estimating the errors on measured entropy and mutual information abstract information entropy and the related quantity mutual information are used extensively as measures of complexity and to identify nonlinearity in dynamical systems expressions for the probability distribution of entropies and mutual informations calculated from finite amounts of data exist in the literature but the expressions have seldom been used in the field of nonlinear dynamics in this paper formulae for estimating the errors on observed information entropies and mutual informations are derived using the standard error analysis familiar to physicists their validity is demonstrated by numerical experiment for illustration the formulae are then used to evaluate the errors on the time lagged mutual information of the logistic map

decomposition of conditional probability for high order symbolic markov chains the main goal of this paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet we elaborate on a decomposition procedure for the conditional probability function of sequences considered to be high order markov chains we represent the conditional probability function as the sum of multilinear memory function monomials of different orders from zero up to the chain order this allows us to introduce a family of markov chain models and to construct artificial sequences via a method of successive iterations taking into account at each step increasingly high correlations among random elements at weak correlations the memory functions are uniquely expressed in terms of the high order symbolic correlation functions the proposed method fills the gap between two approaches namely the likelihood estimation and the additive markov chains the obtained results may have applications for sequential approximation of artificial neural network training

a nonlinear mixed finite eelement method for a degenerate parabolic equation arising in flow in porous media we study a model nonlinear degenerate advection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the true solution is typically lacking in regularity therefore we consider the problem from the point of view of optimal approximation through time integration we develop a mixed variational form that respects the known minimal regularity and then we develop and analyze two versions of a mixed finite element approximation a simpler semidiscrete time continuous version and a fully discrete version our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error the exceptional term is a nonstandard approximation error term we also consider our new formulation for the nondegenerate problem showing the usual optimal l 2 error bounds moreover superconvergence is obtained under special circumstances

estimating functions of probability distributions from a finite set of samples this paper addresses the problem of estimating a function of a probability distribution from a finite set of samples of that distribution a bayesian analysis of this problem is presented the optimal properties of the bayes estimators are discussed and as an example of the formalism closed form expressions for the bayes estimators for the moments of the shannon entropy function are derived then numerical results are presented that compare the bayes estimator to the frequency counts estimator for the shannon entropy we also present the closed form estimators all derived elsewhere for the mutual information mathrm ensuremath chi 2 covariance and some other statistics c 1995 the american physical society

the tsallis entropy of natural information abstract estimating the information contained in natural data such as electroencephalography data is unusually difficult because the relationship between the physical data and the information that it encodes is unknown this unknown relationship is often called the encoding problem the present work provides a solution to this problem by deriving a method to estimate the tsallis entropy in natural data the method is based on two findings the first finding is that the physical instantiation of any information event that is the physical occurrence of a symbol of information must begin and end at a discontinuity or critical point maximum minimum or saddle point in the data the second finding is that in certain data types such as the encephalogram eeg the variance within of an eeg waveform event is directly proportional to its probability of occurrence these two outcomes yield two results the first is the easy binning of data into separate information events the second is the ability to estimate probabilities in two ways frequency counting and computing the variance within of an eeg waveform these results are used to derive a linear estimator of the tsallis entropy functional allowing it to be estimated without deducing the encoding this method for estimating the tsallis entropy is first used to estimate the information in simple signals the amount of information estimated is highly accurate the method is then applied to two problems in electroencephalography the first is distinguishing normal aging from very early alzheimer s disease mild cognitive impairment and the second is medication monitoring of alzheimer s disease treatment the former is done with an accuracy of 92 and the latter with an accuracy of 91 this detection accuracy is the highest published accuracy in the literature which suggests that this method for tsallis entropy estimation is both accurate and useful

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

wavelet analysis and nonlinear dynamics in a nonextensive setting abstract we undertake the study of signals originated in time dependent nonlinear systems by recourse to a wavelet based multiresolution analysis as adapted to a nonextensive tsallis scenario diverse applications are discussed it is shown that the tsallis environment provides one with more detailed information than the conventional shannon one

tsallis conditional mutual information in investigating long range correlation in symbol sequences abstract the presence of long range correlation lrc as opposed to markov chain of a finite order is of primary interest in the study of symbol sequences such as dna among others methods using information theory have been developed for markov chain order estimation in this work we consider the tsallis entropy and define the tsallis conditional mutual information tcmi for markov chain order estimation which is more suitable to identify large orders suggesting effectively lrc the tcmi of order m for a tsallis parameter q is computed for increasing m and a significance test is performed for each m until tcmi is found non significant randomization and parametric significance tests for tcmi are developed for the latter the null distribution of tcmi is approximated with a gamma distribution deriving analytic expressions for its parameters we assess the accuracy of order estimation with the two tests for tcmi and compare them to the respective tests using the shannon entropy extended simulations on markov chains of different orders and structures of the transition probability matrix show that shannon and tsallis tests with parameter q 2 have similar performance for small orders when the problem becomes more demanding for higher markov chain orders and lrcs tsallis tests for q 2 have better approximation to the correct order with the sequence length n finally the tsallis tests are favorably compared to shannon tests to real dna sequences

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

a topological phase transition between small worlds and fractal scaling in urban railway transportation networks abstract fractal and small worlds scaling laws are applied to study the growth of urban railway transportation networks using total length and total population as observational parameters in spite of the variety of populations and urban structures the variation of the total length of the railway network with the total population of conurbations follows similar patterns for large and middle metropolis diachronous analysis of data for urban transportation networks suggests that there is second order phase transition from small worlds behaviour to fractal scaling during their early stages of development

entropy and long range memory in random symbolic additive markov chains the goal of this paper is to develop an estimate for the entropy of random symbolic sequences with elements belonging to a finite alphabet as a plausible model we use the high order additive stationary ergodic markov chain with long range memory supposing that the correlations between random elements of the chain are weak we express the conditional entropy of the sequence by means of the symbolic pair correlation function we also examine an algorithm for estimating the conditional entropy of finite symbolic sequences we show that the entropy contains two contributions i e the correlation and the fluctuation the obtained analytical results are used for numerical evaluation of the entropy of written english texts and dna nucleotide sequences the developed theory opens the way for constructing a more consistent and sophisticated approach to describe the systems with strong short range and weak long range memory

on some criteria for estimating the order of a markov chain tong 1975 has proposed a procedure for estimating the order of a markov chain based on akaike s information criterion aic in this paper the asymptotic distribution of the aic estimator is derived and it is shown that the estimator is inconsistent as an alternative to the aic procedure the bayesian information criterion bic proposed by schwarz 1978 is shown to be consistent these two procedures yield different estimated orders when applied to specific samples of meteorological observations for parameters based on these meteorological examples the aic and bic procedures are compared by means of simulation for finite samples the results obtained have practical implications concerning whether in the routine fitting of precipitation data it is necessary to consider higher than first order markov chains

markov chain order estimation with conditional mutual information abstract we introduce the conditional mutual information cmi for the estimation of the markov chain order for a markov chain of k symbols we define cmi of order m i c m as the mutual information of two variables in the chain being m time steps apart conditioning on the intermediate variables of the chain we find approximate analytic significance limits based on the estimation bias of cmi and develop a randomization significance test of i c m where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence the significance test is applied for increasing m and the markov chain order is estimated by the last order for which the null hypothesis is rejected we present the appropriateness of cmi testing on monte carlo simulations and compare it to the akaike and bayesian information criteria the maximal fluctuation method peres shields estimator and a likelihood ratio test for increasing orders using divergence the order criterion of cmi testing turns out to be superior for orders larger than one but its effectiveness for large orders depends on data availability in view of the results from the simulations we interpret the estimated orders by the cmi testing and the other criteria on genes and intergenic regions of dna chains

on the relation between various entropy concepts and the valoric interpretation abstract the valoric interpretation developed already by clausius helmholtz and ostwald is taken as the basis for a reinterpretation of the various entropy concepts developed by clausius boltzmann gibbs shannon and kolmogorov as the key points we consider the value of energy with respect to work and the value of entropy with respect to informmation processing

human brain dynamics the analysis of eeg signals with tsallis information measure we undertake the study of human eeg signals by recourse to a wavelet based multiresolution analysis as adapted to an information measure scenario dierent information measures are employed it is shown that non extensive ones seem to be of particular usefulness their use opens up perspectives of building up automatic detection devices conjectures concerning general characteristics of focal epilepsy are formulated on the basis of a tsallis type of analysis c 1999 elsevier science b v all rights reserved

method for generating long range correlations for large systems we propose a new method to generate a sequence of random numbers with long range power law correlations that overcomes known difficulties associated with large systems the new method presents an improvement on the commonly used methods we apply the algorithm to generate enhanced diffusion isotropic and anisotropic self affine surfaces and isotropic and anisotropic correlated percolation

