reductions of hidden information sources in all but special circumstances measurements of time dependent processes reflect internal structures and correlations only indirectly building predictive models of such hidden information sources requires discovering in some way the internal states and mechanisms unfortunately there are often many possible models that are observationally equivalent here we show that the situation is not as arbitrary as one would think we show that generators of hidden stochastic processes can be reduced to a minimal form and compare this reduced representation to that provided by computational mechanics the e machine on the way to developing deeper measure theoretic foundations for the latter we introduce a new two step reduction process the first step internal event reduction produces the smallest observationally equivalent algebra and the second internal state reduction removes algebra components that are redundant for optimal prediction for several classes of stochastic dynamical systems these reductions produce representations that are equivalent to e machines

nonclassical properties of semi coherent quantum states we study the properties of semi coherent quantum states introduced by mathews and eswaran in 1973 establishing relations between them and other families of nonclassical states such as displaced number states and multiphoton states we also analyse the photon number distribution photon statistics the wigner and marginal distribution functions and show the ways for possible further generalizations

application of computational mechanics to the analysis of natural data an example in geomagnetism we discuss how the ideal formalism of computational mechanics can be adapted to apply to a noninfinite series of corrupted and correlated data that is typical of most observed natural time series specifically a simple filter that removes the corruption that creates rare unphysical causal states is demonstrated and the concept of effective soficity is introduced we believe that computational mechanics cannot be applied to a noisy and finite data series without invoking an argument based upon effective soficity a related distinction between noise and unresolved structure is also defined noise can only be eliminated by increasing the length of the time series whereas the resolution of previously unresolved structure only requires the finite memory of the analysis to be increased the benefits of these concepts are demonstrated in a simulated times series by a the effective elimination of white noise corruption from a periodic signal using the expletive filter and b the appearance of an effectively sofic region in the statistical complexity of a biased poisson switch time series that is insensitive to changes in the word length memory used in the analysis the new algorithm is then applied to an analysis of a real geomagnetic time series measured at halley antarctica two principal components in the structure are detected that are interpreted as the diurnal variation due to the rotation of the earth based station under an electrical current pattern that is fixed with respect to the sun earth axis and the random occurrence of a signature likely to be that of the magnetic substorm in conclusion some useful terminology for the discussion of model construction in general is introduced

predictive information in a nonequilibrium critical model we propose predictive information that is information between a long past of duration t and the entire infinitely long future of a time series as a general order parameter to study phase transitions in physical systems independently of the underlying dynamics it can be used in particular to study nonequilibrium transitions and other exotic transitions where a simpler order parameter cannot be identified using traditional symmetry arguments as an example we calculate predictive information for a stochastic nonequilibrium dynamics problem that forms an absorbing state under a continuous change of a parameter the information at the transition point diverges as logt and we calculate the expression for a smooth crossover to t0 away from the transition

cumulative residual entropy a new measure of information in this paper we use the cumulative distribution of a random variable to define its information content and thereby develop an alternative measure of uncertainty that extends shannon entropy to random variables with continuous distributions we call this measure cumulative residual entropy cre the salient features of cre are as follows 1 it is more general than the shannon entropy in that its definition is valid in the continuous and discrete domains 2 it possesses more general mathematical properties than the shannon entropy and 3 it can be easily computed from sample data and these computations asymptotically converge to the true values the properties of cre and a precise formula relating cre and shannon entropy are given in the paper finally we present some applications of cre to reliability engineering and computer vision

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

predictive rate distortion for infinite order markov processes predictive rate distortion analysis suffers from the curse of dimensionality clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length the challenge is compounded for infinite order markov processes since conditioning on finite sequences cannot capture all of their past dependencies spectral arguments confirm a popular intuition algorithms that cluster finite length sequences fail dramatically when the underlying process has long range temporal correlations and can fail even for processes generated by finite memory hidden markov models we circumvent the curse of dimensionality in rate distortion analysis of finite and infinite order processes by casting predictive rate distortion objective functions in terms of the forward and reverse time causal states of computational mechanics examples demonstrate that the resulting algorithms yield substantial improvements

entrance and escape dynamics for the typical set according to the asymptotic equipartition property sufficiently long sequences of random variables converge to a set that is typical while the size and probability of this set are central to information theory and statistical mechanics they can often only be estimated accurately in the asymptotic limit due to the exponential growth in possible sequences here we derive a time inhomogeneous dynamics that constructs the properties of the typical set for all finite length sequences of independent and identically distributed random variables these dynamics link the finite properties of the typical set to asymptotic results and allow the typical set to be applied to small and transient systems the main result is a geometric mapping the triangle map relating sequences of random variables of length n to those of length n 1 we show that the number of points in this map needed to quantify the properties of the typical set grows linearly with sequence length despite the exponential growth in the number of typical sequences we illustrate the framework for the bernoulli process and the schlogl model for autocatalytic chemical reactions and demonstrate both the convergence to asymptotic limits and the ability to reproduce exact calculations

observable operator models for discrete stochastic time series a widely used class of models for stochastic systems is hidden markov models systems that can be modeled by hidden markov models are a proper subclass of linearly dependent processes a class of s

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

mixing ergodic and nonergodic processes with rapidly growing information between blocks we construct mixing processes over an infinite alphabet and ergodic processes over a finite alphabet for which shannon mutual information between adjacent blocks of length n grows as n where 0 1 the processes are a modification of nonergodic santa fe processes which were introduced in the context of natural language modeling the rates of mutual information for the latter processes are alike and also established in this paper as an auxiliary result it is shown that infinite direct products of mixing processes are also mixing

a partial ordering of sets making mean entropy monotone consider a state of a system with several subsystems the entropies of the reduced state on different subsystems obey certain inequalities provided there is an equivalence relation and a function measuring volumes or weights of subsystems the entropy per unit volume or unit weight the mean entropy then decreases with respect to an order relation of the subsystems defined in this paper in the context of statistical mechanics a lattice system is studied in detail and a decrease of mean energy is deduced for blow up sequences of regular and irregular octagons examples referring to quantum information with strict relations for mean entropies of subsystems in multipartite systems are also presented

inferring statistical complexity in the dripping faucet experiment abstract we have used topological e machine reconstruction to calculate statistical complexity associated to period five orbits in the dripping faucet experiment basic structures related to the periodic movement and to peripheral tracks in the corresponding directed graphs were identified

characterizing the dynamics of stochastic bistable systems by measures of complexity the dynamics of noisy bistable systems is analyzed by means of lyapunov exponents and measures of complexity we consider both the classical kramers problem with additive white noise and the case when the barrier fluctuates due to additional external colored noise in the case of additive noise we calculate the lyapunov exponents and all measures of complexity analytically as functions of the noise intensity or the mean escape time respectively for the problem of a fluctuating barrier the usual description of the dynamics with the mean escape time is not sufficient the application of the concept of measures of complexity allows us to describe the structures of motion in more detail most complexity measures indicate the value of the correlation time at which the phenomenon of resonant activation occurs with an extremum

analysis of the phase transition in the two dimensional ising ferromagnet using a lempel ziv string parsing scheme and black box data compression utilities in this work we consider information theoretic observables to analyze short symbolic sequences comprising time series that represent the orientation of a single spin in a two dimensional 2d ising ferromagnet on a square lattice of size l 2 128 2 for different system temperatures t the latter were chosen from an interval enclosing the critical point t mathrm c of the model at small temperatures the sequences are thus very regular at high temperatures they are maximally random in the vicinity of the critical point nontrivial long range correlations appear here we implement estimators for the entropy rate excess entropy i e complexity and multi information first we implement a lempel ziv string parsing scheme providing seemingly elaborate entropy rate and multi information estimates and an approximate estimator for the excess entropy furthermore we apply easy to use black box data compression utilities providing approximate estimators only for comparison and to yield results for benchmarking purposes we implement the information theoretic observables also based on the well established m block shannon entropy which is more tedious to apply compared to the first two algorithmic entropy estimation procedures to test how well one can exploit the potential of such data compression techniques we aim at detecting the critical point of the 2d ising ferromagnet among the above observables the multi information which is known to exhibit an isolated peak at the critical point is very easy to replicate by means of both efficient algorithmic entropy estimation procedures finally we assess how good the various algorithmic entropy estimates compare to the more conventional block entropy estimates and illustrate a simple modification that yields enhanced results

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

temporally stable coherent states for infinite well and poschl teller potentials this article is a direct illustration of a construction of coherent states which has been recently proposed by two of us jpg and jk we have chosen the example of a particle trapped in an infinite square well and also in poschl teller potentials of the trigonometric type in the construction of the corresponding coherent states we take advantage of the simplicity of the solutions which ultimately stems from the fact they share a common su 1 1 symmetry a la barut girardello many properties of these states are then studied both from mathematical and from physical points of view

killing vectors in asymptotically flat space times ii asymptotically translational killing vectors and the rigid positive energy theorem in higher dimensions we show that the borderline cases in the proof of the positive energy theorem for initial data sets on spin manifolds in dimensions n 3 are only possible for initial data arising from embeddings in minkowski space time

binless strategies for estimation of information from neural data we present an approach to estimate information carried by experimentally observed neural spike trains elicited by known stimuli this approach makes use of an embedding of the observed spike trains into a set of vector spaces and entropy estimates based on the nearest neighbor euclidean distances within these vector spaces l f kozachenko and n n leonenko probl peredachi inf 23 9 1987 using numerical examples we show that this approach can be dramatically more efficient than standard bin based approaches such as the direct method s p strong r koberle r r de ruyter van steveninck and w bialek phys rev lett 80 197 1998 for amounts of data typically available from laboratory experiments

correlated biased random walk with latency in one and two dimensions asserting patterned and unpredictable movement the correlated biased random walk with latency in one and two dimensions is discussed with regard to the portion of irreducible random movement and structured movement it is shown how a quantitative analysis can be carried out by using computational mechanics the stochastic matrix for both dynamics are reported latency introduces new states in the finite state machine description of the system in both dimensions allowing for a full nearest neighbor coordination in the two dimensional case complexity analysis is used to characterize the movement independently of the set of control parameters making it suitable for the discussion of other random walk models the complexity map of the system dynamics is reported for the two dimensional case

information trimming sufficient statistics mutual information and predictability from effective channel states one of the most basic characterizations of the relationship between two random variables x and y is the value of their mutual information unfortunately calculating it analytically and estimating it empirically are often stymied by the extremely large dimension of the variables one might hope to replace such a high dimensional variable by a smaller one that preserves its relationship with the other it is well known that either x or y can be replaced by its minimal sufficient statistic about y or x while preserving the mutual information while intuitively reasonable it is not obvious or straightforward that both variables can be replaced simultaneously we demonstrate that this is in fact possible the information x s minimal sufficient statistic preserves about y is exactly the information that y s minimal sufficient statistic preserves about x we call this procedure information trimming as an important corollary we consider the case where one variable is a stochastic process past and the other its future in this case the mutual information is the channel transmission rate between the channel s effective states that is the past future mutual information the excess entropy is the amount of information about the future that can be predicted using the past translating our result about minimal sufficient statistics this is equivalent to the mutual information between the forward and reverse time causal states of computational mechanics we close by discussing multivariate extensions to this use of minimal sufficient statistics

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

fractal renewal processes generate 1 f noise 1 mathit f mathit d noise occurs in an impressive variety of physical systems and numerous complex theories have been proposed to explain it we construct two relatively simple renewal processes whose power spectral densities vary as 1 mathit f mathit d i a standard renewal point process with 0d1 and ii a finite valued alternating renewal process with 0d2 the resulting event number statistics coincidence rates minimal coverings and autocorrelation functions are shown also to follow power law forms these fractal characteristics derive from interevent time probability density functions which themselves decay in a power law fashion a number of applications are considered trapping in amorphous semiconductors electronic burst noise movement in systems with fractal boundaries the digital generation of 1 mathit f mathit d noise and ionic currents in cell membranes

killing vectors in asymptotically flat space times i asymptotically translational killing vectors and the rigid positive energy theorem we study killing vector fields in asymptotically flat space times we prove the following result implicitly assumed in the uniqueness theory of stationary black holes if the conditions of the rigidity part of the positive energy theorem are met then in such space times there are no asymptotically null killing vector fields except if the initial data set can be embedded in minkowski space time we also give a proof of the nonexistence of nonsingular in an appropriate sense asymptotically flat space times that satisfy an energy condition and that have a null adm four momentum under conditions weaker than previously considered

recursion theory on the reals and continuous time computation abstract we define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers corresponding to a conceptual analog computer that operates in continuous time this class turns out to be surprisingly large and includes many functions which are uncomputable in the traditional sense we stratify this class of functions into a hierarchy according to the number of uses of the zero finding operator at the lowest level are continuous functions that are differentially algebraic and computable by shannon s general purpose analog computer at higher levels are increasingly discontinuous and complex functions we relate this hierarchy to the arithmetical and analytical hierarchies of classical recursion theory

photon added coherent states as nonlinear coherent states the states m defined as up to a normalization constant and where m is a non negative integer are shown to be the eigenstates of where is a nonlinear function of the number operator the explicit form of is constructed the eigenstates of this operator for negative values of m are introduced the properties of these states are discussed and compared with those of the state m the eigenstates corresponding to the positive and negative values of m are shown to be the result of nonunitarily deforming the number states m and 0 respectively

time series information and unsupervised learning of representations numerous control and learning problems face the situation where sequences of high dimensional highly dependent data are available but no or little feedback is provided to the learner which makes any inference rather challenging to address this challenge we formulate the following problem given a series of observations x 0 ldots x n coming from a large high dimensional space mathcal x find a representation function f mapping mathcal x to a finite space mathcal y such that the series f x 0 ldots f x n preserves as much information as possible about the original time series dependence in x 0 ldots x n we show that for stationary time series the function f can be selected as the one maximizing a certain information criterion that we call time series information some properties of this functions are investigated including its uniqueness and consistency of its empirical estimates implications for the problem of optimal control are presented

practical unitary simulator for non markovian complex processes stochastic processes are as ubiquitous throughout the quantitative sciences as they are notorious for being difficult to simulate and predict in this letter we propose a unitary quantum simulator for discrete time stochastic processes which requires less internal memory than any classical analogue throughout the simulation the simulator s internal memory requirements equal those of the best previous quantum models however in contrast to previous models it only requires a small finite dimensional hilbert space moreover since the simulator operates unitarily throughout it avoids any unnecessary information loss we provide a stepwise construction for simulators for a large class of stochastic processes hence directly opening the possibility for experimental implementations with current platforms for quantum computation the results are illustrated for an example process

automatic filters for the detection of coherent structure in spatiotemporal systems most current methods for identifying coherent structures in spatially extended systems rely on prior information about the form which those structures take here we present two approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures one local sensitivity filtering is a modification of the local lyapunov exponent approach suitable to cellular automata and other discrete spatial systems the other local statistical complexity filtering calculates the amount of information needed for optimal prediction of the system s behavior in the vicinity of a given point by examining the changing spatiotemporal distributions of these quantities we can find the coherent structures in a variety of pattern forming cellular automata without needing to guess or postulate the form of that structure we apply both filters to elementary and cyclical cellular automata eca and cca and find that they readily identify particles domains and other more complicated structures we compare the results from eca with earlier ones based upon the theory of formal languages and the results from cca with a more traditional approach based on an order parameter and free energy while sensitivity and statistical complexity are equally adept at uncovering structure they are based on different system properties dynamical and probabilistic respectively and provide complementary information

an almost machine independent theory of program length complexity sophistication and induction abstract the purpose of this paper is to use a variant of program length complexity to formally define the structure of a binary string where the structure of an object is taken to mean the aggregate of its projectible properties

coherent states for systems with discrete and continuous spectrum for a system with one degree of freedom coherent states that are parametrized by classical canonical action angle variables are introduced these states also possess continuity of labelling a resolution of unity and temporal stability the insistence on canonical action angle variables strongly restricts any remaining arbitrariness in the coherent state definition such states are introduced for semibounded hamiltonian operators having either a discrete or a continuous spectrum hamiltonians that have both discrete and continuous parts in their spectrum are also discussed

new physics in landau levels the most general displaced number coherent states based on the heisenberg su 2 and su 1 1 lie algebras symmetries are constructed they depend on two parameters and can be converted into the well known photon added two variable glauber coherent states and displaced number states respectively depending on which of the parameters is equal to zero the relations of the weyl heisenberg algebra guarantee a corresponding resolution of the identity conditions a discussion of the statistical properties of these states is included significant are their squeezing properties which can be raised by increasing the energy and angular momentum quantum numbers n and m the maximum squeezing is obtained for bext 0 depending on the particular choice of parameters in the above scenarios we are able to determine the status of compliance with poissonian statistics in the limiting case we obtain a major result about the non classical properties of the glauber minimum uncertainty coherent states in other words in addition to the requirement to minimize uncertainty conditions they carry non classical features too finally a theoretical framework is proposed to generate them

minimum description length induction bayesianism and kolmogorov complexity the relationship between the bayesian approach and the minimum description length approach is established we sharpen and clarify the general modeling principles minimum description length mdl and minimum message length mml abstracted as the ideal mdl principle and defined from bayes s rule by means of kolmogorov complexity the basic condition under which the ideal principle should be applied is encapsulated as the fundamental inequality which in broad terms states that the principle is valid when the data are random relative to every contemplated hypothesis and also these hypotheses are random relative to the universal prior the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized if we restrict the model class to finite sets then application of the ideal principle turns into kolmogorov s minimal sufficient statistic in general we show that data compression is almost always the best strategy both in model selection and prediction

informational and causal architecture of continuous time renewal processes we introduce the minimal maximally predictive models epsilon text machines of processes generated by certain hidden semi markov models their causal states are either discrete mixed or continuous random variables and causal state transitions are described by partial differential equations as an application we present a complete analysis of the epsilon text machines of continuous time renewal processes this leads to closed form expressions for their entropy rate statistical complexity excess entropy and differential information anatomy rates

equality in the spacetime positive mass theorem we affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight namely if an asymptotically flat initial data set satisfies the dominant energy condition and has e p then e p 0 where e p is the adm energy momentum vector the dimensional restriction can be removed if we assume the positive mass inequality holds previously the result was only known for spin manifolds beig and chru ciel in j math phys37 4 1939 1961 1996 chru ciel and maerten in j math phys 47 2 2006

compression of two dimensional data distortion free compressibility of individual pictures i e two dimensional arrays of data by finite state encoders is investigated for every individual infinite picture i a quantity rho i is defined called the compressibility of i which is shown to be the asymptotically attainable lower bound on the compression ratio that can be achieved for i by any finite state information lossless encoder this is demonstrated by means of a constructive coding theorem and its converse that apart from their asymptotic significance might also provide useful criteria for finite and practical data compression tasks the proposed picture compressibility is also shown to possess the properties that one would expect and require of a suitably defined concept of two dimensional entropy for arbitrary probabilistic ensembles of infinite pictures while the definition of rho i allows the use of different machines for different pictures the constructive coding theorem leads to a universal compression scheme that is asymptotically optimal for every picture the results are readily extendable to data arrays of any finite dimension

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

identifiability of hidden markov information sources and their minimum degrees of freedom if only a function of the state in a finite state markov chain is observed then the stochastic process is no longer markovian in general this type of information source is found widely and the basic problem of its identifiability remains open that is the problem of showing when two different markov chains generate the same stochastic process the identifiability problem is completely solved by linear algebra where a block structure of a markov transition matrix plays a fundamental role and from which the minimum degree of freedom for a source is revealed

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

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

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

new nonlinear coherent states based on hypergeometric type operators the main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states to do this we first establish a class of hypergeometric type of generalized displacement operators 1fr 0 0 1 r 1 za act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the fock space which admit a resolution of the identity through a positive definite measure on the complex plane furthermore realization of the compact form of these states as functions of the position coordinate x for r 2 leads to a generating function of the hermite polynomials in terms of the modified bessel function finally studying some statistical characters reveals that they have indeed non classical features such as squeezing an anti bunching effect and sub poissonian statistics too

finite statistical complexity for sofic systems we propose a measure of complexity for symbolic sequences which is based on conditional probabilities and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata dfa moreover if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures causal states

extending sdl and lmc complexity measures to quantum states an extension of sdl shiner davison landsberg and lmc lopez ruiz mancini calbet complexity measures is proposed for the quantum information context considering that von neumann entropy is a natural disorder quantifier for quantum states as a first example of application the simple qubit was studied presenting results similar to that obtained by applying sdl and lmc measures to a classical probability distribution then for the werner state a mixture of bell states sdl and lmc measures were calculated depending on the mixing factor providing some conjectures concerning quantum systems

linearization stability of the einstein constraint equations on an asymptotically hyperbolic manifold we study the linearization stability of the einstein constraint equations on an asymptotically hyperbolic manifold in particular we prove that these equations are linearization stable in the neighborhood of vacuum solutions for a nonpositive cosmological constant and of friedman lemaitre robertson walker spaces in a certain range of decays we also prove that this result is no longer true for faster decays the construction of the counterexamples is based on a new construction of transverse traceless tensors on the euclidean space and on positive energy theorems

optimal classical simulation of state independent quantum contextuality simulating quantum contextuality with classical systems requires memory a fundamental yet open question is what is the minimum memory needed and therefore the precise sense in which quantum systems outperform classical ones here we make rigorous the notion of classically simulating quantum state independent contextuality qsic in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite qsic set we obtain the minimum memory needed to simulate arbitrary qsic sets via classical systems under the assumption that the simulation should not contain any oracular information in particular we show that while classically simulating two qubits tested with the peres mermin set requires log 2 24 4 585 bits simulating a single qutrit tested with the yu oh set requires at least 5 740 bits

many roads to synchrony natural time scales and their algorithms we consider two important time scales the markov and cryptic orders that monitor how an observer synchronizes to a finitary stochastic process we show how to compute these orders exactly and that they are most efficiently calculated from the e machine a process s minimal unifilar model surprisingly though the markov order is a basic concept from stochastic process theory it is not a probabilistic property of a process rather it is a topological property and moreover it is not computable from any finite state model other than the e machine via an exhaustive survey we close by demonstrating that infinite markov and infinite cryptic orders are a dominant feature in the space of finite memory processes we draw out the roles played in statistical mechanical spin systems by these two complementary length scales

minimum memory for generating rare events we classify the rare events of structured memoryful stochastic processes and use this to analyze sequential and parallel generators for these events given a stochastic process we introduce a method to construct a process whose typical realizations are a given process rare events this leads to an expression for the minimum memory required to generate rare events we then show that the recently discovered classical quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations