model reduction for dynamical systems with quadratic output summary finite element models for structures and vibrations often lead to second order dynamical systems with large sparse matrices for large scale finite element models the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost pade via krylov methods are widely used and are appreciated projection based model reduction techniques for linear dynamical systems with linear output this paper extends the framework of the krylov methods to systems with a quadratic output arising in linear quadratic optimal control or random vibration problems three different two sided model reduction approaches are formulated based on the krylov methods for all methods the control or right krylov space is the same the difference between the approaches lies thus in the choice of the observation or left krylov space the algorithms and theory are developed for the particularly important case of structural damping we also give numerical examples for large scale systems corresponding to the forced vibration of a simply supported plate and of an existing footbridge in this case a block form of the pade via krylov method is used copyright 2012 john wiley sons ltd received 27 april 2011 revised 28 october 2011 accepted 17 november 2011 in structural dynamics and vibro acoustics frequent use is made of finite element fe models to design and analyze structures subjected to dynamic loading for large scale fe models the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost efficient evaluation of the frequency response function is extremely important in reducing computation time of parametric studies see 1 several methods have been developed for efficient evaluation among the most widely used methods is the model order reduction mor through modal superposition in this case a galerkin projection is performed on a reduced basis consisting of a subset of the eigenvectors of the generalized eigenvalue problem associated with the stiffness and mass matrix of the fe model of the structure this method is particularly efficient in low frequency range where the modal density is small and the structural response is dominated by a few global modes the reduction does not come without a price however as a high number of modes may be required to obtain a good approximation and the computation of these modes may be quite expensive particularly for a large fe model furthermore it is not always easy to select the modes that should be included in the reduced basis other methods which have shown to be extremely useful are related to the ritz vector technique 2 3 this is a krylov subspace method which is also known as pade via krylov 4 7 the system s output is approximated by a low degree rational polynomial such that the first derivatives at the

superstatistical distributions from a maximum entropy principle we deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information a maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter beta of a superstatistical system given certain constraints on the complex system under consideration we apply the theory to three examples the superstatistical quantum mechanical harmonic oscillator the superstatistical classical ideal gas and velocity time series as measured in a turbulent taylor couette flow

fast frequency response computation for rayleigh damping we present a model reduction lanczos method for the computation of frequency response functions of rayleigh damping problems the connection with modal extraction and modal superposition is made the usual model reduction methods require complex arithmetic with the proposed method complex arithmetic is only used for the reduced problem a numerical example from structural damping is given copyright 2007 john wiley sons ltd

extreme value distributions and renormalization group in the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent identically distributed random variables are studied the vast majority of the literature on the subject deals with affine normalization we argue that more general normalizations are natural from a mathematical and physical point of view and work them out the problem is approached using the language of renormalization group transformations in the space of probability densities the limit distributions are fixed points of the transformation and the study of its differential around them allows a local analysis of the domains of attraction and the computation of finite size corrections

a closer look at the indications of q generalized central limit theorem behavior in quasi stationary states of the hmf model abstract we give a closer look at the central limit theorem clt behavior in quasi stationary states of the hamiltonian mean field model a paradigmatic one for long range interacting classical many body systems we present new calculations which show that following their time evolution we can observe and classify three kinds of long standing quasi stationary states qss with different correlations the frequency of occurrence of each class depends on the size of the system the different microscopic nature of the qss leads to different dynamical correlations and therefore to different results for the observed clt behavior

the limit distribution in the q clt for is unique and can not have a compact support in a paper by umarov et al 2008 milan j math 76 307 28 a generalization of the fourier transform called the q fourier transform was introduced and applied for the proof of a q generalized central limit theorem q clt subsequently hilhorst illustrated 2009 braz j phys 39 371 9 2010 j stat mech p10023 that the q fourier transform for is not invertible in the space of density functions indeed using an invariance principle he constructed a family of densities with the same q fourier transform and noted that as a consequence the q clt falls short of achieving its stated goal the distributions constructed there have compact support we prove now that the limit distribution in the q clt is unique and can not have a compact support this result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by hilhorst

escort mean values and the characterization of power law decaying probability densities escort mean values or q moments constitute useful theoretical tools for describing basic features of some probability densities such as those which asymptotically decay like power laws they naturally appear in the study of many complex dynamical systems particularly those obeying nonextensive statistical mechanics a current generalization of the boltzmann gibbs theory they recover standard mean values or moments for q 1 here we discuss the characterization of a non negative probability density by a suitable set of all its escort mean values together with the set of all associated normalizing quantities provided that all of them converge this opens the door to a natural extension of the well known characterization for the q 1 instance of a distribution in terms of the standard moments provided that all of them have finite values this question would be specially relevant in connection with probability densities having divergent values for all nonvanishing standard moments higher than a give

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

diffusive mixing and tsallis entropy brownian motion the classical diffusive process maximizes the boltzmann gibbs entropy the tsallis q entropy which is non additive was developed as an alternative to the classical entropy for systems which are non ergodic a generalization of brownian motion is provided that maximizes the tsallis entropy rather than the boltzmann gibbs entropy this process is driven by a brownian measure with a random diffusion coefficient in addition the distribution of this coefficient is derived as a function of q for 1 q 3 applications to transport in porous media are considered

influence of global correlations on central limit theorems and entropic extensivity abstract we consider probabilistic models of n identical distinguishable binary random variables if these variables are strictly or asymptotically independent then for n i the attractor in distribution space is according to the standard central limit theorem a gaussian and ii the boltzmann gibbs shannon entropy s bgs i 1 w p i ln p i where w 2 n is extensive meaning that s bgs n n if these variables have any nonvanishing global i e not asymptotically independent correlations then the attractor deviates from the gaussian the entropy appears to be more robust in the sense that in some cases s bgs remains extensive even in the presence of strong global correlations in other cases however even weak global correlations make the entropy deviate from the normal behavior more precisely in such cases the entropic form s q 1 q 1 1 i 1 w p i q with s 1 s bgs can become extensive for some value of q 1 this scenario is illustrated with several new as well as previously described models the discussion illuminates recent progress into q describable nonextensive probabilistic systems and the conjectured q central limit theorem q clt which posses a q gaussian attractor

order reduction of large scale second order systems using krylov subspace methods in order reduction of large scale linear time invariant systems krylov subspace methods based on moment matching are among the best choices today however in many technical fields models typically consist of sets of second order differential equations and krylov subspace methods cannot directly be applied two methods for solving this problem are presented in this paper 1 an approach by su and craig is generalized and the number of matching moments is increased 2 a new approach via first order models is presented resulting in an even higher number of matching moments both solutions preserve the specific structure of the second order type model

higher order correlations in common input shapes the output spiking activity of a neural population abstract recent neurophysiological experiments suggest that populations of neurons use a computational scheme in which spike timing is regulated by common non gaussian inputs across neurons the presence of beyond pairwise correlations in the neuronal inputs and the spiking outputs following a non gaussian statistics elicits the need of developing a new theoretical framework taking into account the complexity of synchronous activity patterns to this end we quantify the amount of higher order correlations in the common neuronal inputs and outputs of a population of neurons we provide a novel formalism of easy numerical implementation that can capture the subtle changes of the inputs heterogeneities within our approach correlations across neurons arise from q gaussian inputs into threshold neurons and higher order correlations in the spiking outputs activity are quantified by the parameter q we present an exhaustive analysis of how input statistics are transformed in this threshold process into output statistics and we show under which conditions higher order correlations can lead to either bigger or smaller number of synchronized spikes in the neural population outputs

stretched exponentials from superstatistics abstract distributions exhibiting fat tails occur frequently in many different areas of science a dynamical reason for fat tails can be a so called superstatistics where one has a superposition of local gaussians whose variance fluctuates on a rather large spatio temporal scale after briefly reviewing this concept we explore in more detail a class of superstatistics that hasn t been subject of many investigations so far namely superstatistics for which a suitable power of the local inverse temperature is 2 distributed we show that 0 leads to power law distributions while 0 leads to stretched exponentials the special case 1 corresponds to tsallis statistics and the special case 1 to exponential statistics of the square root of energy possible applications for granular media and hydrodynamic turbulence are discussed

random walks with self similar clusters abstract we construct a random walk on a lattice having a hierarchy of self similar clusters built into the distribution function of allowed jumps the random walk is a discrete analog of a levy flight and coincides with the levy flight in the continum limit the fourier transform of the jump distribution function is the continuous nondifferentiable function of weierstrass we show that for cluster formation it is necessary that the mean squared displacement per jump be infinite and that the random walk be transient we interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker the difference in dimensions is related to the fractal hausdorff besicovitch dimension of the self similar clusters

fisher info and thermodynamics first law abstract we show starting from first principles that thermodynamics first law can be microscopically obtained for fisher s information measure without need of invoking the adiabatic theorem further it is proved that enforcing the fisher first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing fisher s information measure subject to appropriate constraints

jamming hysteresis and oscillation in scalar models for shear thickening we investigate shear thickening and jamming within the framework of a family of spatially homogeneous scalar rheological models these are based on the soft glassy rheology model of sollich et al phys rev lett 78 2020 1997 but with an effective temperature x that is a decreasing function of either the global stress ensuremath sigma or the local strain l for appropriate x x ensuremath sigma it is shown that the flow curves include a region of negative slope around which the stress exhibits hysteresis under a cyclically varying imposed strain rate stackrel ifmmode dot else fi ensuremath gamma a subclass of these x ensuremath sigma have flow curves that touch the stackrel ifmmode dot else fi ensuremath gamma 0 axis for a finite range of stresses imposing a stress from this range jams the system in the sense that the strain ensuremath gamma creeps only logarithmically with time t ensuremath gamma t ensuremath sim mathrm ln t these same systems may produce a finite asymptotic yield stress under an imposed strain in a manner that depends on the entire stress history of the sample a phenomenon we refer to as history dependent jamming in contrast when x x l the flow curves are always monotonic but we show that some x l generate an oscillatory strain response for a range of steady imposed stresses similar spontaneous oscillations are observed in a simplified model with fewer degrees of freedom we discuss this result in relation to the temporal instabilities observed in rheological experiments and stick slip behavior found in other contexts and comment on the possible relationship with delay differential equations that are known to produce oscillations and chaos

means of extreme points and f spaces abstract given a topological space t and a strictly convex real normed space x let c t x be the space of continuous and bounded functions from t into x with its uniform norm this paper is devoted to the study of the relation between the fact of t being an f space and the property that every element in the unit ball of c t x has a representation as a mean of two extreme points

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

statistical modelling of higher order correlations in pools of neural activity abstract simultaneous recordings from multiple neural units allow us to investigate the activity of very large neural ensembles to understand how large ensembles of neurons process sensory information it is necessary to develop suitable statistical models to describe the response variability of the recorded spike trains using the information geometry framework it is possible to estimate higher order correlations by assigning one interaction parameter to each degree of correlation leading to a 2 n 1 dimensional model for a population with n neurons however this model suffers greatly from a combinatorial explosion and the number of parameters to be estimated from the available sample size constitutes the main intractability reason of this approach to quantify the extent of higher than pairwise spike correlations in pools of multiunit activity we use an information geometric approach within the framework of the extended central limit theorem considering all possible contributions from higher order spike correlations the identification of a deformation parameter allows us to provide a statistical characterisation of the amount of higher order correlations in the case of a very large neural ensemble significantly reducing the number of parameters avoiding the sampling problem and inferring the underlying dynamical properties of the network within pools of multiunit neural activity

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

generalized extreme value statistics and sum of correlated variables we show that generalized extreme value statistics the statistics of the kth largest value among a large set of random variables can be mapped onto a problem of random sums this allows us to identify classes of non identical and generally correlated random variables with a sum distributed according to one of the three k dependent asymptotic distributions of extreme value statistics namely the gumbel frechet and weibull distributions these classes as well as the limit distributions are naturally extended to real values of k thus providing a clear interpretation to the onset of gumbel distributions with non integer index k in the statistics of global observables this is one of the very few known generalizations of the central limit theorem to non independent random variables finally in the context of a simple physical model we relate the index k to the ratio of the correlation length to the system size which remains finite in strongly correlated systems

central limit behavior of deterministic dynamical systems we investigate the probability density of rescaled sums of iterates of deterministic dynamical systems a problem relevant for many complex physical systems consisting of dependent random variables a central limit theorem clt is valid only if the dynamical system under consideration is sufficiently mixing for the fully developed logistic map and a cubic map we analytically calculate the leading order corrections to the clt if only a finite number of iterates is added and rescaled and find excellent agreement with numerical experiments at the critical point of period doubling accumulation a clt is not valid anymore due to strong temporal correlations between the iterates nevertheless we provide numerical evidence that in this case the probability density converges to a q gaussian thus leading to a power law generalization of the clt the above behavior is universal and independent of the order of the maximum of the map considered i e relevant for large classes of critical dynamical systems

superstatistics thermodynamics and fluctuations a thermodynamiclike formalism is developed for superstatistical systems based on conditional entropies this theory takes into account large scale variations of intensive variables of systems in nonequilibrium stationary states ordinary thermodynamics is recovered as a special case of the present theory and corrections to it can systematically be evaluated a generalization of einstein s relation for fluctuations is presented using a maximum entropy condition

on a generalization of the binomial distribution and its poisson like limit we examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its poisson like limit such generalizations might be found in quantum optics with imperfect detection we discuss under which conditions this distribution can have a probabilistic interpretation

component mode synthesis cms based on an enriched ritz approach for efficient structural optimization this paper presents a new method to improve the standard superelement reduction methods by taking into account a priori information concerning the potential structural modifications thus allowing the same transformation to be used throughout the optimization process the aim is to create a single enriched model reduction transformation that preserves the precision of the reduced substructure model throughout the optimization process the approach consists firstly in extending the standard condensation basis by a set of optimized static residual and secondly in eliminating the associate coordinates from the reduced system the proposed method can be used with any condensation procedure including both direct reductions and component mode synthesis approaches with any kind of substructure natural modes free free clamped or hybrid modes

extended q gaussian and q exponential distributions from gamma random variables fil budini adrian adolfo comision nacional de energia atomica centro atomico bariloche argentina consejo nacional de investigaciones cientificas y tecnicas argentina

computational applications of nonextensive statistical mechanics computational applications of the nonextensive entropy s q and nonextensive statistical mechanics a current generalization of the boltzmann gibbs bg theory are briefly reviewed the corresponding bibliography is provided as well

generalized superstatistics of nonequilibrium markovian systems abstract the paper is devoted to the construction of the superstatistical description for nonequilibrium markovian systems it is based on kirchhoff s diagram technique and the assumption on the system under consideration to possess a wide variety of cycles with vanishing probability fluxes the latter feature enables us to introduce equivalence classes called channels within which detailed balance holds individually then stationary probability as well as flux distributions are represented as some sums over the channels the latter construction actually forms the superstatistical description which however deals with a certain superposition of equilibrium subsystems rather than a formal expansion of the nonequilibrium steady state distribution into terms of the boltzmann type

using krylov pade model order reduction for accelerating design optimization of structures and vibrations in the frequency domain summary in many engineering problems the behavior of dynamical systems depends on physical parameters in design optimization these parameters are determined so that an objective function is minimized for applications in vibrations and structures the objective function depends on the frequency response function over a given frequency range and we optimize it in the parameter space because of the large size of the system numerical optimization is expensive in this paper we propose the combination of quasi newton type line search optimization methods and krylov pade type algebraic model order reduction techniques to speed up numerical optimization of dynamical systems we prove that krylov pade type model order reduction allows for fast evaluation of the objective function and its gradient thanks to the moment matching property for both the objective function and the derivatives towards the parameters we show that reduced models for the frequency alone lead to significant speed ups in addition we show that reduced models valid for both the frequency range and a line in the parameter space can further reduce the optimization time copyright 2012 john wiley sons ltd

fluctuations correlations and the nonextensivity abstract examples of joint probability distributions are studied in terms of tsallis nonextensive statistics both for correlated and uncorrelated variables in particular it is explicitely shown how correlations in the system can make tsallis entropy additive and that the effective nonextensivity parameter q n decreases towards unity when the number of variables n increases we demonstrate that tsallis distribution of energies of particles in a system leads in a natural way to the negative binomial multiplicity distribution in this system

modelling train delays with q exponential functions we demonstrate that the distribution of train delays on the british railway network is accurately described by q exponential functions we explain this by constructing an underlying superstatistical model

closer look at time averages of the logistic map at the edge of chaos the probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown see u tirnakli c beck and c tsallis phys rev e 75 040106 r 2007 to be numerically consistent with a q gaussian the distribution which under appropriate constraints maximizes the nonadditive entropy sq the basis of nonextensive statistical mechanics this analysis was based on a study of the tails of the distribution we now check the entire distribution in particular its central part this is important in view of a recent q generalization of the central limit theorem which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q gaussian limit distribution we numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q gaussian both in the central region and in the tail region and find a scaling law involving the feigenbaum constant our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy sq and its associated concepts

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

poincare s observation and the origin of tsallis generalized canonical distributions abstract in this paper we present some geometric properties of the maximum entropy tsallis distributions under energy constraint in the case q 1 these distributions are proved to be marginals of uniform distributions on the sphere in the case q 1 they can be constructed as conditional distributions of a cauchy law built from the same uniform distribution on the sphere using a gnomonic projection as such these distributions reveal the relevance of using tsallis distributions in the microcanonical setup an example of such application is given in the case of the ideal gas

interfaces in a random medium and replica symmetry breaking the authors develop a variational approach for studying interfaces and other manifolds in a disordered quenched medium the method may be applied to problems which range from directed polymers to the interface of an ising model in a random magnetic field they find that replica symmetry is spontaneously broken and the results of the flory approximation are recovered in a simple way corrections to this approximation may be computed in a systematic way

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

central limit theorem for anomalous scaling due to correlations we derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables the theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the levy gnedenko uncorrelated case in particular correlated anomalous diffusion is mapped onto levy diffusion under suitable assumptions the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling

inhomogeneous diffusion and ergodicity breaking induced by global memory effects fil budini adrian adolfo comision nacional de energia atomica gerencia del area de investigaciones y aplicaciones no nucleares gerencia de fisica centro atomico constituyentes argentina consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet patagonia norte argentina

fluctuations of entropy and log normal superstatistics nonequilibrium complex systems are often effectively described by the mixture of different dynamics on different time scales superstatistics which is statistics of statistics with two largely separated time scales offers a consistent theoretical framework for such a description here a theory is developed for log normal superstatistics based on the fluctuation theorem for entropy changes as well as the maximum entropy method this gives novel physical insight into log normal statistics other than the traditional multiplicative random processes a comment is made on a possible application of the theory to the fluctuating energy dissipation rate in turbulence

convergence of large deviation estimators we study the convergence of statistical estimators used in the estimation of large deviation functions describing the fluctuations of equilibrium nonequilibrium and manmade stochastic systems we give conditions for the convergence of these estimators with sample size based on the boundedness or unboundedness of the quantity sampled and discuss how statistical errors should be defined in different parts of the convergence region our results shed light on previous reports of phase transitions in the statistics of free energy estimators and establish a general framework for reliably estimating large deviation functions from simulation and experimental data and identifying parameter regions where this estimation converges

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

symmetric generalized binomial distributions in two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers involving asymmetric expressions of probabilities that break the symmetry win loss we present in this article another generalization always associated with a sequence of positive numbers that preserves the symmetry win loss this approach is also based on generating functions and presents constraints of non negativeness similar to those encountered in our previous articles

stationary and dynamical properties of a zero range process on scale free networks we study the condensation phenomenon in a zero range process on scale free networks we show that the stationary state property depends only on the degree distribution of underlying networks the model displays a stationary state phase transition between a condensed phase and an uncondensed phase and the phase diagram is obtained analytically as for the dynamical property we find that the relaxation dynamics depends on the global structure of underlying networks the relaxation time follows the power law ensuremath tau ensuremath sim l z with the network size l in the condensed phase the dynamic exponent z is found to take a different value depending on whether underlying networks have a tree structure or not

central limit theorem for a class of globally correlated random variables the standard central limit theorem with a gaussian attractor for the sum of independent random variables may lose its validity in the presence of strong correlations between the added random contributions here we study this problem for similar interchangeable globally correlated random variables under these conditions a hierarchical set of equations is derived for the conditional transition probabilities this result allows us to define different classes of memory mechanisms that depend on a symmetric way on all involved variables depending on the correlation mechanisms and statistics of the single variables the corresponding sums are characterized by distinct probability densities for a class of urn models it is also possible to characterize their domain of attraction which as in the standard case is parametrized by the probability density of each random variable symmetric and asymmetric q gaussian attractors q 1 arise in a particular two state case of these urn models

maximum relative height of elastic interfaces in random media the distribution of the maximal relative height mrh of self affine one dimensional elastic interfaces in a random potential is studied we analyze the ground state configuration at zero driving force and the critical configuration exactly at the depinning threshold both for the random manifold and random periodic universality classes these configurations are sampled by exact numerical methods and their mrh distributions are compared with those with the same roughness exponent and boundary conditions but produced by independent fourier modes with normally distributed amplitudes using pickands theorem we derive an exact analytical description for the right tail of the latter after properly rescaling the mrh distributions we find that corrections from the gaussian independent modes approximation are in general small as previously found for the average width distribution of depinning configurations in the large size limit all corrections are finite except for the ground state in the random periodic class whose mrh distribution becomes for periodic boundary conditions indistinguishable from the airy distribution we find that the mrh distributions are in general sensitive to changes of boundary conditions

scale free networks with an exponent less than two we study scale free simple graphs with an exponent of the degree distribution ensuremath gamma less than 2 generically one expects such extremely skewed networks which occur very frequently in systems of virtually or logically connected units to have different properties than those of scale free networks with ensuremath gamma g2 the number of links grows faster than the number of nodes and they naturally posses the small world property because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite we discuss a simple prototype model of such networks inspired by real world phenomena which exhibits these properties and allows for a detailed analytical investigation

asymptotically scale invariant occupancy of phase space makes the entropy sq extensive phase space can be constructed for n equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated if they are locally correlated we expect the boltzmann gibbs entropy s bg k i p i ln p i to be extensive i e s bg n n for n in particular if they are independent s bg is strictly additive i e s bg n ns bg 1 n however if the subsystems are globally correlated we expect for a vast class of systems the entropy s q k 1 i p q i q 1 with s 1 s bg for some special value of q 1 to be the one which is extensive i e s q n n for n another concept which is relevant is strict or asymptotic scale freedom or scale invariance defined as the situation for which all marginal probabilities of the n system coincide or asymptotically approach for n the joint probabilities of the n 1 system if each subsystem is a binary one scale freedom is guaranteed by what we hereafter refer to as the leibnitz rule i e the sum of two successive joint probabilities of the n system coincides or asymptotically approaches the corresponding joint probability of the n 1 system the kinds of interplay of these various concepts are illustrated in several examples one of them justifies the title of this paper we conjecture that these mechanisms are deeply related to the very frequent emergence in natural and artificial complex systems of scale free structures and to their connections with nonextensive statistical mechanics summarizing we have shown that for asymptotically scale invariant systems it is s q with q 1 and not s bg the entropy which matches standard clausius like prescriptions of classical thermodynamics

dynamical modelling of superstatistical complex systems abstract we show how to construct the optimum superstatistical dynamical model for a given experimentally measured time series for this purpose we generalise the superstatistics concept and study a langevin equation with a memory kernel whose parameters fluctuate on a large time scale we show how to construct a synthetic dynamical model with the same invariant density and correlation function as the experimental data as a main example we apply our method to velocity time series measured in high reynolds number turbulent taylor couette flow but the method can be applied to many other complex systems in a similar way

relationships and scaling laws among correlation fractality lyapunov divergence and q gaussian distributions abstract we numerically introduce the relationships among correlation fractality lyapunov divergence and q gaussian distributions the scaling arguments between the range of the q gaussian and correlation fractality lyapunov divergence are obtained for periodic windows i e periods 2 3 and 5 of the logistic map as chaos threshold is approached firstly we show that the range of the q gaussian g tends to infinity as the measure of the deviation from the correlation dimension d c o r r 0 5 at the chaos threshold this deviation will be denoted by l approaches to zero moreover we verify that a scaling law of type 1 g l is evident with the critical exponent 0 23 0 01 similarly as chaos threshold is approached the quantity l scales as l a a c where the exponent is 0 84 0 01 secondly we also show that the range of the q gaussian exhibits a scaling law with the correlation length 1 g lyapunov divergence 1 g and the distance to the critical box counting fractal dimension 1 g d d c with the same exponent 0 43 finally we numerically verify that these three quantities d d c scale with the distance to the critical control parameter of the map i e a a c in accordance with the universal huberman rudnick scaling law with the same exponent 0 448 0 003 all these findings can be considered as a new evidence supporting that the central limit behaviour at the chaos threshold is given by a q gaussian

interacting individuals leading to zipf s law institut de physique th eorique universit e de fribourg ch 1700 february 1 2008 we present a general approach to explain the zipf s law of city distribution if the simplestinteraction pairwise is assumed individuals tend to form cities in agreement with the well knownstatisticspacs numbers 89 50 r 05 20 y 05 40 j

complex extremal structure in spaces of continuous functions abstract this paper considers the spacey c t x of all continuous and bounded functions from a topological spacetto a complex normed spacexwith the sup norm the extremal structure of the closed unit ballb y inyhas been intensively studied whenxis strictly convex that is in terms of its unitary functions mappings fromtinto the unit sphere ofx we prove that iftis completely regular andxhas finite dimension then every function inb y is expressible as a convex combination of three unitary functions if and only if the condition dim t r is satisfied where dim tis the covering dimension oftandx r denotesxconsidered as a real normed space ifxis infinite dimensional the above decomposition is always possible without restrictions aboutt these results are interesting whenxis complex strictly convex as a consequence we state a surprising fact the identity function on the unit ball of an infinite dimensional complex normed space can be expressed as the average of three retractions of the unit ball onto the unit sphere really such a representation is the best possible

scaling symmetry renormalization and time series modeling the case of financial assets dynamics we present and discuss a stochastic model of financial assets dynamics based on the idea of an inverse renormalization group strategy with this strategy we construct the multivariate distributions of elementary returns based on the scaling with time of the probability density of their aggregates in its simplest version the model is the product of an endogenous autoregressive component and a random rescaling factor designed to embody also exogenous influences mathematical properties like increments stationarity and ergodicity can be proven thanks to the relatively low number of parameters model calibration can be conveniently based on a method of moments as exemplified in the case of historical data of the s p500 index the calibrated model accounts very well for many stylized facts like volatility clustering power law decay of the volatility autocorrelation function and multiscaling with time of the aggregated return distribution in agreement with empirical evidence in finance the dynamics is not invariant under time reversal and with suitable generalizations skewness of the return distribution and leverage effects can be included the analytical tractability of the model opens interesting perspectives for applications for instance in terms of obtaining closed formulas for derivative pricing further important features are the possibility of making contact in certain limits with autoregressive models widely used in finance and the possibility of partially resolving the long and short memory components of the volatility with consistent results when applied to historical series

asymptotics of superstatistics superstatistics are superpositions of different statistics relevant for driven nonequilibrium systems with spatiotemporal inhomogeneities of an intensive variable e g the inverse temperature they contain tsallis statistics as a special case we develop here a technique that allows us to analyze the large energy asymptotics of the stationary distributions of general superstatistics a saddle point approximation is developed which relates this problem to a variational principle several examples are worked out in detail

statistical mechanics of the chinese restaurant process lack of self averaging anomalous finite size effects and condensation the pitman yor or chinese restaurant process is a stochastic process that generates distributions following a power law with exponents lower than 2 as found in numerous physical biological technological and social systems we discuss its rich behavior with the tools and viewpoint of statistical mechanics we show that this process invariably gives rise to a condensation i e a distribution dominated by a finite number of classes we also evaluate thoroughly the finite size effects finding that the lack of stationary state and self averaging of the process creates realization dependent cutoffs and behavior of the distributions with no equivalent in other statistical mechanical models

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

dynamics of the condensate in zero range processes for stochastic processes leading to condensation the condensate once it is formed performs an ergodic stationary state motion over the system we analyse this motion and especially its characteristic time for zero range processes the characteristic time is found to grow with the system size much faster than the diffusive time scale but not exponentially fast this holds both in the mean field geometry and on finite dimensional lattices in the generic situation where the critical mass distribution follows a power law the characteristic time grows as a power of the system size

quantifying higher order correlations in a neuronal pool abstract recent experiments involving a relatively large population of neurons have shown a very significant amount of higher order correlations however little is known of how these affect the integration and firing behavior of a population of neurons beyond the second order statistics to investigate how higher order inputs statistics can shape beyond pairwise spike correlations and affect information coding in the brain we consider a neuronal pool where each neuron fires stochastically we develop a simple mathematically tractable model that makes it feasible to account for higher order spike correlations in a neuronal pool with highly interconnected common inputs beyond second order statistics in our model correlations between neurons appear from q gaussian inputs into threshold neurons the approach constitutes the natural extension of the dichotomized gaussian model where the inputs to the model are just gaussian distributed and therefore have no input interactions beyond second order we obtain an exact analytical expression for the joint distribution of firing quantifying the degree of higher order spike correlations truly emphasizing the functional aspects of higher order statistics as we account for beyond second order inputs correlations seen by each neuron within the pool we determine how higher order correlations depend on the interaction structure of the input showing that the joint distribution of firing is skewed as the parameter q increases inducing larger excursions of synchronized spikes we show how input nonlinearities can shape higher order correlations and enhance coding performance by neural populations

superpositions of probability distributions probability distributions which can be obtained from superpositions of gaussian distributions of different variances v ensuremath sigma 2 play a favored role in quantum theory and financial markets such superpositions need not necessarily obey the chapman kolmogorov semigroup relation for markovian processes because they may introduce memory effects we derive the general form of the smearing distributions in v which do not destroy the semigroup property the smearing technique has two immediate applications it permits simplifying the system of kramers moyal equations for smeared and unsmeared conditional probabilities and can be conveniently implemented in the path integral calculus in many cases the superposition of path integrals can be evaluated much easier than the initial path integral three simple examples are presented and it is shown how the technique is extended to quantum mechanics

central limit behavior in the kuramoto model at the edge of chaos abstract we study the relationship between chaotic behavior and the central limit theorem clt in the kuramoto model we calculate sums of angles at equidistant times along deterministic trajectories of single oscillators and we show that when chaos is sufficiently strong the pdfs of the sums tend to a gaussian consistently with the standard clt on the other hand when the system is at the edge of chaos i e in a regime with vanishing lyapunov exponents robust q gaussian like limit distributions naturally emerge consistently with recently proved generalizations of the clt

a projection pricing model for non gaussian financial returns abstract stephen leroy jan werner and david luenberger have developed a geometric approach to the capital asset pricing model capm in terms of projections in a hilbert space onto a mean variance efficient frontier using this projection method they were able to elegantly deduce a geometric interpretation of capm and factor asset pricing models in this paper we extend their geometric methods to non euclidean divergence geometries this extension has relevant consequences first it permits to deal with higher order moments of the probability distributions since general statistical divergences could encode global information about these distributions as is the case of the entropy secondly orthogonal euclidean projections and the corresponding least squares problem give place to riemannian projections onto a possibly curved efficient frontier finally our method is flexible enough to deal with huge families of probability distributions in particular there is no need to assume normality of the returns of the financial assets

limit behaviour and scaling relations of two kinds of noisy logistic map in the vicinity of chaos threshold and their robustness abstract in this paper we numerically investigate the distribution of the sums of the iterates of the logistic map and the relationships among the important properties of the nonlinear dynamics in the vicinity of the chaos threshold by adding two kinds of contributions with different densities the first one is the well known white noise whereas the second is a newly defined one named as quartic term which makes contributions from the own structure of the map as the chaos threshold is approached the iterates of the standard logistic map i e noise free have strong correlations and the standard central limit theorem is not valid anymore in a recent work tirnakli 2009 it has been shown that the limit distribution seems to converge to q gaussian distribution which maximizes the nonadditive entropy s q 1 i p i q q 1 under appropriate conditions in this work we investigate the effect of these contributions i e white noise and quartic term on the limit distribution and on the range of the obtained q gaussian distribution as a result of these findings under the existence of white noise and also the quartic term we analyse the validity of the scaling relations among correlation fractality the lyapunov divergence and q gaussian distributions which have recently been observed in afsar 2014 the results obtained here strengthen the argument that the central limit behaviour is given by a q gaussian as the chaos threshold is approached and indicate that the scaling relations obtained for the standard logistic map among the range of the q gaussian the correlation dimension the correlation length the lyapunov exponent fractality and the distance from the chaos threshold are robust under the existence of white noise and the quartic term

central limit theorem and deformed exponentials the central limit theorem clt can be ranked among the most important ones in probability theory and statistics and plays an essential role in several basic and applied disciplines notably in statistical thermodynamics we show that there exists a natural extension of the clt from exponentials to so called deformed exponentials also denoted as q gaussians our proposal applies exactly in the usual conditions in which the classical clt is used

on q gaussians and exchangeability the q gaussian distributions introduced by tsallis are discussed from the point of view of variance mixtures of normals and exchangeability for each there is a q gaussian distribution that maximizes the tsallis entropy under suitable constraints this paper shows that q gaussian random variables can be represented as variance mixtures of normals when q 1 these variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables thereby providing a possible model that has been extensively studied in probability theory the formulation provided has the additional advantage of yielding for each q a process which is naturally the q analog of the brownian motion explicit mixing distributions for q gaussians should facilitate applications to areas such as option pricing the model might provide insight into the study of superstatistics

beyond boltzmann gibbs statistics maximum entropy hyperensembles out of equilibrium what is the best description that we can construct of a thermodynamic system that is not in equilibrium given only one or a few extra parameters over and above those needed for a description of the same system at equilibrium here we argue the most appropriate additional parameter is the nonequilibrium entropy of the system moreover we should not attempt to estimate the probability distribution of the system directly but rather the metaprobability or hyperensemble that the system is described by a particular probability distribution the result is an entropic distribution with two parameters one a nonequilibrium temperature and the other a measure of distance from equilibrium this dispersion parameter smoothly interpolates between certainty of a canonical distribution at equilibrium and great uncertainty as to the probability distribution as we move away from equilibrium we deduce that in general large rare fluctuations become far more common as we move away from equilibrium

statistical properties of randomly broken objects and of multivalley structures in disordered systems the statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common for all these problems if w denotes the weight of the sth piece we show that the probability distributions pi w of the largest piece w p2 w of the second largest piece and n y of y 1 wi always have singularities at w l n w2 l n and y ijn n 1 2 3

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

reduced order modeling of multiscreen frequency selective surfaces using krylov based rational interpolation a method is presented for generating a broad band rational interpolant approximation of the reflection coefficient of multiple screen frequency selective surfaces fsss the technique is structured around a linearization of the system provided by a spectral domain moment method based analysis of the fss followed by a model order reduction of the linearized system using the dual rational arnoldi method this process creates a rational interpolant of the linearized system that matches its transfer function and its derivatives at several expansion points in the laplace domain numerical results indicate that a reduced order model with a system matrix of dimension less than 20 spl times 20 can accurately reproduce the broad band behavior of multiscreen fsss originally modeled with several hundreds or thousands of unknowns

superstatistical generalizations of wishart laguerre ensembles of random matrices using beck and cohen s superstatistics we introduce in a systematic way a family of generalised wishart laguerre ensembles of random matrices with dyson index 1 2 and 4 the entries of the data matrix are gaussian random variables whose variancesfluctuate from one sample to another according to a certain probability density f and a single deformation parameter three superstatistical classes for f are usually considered 2 inverse 2 and log normal distributions while the first class already considered by two of the authors leads to a power law decay of the spectral density we here introduce and solve exactly a superposition of wishart laguerre ensembles with inverse 2 distribution the corresponding macroscopic spectral density is given by a deformation of the semi circle and marcenko pastur laws on a non compact support with exponential tails after discussing in detail the validity of wigner s surmise in the wishart laguerre class we introduce a generalised dependent surmise with stretched exponential tails which well approximates the individual level spacing distribution in the bulk the analytical results are in excellent agreement with numerical simulations to illustrate our findings we compare the 2 and inverse 2 class to empirical data from financial covariance matrices

on the numerical solution of lambda 2 a lambda b c x b and application to structural dynamics in this paper we address the numerical solution of a large linear system depending quadratically on a parameter that varies in a wide range we analyze a solution method whose computational cost grows only sublinearly with the number of parameters that relies on the use of an indefinite inner product important implementation aspects are treated in detail the problem arises in various application areas we shall report on our experience with cases in structural dynamics

global fluctuations and gumbel statistics we explain how the statistics of global observables in correlated systems can be related to extreme value problems and to gumbel statistics this relationship then naturally leads to the emergence of the generalized gumbel distribution ga x with a real index a in the study of global fluctuations to illustrate these findings we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice with local dissipation in which the fluctuations of the global energy are precisely described by the generalized gumbel distribution

improving the efficiency of large scale topology optimization through on the fly reduced order model construction summarytopology optimization of large scale structures is computationally expensive notably because of the cost of solving the equilibrium equations at each iteration reduced order models by projection also known as reduced basis models have been proposed in the past for alleviating this cost we propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures the novel approach is based on constructing the reduced basis on the fly using previously calculated solutions of the equilibrium equations the reduced basis is thus adaptively constructed and enriched based on the convergence behavior of the topology optimization a direct approach and an approach with adjusted sensitivities are described and their algorithms provided the approaches are tested and compared on various 2d and 3d minimum compliance topology optimization benchmark problems computational cost savings by up to a factor of 12 are demonstrated using the proposed methods copyright textcopyright 2014 john wiley sons ltd

a two directional arnoldi process and its application to parametric model order reduction we consider a two directional krylov subspace k k a j b j where besides the dimensionality k of the subspace increases the matrix a j and vector b j which induce the subspace may also augment specifically we consider the case where the matrix a j and the vector b j are augmented by block triangular bordering we present a two directional arnoldi process to efficiently generate a sequence of orthonormal bases q k j of the krylov subspaces the concept of a two directional krylov subspace and an arnoldi process is triggered by the need of a multiparameter moment matching based model order reduction technique for parameterized linear dynamical systems numerical examples illustrate computational efficiency and flexibility of the proposed two directional arnoldi process

density of near extreme events we provide a quantitative analysis of the phenomenon of crowding of near extreme events by computing exactly the density of states dos near the maximum of a set of independent and identically distributed random variables we show that the mean dos converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential faster than pure exponential or as a pure exponential function we argue that some of these results would remain valid even for certain correlated cases and verify it for power law correlated stationary gaussian sequences satisfactory agreement is found between the near maximum crowding in the summer temperature reconstruction data of western siberia and the theoretical prediction

dimension reduction of large scale second order dynamical systems via a second order arnoldi method a structure preserving dimension reduction algorithm for large scale second order dynamical systems is presented it is a projection method based on a second order krylov subspace a second order arnoldi soar method is used to generate an orthonormal basis of the projection subspace the reduced system not only preserves the second order structure but also has the same order of approximation as the standard arnoldi based krylov subspace method via linearization the superior numerical properties of the soar based method are demonstrated by examples from structural dynamics and microelectromechanical systems

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