twenty five years of multifractals in fully developed turbulence a tribute to giovanni paladin the paper on the multifractal nature of fully developed turbulence and chaotic systems by benzi et al 1984 j phys a math gen 17 3521 has been a starting point of many investigations on the different faces of self similarity and intermittency in turbulent phenomena since then the multifractal model has become a useful tool for the study of small scale turbulence in particular for detailed predictions of different eulerian and lagrangian statistical properties on the occasion of the 50th birthday of our unforgettable friend and colleague giovanni paladin 1958 1996 we review here the basic concepts and some applications of the multifractal model for turbulence

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

intermittency and rough pipe turbulence recently by analyzing the measurement data of nikuradze naca tech memo no 1292 1950 it has been proposed n goldenfeld phys rev lett 96 044503 2006 that the friction factor f of rough pipe flow obeys a scaling law in the turbulent regime here we provide a phenomenological scaling argument to explain this law and demonstrate how intermittency modifies the scaling form thereby relating f to the intermittency exponent ensuremath eta by statistically analyzing the measurement data of f we infer a satisfactory estimate for ensuremath eta ensuremath approx 0 02 the inclusion of which is shown to improve the data collapse curve this provides empirical evidence for intermittency other than the direct measurement of velocity fluctuations

cascade structures and scaling exponents in a dynamical model of turbulence measurements and comparison a detailed examination of the cascade statistics and scaling exponents is carried out for a dynamical system model of fully developed turbulence called the goy shell model the convergence in time of the probability density functions and moments of the velocity fluctuations and their scaling exponents is studied with particular care with a large sample size 5 ifmmode times else texttimes fi 10 9 we demonstrate that there exists a finite cutoff for the velocity fluctuations at each inertial range wave number shell and the properties of the cutoff determine the scaling exponents of all moments this cutoff represents the most intermittent structures in the cascade dynamics and exhibits a power law dependence on wave number the accurately determined scaling exponents permit a detailed comparison with various phenomenological models describing the statistics of the energy cascade the consideration of the first and second derivatives of the scaling exponents with respect to the order of the moments p provides the evidence that the hierarchical structure model she and leveque phys rev lett 72 336 1994 predicts the best functional dependence on p of the scaling exponents in the goy shell model

mixing and relaxation dynamics of the henon map at the edge of chaos the mixing properties or sensitivity to initial conditions and relaxation dynamics of the henon map together with the connection between these concepts have been explored numerically at the edge of chaos it is found that the results are consistent with those coming from one dimensional dissipative maps this constitutes the first verification of the scenario in two dimensional cases and obviously reinforces the idea of weak mixing

superstatistical generalization of the work fluctuation theorem abstract we derive a generalized version of the work fluctuation theorem for non equilibrium systems with spatio temporal temperature fluctuations for 2 distributed inverse temperature we obtain a generalized fluctuation theorem based on q exponentials whereas for other temperature distributions more complicated formulae arise since q exponentials have a power law decay the decay rate in this generalized fluctuation theorem is much slower than the conventional exponential decay this implies that work fluctuations can be of relevance for the design of micro or nano structures since the work done on the system is relatively much larger than in the conventional fluctuation theorem

asymmetric unimodal maps at the edge of chaos we numerically investigate the sensitivity to initial conditions of asymmetric unimodal maps x t 1 1 a x t z i i 1 2 correspond to x t 0 and x t 1 0 a or 2 t 0 1 2 at the edge of chaos we employ three distinct algorithms to characterize the power law sensitivity to initial conditions at the edge of chaos namely direct measure of the divergence of initially nearby trajectories the computation of the rate of increase of generalized non extensive entropies s q and multi fractal analysis the first two methods provide consistent estimates for the exponent governing the power law sensitivity in addition to this we verify that the multi fractal analysis does not provide precise estimates of the singularity spectrum f alpha especially near its extremal points such feature prevents to perform a fine check of the accuracy of the scaling relation between f alpha and the entropic index q thus restricting the applicability of the multi fractal analysis for studying the sensitivity to initial conditions in this class of asymmetric maps

anomaly of fractal dimensions observed in stochastically switched systems we studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven by stochastically switched inputs we calculated the dimensions for different switching time lengths in two dimensional linear dynamical systems and found that changes in the dimensions due to the switching time length had a singular point when the system matrix had two different real eigenvalues using partial dimensions along each eigenvector we explicitly derived a generalized dimension d q and a multifractal spectrum f alpha to explain this anomalous property the results from numerical calculations agreed well with those from analytical equations we found that this anomaly is caused by linear independence inhomogeneity of eigenvalues and overlapping conditions the mechanism for the anomaly could be identified for various inhomogeneous systems including nonlinear ones and this reminded us of anomalies in some physical values observed in critical phenomena

