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

nonextensive entropies from information content and stabilities abstract we consider a family of nonextensive entropies derived from information contents which are a basic ingredient in information theory to give a deeper insight into a certain class of stability of generalized entropies and into recent discussion on it we present a criterion for stability which is satisfied by a minimum discrepancy against infinitesimal shift of distributions

complexity analysis of hypergeometric orthogonal polynomials the complexity measures of the cramer rao fisher shannon and lmc lopez ruiz mancini and calvet types of the rakhmanov probability density n x x p n 2 x of the polynomials p n x orthogonal with respect to the weight function x x a b are used to quantify various two fold facets of the spreading of the hermite laguerre and jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways their explicit cramer rao and asymptotical fisher shannon lmc values are given for the three systems of orthogonal polynomials then these complexity type mathematical quantities are numerically examined in terms of the polynomial s degree n and the parameters which characterize the weight function finally several open problems about the generalized hypergeometric functions of lauricella and srivastava daoust types as well as on the asymptotics of weighted l q norms of laguerre and jacobi polynomials are pointed out

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 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

differential escort transformations and the monotonicity of the lmc renyi complexity measure abstract escort distributions have been shown to be very useful in a great variety of fields ranging from information theory nonextensive statistical mechanics till coding theory chaos and multifractals in this work we give the notion and the properties of a novel type of escort density the differential escort densities which have various advantages with respect to the standard ones we highlight the behavior of the differential shannon renyi and tsallis entropies of these distributions then we illustrate their utility to prove the monotonicity property of the lmc renyi complexity measure and to study the behavior of general distributions in the two extreme cases of minimal and very high lmc renyi complexity finally this transformation allows us to obtain the tsallis q exponential densities as the differential escort transformation of the exponential density

a generalized statistical complexity measure applications to quantum systems a two parameter family of complexity measures c based on the renyi entropies is introduced and characterized by a detailed study of its mathematical properties this family is the generalization of a continuous version of the lopez ruiz mancini calbet complexity which is recovered for 1 and 2 these complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system when one of the parameters or goes to infinity one of the global factors becomes a local factor for this special case the complexity is calculated on different quantum systems h atom harmonic oscillator and square well

random differential operational calculus theory and applications in this article we obtain a product rule and a chain rule for mean square derivatives an application of the chain rule to the mean square solution of random differential equations is shown however to achieve such mean square differentiation rules fourth order properties were needed and therefore we first studied a mean fourth order differential and integral calculus results are applied to solve random linear variable coefficient differential problems

the existence of efficient lattice rules for multidimensional numerical integration in this contribution to the theory of lattice rules for multidimensional numerical integration we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants which extends an earlier result of the author for lattice rules of rank 1

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

a novel dynamic model of pseudo random number generator an interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory the main objective of this paper is to produce an ergodic dynamical system which can be implemented in random number generators in order to check the efficacy of pseudo random number generators based on this map we have carried out certain statistical tests on a series of numbers obtained from the introduced hierarchy the results of the tests were promising as the hierarchy passed the tests satisfactorily and offers a great capability to be employed in a pseudo random number generator

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

comment ii on simple measure for complexity the measure of complexity recently proposed by shiner davidson and landsberg phys rev e 59 1459 1999 does not adequately describe the transition from regular to indexed languages observed at the period doubling accumulation points of quadratic maps this comment points to a generic inadequacy of that measure

mean square solution of bessel differential equation with uncertainties this paper deals with the study of a bessel type differential equation where input parameters coefficient and initial conditions are assumed to be random variables using the so called l p random calculus and assuming moment conditions on the random variables in the equation a mean square convergent generalized power series solution is constructed as a result of this convergence the sequences of the mean and standard deviation obtained from the truncated power series solution are convergent as well the results obtained in the random framework extend their deterministic counterpart the theory is illustrated in two examples in which several distributions on the random inputs are assumed finally we show through examples that the proposed method is computationally faster than monte carlo method

statistical measures of complexity for quantum systems with continuous variables abstract the fisher shannon statistical measure of complexity is analyzed for a continuous manifold of quantum observables it is shown that evaluating this measure only in the configuration or in the momentum spaces does not provide an adequate characterization of the complexity of some quantum systems in order to obtain a more complete description of complexity two new measures respectively based on the minimization and the integration of the usual fisher shannon measure over all the parameter space are proposed and compared finally these measures are applied to the concrete case of a free particle in a box

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

a lower bound for rank 2 lattice rules we give a lower bound for a quality measure of rank 2 lattice rules which shows that an existence result of niederreiter is essentially best possible

relativistic effects on information measures for hydrogen like atoms position and momentum information measures are evaluated for the ground state of the relativistic hydrogen like atoms consequences of the fact that the radial momentum operator is not self adjoint are explicitly studied exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances the shannon and renyi entropies the fisher information measure as well as several related information measures are considered as viable alternatives detailed results on the onset of relativistic effects for low nuclear charges and on the extreme relativistic limit are presented the relativistic position density decays exponentially at large r but is singular at the origin correspondingly the momentum density decays as an inverse power of p both features yield divergent renyi entropies away from a finite vicinity of the shannon entropy while the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom this is not the case for the relativistic momentum space some of the results allow interesting insight into the significance of recently evaluated dirac fock vs hartree fock complexity measures for many electron neutral atoms

disequilibrium complexity the schottky effect and q entropies in paramagnetism abstract we investigate connections between statistical quantifiers and paramagnetism more concretely we apply the notions of i disequilibrium and ii statistical complexity to a paramagnetic system of non coupled dipoles interesting insights are thereby obtained in particular we encounter a kind of criticality not associated to the temperature but to the disequilibrium

solving initial and two point boundary value linear random differential equations a mean square approach abstract this paper deals with the construction of mean square real valued solutions to both initial and boundary value problems of linear differential equations whose coefficients are assumed to be stochastic processes and initial and boundary conditions are random variables a key result to conduct our study is the extension of the leibniz integral rule to the random framework taking advantage of the so called random fourth calculus exact expressions for the main statistical functions average and variance associated to the solutions to both problems are also provided illustrative examples computing the average and standard deviation are included

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

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

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

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

on the random gamma function theory and computing abstract this paper deals with the extension in the mean square sense of the deterministic gamma function to the random framework in a first step we provide such extension to a by assuming that the parameter a is a positive random variable satisfying certain conditions related to its exponential moments as a particular case we show that every positive random variable satisfies such conditions if it is bounded and bounded away from zero in a second step we establish the formula a 1 a a that allows us to extend the random gamma function to a class of random variables whose supports lie over the real line with the exception of small neighborhoods of zero and of the negative integers this retains the classical definition of the gamma function when a becomes a deterministic parameter the study is based on the l p stochastic calculus with p 2 and 4 usually referred to as mean square and mean fourth stochastic calculus respectively next we compute the mean and the variance of the random gamma function including several illustrative examples finally with the aid of the random gamma function we define the random bessel function and compute reliable approximations of its mean and variance

differential entropy and dynamics of uncertainty we analyze the functioning of gibbs type entropy functionals in the time domain with emphasis on shannon and kullback leibler entropies of time dependent continuous probability distributions the shannon entropy validity is extended to probability distributions inferred from l 2 r n quantum wave packets in contrast to the von neumann entropy which simply vanishes on pure states the differential entropy quantifies the degree of probability de localization and its time development the associated dynamics of the fisher information functional quantifies nontrivial power transfer processes in the mean both in dissipative and quantum mechanical cases

fisher like atomic divergences mathematical grounds and physical applications abstract two different local divergence measures the fisher fd and the jensen fisher jfd ones are compared in this work by applying them to atomic one particle densities in position and momentum spaces they are defined in terms of the absolute and the relative fisher information functionals the analysis here afforded includes not only neutral atoms but also singly charged cations the results are interpreted and justified according to i shell filling patterns ii short and long range behaviors of the atomic densities and iii the value of the atomic ionization potential the strengths of the fd measure as compared to the jfd one are emphasized

multiscale entropy analysis of complex physiologic time series there has been considerable interest in quantifying the complexity of physiologic time series such as heart rate however traditional algorithms indicate higher complexity for certain pathologic processes associated with random outputs than for healthy dynamics exhibiting long range correlations this paradox may be due to the fact that conventional algorithms fail to account for the multiple time scales inherent in healthy physiologic dynamics we introduce a method to calculate multiscale entropy mse for complex time series we find that mse robustly separates healthy and pathologic groups and consistently yields higher values for simulated long range correlated noise compared to uncorrelated noise

scaling properties of composite information measures and shape complexity for hydrogenic atoms in parallel magnetic and electric fields abstract the scaling properties of various composite information theoretic measures shannon and renyi entropy sums fisher and onicescu information products tsallis entropy ratio fisher shannon product and shape complexity are studied in position and momentum spaces for the non relativistic hydrogenic atoms in the presence of parallel magnetic and electric fields such measures are found to be invariant at the fixed values of the scaling parameters given by s 1 b 3 4 0 2 z 2 m 2 e 3 and s 2 f 4 4 0 3 z 3 e 5 m 2 numerical results which support the validity of the scaling properties are shown by choosing the representative example of the position space shape complexity physical significance of the resulting scaling behavior is discussed

entropy and complexity analysis of hydrogenic rydberg atoms the internal disorder of hydrogenic rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information theoretic spreading quantities the radial and logarithmic expectation values the shannon entropy and the fisher information as well the complexity measures of cramer rao fisher shannon and lopez ruiz mancini calvet types are investigated in both reciprocal spaces the leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the laguerre and gegenbauer orthogonal polynomials which control the wavefunctions of the rydberg states in both position and momentum spaces the associated generalized heisenberg like logarithmic and entropic uncertainty relations are also given finally application to linear l 0 circular l n 1 and quasicircular l n 2 states is explicitly done

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

lattice rules for multiple integration and discrepancy upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established in this way the applicability of lattice rules is extended to nonperiodic integrands

tendency towards maximum complexity in a nonequilibrium isolated system the time evolution equations of a simplified isolated ideal gas the open quotes tetrahedral close quotes gas are derived the dynamical behavior of the lopez ruiz endash mancini endash calbet complexity r lopez ruiz h l mancini and x calbet phys lett a 209 321 1995 is studied in this system in general it is shown that the complexity remains within the bounds of minimum and maximum complexity we find that there are certain restrictions when the isolated open quotes tetrahedral close quotes gas evolves towards equilibrium in addition to the well known increase in entropy the quantity called disequilibrium decreases monotonically with time furthermore the trajectories of the system in phase space approach the maximum complexity path as it evolves toward equilibrium

features of the extension of a statistical measure of complexity to continuous systems we discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by l opez ruiz mancini and calbet phys lett a 209 321 1995 in general the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some kind of choice in the present study several possibilities appear available one of them is examined in detail some interesting properties desirable for any magnitude of complexity are discovered on this particular extension

on a q generalization of circular and hyperbolic functions a generalization of the circular and hyperbolic functions is proposed based on the tsallis statistics and on a consistent generalization of the euler formula some properties of the presently proposed q trigonometry are then investigated the generalized functions are exact solutions of a nonlinear oscillator original circular and hyperbolic relations are recovered as the limiting case

reply to comments on simple measure for complexity we respond to the comment by crutchfield feldman and shalizi comment in this issue phys rev e 62 2996 2000 and that by binder and perry preceding comment phys rev e 62 2998 2000 pointing out that there may be many maximum entropies and therefore disorders and simple complexities which ones are appropriate depend on the questions being addressed disorder is not restricted to be the ratio of a nonequilibrium entropy to the corresponding equilibrium entropy therefore simple complexity need not vanish for all equilibrium systems nor must it be nonvanishing for a nonequilibrium system