on the p and hp extension of nedelec s curl conforming elements abstract we establish interpolation error estimates for the hp extension of nedelec s curl and divergence conforming three dimensional finite elements using these interpolation estimates we investigate the use of higher order edge finite elements for the linear three dimensional magnetostatic problem our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p

on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

zero temperature dynamics of ising models on the triangular lattice we consider the nature of spin flips of zero temperature dynamics for ferromagnetic ising models on the triangular lattice with nearest neighbor interactions and an initial configuration chosen from a symmetric bernoulli distribution we prove that all spins flip infinitely many times for almost every realization of the dynamics and initial configuration

analysis of a finite element method for maxwell s equations the use of finite elements to discretize the time dependent maxwell equations on a bounded domain in three dimensional space is analyzed energy norm error estimates are provided when general finite element methods are used to discretize the equations in space in addition it is shown that if some curl conforming elements due to nedelec are used error estimates may also be proved in the l 2 norm

binary generalized post correspondence problem we give a new proof for the decidability of the binary post correspondence problem pcp originally proved in 1982 by ehrenfeucht karhumki and rozenberg our proof is complete and somewhat shorter than the original proof although we use the same basic copyright 2002 elsevier science b v all rights reserved

an analysis of ne de lec s method for the spatial discretization of maxwell s equations abstract in 1980 nedelec developed a family of curl and divergence conforming finite elements in r 3 he proposed the use of these elements to discretize the time dependent maxwell equations noting that the elements have the advantage that the discrete magnetic displacement can be made exactly divergence free in this paper we shall analyze a slight generalization of nedelec s scheme and prove essentially optimal order convergence estimates in a variety of situations we also demonstrate that the nedelec method can be superconvergent at certain special points and we relate the method to yee s finite difference scheme a by product of our analysis will be a convergence proof for yee s method

a method for incorporating gauss lasw into electromagnetic pic codes abstract an algorithm is presented which solves a modified set of maxwell s equations none of them elliptic in an electromagnetic pic type code the algorithm prevents large violations of gauss law by diffusing away numerical errors arising from the particle to grid apportionment method there are no restrictions on the manner in which charge and current densities are determined from the particle distribution in particular point particles and simple linear interpolation are used

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

message passing approach for recurrent state epidemic models on networks epidemic processes are common out of equilibrium phenomena of broad interdisciplinary interest recently dynamic message passing dmp has been proposed as an efficient algorithm for simulating epidemic models on networks and in particular for estimating the probability that a given node will become infectious at a particular time to date dmp has been applied exclusively to models with one way state changes as opposed to models like sis and sirs where nodes can return to previously inhabited states because many real world epidemics can exhibit such recurrent dynamics we propose a dmp algorithm for complex recurrent epidemic models on networks our approach takes correlations between neighboring nodes into account while preventing causal signals from backtracking to their immediate source and thus avoids echo chamber effects where a pair of adjacent nodes each amplify the probability that the other is infectious we demonstrate that this approach well approximates results obtained from monte carlo simulation and that its accuracy is often superior to the pair approximation which also takes second order correlations into account moreover our approach is more computationally efficient than the pair approximation especially for complex epidemic models the number of variables in our dmp approach grows as 2mk where m is the number of edges and k is the number of states as opposed to mk 2 for the pair approximation we suspect that the resulting reduction in computational effort as well as the conceptual simplicity of dmp will make it a useful tool in epidemic modeling especially for high dimensional inference tasks

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

elements finis mixtes incompressibles pour l equation de stokes dans ℝ3 we introduce some new families of finite element approximation for the stationary stokes and navier stokes equations in a bounded domain in 3 these elements can used tetahedrons or cubes the approximation satisfie exactly the incompressibility condition

phase ordering after a deep quench the stochastic ising and hard core gas models on a tree consider a low temperature stochastic ising model in the phase coexistence regime with markov semigroup p t a fundamental and still largely open problem is the understanding of the long time behavior of p t when the initial configuration is sampled from a highly disordered state e g a product bernoulli measure or a high temperature gibbs measure exploiting recent progresses in the analysis of the mixing time of monte carlo markov chains for discrete spin models on a regular b ary tree open image in new window we study the above problem for the ising and hard core gas independent sets models on open image in new window if is a biased product bernoulli law then under various assumptions on the bias and on the thermodynamic parameters we prove almost sure weak convergence of p t to an extremal gibbs measure pure phase and show that the limit is approached at least as fast as a stretched exponential of the time t in the context of randomized algorithms and if one considers the glauber dynamics on a large finite tree our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution

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

a homomorphic characterization of principal semiafls without using intersection with regular sets abstract if one is asked to consider which one of the semiafl or afl operations differs most from the remaining ones a set theoretically well grounded conclusion would be intersection with regular sets inspired by the result of culik fich and salomaa 1 we shall show that this operation can be replaced by catenation with an endmarker the basis of the paper is the result that any free principal semiafl m l can be presented in the form m l h 4 h 3 1 h 2 h 1 1 l each h 1 a homomorphism h 2 and h 3 free h 4 length preserving where is a symbol not in the alphabet of l if m l is not free we get the same representation but now h 4 erases the last letter from each word in h 3 1 h 2 1 h 1 1 l and is length preserving for other letters for the family of free regular languages we may choose l the last section contains a homomorphic characterization of regular star languages

a high order mixed vector finite element method for solving the time dependent maxwell equations on unstructured grids we present a mixed vector finite element method for solving the time dependent coupled ampere and faraday laws of maxwell s equations on unstructured hexahedral grids that employs high order discretization in both space and time the method is of arbitrary order accuracy in space and up to 4th order accurate in time making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods in addition the method correctly models both the jump discontinuities and the divergence free properties of the electric and magnetic fields is charge and energy conserving conditionally stable and free of spurious modes several computational experiments are performed to demonstrate the accuracy efficiency and benefits of the method

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

extension of the decidability of the marked pcp to instances with unique blocks in the post correspondence problem pcp an instance h g consists of two morphisms h and g and the problem is to determine whether or not there exists a nonempty word w such that h w g w here we prove that the pcp is decidable for instances with unique blocks using the decidability of the marked pcp also we show that it is decidable whether an instance satisfying the uniqueness condition for continuations has an infinite solution these results establish a new and larger class of decidable instances of the pcp including the class of marked instances

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

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

refining the hierarchy of blind multicounter languages and twist closed trios we introduce the new families k r rbc of languages accepted in quasi realtime by one way counter automata having k blind counters of which at least r are reversal bounded it is proved that these families form a strict and linear hierarchy of semi afls within the the family blind m c1 of blind multicounter languages with generator c1 w a1 b1 w a1 w b1 this thereby combines the families blind and rbc from 13 to one strict hierarchy and generalizes and sharpens greibachs results the strict inclusions between the k counter families k r rbc are proved using linear algebra techniques we also study the language theoretic monadic operation twist 18 20 in connection with the semi afls of languages accepted by multicounter and multipushdown acceptors all restricted to reversal bounded behavior it is verified that the family k r rbc is twist closed if and only if r 0 in which case k 0 rbc m ck ck being the k fold shuffle of disjoint copies of c1 we characterize the family m pal of languages accepted in quasi realtime by nondeterministic one way reversal bounded multipushdown acceptors as the least twist closed trio mtwist pal generated by the set of palindromes pal w a b w wrev

