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

symbolic dynamics and characterization of complexity abstract symbolic dynamics deals with robust properties of dynamics without digging into numbers for one dimensional maps there have been some new results periodic window theorem generalized composition rule construction of median words without using harmonics and antiharmonics incorporation of discrete symmetry to analyze symmetry breaking and symmetry restoration etc this new understanding has been applied to ordinary differential equations in particular the systematics of stable periodic solutions in the lorenz model has been shown to be given basically by symbolic dynamics of the cubic map symbolic dynamics may be used to extract invariant characteristics from time series its relation with grammatical complexity will also be commented on

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

the attractor basin portrait of a cellular automaton local space time structures such as domains and the intervening dislocations dominate a wide class of cellular automaton ca behavior for such spatially extended dynamics regular domains vicinities and attractors are introduced as organizing principles to identify the discretized analogs of attractors basins and separatrices structures used in classifying dissipative continuous state dynamical systems we describe the attractor basin portrait of nonlinear elementary ca rule 18 whose global dynamics is largely determined by a single regular attracting domain the latter s basin is analyzed in terms of subbasin and portal structures associated with particle annihilation the conclusion is that the computational complexity of such ca is more apparent than real transducer machines are constructed that automatically identify domain and dislocation structures in space time count the number of dislocations in a spatial pattern and implement an isomorphism between rule 18 and rule 90 we use a transducer to trace dislocation trajectories and confirm that in rule 18 isolated dislocation trajectories as well as a dislocation gas agree extremely well with the classical model of annihilating diffusive particles the ca efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space time

statistical analysis of a multiply twisted helix abstract a multiply twisted helix in which a one dimensional array of components is twisted producing a helix which if twisted itself produces a doubly twisted helix and so on in which there are couplings between adjacent rounds of helices was investigated using a cellular automaton and the spectral statistics of a quantum particle a useful distance between the components of the structure is measured using a cellular automaton the dynamics of which is simulated for a number of initial conditions in order to determine the degree of connectivity between the components the area of the sphere with radius r on the basis of the above distance is shown to be proportional to r2 when r is small the anderson transition was investigated based on the spectral statistics of a quantum particle in a multiply twisted helix with on site random potentials as the strength of the on site random potentials increases the anderson transition occurs both results support our conclusion that the number of dimensions is always three for every multiply twisted helix if the couplings between adjacent rounds are strong enough

universal singular sets in the calculus of variations for regular one dimensional variational problems ball and nadirashvilli introduced the notion of the universal singular set of a lagrangian l and established its topological negligibility this set is defined to be the set of all points in the plane through which the graph of some absolutely continuous l minimizer passes with infinite derivative motivated by tonelli s partial regularity results the question of the size of the universal singular set in measure naturally arises here we show that universal singular sets are characterized by being essentially purely unrectifiable that is they intersect most lipschitz curves in sets of zero length and any compact purely unrectifiable set is contained within the universal singular set of some smooth lagrangian with given superlinear growth this gives examples of universal singular sets of hausdorff dimension two filling the gap between previously known one dimensional examples and sychev s result that universal singular sets are lebesgue null we show that some smoothness of the lagrangian is necessary for the topological size estimate and investigate the relationship between growth of the lagrangian and the existence of pathological rectifiable pieces in the universal singular set we also show that tonelli s partial regularity result is stable in that the energy of a near minimizer u over the set where it has large derivative is controlled by how far u is from being a minimizer

recursive definition of global cellular automata mappings a method for a recursive definition of global cellular automata mappings is presented the method is based on a graphical representation of global cellular automata mappings for a given cellular automaton rule the recursive algorithm defines the change of the global cellular automaton mapping as the number of lattice sites is incremented a proof of lattice size invariance of global cellular automata mappings is derived from an approximation to the exact recursive definition the recursive definitions are applied to calculate the fractal dimension of the set of reachable states and of the set of fixed points of cellular automata on an infinite lattice

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

fractals from genomes exact solutions of a biology inspired problem abstract this is a review of a few recent papers with some new results added after a brief biological introduction a visualization scheme of the string composition of long dna sequences in particular of bacterial complete genomes will be described this scheme leads to a class of self similar and self overlapping fractals in the limit of infinitely long constituent strings the calculation of their exact dimensions and the counting of true and redundant avoided strings at different string lengths turn out to be one and the same problem we give exact solution of the problem using two independent methods the goulden jackson cluster method in combinatorics and the method of formal language theory

block transformations of one dimensional deterministic cellular automaton rules given a one dimensional cellular automaton rule f a block transform of f is a rule tbf such that there exists between the limit sets of both rules a bijection that replaces each site value x in a configuration belonging to the limit set of f with a string xb xx x of length b in the corresponding configuration belonging to the limit set of tbf if f is totalistic there exists a unique totalistic block transform and a large number of nontotalistic block transforms tbf if f is not totalistic there are no totalistic block transforms but there still exists a large number of non totalistic block transforms their number increases very rapidly with the block size b and the range of f the range of tbf may be any integer greater than or equal to rb many block transforms are studied the evolution according to rule tbf towards its limit set is discussed in terms of the annihilation of defects these defects are often simply related to the defects characterizing the evolution according to rule f

chaos and diffusion in deterministic cellular automata abstract it is shown that some of the deterministic one dimensional cellular automata studied recently by wolfram exhibit a kind of spontaneous symmetry breaking the associated kinks bloch walls perform annihilating diffusive walks for infinite lattices and large times in addition close analogies between these automata and strange attractors are exhibited by mapping the automata onto dynamical systems defined by iterations of the unit square

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

bulking ii classifications of cellular automata this paper is the second part of a series of two papers dealing with bulking a way to define quasi order on cellular automata by comparing space time diagrams up to rescaling in the present paper we introduce three notions of simulation between cellular automata and study the quasi order structures induced by these simulation relations on the whole set of cellular automata various aspects of these quasi orders are considered induced equivalence relations maximum elements induced orders etc providing several formal tools allowing to classify cellular automata

nonlinear continuous integrable hamiltonian couplings based on a kind of special non semisimple lie algebras a scheme is presented for constructing nonlinear continuous integrable couplings variational identities over the corresponding loop algebras are used to furnish hamiltonian structures for the resulting continuous integrable couplings the application of the scheme is illustrated by an example of nonlinear continuous integrable hamiltonian couplings of the akns hierarchy of soliton equations

grammatical complexity for two dimensional maps we calculate the grammatical complexity of the symbol sequences generated from the henon map and the lozi map using the recently developed methods to construct the pruning front when the map is hyperbolic the language of symbol sequences is regular in the sense of the chomsky hierarchy and the corresponding grammatical complexity takes finite values it is found that the complexity exhibits a self similar structure as a function of the system parameter and the similarity of the pruning fronts is discussed as an origin of such self similarity for non hyperbolic cases it is observed that the complexity monotonically increases as we increase the resolution of the pruning front

the multi component coupled burgers hierarchy of soliton equations and its multi component integrable couplings system with two arbitrary functions abstract a new loop algebra g m is constructed which is devoted to establishing an isospectral problem by making use of tu scheme the multi component coupled burgers hierarchy is obtained furthermore an expanding loop algebra f m of the loop algebra g m is presented it follows that the multi component integrable coupling system of the multi component coupled burgers hierarchy is worked out in general the method can be applied to other evolution equations hierarchies

thermodynamics of systems of finite sequences it is proposed that the hamming distance between bit strings plays the role of interaction energy the statistical sum and thermodynamic functions are calculated for the canonical ensemble of bit sequences the method is illustrated with an ensemble of two sequence systems which evolve according to cellular automata rules

percolation and disorder resistance in cellular automata we rigorously prove a form of disorder resistance for a class of one dimensional cellular automaton rules including some that arise as boundary dynamics of two dimensional solidification rules specifically when started from a random initial seed on an interval of length l with probability tending to one as l the evolution is a replicator that is a region of space time of density one is filled with a spatially and temporally periodic pattern punctuated by a finite set of other finite patterns repeated at a fractal set of locations on the other hand the same rules exhibit provably more complex evolution from some seeds while from other seeds their behavior is apparently chaotic a principal tool is a new variant of percolation theory in the context of additive cellular automata from random initial states

training neural nets with the reactive tabu search in this paper the task of training subsymbolic systems is considered as a combinatorial optimization problem and solved with the heuristic scheme of the reactive tabu search rts an iterative optimization process based on a modified local search component is complemented with a meta strategy to realize a discrete dynamical system that discourages limit cycles and the confinement of the search trajectory in a limited portion of the search space the possible cycles are discouraged by prohibiting i e making tabu the execution of moves that reverse the ones applied in the most recent part of the search the prohibition period is adapted in an automated way the confinement is avoided and a proper exploration is obtained by activating a diversification strategy when too many configurations are repeated excessively often the rts method is applicable to nondifferentiable functions is robust with respect to the random initialization and effective in continuing the search after local minima three tests of the technique on feedforward and feedback systems are presented

an algorithm to prune the area preserving henon map an explicit algorithm to provide the pruning front for the area preserving henon map is presented the procedure terminates within finitely many steps when the map has hyperbolic structure the only information required to specify the pruning front is a bifurcation diagram of homoclinic orbits and it is obtained by tracking orbits from the anti integrable limit the pruned region thus determined is used to construct the markov partition of the map and the topological entropy is evaluated as an application

constructing nonlinear discrete integrable hamiltonian couplings beginning with lax pairs from special non semisimple matrix lie algebras we establish a scheme for constructing nonlinear discrete integrable couplings discrete variational identities over the associated loop algebras are used to build hamiltonian structures for the resulting integrable couplings we illustrate the application of the scheme by means of an enlarged volterra spectral problem and present an example of nonlinear discrete integrable hamiltonian couplings for the volterra lattice equations

global properties of cellular automata cellular automata are discrete mathematical systems that generate diverse often complicated behavior using simple deterministic rules analysis of the local structure of these rules makes possible a description of the global properties of the associated automata a class of cellular automata that generate infinitely many aperiodic temporal sequences is defined as is the set of rules for which inverses exist necessary and sufficient conditions are derived characterizing the classes of nearest neighbor rules for which arbitrary finite initial conditions i evolve to a homogeneous state ii generate at least one constant temporal sequence

a note on inherent replication properties of local cellular automata transition functions local transition functions of elementary cellular automata show different tendencies to replicate parts of a configuration in a later generation this is seen as regularities in the time space diagram this replication depends on both the configuration and the local transition function a possibility to isolate the influence of the local transition function is shown

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

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

1 f fluctuation in the game of life abstract the fourier power spectra are examined for the game of life and its variants the results show that only the game of life exhibits fluctuation by means of observing patterns and measuring entropy and difference pattern spreading rate we conclude that the game of life only belongs to wolfram s class iv cellular automata which are thought to be capable of supporting universal computation this predicts a connection between fluctuation and computational universality

predecessors of cellular automata states ii pre images of finite sequences abstract graph theoretic techniques are used to derive algorithms which determine all pre images of any given finite sequence s under one dimensional cellular automata rules defined over z 2 it is also shown that for a given rule q the number of pre images of any finite sequence is determined by an algebraic form in the components of q this extends results of jen on the enumeration of pre images we also show that the set of k site rules for which a given sequence s has no pre images is determined by the solution set of a non linear diophantine equation

another theorem of classical solvability in small for one dimensional variational problems in this paper we suggest a direct method for studying local minimizers of one dimensional variational problems which naturally complements the classical local theory this method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable the basis of these results is a regularity theory a priori estimates and compactness in c1 for solutions of obstacle problems with sufficiently close obstacles in these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

a condition on the value function both necessary and sufficient for full regularity of minimizers of one dimensional variational problems we study two point lagrange problems for integrands l l t u v p b f u y l t u t u t dt inf a u c v c w1l a b in lv a avv b 3 under very weak regularity hypotheses l is holder continuous and locally el liptic on each compact subset of ilt x iltn x iltn we obtain when l is of superlinear growth in v a characterization of problems in which the minimizers of p are c1 regular for all boundary data this characterization involves the behavior of the value function s ilt x ilt x r x in ilt defined by s a a b b inf a f namely all minimizers for p are c1 regular in neighborhoods of a and b if and only if s is lipschitz continuous at a a b b consequently problems p possessing no singular minimizers are characterized in cases where not even a weak form of the euler lagrange equations is available for guidance full reg ularity results for problems where l is nearly autonomous nearly independent of u or jointly convex in u v are presented

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

upper bound on the products of particle interactions in cellular automata abstract particle like objects are observed to propagate and interact in many spatially extended dynamical systems for one of the simplest classes of such systems one dimensional cellular automata we establish a rigorous upper bound on the number of distinct products that these interactions can generate the upper bound is controlled by the structural complexity of the interacting particles a quantity which is defined here and which measures the amount of spatio temporal information that a particle stores along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata thereby elucidating why that approach is of general utility the upper bound is tested against several relatively complex domain particle cellular automata and found to be tight

computing fractal dimensions for additive cellular automata abstract efficient methods are given for finding the statistics of symbol blocks occuring in the graphs of additive cellular automata after large numbers of iterations in particular the methods yield fractal dimensions entropies and a recurrence relation for the number of nonzero symbols the methods are based on the construction of a nonnegative integer matrix a closely related to the automation the largest eigenvalue of a tells the fractal dimension while the corresponding eigenvector gives the relative frequencies of blocks

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

possible origin of power law behavior in n tuple zipf analysis in n tuple zipf analysis words are defined as strings of n digits and their normalized frequency of occurrence v is measured for a given text sequence of digits in the case of various non markovian sequences the probability density of the frequencies p v has a power law tail here we argue that a broad class of unbiased binary texts exhibiting a nonexponential distribution of cluster sizes can indeed yield a power law behavior of p v where we define clusters to be strings of identical digits we support this result by numerical studies of long range correlated sequences generated by three different methods that result in

excess action and broken characteristics for hamilton jacobi equations abstract we study propagation of singularities for hamilton jacobi equations s t h t x s 0 t x 0 r n by means of the excess lagrangian action and a related class of characteristics in a sense the excess action gauges how far a curve x t is from being action minimizing for a given viscosity solution s t x of the hamilton jacobi equation broken characteristics are defined as curves along which the excess action grows at the slowest pace possible in particular we demonstrate that broken characteristics carry the singularities of the viscosity solution

robust periodic solutions and evolution from seeds in one dimensional edge cellular automata we study one dimensional cellular automata ca with values 0 and 1 we assume that such ca are started from semi infinite configurations those that have 0 s to the left of some site and we focus on the identification of robust periodic solutions rps which when observed from the left edge of the light cone advance into any environment with positive velocity we then utilize rps and related concepts to analyze ca dynamics from seeds i e initial configurations with finitely many 1 s

long range correlations in chaotic cellular automata abstract one dimensional cellular automata of class 3 are studied as models for spatially extended chaotic systems their irreversible dynamics can generate weak but long range spatial correlations the one labelled 22 by wolfram is treated in an exemplary way its stationary state is approximated by a markov chain the regular grammar underlying the markov chain is chosen with care so that it includes a maximum amount of information on the stationary state the approximation can be performed without simulating the dynamics of the cellular automaton 22 and reproduces its long range correlations qualitatively their origin is explained intuitively and they are argued to be a frequent feature of cellular automata with more complicated rules

cellular automaton rules conserving the number of active sites this paper shows how to determine all of the unidimensional two state cellular automaton rules of a given number of inputs which conserve the number of active sites these rules have to satisfy a necessary and sufficient condition if the active sites are viewed as cells occupied by identical particles these cellular automaton rules represent evolution operators of systems of identical interacting particles whose total number is conserved some of these rules which allow motion in both directions mimic ensembles of one dimensional pseudorandom walkers numerical evidence indicates that the corresponding stochastic processes might be non gaussian

edge of chaos in rule changing cellular automata abstract we present a model of rule changing cellular automata ca to study dynamical structure of ca in detail in our model calculation the edge of chaos is observed on the basis of the source rule 22 which yields the most interesting behavior we find that the edge of chaos is generated through competition between two rule changing behaviors these rule changing behaviors are analyzed quantitatively by using the mutual information as a measure ofcomplexity it is noted that the mutual information has the maximum value near the edge of chaos because of competition

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

nonoccurrence of the lavrentiev phenomenon for many optimal control problems in this paper we study nonoccurrence of the lavrentiev phenomenon for a large class of nonconvex optimal control problems identified with the corresponding complete metric space of integrands mathcal m which satisfy a growth condition common in the literature and are lipschitzian on bounded sets we establish that for most elements of mathcal m in the sense of baire category the infimum on the full admissible class of trajectory control pairs is equal to the infimum on a subclass of trajectory control pairs whose controls are bounded by a certain constant

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

computational mechanics of cellular automata an example abstract we illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the space time behavior of elementary one dimensional cellular automaton rule 54 the dominant regular domain of the cellular automation is identified and a domain filter is constructed to locate and classify defects in the domain the primary particles are identified and a range of interparticle interactions is studied the deterministic equation of motion of the filtered space time behavior is derived filters of increasing sophistication are constructed for the efficient gathering of particle statistics and for the identification of higher level defects particle interactions and secondary domains we define the emergence time at which the space time behavior condenses into configurations consisting only of domains particles and particle interactions taken together these techniques serve as the basis for the investigation of pattern evolution and self organization in this representative system

a powerful approach to generate new integrable systems in this paper an effective algorithm to generate integrable systems is given as a result many new integrable equations are derived in a systematic way

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

existence and regularity in the small in the calculus of variations abstract a local existence theorem is proved for the basic problem in the calculus of variations that of minimizing l t x x dt over a class of functions x assuming given boundary conditions the lagrangian l is only assumed to be locally lipschitz and strictly convex in its x variable

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

a nonlinear integrable couplings of c kdv soliton hierarchy and its infinite conservation laws abstract a kind of nonlinear integrable couplings of c kdv soliton hierarchy is constructed then we present the infinitely many conservation laws for the nonlinear integrable couplings of the c kdv soliton hierarchy

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

a direct method for integrable couplings of td hierarchy a direct method for establishing integrable couplings is proposed in this paper as an example illustration integrable couplings of td hierarchy are obtained by constructing a suitable transformation of lax pairs and a new lie algebra

hierarchy of fuzzy cellular automata abstract we elaborate notion of cellular automata in the directions of synchroneity asynchroneity stationarity nonstationarity and fuzzify the principle features of cellular automata as cell state transition delay of transition and choice of a local transitions function through analyzing the properties in various combinations we divide a set of fuzzy cellular automata into 14 classes to pelucide the whole structure of the classes and precise relations between them we prove the complexity hierarchy

predecessor states for certain cellular automata evolutions this paper reports the solution of the problem of finding which inputs for a function of discrete variables will result in any specified output for the set of binomially determined nearest neighbor additive cellular automata defined on finite or half infinite sequences in computing the solution to this problem a process which is the discrete analog to backward integration is defined this process is determined in terms of an operator which exhibits an interesting period multiplying property

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

cellular automata and nonlinear dynamical models abstract a point of view is pursued in which cellular automata ca are viewed as a laboratory to investigate nonlinear dynamics we introduce an irreversible cellular automaton ica with minimal coupling which exhibit class 4 behavior periodic structures phases are studied along with their stability properties we observe topological conserved quantities and introduce a classification of structures via a topological number time reversal invariant cellular automata trca are also investigated we discuss stability of phases and use a concept of local entropy to measure the growth of chaos in slightly perturbed phases a classification of approach to chaos into 3 classes is proposed for trca

complexity of two dimensional patterns in dynamical systems such as cellular automata and iterated maps it is often useful to look at a language or set of symbol sequences produced by the system there are well established classification schemes such as the chomsky hierarchy with which we can measure the complexity of these sets of sequences and thus the complexity of the systems which produce them in this paper we look at the first few levels of a hierarchy of complexity for two or more dimensional patterns we show that several definitions of regular language or local rule that are equivalent in d 1 lead to distinct classes in d 2 we explore the closure properties and computational complexity of these classes including undecidability and l nl and np completeness results we apply these classes to cellular automata in particular to their sets of fixed and periodic points finite time images and limit sets we show that it is undecidable whether a ca in d 2 has a periodic point of a given period and that certain local lattice languages are not finite time images or limit sets of any ca we also show that the entropy of a d dimensional ca s finite time image cannot decrease faster than t d unless it maps every initial condition to a single homogeneous state

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

statistics and meta statistics in the concept of complexity abstract a crucial distinction between different classes of complexity measures is found to consist in their statistical structure while first order statistics always provides one or another kind of a measure of randomness second order meta statistics provides more subtle measures of complexity an example of a second order measure is constructed and applied to specific cases some formal and conceptual issues concerning the general relevance of meta statistical concepts of complexity are pointed out

phenomenology of nonlocal cellular automata dynamical systems with nonlocal connections have potential applications to economic and biological systems this paper studies the dynamics of nonlocal cellular automata in particular all two state three input nonlocal cellular automata are classified according to the dynamical behavior starting from random initial configurations and random wirings although it is observed that sometimes a rule can have different dynamical behaviors with different wirings the nonlocal cellular automata rule space is studied using a mean field parametrization which is ideal for the situation of random wiring nonlocal cellular automata can be considered as computers carrying out computation at the level of each component their computational abilities are studied from the point of view of whether they contain many basic logical gates in particular i ask the question of whether a three input cellular automaton rule contains the three fundamental logical gates two input rules and and or and one input rule not a particularly interesting edge of chaos nonlocal cellular automaton the rule 184 is studied in detail it is a system of coupled selectors or multiplexers it is also part of the fredkin s gate a proposed fundamental gate for conservative computations this rule exhibits irregular fluctuations of density large coherent structures and long transient times

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

hamiltonian and quasi hamiltonian structures associated with semi direct sums of lie algebras the trace variational identity is generalized to zero curvature equations associated with non semi simple lie algebras or equivalently lie algebras possessing degenerate killing forms an application of the resulting generalized variational identity to a class of semi direct sums of lie algebras in the akns case furnishes hamiltonian and quasi hamiltonian structures of the associated integrable couplings three examples of integrable couplings for the akns hierarchy are presented one hamiltonian and two quasi hamiltonian

invariant strings and pattern recognizing properties of one dimensional cellular automata a cellular automaton is a discrete dynamical system whose evolution is governed by a deterministic rule involving local interactions it is shown that given an arbitrary string of values and an arbitrary neighborhood size representing the range of interaction a simple procedure can be used to find the rules of that neighborhood size under which the string is invariant the set of nearestneighbor rules for which invariant strings exist is completely specified as is the set of strings invariant under each such rule for any automaton rule an associated filtering rule is defined for which the only attractors are spatial sequences consisting of concatenations of invariant strings a result is provided defining the rule of minimum neighborhood size for which an arbitrarily chosen string is the unique invariant string the applications of filtering rules to pattern recognition problems are discussed

computations on nondeterministic cellular automata this work concerns the trade offs between the dimension and the time and space complexity of computations on nondeterministic cellular automata we assume that the space complexity is the diameter of area in space involved in computation it is proved that 1 every nondeterministic cellular automata nca a of dimensionr computing a predicatepwith time complexityt n and space complexitys n can be simulated byr dimensional nca with time and space complexityo t1 r 1 sr r 1 and byr 1 dimensional nca with time and space complexityo t1 2 s wheretandsare functions constructible in time 2 for any predicatepand integerr 1 if a is a fastestr dimensional nca computingpwith time complexityt n and space complexitys n thent o s and 3 iftr pis the time complexity of a fastestr dimensional nca computing predicatepthentr 1 p o tr p 1 r r 1 2 tr 1 p o tr p 1 2 r similar problems for deterministic cellular automata ca are discussed

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