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2

analysis of self organized criticality in ehrenfest s dog flea model the self organized criticality in ehrenfest s historical dog flea model is analyzed by simulating the underlying stochastic process the fluctuations around the thermal equilibrium in the model are treated as avalanches we show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q gaussian the distribution which is obtained naturally in the context of nonextensive statistical mechanics if one avoids the finite size effects by increasing the system size we provide clear numerical evidence that the relation between the exponent ensuremath tau of avalanche size distribution obtained by maximum likelihood estimation and the q value of appropriate q gaussian obeys the analytical result recently introduced by caruso et al phys rev e 75 055101 r 2007 this allows us to determine the value of q parameter a priori from one of the well known exponents of such dynamical systems

nelkin scaling for the burgers equation and the role of high precision calculations nelkin scaling the scaling of moments of velocity gradients in terms of the reynolds number is an alternative way of obtaining inertial range information it is shown numerically and theoretically for the burgers equation that this procedure works already for reynolds numbers of the order of 100 or even lower when combined with a suitable extended self similarity technique at moderate reynolds numbers for the accurate determination of scaling exponents it is crucial to use higher than double precision similar issues are likely to arise for three dimensional navier stokes simulations

diffusion on random hierarchical structures the authors study diffusion on a statistically self similar fractal substrate with random transition rates possessing a hierarchical structure the master equation is solved analytically by means of a recursion technique and different choices of random hierarchies for the transition rates are considered leading to algebraic stretched exponential or exponential logarithmic behaviour for the moments and the correlation decay making use of the central limit theorem analytical expressions for the fluctuation corrections to lowest order in the relative variances are derived for the first two cases where the behaviour is qualitatively the same as in the systems without fluctuations the sign of the corrections turns out to depend on the temporal and spatial scale at which fluctuations are externally suppressed they are negative if the diffusion is taking place at smaller scales positive otherwise the relevance to diffusion in a turbulent medium intermittency corrections is discussed

turbulent self organized criticality abstract in the prototype sandpile model of self organized criticality time series obtained by decomposing avalanches into waves distinguished by the number of times the avalanche seed site has toppled show intermittent fluctuations the q th moments of wave size differences possess local multiscaling and global simple scaling regimes analogous to those holding for velocity structure functions in fluid turbulence the correspondence involves identity of a basic scaling relation and of the form of relevant probability distributions the sandpile provides a qualitative analog of many features of turbulent phenomena

waves of topplings in an abelian sandpile abstract we study the structure of the avalanche process in the 2d abelian sandpile model an avalanche may be represented as a sequence of waves each consisting of sites that toppled only once in that wave it is shown that the waves are in one to one correspondence with the set of two rooted spanning trees and can be described in terms of green functions we argue that the probability distribution of waves of size s varies as 1 s for large s we prove also that the avalanches started at the open boundary consist of only one wave and find the asymptotic distribution of their sizes which varies as s 3 2 we establish the equivalence between waves and inverse avalanches introduced by dhar and manna and reproduce their result 11 8 for the critical exponent of the size distribution of the first inverse avalanche or in our terms the last wave

directed fixed energy sandpile model we numerically study the directed version of the fixed energy sandpile on a closed square lattice the dynamical evolution of a fixed density of sand grains is studied the activity of the system shows a continuous phase transition around a critical density while the deterministic version has the set of nontrivial exponents the stochastic model is characterized by mean field like exponents

generalization of the maier saupe theory of the nematics within tsallis thermostatistics abstract in this study one of the mean field theories in nematics the maier saupe theory mst is generalized within tsallis thermostatistics tt the variation of the order parameter versus temperature has been investigated and compared with experimental data for p azoxyanisole paa so far we believe that this is the first attempt at the application of tt in liquid crystals it is well known that mst fails to explain the experimental data for some of the nematics one of which is paa however generalized mst is able to account for the experimental data for a long range of temperatures also in this study the effect of nonextensivity is shown for various values of the entropic index

continuous transforming the btw to the manna model abstract in this paper we define some stochastic perturbations of the btw model which make it into manna model these models have a continuous parameter p where p 0 and 1 correspond to the btw and manna models respectively we have investigated the properties of the statistical observables of the waves of avalanches for various values of p our data supports the expectation of a crossover in such systems at large scales manna model is dominant therefore we find strong evidence in favor of the universality classes being distinct also it is observed that the btw fixed point is unstable when manna type perturbation is added to the model

on a discrete dynamical model for local turbulence works towards an explanation of the observed experimental deviation of k41 exponent law are generally based on an energy cascade process an alternative approach based on the analysis of turbulent sequences allows us to construct a sequence model where this deviation has a dynamical origin we finally show why we were forced to go beyond the markovian model and incorporate into the model the frequent long flights an intermittent effect also present in the experimental data

dissipative sandpile models with universal exponents we consider a dissipative variant of the stochastic abelian sandpile model on a two dimensional lattice the boundaries are closed and the dissipation is due to the fact that each toppled grain is removed from the lattice with probability epsilon it is shown that the scaling properties of this model are in the universality class of the stochastic abelian models with conservative dynamics and open boundaries in particular the dissipation rate epsilon can be adjusted according to a suitable function epsilon f l such that the avalanche size distribution will coincide with that of the conservative model on a finite lattice of size l

a nonextensive statistical approach to the kinetics of phase transformation abstract some limitations of johnson mehl avrami kolmogorov jmak equation which is used widely for describing kinetics of phase transformation were demonstrated using probabilistic analysis and monte carlo simulations acta mater 48 2000 4217 as well known jmak equation predicts correctly the real transformed fraction only if the number of the growing nuclei in the controlled volume is large however jmak equation seems to fail if the number of growing nuclei is small no matter how large the volume of the controlled volume is in this study we propose a different equation for describing kinetics of phase transformation using a nonextensive formalism namely tsallis thermostatistics which has been commonly employed to study the various physical systems for a decade

power law sensitivity to initial conditions in a time series with applications to epileptic seizure detection recently the concept of exponential sensitivity to initial conditions esic of deterministic chaos is generalized to power law sensitivity to initial conditions psic we describe a general computational procedure to examine psic from a time series study of noise free and noisy logistic and henon maps at the edge of chaos finds that psic cannot be shown from clean scalar time series however when there is dynamic noise motions around the edges of chaos all collapse onto the psic attractor regardless whether they are simply regular or truly chaotic when noise is absent hence dynamic noise makes psic observable the psic concept is further applied to the analysis of long continuous eeg signals with epileptic seizures it is shown that measures from the psic framework is quite effective in detecting seizures often better than the lyapunov exponent based methods from the conventional esic framework

entropy optimization models with convex constraints in this paper we study the minimum cross entropy optimization problem subject to a general class of convex constraints using a simple geometric inequality and the conjugate inequality we demonstrate how to directly construct a partial geometric dual program which allows us to apply the dual perturbation method to derive the strong duality theorem and a dual to primal conversion formula this approach generalizes the previous results of linearly quadratically and entropically constrained cross entropy optimization problems and provides a platform for using general purpose optimizers to generate optimal solution pair to the problem

linking eulerian and lagrangian structure functions scaling exponents in turbulence abstract in fully developed turbulence intermittency is classically characterized by e q the eulerian scaling exponent of structure functions the same approach can be used in a lagrangian framework using l q to characterize the temporal intermittency of the velocity of a particle advected by a turbulent intermittent field an interesting question is then to know how to relate the scaling functions and explore the links between l q and e q we first provide different transformations between these functions associated to different transformations linking space and time scales we test these relations using experimental estimates for the two functions for small and medium order of moments they are both close to data for larger moments experimental estimates have still too much scatter to conclude the present paper underlines the need of more precise estimates and provides a methodology for future comparisons of eulerian and lagrangian scaling exponents

statistical properties of a turbulent cascade abstract statistical properties of a turbulent cascade are evaluated by considering the joint probability distribution p 1 l1 2 l2 for two velocity increments 1 2 of different length scales l1 l2 we present experimental evidence that the conditional probability distribution p 2 l2 1 l1 obeys a chapman kolmogorov equation we evaluate the kramers moyal coefficient and show evidence that higher order coefficients vanish except for the drift and diffusion coefficient as a result the joint probability distributions obeys a fokker planck equation we calculate drift and diffusion coefficients and discuss their relationship to universal behaviour in the scaling region and to intermittency of the turbulent cascade

statistics of toppling wave boundaries in deterministic and stochastic sandpile models we study numerically the statistics of curves which form the boundaries of toppling wave clusters in the deterministic bak tang and wiesenfeld sandpile model and stochastic manna model on a square lattice we consider the abelian version of each model multiple tests show that the boundary of toppling wave clusters in both deterministic and stochastic models can be described by sle curves with diffusivity 2

intermittency and predictability in a shell model for three dimensional turbulence abstract we review some recent work on scaling laws in turbulence the navier stokes equations and the equation for the transport of passive scalar fields are formulated in terms of shell models in fourier space which incorporate the appropriate conservation laws we find that the kolmogorov scaling for the energy cascade and the obukhov corrsin scaling for the passive scalar cascade are dynamically unstable this fact gives rise to corrections to classical values of the scaling exponents we relate these corrections to the dynamical behavior of the intermittency bursts present in the shell models our numerical results are in good quantitative agreement with the experimental results in fluids finally we discuss the problem of predictability

quasi additivity of tsallis entropies and correlated subsystems abstract we use beck s quasi additivity of tsallis entropies for n independent subsystems to show that like the case of n 2 the entropic index q approaches 1 by increasing system size then we will generalize that concept to correlated subsystems to find that in the case of correlated subsystems when system size increases q also approaches a value corresponding to the additive case

non extensive random walks stochastic variables whose addition leads to q gaussian distributions gq x 1 q 1 bx 2 1 1 q with b40 1pqo3 and where f x maxff x 0g as limit law for a large number of terms are investigated random walk sequences related to this problem possess a simple additive multiplicative structure commonly found in several contexts thus justifying the ubiquity of those distributions a characterization of the statistical properties of the random walk step lengths is performed moreover a connection with non linear stochastic processes is exhibited q gaussian distributions have special relevance within the framework of non extensive statistical mechanics a generalization of the standard boltzmann gibbs formalism introduced by tsallis over one decade ago therefore the present findings may give insights on the domain of applicability of such generalization r 2005 elsevier b v all rights reserved

steady state of stochastic sandpile models we study the steady state of the abelian sandpile models with stochastic toppling rules the particle addition operators commute with each other but in general these operators need not be diagonalizable we use their abelian algebra to determine their eigenvalues and the jordan block structure these are then used to determine the probability of different configurations in the steady state we illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example for systems of size 12 and also study the density profile in the steady state

characteristics of deterministic and stochastic sandpile models in a rotational sandpile model rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena in order to study the effect of rotational bias in an out of equilibrium situation like self organized criticality a two state quasideterministic rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains an extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model the probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves the model exhibits characteristics of both deterministic and stochastic sandpile models

scaling exponents in fully developed turbulence from nonextensive statistical mechanics abstract we combine tsallis statistics with extended selfsimilarity to calculate scaling exponents of moments in a nonextensive statistical model of fully developed turbulence one obtains numerical values of scaling exponents that agree very well with experimental data our approach is based on a scale dependent nonextensitivity parameter q 1 and can be embedded into previously successful models for probability densities of velocity differences

scale dependent dimension in the forest fire model the forest fire model is a reaction diffusion model where energy in the form of trees is injected uniformly and burned dissipated locally we show that the spatial distribution of fires forms a geometric structure where the fractal dimension varies continuously with the length scale in the three dimensional model the dimensions vary from zero to three proportional with mathrm ln l as the length scale increases from l ensuremath sim 1 to a correlation length l ensuremath xi beyond the correlation length which diverges with the growth rate p as ensuremath xi ensuremath propto p ensuremath 2 3 the distribution becomes homogeneous we suggest that this picture applies to the intermediate range of turbulence where it provides a natural interpretation of the extended scaling that has been observed at small length scales unexpectedly it might also be applicable to the spatial distribution of luminous matter in the universe in the two dimensional version the dimension increases to d 1 at a length scale l ensuremath sim 1 p where there is a crossover to homogeneity i e a jump from d 1 to d 2

log poisson statistics and extended self similarity in driven dissipative systems abstract the bak chen tang forest fire model phys lett a 147 1990 297 was proposed as a toy model of turbulent systems where energy in the form of trees is injected uniformly and globally but is dissipated burns locally we review our previous results on the model phys rev e 62 2000 1613 phys rev lett 86 2000 4215 and present our new results on the statistics of the higher order moments for the spatial distribution of fires we show numerically that the spatial distribution of dissipation can be described by log poisson statistics which leads to extended self similarity phys rev e 48 1993 r29 phys rev lett 73 1994 959 similar behavior is also found in models based on directed percolation this suggests that the concept of log poisson statistics of appropriately normalized variables can be used to describe scaling not only in turbulence but also in a wide range of driven dissipative systems

extended quasi additivity of tsallis entropies abstract we consider statistically independent non identical subsystems with different entropic indices q 1 and q 2 a relation between q 1 q 2 and q for the entire system extends a power law for entropic index as a function of distance r a few examples illustrate a role of the proposed constraint q min q 1 q 2 for the beck s concept of quasi additivity

extended self similarity in kinetic surface roughening we show from simulations that a limited mobility solid on solid model of kinetically rough surface growth exhibits extended self similarity analogous to that found in fluid turbulence the range over which scale independent power law behavior is observed is significantly enhanced if two correlation functions of different order such as those representing two different moments of the difference in height between two points are plotted against each other this behavior found in both one and two dimensions suggests that the relative exponents may be more fundamental than the absolute ones

metastable states anomalous distributions and correlations in the hmf model abstract we study the microscopic dynamics of the metastable quasi stationary states qss in the hamiltonian mean field hmf model a hamiltonian system of n classical inertial spins with infinite range interactions which shows a second order phase transition in order to understand the origin of metastability which appears in an energy region below the critical point we consider two different classes of out of equilibrium initial conditions both leading to qss and having respectively initial magnetization equal to one m1 ic and equal to zero m0 ic we compare the corresponding space the resulting velocity pdfs and correlations and the eventual aging features of the microscopic dynamics in both cases the model exhibits nongaussian pdfs though anomalous correlations are present only when the system is started with an initial magnetization equal to one in the m0 ic case the relaxation to equilibrium is almost exponential while for m1 ic when correlations and aging are found the decay is a power law and the overall behavior can be very well reproduced by a tsallis q exponential function these results contribute to clarify the overall scenario which is more complex than previously expected and stress the importance of the dynamics in the relaxation process the nonextensive statistical mechanics formalism proposed by tsallis seems to be valid in the out of equilibrium phase when correlations and strong long term memory effects emerge this regime becomes stable if the n limit is performed before the t limit

beyond scaling and locality in turbulence an analytic perturbation theory is suggested in order to find finite size corrections to the scaling power laws in the frame of this theory it is shown that the first order finite size correction to the scaling power laws has following form s r cong cr alpha 0 ln r eta alpha 1 where is a finite size scale in particular for turbulence it can be the kolmogorov dissipation scale using data of laboratory experiments and numerical simulations it is shown shown that a degenerate case with 0 0 can describe turbulence statistics in the near dissipation range r where the ordinary power law scaling does not apply for moderate reynolds numbers the degenerate scaling range covers almost the entire range of scales of velocity structure functions the log corrections apply to finite reynolds number interplay between local and non local regimes has been considered as a possible hydrodynamic mechanism providing the basis for the degenerate scaling of structure functions and extended self similarity these results have been also expanded on passive scalar mixing in turbulence overlapping phenomenon between local and non local regimes and a relation between position of maximum of the generalized energy input rate and the actual crossover scale between these regimes are briefly discussed

tsallis statistics and gradually truncated levy flight distribution of an economical index abstract tsallis postulated a generalized form for entropy and give rise to a new statistics now known as tsallis statistics in the present work we compare the tsallis statistics with the gradually truncated levy flight and discuss the distribution of an economical index the standard and poor s 500 using the values of standard deviation as calculated by our model we find that both statistics give almost the same distribution thus we feel that gradual truncation of levy distribution after certain critical step size for describing complex systems is a requirement of generalized thermodynamics or similar the gradually truncated levy flight is based on physical considerations and bring a better physical picture of the dynamics of the whole system tsallis statistics gives a theoretical support both statistics together can be utilized for the development of a more exact portfolio theory or to understand better the complexities in human and financial behaviors a comparison of both statistics is made

application of the beck model to stock markets value at risk and portfolio risk assessment abstract we apply the beck model developed for turbulent systems that exhibit scaling properties to stock markets our study reveals that the beck model elucidates the properties of stock market returns and is applicable to practical use such as the value at risk estimation and the portfolio analysis we perform empirical analysis with daily intraday data of the s p500 index return and find that the volatility fluctuation of real markets is well consistent with the assumptions of the beck model the volatility fluctuates at a much larger time scale than the return itself and the inverse of variance or inverse temperature obeys distribution as predicted by the beck model the distribution of returns is well fitted by q gaussian distribution of tsallis statistics the evaluation method of value at risk var one of the most significant indicators in risk management is studied for q gaussian distribution our proposed method enables the var evaluation in consideration of tail risk which is underestimated by the variance covariance method a framework of portfolio risk assessment under the existence of tail risk is considered we propose a multi asset model with a single volatility fluctuation shared by all assets named the single model and empirically examine the agreement between the model and an imaginary portfolio with dow jones indices it turns out that the single model gives good approximation to portfolios composed of the assets with non gaussian and correlated returns

scaling of waves in the bak tang wiesenfeld sandpile model n with aen ae0 n a similar ansatzes should according to this view hold for the number of topplings s which difiers from a because sites can topple more than once in an avalanche and for the radius and duration of avalanches but recent inves tigations 10 12 show that the two dimensional btw model may be characterized by a multifractal behavior where difierent moments of a are governed by exponents aen which are not linear in n and are indeed not related to each other for difierent n difierent reasons for this have been proposed in 10 11 and in 12 notice that multi fractality of avalanches can be proven for certain variants of the 1 d sandpile model 14 deviations from pure power laws had been seen already in early simulations but were usually interpreted as fl nite size efiects due to avalanches which touch the bound ary of the lattice to illustrate that this is most likely not true and that there is a real problem with simple scaling we show in fig 1 the ratio ps x l pa x l of the integrated distributions ps x l r 1 x dx 0 ps x 0 l for d 2 and for difierent values of l in these simula tions cylindrical boundary conditions were used open at y l 2 and periodic at x 0 l and data were col lected only for avalanches starting at y 0 in this way we hope to have minimized boundary efiects also since we do not make separate flts to ps x l and pa x l we have none of the uncertainties inherent in such flts due

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

constructing a statistical mechanics for beck cohen superstatistics the basic aspects of both boltzmann gibbs bg and nonextensive statistical mechanics can be seen through three different stages first the proposal of an entropic functional s bg k sigma i p i ln p i for the bg formalism with the appropriate constraints sigma i p i 1 and sigma i p i e i u for the bg canonical ensemble second through optimization the equilibrium or stationary state distribution p i e betae i z bg with z bg sigma j e betae j for bg third the connection to thermodynamics e g f bg 1 beta ln z bg and u bg partial differential partial differential beta ln z bg assuming temperature fluctuations beck and cohen recently proposed a generalized boltzmann factor b e integral infinity 0 dbetaf beta e betae this corresponds to the second stage described above in this paper we solve the corresponding first stage i e we present an entropic functional and its associated constraints which lead precisely to b e we illustrate with all six admissible examples given by beck and cohen

rare events and breakdown of simple scaling in the abelian sandpile model due to intermittency and conservation the abelian sandpile in two dimensions obeys multifractal rather than finite size scaling in the thermodynamic limit a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution thus rare events shape most of the scaling pattern and preserve a meaning for effective exponents which can be determined on the basis of numerical and exact results

nonintegrability chaos and complexity abstract two dimensional driven dissipative flows are generally integrable via a conservation law that is singular at equilibria nonintegrable dynamical systems are confined to n 3 dimensions even driven dissipative deterministic dynamical systems that are critical chaotic or complex have n 1 local time independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes those conservation laws generally have either branch cuts phase singularities or both the consequence of the existence of singular conservation laws for experimental data analysis and also for the search for scale invariant critical states via uncontrolled approximations in deterministic dynamical systems is discussed finally the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling nonuniversal and to some degree computationally complex

critical properties of a dissipative sandpile model on small world networks a dissipative sandpile model is constructed and studied on small world networks swns swns are generated by adding extra links between two arbitrary sites of a two dimensional square lattice with different shortcut densities ensuremath phi three regimes are identified regular lattice rl for ensuremath phi ensuremath lesssim 2 ensuremath 12 swn for 2 ensuremath 12 l ensuremath phi l0 1 and random network rn for ensuremath phi ensuremath ge 0 1 in the rl regime the sandpile dynamics is characterized by the usual bak tang and weisenfeld btw type correlated scaling whereas in the rn regime it is characterized by mean field scaling on swns both scaling behaviors are found to coexist small compact avalanches below a certain characteristic size s c are found to belong to the btw universality class whereas large sparse avalanches above s c are found to belong to the mean field universality class a scaling theory for the coexistence of two scaling forms on a swn is developed and numerically verified though finite size scaling is not valid for the dissipative sandpile model on rls or on swns it is found to be valid on rns for the same model finite size scaling on rns appears to be an outcome of super diffusive sand transport and uncorrelated toppling waves

lagrangian single particle turbulent statistics through the hilbert huang transform the hilbert huang transform is applied to analyze single particle lagrangian velocity data from numerical simulations of hydrodynamic turbulence the velocity trajectory is described in terms of a set of intrinsic mode functions c i t and of their instantaneous frequency i t on the basis of this decomposition we define the conditioned statistical moments of the c i modes named q order hilbert spectra hs we show that such quantities have enhanced scaling properties as compared to traditional fourier transform or correlation based structure functions statistical indicators thus providing better insights into the turbulent energy transfer process we present clear empirical evidence that the energylike quantity i e the second order hs displays a linear scaling in time in the inertial range as expected from a dimensional analysis we also measure high order moment scaling exponents in a direct way without resorting to the extended self similarity procedure this leads to an estimate of the lagrangian structure function exponents which are consistent with the multifractal prediction in the lagrangian frame as proposed by biferale et al

measuring nonextensitivity parameters in a turbulent couette taylor flow we investigate probability density functions of velocity differences at different distances r measured in a couette taylor flow for a range of reynolds numbers re there is good agreement with the predictions of a theoretical model based on nonextensive statistical mechanics where the entropies are nonadditive for independent subsystems we extract the scale dependent nonextensitivity parameter q r mathrm re from the laboratory data

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

quadratically constrained information theoretic anaylsis in this paper we extend the framework of minimum discrimination information estimation mdi techniques to include the quadratic and linear inequality constrained problems duality theory and computational aspects of the technique are exposed and the methodology is illustrated by a new solution procedure for an important problem in actuarial science

a simple duality proof in convex quadratic programming with a quadratic constraint and some applications abstract in this paper a simple derivation of duality is presented for convex quadratic programs with a convex quadratic constraint this problem arises in a number of applications including trust region subproblems of nonlinear programming regularized solution of ill posed least squares problems and ridge regression problems in statistical analysis in general the dual problem is a concave maximization problem with a linear equality constraint we apply the duality result to 1 the trust region subproblem 2 the smoothing of empirical functions and 3 to piecewise quadratic trust region subproblems arising in nonlinear robust huber m estimation problems in statistics the results are obtained from a straightforward application of lagrange duality

extended self similarity in turbulent flows we report on the existence of a hitherto undetected form of self similarity which we call extended self similarity ess ess holds at high as well as at low reynolds number and it is characterized by the same scaling exponents of the velocity differences of fully developed turbulence

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

tsallis distributions and 1 f noise from nonlinear stochastic differential equations probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems in this article we unite modeling of such distributions with the model of widespread 1 f noise we propose a class of nonlinear stochastic differential equations giving both the q exponential or q gaussian distributions of signal intensity revealing long range correlations and 1 f behavior of the power spectral density the superstatistical framework to get 1 f noise with q exponential and q gaussian distributions of the signal intensity is proposed as well

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

intermittency in turbulence abstract the phenomenon of turbulent intermittency and its characteristic features are briefly reviewed intermittent spottiness of turbulent activity has been modelled in various ways including dynamical models here we report on results from approximate solutions of the navier stokes equation itself based on a fourier superposition with a geometrically scaling subset of wave vector fourier weierstrass ansatz turbulent flow with reynolds numbers up to 10 7 can be treated the resolution comprises more than 3 orders of magnitude the probability density function for the velocity fluctuations becomes streched exponential with decreasing scales intermittency in time and structures in space show up in the solution remarks on the restricted usefulness of kolmogorov s refined similarity hypothesis are added we furthermore offer a new type of scaling law for the stretching exponent for the 1 u 1 probability density in terms of an algebraic decrease in powers of the logarithm of re

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

differentiable classification of 4 manifolds with singular riemannian foliations in this paper we first prove that any closed simply connected 4 manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4 manifolds mathbb s 4 mathbb cp 2 mathbb s 2 times mathbb s 2 or mathbb cp 2 pm mathbb cp 2 as an application we prove that any closed simply connected 4 manifold admitting a nontrivial singular riemannian foliation is diffeomorphic to a connected sum of copies of standard mathbb s 4 pm mathbb cp 2 and mathbb s 2 times mathbb s 2 a classification of singular riemannian foliations of codimension 1 on all closed simply connected 4 manifolds is obtained as a byproduct in particular there are exactly 3 non homogeneous singular riemannian foliations of codimension 1 complementing the list of cohomogeneity one 4 manifolds

pdf of velocity fluctuation in turbulence by a statistics based on generalized entropy an analytical formula for the probability density function pdf of the velocity fluctuation in fully developed turbulence is derived non perturbatively by assuming that its underlying statistics is the one based on the generalized measures of entropy the renyi entropy or the tsallis entropy the parameters appearing in the pdf including the index q in the generalized measures are determined self consistently with the help of the observed value of the intermittency exponent the derived pdf explains quite well the experimentally observed density functions

highly turbulent solutions of the lagrangian averaged navier stokes alpha model and their large eddy simulation potential we compute solutions of the lagrangian averaged navier stokes alpha lans alpha model for significantly higher reynolds numbers up to re approximately 8300 than have previously been accomplished this allows sufficient separation of scales to observe a navier stokes inertial range followed by a second inertial range specific to the lans alpha model both fully helical and nonhelical flows are examined up to reynolds numbers of approximately 1300 analysis of the third order structure function scaling supports the predicted l3 scaling it corresponds to a k 1 scaling of the energy spectrum for scales smaller than alpha the energy spectrum itself shows a different scaling which goes as k1 this latter spectrum is consistent with the absence of stretching in the subfilter scales due to the taylor frozen in hypothesis employed as a closure in the derivation of the lans alpha model these two scalings are conjectured to coexist in different spatial portions of the flow the l3 e k approximately k 1 scaling is subdominant to k1 in the energy spectrum but the l3 scaling is responsible for the direct energy cascade as no cascade can result from motions with no internal degrees of freedom we demonstrate verification of the prediction for the size of the lans alpha attractor resulting from this scaling from this we give a methodology either for arriving at grid independent solutions for the lans alpha model or for obtaining a formulation of the large eddy simulation optimal in the context of the alpha models the fully converged grid independent lans alpha model may not be the best approximation to a direct numerical simulation of the navier stokes equations since the minimum error is a balance between truncation errors and the approximation error due to using the lans alpha instead of the primitive equations furthermore the small scale behavior of the lans alpha model contributes to a reduction of flux at constant energy leading to a shallower energy spectrum for large alpha these small scale features however do not preclude the lans alpha model from reproducing correctly the intermittency properties of the high reynolds number flow

the statistical distributions of one dimensional turbulence we study a one dimensional discrete analog of the von karman flow widely investigated in turbulence made of a lattice of anharmonic oscillators excited by both ends in the presence of a dissipative term proportional to the second order finite difference of the velocities similar to the viscous term in a fluid the dynamics of the model shows striking similarities with an actual turbulent flow both at local and global scales calculations of the probability distribution function of velocity increments extensively studied in turbulence with a very large number of points in order to determine accurately the statistics of rare events allow us to provide a meaningful comparison of different theoretical expressions of the pdfs

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

a simple duality proof for quadratically constrained entropy functional and extension to convex constraints in this paper a simple proof of duality results is presented that was recently derived by zhang and brockett siam j appl math 47 1987 pp 871 885 for the problem of minimizing the kullback liebler discrimination information measure subject to quadratic and linear inequality constraints the proof is a direct and simple application of lagrangian duality theory and is extended to a more general class of entropy optimization problems with composite convex constraints

from waves to avalanches two different mechanisms of sandpile dynamics time series resulting from wave decomposition show the existence of different correlation patterns for avalanche dynamics for the d 2 bak tang wisenfeld model long range correlations determine a modification of the wave size distribution under coarse graining in time and multifractal scaling for avalanches in the manna model the distribution of avalanche coincides with that of waves which are uncorrelated and obey finite size scaling a result expected also for the d 3 bak tang wiesenfeld sandpile

moduli spaces of anti self dual connections on ale gravitational instantons we study the framed moduli space m of anti self dual connections on a principal bundle over an ale hyperkahler 4 manifold it has the natural riemannian metric g m and the hyperkahler structure i m j m k m induced from those on the base manifold

fractional kinetics from pseudochaotic dynamics to maxwell s demon abstract in hamiltonian dynamics chaotic trajectories can be characterized by a non zero lyapunov exponent in general case of random dynamics the lyapunov exponent can be close to zero because of the stickiness or simply zero as in the case of pseudochaos kinetic description of such situations is based on scaling properties of the dynamics in both space and time it is shown for different models that the ergodic theorem cannot be applied for the observed data and that weak mixing leads to unusual macroscopic behavior such phenomenon as maxwell s demon obtains a natural realization as a persistent fluctuation that does not decay in an exponential way as in the kinetics of the gaussian type

minimum cross entropy spectral analysis of multiple signals this paper presents a new information theoretic method for simultaneously estimating a number of power spectra when a prior estimate of each is available and new information is obtained in the form of values of the autocorrelation function of their sum one application of this method is the separate estimation of the spectra of a signal and additive noise based on autocorrelations of the signal plus noise a derivation of the method from the principle of minimum cross entropy is given and the method is compared to minimum cross entropy spectral analysis of which it is a generalization some basic mathematical properties are discussed three numerical examples are included two based oil synthetic spectra and one based on actual speech data

on the relevance of q distribution functions the return time distribution of restricted random walker there exists a large literature on the application of q statistics to the out of equilibrium non ergodic systems in which some degree of strong correlations exists here we study the distribution of first return times to zero pr 0 t of a random walk on the set of integers 0 1 2 l with a position dependent transition probability given by we find that for all values of pr 0 t can be fitted by q exponentials but only for a 1 is pr 0 t given exactly by a q exponential in the limit this is a remarkable result since the exact analytical solution of the corresponding continuum model represents pr 0 t as a sum of bessel functions with a smooth dependence on a from which we are unable to identify a 1 as of special significance however from the high precision numerical iteration of the discrete master equation we do verify that only for a 1 is pr 0 t exactly a q exponential and that a tiny departure from this parameter value makes the distribution deviate from q exponential further research is certainly required to identify the reason for this result and also the applicability of q statistics and its domain

a new nonlinear quantizer for image processing within nonextensive statistics abstract in this study we introduce a new nonlinear quantizer for image processing by using tsallis entropy lloyd max quantizer is commonly used in minimizing the quantization errors we report that the new introduced technique works better than lloyd max one for selected standard images and could be an alternative way to minimize the quantization errors for image processing we therefore hopefully expect that the new quantizer could be a useful tool for all the remaining process after image quantization such as coding lossy and lossless compression

nonlinear statistical coupling abstract by considering a nonlinear combination of the probabilities of a system a physical interpretation of tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed the escort probability is interpreted as the coupled probability with q 1 q defined as the degree of nonlinear coupling between the statistical states positive values of q have coupled statistical states a larger entropy metric and a maximum coupled entropy distribution of compact support coupled gaussians negative values of q have decoupled statistical states and for 2 q 0 a maximum coupled entropy distribution of heavy tail coupled gaussians the conjugate transformation between the heavy tail and compact support domains is shown to be q 2 q 2 q for coupled gaussian distributions this conjugate relationship has been used to extend the generalized fourier transform to the compact support domain and to define a scale invariant correlation structure with heavy tail limit distribution in the present paper we show that the conjugate is a mapping between the source of nonlinearity in non stationary stochastic processes and the nonlinear coupling which defines the coupled gaussian limit distribution the effects of additive and multiplicative noise are shown to be separable into the coupled variance and the coupling parameter q providing further evidence of the importance of the generalized moments