ergodicity of probabilistic cellular automata a constructive criterion we give a sequence of criteria of increasing complexity for the exponential ergodicity of discrete time interacting particle systems each criterion involves estimating the dependence on initial conditions of the process on finite space time volumes it generalizes and improves the existing single site condition and is the analog of the dobrushin shlosmanc v condition in equilibrium statistical mechanics our dynamic criterion may also be used to prove the uniqueness of gibbs state in situations where thec v condition fails as a converse we prove that if there is a certain form of convergence to the stationary measure faster thann d wheren is the time andd is the dimension of the lattice then our condition holds for some space time volumes and hence the convergence must be exponentially fast

a reader s guide to gacs s positive rates paper peter gacs s monograph which follows this article provides a counterexample to the important positive rates conjecture this conjecture which arose in the late 1960 s was based on very plausible arguments some of which come from statistical mechanics during the long gestation period of the gacs example there has been a great deal of skepticism about the validity of his work the construction and verification of gacs s counterexample are unavoidably complex and as a consequence his paper is quite lengthy but because of the novelty of the techniques and the significance of the result his work deserves to become widely known this reader s guide is intended both as a cheap substitute for reading the whole thing as well as a warm up for those who want to plumb its depths

when is an interacting particle system ergodic we consider stochastic flip dynamics for an infinite number of ising spins on the lattice d we find a sequence of constructive criteria for the system to be exponentially ergodic the main idea is to approximate the continuous time process with discrete time processes itseuler polygon and to use an improved version of previous results ms about constructive ergodicity of discrete time processes

exponential decay of correlations for strongly coupled toom probabilistic cellular automata we investigate the low noise regime of a large class of probabilistic cellular automata including the north east center model of toom they are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion we prove for a set of initial conditions exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value we also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing

density classification performance and ergodicity of the gacs kurdyumov levin cellular automaton model iv almost four decades ago gacs kurdyumov and levin introduced three different cellular automata to investigate whether one dimensional nonequilibrium interacting particle systems are capable of displaying phase transitions and as a byproduct they introduced the density classification problem the ability to classify arrays of symbols according to their initial density in the cellular automata literature their model ii became a well known model in theoretical computer science and statistical mechanics the other two models however did not receive much attention here we characterize the density classification performance of gacs kurdyumov and levin s model iv a four state cellular automaton with three absorbing states only two of which are attractive by numerical simulations we show that model iv compares well with its sibling model ii in the density classification task the additional states slow down the convergence to the majority state but confer a slight advantage in classification performance we also show that unexpectedly initial states diluted in one of the nonclassifiable states are more easily classified the performance of model iv under the influence of noise was also investigated and we found signs of an ergodic nonergodic phase transition at some small finite positive level of noise although the evidence is not entirely conclusive we set an upper bound on the critical point for the transition if any

structure in asynchronous cellular automata abstract the properties of synchronous cellular automata are compared with two types of cellular automata which iterate asynchronously either at random or with clocks of slightly different periods it is shown that some of the apparent self organization of cellular automata is an artifact of the synchronization of the clocks the structure which does exist in asynchronous cellular automata is compared with that found in the synchronous case and some apparently new forms of self organization which appear are studied

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

percolation games probabilistic cellular automata and the hard core model let each site of the square lattice mathbb z 2 be independently assigned one of three states a trap with probability p a target with probability q and open with probability 1 p q where 0 p q 1 consider the following game a token starts at the origin and two players take turns to move where a move consists of moving the token from its current site x to either x 0 1 or x 1 0 a player who moves the token to a trap loses the game immediately while a player who moves the token to a target wins the game immediately is there positive probability that the game is drawn with best play i e that neither player can force a win this is equivalent to the question of ergodicity of a certain family of elementary one dimensional probabilistic cellular automata pca these automata have been studied in the contexts of enumeration of directed lattice animals the golden mean subshift and the hard core model and their ergodicity has been noted as an open problem by several authors we prove that these pca are ergodic and correspondingly that the game on mathbb z 2 has no draws on the other hand we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions including an oriented version of the even sublattice of mathbb z d in all d ge 3 this is proved via a dimension reduction to a hard core lattice gas in dimension d 1 we show that draws occur whenever the corresponding hard core model has multiple gibbs distributions we conjecture that draws occur also on the standard oriented lattice mathbb z d for d ge 3 but here our method encounters a fundamental obstacle

pattern generation using likelihood inference for cellular automata cellular automata are discrete dynamical systems which evolve on a discrete grid recent studies have shown that cellular automata with relatively simple rules can produce highly complex patterns we develop likelihood based methods for estimating rules of cellular automata aimed at the re generation of observed regular patterns under noisy data our approach is equivalent to estimating the local map of a stochastic cellular automaton direct computations of the maximum likelihood estimates are possible for regular binary patterns the likelihood formulation of the problem is congenial with the use of the minimum description length principle as a model selection tool we illustrate our method with a series of examples using binary images

low order dpg fem for an elliptic pde this paper introduces a novel lowest order discontinuous petrovgalerkin dpg finite element method fem for the poisson model problem the ultra weak formulation allows for piecewise constant and affine ansatz functions and for piecewise affine and lowest order raviartthomas test functions this lowest order discretization for the poisson model problem allows for a direct proof of the discrete infsup condition and a complete apriori and aposteriori error analysis numerical experiments investigate the performance of the method and underline the quasi optimal convergence

phase diagrams of majority voter probabilistic cellular automata we present a detailed numerical analysis of the phase diagrams for some majority voter probabilistic cellular automata and connect the results with theory enabling us to prove many of the observed features

fully asynchronous behavior of double quiescent elementary cellular automata in this paper we propose a probabilistic analysis of the fully asynchronous behavior i e two cells are never simultaneously updated as in a continuous time process of elementary finite cellular automata i e 0 1 states radius 1 and unidimensional for which both states are quiescent i e 0 0 0 0 and 1 1 1 1 it has been experimentally shown in previous works that introducing asynchronism in the global function of a cellular automata was perturbing its behavior but as far as we know only few theoretical work exists on the subject the cellular automata we consider live on a ring of size n and asynchronism is introduced as follows at each time step one cell is selected uniformly at random and the transition is made on this cell while the others stay in the same state among the 64 cellular automata belonging to the class we consider we show that 9 of them diverge on all non trivial configurations while the 55 other converge almost surely to a random fixed point we show that the exact convergence time of these 55 automata can only take the following values either 0 n ln n n2 n3 or n2n furthermore the global behavior of each of these cellular automata is fully determined by reading its code

axioms of adaptivity with separate marking for data resolution mixed finite element methods with flux errors in h div norms and div least squares finite element methods require a separate marking strategy in obligatory adaptive mesh refining the refinement indicator sigma 2 mathcal t k eta 2 mathcal t k mu 2 k of a finite element domain k in an admissible triangulation mathcal t consists of some residual based error estimator eta mathcal t k with some reduction property under local mesh refining and some data approximation error mu k separate marking means either dorfler marking if mu 2 mathcal t leq kappa eta 2 mathcal t or otherwise an optimal data approximation algorithm with controlled accuracy the axioms are sufficient conditions on the estimators eta mathcal t k and data approximation errors mu k for optimal asymptotic convergence rates the enfolded set of axioms of this paper simplifies c carstensen m feischl m page and d praetorius comput math appl 67 2014 pp 1195 1253 for collective m

spatial pattern formation in asynchronous cellular automata with mass conservation abstract this paper proposes a class of two dimensional asynchronous cellular automata with conservation of mass for the formation of patterns in groups and describes the merits given by this methodology a cellular automaton rule causing a specified kind of pattern was designed manually thanks to this realistic modeling method reflecting nature the mechanism of pattern formation was found to be similar to real chemical processes because of the conservation of mass some boring cellular automata which evolve for instance into homogeneity are automatically excluded in this scheme this is greatly advantageous for searching automatically for pattern forming cellular automata a search technique using genetic algorithm is proposed and it successfully find a few types of pattern spontaneously that is without giving a priori information on the type of pattern this technique is expected to be applied to a wide range of potential studies related to self organization

exact solution of a directed site animals enumeration problem in three dimensions an exact equivalence is established between the directed site animals enumeration dsae problem in d dimensions and a crystal growth model defined on the same lattice in special cases the latter reduces to the calculation of the free energy of a d ensuremath 1 dimensional lattice gas with extended hard cores the author solves a dsae problem with d 3 exactly by using its equivalence to the hard hexagon problem and shows that the exponent ensuremath theta frac 5 6 a new and simpler solution to the dsae problem on the square and triangular lattices is given

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

asymptotic stationarity of queues in series and the heavy traffic approximation one deals with a system of queues in series that is to say a number of queues through which a unit passes in turn spending a waiting time including service in any particular one and proceeding to the next succeeding queue immediately the service time is completed the first main result deals with the asymptotic behaviour of the vector of waiting times of a unit in each queue and of the time which a unit spends in the tandem queue the second main result deals with the heavy traffic approximation of the mentioned characteristics

progresses in the analysis of stochastic 2d cellular automata a study of asynchronous 2d minority cellular automata are often used to model systems in physics social sciences biology that are inherently asynchronous over the past 20 years studies have demonstrated that the behavior of cellular automata drastically changes under asynchronous updates still the few mathematical analyses of asynchronism focus on 1d probabilistic cellular automata either on single examples or on specific classes as for other classic dynamical systems in physics extending known methods from 1d to 2d systems is a long lasting challenging problem in this paper we address the problem of analyzing an apparently simple 2d asynchronous cellular automaton 2d minority where each cell when fired updates to the minority state of its neighborhood our simulations reveal that in spite of its simplicity the minority rule exhibits a quite complex response to asynchronism by focusing on the fully asynchronous regime we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints besides these technical results we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata

convergence of natural adaptive least squares finite element methods the first order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional in this paper we prove q linear convergence of the associated adaptive mesh refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right hand sides in a poisson model problem the proof relies on some modification of known supercloseness results to non convex polygonal domains plus the flux representation formula the analysis is carried out for the lowest order case in two dimensions for the simplicity of the presentation

around probabilistic cellular automata we survey probabilistic cellular automata with approaches coming from combinatorics statistical physics and theoretical computer science each bringing a different viewpoint some of the questions studied are specific to a domain and some others are shared most notably the ergodicity problem

dynamic aspects of the flip annihilation process a one dimensional interacting particle system is revisited it has discrete time and its components are located in the set of integers these components can disappear in the functioning process each component assumes two possible states called plus and minus and interacts at every time step only with its nearest neighbors the following two transformations happen the first one is called flip under its action a component in state minus turns into a plus with probability the second one is called annihilation under its action whenever a component in state plus is a left neighbor of a component in state minus both components disappear with probability let us consider a set of initial measures to the process for these measures we show the upper bound for the mean time of convergence which is a function of the initial measure moreover we obtain the upper bound to the mean quantity of minuses on the process in each time step considering the initial measure concentrated at the configuration whose components are in the state minus we improved a well known result that the process is non ergodic when 2 250 now we are able to offer non ergodicity when 9 2 1000 we also established new conditions to the ergodicity of the process finally we performed some monte carlo simulations for this process

asymptotic results on infinite tandem queueing networks we consider an infinite tandem queueing network consisting of gi 1 stations with i i d service times we investigate the asymptotic behavior of t n k the inter arrival times between customers n and n 1 at station k and that of w n k the waiting time of customer n at station k we establish a duality property by which w n k and the idle times y n k play symmetrical roles this duality structure interesting by itself is also instrumental in proving some of the ergodic results we consider two versions of the model the quadrant and the half plane in the quadrant version the sequences of boundary conditions w 0 k k and t n 0 n are given in the half plane version the sequence t n 0 n is given under appropriate assumptions on the boundary conditions and on the services we obtain ergodic results for both versions of the model for the quadrant version we prove the existence of temporally ergodic evolutions and of spatially ergodic ones furthermore the process t n k n converges weakly with k to a limiting distribution which is invariant for the queueing operator in the more difficult half plane problem the aim is to obtain evolutions which are both temporally and spatially ergodic we prove that 1 n k 1 nw 0 k converges almost surely and in l1 to a finite constant this constitutes a first step in trying to prove that t n k n converges weakly with k to an invariant limiting distribution

on cesaro limit distribution of a class of permutative cellular automata we study cesaro means time averages of the evolution measures of the class of permutative cellular automata over 0 1 defined by mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepekkfr0xfr x fr xb9adbaqaaegacigaaiaabeqaamaabaabaagcbagaaiikaiabew 9aqnaabaaaleaacawgcbaabeaakiaadihacaggpawaasbaasqaaiaa d6gaaeqaaogaeyypa0jaamieamaabaaaleaacawgubgaagjcvlaayi w7cqghrawkcaami8uaagjcvlaadkfaaeqaaogaey4kasiaeuioda1a a0baasqaaiaadqgacaami8uaagjcvlabg2da9iaayiw7caami8uaag imaaqaaiaadkfacaami8uaagjcvlabgkhitiaayiw7caami8uaagym aaaakiaayiw7caami8uaagjcvlaayiw7caami8uaagjcvlaacicaca aixagaey4kasiaamoyamaabaaaleaacawgqbaabeaakiabgucariaa dihadawgaawcbagaamobaiaayiw7caami8uaey4kasiaagjcvlaayi w7cawgqbaabeaakiaacmcaaaa 7530 varphi b x n x n kern 1pt kern 1pt kern 1pt kern 1pt r pi j kern 1pt kern 1pt kern 1pt kern 1pt 0 r kern 1pt kern 1pt kern 1pt kern 1pt 1 kern 1pt kern 1pt kern 1pt kern 1pt kern 1pt kern 1pt 1 b j x n kern 1pt kern 1pt kern 1pt kern 1pt j where b b 0 b r 1is an aperiodic block in 0 1 r and operations are taken mod 2 if the initial measure is bernoulli we prove that the limit of the cesaro mean of the first column distribution exists when r 1 and b 1 b is the mod 2 sum automaton for this automaton we show that the limit is the 1 2 1 2 bernoulli measure and if the initial measure is markov we show that the limit of cesaro mean of the one site distribution is equidistributed

cellular automata as a paradigm for ecological modeling we review cellular automata as a modeling formalism and discuss how it can be used for modeling spatial ecological processes the implications of this modeling paradigm for ecological observation are stressed finally we discuss some shortcomings of the cellular automaton formalism and mention some extensions and generalizations which may remedy these shortcomings

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

standing waves in a counter rotating vortex filament pair abstract the distance among two counter rotating vortex filaments satisfies a beam type of equation according to the model derived in 15 this equation has an explicit solution where two straight filaments travel with constant speed at a constant distance the boundary condition of the filaments is 2 periodic using the distance of the filaments as bifurcating parameter an infinite number of branches of periodic standing waves bifurcate from this initial configuration with constant rational frequency along each branch

a low order discontinuous petrov galerkin method for the stokes equations this paper introduces a low order discontinuous petrov galerkin dpg finite element method fem for the stokes equations the ultra weak formulation utilizes piecewise constant and affine ansatz functions and piecewise affine and discontinuous lowest order raviart thomas test search functions this low order discretization for the stokes equations allows for a direct proof of the discrete inf sup condition with explicit constants the general framework of carstensen et al siam j numer anal 52 3 1335 1353 2014 then implies a complete a priori and a posteriori error analysis of the dpg fem in the natural norms numerical experiments investigate the performance of the method and underline its quasi optimal convergence

guaranteed lower eigenvalue bounds for the biharmonic equation the computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate s vibrations this paper shows that the eigenvalue provided by the nonconforming morley finite element analysis which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense is not always a lower bound a fully explicit error analysis of the morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound this paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower order terms

monte carlo investigation of the critical behavior of stavskaya s probabilistic cellular automaton stavskaya s model is a one dimensional probabilistic cellular automaton pca introduced in the end of the 1960s as an example of a model displaying a nonequilibrium phase transition although its absorbing state phase transition is well understood nowadays the model never received a full numerical treatment to investigate its critical behavior in this brief report we characterize the critical behavior of stavskaya s pca by means of monte carlo simulations and finite size scaling analysis the critical exponents of the model are calculated and indicate that its phase transition belongs to the directed percolation universality class of critical behavior as would be expected on the basis of the directed percolation conjecture we also explicitly establish the relationship of the model with the domany kinzel pca on its directed site percolation line a connection that seems to have gone unnoticed in the literature so far

nonlinear discontinuous petrov galerkin methods the discontinuous petrov galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper its lowest order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation a reduced formulation and a weighted nonlinear least squares method quasi optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dpg framework for the approximation of a regular solution the variational model example allows for a built in guaranteed error control despite inexact solve the subtle uniqueness of discrete minimizers is monitored in numerical examples

