additive one dimensional cellular automata are chaotic according to devaney s definition of chaos abstract we study the chaotic behavior of a particular class of dynamical systems cellular automata we specialize the definition of chaos given by devaney for general dynamical systems to the case of cellular automata a dynamical system x f is chaotic according to devaney s definition of chaos if its transition map f is sensitive to the initial conditions topologically transitive and has dense periodic orbits on x our main result is the proof that all the additive one dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of devaney

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

reversibility of general 1d linear cellular automata over the binary field z 2 under null boundary conditions the reversibility of cellular automata ca has long been studied but there are still many problems untouched this paper tackles the reversibility problem of the general case concerning 1d ca under null boundary conditions defined by linear rules over the binary field z 2 although transition matrix has been widely used for some special linear ca lca rules it has severe limitations its complexity depends on the number of cells and determining its reversibility becomes another unsolved tough problem for general rules by constructing deterministic finite automata dfa of the de bruijn graph presentation of ca we conclude the reversibility problem of all 1d linear rules over z 2 under null boundary conditions it turns out that any 1d lca except for some unilateral rules over z 2 under null boundary conditions can be reversible once having a proper number of cells and dfa provides an efficient way to find those numbers moreover the complexity of the dfa solution is independent of the number of cells in addition the relative low complexity of dfa can be further significantly reduced by using only a small part of it

nonlinear self adjointness of a 2d generalized second order evolution equation abstract we study the nonlinear self adjointness of a general class of quasilinear 2d second order evolution equations which do not possess variational structure for this purpose we use the method of ibragimov devised and developed recently this approach enables one to establish the conservation laws for any differential equation we first obtain conditions determining the self adjoint sub class in the general case then we establish the conservation laws for important particular cases the ricci flow equation the modified ricci flow equation and the nonlinear heat equation

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

cellular automata between sofic tree shifts we study the sofic tree shifts of a s where s is a regular rooted tree of finite rank in particular we give their characterization in terms of unrestricted rabin automata we show that if x a s is a sofic tree shift then the configurations in x whose orbit under the shift action is finite are dense in x and as a consequence of this we deduce that every injective cellular automata t x x is surjective moreover a characterization of sofic tree shifts in terms of general rabin automata is given we present an algorithm for establishing whether two unrestricted rabin automata accept the same sofic tree shift or not this allows us to prove the decidability of the surjectivity problem for cellular automata between sofic tree shifts we also prove the decidability of the injectivity problem for cellular automata defined on a tree shift of finite type

nondeterministic cellular automata abstract using purely topological concepts we characterize the class of nondeterministic mappings between shift spaces that give rise to nondeterministic sliding block codes we use such topological notions to investigate basic properties of surjective and reversible nondeterministic cellular automata

when and how can a cellular automaton be rewritten as a lattice gas both cellular automata ca and lattice gas automata lg provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics it is customary to use the terms cellular automaton and lattice gas for a dynamic system itself as well as for its presentation the two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability following a conjecture by toffoli and margolus it had been proved by kari that any invertible ca at least up to two dimensions can be rewritten as an isomorphic lg but until now it was not known whether this is possible in general for noninvertible ca which comprise almost all ca and represent the bulk of examples in theory and applications even circumstantial evidence whether in favor or against was lacking here for noninvertible ca a we prove that an lg presentation is out of the question for the vanishingly small class of surjective ones we then turn our attention to all the rest noninvertible and nonsurjective which comprise all the typical ones including conway s game of life for these b we prove by explicit construction that all the one dimensional ones are representable as lg and c we present and motivate the conjecture that this result extends to any number of dimensions the tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale

variational integrators for nonvariational partial differential equations abstract variational integrators for lagrangian dynamical systems provide a systematic way to derive geometric numerical methods these methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the lagrangian via noether s theorem an inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem even though for a large class of systems this requirement is fulfilled there are many interesting examples which do not belong to this class e g equations of advection diffusion type frequently encountered in fluid dynamics or plasma physics on the other hand it is always possible to embed an arbitrary dynamical system into a larger lagrangian system using the method of formal or adjoint lagrangians we investigate the application of the variational integrator method to formal lagrangians and thereby extend the application domain of variational integrators to include potentially all dynamical systems the theory is supported by physically relevant examples such as the advection equation and the vorticity equation and numerically verified remarkably the integrator for the vorticity equation combines arakawa s discretisation of the poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved in the presentation of the results we try to make the geometric framework of variational integrators accessible to non specialists

how to turn a second order cellular automaton into a lattice gas a new inversion scheme in second order cellular automata the signals made available to the local transition function include besides the current state of a site s neighbors also the previous state of the site itself some similarities had been noted especially in physics simulations between certain second order cellular automata and certain lattice gases here we show how to construct for any second order cellular automaton a lattice gas having isomorphic functional behavior paradoxically this isomorphism of function is achieved by compromising on the isomorphism of structure namely the group of translation symmetries of the resulting lattice gas is coarser than that of the original cellular automaton the advantage of our construction is that while invertibility in cellular automata is not directly deducible from the local map indeed it is in general undecidable in the second order case the invertibility of the cellular automaton goes hand in hand with that of the corresponding lattice gas and for lattices gases invertibility is trivially decidable on the basis of the local structure from a physical viewpoint our construction illustrates a trade off between two ways of achieving a force of a given range in a cellular automaton multiple copies of a signal are made and distributed in parallel to several sites explicitly providing the desired fanout width in a lattice gas with our construction one can let a single signal serially service a number of sites along an appropriate spacetime route the latter method is better suited to a nondissipative implementation of the dynamics

invertible cellular automata a review abstract in the light of recent developments in the theory of invertible cellular automata we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics

percolation in the random cluster process and q state potts model the critical probabilities pt pt q and ph ph q of the random cluster model with parameter q or 1 on the square lattice are shown to satisfy pt or square root q 1 square root q or ph ph or q 1 pt pt q 1 pt the last inequality is the q analogue of the relation pt ph or 1 of ordinary q 1 percolation though of course in this case equality holds

differential invariants for quasi linear and semi linear wave type equations abstract in this paper we consider the class of wave type equations u tt f x t u u xx g x t u u x u t and two special cases of it we derive the equivalence transformations for these equations and using these transformations we obtain differential invariants and invariant equations we employ these invariants or and invariant equations to classify all equations of this general class that can be mapped into a simple linear hyperbolic or into an elliptic equation additional applications are presented

garden of eden configurations for 2 d cellular automata with rule 2460n an important problem in cellular automata theory is the reversibility of a cellular automaton which is related to the existence of garden of eden configurations in cellular automata in this paper we study new local rules for two dimensional cellular automata over the ternary field z 3 the set of integers modulo three with some of their important characteristics we obtain necessary and sufficient conditions for the existence of garden of eden configurations for two dimensional ternary cellular automata also by making use of the matrix representation of two dimensional cellular automata we provide an algorithm to obtain the number of garden of eden configurations for two dimensional cellular automata defined by rule 2460n we present an application of the reversible two dimensional ternary cellular automata to cryptography

efficient exhaustive listings of reversible one dimensional cellular automata this paper looks at an algebraic formulation of one dimensional cellular automata using the formulation connections to combinatorial structures and graph theory become clear strong results about uniqueness and isomorphism allows us to outline effective algorithms for the generation of exhaustive lists of reversible one dimensional cellular automata and to count the number of distinct examples that exist these algorithms use the orderly algorithm methods to avoid the pitfalls of brute force searches

reversible simulation of one dimensional irreversible cellular automata abstract a cellular automaton ca is called reversible or injective iff its global function is one to one it has been shown by toffoli that any irreversible k dimensional ca is simulated by a k 1 dimensional rca in this paper we show that any one dimensional ca with finite configurations can be simulated by a one dimensional reversible ca this is proved by using the framework of partitioned ca pca

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

conservation laws of the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms in this paper conservation laws for the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms are studied due to the existence of cross terms the new conservation theorem given by ibragimov cannot be applied to the two equations directly we propose two rules of modifications which ensure the theorem can be applied to nonlinear evolution equations with cross terms formulas of conservation laws for the kp equation and burgers equation are given using these formulas non trivial and time dependent conservation laws for these equations are derived according to their different lie symmetries

reversible space time simulation of cellular automata abstract the goal of this paper is to design a reversible d dimensional cellular automaton which is capable of simulating the behavior of any given d dimensional cellular automaton over any given configuration even infinite with respect to a well suited notion of simulation we introduce we generalize a problem which was originally addressed in a paper by toffoli in 1977 he asked whether a d dimensional reversible cellular automaton could simulate d dimensional cellular automata in the same paper he proved that there exists a d 1 dimensional reversible cellular automaton which can simulate a given d dimensional cellular automaton to prove our result we use as an intermediate model partition cellular automata defined by morita et al in 1989

conservation laws for some equations that admit compacton solutions induced by a non convex convection abstract in this study we consider some equations that admit compacton solutions induced by a non convex convection some conservation laws are derived for the nonlinearly self adjoint equations based on differential substitutions we also derive some conservation laws using the multipliers method

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

graph theoretical characterization of invertible cellular automata abstract a cellular automaton is defined to be invertible if its induced global rule is surjective on the set of all finite configurations it is shown that all one dimensional invertible cellular automata are obtainable from invertible nearest pair ones i e cellular automata given by a rule matrix for these an algorithm is proposed which runs in a time which is a polynomial function of the number of letters of the alphabet the result is expressed in a graph theoretical way the cellular automaton is invertible if and only if two graphs constructed by the algorithm have no edges in common it follows that if the number of letters in the alphabet is a prime the only invertible cellular automata are those for which every row or column of the rule matrix is a permutation of the alphabet for composite alphabet cardinalities several methods analytical and computational ones to obtain more complicated invertible cellular automata are proposed and used to construct examples all four letter invertible automata are constructed

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

the undecidability of the infinite ribbon problem implications for computing by self assembly self assembly the process by which objects autonomously come together to form complex structures is omnipresent in the physical world recent experiments in self assembly demonstrate its potential for the parallel creation of a large number of nanostructures including possibly computers a systematic study of self assembly as a mathematical process has been initiated by l adleman and e winfree the individual components are modeled as square tiles on the infinite two dimensional plane each side of a tile is covered by a specific glue and two adjacent tiles will stick iff they have matching glues on their abutting edges tiles that stick to each other may form various two dimensional structures such as squares and rectangles or may cover the entire plane in this paper we focus on a special type of structure called a ribbon a non self crossing rectilinear sequence of tiles on the plane in which successive tiles are adjacent along an edge and abutting edges of consecutive tiles have matching glues we prove that it is undecidable whether an arbitrary finite set of tiles with glues infinite supply of each tile type available can be used to assemble an infinite ribbon while the problem can be proved undecidable using existing techniques if the ribbon is required to start with a given seed tile our result settles the unseeded case an open problem formerly known as the unlimited infinite snake problem the proof is based on a construction due to r robinson of a special set of tiles that allow only aperiodic tilings of the plane this construction is used to create a special set of directed tiles tiles with arrows painted on the top with the strong plane filling property a variation of the plane filling property previously defined by j kari a construction of sandwich tiles is then used in conjunction with this special tile set to reduce the well known undecidable tiling problem to the problem of the existence of an infinite directed zipper a special kind of ribbon a motif construction is then introduced that allows one tile system to simulate another by using geometry to represent glues using motifs the infinite directed zipper problem is reduced to the infinite ribbon problem proving the latter undecidable an immediate consequence of our result is the undecidability of the existence of arbitrarily large structures self assembled using tiles from a given tile set

cellular automata as an alternative to rather than an approximation of differential equations in modeling physics abstract cellular automata are models of distributed dynamical systems whose structure is particularly well suited to ultrafast exact numerical simulation on the other hand they constitute a radical departure from the traditional partial differential equation approach to distributed dynamics here we discuss the problem of encoding the state variables and evolution laws of a physical system into this new setting and of giving suitable correspondence rules for interpreting the model s behavior

uniqueness of the infinite component in a random graph with applications to percolation and spin glasses we extend the theorem of burton and keane on uniqueness of the infinite component in dependent percolation to cover random graphs on d or d with long range edges we also study a short range percolation model related to nearest neighbor spin glasses on d or on a slab d 0 k and prove both that percolation occurs and that the infinite component is unique forv 2 0 1 or larger

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

emergence of universal global behavior from reversible local transitions in asynchronous systems reversible computing usually focuses on how to establish a valid equivalence between the global reversibility and local reversibility in computational systems hitherto the equivalence has been precisely developed in cellular automata combinational circuits and quantum computers but it implicitly assumes that the underlying systems are synchronously timed alternative systems include the delay insensitive di circuits and asynchronous cellular automata acas where the local operations of each component cell may be executed independently at random times despite the randomness associated with asynchronous timings equivalence between the global and local reversibility can be simply achieved in both di circuits morita 2001 and acas lee et al 2003 provided that their local operations transitions are thoroughly serialized the complete exclusion of concurrency in local behavior however will profoundly depress the parallel processing efficiency of di circuits as well as the intrinsic massive parallelism of acas this paper aims at exploring what kind of complex global behavior may arise from the concurrent operations that are invertible at local level to this end we show that di circuits composed of reversible elements can actually exhibit universal input and output behavior with the universality emerging from the concurrency in reversible local operations likewise by further embedding all circuits into the cellular space the emergence of universal global transitions from reversible local transitions can be exactly identified in an aca which due to the bijectivity of its local function has significantly lower complexity as compared to other models

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

investigating topological chaos by elementary cellular automata dynamics we apply the two different definitions of chaos given by devaney and by knudsen for general discrete time dynamical systems dtds to the case of elementary cellular automata i e 1 dimensional binary cellular automata with radius 1 a dtds is chaotic according to the devaney s definition of chaos iff it is topologically transitive has dense periodic orbits and it is sensitive to initial conditions a dtds is chaotic according to the knudsen s definition of chaos iff it has a dense orbit and it is sensitive to initial conditions we enucleate an easy to check property left or rightmost permutivity of the local rule associated with a cellular automaton which is a sufficient condition for d chaotic behavior it turns out that this property is also necessary for the class of elementary cellular automata finally we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata

structure and reversibility of 2d hexagonal cellular automata cellular automata are used to model dynamical phenomena by focusing on their local behavior which depends on the neighboring cells in order to express their global behavior the geometrical structure of the models suggests the algebraic structure of cellular automata after modeling the dynamical phenomena it is sometimes an important problem to be able to move backwards in order to understand it better this is only possible if cellular automata is reversible in this paper 2d finite cellular automata defined by local rules based on hexagonal cell structure are studied rule matrix of the hexagonal finite cellular automaton is obtained the rank of rule matrices representing the 2d hexagonal finite cellular automata via an algorithm is computed it is a well known fact that determining the reversibility of a 2d cellular automata is a very difficult problem in general here the reversibility problem of this family of 2d hexagonal cellular automata is also resolved completely

relaxation properties of elementary reversible cellular automata abstract the relaxation properties of a number of elementary reversible cellular automata are studied different rules have different types of relaxation behavior possible relationships between the thermodynamic behavior and phase space structure are discussed

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics abstract exact solutions of some classical pdes with two independent variables are achieved by exploiting a double reduction method which is applied after deducing conserved vectors associated with lie point symmetries these are generally obtained through the application of noether s theorem after introducing the notions of adjoint equation and extended lagrangian alternatively associated conserved vectors can be obtained by a direct method

on the nonlinear self adjointness of a class of fourth order evolution equations in this paper we study the property of nonlinear self adjointness for a class of nonlinear fourth order equation it is shown that if an equation itself is a conservation law then it possesses the property of nonlinear self adjointness and hence it can be rewritten in an equivalent strictly self adjoint form the property of nonlinear self adjointness is used to obtain all nontrivial conservation laws for the class under consideration

σ automata and chebyshev polynomials a automaton is an additive binary cellular automaton on a graph for product graphs such as a grids and cylinders reversibility and periodicity properties of the corresponding automaton can be expressed in terms of a binary version of chebyshev polynomials we will give a detailed analysis of the divisibility properties of these polynomials and apply our results to the study of automata

nonlinear self adjointness of a generalized fifth order kdv equation the new concepts of self adjoint equations formulated by ibragimov and gandarias are applied to a class of fifth order evolution equations then from ibragimov s theorem on conservation laws conservation laws for the generalized kawahara equation simplified kahawara equation and modified simplified kawahara equation are established

integrating factors adjoint equations and lagrangians integrating factors and adjoint equations are determined for linear and non linear differential equations of an arbitrary order the new concept of an adjoint equation is used for construction of a lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables the method is illustrated by considering several equations traditionally regarded as equations without lagrangians noether s theorem is applied to the maxwell equations

an order preserving property of additive invariants for takesue type reversible cellular automata we show that for a large and important class of reversible one dimensional cellular automata the set of additive invariants exhibits an algebraic structure more precisely if f and g are one dimensional reversible cellular automata of the kind considered by takesue 1989 1 we show that the component wise maximum on these automata is such that j f email protected email protected g where j f denotes the set of additive invariants of f and denotes the inclusion relation between real subspaces

comment on the paper on conservation laws by lie symmetry analysis for 2 1 dimensional bogoyavlensky konopelchenko equation in wave propagation by s saha ray abstract in a recent paper referred to in the title the author used the concept of quasi self adjointness to obtain conservation laws for a system of the 2 1 dimensional bogoyavlensky konopelchenko equation apart from the adjoint system of equations all the results on the quasi self adjointness and the subsequent conservation laws obtained are inaccurate in this comment we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through noether theorem

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

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

local structure theory for cellular automata abstract the mean field theory for cellular automata wolfram and schulman and seiden is generalized to the local structure theory the local structure theory is a sequence of finitely parameterized models of the statistical features of a cellular automaton s evolution the n th model in the sequence takes into account correlations in terms of the probability of blocks of n states a class of measures the n block measures is introduced the local structure operator of order n maps n block measures to n block measures in a manner which reflects the cellular automaton map on blocks of states the fixed points of the map on measures approximate the invariant measures of the cellular automaton the ability of the local structure theory to model evolution from uncorrelated initial distributions is studied the theory gives exact results in simple cases in more complex cases monte carlo numerical experiments suggest that an accurate statistical portrait of cellular automaton evolution is obtained the invariant measures of a cellular automaton and the stability of these measures may be obtained from the local structure theory the local structure theory appears to be a powerful method for characterization and classification of cellular automata nearest neighbor cellular automata with two states per cell are studied using this method

nonlinear self adjointness a criterion for linearization of pdes in this communication we show that the property of nonlinear self adjointness is conserved by an invertible point transformation we use this property to give necessary conditions for the existence of an invertible point transformation mapping of a given partial differential equation pde to a linear pde

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

invariant in cellular automata for a subclass of the set of reversible cellular automata the author gives the form of an exact time invariant quantity which can be seen as a kind of energy with the ergodic assumption this is a model of two interacting ising spin systems

computational processes observers and turing incompleteness we propose a formal definition of wolfram s notion of computational processes based on iterated transducers together with a weak observer a model of computation that captures some aspects of physics like computation these processes admit a natural classification into decidable intermediate and complete where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting it is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process

inversion of 2d cellular automata some complexity results abstract in this paper we prove the co np completeness of the following decision problem given a two dimensional cellular automaton a even with von neumann neighborhood is a injective when restricted to finite configurations not greater than its length in order to prove this result we introduce two decision problems concerning respectively turing machines and tilings that we prove np complete then we present a transformation of problems concerning tilings into problems concerning cellular automata

statistics of collapsed lattice animals rigorous results and monte carlo simulations in this paper a two dimensional model of collapsed lattice animals with nearest neighbour contact interactions is introduced and the limiting free energy for the model is proved to exist we also establish that the partition function for collapsed lattice animals is related to the reliability polynomial for the m m grid graph and to the generating function for spanning forests on the m m grid graph from these relationships and using results about directed animals we are able to obtain rigorous upper and lower bounds on the free energy for collapsed lattice animals in addition we present monte carlo evidence that the heat capacity associated with this model remains bounded in the thermodynamic limit in fact we do not find any evidence for a second order phase transition for collapsed lattice animals

