statistical complexity measure of pseudorandom bit generators abstract pseudorandom number generators prng are extensively used in monte carlo simulations gambling machines and cryptography as substitutes of ideal random number generators rng each application imposes different statistical requirements to prngs as l ecuyer clearly states the main goal for monte carlo methods is to reproduce the statistical properties on which these methods are based whereas for gambling machines and cryptology observing the sequence of output values for some time should provide no practical advantage for predicting the forthcoming numbers better than by just guessing at random in accordance with different applications several statistical test suites have been developed to analyze the sequences generated by prngs in a recent paper a new statistical complexity measure phys lett a 311 2003 126 has been defined here we propose this measure as a randomness quantifier of a prngs the test is applied to three very well known and widely tested prngs available in the literature all of them are based on mathematical algorithms another prngs based on lorenz 3d chaotic dynamical system is also analyzed prngs based on chaos may be considered as a model for physical noise sources and important new results are recently reported all the design steps of this prng are described and each stage increase the prng randomness using different strategies it is shown that the mpr statistical complexity measure is capable to quantify this randomness improvement the prng based on the chaotic 3d lorenz dynamical system is also evaluated using traditional digital signal processing tools for comparison

minimum connected transversals in graphs new hardness results and tractable cases using the price of connectivity abstract we perform a systematic study in the computational complexity of the connected variant of three related transversal problems vertex cover feedback vertex set and odd cycle transversal just like their original counterparts these variants are np complete for general graphs a graph g is h free for some graph h if g contains no induced subgraph isomorphic to h it is known that connected vertex cover is np complete even for h free graphs if h contains a claw or a cycle we show that the two other connected variants also remain np complete if h contains a cycle or claw in the remaining case h is a linear forest we show that connected vertex cover connected feedback vertex set and connected odd cycle transversal are polynomial time solvable for s p 2 free graphs for every constant s 1 for proving these results we use known results on the price of connectivity for vertex cover feedback vertex set and odd cycle transversal this is the first application of the price of connectivity that results in polynomial time algorithms

rigorous properties and uncertainty like relationships on product complexity measures application to atomic systems most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework valid for any arbitrary n dimensional distribution in the present work different product complexities constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated different results on product complexities such as rigorous bounds uncertainty like inequalities relationships among themselves and also statistical interpretations are given for the sake of completeness some of these analytical results on product complexities are numerically checked for the one particle densities of atomic systems in both conjugated spaces

a method to discern complexity in two dimensional patterns generated by coupled map lattices complex patterns generated by the time evolution of a one dimensional digitalized coupled map lattice are quantitatively analyzed a method for discerning complexity among different patterns is implemented the quantitative results indicate two zones in parameter space where the dynamics show the most complex patterns these zones are located on the two edges of an absorbent region where the system displays spatio temporal intermittency

simple measure for complexity a measure of complexity is proposed based on appropriately defined notions of order and disorder which has a considerable degree of flexibility in its dependence on these concepts the possible functional dependencies which result encompass those of many earlier definitions of complexity the proposed measure is in principle easy to calculate and has the property of an intensive thermodynamic quantity with appropriate choices of parameters it behaves similarly to effective measure complexity for the logistic map it is also a generalization of the normalized complexity of lopez ruiz et al but does not suffer from over universality

a maximum principle for differential inclusion problems with state constraints we study boundary trajectories of differential inclusions with state constraints and prove a maximum principle of the pontryagin type with the adjoint equation and the maximum condition separated we obtain a more general theorem and use a different method of proof than in previous results of this type

a comparison of shannon kullback leibler and renormalized entropies within successive bifurcations abstract an issue in the context of self organization is the existence of bifurcation processes that are often observed in dissipative dynamical systems the first bifurcation occurs when a stable fixed point becomes unstable as the parameter set of the system slightly changes then the system eventually paves their way to new stable branches when the most ordered spatial patterns emerge the system implies a high level of self organization in the phase space exchange of energy matter with its surroundings is allowed for in such open systems open systems cannot be analyzed in terms of usual h theorem of boltzmann since it is only valid for isolated systems thereby the boltzmann gibbs bg entropy or shannon is not able to explain such systems which increase their order as a signature of self organization however klimontovich showed that the bg entropy can still be used in such systems after a renormalization procedure and he also introduced the renormalized entropy as a measure of the relative degree of order by using his s theorem in this paper we analyze the henon map and the rossler oscillator as high dimensional dissipative systems we show that the renormalized entropy detects transition points of the systems and explains changes in relative order through successive bifurcations or chaotic band mergings the correlation analysis also shows that the renormalized entropy is a better measure than the shannon or kullback leibler k l entropies

a comparison of lmc and sdl complexity measures on binomial distributions abstract the concept of complexity has been widely discussed in the last forty years with a lot of thinking contributions coming from all areas of the human knowledge including philosophy linguistics history biology physics chemistry and many others with mathematicians trying to give a rigorous view of it in this sense thermodynamics meets information theory and by using the entropy definition lopez ruiz mancini and calbet proposed a definition for complexity that is referred as lmc measure shiner davison and landsberg by slightly changing the lmc definition proposed the sdl measure and the both lmc and sdl are satisfactory to measure complexity for a lot of problems here sdl and lmc measures are applied to the case of a binomial probability distribution trying to clarify how the length of the data set implies complexity and how the success probability of the repeated trials determines how complex the whole set is

binary indices of time series complexity measures and entropy plane abstract complexity of time series is an important feature of dynamical systems such as financial systems in order to bring out the complete non linear behavior of financial time series non linear tools of complexity measurement like entropy measures are indispensable sample entropy sampen and distribution entropy disten are popular methods of assessing the complexity in various fields however both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets therefore we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane in order to further improve the accuracy of the research results we take the embedding dimension m as the variable expand the entropy point into the entropy curve for further research furthermore considering the shortcomings of sample entropy and distribution entropy in practical application we generalize them by i replacing sample entropy and distribution entropy with multi scale sample entropy and multi scale distribution entropy ii generalizing shannon entropy as tsallis entropy also scale factor and entropic index q are taken respectively as variables to get the entropy curves by using the artificial data we confirm the rationality of using entropy points and entropy curves to study the complexity of time series finally we apply this method to measure the complexity of real world financial time series the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences

a statistical complexity measure with nonextensive entropy and quasi multiplicativity the properties of a statistical complexity measure that are characterized by nonextensivity in entropy have been investigated which is of so called disequilibrium type considering the composition law for two systems with different nonextensivities quasi multiplicativity a nontrivial relation between the nonextensive parameters and the fluctuating bit number in information theory has been mentioned to see the time evolution of the nonextensive complexity measure we examine systems having a lognormal distribution the underlying dynamics for which is known to obey a random multiplicative process in the presence of a boundary constraint

a high accuracy method for solving odes with discontinuous right hand side ordinary differential equations with discontinuities in the state variables require a differential inclusion formulation to guarantee existence 8 from this formulation a high accuracy method for solving such initial value problems is developed which can give any order of accuracy and tracks the discontinuities the method uses an active set approach and determines appropriate active sets from solutions to linear complementarity problems convergence results are established under some non degeneracy assumptions the method has been implemented and results compare favourably with previously published methods 7 21

complex and composite entropy fluctuation behaviors of statistical physics interacting financial model abstract in an attempt to investigate and reproduce the fluctuation dynamics of price changes in financial markets a novel complex random interacting financial price model is developed by combining voter interacting system with compound poisson process in this model the dissemination of investors investment attitudes can simulate normal and frequent small price fluctuations of financial markets while the random drastic jumps can capture undue and rare large price fluctuations as results of sudden economic events and political events in the markets in order to verify the rationality of the model some statistical characteristics of the model returns including fat tail power law scaling complexity and multifractal are compared with real returns by probability density functions composite distance fuzzy entropy c fuzzyen matching energy and multifractal detrended fluctuation analysis the c fuzzyen is a new entropy based approach based on fuzzy entropy and complexity invariant to measure the complexity behaviors of return series the effectiveness analysis of c fuzzyen indicates that it has a higher estimation accuracy and stronger robustness than fuzzy entropy the empirical results show that the price model is able to reproduce some important statistical properties of financial markets to a certain extent and the complexity of returns for the proposed model increases when infection intensity or jump intensity increases

evidence of self organization in brain electrical activity using wavelet based informational tools abstract in the present work we show that appropriate information theory tools based on the wavelet transform relative wavelet energy normalized total wavelet entropy h generalized wavelet complexity c w when applied to tonic clonic epileptic eeg data provide one with valuable insights into the dynamics of neural activity twenty tonic clonic secondary generalized epileptic records pertaining to eight patients have been analyzed if the electromyographic activity is excluded the difference between the ictal and pre ictal mean entropic values h h ictal h pre ictal is negative in 95 of the cases p 0 0001 and the mean complexity variation c w c w ictal c w pre ictal is positive in 85 of the cases p 0 0002 thus during the seizure entropy diminishes while complexity grows this is construed as evidence supporting the conjecture that an epileptic focus in this kind of seizures triggers a self organized brain state characterized by both order and maximal complexity

