growth rates and explosions in sandpiles we study the abelian sandpile growth model where n particles are added at the origin on a stable background configuration in zd any site with at least 2d particles then topples by sending one particle to each neighbor we find that with constant background height h 2d 2 the diameter of the set of sites that topple has order n1 d this was previously known only for h d our proof uses a strong form of the least action principle for sandpiles and a novel method of background modification we can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d 1 on the other hand we show that if the background height 2d 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random then adding finitely many particles creates an explosion a sandpile that never stabilizes

three dimensional non planar crack growth by a coupled extended finite element and fast marching method a numerical technique for non planar three dimensional linear elastic crack growth simulations is proposed this technique couples the extended finite element method x fem and the fast marching method fmm in crack modeling using x fem the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two dimensional asymptotic crack tip displacement fields the initial crack geometry is represented by two level set functions and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation crack modeling is performed without the need to mesh the crack and crack propagation is simulated without remeshing crack growth is conducted using fmm unlike a level set formulation for interface capturing no iterations nor any time step restrictions are imposed in the fmm planar and non planar quasi static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique copyright 2008 john wiley sons ltd

metastable behavior of stochastic dynamics a pathwise approach in this paper a new approach to metastability for stochastic dynamics is proposed the basic idea is to study the statistics of each path performing time averages along the evolution metastability would be characterized by the fact that the process of these time averages converges under a suitable rescaling to a measure valued markov jump process here this convergence is shown for the curie weiss mean field dynamics and also for a model with spatial structure harris contact process

stress intensity factor computation along a non planar curved crack in three dimensions abstract in this paper a numerical procedure incorporated with the finite element method is developed for calculation of the mixed mode stress intensity factors along a 3d curved crack with non planar surfaces the approach is formulated by modifying the concept of the jk and giii integrals note that the property of path independence for curved cracks does not hold in a manner as that for straight cracks and needs to be properly modified as a consequence the near front region is always included in the integration an appropriate numerical method is therefore developed for simulating the associated singular behavior since no extra auxiliary solutions are required for the computation the proposed method appears to be versatile and can be used for evaluation of the stress intensity factors associated with crack front and crack surfaces of arbitrary curvatures

finite size effects for anisotropic bootstrap percolation logarithmic corrections in this note we analyse an anisotropic two dimensional bootstrap percolation model introduced by gravner and griffeath we present upper and lower bounds on the finite size effects we discuss the similarities with the semi oriented model introduced by duarte

simulation of a cellular automat with an oriented bootstrap rule abstract the evolution of a cellular automat with a culling or bootstrap condition on the minimum number of neighbours for survival has been simulated the introduction of an orientational constraint converts the lattice from isotropic to semidirected simulations of systems of up to 230 million sites are found to be compatible with a threshold value of p 0 966 0 01

noisy collective behaviour in deterministic cellular automata we investigate cellular automata in four and five dimensions for which chate and manneville recently have found nontrivial collective behaviour more precisely though being fully deterministic the average magnetization seems to be periodic respectively quasiperiodic with superimposed noise whose amplitude decreases with system size we confirm this behaviour on very large systems and over very large times we analyse in detail the statistical properties of the noise systems on small lattices and or subject to additional external noise are metastable arguments by grinstein et al suggest that in the periodic case the infinite deterministic systems should be metastable too these arguments are generalized to quasiperiodic systems we find evidence that they do indeed apply but we find no direct evidence for metastability of large systems

comparison of bootstrap percolation models new simulations involving up to nearly 109 sites for square lattice bootstrap percolation give a percolation threshold compatible with unity in this model sites in a square lattice are initially occupied randomly and then are removed if they have less than three occupied neighbours the results are compared with those of similar models such as that studied by frobose 1989 and the earlier bootstrap and diffusion percolation models in two three and four dimensions of adler and aharony 1986 8 in the latter models the effective threshold approaches its asymptotic limit one or zero as lambda ln l d 1 for samples of size ld in agreement with theoretical predictions the coefficient lambda decreases monotonically with the parameters which facilitate the removal addition of sites

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

energy release rate along any crack front in the thickness direction of shell elements abstract this paper deals with a numerical method for calculating the profile of the energy release rate g or j integral along any crack front in the thickness direction of shell elements for this purpose the virtual crack extension technique is firstly used for its average evaluation throughout the entire shell thickness secondly local values for points on the crack front are derived by assuming that the g parameter at any level in the shell s thickness direction is linearly proportional to the strain energy density of points at the same level in the region where hrr singular fields are dominant example computations show that the g parameter profile by the present method is in good agreement with that by alternative methods throughout a large range of elastic plastic deformation

application of pht splines in bending and vibration analysis of cracked kirchhoff love plates abstract in this work we present an extended geometry independent field approximation x gift formulation for cracked kirchhoff love plates the plate geometry is modeled by non uniform rational b splines nurbs while the solution is approximated by polynomial splines over hierarchical t meshes pht splines and enriched by the heaviside function and crack tip asymptotic expansions the adaptive refinement is driven by a recovery based error estimator the formulation is employed for bending and vibration analysis we compare different strategies for refinement enrichment and evaluation of fracture parameters the obtained results are shown to be in a good agreement with the reference solutions

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

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

an improved numerical method for computation of stress intensity factors along 3d curved non planar cracks in fgms abstract a simplified strategy based on the interaction energy integral is implemented in the finite element framework to evaluate mixed mode stress intensity factors sifs in 3d non planar cracks the proposed approach does not require any a priori information about crack front and crack surface curvatures therefore different arbitrary non planar cracks can be easily investigated in particular both conical and lens shaped cracks in homogeneous materials are considered as case studies in order to demonstrate the accuracy of the present approach finally the computational strategy is extended to functionally graded materials fgms and the effect of graded material properties young s modulus and poisson s ratio on the sifs is studied in detail

k core percolation in four dimensions the k core percolation on the bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first order second order nature we investigate numerically k core percolation on the four dimensional regular lattice for k 4 the presence of a discontinuous transition is clearly established but its nature is strictly first order in particular the k core density displays no singular behavior before the jump and its correlation length remains finite for k 3 the transition is continuous

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

probabilistic bootstrap percolation in bootstrap percolation sites are occupied with probabilityp but those with less thanm occupied first neighbors are removed this culling process is repeated until a stable configuration all occupied sites have at leastm occupied first neighbors or the whole lattice is empty is achieved form m 1 the transition is first order while form m 1 it is second order withm dependent exponents in probabilistic bootstrap percolation sites have probabilityr or 1 r of beingm orm sites respectively m sites are those which need at leastm occupied first neighbors to remain occupied we have studied the model on bethe lattices where an exact solution is available form 2 andm 3 the transition changes from second to first order atr 1 1 2 and the exponent is different forr 1 2 the same qualitative behavior is found form 1 andm 3 on the other hand form 1 andm 2 the transition is always second order with the same exponents ofm 1 for any value ofr 0 we found form z 1 andm z wherez is the coordination number of the lattice thatp c 1 for a value ofr which depends onz but is always above zero finally we argue that for bootstrap percolation on real lattices the exponents and form 2 andm 1 are equal for dimensions below 6

critical points of two dimensional bootstrap percolation like cellular automata cellular automata in two dimensions that generalize the bootstrap percolation dynamics are considered focusing on the thresholdp c of the initial density for convergence to total occupancy to occur these models are classified according top c being 0 1 or strictly between these extreme values explicit upper and lower bounds are provided in the third case

percolation of the loss of tension in an infinite triangular lattice we introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation our geometric bootstrap percolation comes from rigidity theory and convex geometry we outline two percolation models a poisson model and a lattice model our poisson model describes how defects holes is one of the possible interpretations of these defects imposed on a tensed membrane result in a redistribution or loss of tension in this membrane the lattice model is motivated by applications of hooke spring networks to problems in material sciences an analysis of the poisson model is given by menshikov et al 4 in the discrete set up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1 p the problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses we show that both regular and perturbed lattices cannot support tension for any p 1 moreover the complete relaxation of tension as defined in section 4 occurs in a finite time almost surely furthermore we underline striking similarities in the properties of the poisson and lattice models

domain integral formulation for stress intensity factor computation along curved three dimensional interface cracks new domain integrals for extracting mixed mode stress intensity factors along curved three dimensional bimaterial interface cracks are derived in the derivation the asymptotic auxiliary fields for the plane problem of a bimaterial interface crack are imposed along a curved crack front the general crack tip interaction integral a contour integral surrounding a point on the crack front which is evaluated in the limit as the contour is shrunk onto the crack tip is then expressed in domain form which is more suitable for numerical computation a consequence of imposing the auxiliary fields along a curved crack front is that the auxiliary stress fields do not satisfy equilibrium and the auxiliary strain fields do not satisfy compatibility the terms which arise due to the lack of equilibrium and compatibility are not sufficiently singular to contribute to the crack tip interaction integral or to affect its path independence however these terms become important when domain integral representations are introduced because they involve fields which are not asymptotically close to the crack tip in order to compute the pointwise stress intensity factors along the crack front the domain integrals are evaluated as a post processing step in the finite element method in the numerical results it is demonstrated that it is crucial to incorporate the terms that arise due to lack of equilibrium and compatibility of the auxiliary fields especially in regions where the local crack front curvature is high in the paper we present two numerical examples as a benchmark we first consider the problem of a penny shaped interface crack embedded in a cylinder the results for the complex stress intensity factor and phase angle are found to be in excellent agreement with the analytical solution the problem of an elliptical crack embedded between two dissimilar isotropic materials is also considered and the results are discussed

modelling of cohesive crack growth in concrete structures with the extended finite element method in this paper the extended finite element method xfem is used for a discrete crack simulation of concrete using an adaptive crack growth algorithm different criteria for predicting the direction of the extension of a cohesive crack are investigated in the context of the xfem the robustness of these criteria is analyzed with reference to three examples and the results are compared to experiments furthermore the efficient application of the quadtree data structure for an iterative mesh refinement in combination with the xfem is shown considerably to reduce the computational effort

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

chaos order statistics and unstable periodic orbits we present an interesting connection between order statistics and unstable periodic orbits of chaotic maps this can be used to locate all the unstable periodic points of a given order of one dimensional chaotic maps with continuous invariant density the densities of the ordered variates of the iterates are discontinuous exactly at unstable periodic points of the map this is illustrated using the logistic map where densities corresponding to a small number of iterates have been obtained in closed form this scheme can also be applied to a class of continuous time systems where the successive maxima of the time series behave as if they were generated from a unimodal map we demonstrate this by using the lorenz model

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

bootstrap percolation in a polluted environment let a low densityp of sites on the lattice z2 be occupied remove a proportionq of them and call the remaining sites empty then update this configuration in discrete time by iteration of the following synchronous rule an empty site becomes occupied by contact with at least two occupied nearest neighbors while occupied and removed sites nerver change their states ifq p2 is large most sites remain unoccupied forever while ifq p2 is small this dynamics eventually makes most sites occupied this demonstrates how sensitive the usual bootstrap percolation rule theq 0 case is to the pollution of space

1 d expansion for k core percolation the physics of k core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon examples of such a phenomenon range from orientational ordering in solid ortho para mathrm h 2 mixtures to the onset of rigidity in bar joint networks to dynamical arrest in glass forming liquids unlike ordinary k 1 and biconnected k 2 percolation the mean field k ensuremath geqslant 3 core percolation transition is both continuous and discontinuous i e there is a jump in the order parameter accompanied with a diverging length scale to determine whether or not this hybrid transition survives in finite dimensions we present a 1 d expansion for k core percolation on the d dimensional hypercubic lattice we show that to order 1 d 3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability this result suggests that the unusual hybrid nature of the mean field k core transition survives in high dimensions

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

sharp thresholds in bootstrap percolation in the standard bootstrap percolation on the d dimensional grid gnd in the initial position each of the nd sites is occupied with probability p and empty with probability 1 p independently of the state of every other site once a site is occupied it 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 configuration percolates by making use of a theorem of friedgut and kalai proc amer math soc 124 1996 2993 we shall show that the threshold function of the percolation is sharp we shall prove similar results for three other models of bootstrap percolation as well

mode decomposition of three dimensional mixed mode cracks via two state integrals a numerical scheme is proposed to obtain the individual stress intensity factors in an axisymmetric crack and in a three dimensional mixed mode crack the procedures presented here are based on the path independence of j and m integrals and mutual or two state conservation integrals which involve two elastic fields a useful method to decompose the stress intensity factors along curved three dimensional cracks under mixed mode is derived by using appropriate auxiliary fields for the plane problems the choice of the auxiliary fields available is critical to success of the present scheme and in this study it is made of not only the asymptotic plane strain solution which requires some remedy in application of the two state integral due to the lack of equilibrium and compatibility but a numerical solution with a given stress intensity as well some numerical examples of penny shaped cracks are presented to investigate the applicability and effectiveness of the method for problems of axisymmetric and three dimensional cracks

bootstrap automata the dynamics of destruction abstract simulations for bootstrap cellular automata on an oriented square lattice are shown to be consistent with exact results on the critical threshold p c 1 and a finite size scaling bound regimes above below and close to p c for finite samples are detailed and characterized by a wide choice of updating and coding algorithms

rates for the probability of large cubes being non internally spanned in modified bootstrap percolation we find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation it is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line

on entropy and monotonicity for real cubic maps consider real cubic maps of the interval onto itself either with positive or with negative leading coefficient this paper completes the proof of the monotonicity conjecture which asserts that each locus of constant topological entropy in parameter space is a connected set the proof makes essential use of the thesis of christopher heckman and is based on the study of bones in the parameter triangle as defined by tresser and r mackay

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

accurate fracture modelling using meshless methods the visibility criterion and level sets formulation and 2d modelling fracture modelling using numerical methods is well advanced in 2d using techniques such as the extended finite element method xfem the use of meshless methods for these problems lags somewhat behind but the potential benefits of no meshing particularly in 3d prompt continued research into their development in methods where the crack face is not explicitly modelled as the edge of an element for instance two procedures are instead used to associate the displacement jump with the crack surface the visibility criterion and the diffraction method the visibility criterion is simple to implement and efficient to compute especially with the help of level set coordinates however spurious discontinuities have been reported around crack tips using the visibility criterion whereas implementing the diffraction method in 3d is much more complicated than the visibility criterion in this paper a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved the formulation is based on the use of level set coordinates and the element free galerkin method and is generally applicable for single or multiple crack problems in 2d or 3d the paper explains the formulation and provides verification of the method against a number of 2d crack problems copyright 2011 john wiley sons ltd

on the study of jamming percolation we investigate kinetically constrained models of glassy transitions and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales the models we investigate have constraints similar to that of the knights model introduced by toninelli biroli and fisher tbf but differing neighbor relations we find that such knights like models otherwise known as models of jamming percolation need a no parallel crossing rule for the tbf proof of a glassy transition to be valid furthermore most knights like models fail a no perpendicular crossing requirement and thus need modification to be made rigorous we also show how the no parallel crossing requirement can be used to evaluate the provable glassiness of other correlated percolation models by looking at models with more stable directions than the knights model finally we show that the tbf proof does not generalize in any straightforward fashion for three dimensional versions of the knights like models

derivation of a new analytical solution for a general two dimensional finite part integral applicable in fracture mechanics an exact expression is derived for the general finite part integral over an inclined ellipticaldomain r denotes the distance of a point in to the singular point is a general function of the cartesian co ordinates x0 y0 the boundary of the region represents the equation z x0 y0 o these integrals appear during the numerical solution of plane crack problems in three dimensional elasticity where they are the dominant part of a hypersingular integral equation the availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and simultaneously lead to quicker numerical results the considered finite part integral can be expressed in closed form as function of complete elliptical integrals or gauss hypergeometric functions respectively formuias for special cases and some i j values and their numerical verification are given in appendices ii and iii

force balance percolation we study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance i e for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions we prove rigorously that p c 1 for the two dimensional models studied numerical data indicate that the force balance percolation transition is discontinuous with a growing crossover length with perhaps the same form as the jamming percolation models suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases we find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition finally we study the dynamics of the culling procedure invoked to obtain the force balance configurations and find a dynamical exponent similar to that found in sandpile models

rigorous computation of topological entropy with respect to a finite partition we present a method to compute rigorous upper bounds for the topological entropy h t a of a continuous map t with respect to a fixed coarse partition of the phase space a long trajectories are not used rather a single application of t to the phase space produces a topological markov chain which contains all orbits of t plus some additional spurious orbits by considering the markov chain as a directed graph and labelling the arcs according to the fixed partition one constructs a sofic shift with topological entropy greater than or equal to h t a to exactly compute the entropy of the sofic shift we produce a subshift of finite type with equal entropy via a standard technique the exact entropy calculation for subshifts is then straightforward we prove that the upper bounds converge monotonically to h t a as the topological markov chains become increasingly accurate the entire procedure is completely automatic 2001 elsevier science b v all rights reserved msc 37b40 94a17 37m25

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

calculating topological entropy the attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem the algorithm introduced here is motivated by the fact that if f 0 1 0 1 is a piecewise monotone map of the unit interval into itself thenh f limn 1 n log var f n where h f is the topological entropy off and var f n is the total variation off n we show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function because of the slow convergence however a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper we prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices with shift of origin in those cases for which the function is markov and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one parameter families of test functions

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

improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation we generalize and improve results of andrews gravner holroyd liggett and romik on metastability thresholds for generalized two dimensional bootstrap percolation models and answer several of their open problems and conjectures specifically we prove slow convergence and localization bounds for holroyd liggett and romik s k percolation models and in the process provide a unified and improved treatment of existing results for bootstrap modified bootstrap and frobose percolation furthermore we prove improved asymptotic bounds for the generating functions of partitions without k gaps which are also related to certain infinite probability processes relevant to these percolation models one of our key technical probability results is also of independent interest we prove new upper and lower bounds for the probability that a sequence of independent events with monotonically increasing probabilities contains no k gap patterns which interpolates the general markov chain solution that arises in the case that all of the probabilities are equal

numerical simulation of multiple 3d fracture propagation using arbitrary meshes this paper describes a mesh independent finite element based method for propagating fractures in three dimensions the iterative algorithm automatically grows fractures in a 3d brittle medium represented by an isotropic linear elastic matrix growth is controlled by an input failure and propagation criterion the geometry and mesh are stored separately and mesh refinement is topologically guided propagation results in the modification of crack geometry as opposed to changes in the mesh as the arbitrary tetrahedral mesh adapts to the evolving geometry stress intensity factors are computed using the volumetric j integral on a virtual piecewise cylinder modal stress intensity factors are computed using the decomposition method mesh and cylinder size effects are studied as is computational efficiency a through going crack embedded in a thick slab and a horizontal and inclined penny shape crack are used to validate the accuracy of the method the predicted stress intensity factors are in good agreement with analytical solutions for six integration points per tip segment integration local to single tips and a cylinder radius that adapts to the local geometric conditions results agree with analytical solutions with less than 5 deviation from experimental results

scaling limit and critical exponents for two dimensional bootstrap percolation consider a cellular automaton with state space 0 1 2 where the initial configuration 0 is chosen according to a bernoulli product measure 1 s are stable and 0 s become 1 s if they are surrounded by at least three neighboring 1 s in this paper we show that the configuration n at time n converges exponentially fast to a final configuration bar omega and that the limiting measure corresponding to bar omega is in the universality class of bernoulli independent percolation more precisely assuming the existence of the critical exponents and and of the continuum scaling limit of crossing probabilities for independent site percolation on the close packed version of 2 i e for independent percolation on we prove that the bootstrapped percolation model has the same scaling limit and critical exponents this type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given at each time step by a monotonic and nonessential enhancement aizenman and grimmett j stat phys 63 817 835 1991 grimmett percolation 2nd ed springer berlin 1999

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

computing the topological entropy for piecewise monotonic maps on the interval a new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented it uses a transition matrix associated with the transformation for this matrix we give a spectral theorem this can be used for an estimation of the accuracy of the algorithm

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

finite size scaling behavior of a biased majority rule cellular automaton abstract we consider the biased majority rule cellular automaton in which the sites take the majority of the states 0 or 1 of their neighbors to update their states with a bias towards 1 in case of ties we observe that in any dimension for any initial density of 1 s the probability of final complete occupancy by 1 s converges to 1 as the lattice grows in two dimensions the value p l of the initial density for which the probability of eventual complete occupancy of a lattice l l is 0 1 is shown to satisfy for fixed the bounds a log l b log l for large l where a and b are positive and finite constants

boundary element analysis of three dimensional mixed mode cracks via the interaction integral a three dimensional boundary element method bem implementation of the interaction integral methodology for the numerical analysis of mixed mode three dimensional cracks is presented in this paper the interaction integral is evaluated from a domain representation naturally compatible with the bem since stresses strains and derivatives of displacements at internal points can be evaluated using their appropriate boundary integral equations special emphasis is put in the selection of the auxiliary function that represents the virtual crack advance in the domain integral this is found to be a key feature to obtain reliable results at the intersection of crack fronts with free surfaces several examples are analysed to demonstrate the efficiency and accuracy of the implementation

a disk shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes a novel domain integral approach is introduced for the accurate computation of pointwise j integral and stress intensity factors sifs of 3d planar cracks using tetrahedral elements this method is efficient and easy to implement and does not require a structured mesh around the crack front the method relies on the construction of virtual disk shaped integral domains at points along the crack front and the computation of domain integrals using a series of virtual triangular and line elements the accuracy of the numerical results computed for through the thickness penny shaped and elliptical crack configurations has been validated by using the available analytical formulations the average error of computed sifs remains below 1 for fine meshes and 2 3 for coarse ones the results of an extensive parametric study suggest that there exists an optimum mesh dependent domain radius at which the computed sifs are the most accurate furthermore the results provide evidence that tetrahedral elements are efficient reliable and robust instruments for accurate linear elastic fracture mechanics calculations

decomposition of the mixed mode j integral revisited in this paper the correct decomposition of three dimensional stress and displacement fields for decoupling the mixed mode j integral into mode i ii and iii is presented it is believed that this has not been achieved before as certain components for the correct derivation have been missing it is shown that by using the correct decomposition method a different area integral for the mode ii and iii j integrals is obtained several test examples are presented to demonstrate the accuracy of the method

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

computing topological entropy in a space of quartic polynomials this paper adds a computational approach to a previous theoretical result illustrating how the complexity of a simple dynamical system evolves under deformations the algorithm targets topological entropy in the 2 dimensional family p q of compositions of two logistic maps estimation of the topological entropy is made possible by the correspondence between p q and a subfamily of sawtooth maps p t and is based on the well known fact that the kneading data of a map determines its entropy a complex search for kneading data in p t turns out to be computationally fast and reliable delivering good entropy estimates finally the algorithm is used to produce a picture of the entropy level sets in p q as illustration to theoretical results such as hu ph d thesis cuny 1995 and radulescu discrete cont dyn syst 19 1 139 175 2007

the time of bootstrap percolation with dense initial sets let r n in r neighbour bootstrap percolation on the vertex set of a graph g vertices are initially infected independently with some probability p at each time step the infected set expands by infecting all uninfected vertices that have at least r infected neighbours when p is close to 1 we study the distribution of the time at which all vertices become infected given t t n o logn loglogn we prove a sharp threshold result for the probability that percolation occurs by time t in d neighbour bootstrap percolation on the d dimensional discrete torus t d n moreover we show that for certain ranges of p p n the time at which percolation occurs is concentrated either on a single value or on two consecutive values we also prove corresponding results for the modified d neighbour rule

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

diffusion percolation i infinite time limit and bootstrap percolation several models for the dynamic growth of percolation clusters or diffusion percolation dp are introduced and analysed in these models a random walker an ant walks on percolation clusters which are occupied with initial site concentration pi the ant is allowed to step off such clusters and add new sites to them if certain conditions are met some of these models are shown to have a one to one correspondence with models of bootstrap percolation bp in which sites which do not have a required number of neighbours are successively culled two new percolation thresholds have been calculated for two diffusion percolation models on the square lattice

on the jx1 integral and the out of plane constraint in a 3d elastic cracked plate loaded in tension abstract in this study the behaviour of the j x 1 integral and its components j p integral and j a integral is numerically investigated for 3d linear elastic cracked plates of different thicknesses loaded in tension unlike the path independent j x 1 integral the components j p and j a are shown to be path dependent in a region that coincides with the out of plane constrained zone due to the thickness effect enabling the characterization of this zone with such integrals the distribution of the out of plane components 33 and e 33 in the vicinity of the crack front is also analyzed for different thicknesses and confronted to 2d solutions in addition a novel tensor concept of the stress intensity k ij is introduced showing a unique structure depending on k i irrespective of the specimen thickness it is confirmed that a pure plane strain condition close and far from the crack front is only asymptotically reached when the specimen thickness is very large when compared to the in plane dimensions for thin plates it is confirmed that the 2d plane stress condition is meaningless in the neighborhood of a 3d crack front under elastic behaviour

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

an extended isogeometric thin shell analysis based on kirchhoff love theory abstract an extended isogeometric element formulation xiga for analysis of through the thickness cracks in thin shell structures is developed the discretization is based on non uniform rational b splines nurbs the proposed xiga formulation can reproduce the singular field near the crack tip and the discontinuities across the crack it is based on the kirchhoff love theory where c 1 continuity of the displacement field is required this condition is satisfied by the nurbs basis functions hence the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy the performance and validity of the formulation is tested by several benchmark examples

finite size effects for some bootstrap percolation models the consequences of schonmann s new proof that the critical threshold is unity for certain bootstrap percolation models are explored it is shown that this proof provides an upper bound for the finite size scaling in these systems comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics we show that the threshold for a finite system of sizel scales aso ln lnl for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled

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

domain integrals for axisymmetric interface crack problems new domain integrals for axisymmetric interface crack problems are derived special attention is given to the specification and subsequent treatment of auxiliary fields for the extraction of mixed mode stress intensity factors using interaction energy integrals the effect of crack front curvature is shown to play an important role in the derivation of the integrals numerical examples include an interface penny shaped crack and a longitudinal interface crack fiber pull out problem and yield very accurate and consistent results when compared with analytical solutions from the literature the method is particularly suited for the analysis of fiber pull out experiments and the accurate determination of fracture parameters such as energy release rate g mixed mode stressintensity factors ki and kii and phase angle

metastability thresholds for anisotropic bootstrap percolation in three dimensions in this paper we analyze several anisotropic bootstrap percolation models in three dimensions we present the order of magnitude for the metastability thresholds for a fairly general class of models in our proofs we use an adaptation of the technique of dimensional reduction we find that the order of the metastability threshold is generally determined by the easiest growth direction in the model in contrast to anisotropic bootstrap percolation in two dimensions in three dimensions the order of the metastability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation

topological entropy of one dimensional maps approximations and bounds we present a method for computing the topological entropy of one dimensional maps as an approximation scheme the algorithm converges rapidly and provides both upper and lower bounds

calculation of fracture mechanics parameters for an arbitrary three dimensional crack by the equivalent domain integral method summary in this paper an equivalent domain integral edi method and the attendant numerical algorithms arc presented for the computation of a near crack tip field parameter the vector je integral and its variation along the front of an arbitrary three dimensional crack in a structural component account is taken of possible non elastic strains present in the structure in this case the near tip je values may be significantly different from the far field values jf especially under non proportional loading

isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth abstract we present a novel numerical method to simulate crack growth in 3d directly from the computer aided design cad geometry of the component without any mesh generation the method is an isogeometric boundary element method igabem based on non uniform rational b splines nurbs nurbs basis functions are used for the domain and crack representation as well as to approximate the physical quantities involved in the simulations a stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted elements convergence studies in the crack opening displacement is performed for a penny shaped crack and an elliptical crack two approaches to extract stress intensity factors sifs the contour m integral and the virtual crack closure integral are compared using dual integral equations the results show remarkable accuracy in the computed sifs leading to smooth crack paths and reliable fatigue lives without requiring the generation of any mesh from the cad model of the component under consideration

characteristic exponents for two dimensional bootstrap percolation bootstrap percolation is a model in which an element of mathbf z 2 becomes occupied in one time unit if two appropriately chosen neighbors are occupied schonmann 4 proved that starting from a bernoulli product measure of positive density the distribution of the time needed to occupy the origin decays exponentially we show that for alpha 1 the exponent can be taken as delta p 2 alpha for some delta 0 thus showing that the associated characteristic exponent is at most two another characteristic exponent associated to this model is shown to be equal to one

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