pattern growth in elementary cellular automata a classification of elementary cellular automata ca based on their pattern growth is introduced it is shown that this classification is effective that is there exists an algorithm to determine to which class a given ca belongs this algorithm is based on the properties of the local rule of cas not requiring the observation of their evolution furthermore necessary and sufficient conditions to detect all the elementary cas exhibiting a shift like behavior are given these cas have interesting dynamical properties and chaotic characteristics

cellular automata in fuzzy backgrounds abstract the main purpose of this work is to understand some limitations introduced by the classical definitions of cellular automata ca to this end we have defined a new model of cas fuzzy ca s which allows the observation of interesting chaotic properties of elementary cas to date neither a formal nor a precise definition of chaos in cas exists we believe that the proposed model provides a sharper tool to detect which properties can be associated to a chaotic behavior we also define a measure rule entropy which gives information about the ca s dynamics solely on the basis of the rule table and provides theoretical explanations to some of the empirical observations

explicit isoperimetric constants and phase transitions in the random cluster model the random cluster model is a dependent percolation model that has applications in the study of ising and potts models in this paper several new results are obtained for the random cluster model on nonamenable graphs with cluster parameter q geq 1 among these the main ones are the absence of percolation for the free random cluster measure at the critical value and examples of planar regular graphs with regular dual where p mathrm c mathrm free q p mathrm u mathrm wired q for q large enough the latter follows from considerations of isoperimetric constants and we give the first nontrivial explicit calculations of such constants such considerations are also used to prove nonrobust phase transition for the potts model on nonamenable regular graphs

the complexity of reversible cellular automata we study the orbits of reversible one dimensional cellular automata it is shown that the turing degree structure of the orbits of these automata is the same as for general cellular automata in particular there are reversible cellular automata whose orbits have arbitrary recursively enumerable degree

weak self adjoint differential equations the concepts of self adjoint and quasi self adjoint equations were introduced by ibragimov 2006 j math anal appl 318 742 57 2007 arch alga 4 55 60 in ibragimov 2007 j math anal appl 333 311 28 a general theorem on conservation laws was proved in this paper we generalize the concept of self adjoint and quasi self adjoint equations by introducing the definition of weak self adjoint equations we find a class of weak self adjoint quasi linear parabolic equations the property of a differential equation to be weak self adjoint is important for constructing conservation laws associated with symmetries of the differential equation

the magic rule spaces of neural like elementary cellular automata abstract a subclass of elementary cellular automata ca equivalent to a particular class of neural networks nn is studied the nn are dependent on four real parameters a b c and the subspace of the rule space which is obtained by varying the parameters and the dynamical characteristics of the rules belonging to this subset are determined finally a few nice and magic regularities in this subset are noted and explained

quasi self adjoint nonlinear wave equations in this paper we generalize the classification of self adjoint second order linear partial differential equation to a family of nonlinear wave equations with two independent variables we find a class of quasi self adjoint nonlinear equations which includes the self adjoint linear equations as a particular case the property of a differential equation to be quasi self adjoint is important e g for constructing conservation laws associated with symmetries of the differential equation

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

self adjointness and conservation laws of a generalized burgers equation a 2 1 dimensional generalized burgers equation is considered having written this equation as a system of two dependent variables we show that it is quasi self adjoint and find a nontrivial additional conservation law

linear speed up for cellular automata synchronizers and applications abstract simple cellular automata can be used as language recognizers or function calculators there exist several proofs of the linear speed up theorem in strong form for recognizers 7 6 11 but not for calculators in this paper we design a linear speed up method for a special kind of calculators namely the synchronizers which are different from recognizers as a consequence we extend a result from j mazoyer by proving that any polynomial with rational coefficients of a synchronization time is also a synchronization time provided that it be greater than a certain minimal time

predecessors of cellular automata states iii garden of eden classification of cellular automata abstract a procedure is given which determines all finite sequences without pre images for any one dimensional cellular automata with entries in z 2 on this basis we determine some properties of such sequences and classify all three site rules categories defined by this classification are compared to other phenomenological classifications introduced by wolfram gutowitz and jen