be a second discontinuous hybrid transition at a higher threshold avalanches of activations increase in size as this second critical point is approached finally diverging at this threshold we describe the existence of a special critical point at which this second transition first appears in networks with degree distributions whose second moment diverges but whose first moment does not we find a qualitatively different behavior in this case the giant active component appears for any fg0 and pg0 and the discontinuous transition is absent this means that the giant active component is robust to damage and also is very easily activated we also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold

bootstrap percolation in three dimensions by bootstrap percolation we mean the following deterministic process on a graph g given a set a of vertices infected at time 0 new vertices are subsequently infected at each time step if they have at least n previously infected neighbors when the set a is chosen at random the main aim is to determine the critical probability p c g r at which percolation infection of the entire graph becomes likely to occur this bootstrap process has been extensively studied on the d dimensional grid n d with 2 r d fixed it was proved by cerf and cirillo for d r 3 and by cerf and manzo in general that p c n d r 1 log r 1 n d r 1 where log r is an r times iterated logarithm however the exact threshold function is only known in the case d r 2 where it was shown by holroyd to be 1 o 1 2 18 log n in this paper we shall determine the exact threshold in the crucial case d r 3 and lay the groundwork for solving the problem for all fixed d and r

integrals partitions and cellular automata we prove that 1 0 log f x x dx 2 3ab where f x is the decreasing function that satisfies f a f b x a x b for 0 a b when a is an integer and b a 1 we deduce several combinatorial results these include an asymptotic formula for the number of integer partitions not having a consecutive parts and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation

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

fixed points and equilibria in nonconvex sets tcentre de recherches mathtmatiques universite de montreal montreal quebec h3c 357 canada psteklov institute of mathematics moscow 117966 russia and ildepartment of mathematics and statistics concordia university montreal quebec h4b lr6 canada received 30 september 1993 received forpublication 18 august 1994 key words and phrases multivalued mapping tangent cones zeros fixed points differential inclusion weak invariance nonsmooth analysis 1 introduction

bootstrap percolation on homogeneous trees has 2 phase transitions we study the threshold bootstrap percolation model on the homogeneous tree with degree b 1 2 b and initial density p it is known that there exists a nontrivial critical value for p which we call pf such that a for p pf the final bootstrapped configuration is fully occupied for almost every initial configuration and b if p pc then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration moreover we show that 3 for p pc the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail 4 at p pc the expected occupied cluster size in the final bootstrapped configuration is infinite 5 the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on 0 pf and analytic on pc pf admitting an analytic continuation from the right at pc and only in the case b also from the left at pf

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

newman ziff algorithm for the bootstrap percolation application to the archimedean lattices abstract we propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the newman ziff algorithm of the classical percolation m e j newman and r m ziff 2000 27 using these algorithms and the finite size scaling we calculated with high precision the percolation threshold and critical exponents in the eleven two dimensional archimedean lattices we present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation and show that they have the same critical exponents as the classical percolation within error bars in two dimensions we conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation

the constrained degree and fixed point index theory for set valued maps abstract in the first part of the paper we give a construction of a topological degree theory for set valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds in the second part we propose a fixed point index for inward maps on arbitrary closed convex sets

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

fixed point index for weakly inward mappings abstract the fixed point index for weakly inward mappings which are not cone mappings is defined in this paper its properties are also investigated consequently we study the derivatives of weakly inward mappings and nonzero fixed points of such mappings

constrained topological degree and positive solutions of fully nonlinear boundary value problems abstract in the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form a f where a d a e is a densely defined m accretive operator in a banach space e and f u e is a tangent field defined on an open subset u of a neighborhood retract m being invariant with respect to the resolvents of a the construction is performed under the assumption that resolvents of a are completely continuous in the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction diffusion equations and equations with one dimensional p laplacian with possibly non positive perturbations as well as some controlled neumann like problems

higher order corrections for anisotropic bootstrap percolation we study the critical probability for the metastable phase transition of the two dimensional anisotropic bootstrap percolation model with 1 2 neighbourhood and threshold r 3 the first order asymptotics for the critical probability were recently determined by the first and second authors here we determine the following sharp second and third order asymptotics begin aligned p c big l 2 mathcal n scriptscriptstyle 1 2 3 big frac log log l 2 12 log l frac log log l log log log l 3 log l frac left log frac 9 2 1 pm o 1 right log log l 6 log l end aligned we note that the second and third order terms are so large that the first order asymptotics fail to approximate p c even for lattices of size well beyond 10 10 1000

polluted bootstrap percolation with threshold two in all dimensions in the polluted bootstrap percolation model the vertices of a graph are independently declared initially occupied with probability p or closed with probability q at subsequent steps a vertex becomes occupied if it is not closed and it has at least r occupied neighbors on the cubic lattice mathbb z d of dimension d ge 3 with threshold r 2 we prove that the final density of occupied sites converges to 1 as p and q both approach 0 regardless of their relative scaling our result partially resolves a conjecture of morris and contrasts with the d 2 case where gravner and mcdonald proved that the critical parameter is q p 2

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

exact solution of a jamming transition closed equations for a bootstrap percolation problem jamming or dynamical arrest is a transition at which many particles stop moving in a collective manner in nature it is brought about by for example increasing the packing density changing the interactions between particles or otherwise restricting the local motion of the elements of the system the onset of collectivity occurs because when one particle is blocked it may lead to the blocking of a neighbor that particle may then block one of its neighbors these effects propagating across some typical domain of size named the dynamical correlation length when this length diverges the system becomes immobile even where it is finite but large the dynamics is dramatically slowed such phenomena lead to glasses gels and other very long lived nonequilibrium solids the bootstrap percolation models are the simplest examples describing these spatio temporal correlations we have been able to solve one such model in two dimensions exactly exhibiting the precise evolution of the jamming correlations on approach to arrest we believe that the nature of these correlations and the method we devise to solve the problem are quite general both should be of considerable help in further developing this field

topological degree methods for perturbations of operators generating compact c 0 semigroups abstract the main aim of this paper is to construct a topological degree for maps a f m d a e where a densely defined closed operator a d a e of a banach space e is such that a is the generator of a compact c 0 semigroup and f m e is a locally lipschitz map defined on a neighborhood retract m e if m is a closed convex cone then a degree formula allowing an effective computation of the degree is proved this formula provides an infinite dimensional counterpart of the well known krasnosel skii theorem by the use of the introduced topological degree and an abstract result concerning branching of fixed points the bifurcation of periodic points of the parameterized boundary value problem u au f t u 0 u t m u 0 u t is studied examples of applications to partial differential equations are discussed

finite size scaling in three dimensional bootstrap percolation we consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior bootstrap percolation is an example of cellular automata defined on the d dimensional lattice 1 2 l d in which each site can be empty or occupied by a single particle in the starting configuration each site is occupied with probability p occupied sites remain occupied forever while empty sites are occupied by a particle if at least among their 2d nearest neighbor sites are occupied when d is fixed the most interesting case is the one d this is a sort of threshold in the sense that the critical probability p c for the dynamics on the infinite lattice z d switches from zero to one when this limit is crossed finite size effects in the three dimensional case are already known in the cases 2 in this paper we discuss the case 3 and we show that the finite size scaling function for this problem is of the form f l const in in l we prove a conjecture proposed by a c d van enter

zero temperature glauber dynamics on mathbb z d we study zero temperature glauber dynamics on mathbb z d which is a dynamic version of the ising model of ferromagnetism spins are initially chosen according to a bernoulli distribution with density p and then the states are continuously and randomly updated according to the majority rule this corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field to one at zero temperature with no external field define p c mathbb z d to be the infimum over p such that the system fixates at with probability 1 it is a folklore conjecture that p c mathbb z d 1 2 for every 2 le d in mathbb n we prove that p c mathbb z d to 1 2 as d

on the solution sets of differential inclusions and the periodic problem in banach spaces abstract in this paper the topological structure of the solution set of a constrained semilinear differential inclusion in a banach space e is studied it is shown that the set of all mild solutions with values in a closed and in general thin subset d e is an r set provided natural boundary conditions and appropriate geometrical assumptions on d which hold e g when d is convex are satisfied applications to the periodic problem and to the existence of equilibria are given

bootstrap percolation on the hypercube in the bootstrap percolation on the n dimensional hypercube in the initial position each of the 2n sites is occupied with probability p and empty with probability 1 p independently of the state of the other sites every occupied site remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial position spans the hypercube we shall show that there are constants c1 c2 0 such that for open image in new window the probability of spanning tends to 1 as n while for open image in new window the probability tends to 0 furthermore we shall show that for each n the transition has a sharp threshold function

fixed points of weakly inward multis soit x un espace de banach reel k x convexe ferme borne f k 2 x semicontinue superieurement et condensant f x convexe ferme x k et f faiblement interieur sur k alors f a un point fixe

sharp metastability threshold for an anisotropic bootstrap percolation model bootstrap percolation models have been extensively studied during the two past decades in this article we study the following anisotropic boot strap percolation model the neighborhood of a point m n is the set m 2 n m 1 n m n 1 m 1 n m 2 n m n 1 at time 0 sites are occupied with probability p at each time step sites that are occupied remain occupied while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied we prove that it exhibits a sharp metastability threshold this is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model

on some growth models with a small parameter we consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models when a certain parameter becomes small the basic example treated is the variant of richardson s growth model on d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors at rate 1 if it has exactly 1 occupied neighbor and of course at rate 0 if it has no occupied neighbor occupied sites remain occupied forever starting from a single occupied site this model has asymptotic speeds of growth in each direction as time goes to infinity and these speeds determine an asymptotic shape in the usual sense it is proven that as tends to 0 the asymptotic speeds scale as 1 d and the asymptotic shape when renormalized by dividing it by 1 d converges to a cube other similar models which are partially oriented are also studied

a nucleation and growth model we consider the following simple nucleation and growth model on the lattice d starting with all sites unoccupied a site becomes occupied at rate e if it has no occupied neighbors at rate e if it has 1 occupied neighbor and at rate 1 if it has 2 or more occupied neighbors occupied sites remain occupied forever the parameters are fixed and we are interested in the behavior of the system as we show that the relaxation time of this system scales as e c where c max d 1