preface to the special issue recent advances in numerical methods for hyperbolic partial differential equations hyperbolic partial differential equations pde are a very powerful mathematical tool to describe complex dynamic processes in science and engineering very often hyperbolic pde can be derived directly from first principles in physics such as the conservation of mass momentum and energy these principles are universally accepted to be valid and can be used for the simulation of a very wide class of different problems ranging from astrophysics rotating gas clouds the merger of a binary neutron star system into a black hole and the associated generation and propagation of gravitational waves over geophysics generation and propagation of seismic waves after an earthquake landslides avalanches tidal waves storm surges flooding and morphodynamics of rivers to engineering turbulent flows over aircraft and the associated noise generation and propagation rotating flows in turbo machinery multi phase liquid gas flows in internal combustion engines and computational biology blood flow in the human cardio vascular system it is of course impossible to cover all the above topics in a single special issue however all these apparently different applications have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations possibly containing also higher order derivative terms non conservative products and nonlinear potentially stiff source terms from the mathematical point of view the major difficulties in these systems arise due to the inherent nonlinearities and the formation of non smooth solution features such as e g shock waves the construction of robust and accurate numerical methods for such type of problems is even after decades of successful research an ongoing quest this quest is fueled by recent advances in the development of novel methods with promising additional properties such as e g high spatial order on unstructured meshes algorithmic simplicity for modern multi core architectures and automatic mesh and or trial function adaptation the final goal is to construct methods that efficiently produce reliable results for such type of problems this special issue is dedicated to recent advances in numerical methods for such nonlinear systems of hyperbolic pde and tries to cover a wide spectrum of different problems and numerical approaches

glauber dynamics on trees boundary conditions and mixing time we give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the glauber dynamics in the so called bethe approximation specifically we show that the spectral gap and the log sobolev constant of the glauber dynamics for the ising model on an n vertex regular tree with boundary are bounded below by a constant independent of n at all temperatures and all external fields this implies that the mixing time is o logn in contrast to the free boundary case where it is not bounded by any fixed polynomial at low temperatures in addition our methods yield simpler proofs and stronger results for the spectral gap and log sobolev constant in the regime where the mixing time is insensitive to the boundary condition our techniques also apply to a much wider class of models including those with hard core constraints like the antiferromagnetic potts model at zero temperature proper colorings and the hard core lattice gas independent sets

zero temperature glauber dynamics on the 3 regular tree and the median process in zero temperature glauber dynamics vertices of a graph are given i i d initial spins sigma x 0 from 1 1 with mathbb p p sigma x 0 1 p and they update their spins at the arrival times of i i d poisson processes to agree with a majority of their neighbors we study this process on the 3 regular tree mathbb t 3 where it is known that the critical threshold p c below which mathbb p p a s all spins fixate to 1 is strictly less than 1 2 defining theta p to be the mathbb p p probability that a vertex fixates to 1 we show that theta is a continuous function on 0 1 so that in particular theta p c 0 to do this we introduce a new continuous spin process we call the median process which gives a coupling of all the measures mathbb p p along the way we study the time infinity agreement clusters of the median process show that they are a s finite and deduce that all continuous spins flip finitely often in the second half of the paper we show a correlation decay statement for the discrete spins under mathbb p p for a e value of p the proof relies on finiteness of a vertex s trace in the median process to derive a stability of discrete spins under finite resampling last we use our methods to answer a question of howard j appl probab 37 736 747 2000 on the emergence of spin chains in mathbb t 3 in finite time

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

on the combinatorial problem of codes composed of two words resume nous montrons que si x est un code compose de deux mots x et y alors un mot de x suffisamment long admettant deux x interp etations disjointes est necessairement facteur d un mot de x ou de y ou de y ou d une puissance d un mot imprimitif appartenant a x y xy

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

self organized segregation on the grid we consider an agent based model with exponentially distributed waiting times in which two types of agents interact locally over a graph and based on this interaction and on the value of a common intolerance threshold tau decide whether to change their types this is equivalent to a zero temperature ising model with glauber dynamics an asynchronous cellular automaton with extended moore neighborhoods or a schelling model of self organized segregation in an open system and has applications in the analysis of social and biological networks and spin glasses systems some rigorous results were recently obtained in the theoretical computer science literature and this work provides several extensions we enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size epsilon 0 to size approx 0 134 namely we show that for 0 433 tau 1 2 and by symmetry 1 2 tau 0 567 the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood we further extend the interval leading to expected large segregated regions to size approx 0 312 considering almost segregated regions namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows in this case we show that for 0 344 tau le 0 433 and by symmetry for 0 567 le tau 0 656 the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood this behavior is reminiscent of supercritical percolation where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster the exponential bounds that we provide also imply that complete segregation where agents of a single type cover the whole grid does not occur with high probability for p 1 2 and the range of intolerance considered

the time of bootstrap percolation in two dimensions we study the distribution of the percolation time t of 2 neighbour bootstrap percolation on n 2 with initial set a sim text bin n 2 p we determine t up to a constant factor with high probability for all p above the critical probability for percolation and to within a 1 o 1 factor for a large range of p

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

eigenvalue approximation by a mixed method for resonant inhomogeneous cavities with metallic boundaries for an inhomogeneous cavity bounded by a perfect conductor we prove that the approximation of the eigenvalues for the maxwell problem leads to a second order rate of convergence when using mixed finite elements if the cavity has a disconnected boundary the problem has null eigenvalues we verify the existence of null eigenvalues for the approximate problem they do not mix with the others that still converge at the same rate

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

discontinuous galerkin algorithms for fully kinetic plasmas abstract we present a new algorithm for the discretization of the non relativistic vlasov maxwell system of equations for the study of plasmas in the kinetic regime using the discontinuous galerkin finite element method for the spatial discretization we obtain a high order accurate solution for the plasma s distribution function time stepping for the distribution function is done explicitly with a third order strong stability preserving runge kutta method since the vlasov equation in the vlasov maxwell system is a high dimensional transport equation up to six dimensions plus time we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution including the use of a reduced high order basis set a series of benchmarks from simple wave and shock calculations to a five dimensional turbulence simulation are presented to verify the efficacy of our set of numerical methods as well as demonstrate the power of the implemented features

unary multiple equality sets the languages of rational matrices unary multiple equality sets are equality sets of finite sets of homomorphisms which map into the free monoids over single letter alphabets these languages can equally well be defined by rational matrices unary multiple equality sets have extraordinary language theoretic properties they coincide with the commutative multiple equality sets each unary multiple equality set is precisely classified by the rank of its defining matrices and all fundamental decision questions are polynomially decidable the least trio containing all unary multiple equality sets equals the class of languages accepted by blind multicounter machines and by simple multihead finite automata finally crossing free unary multiple equality sets are introduced which vary from unary multiple equality sets in a similar way as the one sided dyck set varies from the two sided dyck set

binary equality sets are generated by two words we show that the equality set eq g h of two non periodic binary morphisms g h a is generated by at most two words if the rank of eq g h is two then and begin and end with different letters this in particular implies that any binary language has a test set of cardinality at most two

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

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

zero temperature relaxation of three dimensional ising ferromagnets we investigate the properties of the ising glauber model on a periodic cubic lattice of linear dimension l after a quench to zero temperature the resulting evolution is extremely slow with long periods of wandering on constant energy plateaus punctuated by occasional energy decreasing spin flip events the characteristic time scale for this relaxation grows exponentially with the system size we provide a heuristic and numerical evidence that exp l 2 for all but the smallest size systems the long time state is almost never static instead the system contains a small number of blinker spins that continue to flip forever with no energy cost thus the system wanders ad infinitum on a connected set of equal energy blinker states these states are composed of two topologically complex interwoven domains of opposite phases the average genus g l of the domains scales as l with 1 7 thus domains typically have many holes leading to a plumber s nightmare geometry

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

the second term for two neighbour bootstrap percolation in two dimensions in the r neighbour bootstrap process on a graph g vertices are infected in each time step if they have at least r already infected neighbours motivated by its close connections to models from statistical physics such as the ising model of ferromagnetism and kinetically constrained spin models of the liquid glass transition the most extensively studied case is the two neighbour bootstrap process on the two dimensional grid n 2 around 15 years ago in a major breakthrough holroyd determined the sharp threshold for percolation in this model and his bounds were subsequently sharpened further by gravner and holroyd and by gravner holroyd and morris in this paper we strengthen the lower bound of gravner holroyd and morris by proving that the critical probability p c big n 2 2 big for percolation in the two neighbour model on n 2 satisfies p c big n 2 2 big frac pi 2 18 log n frac theta 1 log n 3 2 the proof of this result requires a very precise understanding of the typical growth of a critical droplet and involves a number of technical innovations we expect these to have other applications for example to the study of more general two dimensional cellular automata and to the r neighbour process in higher dimensions

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

marked pcp is decidable we show that the marked version of the post correspondence problem where the words on a list are required to differ in the first letter is decidable on the other hand we prove that the pcp remains undecidable if we only require the words to differ in the first two letters thus we locate the decidability undecidability boundary between marked and 2 marked pcp

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

handling the divergence constraints in maxwell and vlasov maxwell simulations the aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the vlasov maxwell equations and the maxwell equations with given sources these methods can be classified in two types field correction methods and source correction methods the field correction methods introduce new unknowns in the equations for which additional boundary conditions are in some cases non trivial to find the source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete gauss law that needs to be defined in accordance with the discretization of the maxwell propagation operator

representations of language families by homomorphic equality operations and generalized equality sets abstract in this paper the operations of homomorphic equality and inverse homomorphic equality are introduced and are studied in all detail as part of formal language theory these operations are defined from n tuples of homomorphisms by a matching of the homomorphic images they model a simple and powerful equality mechanism and incorporate the notion of generalized equality sets single homomorphisms on free monoids are natural and simple operations however combining n tuples of homomorphisms into a homomorphic equality or inverse homomorphic equality may result in very complex morphic mappings which map e g to the set of solutions of an instance of the post correspondence problem in this paper particular emphasis is laid upon strong representation theorems in the general case every recursively enumerable set can be obtained from the trivial set by applying an inverse homomorphism a inverse homomorphic equality of two nonerasing homomorphisms and an erasing homomorphism this is a uniform morphic representation of the recursively enumerable sets with special regular sets as a basis and it is minimal in its constituents hence the recursively enumerable sets are the smallest class containing all regular sets of the form and closed under homomorphic equality or under inverse homomorphic equality and homomorphism every nonempty language is a generator of the recursively enumerable sets when the full trio operations are extended to include homomorphic equality or inverse homomorphic equality in a similar manner the regular sets can be characterized in terms of homomorphic equality and inverse homomorphic equality operations with bounded balance finally in the nonerasing case the target are classes of the form h l bnp for a class of languages l h l bnp is the smallest class containing l and closed under nonerasing homomorphic equality or under inverse homomorphic equality and nonerasing homomorphism it includes the intersection closure of l every language in h l bnp can be obtained from a language in l by a nonerasing homomorphic equality or an inverse homomorphic equality of three homomorphisms and followed by a nonerasing homomorphism provided that l is closed under inverse homomorphism and endmarking this representation goes through to trio generators of l which are the generators of h l bnp if the trio operations are extended to include nonerasing homomorphic equality or inverse homomorphic equality the operations of homomorphic equality and inverse homomorphic equality are based on generalized equality sets these are equality sets of finite sets of homomorphisms which are used forwards or backwards we investigate basic language theoretic properties of these sets including the star event property commutativity and hardest sets and provide a complete classification of the generalized equality sets according to the numbers of homomorphisms that are used forwards or backwards

decision problems for semi thue systems with a few rules we show that the accessibility problem the common descendant problem the termination problem and the uniform termination problem are undecidable for 3 rules semi thue systems as a corollary we obtain the undecidability of the post correspondence problem for 7 rules

the percolation transition for the zero temperature stochastic ising model on the hexagonal lattice on the planar hexagonal lattice mathbb h we analyze the markov process whose state t in 1 1 mathbb h updates each site v asynchronously in continuous time t 0 so that v t agrees with a majority of its three neighbors the initial v 0 s are i i d with p v 0 1 p 0 1 we study both rigorously and by monte carlo simulation the existence and nature of the percolation transition as t and p 1 2 denoting by t p the expected size of the plus cluster containing the origin we 1 prove that 1 2 and 2 study numerically critical exponents associated with the divergence of p as p 1 2 a detailed finite size scaling analysis suggests that the exponents and of this t dependent percolation model have the same values 4 3 and 43 18 as standard two dimensional independent percolation we also present numerical evidence that the rate at which t as t is exponential