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

incomparability entropy and mixing dynamics abstract boltzmann states are considered in a fundamentally new way specifically their mixing character the mixing character of these states is known to be partially ordered by majorization and thus contains information regarding their incomparability previously we showed that boltzmann entropy states had a huge range in the number of incomparable states here we phase average incomparability across boltzmann states we propose the consequent function as a new state variable complimentary to entropy that provides new insights to boltzmann systems this function can be related to a traditional complexity viewpoint and so we call it average boltzmann complexity abc the evolution of boltzmann complexity is explored via a monte carlo lattice dynamics approach and is shown to be consistent with the view of entropy as the arrow of time our report concludes with a discussion of the information contained in boltzmann complexity and offers suggestions for further studies

optimal control of systems with discontinuous differential equations in this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right hand side in our work we are motivated by optimal control of mechanical systems with coulomb friction which exhibit such a right hand side notwithstanding the impressive development of nonsmooth and set valued analysis these systems have not been closely studied either computationally or analytically we show that even when the solution crosses and does not stay on the discontinuity differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize this means that the strategy of optimize the discretization will usually fail for problems of this kind we approximate the discontinuous right hand side for the differential equations or inclusions by a smooth right hand side for these smoothed approximations we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that euler s method is used where the step size is sufficiently small in comparison with the smoothing parameter numerical results are presented for a crude model of car racing that involves coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity

a model for competitiveness level analysis in sports competitions application to basketball abstract the degree of overall competitiveness of a sport league is a complex phenomenon it is difficult to assess and quantify all elements that yield the final standing in this paper we analyze the general behavior of the result matrices of each season and we use the corresponding results as a probably density thus the results of previous seasons are a way to investigate the probability that each team has to reach a certain number of victories we developed a model based on shannon entropy using two extreme competitive structures a hierarchical structure and a random structure and applied this model to investigate the competitiveness of two of the best professional basketball leagues the nba usa and the acb spain both leagues entropy levels are high nba mean 0 983 acb mean 0 980 indicating high competitiveness although the entropy of the acb from 0 986 to 0 972 demonstrated more seasonal variability than that of the nba from 0 985 to 0 990 a possible result of greater sporting gradients in the acb the use of this methodology has proven useful for investigating the competitiveness of sports leagues as well as their underlying variability across time

complexity analysis of the turbulent environmental fluid flow time series abstract we have used the kolmogorov complexities sample and permutation entropies to quantify the randomness degree in river flow time series of two mountain rivers in bosnia and herzegovina representing the turbulent environmental fluid for the period 1926 1990 in particular we have examined the monthly river flow time series from two rivers the miljacka and the bosnia in the mountain part of their flow and then calculated the kolmogorov complexity kl based on the lempel ziv algorithm lza lower kll and upper klu sample entropy se and permutation entropy pe values for each time series the results indicate that the kll klu se and pe values in two rivers are close to each other regardless of the amplitude differences in their monthly flow rates we have illustrated the changes in mountain river flow complexity by experiments using i the data set for the bosnia river and ii anticipated human activities and projected climate changes we have explored the sensitivity of considered measures in dependence on the length of time series in addition we have divided the period 1926 1990 into three subintervals a 1926 1945 b 1946 1965 c 1966 1990 and calculated the kll klu se pe values for the various time series in these subintervals it is found that during the period 1946 1965 there is a decrease in their complexities and corresponding changes in the se and pe in comparison to the period 1926 1990 this complexity loss may be primarily attributed to i human interventions after the second world war on these two rivers because of their use for water consumption and ii climate change in recent times

hidden discontinuities and parametric sensitivity calculations parametric sensitivity analysis is becoming a very important tool when studying differential equations however potential pitfalls exist if this analysis is applied naively in particular if the model contains discontinuities then the application of standard numerical codes will typically result in incorrect sensitivity trajectories the purpose of this paper is to demonstrate the problems associated with hidden discontinuities during sensitivity analysis i e discontinuities not handled explicitly and to present a code analysis and transformation approach through which differential equations containing discontinuities can be solved efficiently and correctly with minimal effort required by the modeler

tsallis deformation parameter q quantifies the classical quantum transition abstract we investigate the classical limit of a type of semiclassical evolution the pertinent system representing the interaction between matter and a given field on using tsallis q entropy as a quantifier of the ensuing dynamics we find that it not only appropriately describes the quantum classical transition but that the associated deformation parameter q itself characterizes the different regimes involved in the process detecting the most salient fine details of the changeover

entropic descriptor of a complex behaviour abstract we propose a new type of entropic descriptor that is able to quantify the statistical complexity a measure of complex behaviour by taking simultaneously into account the average departures of a system s entropy s from both its maximum possible value s max and its minimum possible value s min when these two departures are similar to each other the statistical complexity is maximal we apply the new concept to the variability over a range of length scales of spatial or grey level pattern arrangements in simple models the pertinent results confirm the fact that a highly non trivial length scale dependence of the entropic descriptor makes it an adequate complexity measure able to distinguish between structurally distinct configurational macrostates with the same degree of disorder a feature that makes it a good tool for discerning structures in complex patterns

complexity of dynamics as variability of predictability we construct a complexity measure from first principles as an average over the obstruction against prediction of some observable that can be chosen by the observer our measure evaluates the variability of the predictability for characteristic system behaviors which we extract by means of the thermodynamic formalism using theoretical and experimental applications we show that complex and chaotic are different notions of perception in comparison to other proposed measures of complexity our measure is easily computable non divergent for the classical 1 d dynamical systems and has properties of non overuniversality the measure can also be computed for higher dimensional and experimental systems including systems composed of different attractors moreover the results of the computations made for classical 1 d dynamical systems imply that it is not the nonhyperbolicity but the existence of a continuum of characteristic system length scales that is at the heart of complexity

measuring randomness by leave one out prediction error analysis of eeg after painful stimulation abstract a parametric approach to measure randomness in time series is presented time series are modelled by a kernel machine performing regularized least squares and the leave one out loo error is used to quantify unpredictability on analyzing simulated data sets we find that structure in data leads to a minimum of the loo error as the regularizing parameter is varied we consider electroencephalographic signals from migraineurs and healthy humans after painful stimulation and use the proposed approach to detect changes of physiological state and to find differences between the response from patients and healthy subjects as painful stimulus causes organization of the local activity in the cortex eeg series become more predictable after stimulation this phenomenon is less evident in patients the inadequate cortical response to pain in migraineurs separates patients from controls with a probability close to 0 005

extremum complexity distribution of a monodimensional ideal gas out of equilibrium abstract the extremum complexity momentum distribution for an isolated monodimensional ideal gas out of equilibrium is analytically approximated in a first approximation it consists of a double non overlapping gaussian distribution in good agreement with this result the numerical simulations of a particular isolated monodimensional gas which is abruptly pushed far from equilibrium shows the extremum complexity distribution in the decay of the system toward equilibrium

the maximum principle for an optimal solution to a differential inclusion with end points constraints we derive pontryagin s maximum principle for a general optimal control problem using the set valued version of variational equation we achieve this aim by exploiting an adequate differential calculus of set valued maps furthermore the calmness condition is replaced by a surjectivity condition involving reachable sets of the set valued linearization of the initial control problem duality then provides both the adjoint differential inclusion and the maximum principle

a statistical measure of complexity with nonextensive entropy abstract a statistical measure of complexity utilising the concept of entropy or information is proposed our way in this study is to use a nonextensive entropy instead of an extensive additive shannon entropy in the definition but can be characterised as a difference between the qth order renyi entropy and the second one furthermore we devise a conditional joint and mutual complexity measure as a coherent possibility the behavior of the measure for the logistic map shows that it is more sensitive to nonextensivity at the transition point ac 3 8284 than any other values when 0

comment i on simple measure for complexity we critique the measure of complexity introduced by shiner davison and landsberg phys rev e 59 1459 1999 in particular we point out that it is over universal in the sense that it has the same dependence on disorder for structurally distinct systems we then give counterexamples to the claim that complexity is synonymous with being out of equilibrium equilibrium systems can be structurally complex and nonequilibrium systems can be structurally simple we also correct a misinterpretation of a result given by two of the present authors j p crutchfield and d p feldman phys rev e 55 r1239 1997

inhomogeneity and complexity measures for spatial patterns abstract in this work we examine two different measures for inhomogeneity and complexity that are derived from non extensive considerations a la tsallis their performance is then tested on theoretically generated patterns all measures are found to exhibit a most sensitive behaviour for sierpinski carpets the procedures here introduced provide us with new powerful tsallis tools for analysing the inhomogeneity and complexity of spatial patterns

connected vertex covers in dense graphs we consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph we give new approximation results for this problem in dense graphs in which either the minimum or the average degree is linear in particular we prove tight parameterized upper bounds on the approximation returned by savage s algorithm and extend a vertex cover algorithm from karpinski and zelikovsky to the connected case the new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs all these results are shown to be tight finally we introduce the price of connectivity for the vertex cover problem defined as the worst case ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover we prove that the price of connectivity is bounded by 2 1 e in graphs with average degree en and give a family of near tight examples

computation of time optimal controls applied to rigid manipulators with friction abstract we present a numerical procedure to compute non singular time optimal solutions for non linear systems that are linear in the control and have fixed initial and final states and bounded control part of our procedure is a new numerically computable test that determines whether bang bang solutions satisfy pontryagin s minimum principle this test reveals the new important fact that for non linear systems with linear control and dimension n the probability that a bang bang solution with more than n 1 switches satisfies pontryagin s mini mum principle is almost zero using a parameter optimization procedure we search for bang bang solutions with up to n 1 switches which transfer the system from the initial to the final state if no controls with up to n 1 switches can be found to satisfy pontryagin s minimum principle the problem is very likely singular we apply our procedure to the time optimal control problem for rigid manipulators where friction may be included in the dynamics we will

approximating edge dominating set in dense graphs we study the approximation complexity of the minimum edge dominating set problem in everywhere e dense and average e dense graphs more precisely we consider the computational complexity of approximating a generalization of the minimum edge dominating set problem the so called minimum subset edge dominating set problem as a direct result we obtain for the special case of the minimum edge dominating set problem in everywhere e dense and average e dense graphs by using the techniques of karpinski and zelikovsky the approximation ratios of min 2 3 1 2 e and of min 2 3 3 21 e respectively on the other hand we give new approximation lower bounds for the minimum edge dominating set problem in dense graphs assuming the unique game conjecture we show that it is np hard to approximate the minimum edge dominating set problem in everywhere e dense graphs with a ratio better than 2 1 e with e 1 3 and 2 2 1 e with e 5 9 in average e dense graphs

on the simulation and control of some friction constrained motions in this paper some issues involved with numerical simulation and control of some elasto dynamic systems are discussed the motivation is the simulation of dry or coulomb friction in the joints that link together remote manipulator systems used in aerospace operations for example space shuttle remote anipulator systems the goal here is to develop numerical techniques to simulate and control these systems while properly modeling the coulomb friction the numerical procedure described employs a finite difference time discretization in conjunction with a vector of multipliers that predicts the friction effect for all time in addition to this discrete multiplier technique an associated regularization procedure that greatly improves the behavior of these multipliers is also presented numerical examples conclude the paper

extracting features of gaussian self similar stochastic processes via the bandt pompe approach by recourse to appropriate information theory quantifiers normalized shannon entropy and martin plastino rosso intensive statistical complexity measure we revisit the characterization of gaussian self similar stochastic processes from a bandt pompe viewpoint we show that the ensuing approach exhibits considerable advantages with respect to other treatments in particular clear quantifiers gaps are found in the transition between the continuous processes and their associated noises

symbolic dynamics of event related brain potentials we apply symbolic dynamics techniques such as word statistics and measures of complexity to nonstationary and noisy multivariate time series of electroencephalograms eeg in order to estimate event related brain potentials erp their significance against surrogate data as well as between different experimental conditions is tested these methods are validated by simulations using stochastic dynamical systems with time dependent control parameters and compared with traditional erp analysis techniques continuous eeg data are cut into epochs according to stimuli events presented to the subjects these ensembles of time series can be considered as ensembles of trajectories given by some dynamical systems we employ a statistical mechanics approach motivated by the frobenius perron equation and apply it to coarse grained symbolic descriptions of the dynamics we develop time dependent measures of complexity founded on running cylinder sets and show that these quantities are able to distinguish simulated data obtained by different control parameters as well as experimental data between different experimental conditions as a first finding our approach restores the well known erp components and it reveals additionally qualitative changes in the eeg that cannot be detected by means of the traditional techniques we criticize the prerequisites of the traditional approach to erp analysis and propose to consider erp instead in terms of dynamical system theory and information theory

an information based tool for inferring the nature of deterministic sources in real data the scope of the paper is to find signatures of the forces controlling complex systems modeled by langevin equations by recourse to information theory quantifiers we evaluate in detail the permutation entropy pe and the permutation statistical complexity psc measures for two similarity classes of stochastic models characterized by either drifting or reversion properties and employ them as a reference basis for the inspection of real series new relevant model parameters arise as compared to standard entropy measures we determine the normalized pe and psc curves according to them over a range of permutation orders n and infer the limiting measures for arbitrary large order we found that the psc measure is strongly scale dependent with systems of the drifting class showing crossovers as n increases this result gives warning signs about the proper interpretation of finite scale analysis of complexity in general processes conversely a key n invariant outcome arises that is the normalized pe values for both classes of models keep complementary for any n we argue that both pe and psc measures enable one to unravel the nature drifting or restoring of the deterministic sources underlying complexity we conclude by investigating the presence of local trends in stock price series