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

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

deducing local rules for solving global tasks with random boolean networks abstract it has been shown that uniform as well as non uniform cellular automata ca can be evolved to perform certain computational tasks random boolean networks are a generalization of two state cellular automata where the interconnection topology and the cell s rules are specified at random here we present a novel analytical approach to find the local rules of random boolean networks rbns to solve the global density classification and the synchronization task from any initial configuration we quantitatively and qualitatively compare our results with previously published work on cellular automata and show that randomly interconnected automata are computationally more efficient in solving these two global tasks our approach also provides convergence and quality estimates and allows the networks to be randomly rewired during operation without affecting the global performance finally we show that rbns outperform small world topologies on the density classification task and that they perform equally well on the synchronization task our novel approach and the results may have applications in designing robust complex networks and locally interacting distributed computing systems for solving global tasks

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

intermediate model between majority voter pca and its mean field model probabilistic cellular automata pca are simple models used to study dynamical phase transitions there exist mean field approximations to pca that can be shown to exhibit a phase transition we introduce a model interpolating between a class of pca called majority voters and their corresponding mean field models using graphical methods we prove that this model undergoes a phase transition

eroders on a plane with three states at a point part i deterministic imagine a huge in theory infinite space whose elements are called components we say that we have a configuration if for every component we have specified its state all components have one and the same finite set m of possible states and one of the elements of m is called zero if all the components of a configuration are zeros we call it all zeros we are especially interested in those configurations in which only a bounded set of components are in a state different from zero such a configuration is called an island our time is discrete and we may imagine that due to the forces of nature at every time step the whole configuration landscape is subject to a deterministic uniform local rule d such that non zero components may appear only in the vicinity of already existing non zero components like in percolation or contact processes we say that an operator d erodes an island x if there is a natural t such that x d t all zeros that is t iterative applications of d turn x into all zeros we call d an eroder if it erodes all islands we look for an algorithm which decides for any d whether it is an eroder or not and does it make islands grow or not in general this problem is algorithmically unsolvable so we need to restrict our scope in addition to the afore mentioned conditions we assume that d is monotonic studying such processes one has to decide whether time and space are discrete or continuous and in this study we choose discrete time and continuous space just because this case is underrepresented in the literature but especially important is the set of possible states of every single component years ago one of us a toom presented a rule to decide whether d is an eroder for the case when every component has only two possible states a toom also showed that if d is an eroder then it erodes any island in time which is linear in the diameter of the island in this work every component has three possible states it turned out that the difference between two and three states which may seem trivial in fact leads to a qualitative difference in the three states case an eroder may take non linear time to reach all zeros also unlike two state case in the three state case we have in addition to eroders to introduce degraders which reduce any island to an island where states of all components take only two possible values our main results are stated as three theorems theorem 1 gives sufficient but regretfully not necessary conditions for a linear degrader and an eroder theorems 2 and 3 concentrate on the case when d needs a non linear time to erode an island a case impossible if states of components take only two values in addition theorem 3 presents a sufficient condition for islands to grow

variable length analog of stavskaya process a new example of misleading simulation this article presents a new example intended to showcase limitations of computer simulations in the study of random processes with local interaction for this purpose we examine a new version of the well known stavskaya process which is a discrete time analog of the well known contact processes like the bulk of random processes studied till now the stavskaya process is constant length that is its components do not appear or disappear in the course of its functioning the process which we study here and call variable stavskaya vs is similar to stavskaya it is discrete time its states are bi infinite sequences whose terms take only two values denoted here as minus and plus and the measure concentrated in the configuration all pluses is invariant however it is a variable length which means that its components also called particles may appear and disappear under its action the operator vs is a composition of the following two operators the first operator called birth depends on

theory of cellular automata a survey this article surveys some theoretical aspects of cellular automata ca research in particular we discuss classical and new results on reversibility conservation laws limit sets decidability questions universality and topological dynamics of ca the selection of topics is by no means comprehensive and reflects the research interests of the author the main goal is to provide a tutorial of ca theory to researchers in other branches of natural computing to give a compact collection of known results with references to their proofs and to suggest some open problems

convergence rates of the dpg method with reduced test space degree this paper presents a duality theorem of the aubin nitsche type for discontinuous petrov galerkin dpg methods this explains the numerically observed higher convergence rates in weaker norms considering the specific example of the mild weak or primal dpg method for the laplace equation two further results are obtained first the dpg method continues to be solvable even when the test space degree is reduced provided it is odd second a non conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree

discrete stochastic models for traffic flow we investigate a probabilistic cellular automaton model which has been introduced recently this model describes single lane traffic flow on a ring and generalizes the asymmetric exclusion process models we study the equilibrium properties and calculate the so called fundamental diagrams flow versus density for parallel dynamics this is done numerically by computer simulations of the model and by means of an improved mean field approximation which takes into account short range correlations for cars with a maximum velocity of 1 the simplest nontrivial approximation gives the exact result for higher velocities the analytical results obtained by iterated application of the approximation scheme are in excellent agreement with the numerical simulations

the nilpotency problem of one dimensional cellular automata the limit set of a celullar automaton consists of all the configurations of the automaton that can appear after arbitrarily long computations it is known that the limit set is never empty it contains at least one homogeneous configuration a ca is called nilpotent if its limit set contains just one configuration the present work proves that it is algorithmically undecidable whether a given one dimensional cellular automaton is nilpotent the proof is based on a generalization of the well known result about the undecidability of the tiling problem of the plane the generalization states that the tiling problem remains undecidable even if one considers only so called nw deterministic tile sets that is tile sets in which the left and upper neighbors of each tile determine the tile uniquely the nilpotency problem is known to be undecidable for d dimensional ca for d geq 2 the result is the basis of the proof of rice s theorem for ca limit sets which states that every nontrivial property of limit set

nondeterministic density classification with diffusive probabilistic cellular automata we present a probabilistic cellular automaton ca with two absorbing states which performs classification of binary strings in a nondeterministic sense in a system evolving under this ca rule empty sites become occupied with a probability proportional to the number of occupied sites in the neighborhood while occupied sites become empty with a probability proportional to the number of empty sites in the neighborhood the probability that all sites become eventually occupied is equal to the density of occupied sites in the initial string

non ergodicity in a 1 d particle process with variable length we present a 1 d random particle process with uniform local interaction which displays some form of non ergodicity similar to contact processes but more unexpected particles enumerated by integer numbers interact at every step of the discrete time only with their nearest neighbors every particle has two possible states called minus and plus at every time step two transformations occur the first one turns every minus into plus with probability independently from what happens at other places and thereby favors pluses against minuses the second one is impartial under its action whenever a plus is a left neighbor of a minus both disappear with probability independently from presence and fate of other pairs of this sort if is small enough by comparison with 2 and we start with all minuses the minuses never die out

low order discontinuous petrov galerkin finite element methods for linear elasticity this paper analyzes lowest order discontinuous petrov galerkin dpg finite element methods fem for the navier lame equations with different norms and side restrictions the focus is on the direct proof of a discrete inf sup condition for a low order test search space the low order finite element spaces in the ultraweak formulation involve the piecewise constant and affine ansatz functions and discontinuous piecewise affine test functions in two and three space dimensions with neumann boundary conditions or the pure dirichlet problem those lowest order discretizations for linear elasticity allow for a direct proof of the discrete inf sup condition and a complete a priori and a posteriori error analysis which is robust in the incompressible limit as lambda rightarrow infty numerical experiments with uniform and adaptive mesh refinements investigate lambda robustness and confirm that one scheme is locking free

a posteriori error control for dpg methods a combination of ideas in least squares finite element methods with those of hybridized methods recently led to discontinuous petrov galerkin dpg finite element methods they minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard locally computable dual norm this paper establishes a general a posteriori error analysis for the natural norms of the dpg schemes under conditions equivalent to a priori stability estimates it is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors the presented abstract framework for a posteriori error analysis applies to known dpg discretizations of laplace and lame equations and to a novel dpg method for the stress velocity formulation of stokes flow with symmetric stress approximations since the error control does not rely on the discrete equations it applies to inexactly computed or otherwise perturbed solutions within the discrete space

convergence and optimality of adaptive least squares finite element methods the first order div least squares finite element methods lsfems allow for an immediate a posteriori error control by the computable residual of the least squares functional this paper establishes an adaptive refinement strategy based on some equivalent refinement indicators since the first order div lsfem measures the flux errors in h div the data resolution error measures the l 2 norm of the right hand side f minus the piecewise polynomial approximation pi f without a mesh size factor hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment e g by the thresholding second algorithm due to binev and devore the resulting novel adaptive lsfem with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class

an unconditionally stable space time fe method for the korteweg de vries equation abstract we introduce an unconditionally stable finite element fe method the automatic variationally stable fe avs fe method for the numerical analysis of the korteweg de vries kdv equation the avs fe method is a petrov galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous petrov galerkin dpg method however since avs fe method is a minimum residual method we establish a global saddle point system instead of computing optimal test functions element by element this system allows us to seek both the approximate solution of the kdv initial boundary value problem ibvp and a riesz representer of the approximation error the avs fe method distinguishes itself from other minimum residual methods by using globally continuous hilbert spaces such as h 1 while at the same time using broken hilbert spaces for the test consequently the avs fe approximations are classical c 0 continuous fe solutions the unconditional stability of this method allows us to solve the kdv equation space and time without having to satisfy a cfl condition we present several numerical verifications for both linear and nonlinear versions of the kdv equation leading to optimal convergence behavior finally we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear kdv equation

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

a first order system least squares method for hyperelasticity a least squares mixed finite element method for deformations of hyperelastic materials using stress and displacement as process variables is presented and studied the method is investigated in detail for the specific case of a neo hookean material law and is based on the representation of the strain stress relation a formulation is derived for compressible materials and shown to remain valid in the incompressible limit automatically enforcing the incompressibility constraint the mapping properties of the first order system operator are studied in appropriate sobolev spaces under the assumption of a locally unique solution with sufficient regularity it is proved that the first order least squares residual constitutes an upper bound for the error measured in a suitable norm provided that the finite element approximation is sufficiently close the method is tested numerically in a plane strain situation using next to lowest order raviart thomas elements for the stress tensor and conforming quadratic

a bifurcation theorem for potential operators abstract a sufficient condition for bifurcation of real solutions of g u 0 where g r d e d e is given in terms of the crossing number of the eigenvalue perturbation of the linearization along the trivial solutions 0 this number describes how many eigenvalues leave the negative real axis wen the parameter varies from negative to positive values whereas in general only an odd crossing number entails bifurcation for potential operators in a hilbert space e a nonzero crossing number is sufficient for bifurcation the proof uses a lyapunov schmidt reduction and conley s index theory

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

consistency stability a priori and a posteriori errors for petrov galerkin methods applied to nonlinear problems in an abstract framework we present a formalism which specifies the notions of consistency and stability of petrov galerkin methods used to approximate nonlinear problems which are in many practical situations strongly nonlinear elliptic problems this formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems this theory is illustrated with the example div k u u c u f in a two dimensional domain with dirichlet boundary conditions

stationary states of toom cellular automata in simulations stationary states of toom probabilistic cellular automata are tested in computer experiments the aim of the tests is to identify the features characterizing the equilibrium states understood as in statistical mechanics namely we investigate the following scaling laws that involve critical parameters beta gamma and nu locality of the interactions and density of the relative entropy between stationary states the arguments showing that stationary toom states are not the equilibrium ones are provided

chaos and monte carlo approximations of the flip annihilation process the flip annihilation process is a random particle process with one dimensional local interaction in discrete time initially presented by one of us namely toom in 2004 its components are enumerated by integer numbers and every component has two states minus and plus at every time step two transformations occur the first one called flip independently turns every minus into plus with probability the second one called annihilation acts thus whenever a plus is a left neighbor of a minus both disappear with probability independently from other components what is interesting about this process is that it is ergodic for 2 and non ergodic for 2 250 it is natural to conjecture that there is some transition curve which we call the true curve and denote by beta mathsf true alpha which separates the areas of ergodicity and non ergodicity of this process from each other the estimates mentioned above albeit rigorous leave a large gap between them and the present article s purpose is to obtain some closer albeit non rigorous approximations of the true curve we do it in two ways one of which is a chaos approximation and the other is a monte carlo simulation thus we obtain two curves which are much closer to each other than the rigorous estimations also we fill in albeit only numerically another shortcoming of the rigorous estimation 2 250 namely that it leaves us uncertain whether the true curve has a zero or positive slope at the point 0 both approximate curves have a positive slope at 0 as we hoped

the gacs kurdyumov levin automaton revisited we study a one dimensional cellular automaton that was originally proposed as a candidate for exhibiting nonergodic behavior under noise we prove that the deterministic model has the eroder property for two and only two invariant states moreover we give the best possible estimates for the corresponding erosion times we then review the results we have obtained from extensive computer simulations for the stochastic model and for a mixed model and argue that they suggest numerical and heuristic evidence in favour of ergodic behavior for all nonzero values of the noise parameter

a primal dpg method without a first order reformulation abstract we show that it is possible to apply the dpg methodology without reformulating a second order boundary value problem into a first order system by considering the simple example of the poisson equation the result is a new weak formulation and a new dpg method for the poisson equation which has no numerical trace variable but has a numerical flux approximation on the element interfaces in addition to the primal interior variable

asynchronous game of life abstract this paper proposes the 8 state asynchronous equivalence of the well known game of life gl our cellular automaton called asynchronous game of life simulates exactly the behavior of the gl such as universal computation and self organization no matter whether the update of cells is simultaneous or independent according to some updating scheme like a step driven or time driven method see biosystems 51 1999 123 we employ the updating scheme of blok and bergersen phys rev e 59 1999 3876 such that at every time step each cell has a certain probability to be updated and investigate the statistical properties of our model through power spectral analyses

nonlinear standing and rotating waves on the sphere abstract we establish the existence of time periodic solutions of semi linear wave equations on the unit sphere in r 3 the problem has been studied previously in 1982 v benci and fortunato ann mat pura appl 132 215 242 using variational techniques our results here are much sharper we employ delicate methods of bifurcation theory 1979 h kielhofer j math anal appl 68 408 420 1987 h kielhofer and p kotzner j appl math phys 38 201 212 combined with well known group theoretic ideas to find branches of small amplitude solutions precise spatio temporal patterns of solutions are uncovered as a by product of the analysis moreover in certain cases we prove the existence of global solution branches in the sense of p rabinowitz

from pca s to equilibrium systems and back stationary measures for probabilistic cellular automata pca s ind dimensions give rise to space time histories whose statistics may naturally be described by gibbs states ind 1 dimensions for an interaction energy obtained from the pca in this note we study the converse question do all gibbs states for this correspond to statistical space time histories for the pca our main result states that the answer is yes at least for translation invariant or periodic gibbs states thus ergodicity questions for pca s can at least partially be formulated as questions of uniqueness of gibbs states

convergence time and phase transition in a non monotonic family of probabilistic cellular automata in dynamical systems some of the most important questions are related to phase transitions and convergence time we consider a one dimensional probabilistic cellular automaton where their components assume two possible states zero and one and interact with their two nearest neighbors at each time step under the local interaction if the component is in the same state as its two neighbors it does not change its state in the other cases a component in state zero turns into a one with probability alpha and a component in state one turns into a zero with probability 1 beta for certain values of alpha and beta we show that the process will always converge weakly to delta 0 the measure concentrated on the configuration where all the components are zeros moreover the mean time of this convergence is finite and we describe an upper bound in this case which is a linear function of the initial distribution we also demonstrate an application of our results to the percolation pca finally we use mean field approximation and monte carlo simulations to show coexistence of three distinct behaviours for some values of parameters alpha and beta

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions