strong desynchronizing effects of weak noise in globally coupled systems in assemblies of globally coupled dynamical units weak noise perturbing independently the individual units can cause anomalous dispersion in the synchronized cloud of the units in the phase space when the noise free dynamics of the synchronized assembly is nonperiodic various moments of the linear dimension of the cloud as a function of the noise strength exhibit multiscaling properties with parameter dependent scaling exponents some numerical evidence of this peculiar behavior as well as its interpretation in terms of a multiplicative stochastic process with small additive noise is provided universality of the phenomenon is also discussed

a theorem on the equilibrium thermodynamics of hamiltonian systems we prove a theorem stating that within an interesting class of stationary solutions of the liouville equation the tsallis q distributions are the only ones that satisfy a condition which guarantees the existence of an integrating factor for the heat the functional forms of the distribution as well as that of entropy temperature and heat capacity are derived we then explain why the gibbs distribution as a limit of the tsallis one plays a privileged role for large size ordinary molecular systems

entanglement rates for renyi tsallis and other entropies we provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution a large class of entropy measures in considered which includes renyi and tsallis entropies the result is derived from a general bound on the trace norm of a commutator which can be expected to find other implementations we apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with renyi and tsallis entanglement entropies taken as measures of entanglement in the system we provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution a large class of entropy measures in considered which includes renyi and tsallis entropies the result is derived from a general bound on the trace norm of a commutator which can be expected to find other implementations we apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with renyi and tsallis entanglement entropies taken as measures of entanglement in the system

towards a dynamical temperature of finite hamiltonian systems abstract with the aid of numerical simulations of the fermi pasta ulam fpu system we compare the different definitions of dynamical temperature for hamiltonian systems we have shown that each definition gives different values of temperature for a system with a small number of degrees of freedom dof only for systems with a sufficiently large number of dof do all the definitions of dynamical temperature approach the same value

on the definition of temperature using time averages abstract this paper is a natural continuation of a previous one by the author which was concerned with the foundations of statistical thermodynamics far from equilibrium one of the problems left open in that paper was the correct definition of temperature in the literature temperature is in general defined through the mean kinetic energy of the particles of a given system in this paper instead temperature is defined a la caratheodory the system being coupled to a heat bath and temperature being singled out as the right integrating factor of the exchanged heat as a byproduct the right expression for the entropy is also obtained in particular in the case of a q distribution the entropy turns out to be that of tsallis

integrable schrodinger operators with magnetic fields factorization method on curved surfaces the factorization method for schrodinger operators with magnetic fields on a two dimensional surface m2 with nontrivial metric is investigated this leads to the new integrable examples of such operators and brings a new look at some classical problems such as the dirac magnetic monopole and the landau problem the global geometric aspects and related spectral properties of the operators from the factorization chains are discussed in detail we also consider the laplace transformations on a curved surface and extend the class of schrodinger operators with two integrable levels introduced in the flat case by s p novikov and one of the authors

spectral properties of the cauchy transform on l2 c e z 2λ z let hm p z m p z z be the landau orthogonal basis of the hilbert space on l2 c e z 2d z where z is the usual lebesgue measure on the complex plane in this paper we give some spectral properties of the cauchy transform on the orthogonal complement of bargmann space 0 c in l2 c e z 2d z in particular for m fixed we consider the orthogonal projection operator on the hilbert subspace spanned by hm p z p 0 1 2 and we give explicitly the sequence of singular values of its composition with the cauchy transform in l2 c e z 2d z as application of these of the cauchy transform we get some identities for special functions which could be of independent interest

mathematical structures derived from the q multinomial coefficient in tsallis statistics abstract we present q stirling s formula q multinomial coefficient one to one correspondence between q multinominal coefficient and tsallis entropy q pascal s triangle and a conjecture on the q central limit theorem in tsallis statistics for the generalization of the well known fundamental formulas to systems exhibiting power law behaviors the main approach is based on the q product uniquely determined by tsallis entropy which has already been successfully applied to our recent proof of the law of error in tsallis statistics

temporal extensivity of tsallis entropy and the bound on entropy production rate the tsallis entropy which is a generalization of the boltzmann gibbs entropy plays a central role in nonextensive statistical mechanics of complex systems a lot of efforts have recently been made on establishing a dynamical foundation for the tsallis entropy they are primarily concerned with nonlinear dynamical systems at the edge of chaos here it is shown by generalizing a formulation of thermostatistics based on time averages recently proposed by carati a carati physica a 348 110 2005 that whenever relevant the tsallis entropy indexed by q is temporally extensive linear growth in time i e finite entropy production rate then the universal bound on the entropy production rate is shown to be 1 absolute value 1 q the property of the associated probabilistic process i e the sojourn time distribution determining randomness of motion in phase space is also analyzed

effectiveness of the kozachenko leonenko estimator for generalized entropic forms in this brief report we discuss the effectiveness of the kozachenko leonenko entropy estimator when generalized to cope with entropic forms customarily applied to study systems evincing asymptotic scale invariance and dependence either of linear or nonlinear kind we show that when the variables are independently and identically distributed the estimator is only valuable along the whole domain if the data follow the uniform distribution whereas for other distributions the estimator is only effectual in the limit of the boltzmann gibbs shanon entropic form we also analyze the influence of the dependence linear and nonlinear between variables on the accuracy of the estimator between variables as expected in the last case the estimator loses efficiency for the boltzmann gibbs shanon entropic form as well

q gaussian diffusion in stock markets we analyze the standard poor s 500 stock market index from the past 22 years the probability density function of price returns exhibits two well distinguished regimes with self similar structure the first one displays strong superdiffusion together with short time correlations and the second one corresponds to weak superdiffusion with weak time correlations both regimes are well described by q gaussian distributions the porous media equation a special case of the tsallis bukman equation is used to derive the governing equation for these regimes and the black scholes diffusion coefficient is explicitly obtained from the governing equation

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

explicit formulas for reproducing kernels of generalized bargmann spaces on cn we introduce a class of generalized bargmann spaces on cn for which we establish explicit formulas of their reproducing kernels some applications of the obtained formulas are given

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

spectral analysis on planar mixed automorphic forms abstract we introduce and we carry out some concrete spectral properties of a class of magnetic schrodinger operators leaving invariant the space of the planar mixed automorphic forms of type with respect to an equivariant pair and given lattice c the associated polynomials constitute classes of generalized complex polynomials of hermite type

spectral analysis of schrodinger operators with magnetic fields abstract the spectra of quadratic schrodinger operators in general dimensional euclidean spaces are determined by considering the corresponding classical mechanics and the corresponding dirac operators the results are applied to the study on the semiclassical approximation of the eigenvalues of schrodinger operators with magnetic fields

on generalisations of the log normal distribution by means of a new product definition in the kapteyn process abstract we discuss the modification of the kapteyn multiplicative process using the q product of borges e p borges a possible deformed algebra and calculus inspired in nonextensive thermostatistics physica a 340 2004 95 depending on the value of the index q a generalisation of the log normal distribution is yielded namely the distribution increases the tail for small when q 1 or large when q 1 values of the variable upon analysis the usual log normal distribution is retrieved when q 1 which corresponds to the traditional kapteyn multiplicative process the main statistical features of this distribution as well as related random number generators and tables of quantiles of the kolmogorov smirnov distance are presented finally we illustrate the validity of this scenario by describing a set of variables of biological and financial origin

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

coherent states and hermite polynomials on quaternionic hilbert spaces coherent states similar to the canonical coherent states of the harmonic oscillator labeled by quaternions are established on the right and left quaternionic hilbert spaces on the left quaternionic hilbert space reproducing kernels are established as was claimed by adler and millard j math phys 1997 38 2117 26 it is proved that these coherent states cannot be realized as a group action on a quaternionic hilbert space as an extension of the complex hermite polynomials quaternionic hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions

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

essential self adjointness for semi bounded magnetic schrodinger operators on non compact manifolds abstract we prove essential self adjointness for semi bounded below magnetic schrodinger operators on complete riemannian manifolds with a given positive smooth measure which is fixed independently of the metric some singularities of the scalar potential are allowed this is an extension of the povzner wienholtz simader theorem the proof uses the scheme of wienholtz but requires a refined invariant integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

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

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

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

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

nonadditive entropies yield probability distributions with biases not warranted by the data different quantities that go by the name of entropy are used in variational principles to infer probability distributions from limited data shore and johnson showed that maximizing the boltzmann gibbs form of the entropy ensures that probability distributions inferred satisfy the multiplication rule of probability for independent events in the absence of data coupling such events other types of entropies that violate the shore and johnson axioms including nonadditive entropies such as the tsallis entropy violate this basic consistency requirement here we use the axiomatic framework of shore and johnson to show how such nonadditive entropy functions generate biases in probability distributions that are not warranted by the underlying data

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

reflections on the q fourier transform and the q gaussian function abstract the standard q fourier transform qft of a constant diverges which begs for a better treatment in addition hilhorst has conclusively proved that the ordinary qft is not of a one to one character for an infinite set of functions h j hilhorst j stat mech 2010 p10023 generalizing the ordinary qft analyzed in s umarov c tsallis s steinberg milan j math 76 2008 307 we appeal here to a complex q fourier transform and show that the problems above mentioned are overcome

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

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

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

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

generalized boltzmann factors and the maximum entropy principle entropies for complex systems abstract we generalize the usual exponential boltzmann factor to any reasonable and potentially observable distribution function b e by defining generalized logarithms as inverses of these distribution functions we are led to a generalization of the classical boltzmann gibbs entropy s bg d e e b e ln b e to the expression s d e e 0 b e d x x which contains the classical entropy as a special case we show that this is the unique modification of entropy which is compatible with the maximum entropy principle for arbitrary non exponential distribution functions we demonstrate that this entropy has two important features first it describes the correct thermodynamic relations of the system and second the observed distributions are straightforward solutions to the jaynes maximum entropy principle with the ordinary not escort constraints tsallis entropy is recovered as a further special case

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

the gln q module structure of the symmetric algebra around the steinberg module abstract we determine the graded composition multiplicity in the symmetric algebra s v of the natural gl n q module v or equivalently in the coinvariant algebra of v for a large class of irreducible modules around the steinberg module this was built on a computation via connections to algebraic groups of the steinberg module multiplicity in a tensor product of s v with other tensor spaces of fundamental weight modules

composite systems with extensive sq power law entropies abstract the problem of characterizing the kind of correlations leading to an extensive behaviour of the s q power law entropic measure has recently been considered by tsallis gell mann and sato tgs complexity and non extensivity new trends in statistical mechanics yukawa institute kyoto 14 18 march 2005 we propose a family of models for the probability occupancy of phase space exhibiting an extensive behaviour of s q and allowing for an explicit analysis of the n thermodynamic limit

real and spurious contributions for the shannon renyi and tsallis entropies abstract a two parameter probability distribution is constructed by dilatation or contraction of the escort probability distribution this transformation involves a physical probability distribution p associated with the system under study and an almost arbitrary reference probability distribution p in contrast to the shannon and renyi entropies the tsallis entropy does not decompose as the sum of the physical contribution due to p and the reference or spurious part owing to p for solving this problem a slight modification to the relation between tsallis and renyi entropies must be introduced the procedure in this paper gives rise to a nonconventional one parameter shannon entropy and to two parameter renyi and tsallis entropies associated with p it also contributes to clarify the meaning and role of the escort probabilities set

multiplicative duality q triplet and μ ν q relation derived from the one to one correspondence between the μ ν multinomial coefficient and tsallis entropy sq abstract we derive the multiplicative duality q 1 q and other typical mathematical structures as the special cases of the q relation behind tsallis statistics by means of the multinomial coefficient recently the additive duality q 2 q in tsallis statistics is derived in the form of the one to one correspondence between the q multinomial coefficient and tsallis entropy a slight generalization of this correspondence for the multiplicative duality requires the multinomial coefficient as a generalization of the q multinomial coefficient this combinatorial formalism provides us with the one to one correspondence between the multinomial coefficient and tsallis entropy s q which determines a concrete relation among three parameters and q i e 1 1 q which is called q relation in this paper as special cases of the q relation the additive duality and the multiplicative duality are recovered when 1 and q respectively as other special cases when 2 q a set of three parameters q is identified with the q triplet q sen q rel q stat recently conjectured by tsallis moreover when 1 q the relation 1 1 q sen 1 min 1 max in the multifractal singularity spectrum f is recovered by means of the q relation

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

modular invariants of parabolic subgroups of general linear groups abstract let p be a prime number and let a be an elementary abelian p group of rank m the purpose of this paper is to determine a full system for the invariants of parabolic subgroups of the general linear group gl m f p in h a f p a relation between these invariants and dickson ones is also obtained

a class of generalized complex hermite polynomials a class of generalized complex polynomials of hermite type suggested by a special magnetic schrodinger operator is introduced and some related basic properties are discussed

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

on alternative q weibull and q extreme value distributions properties and applications abstract tsallis statistics and tsallis distributions have been attracting a significant amount of research work in recent years importantly the tsallis statistics q distributions have been applied in different disciplines yet a relationship between some existing q weibull distributions and q extreme value distributions that is parallel to the well established relationship between the conventional weibull and extreme value distributions through a logarithmic transformation has not be established in this paper we proposed an alternative q weibull distribution that leads to a q extreme value distribution via the q logarithm transformation some important properties of the proposed q weibull and q extreme value distributions are studied maximum likelihood and least squares estimation methods are used to estimate the parameters of q weibull distribution and their performances are investigated through a monte carlo simulation study the methodologies and the usefulness of the proposed distributions are illustrated by fitting the 2014 traffic fatalities data from the national highway traffic safety administration

a shannon tsallis transformation we determine a general link between two different solutions of the maxent variational problem namely the ones that correspond to using either shannon s or tsallis entropies in the concomitant variational problem it is shown that the two variations lead to equivalent solutions that take different appearances but contain the same information these solutions are linked by our transformation

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

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

asymptotic of complex hyperbolic geometry and l2 spectral analysis of landau like hamiltonians in this paper we show that the flat hermitian complex geometry of cn n 1 is approximated by the complex hyperbolic geometry of the bergman complex balls b n cn of radius 0 furthermore it will be shown that some elements of the l2 spectral analysis such as the spectrum the l2 eigenprojector and the resolvent kernels associated to the so called landau like hamiltonian hb on b n give rise to their analogous of the landau like hamiltonian hb on cn by letting tend to infinity

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

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

a possible deformed algebra and calculus inspired in nonextensive thermostatistics we present a deformed algebra related to the q exponential and the q logarithm functions that emerge from nonextensive statistical mechanics we also develop a q derivative and consistently a q integral for which the q exponential is an eigenfunction the q derivative and the q integral have a dual nature that is also presented

on the extended kolmogorov nagumo information entropy theory the q 1 q duality and its possible implications for a non extensive two dimensional ising model abstract the aim of this paper is to investigate the q 1 q duality in an information entropy theory of all q generalized entropy functionals tsallis renyi and sharma mittal measures in the light of a representation based on generalized exponential and logarithm functions subjected to kolmogorov s and nagumo s averaging we show that it is precisely in this representation that the form invariance of all entropy functionals is maintained under the action of this duality the generalized partition function also results to be a scalar invariant under the q 1 q transformation which can be interpreted as a non extensive two dimensional ising model duality between systems governed by two different power law long range interactions and temperatures this does not hold only for tsallis statistics but is a characteristic feature of all stationary distributions described by q exponential boltzmann factors

complex and real hermite polynomials and related quantizations it is known that the anti wick or standard coherent state quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator up to the addition of a constant in this work we show that these two issues are not necessarily coupled there exists a family of separable hilbert spaces including the usual fock bargmann space and in each element in this family there exists an overcomplete set of unit norm states resolving the unity with the exception of the fock bargmann case they all produce non canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant the statistical aspects of these non equivalent coherent state quantizations are investigated we also explore the localization aspects in the real line yielded by similar quantizations based on real hermite polynomials

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

extensions for finite chevalley groups iii rational and generic cohomology abstract let g be a connected reductive algebraic group and b be a borel subgroup defined over an algebraically closed field of characteristic p 0 in this paper the authors study the existence of generic g cohomology and its stability with rational g cohomology groups via the use of methods from the authors earlier work new results on the vanishing of g and b cohomology groups are presented furthermore vanishing ranges for the associated finite group cohomology of g f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

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

information theoretic approach to statistical properties of multivariate cauchy lorentz distributions the cauchy lorentz cl distribution has divergent lowest moments this necessarily leads to the fact that the information theoretic approach is essential for the study of its statistical properties here correlation measured by the mutual entropy is discussed for the multivariate cl distribution it is found that correlation obeys a simple scaling law with respect to the dimensionality of the distribution then regarding the cl distribution as a power law quantum wavepacket the information entropic uncertainty relation is also discussed both analytically and numerically it is found that the sum of the position and momentum information entropies tends to the value of the lower bound for large dimensions

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

renyi and tsallis entropies for incomplete or overcomplete systems of events abstract in this article shannon renyi and tsallis entropies are considered for a system of events characterized by an arbitrary probability distribution p that can be incomplete complete or overcomplete after a suitable transformation that leads to the escort probabilities of p these can be written as the canonical probability distribution for a set of pseudo energies hartley information e n ln p n and a dimensionless parameter q that plays the role of thermodynamics several relations between the entropies are presented including the analysis of compound systems the method is illustrated with an example

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

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

physical meaning of the time correlation length obtained in a computer simulation a comparison is made of three statistical methods of determining the autocorrelation coefficients and correlation length in a time data series obtained by a computer simulation three points of state were chosen where the fluctuations and correlations play different roles in the liquid in the solid near melting and in the melting zone taking account of the physical meaning of the results it is found that there is always a limiting value of the lag at which the series must be cut off

eigenvalue problems for the schrodinger operator with the magnetic field on a compact riemannian manifold abstract we consider the eigenvalue problem of the schrodinger operator with the magnetic field on a compact riemannian manifold first we discuss the least eigenvalue we give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the schrodinger operator without the magnetic field second we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup here we use the theory of asymptotic expansion for wiener functionals

two long standing problems in tsallis statistics abstract on the basis of tsallis entropy and the joint probability factorization condition two controversial problems existing in tsallis statistics are investigated where one is whether energy is extensive or not and the other is whether it is necessary to introduce the so called generalized zeroth law of thermodynamics or not the results obtained show clearly that like entropy energy is also nonextensive in tsallis statistics and that the zeroth law of thermodynamics has been implicitly used in tsallis statistics since 1988 moreover it is expounded that the standard energy additivity rule adopted by a great number of researchers is not suitable in tsallis statistics because its corollary is in contradiction with the zeroth law of thermodynamics

inversion of umarov tsallis steinberg s q fourier transform and the complex plane generalization we introduce a complex q fourier transform as a generalization of the real one analyzed in s umarov c tsallis s steinberg milan j math 307 2008 by recourse to tempered ultradistributions we show that this complex plane generalization overcomes all the troubles that afflict its real counterpart

superstatistics in nanoscale electrochemical systems stochastic electrochemical reaction steps on nanosized electrodes are non markovian when externally driven by an applied voltage we show that compared to the markovian case when external driving is absent nanoscale electrochemical systems obey a superstatistics characterized by a superposition of tsallis q indices the distribution of tsallis q indices along stochastic trajectories can be calculated from the electrochemical master equation and normal distributions from boltzmann gibbs thermostatistics are recovered in the thermodynamic limit the infinite electrode size limit although on the nanoscale the external control makes intricate correlations between the microstates in the superstatistical frame one can still address the microstates as if they were uncorrelated the resulting superstatistical entropic form is additive in this frame and tsallis indices have on the time average values which is indeed an example of a superstatistical system where no ad hoc distribution has to be assumed for the fluctuations rather the distribution is directly calculated from a mesoscopic master equation without freely adjustable parameters

information theoretical properties of tsallis entropies a chain rule and a subadditivity for the entropy of type which is one of the nonadditive entropies were derived by daroczy in this paper we study the further relations among tsallis type entropies which are typical nonadditive entropies the chain rule is generalized by showing it for tsallis relative entropy and the nonadditive entropy we show some inequalities related to tsallis entropies especially the strong subadditivity for tsallis type entropies and the subadditivity for the nonadditive entropies the subadditivity and the strong subadditivity naturally lead to define tsallis mutual entropy and tsallis conditional mutual entropy respectively and then we show again chain rules for tsallis mutual entropies we give properties of entropic distances in terms of tsallis entropies finally we show parametrically extended results based on information theory

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

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

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

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators

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