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

finite element approximation of the transport of reactive solutes in porous media part 1 error estimates for nonequilibrium adsorption processes in this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration backward euler time discretization and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media find u x t v x t such that begin eqnarray partial t u partial t v delta u f quad mbox in omega times 0 t qquad u 0 quad mbox on partial omega times 0 t partial tv k varphi u v quad mbox in omega times 0 t u cdot 0 g 1 cdot quad v cdot 0 g 2 cdot quad mbox in omega subset real d quad 1 le d le 3 end eqnarray for given data k in real f g 1 g 2 and a monotonically increasing varphi in c 0 real cap c 1 infty 0 cup 0 infty satisfying varphi 0 0 which is only locally holder continuous with exponent p in 0 1 at the origin e g varphi s equiv s p this lack of lipschitz continuity at the origin limits the regularity of the unique solution u v and leads to difficulties in the finite element error analysis

guessing probability distributions from small samples we propose a new method for the calculation of the statistical properties e g the entropy of unknown generators of symbolic sequences the probability distributionp k of the elementsk of a population can be approximated by the frequenciesf k of a sample provided the sample is long enough so that each elementk occurs many times our method yields an approximation if this precondition does not hold for a givenf k we recalculate the zipf ordered probability distribution by optimization of the parameters of a guessed distribution we demonstrate that our method yields reliable results

two general models that generate long range correlation abstract in this paper we study two models that generate sequences with lrc long range correlation for the ift inverse fourier transform model our conclusion is the low frequency part leads to lrc while the high frequency part tends to eliminate it therefore a typical method to generate a sequence with lrc is multiplying the spectrum of a white noise sequence by a decaying function a special case is analyzed the linear combination of a smooth curve and a white noise sequence in which the dfa plot consists of two line segments for the patch model our conclusion is long subsequences leads to lrc while short subsequences tend to eliminate it therefore we can generate a sequence with lrc by using a fat tailed pdf probability distribution function of the length of the subsequences a special case is also analyzed if a patch model with long subsequences is mixed with a white noise sequence the dfa plot will consist of two line segments we have checked known models and actual data and found they are all consistent with this study

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

some applications of the useful mutual information abstract the useful mutual information statistic is considered and its asymptotic properties are investigated either considering random sampling or stratified random sampling under different assumptions it is shown that the asymptotic distribution of this statistic is either normal or a linear form in gc 2 variables finally its application to testing hypotheses of independence in multinomial populations is given

statistical analysis of gene and intergenic dna sequences abstract much of the on going statistical analysis of dna sequences is focused on the estimation of characteristics of coding and non coding regions that would possibly allow discrimination of these regions in the current approach we concentrate specifically on genes and intergenic regions to estimate the level and type of correlation in these regions we apply various statistical methods inspired from nonlinear time series analysis namely the probability distribution of tuplets the mutual information and the identical neighbour fit the methods are suitably modified to work on symbolic sequences and they are first tested for validity on sequences obtained from well known simple deterministic and stochastic models then they are applied to the dna sequence of chromosome 1 of arabidopsis thaliana the results suggest that correlations do exist in the dna sequence but they are weak and that intergenic sequences tend to be more correlated than gene sequences the use of statistical tests with surrogate data establish these findings in a rigorous statistical manner

a model for high order markov chains on introduit un modele pour des chaines de markov du leme ordre qui comporte seulement un parametre additionnel pour chaque retard

dynamical complexity detection in pre seismic emissions using nonadditive tsallis entropy abstract we investigate within the framework of nonextensive tsallis entropy particular aspects of the relation between signal analysis and signals originating in a very special nonlinear system the focal area of an impending earthquake the tsallis like time dependent entropy is shown to be a rather powerful tool for providing a novel quantitative strategy for monitoring the focal area states just before the earthquake occurrence

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

error analysis for implicit approximations to solutions to cauchy problems exact error analyses for two implicit approximation schemes to the cauchy problem du t dt mathcal a u t mathrel backepsilon 0 u 0 u 0 in d mathcal a are presented the operator mathcal a is required to be maximal monotone on a hilbert space mathcal h the approximations considered are u h kh i h mathcal a k u 0 and solutions u lambda to the approximating problems du lambda t dt mathcal a lambda u lambda 0 u lambda 0 u 0 where a lambda is the yosida approximation of mathcal a the special case of a being a subgradient is considered and tight estimates of the convergence of both u h and mathcal a h u h or u h and a h u h are given the new results in this case are the linear rates of convergence of u h in l infty 0 infty mathcal h and in the energy norm and the sublinear rate of convergence of a h u h in l 2 0 infty mathcal h applications to both linear and

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

sumudu transform in fractal calculus abstract the c calculus includes functions on fractal sets which are not differentiable or integrable using ordinary calculus sumudu transforms have an important role in control engineering problems because of preserving units the scaling property of domains easy visualization and transforming linear differential equations to algebraic equations that can be easily solved analogues of the laplace and sumudu transforms in c calculus are defined and the corresponding theorems are proved the generalized laplace and sumudu transforms involve functions with totally disconnected fractal sets in the real line linear differential equations on cantor like sets are solved utilizing fractal sumudu transforms the results are summarized in tables and figures illustrative examples are solved to give more details

a robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media we study a nonlinear degenerate convection diffusion model problem having an application in groundwater aquifer and petroleum reservoir simulation the true solution typically possesses low regularity and therefore special numerical techniques for its approximation are needed we design a robust efficient and reliable linear relaxation approximation scheme we prove the convergence of iterations at each time step in the h 1 omega norm finally the convergence of the approximate solution in corresponding functional spaces to its exact counterpart for the parabolic problem is shown

change in order of phase transitions on fractal lattices abstract we re examine a population model which exhibits a continuous absorbing phase transition belonging to directed percolation in 1d and a first order transition in 2d and above studying the model on sierpinski carpets of varying fractal dimensions we examine at what fractal dimension 1 d f 2 the change in order occurs as well as commenting on the order of the transitions we produce estimates for the critical points and for continuous transitions some critical exponents

long range memory elementary 1d cellular automata dynamics and nonextensivity abstract we numerically study the dynamics of elementary 1d cellular automata ca where the binary state i t 0 1 of a cell i does not only depend on the states in its local neighborhood at time t 1 but also on the memory of its own past states i t 2 i t 3 i t we assume that the weight of this memory decays proportionally to with 0 the limit corresponds to the usual ca since the memory function is summable for 1 and nonsummable for 0 1 we expect pronounced changes of the dynamical behavior near 1 this is precisely what our simulations exhibit particularly for the time evolution of the hamming distance h of initially close trajectories we typically expect the asymptotic behavior h t t 1 1 q where q is the entropic index associated with nonextensive statistical mechanics in all cases the function q exhibits a sensible change at 1 we focus on the class ii rules 61 99 and 111 for rule 61 q 0 for 0 c 1 3 and q 0 for c whereas the opposite behavior is found for rule 111 for rule 99 the effect of the long range memory on the spread of damage is quite dramatic these facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory finite size scaling studies varying system size n indicate that the range of the power law regime for h t typically diverges n z with 0 z 1

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

tsallis like information measures and the analysis of complex signals abstract a tsallis like information measure that elaborates upon those of the tsallis type is shown to be a rather powerful tool for the analysis of complex non stationary signals as exemplified by the henon map and eegs the ensuing results could be of importance concerning the automatic detection of epileptic spikes

directed percolation in 2 1 dimensions the author presents results of monte carlo simulations for directed percolation in 2 1 dimensions very close to the percolation threshold his values for pc for bond and site percolation on the bcc and simple cubic lattices are more precise than previous estimates by one to two orders of magnitude the improvement on the critical exponents describing the behaviour at pc is less impressive but they are also substantially more precise than previous ones

symbolic stochastic dynamical systems viewed as binary n step markov chains a theory of systems with long range correlations based on the consideration of binary n step markov chains is developed in the model the conditional probability that the ith symbol in the chain equals zero or unity is a linear function of the number of unities among the preceding n symbols the correlation and distribution functions as well as the variance of the number of symbols in the words of arbitrary length l are obtained analytically and numerically a self similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented the diffusion fokker planck equation governing the distribution function of the l words is explored if the persistent correlations are not extremely strong the distribution function is shown to be the gaussian with the variance being nonlinearly dependent on l the applicability of the developed theory to the coarse grained written and dna texts is discussed

on a numerical approach to stefan like problems this paper is devoted to the numerical analysis of a bidimensional two phase stefan problem we approximate the enthalpy formulation byc 0 piecewise linear finite elements in space combined with a semi implicit scheme in time under some restrictions related to the finite element mesh and to the timestep we prove positivity stability and convergence results various numerial tests are presented and discussed in order to show the accuracy of our scheme

phase transitions in a lattice population model we introduce a model for a population on a lattice with diffusion and birth death according to and for a particle a we find that the model displays a phase transition from an active to an absorbing state which is continuous in 1 1 dimensions and of first order in higher dimensions in agreement with the mean field equation for the 1 1 dimensional case we examine the critical exponents and a scaling function for the survival probability and show that it belongs to the universality class of directed percolation in higher dimensions we look at the first order phase transition by plotting a histogram of the population density and use the presence of phase coexistence to find an accurate value for the critical point in 2 1 dimensions

analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media we present an analysis of expanded mixed finite element methods applied to richards equation a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium we consider the full range of saturated to completely unsaturated media in the case of the lowest order raviart thomas spaces and the range of all possible saturations we bound the h 1 norm of the error in capacity in terms of approximation error this estimate uses a time integrated scheme and the kirchhoff transformation to handle a degeneracy in the case of completely unsaturated flow optimal convergence is then shown for a nonlinear form related to the error in the capacity for the case of saturated to partially saturated flow convergence rates depending on the holder continuity of the capacity term are derived last optimal convergence of pressures and fluxes is stated for the case of strictly partially saturated flow

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry