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

how porous is the graph of brownian motion we prove that the graph of brownian motion is almost surely porous and determine the hausdorff dimension of sets with a given porosity index in particular we show that the porosity index of the graph is yo 0 6948

a lower bound for nodal count on discrete and metric graphs we study the number of nodal domains maximal connected regions on which a function has constant sign of the eigenfunctions of schrodinger operators on graphs under a certain genericity condition we show that the number of nodal domains of the nth eigenfunction is bounded below by n l where l is the number of links that distinguish the graph from a tree

slow points in the support of historical brownian motion a slow point from the left for brownian motion is a time during a given interval for which the oscillations of the path immediately to the left of this time are smaller than the typical ones that is those given by the local lil these slow points occur at random times during a given interval for historical super brownian motion the support at a fixed time contains an infinite collection of paths this paper makes use of a branching process description of the support to investigate the slowness of these paths at the fixed time the upper function found is the same as that found for slow points in the brownian motion case

mouvement brownien cones et processus stables letw wt t 0 denote a two dimensional brownian motion starting at 0 and for 0 4 here we construct a measure a local time supported onh for 4 2 the brownian motionw time changed by the inverse of this local time is shown to be a two dimensional stable process with index 2 2 this results extends spitzer s construction of the cauchy process which is recovered by taking 2 a formula which describes the behaviour ofw before a timet h is established and applied to the proof of a conjecture of burdzy we also obtain a two dimensional version of the famous theorem of levy concerning the maximum process of linear brownian motion precisely for 0 2 letst denote the vertex of the smallest wedge of the typez c which contains the path ofw up to timet the processst wt is shown to be a reflected brownian motion in the wedgec with oblique reflection on the sides finally we investigate various extensions of the previous results to brownian motion inrd d 3 letc be the cone associated with an open subset of the spheresd 1 and leth be defined ash above sufficient conditions are given forh to contain nonzero times in terms of the first eigenvalue of the dirichlet laplacian on

spectral gap for quantum graphs and their edge connectivity the spectral gap for laplace operators on metric graphs and the relation between the graph s edge connectivity is investigated in particular what happens to the gap if an edge is added to or deleted from a graph it is shown that in contrast to discrete graphs the connection between the connectivity and the spectral gap is not one to one the size of the spectral gap depends not only on the topology of the metric graph but on its geometric properties as well it is shown that adding sufficiently large edges as well as cutting away sufficiently small edges leads to a decrease of the spectral gap corresponding explicit criteria are given

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

estimating generating partitions of chaotic systems by unstable periodic orbits an outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1 it has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition here we present a general dimension independent and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits our algorithm allows us to obtain the approximate location of the generating partition for the ikeda hammel jones moloney map

the number of nodal domains on quantum graphs as a stability index of graph partitions the courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of laplacian type operators in particular it holds for generic eigenfunctions of a quantum graph the theorem stipulates that after ordering the eigenvalues as a non decreasing sequence the number of nodal domains n of the n th eigenfunction satisfies n n here we provide a new interpretation for the courant nodal deficiency d n n n in the case of quantum graphs it equals the morse index at a critical point of an energy functional on a suitably defined space of graph partitions thus the nodal deficiency assumes a previously unknown and profound meaning it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction to demonstrate this connection the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail

negative energy ground states for the l2 critical nlse on metric graphs we investigate the existence of ground states with prescribed mass for the focusing nonlinear schrodinger equation with l2 critical power nonlinearity on noncompact quantum graphs we prove that unlike the case of the real line for certain classes of graphs there exist ground states with negative energy for a whole interval of masses a key role is played by a thorough analysis of gagliardo nirenberg inequalities and by estimates of the optimal constants most of the techniques are new and suited to the investigation of variational problems on metric graphs

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

star graphs and seba billiards we derive an exact expression for the two point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity this turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit the reasons for this are discussed the formula we derive is also shown to be equivalent to a series expansion for the form factor the fourier transform of the two point correlation function previously calculated using periodic orbit theory

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

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

information theory explanation of the fluctuation theorem maximum entropy production and self organized criticality in non equilibrium stationary states jaynes information theory formalism of statistical mechanics is applied to the stationary states of open non equilibrium systems first it is shown that the probability distribution p of the underlying microscopic phase space trajectories over a time interval of length satisfies p exp 2kb where is the time averaged rate of entropy production of three consequences of this result are then derived 1 the fluctuation theorem which describes the exponentially declining probability of deviations from the second law of thermodynamics as 2 the selection principle of maximum entropy production for non equilibrium stationary states empirical support for which has been found in studies of phenomena as diverse as the earth s climate and crystal growth morphology and 3 the emergence of self organized criticality for flux driven systems in the slowly driven limit the explanation of these results on general information theoretic grounds underlines their relevance to a broad class of stationary non equilibrium systems in turn the accumulating empirical evidence for these results lends support to jaynes formalism as a common predictive framework for equilibrium and non equilibrium statistical mechanics

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

anomalous nodal count and singularities in the dispersion relation of honeycomb graphs we study the nodal count of the so called bi dendral graphs and show that it exhibits an anomaly the nodal surplus is never equal to 0 or the first betti number of the graph according to the nodal magnetic theorem this means that bands of the magnetic spectrum dispersion relation of such graphs do not have maxima or minima at the usual symmetry points of the fundamental domain of the reciprocal space of magnetic parameters in search of the missing extrema we prove a necessary condition for a smooth critical point to happen inside the reciprocal fundamental domain using this condition we identify the extrema as the singularities in the dispersion relation of the maximal abelian cover of the graph the honeycomb graph being an important example in particular our results show that the anomalous nodal count is an indication of the presence of conical points in the dispersion relation of the maximal universal cover we also discover that the conical points are present in the dispersion relatio

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

brownian motion at a slow point on rappelle des resultats de greenwood et perkins 1983 84 et on etudie les proprietes probabilistes et de trajectoire du processus x t b t t b t on montre en particulier que x est un processus de markov non homogene satisfaisant une certaine equation differentielle stochastique

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

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

brownian motion reflected on brownian motion we study brownian motion reflected on an independent brownian path we prove results on the joint distribution of both processes and the support of the parabolic measure in the space time domain bounded by a brownian path we show that there exist two different natural local times for a brownian path reflected on a brownian path

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

a sandwich proof of the shannon mcmillan breiman theorem let x t be a stationary ergodic process with distribution p admitting densities p x 0 ldots x n 1 relative to a reference measure m that is finite order markov with stationary transition kernel let i m p denote the relative entropy rate then n 1 log p x 0 ldots x n 1 rightarrow i m p mathrm a s p we present an elementary proof of the shannon mcmillan breiman theorem and the preceding generalization obviating the need to verify integrability conditions and also covering the case i m p infty a sandwich argument reduces the proof to direct applications of the ergodic theorem

a characteristic equation associated to an eigenvalue problem on c2 networks abstract a characteristic equation for the spectrum of an eigenvalue problem on c 2 networks is deduced and those special cases are investigated in which the spectrum depends mainly on the structure of the abstract graph belonging to the c 2 network

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

large deviations for markov processes with resetting markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches foraging and population dynamics here we study the large deviations of time additive functions or observables of markov processes with resetting by deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables characterizing the likelihood of their fluctuations in the long time limit we consider as an illustration the large deviations of the area of the ornstein uhlenbeck process with resetting other applications involving diffusions random walks and jump processes with resetting or catastrophes are discussed

edge connectivity and the spectral gap of combinatorial and quantum graphs we derive a number of upper and lower bounds for the first nontrivial eigenvalue of laplacians on combinatorial and quantum graph in terms of the edge connectivity i e the minimal number of edges which need to be removed to make the graph disconnected on combinatorial graphs one of the bounds corresponds to a well known inequality of fiedler of which we give a new variational proof on quantum graphs the corresponding bound generalizes a recent result of band and levy all proofs are general enough to yield corresponding estimates for the p laplacian and allow us to identify the minimizers based on the betti number of the graph we also derive upper and lower bounds on all eigenvalues which are asymptotically correct i e agree with the weyl asymptotics for the eigenvalues of the quantum graph in particular the lower bounds improve the bounds of friedlander on any given graph for all but finitely many eigenvalues while the upper bounds improve recent results of ariturk our estimates are also used to derive bounds on the eigenvalues of the normalized laplacian matrix that improve known bounds of spectral graph theory

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

on the length of the longest excursion summary a lower limit of the length of the longest excursion of a symmetric random walk is given certain related problems are also discussed it is shown e g that for any a 0 and all sufficiently large n there are c e loglogn excursions in the interval 0 n with total length greater than n l a with probability 1

stable processes have thorns let x t be the symmetric stable process in r 0 2 d 2 for f dvtx 0 1 0 let d f be the thorn x r dvtxx1 0 1 allowbreak x2 xd f x1 we give an integral criterion in terms of f for the existence of a random time s such that x t remains in x s d f for all t s s 1

uniqueness and non uniqueness of prescribed mass nls ground states on metric graphs abstract we consider the problem of uniqueness of ground states of prescribed mass for the nonlinear schrodinger energy with power nonlinearity on noncompact metric graphs we first establish that the lagrange multiplier appearing in the nls equation is constant on the set of ground states of mass up to an at most countable set of masses then we apply this result to obtain uniqueness of ground states on two specific noncompact graphs finally we construct a graph that admits at least two ground states with the same mass having different lagrange multipliers our proofs are based on careful variational arguments and rearrangement techniques and hold both for the subcritical range p 2 6 and in the critical case p 6

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

on brownian slow points it is shown that for a wiener process x t both the quantities mathop inf limits t mathop overline lim limits h text to text 0 x t text text h x t sqrt h text and mathop text sup limits t mathop underline lim limits h text to text 0 text x t text text h x t sqrt h are almost surely equal to 1

simplicity of eigenvalues and non vanishing of eigenfunctions of a quantum graph we prove that after an arbitrarily small adjustment of edge lengths the spectrum of a compact quantum graph with type vertex conditions can be simple we also show that the eigenfunctions with the exception of those living entirely on a looping edge can be made to be non vanishing on all vertices of the graph as an application of the above result we establish that the secular manifold also called determinant manifold of a large family of graphs has exactly two smooth connected components

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

a note on an inequality involving the normal distribution the following inequality is useful in studying a variation of the classical isoperimetric problem let x be normally distributed with mean 0 and variance 1 if g is absolutely continuous and g x has finite variance then e lbrack g x rbrack 2 geq operatorname var lbrack g x rbrack with equality if and only if g x is linear in x the proof involves expanding g x in hermite polynomials

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

surgery principles for the spectral analysis of quantum graphs we present a systematic collection of spectral surgery principles for the laplacian on a compact metric graph with any of the usual vertex conditions natural dirichlet or delta type which show

non extinction of a fleming viot particle model we consider a branching particle model in which particles move inside a euclidean domain according to the following rules the particles move as independent brownian motions until one of them hits the boundary this particle is killed but another randomly chosen particle branches into two particles to keep the population size constant we prove that the particle population does not approach the boundary simultaneously in a finite time in some lipschitz domains this is used to prove a limit theorem for the empirical distribution of the particle family

la filtration de b l let btbe a one dimensional brownian motion and let ltbe its local time at zero we answer a question of j pitman by showing that the canonical filtrations of b and b cl are equal for all real constants c it is also shown that the filtrations of b and b cl are equal if and only if c 1

a critical case for brownian slow points letxt be a brownian motion and lets c be the set of realsr 0 such that uxr t xru c t 0 t h for someh h r 0 it is known thats c is empty ifc 1 a s in this paper we prove thats 1 is empty a s

spectral preorder and perturbations of discrete weighted graphs in this article we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential the first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights the second preorder refers to the spectrum of the associated laplacian of the magnetic weighted graph these relations give a quantitative control of the effect of elementary and composite perturbations of the graph deleting edges contracting vertices etc on the spectrum of the corresponding laplacians generalising interlacing of eigenvalues we give several applications of the preorders we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d clique moreover we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic cheeger constants with respect to the geometric preorder finally we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph

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

forward brownian motion we consider processes which have the distribution of standard brownian motion in the forward direction of time starting from random points on the trajectory which accumulate at infty we show that these processes do not have to have the distribution of standard brownian motion in the backward direction of time no matter which random time we take as the origin we study the maximum and minimum rates of growth for these processes in the backward direction we also address the question of which extra assumptions make one of these processes a two sided brownian motion

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

flat points of two dimensional brownian motion supposez is a two dimensional brownian motion it is shown that a s there existt0 and 0 such thatz t0 is an extremal point of the convex hull of z t t0 t t0 and also an extremal point of the convex hull of z t t0 t t0 and moreover the tangent lines to the convex hulls atz t0 form a non zero angle

a conditioned limit theorem for random walk and brownian local time on square root boundaries we count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a local time whose inverse is a stable subordinator of a known index and a brownian motion the study of this limit process leads to some precise sample path properties of brownian motion these properties improve earlier results of dvoretsky and kahane on the existence of small oscillations in the brownian path

a pure jump markov process with a random singularity spectrum we construct a nondecreasing pure jump markov process whose jump measure heavily depends on the values taken by the process we determine the singularity spectrum of this process which turns out to be random and to depend locally on the values taken by the process the result relies on fine properties of the distribution of poisson point processes and on ubiquity theorems

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

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

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

nonstandard construction of the stochastic integral and applications to stochastic differential equations ii h j keisler has recently used a nonstandard theory of lto integration due to r m anderson to construct solutions of ito integral equations by solving an associated internal difference equation in this paper we use the same general approach to find solutions y t of semimartingale integral equations of the form y t c h t c f s y w dz s where z is a given semimartingale h is a right continuous process and f s w is continuous on the space of right continuous functions with left limits with the topology of uniform convergence on compacts in addition we generalize keisler s continuity theorem and give necessary and sufficient conditions for an internal martingale to be s continuous introduction this paper is a continuation of hoover and perkins 5 notation and definitions presented in 5 are used without further introduction and references to that paper are made by simply citing the number of the theorem or lemma due to the consecutive numbering of the two papers no ambiguity will arise finally the reader is referred to 0 of 5 for a brief description of the contents of this work 8 continuous local martingales we assume throughout this section that all processes are r or r valued since the main results theorems 8 1 8 5 and 8 6 then follow immediately for higher dimensions we will show theorem 8 1 that a continuous local martingale has a q1 local martingale lifting for any internal filtration t recall that in general theorem 5 6 asserted only that such a lifting exists for some internal filtration for this reason unless otherwise stated the internal filtration under consideration will be it i t e t which was used in the original definition of t note that if x t x q m is an s continuous process i e x w is s continuous a s then st x has continuous paths a s and x is a uniform lifting of st x in the sense of keisler 6 the following lifting theorem was first proved in a more specialized setting in panetta 9 the proof given here is different received by the editors january 22 1982 the results of this paper were presented at the second victoria symposium on nonstandard analysis victoria canada june 16 20 1980 a ms mos subject cllassifications 1970 primary 60h20 02h25 secondary 60g45

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

one more square root law for brownian motion and its application to spdes it is proved that there is a function p c 0 such that p c 0 if c is large enough and a s for any t 0 1 the trajectory of brownian motion after time t is contained in a parallel shift of the box 0 2 k 0 c2 k 2 for all k belonging to a set with lower density p c this law of square root helps show that solutions of one dimensional spdes are holder continuous up to the boundary

on the hausdorff dimension of the brownian slow points for each c1 c2 consider the set of one sided slow points of the brownian path b defined by s c 1 c 2 omega t exists delta 0 text b t h b t in c 1 h 1 2 c 2 h 1 2 forall 0 underset raise0 3em hbox smash scriptscriptstyle leqslant h underset raise0 3em hbox smash scriptscriptstyle leqslant delta we find the hausdorff dimension of s c1 c2 as well as s c1 c2 a where a a progressively measurable set satisfying certain hypotheses in particular the hausdorff dimension of the two sided slow points s c1 c2 and the one sided and two sided slow points in the zero set of b are obtained by making an appropriate choice of a these results show for example that s c c o or o according as c is greater than or less than the smallest positive zero of m 1 2 1 2 x2 2 1 3069 where m is the confluent hypergeometric function this confirms a conjecture of burgess davis whose methods play a major role in the proofs the results are also extended to higher dimensions

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

multifractal spectra of fragmentation processes let s t t 0 be a homogeneous fragmentation of 0 1 with no loss of mass for x 0 1 we say that the fragmentation speed of x is v if and only if as time passes the size of the fragment that contains x decays exponentially with rate v we show that there is v typ 0 such that almost every point x 0 1 has speed v typ nonetheless for v in a certain range the random set g v of points of speed v is dense in 0 1 and we compute explicitly the spectrum v dim g v where dim is the hausdorff dimension

perturbed brownian motions we study perturbed brownian motions that can be loosely speaking described as follows they behave exactly as linear brownian motion except when they hit their past maximum or and maximum where they get an extra push we define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain natural class of processes in the case where both perturbations at the maximum and at the minimum are self repelling we show that in fact more is true such a process can almost surely be constructed from brownian paths by a one to one measurable transformation this generalizes some results of carmona petit yor and davis we also derive some fine properties of perturbed brownian motions hausdorff dimension of points of monotonicity for example

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