information theoretical analysis of the statistical dependencies among three variables applications to written language we develop the information theoretical concepts required to study the statistical dependencies among three variables some of such dependencies are pure triple interactions in the sense that they cannot be explained in terms of a combination of pairwise correlations we derive bounds for triple dependencies and characterize the shape of the joint probability distribution of three binary variables with high triple interaction the analysis also allows us to quantify the amount of redundancy in the mutual information between pairs of variables and to assess whether the information between two variables is or is not mediated by a third variable these concepts are applied to the analysis of written texts we find that the probability that a given word is found in a particular location within the text is not only modulated by the presence or absence of other nearby words but also on the presence or absence of nearby pairs of words we identify the words enclosing the key semantic concepts of the text the triplets of words with high pairwise and triple interactions and the words that mediate the pairwise interactions between other words

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

on the role of pattern matching in information theory in this paper the role of pattern matching in information theory is motivated and discussed we describe the relationship between a pattern s recurrence time and its probability under the data generating stochastic source we show how this relationship has led to great advances in universal data compression we then describe nonasymptotic uniform bounds on the performance of data compression algorithms in cases where the size of the training data that is available to the encoder is not large enough so as to yield the asymptotic compression the shannon entropy we then discuss applications of pattern matching and universal compression to universal prediction classification and entropy estimation

a note on the ziv lempel model for compressing individual sequences corresp the ziv lempel compression algorithm is a string matching and parsing approach to data compression the symbolwise equivalent for parsing models has been defined by rissanen and langdon and gives the same ideal codelength at the same cost in coding parameters by describing the context and coding parameter for each symbol an insight is provided into how the ziv lempel method achieves compression this treatment does not employ a probabilistic source for the data string the ziv lempel method effectively counts symbol instances within parsed phrases the coding parameter for each symbolwise context is determined by cumulative count ratios the code string length increase for a symbol y following substring s under the symbolwise equivalent is the log of the ratio of node counts in subtrees s and s cdot y of the ziv lempel parsing tree to demonstrate the symbolwise equivalent of the ziv lempel algorithm we extend the work of rissanen and langdon to incomplete parse trees the result requires the proper handling of the comma when one phrase is the prefix of another phrase

structuring the elementary components of graphs having a perfect internal matching graphs with perfect internal matchings are decomposed into elementary components and these components are given a structure reflecting the order in which they can be reached by external alternating paths it is shown that the set of elementary components can be grouped into pairwise disjoint families determined by the two way accessible relationship among them a family free is established by which every family member except the root has a unique father and mother identified as another elementary component and one of its canonical classes from which the given member is two way accessible it is proved that every member of the family is only accessible through a distinguished canonical class of the root by external alternating paths the families themselves are arranged in a partial order according to the order they can be covered by external alternating paths and a complete characterization of the graph s forbidden and impervious edges is elaborated

low density series expansions for directed percolation i a new efficient algorithm with applications to the square lattice a new algorithm for the derivation of low density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems numerical evidence shows that the computational complexity grows exponentially but with a growth factor which is much smaller than the growth factor of the previous best algorithm for bond site percolation on the directed square lattice the series has been extended to order 171 158 analysis of the series yields sharper estimates of the critical points and exponents

scaling behaviour of entropy estimates entropy estimation of information sources is highly non trivial for symbol sequences with strong long range correlations the rabbit sequence related to the symbolic dynamics of the nonlinear circle map at the critical point as well as the logistic map at the feigenbaum point is known to produce long memory tails for both dynamical systems the scaling behaviour of the block entropy of order n has been shown to increase log n in contrast to such probabilistic concepts we investigate the scaling behaviour of certain non probabilistic entropy estimation schemes suggested by lempel and ziv lz in the context of algorithmic complexity and data compression these are applied in a sequential manner with the scaling variable being the length n of the sequence we determine the scaling law for the lz entropy estimate applied to the case of the critical circle map and the logistic map at the feigenbaum point in a binary partition

string matching bounds via coding it is known that the length l x 1 n of the longest block appearing at least twice in a randomly chosen sample path of length n drawn from an i i d process is asymptotically almost surely equal to c log n where the constant c depends on the process a simple coding argument will be used to show that for a class of processes called the finite energy processes l x 1 n is almost surely upper bounded by c log n where c is a constant while the coding technique does not yield the exact constant c it does show clearly what is needed to obtain log n bounds

study of long range correlations in pc binary files we employ fractal techniques to investigate long range correlations in some commonly used pc binary files we find that strong long range correlations exist in most of those files and argue that traditional models of information theory are not appropriate for calculating the information content of computer files in the real world

directed compact percolation cluster size and hyperscaling exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice the corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained the work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice a duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve the moments of both distributions have critical exponents which satisfy a constant gap hypothesis with gap exponents nu 2 and delta 3 corresponding to the scaling length and scaling size respectively the usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by delta nu d nu perpendicular to where nu perpendicular to is the exponent corresponding to the cluster width and d is the number of transverse dimensions 1 for the square lattice

self avoiding walks in four dimensions logarithmic corrections we present simulation results for long n or 4000 self avoiding walks in four dimensions we find definite indications of logarithmic corrections but the data are poorly described by the asymptotically leading terms detailed comparisons are presented with renormalization group flow equations derived by direct renormalization and with the results of a field theoretic calculation

equation of state for directed percolation using field theoretic renormalization group methods we calculate the equation of state for non equilibrium systems belonging to the universality class of directed percolation to second order in by introducing a parametric representation the result can be written to this order in a very simple form we use our result to obtain a universal amplitude ratio to second order in

finite size scaling of directed percolation in the steady state recently considerable progress has been made in understanding finite size scaling in equilibrium systems here we study finite size scaling in non equilibrium systems at the instance of directed percolation dp which has become the paradigm of non equilibrium phase transitions into absorbing states above at and below the upper critical dimension we investigate the finite size scaling behavior of dp analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d dimensional hypercube of finite edge length l with periodic boundary conditions near the bulk critical point in particular we study the order parameter and its higher moments using renormalized field theory we derive finite size scaling forms of the moments in a one loop calculation moreover we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous binder cumulant in equilibrium systems and that in particular provides a new signature of the dp universality class to complement our analytical work we perform monte carlo simulations which confirm our analytical results

separation of trajectories and its relation to entropy for intermittent systems with a zero lyapunov exponent i t where is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points the mean of is determined by the infinite invariant density using semianalytical arguments we calculate the infinite invariant density for the pomeaumanneville map and with it we obtain excellent agreement between numerical simulation and theory we show that is equal to krengel s entropy and to the complexity calculated by the lempel ziv compression algorithm this generalized pesin s identity shows that and krengel s entropy are the natural generalizations of usual lyapunov exponent and entropy for these systems

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

logarithmic corrections to scaling in critical percolation and random resistor networks we study the critical behavior of various geometrical and transport properties of percolation in six dimensions by employing field theory and renormalization group methods we analyze fluctuation induced logarithmic corrections to scaling up to and including the next to leading order correction our study comprehends the percolation correlation function i e the probability that two given points are connected and some of the fractal masses describing percolation clusters to be specific we calculate the mass of the backbone the red bonds and the shortest path moreover we study key transport properties of percolation as represented by the random resistor network we investigate the average two point resistance as well as the entire family of multifractal moments of the current distribution

universal scaling behaviour of directed percolation and the pair contact process in an external field we consider the scaling behaviour of directed percolation and of the pair contact process with a conjugated field in particular we determine numerically the equation of state and show that both models are characterized by the same universal scaling function furthermore we derive the equation of state for the pair contact process within a mean field approach which again agrees with the mean field equation of state of the directed percolation universality class

long range effects in an elementary cellular automaton we present evidence that one of the elementary one dimensional cellular automata in the sense of wolfram rule 22 in wolfram s notation involves very complex long range effects similar to a critical phenomenon this is in contrast to superficial evidence that would suggest that this rule leads to fairly simple behavior

on a representation of tree automata sequential compositions of tree automata called selective products are introduced as a connection of processors computing functions decidability results are established for the unary case these results are related to classical automata theory

prediction and entropy of nonlinear dynamical systems and symbolic sequences with lro abstract following shannon we introduce higher order entropies and derive dynamic entropies the nth order dynamic entropy conditional entropy is a measure of the uncertainty of the next state which follows after the observation of n foregoing states the asymptotic behaviour of the dynamic entropies at large n is studied for several nonlinear model systems and for symbolic sequences with long range order lro for example we investigate 1d maps texts dna strings and time series it is shown that the existence of long correlations improves the possibility of predictions characteristic scaling laws for the higher order shannon entropies and the conditional entropies are derived and a new interpolation formula is tested finally instead of the dynamic entropies which yield mean values of the uncertainty predictability we investigate the local values of the uncertainty predictability

universal scaling behavior of directed percolation around the upper critical dimension in this work we consider the steady state scaling behavior of directed percolation around the upper critical dimension in particular we determine numerically the order parameter its fluctuations as well as the susceptibility as a function of the control parameter and the conjugated field additionally to the universal scaling functions several universal amplitude combinations are considered we compare our results with those of a renormalization group approach

pointwise redundancy in lossy data compression and universal lossy data compression we characterize the achievable pointwise redundancy rates for lossy data compression at a fixed distortion level pointwise redundancy refers to the difference between the description length achieved by an nth order block code and the optimal nr d bits for memoryless sources we show that the best achievable redundancy rate is of order o spl radic n in probability this follows from a second order refinement to the classical source coding theorem in the form of a one sided central limit theorem moreover we show that along almost any source realization the description lengths of any sequence of block codes operating at distortion level d exceed nr d by at least as much as c spl radic nloglogn infinitely often corresponding direct coding theorems are also given showing that these rates are essentially achievable the above rates are in sharp contrast with the expected redundancy rates of order o log n reported by various authors our approach is based on showing that the compression performance of an arbitrary sequence of codes is essentially bounded below by the performance of shannon s random code we obtain partial generalizations of the above results for arbitrary sources with memory and we prove lossy analogs of barron s lemma barron 1985

average profile and limiting distribution for a phrase size in the lempel ziv parsing algorithm consider the parsing algorithm developed by lempel and ziv 1978 that partitions a sequence of length n into variable phrases blocks such that a new block is the shortest substring not seen in the past as a phrase in practice the following parameters are of interest number of phrases the size of a phrase the number of phrases of given size and so forth in this paper we focus on the size of a randomly selected phrase and the average number of phrases of a given size the so called average profile of phrase sizes these parameters can be efficiently analyzed through a digital search tree representation for a memoryless source with unequal probabilities of symbols generation the so called asymmetric bernoulli model we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed in terms of digital search trees we prove the normal limiting distribution of the typical depth i e the length of a path from the root to a randomly selected node the latter finding is proved by a technique that belongs to the toolkit of the analytical analysis of algorithms and it seems to be novel in the context of data compression

number of first passage times as a measurement of information for weakly chaotic systems we consider a general class of maps of the interval having lyapunov subexponential instability xt x0 exp t x0 t where t grows sublinearly as t we outline here a scheme j stat phys 154 988 2014 whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate t this matching approach is based on the infinite measure property of such systems we show that the average information that is necessary to record without ambiguity a trajectory of the system tends to t suitably extending the kolmogorov sinai entropy and pesin s identity for such systems information behaves like a random variable for random initial conditions its statistics obeying a universal mittag leffler law we show that for individual trajectories information can be accurately inferred by the number of first passage times through a given turbulent phase space cell this enables us to calculate far more efficiently lyapunov exponents for such systems lastly we also show that the usual renewal description of jumps to the turbulent cell usually employed in the literature does not provide the real number of entrances there our results are supported by exhaustive numerical simulations

long range interactions in sequences of human behavior measures derived from the thermodynamic formalism for dynamical systems are applied to human behavior to determine the degree of interaction within sequences of choices sequences of 500 binary choices generated by 22 human subjects are analyzed using autocorrelation and mutual information functions as well as the fluctuation spectrum of local dynamical entropies or local spatial scaling exponents the main results are as follows 1 choices generated by the subjects are interdependent on short midrange and large scales relative to the 500 choices generated in the paradigm 2 human behavior is characterized by large fluctuations of dynamical and geometrical choice characteristics and 3 these geometric and dynamical characteristics can be related to a few competing principles s1063 651x 97 01203 8

finite size scaling of directed percolation above the upper critical dimension critical phenomena such as second order phase transitions are characterized by singularities causing a discontinuous behavior of various quantities at the transition point e g the speciflc heat the susceptibility and the correlation length these singularities are described by power laws deflning the well known critical exponents studying the phase transition of a given system the aim of investigation is to identify the set of critical exponents which characterizes together with certain universal scaling functions the so called universality class since most systems are not analytically tractable their critical behavior is often investigated via numerical methods for example monte carlo simulations or transfer matrix calculations in these cases the obtained data are limited to flnite system sizes therefore flnite size scaling fss is widely used to extrapolate to the behavior of the inflnite systems in particular fss is an e cient method

maximal repetition and zero entropy rate maximal repetition of a string is the maximal length of a repeated substring this paper investigates maximal repetition of strings drawn from stochastic processes strengthening previous results two new bounds for the almost sure growth rate of maximal repetition are identified an upper bound in terms of conditional renyi entropy of order gamma 1 given a sufficiently long past and a lower bound in terms of unconditional shannon entropy gamma 1 both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time we also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity in particular it is shown that a power law logarithmic growth of maximal repetition with respect to the string length recently observed for texts in natural language may hold only if the conditional renyi entropy rate given a sufficiently long past equals zero according to this observation natural language cannot be faithfully modeled by a typical hidden markov process which is a class of basic language models used in computational linguistics

scaling of waves in the bak tang wiesenfeld sandpile model n with aen ae0 n a similar ansatzes should according to this view hold for the number of topplings s which difiers from a because sites can topple more than once in an avalanche and for the radius and duration of avalanches but recent inves tigations 10 12 show that the two dimensional btw model may be characterized by a multifractal behavior where difierent moments of a are governed by exponents aen which are not linear in n and are indeed not related to each other for difierent n difierent reasons for this have been proposed in 10 11 and in 12 notice that multi fractality of avalanches can be proven for certain variants of the 1 d sandpile model 14 deviations from pure power laws had been seen already in early simulations but were usually interpreted as fl nite size efiects due to avalanches which touch the bound ary of the lattice to illustrate that this is most likely not true and that there is a real problem with simple scaling we show in fig 1 the ratio ps x l pa x l of the integrated distributions ps x l r 1 x dx 0 ps x 0 l for d 2 and for difierent values of l in these simula tions cylindrical boundary conditions were used open at y l 2 and periodic at x 0 l and data were col lected only for avalanches starting at y 0 in this way we hope to have minimized boundary efiects also since we do not make separate flts to ps x l and pa x l we have none of the uncertainties inherent in such flts due

non universality of the absorbing state phase transition in a linear chain with power law diluted long range connections abstract in this work we study the critical behavior of the absorbing state phase transition exhibited by the contact process in a linear chain with power law diluted long range connections each pair of sites is connected with a probability p r that decays with the distance between the sites r as 1 r the model allows for a continuous tuning between a standard one dimensional chain with only nearest neighbor couplings to a fully connected network 0 we develop a finite size scaling analysis to obtain the critical point and a set of dynamical and stationary critical exponents for distinct values of the decay exponent 2 corresponding to finite average bond lengths and low average site connectivity data for the order parameter collapse over a universal curve when plotted after a proper rescaling of parameters we show further that the critical exponents depend on in the regime of diverging bond length fluctuations 3

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

second order noiseless source coding theorems shannon s celebrated source coding theorem can be viewed as a one sided law of large numbers we formulate second order noiseless source coding theorems for the deviation of the codeword lengths from the entropy for a class of sources that includes markov chains we prove a one sided central limit theorem and a law of the iterated logarithm

nonparametric entropy estimation for stationary processes and random fields with applications to english text we discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a doeblin type mixing condition the estimators are cesaro averages of longest match lengths and their consistency follows from a generalized ergodic theorem due to maker 1940 we provide examples of their performance on english text and we generalize our results to countable alphabet processes and to random fields

regular hilberg processes an example of processes with a vanishing entropy rate a regular hilberg process is a stationary process that satisfies both a hyperlogarithmic growth of maximal repetition and a power law growth of topological entropy which are a kind of dual conditions the hyperlogarithmic growth of maximal repetition has been experimentally observed for texts in natural language whereas the power law growth of topological entropy implies a vanishing shannon entropy rate and thus probably does not hold for natural language in this paper we provide a constructive example of regular hilberg processes which we call random hierarchical association rha processes our construction does not apply the standard cutting and stacking method for the constructed rha processes we demonstrate that the expected length of any uniquely decodable code is the orders of magnitude larger than the shannon block entropy of the ergodic component of the rha process our proposition supplements the classical result by shields concerning nonexistence of universal redundancy rates

tricritical directed percolation we consider a modification of the contact process incorporating higher order reaction terms the original contact process exhibits a non equilibrium phase transition belonging to the universality class of directed percolation the incorporated higher order reaction terms lead to a non trivial phase diagram in particular a line of continuous phase transitions is separated by a tricritical point from a line of discontinuous phase transitions the corresponding tricritical scaling behavior is analyzed in detail i e we determine the critical exponents various universal scaling functions as well as universal amplitude combinations

a preadapted universal switch distribution for testing hilberg s conjecture hilberg s conjecture about natural language states that the mutual information between two adjacent long blocks of text grows like a power of the block length the exponent in this statement can be upper bounded using the pointwise mutual information estimate computed for a carefully chosen code the bound is the better the lower the compression rate is but there is a requirement that the code be universal so as to improve a received upper bound for hilberg s exponent in this paper we introduce two novel universal codes called the plain switch distribution and the preadapted switch distribution generally speaking switch distributions are certain mixtures of adaptive markov chains of varying orders with some additional communication to avoid the so called catch up phenomenon the advantage of these distributions is that they both achieve a low compression rate and are guaranteed to be universal using the switch distributions we obtain that a sample of a text in english is non markovian with hilberg s exponent being 0 83 which improves over the previous bound 0 94 obtained using the lempel ziv code

deterministic soliton automata with a single exterior node abstract soliton valves have been proposed as molecular switching elements their mathematical model is the soliton graph and the soliton automaton introduced in dassow jurgensen 1991 in this paper we continue the study of the logic aspects of soliton switching there are two cases of special importance those of deterministic and of strongly deterministic soliton automata the former have deterministic state transitions in the usual sense of automaton theory the latter do not only have deterministic state transitions but also deterministic soliton paths a much stronger property as it turns out in dassow jurgensen 1991 a characterization of indecomposable strongly deterministic soliton automata was proved and it was shown that their transition monoids are primitive groups of permutations roughly speaking the main difference between deterministic and strongly deterministic soliton automata is that in the former the underlying soliton graphs may contain cycles of odd lengths while such cycles are not permitted in the soliton graphs belonging to strongly deterministic soliton automata the presence of such cycles renders the analysis quite complicated in the present paper we focus on a special class of deterministic soliton automata that of deterministic soliton automata whose underlying graphs have a single exterior node such graphs can be thought of as one kind of basic building blocks in the construction of soliton graphs we show that these soliton automata have at most two states this statement is not correct if the assumption of determinism is dropped

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

on the vocabulary of grammar based codes and the logical consistency of texts this paper presents a new interpretation for zipf mandelbrot s law in natural language which rests on two areas of information theory firstly we construct a new class of grammar based codes and secondly we investigate properties of strongly nonergodic stationary processes the motivation for the joint discussion is to prove a proposition with a simple informal statement if a text of length n describes n independent facts in a repetitive way then the text contains at least n log n different words under suitable conditions on n in the formal statement two modeling postulates are adopted firstly the words are understood as nonterminal symbols of the shortest grammar based encoding of the text secondly the text is assumed to be emitted by a finite energy strongly nonergodic source whereas the facts are binary iid variables predictable in a shift invariant way

self organized formulation of standard percolation phenomena abstract we consider standard percolation processes such as epidemic processes with or without immunization we show that their dynamics can be formulated so that they mimic self organized critical phenomena the wetting probability p needs not to be fine tuned to its critical value p c in order to arrive at criticality but it rather emerges as a singularity in some time dependent distribution on the one hand this casts doubts on the significance of self organized as opposed to ordinary criticality on the other hand it suggests very efficient algorithms where percolation problems are studied at several values of p in a single run as an example we apply such an algorithm to directed percolation in 2 1 dimensions where it allows a very precise determination of critical behavior

automata represented by products of soliton automata abstract soliton valves have been proposed as molecular switching elements a mathematical model of the logic aspects of soliton switching called soliton automaton was introduced by dassow and jurgensen they analysed the simulation power of strongly deterministic soliton automata with respect to transition semigroups here a more detailed analysis in terms of homomorphic and isomorphic representations with respect to various automaton products is given

estimating the information content of symbol sequences and efficient codes several variants of an algorithm for estimating shannon entropies of symbol sequences are presented they are all related to the lempel ziv algorithm 1976 1977 and to recent algorithms for estimating hausdorff dimensions the average storage and running times increase as n and nlog n respectively with the sequence length n these algorithms proceed basically by constructing efficient codes they seem to be the optimal algorithms for sequences with strong long range correlations e g natural languages an application to written english illustrates their use

how sampling reveals a process a series of observations xi 1 xi 2 xi 3 ldots is presented to us and at each time n when we have observed the first n of them we are called upon to give our guess for what stochastic process produced the data a universal scheme is given which for any bernoulli process not necessarily independent gives a sequence of processes that converges in a strong sense the bar d metric to the real process in addition to this main result many others are given which put it into proper perspective in particular it is shown that in a certain sense the class of bernoulli processes is the largest one for which such a universal scheme is possible

homoclinic tangencies generating partitions and curvature of invariant manifolds a method to compute the curvature of the unstable manifold is introduced and applied to henon map and duffing attractor showing that it allows the authors to locate the homoclinic tangencies and in turn to construct a generating partition the probability distribution of curvature values is investigated showing a power law decay finally the shape of the multifractal spectrum of effective liapunov exponents in nonhyperbolic systems is discussed

critical percolation in high dimensions we present monte carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 13 dimensions for d 6 they are preliminary for d 6 they are between 20 and 10 4 times more precise than the best previous estimates this was achieved by three ingredients i simple and fast hashing that allowed us to simulate clusters of millions of sites on computers with less than 500 mbytes memory ii a histogram method that allowed us to obtain information for several p values from a single simulation and iii a variance reduction technique that is especially efficient at high dimensions where it reduces error bars by a factor of up to 30 and more based on these data we propose a scaling law for finite cluster size corrections

crossover from isotropic to directed percolation we generalize the directed percolation dp model by relaxing the strict directionality of dp such that propagation can occur in either direction but with anisotropic probabilities we denote the probabilities as p p pd and p p 1 pd with p representing the average occupation probability and pd controlling the anisotropy the leath alexandrowicz method is used to grow a cluster from an active seed site we call this model with two main growth directions biased directed percolation bdp standard isotropic percolation ip and dp are the two limiting cases of the bdp model corresponding to pd 1 2 and pd 0 1 respectively in this work besides ip and dp we also consider the 1 2 pd 1 region extensive monte carlo simulations are carried out on the square and the simple cubic lattices and the numerical data are analyzed by finite size scaling we locate the percolation thresholds of the bdp model for pd 0 6 and 0 8 and determine various critical exponents these exponents are found to be consistent with those for standard dp we also determine the renormalization exponent associated with the asymmetric perturbation due to pd 1 2 6 0 near ip and confirm that such an asymmetric scaling field is relevant at ip

logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension we study numerically the dynamical properties of the btw model on a square lattice for various dimensions the aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean field exponents our results are consistent with the assumption that the scaling behavior of the four dimensional btw model is characterized by the mean field exponents with additional logarithmic corrections we benefit in our analysis from the exact solution of the directed btw model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension similar logarithmic corrections forms fit the numerical data for the four dimensional btw model strongly suggesting that the value of the upper critical dimension is four

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