on the directional dynamics of additive cellular automata we continue the study of cellular automata ca directional dynamics i e the behavior of the joint action of ca and shift maps this notion has been investigated for general ca in the case of expansive dynamics by boyle and lind and by sablik for sensitivity and equicontinuity in this paper we give a detailed classification for the class of additive ca providing non trivial examples for some classes of sablik s classification moreover we extend the directional dynamics studies by considering also factor languages and attractors

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

complex dynamics of cellular automata rule 119 abstract in this paper the dynamical behaviors of cellular automata rule 119 are studied from the viewpoint of symbolic dynamics in the bi infinite symbolic sequence space 2 it is shown that there exists one bernoulli measure global attractor of rule 119 which is also the nonwandering set of the rule moreover it is demonstrated that rule 119 is topologically mixing on the global attractor and possesses the positive topological entropy therefore rule 119 is chaotic in the sense of both li yorke and devaney on the global attractor it is interesting that rule 119 a member of wolfram s class ii which was said to be simple as periodic before actually possesses a chaotic global attractor in 2 finally it is noted that the method presented in this work is also applicable to studying the dynamics of other rules especially the 112 bernoulli shift rules therein

the ring of conditions of a semisimple group abstract in this paper we are going to give an explicit description of the so called ring of conditions of an adjoint semisimple algebraic group g originally introduced in c de concini c procesi complete symmetric varieties ii intersection theory in algebraic groups and related topics kyoto nagoya 1983 in adv stud pure math vol 6 north holland amsterdam 1985 pp 481 513 in order to study intersection theory in g

kolmogorov complexity and cellular automata classification we present a new approach to cellular automata ca classification based on algorithmic complexity we construct a parameter which is based only on the transition table of ca and measures the randomness of evolutions is better in a certain sense than any other parameter recursively definable on ca tables we investigate the relations between the classical topological approach and ours our parameter is compared with langton s parameter turns out to be theoretically better and also agrees with some practical evidences reported in literature finally we propose a protocol to approximate and make experiments on ca dynamical behavior

solution of some conjectures about topological properties of linear cellular automata we study two dynamical properties of linear d dimensional cellular automata over zm namely denseness of periodic points and topological mixing for what concerns denseness of periodic points we complete the work initiated in theoret comput sci 174 1997 157 theoret comput sci 233 1 2 2000 147 14th annual symp on theoretical aspects of computer science stacs 97 lncs n 1200 springer berlin 1997 pp 427 438 by proving that a linear cellular automata has dense periodic points over the entire space of configurations if and only if it is surjective as conjectured in cattaneo et al 2000 for non surjective linear ca we give a complete characterization of the subspace where periodic points are dense for what concerns topological mixing we prove that this property is equivalent to transitivity and then easily checkable finally we classify linear cellular automata according to the definition of chaos given by devaney in an introduction to chaotic dynamical systems 2nd ed addison wesley reading ma usa 1989

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

an efficient hydrodynamic cellular automata for simulating fluids with large viscosities a hydrodynamic cellular automata hdca for simulating two dimensional fluids with large viscosities is proposed the model is characterized by a mean free path which is of the same size as in the fhp ii model but with a viscosity more than 10 times larger this new model should make simulations of flows at low reynolds number more efficient

stationary turbulent closure via the hopf functional equation exact closed form solutions are exhibited for the hopf equation for stationary incompressible 3d navier stokes flow for the cases of homogeneous forced flow including a solution with depleted nonlinearity and inhomogeneous flow with arbitrary boundary conditions this provides an exact method for computing two and higher point moments given the mean flow

number conserving cellular automata i decidability we prove that definitions of number conserving cellular automata found in literature are equivalent a necessary and sufficient condition for cellular automata to be number conserving is proved using this condition we give a quasi linear time algorithm to decide number conservation

on the undecidability of the limit behavior of cellular automata cellular automata ca are discrete dynamical systems and an abstract model of parallel computation the limit set of a cellular automaton is its maximal topological attractor a well known result due to kari says that all nontrivial properties of limit sets are undecidable in this paper we consider the properties of limit set dynamics i e properties of the dynamics of ca restricted to their limit sets there can be no equivalent of kari s theorem for limit set dynamics anyway we show that there is a large class of undecidable properties of limit set dynamics namely all properties of limit set dynamics which imply stability or the existence of a unique subshift attractor as a consequence we have that it is undecidable whether the cellular automaton map restricted to the limit set is the identity map and whether it is closing injective expansive positively expansive and transitive

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

combinatorial constructions associated to the dynamics of one sided cellular automata in this paper we study combinatorial constructions which lead to one sided invertible cellular automata with different dynamical behavior equicontinuity existence of equicontinuous points but not equicontinuity sensitivity and expansivity in particular we provide a simple characterization of the class of equicontinuous invertible one sided cellular automata and we construct families of expansive one sided cellular automata

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

exact results for deterministic cellular automata with additive rules deterministic cellular automata ca with additive rules are studied by exploiting the properties of circulant matrices on finite fields complete state transition diagrams for higher order and multidimensional ca on finite lattices are analyzed conditions on the rules which make them reversible are obtained it is shown that all state transition diagrams of the ca have identical trees rooted on cycles general formulae for cycle lengths and multiplicities are given

instanton filtering for the stochastic burgers equation we address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven burgers equation for this purpose we first solve the instanton equations using the chernykh stepanov method 2001 phys rev e 64 026306 these results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of chernykh and stepanov with remarkable agreement

a complete and efficiently computable topological classification of d dimensional linear cellular automata over z m we study the dynamical behavior of d dimensional linear cellular automata over zm we provide easy to check necessary and sufficient conditions for a d dimensional linear cellular automata over zm to be sensitive to initial conditions positively expansive strongly transitive and equicontinuous as a consequence of our results we have a complete and efficiently computable topological classification of d dimensional linear cellular automata over zm according to the most important dynamical properties studied in the theory of discrete time dynamical systems

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

dynamical behavior of additive cellular automata over finite abelian groups abstract we study the dynamical behavior of additive d dimensional d 1 cellular automata where the alphabet is any finite abelian group this class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list 38 44 41 among our major contributions there is the proof that topologically transitive additive d dimensional cellular automata over a finite abelian group are ergodic this result represents a solid bridge between the world of measure theory and that of topology and greatly extends previous results obtained in 12 44 for linear ca over z m z i e additive ca in which the alphabet is the cyclic group z m z and the local rules are linear combinations with coefficients in z m z in our scenario the alphabet is any finite abelian group and the global rule is any additive map this class of ca strictly contains the class of linear ca over z m z n i e with the local rule defined by n n matrices with elements in z m z which in turn strictly contains the class of linear ca over z m z in order to further emphasize that finite abelian groups are more expressive than z m z we prove that contrary to what happens in z m z there exist additive ca over suitable finite abelian groups which are roots with arbitrarily large indices of the shift map as a relevant consequence of our results we have that for additive d dimensional ca over a finite abelian group ergodic mixing weak ergodic mixing ergodicity topological mixing weak topological mixing topological total transitivity and topological transitivity are all equivalent properties as a corollary we see that invertible transitive additive ca are isomorphic to bernoulli shifts furthermore we prove that surjectivity implies openness for additive d dimensional ca over a finite abelian group hence we get that topological transitivity is equivalent to the well known devaney notion of chaos when d 1 moreover we provide a first characterization of strong transitivity for additive ca which we suspect to be true also for the general case

cohomology of regular embeddings we give in this paper an explicit description of the rational cohomology ring of a complete symmetric variety or regular compactification of an algebraic symmetric variety the motivation for this computation comes from the desire to describe an explicit schubert calculus for these varieties which have been considered since the work of chasles and schubert in several special cases the technique we use is the result of our attempts to understand better the work of jurkiewicz and danilov in the case of torus embeddings the combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology the key point is that the reisner stanley algebra of the torus embedding coincides with its equivariant cohomology ring we consider a class of spaces regular embeddings that contains both complete symmetric varieties and smooth torus embeddings by suitably generalizing the notion of reisner stanley algebra we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way

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

recovery of the absorption coefficient from diffused reflected light using a discrete diffusive model an algorithm based on a discrete model for photon migration is proposed for recovering the absorption coefficient in a semi infinite slab a pulsed light source is modeled and only reflected signals are used in the reconstruction the diffusion coefficient and speed of light are assumed constant and known the algorithm is based on the result that the earliest reflected signals obtained from a particular voxel depend either in a linear or in a quadratical fashion on the square of the absorption coefficient in that voxel by solving the discrete model in the forward direction in a region where the absorption coefficient has been calculated already the absorption coefficients in the voxels on the boundary of the region are calculated using interpolation results show that simple absorptive regions can be accurately recovered

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

generalized sub shifts in elementary cellular automata the strange case of chaotic rule 180 abstract we study the dynamical behavior of elementary cellular automaton 180 this rule gives rise to a global dynamics on the phase space of all one dimensional bi infinite configurations which is devaney topologically chaotic the dense sub dynamical system of configurations in background of 0s is a generalized sub shift i e multiple sub shift whose multiplicity constant depends on the initial configuration this sub dynamical system is deeply stable in the sense that the null configuration is a global attractor but with some components of the chaotic behaviour transitivity and unpredictability the dense sub dynamical system of configurations in background of 1s is a fractal like system with strongly chaotic components expansivity

on the limit sets of cellular automata the limit sets of cellular automata defined by wolfram play an important role in applications of cellular automata to complex systems a number of results on limit sets are proved considering both finite and infinite configurations of cellular automata the main concern of this paper is with testing membership and essential emptiness of limit sets for linear and two dimensional cellular automata

pre expansivity in cellular automata we introduce the property of pre expansivity for cellular automata ca it is the property of being expansive on asymptotic pairs of configurations i e configurations that differ in only finitely many positions pre expansivity therefore lies between expansivity and pre injectivity two important notions of ca theory we show that there exist one dimensional positively pre expansive cas which are not positively expansive and they can be chosen reversible while positive expansivity is impossible for reversible cas we show however that no bi dimensional ca which is linear over an abelian group can be pre expansive we also consider the finer notion of k expansivity expansivity over pairs of configurations with exactly k differences and show examples of linear ca in dimension 2 and on the free group that are k expansive depending on the value of k whereas no positively expansive ca exists in this setting

ergodicity transitivity and regularity for linear cellular automata over z m 1 we study the dynamical behavior of d dimensional linear cellular automata over zm we provide an easy to check necessary and sufficient condition for a d dimensional linear cellular automata over zm to be ergodic and topologically transitive as a byproduct we get that for linear cellular automata ergodicity is equivalent to topological transitivity finally we prove that for 1 dimensional linear cellular automata over zm regularity denseness of periodic orbits is equivalent to surjectivity

classification of sofic projective subdynamics of multidimensional shifts of finite type motivated by hochman s notion of subdynamics of a z subshift 8 we define and examine the projective subdynamics of z shifts of finite type sfts where we restrict not only the action but also the phase space we show that any z sofic shift of positive entropy is the projective subdynamics of a z2 z sft and that there is a simple condition characterizing the class of zero entropy z sofic shifts which are not the projective subdynamics of any z2 sft we define notions of stable and unstable subdynamics in analogy with the notions of stable and unstable limit sets in cellular automata theory and discuss how our results fit into this framework one dimensional strictly sofic shifts of positive entropy admit both a stable and an unstable realization whereas a particular class of zero entropy z sofics only allows for an unstable realization finally we prove that the union of z subshifts all of which are realizable in z sfts is again realizable when it contains at least two periodic points that the projective subdynamics of z2 sfts with the uniform filling property ufp are always sofic and we exhibit a class of non sofic z subshifts which are not the subdynamics of any z sft

multidimensional cellular automata closing property quasi expansivity and un decidability issues in this paper we study the dynamics of d dimensional cellular automata ca by considering them as one dimensional 1d ca along some direction slicing constructions these constructions allow to give the d dimensional version of important notions as 1d closing property and lift well known one dimensional results to the d dimensional settings indeed like in one dimensional case closing d dimensional ca have jointly dense periodic orbits and biclosing d dimensional ca are open by the slicing constructions we further prove that for the class of closing d dimensional ca surjectivity implies surjectivity on spatially periodic configurations old standing open problem we also deal with the decidability problem of the d dimensional closing by extending the kari s construction from 31 based on tilings we prove that the two dimensional and then d dimensional closing property is undecidable in such a way we add one further item to the class of dimension sensitive properties i e properties that are decidable in dimension 1 and are undecidable in higher dimensions it is well known that there are not positively expansive ca in dimension 2 and higher as a meaningful replacement we introduce the notion of quasi expansivity for d dimensional ca which shares many global properties in the d dimensional settings with the 1d positive expansivity we also prove that for quasi expansive d dimensional ca the topological entropy which is an uncomputable property for general ca has infinite value in a similar way as quasi expansivity the notions of quasi sensitivity and quasi almost equicontinuity are introduced and studied

quasi periodic configurations and undecidable dynamics for tilings infinite words and turing machines we describe turing machines tilings and infinite words as dynamical systems and analyze some of their dynamical properties it is known that some of these systems do not always have periodic configurations we prove that they always have quasi periodic configurations and we quantify quasi periodicity we then study the decidability of dynamical properties for these systems in analogy to rice s theorem for computable functions we derive a theorem that characterizes dynamical system properties that are undecidable as an illustration of this result we prove that topological entropy is undecidable for turing machines and for tilings

the one dimensional exactly 1 cellular automaton replication periodicity and chaos from finite seeds in the exactly 1 cellular automaton also known as rule 22 every site of the one dimensional lattice is either in state 0 or in state 1 and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three site neighborhood at the current time we analyze this rule started from finite seeds i e those initial configurations that have only finitely many 1 s three qualitatively different types of evolution are observed replication periodicity and chaos we focus on rigorous results assisted by algorithmic searches for the first two behaviors in particular we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds some empirical observations about chaotic seeds are also presented

markovian solutions of inviscid burgers equation for solutions of inviscid forceless one dimensional burgers equation with random initial condition it is heuristically shown that a stationary feller markov property with respect to the space variable at some time is conserved at later times and an evolution equation is derived for the infinitesimal generator previously known explicit solutions such as frachebourg martin s white noise initial velocity and carraro duchon s levy process intrinsic statistical solutions including brownian initial velocity are recovered as special cases

on the bernoulli automorphism of reversible linear cellular automata this investigation studies the ergodic properties of reversible linear cellular automata over z m for m n we show that a reversible linear cellular automaton is either a bernoulli automorphism or non ergodic this gives an affirmative answer to an open problem proposed by pivato 20 for the case of reversible linear cellular automata

on computing the entropy of cellular automata we study the topological entropy of a particular class of dynamical systems cellular automata the topological entropy of a dynamical system x f is a measure of the complexity of the dynamics of f over the space x the problem of computing or even approximating the topological entropy of a given cellular automata is algorithmically undecidable ergodic theory dynamical systems 12 1992 255 in this paper we show how to compute the entropy of two important classes of cellular automata namely linear and positively expansive cellular automata in particular we prove a closed formula for the topological entropy of d dimensional d 1 linear cellular automata over the ring zm m 2 and we provide an algorithm for computing the topological entropy of positively expansive cellular automata

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

equivariant cohomology of the wonderful group compactification in this paper we compute the rational g g equivariant cohomology ring of the so called wonderful compactification of g see c de concini c procesi complete symmetric varieties in invariant theory montecatini 1982 in lecture notes in math vol 996 springer verlag berlin 1983 pp 1 44 this is obtained as an application of the results in e bifet c de concini c procesi cohomology of regular embeddings adv math 82 1 1990 1 34 e strickland computing the equivariant cohomology of group compactifications math ann 291 2 1991 275 280 by a careful analysis of the relevant stanley reisner systems

the topological entropy of invertible cellular automata this paper is concerned with the topological entropy of invertible one dimensional linear cellular automata i e the maps t f r r z m z z m z which are given by t f r r x y n n y n f x n r x n r email protected i r email protected ix n i modm x x n n email protected z m z and f z m 2 r 1 z m over the ring z m m 2 by means of algorithm defined by d amica et al on computing the entropy of cellular automa theoret comput sci 290 2003 1629 1646 we prove that if a one dimensional linear cellular automata is invertible then the topological entropies of this cellular automata and its inverse are equal

exact theory for the self similarity and decay of homogeneous turbulence it is shown that space time dilatation invariance x 1x t 2t with concomitant transformations for dependent variables and linearity of the equation engender an exact time explicit generic form for the solution applicable to freely decaying homogeneous incompressible fluid turbulence this solution features a summation over mutually independent dynamical modes labeled by the dilatation scaling index n 1 without the assumption of isotropy nor introduction of a closure approximation procedure the theory provides an explanation for the experimentally observed self similarity of the correlation tensors and the decay laws u x t 2 t n for the different types and decay stages of homogeneous turbulence

the reisner stanley system and equivariant cohomology for a class of wonderful varieties in this paper we are going to determine the equivariant cohomology of the wonderful compactification of a symmetric variety g h and its equivariant ring of conditions under the assumption that rk h rk g h rk g

on the dynamics and recursive properties of multidimensional symbolic systems we study the sub dynamics of multidimensional shifts of finite type and sofic shifts and the action of cellular automata on their limit sets such a subaction is always an effective dynamical system i e it is isomorphic to a subshift over the cantor set the complement of which can be written as the union of a recursive sequence of basic sets our main result is that to varying degrees this recursive theoretic condition is also sufficient we show that the class of expansive subactions of multidimensional sofic shifts is the same as the class of expansive effective systems and that a general effective system can be realized modulo a small extension as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set after removing a dynamically trivial set as applications we characterize in terms of their computational properties the numbers which can occur as the entropy of cellular automata and construct sfts and cas with various interesting properties

number conserving cellular automata ii dynamics in this second part we study the dynamics of the number conserving cellular automata we give a classification which focuses on pattern divergence and chaoticity moreover we prove that in the case of number conserving cellular automata surjectivity is equivalent to regularity as a byproduct we obtain a strong characterization of the class of cellular automata with bounded evolutions on finite configurations

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

the inviscid burgers equation with brownian initial velocity the law of the hopf cole solution of the inviscid burgers equation with brownian initial velocity is made explicit as examples of applications we investigate the smoothness of the solution the statistical distribution of the shocks we determine the exact hausdorff function of the lagrangian regular points and investigate the existence of lagrangian regular points in a fixed borel set

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

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

some mathematical results for cellular automata abstract many mathematical concepts from different branches of mathematics are united to study some properties of cellular automata ca periodicity and chaos in the sense of sensitive dependence on initial conditions are studied for 1 dimensional ca the method of the characteristic polynomial of the evolution matrix is applied then this study is generalized to extended nonlocal ca and to higher dimensions three state ca and nonlinear ca are also studied some results for the deterministic limits of domany kinzel and bagnoli et al models are derived

near invariance under dynamic scaling for navier stokes equations in critical spaces a probabilistic approach to regularity problems we make a detailed comparison between the navier stokes equations and their dynamically scaled counterpart the so called leray equations the navier stokes equations are invariant under static scaling transforms but are not generally invariant under dynamic scaling transforms we will study how close they can be brought together using the critical dependent variables and discuss the implications on the regularity problems assuming that the navier stokes equations written in the vector potential have a solution that blows up at t 1 we derive the leray equations by dynamic scaling we observe 1 the leray equations have only one term extra on top of those of the navier stokes equations 2 we can recast the navier stokes equations as a wiener path integral and the leray equations as another ornstein uhlenbeck path integral by the maruyama girsanov theorem both equations take the identical form modulo the maruyama girsanov density which is valid up to t 2 2 by the novikov condition 3 the global solution of the leray equations is given by a finite dimensional projection r of a functional of an ornstein uhlenbeck process and a probability measure if r remains smooth beyond t 1 under an absolute continuous change of the probability measure we can rule out finite time blowup by contradiction there are two cases a r given by a finite number of wiener integrals and b otherwise ruling out blowup in a is straightforward for b a condition based on a limit passage in the picard iterations is identified for such a contradiction to come out the whole argument equally holds in r d for any d 2

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

a family of sand automata we study some dynamical properties of a family of two dimensional cellular automata those that arise from an underlying one dimensional sand automaton whose local rule is obtained using a latin square we identify a simple sand automaton whose local rule is algebraic and classify this automaton as having equicontinuity points but not being equicontinuous we also show that it is not surjective we generalise some of these results to a wider class of sand automata

computation theoretic aspects of cellular automata abstract cellular automata may be viewed as computational models of complex systems they also can be seen as continuous functions on a particular family of compact metric spaces this paper surveys results largely motivated by an attempt to answer questions posed by the latter approach by means of techniques from the former

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

realization problems for nonuniform cellular automata given a finite set of local rules the sequences built from them give a set of nonuniform cellular automata it is known 1 that for any such set of local rules the subset of nonuniform cellular automata built from them which give a surjective global map is in fact a sofic subshift for the injective sequences this set is only automatic accepted by a buchi automaton and it is not necessarily open or closed we prove the first corresponding realization result we provide for any sft a set of local rules such that the surjective injective nonuniform cellular automata over them are precisely the sequences in the sft up to a renaming of symbols in fact we prove that sfts are exactly the subshifts which are the set of injective sequences of nonuniform cellular automata for some set of local rules we also consider surjectivity of subclasses of nonuniform cellular automata and realizability questions for other properties in particular number conservation and chain transitivity

statistical mechanics of surjective cellular automata reversible cellular automata are seen as microscopic physical models and their states of macroscopic equilibrium are described using invariant probability measures we establish a connection between the invariance of gibbs measures and the conservation of additive quantities in surjective cellular automata namely we show that the simplex of shift invariant gibbs measures associated to a hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the hamiltonian a special case is the well known invariance of the uniform bernoulli measure under surjective cellular automata which corresponds to the conservation of the trivial hamiltonian as an application we obtain results indicating the lack of non trivial gibbs or markov invariant measures for sufficiently chaotic cellular automata we discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium and pose several open questions as an aside a shift invariant pre image of a gibbs measure under a pre injective factor map between shifts of finite type turns out to be always a gibbs measure we provide a sufficient condition under which the image of a gibbs measure under a pre injective factor map is not a gibbs measure we point out a potential application of pre injective factor maps as a tool in the study of phase transitions in statistical mechanical models

a second step towards complexity theoretic analogs of rice s theorem abstract rice s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable borchert and stephan 1997 initiated the search for complexity theoretic analogs of rice s theorem in particular they proved that every nontrivial counting property of circuits is up hard and that a number of closely related problems are spp hard the present paper studies whether their up hardness result itself can be improved to spp hardness we show that their up hardness result cannot be strengthened to spp hardness unless unlikely complexity class containments hold nonetheless we prove that every p constructibly bi infinite counting property of circuits is spp hard we also raise their general lower bound from unambiguous nondeterminism to constant ambiguity nondeterminism

replication in one dimensional cellular automata abstract in a cellular automaton ca replication is the ability to indefinitely generate copies of a finite collection of patterns starting from finite seeds a transparent feature of additive ca replication mechanisms are less clear in the absence of additivity this paper investigates such dynamics through several examples for the 1 or 2 rule and its generalizations replication is inevitable and we investigate self organization properties in the perturbed exactly 1 rule we study frequency of replicators and the new phenomenon is called quasireplication the last ca is the extended 1 or 3 rule which allows for replication on different backgrounds we employ a mixture of rigorous and empirical techniques

chaos and ergodicity are decidable for linear cellular automata over z mz n abstract we prove that important properties describing complex behaviours as ergodicity chaos topological transitivity and topological mixing are decidable for one dimensional linear cellular automata lca over z m z n theorem 6 and corollary 7 a large and important class of cellular automata ca which are able to exhibit the complex behaviours of general ca and are used in applications in particular we provide a decidable characterization of ergodicity which is known to be equivalent to all the above mentioned properties in terms of the characteristic polynomial of the matrix associated with lca we stress that the setting of lca over z m z n with n 1 is more expressive gives rise to much more complex dynamics and is more difficult to deal with than the already investigated case n 1 the proof techniques from 23 25 used when n 1 for obtaining decidable characterizations of dynamical and ergodic properties can no longer be exploited when n 1 for achieving the same goal indeed in order to get the decision algorithm algorithm 1 we need to prove a non trivial result of abstract algebra theorem 5 which is also of interest in its own we also illustrate the impact of our results in real world applications concerning the important and growing domain of cryptosystems which are often based on one dimensional lca over z m z n with n 1 as a matter of facts since cryptosystems have to satisfy the so called confusion and diffusion properties ensured by ergodicity and chaos respectively of the involved lca algorithm 1 turns out to be an important tool for building chaotic ergodic one dimensional linear ca over z m z n and hence for improving the existing methods based on them

an exact representation of the space time characteristic functional of turbulent navier stokes flows with prescribed random initial states and driving forces by adopting a formal operator viewpoint the space time characteristic functional associated with navier stokes turbulence is expressed in terms of a linear operator acting on the space of functionals obtained by a simple similarity transformation of the local translation operator generated by the nonlinear terms in the navier stokes equation this operator is unitary with respect to the formal scalar product of functionals the equivalence of this operator representation to the functional integral representation of rosen is shown and for gaussian initial velocity and external force fields some consequences of this representation are presented

rice s theorem for the limit sets of cellular automata abstract rice s theorem is a well known result in the theory of recursive functions a corresponding theorem for cellular automata limit sets is proved all nontrivial properties of limit sets of cellular automata cas are shown undecidable the theorem remains valid even if only one dimensional cas are considered

inversion of circulant matrices over z m in this paper we consider the problem of inverting an n n circulant matrix with entries over zm we show that the algorithm for inverting circulants based on the reduction to diagonal form by means of fft has some drawbacks when working over zm we present three different algorithms which do not use this approach our algorithms require different degrees of knowledge of m and n and their costs range roughly from n log n log log n to n log2 n log log n log m operations over zm we also present an algorithm for the inversion of finitely generated bi infinite toeplitz matrices the problems considered in this paper have applications to the theory of linear cellular automata

formal languages and global cellular automaton behavior abstract a connection is made between the description of one dimensional cellular automata as dynamical systems acting on a compact metric space and as computational systems acting on strings operators are defined which determine a bijective correspondence between subshifts of infinite configurations and languages of finite strings under this correspondence families of languages give rise to families of subshifts regular languages correspond to sofic subshifts families of subshifts corresponding to context free context sensitive and recursively enumerable languages are introduced and their closure properties under forward and backward cellular automaton images studied