polynomial lienard equations near infinity abstract we study the behaviour near infinity of the generalized lienard equations x y y m k 0 a k x k y n k 0 b k x k providing a complete classification using poincare compactification as well as poincare lyapunov compactification we show that all the necessary information is contained in a m and b n except for the so called center focus problem occuring in case m 2 n 1 with m n odd and where the behaviour also depends on the value of the other coefficients a i and b j we also indicate how to take care about obtaining uniform information near infinity

non gaussian effects on quantum entropies abstract a deduction of generalized quantum entropies within the non gaussian frameworks tsallis and kaniadakis is derived using a generalized combinatorial method and the so called q and calculus in agreement with previous results we also show that for the tsallis formulation the q quantum entropy is well defined for values of the nonextensive parameter q lying in the interval 0 2

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

robustness in quantum measurements conditions are formulated under which a representation of an intrinsic c algebra of often quasilocal observables of an infinite system is appropriate to describe measurement type processes such a representation should allow for the description of robust experiments it should be separable and the pointer observable should be in its weak closure if the pointer values are discrete the existence of such a measurement representation can be proven if the pointer can take continuously many values then the existence can only be proven under the additional assumptions of having an asymptotically abelian system or dealing with type i representations in the constructed measurement representations the pointer observable turns out to be classical the structure of the representation suggests that spontaneous symmetry breaking might be a physical explanation of the emergence of the classical pointer

lesche stability of κ entropy abstract the lesche stability condition for the shannon entropy j stat phys 27 1982 419 represents a fundamental test for its experimental robustness for systems obeying the maxwell boltzmann statistical mechanics of course this stability condition must be satisfied by any entropic functional candidate to generate non conventional statistical mechanics in the present effort we show that the entropy recently introduced in literature phys rev e 66 2002 056125 satisfies the lesche stability condition

phase spaces and dynamical descriptions of infinite mean field quantum systems following the work of p bona j math phys 29 2223 1988 a class of infinite mean field quantum lattice systems is considered within the framework of algebraic quantum mechanics and the following question is discussed which are the possible dynamical descriptions of the system i e in which representations of the system s quasilocal c algebra a do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged c algebra c containing a and some of the classical observables of the system in to this end we associate a classical state space e to any representation which is a subset of the phase space e rl defined by bona for the largest possible representation and show how it is characterized by the normal states thereby obtaining a way to determine e explicitly in many cases the answer to the question posed then reads the limiting heisenberg dynamics exists in on c if and only if the set e is invariant under the flow

properties of the non extensive gaussian entropy the present work investigates the lesche stability experimental robustness the thermodynamic stability the legendre structure of thermodynamics and derives the maximum entropy distribution of the one parametric non extensive gaussian entropy we show that this entropy definition fulfills both stability conditions for all values of its parameter q r the entropy maximizer contains the lambert w function which allows the preservation of the legendre transformations

relaxation functions in dipolar materials we compare two simple cartoons of relaxation processes in dipolar materials the first passage relaxation function introduced by k weron 1991 and the average relaxation function expressing the proportion of dipoles which did not change their imposed aligned orientation up to a certain time the latter providing a description closer to what is experimentally measured in some cases the two relaxation functions are proved to coincide

generalized thermostatistics based on deformed exponential and logarithmic functions abstract the equipartition theorem states that inverse temperature equals the log derivative of the density of states this relation can be generalized by introducing a proportionality factor involving an increasing positive function x it is shown that this assumption leads to an equilibrium distribution of the boltzmann gibbs form with the exponential function replaced by a deformed exponential function in this way one obtains a formalism of generalized thermostatistics introduced previously by the author it is shown that tsallis thermostatistics with a slight modification is the most obvious example of this formalism and corresponds with the choice x x q

self similarity degree of deformed statistical ensembles abstract we consider self similar statistical ensembles with the phase space whose volume is invariant under the deformation that squeezes expands the coordinate and expands squeezes the momentum the related probability distribution function is shown to possess a discrete symmetry with respect to manifold action of the jackson derivative to be a homogeneous function with a self similarity degree q fixed by the condition of invariance under n 1 fold action of the related dilatation operator in slightly deformed phase space we find the homogeneous function is defined with the linear dependence at n 0 whereas the self similarity degree equals the gold mean at n 1 and q n in the limit n dilatation of the homogeneous function is shown to decrease the self similarity degree q at n 0

an axiomatic characterization of a two parameter extended relative entropy the uniqueness theorem for a two parameter extended relative entropy is proven this result extends our previous one the uniqueness theorem for a one parameter extended relative entropy to a two parameter case in addition the properties of a two parameter extended relative entropy are studied

on the relationship among three theories of relaxation in disordered systems abstract empirical evidence has accumulated showing the wide occurrence of a stretched exponential relaxation decay law for many diverse condensed matter systems several theories based on different physical mechanisms have been successful in deriving the stretched exponential decay law in a natural way three of these theories the direct transfer model the hierarchically constrained dynamics model and the defect diffusion model are shown here to have an underlying common mathematical structure

macroscopic thermodynamics based on composable nonextensive entropies abstract composability of entropy implies that entropy of the total system composed of independent subsystems is given by a function of entropies of the subsystems it is shown that the zeroth law of thermodynamics naturally leads to tsallis type pseudoadditivity of the composable entropies then macroscopic thermodynamics is discussed based on the simplest example of such entropies it is found that even if nonextensive entropy is initially employed the resulting thermodynamic entropy is necessarily extensive at the macroscopic level in accordance with caratheodory s theorem this result depends crucially on the assumptions that the subsystems are independent and the total energy is the sum of the energies of the subsystems

legendre structure of κ thermostatistics revisited in the framework of information geometry information geometry is a powerful framework in which to study families of probability distributions or statistical models by applying differential geometric tools it provides a useful framework for deriving many important structures in probability theory by identifying the space of probability distributions with a differentiable manifold endowed with a riemannian metric in this paper we revisit some aspects concerning the thermostatistics based on the entropy s in the framework of information geometry after introducing the dually flat structure associated with the distribution we show that the dual potentials derived in the formalism of information geometry correspond to the generalized massieu function and the generalized entropy s characterizing the legendre structure of the deformed statistical mechanics in addition we obtain several quantities such as escort distributions and canonical divergence relevant for the further development of the theory

application of generalized thermostatistics to fully developed turbulence abstract we apply the formalism of non extensive statistical mechanics to fully developed turbulent flows we obtain analytical formulas for probability density functions of velocity differences depending on distance r and reynolds number r which are in very good agreement with turbulence experiments the skewness is taken into account by analysing universal contributions due to higher order correlations of the chaotic dynamics we also calculate the moment scaling exponents m which also agree well with experimental data

nondistributive algebraic structures derived from nonextensive statistical mechanics we propose a two parametric nondistributive algebraic structure that follows from q q logarithm and q q exponential functions properties of generalized q q operators are analyzed we also generalize the proposal into a multiparametric structure generalization of logarithm and exponential functions and the corresponding algebraic operators all n parameter expressions recover n 1 generalization when the corresponding qn 1 nonextensive statistical mechanics has been the source of successive generalizations of entropic forms and mathematical structures in which this work is a consequence

a unified characterization of generalized information and certainty measures abstract in this paper we consider the axiomatic characterization of information and certainty measures in a unified way we present the general axiomatic system which captures the common properties of a large number of the measures previously considered by numerous authors we provide the corresponding characterization theorems and define a new generalized measure called the inforcer which is the quasi linear mean of the function associated with the event probability following the general composition law in particular we pay attention to the polynomial composition and the corresponding polynomially composable inforcer measure the most common measures appearing in literature can be obtained by specific choice of parameters appearing in our generic measure and they are listed in tables

deformed exponentials and logarithms in generalized thermostatistics abstract criteria are given that deformed logarithmic and exponential functions should satisfy with a pair of such functions one can associate another function called the deduced logarithmic function it is shown that generalized thermostatistics can be formulated in terms of deformed exponential functions together with the associated deduced logarithmic functions

a set of new three parameter entropies in terms of a generalized incomplete gamma function abstract in this paper by considering the first three khinchin axioms k1 k3 and neglecting the fourth k4 a set of new three parameter entropies will be introduced which are expressed in terms of a generalized incomplete gamma function as s d c 1 c 2 p i d 1 1 c 1 ln p i 1 c 2 ln p i also its asymptotic behavior will be found and it will be shown that for some special values of parameters some known entropies like tsallis kaniadakis and abe entropies are recovered

a relationship between asymmetric levy stable distributions and the dielectric susceptibility this paper as a complement to the work of montroll and bendler is concerned with the levy stable distributions and their connection to the dielectric response of dipolar materials in the frequency domain the necessary and sufficient condition for this connection is found the presented probabilistic analysis is based on the mathematically correct representation of the meaning of the relaxation function of a system of dipoles and shows why the same form of a distribution of relaxation rates namely the completely asymmetric levy stable distribution should apply in all different relaxing systems this is in contrast to the traditional definition of the relaxation function expressed as a weighted average of exponential relaxation functions which does not explain the universality of the dielectric relaxation law it also follows from the present considerations that not only is the imaginary part of the dielectric susceptibility directly related to the levy stable distribution as was found by montroll and bendler but so is the real part as a consequence the relation cot n 2 for p and 0 n 1 implied by experimental results is obtained

general pseudoadditivity of composable entropy prescribed by the existence of equilibrium the concept of composability states that entropy of the total system composed of independent subsystems is a function of entropies of the subsystems here the most general pseudoadditivity rule for composable entropy is derived based only on the existence of equilibrium

the two parameters κ r in the generalized statistics abstract based on the generalized boltzmann equation and the reverse function of the distribution function we investigate the two parameter generalized statistics and get an expression between the two parameters r and the physical quantities about the system considered we find that the two parameters can define some characteristics of the system as examples this result can just return to the previous one obtained for tsallis and statistics for some complex systems we may need the two parameter statistics to describe

scaling in currency exchange a conditionally exponential decay approach abstract we use the conditionally exponential decay ced model to explain the scaling behavior in currency exchange fx rates this approach enables us not only to show that fx returns satisfy scaling with an exponent qualitatively different from that of a random walk but also to identify the distributions of these returns corresponding to the empirical scaling laws the study is conducted via three different estimation methods and using intra daily fx data which offers the great advantage of large samples and high significance

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

configurations of limit cycles in lienard equations abstract we show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial lienard equation the related vector field x is morse smale moreover it has the minimum number of singularities required for realizing the configuration in a lienard equation we provide an explicit upper bound on the degree of x which is lower than the results obtained before obtained in the context of general polynomial vector fields

two parameter entropies s k r and their dualities this paper investigates the dualities of two parameter entropies sk r kaniadakis et al 2005 two kinds of dualities are found namely inverse r r and inverse scaling r rs k ks which recovers the additive q 2 q and multiplicative q 1 q dualities in the tsallis limit

two parameter deformations of logarithm exponential and entropy a consistent framework for generalized statistical mechanics a consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace form entropy and by requiring that physically motivated mathematical properties are preserved the emerging differential functional equation yields a two parameter class of generalized logarithms from which entropies and power law distributions follow these distributions could be relevant in many anomalous systems within the specified range of parameters these entropies possess positivity continuity symmetry expansibility decisivity maximality concavity and are lesche stable the boltzmann shannon entropy and some one parameter generalized entropies already known belong to this class these entropies and their distribution functions are compared and the corresponding deformed algebras are discussed

existence of the limiting mean field dynamics in general equilibrium representations abstract the existence of the limiting finite time transformations is proved for a class of discrete mean field models in all states which are asymptotically permutation invariant to this end detailed approximation and analyticity properties of the complex many time green s functions are worked out for a rather general class of grand canonical equilibrium states the limiting time transformations are shown to constitute a w automorphism group in the corresponding gns von neumann algebra in spite of the extremely long range interaction potential and in spite of the center not necessarily being point wise invariant approximation properties for the finite temperature heisenberg generators are obtained by complex differentiation

maximum entropy generation and κ exponential model abstract an increasing number of natural phenomena appear to deviate from standard statistical distributions it has kindled interest in alternative formulation of statistical mechanics which should preserve most of the mathematical structures of the boltzmann gibbs theory while reproducing the phenomenology of the anomalous systems the theory has obtained a lot of results in the analysis of natural phenomena but it was not extended to irreversible open systems a modified theory is proposed for its use in irreversible open systems

a new one parameter deformation of the exponential function abstract recently in kaniadakis physica a 296 2001 405 a new one parameter deformation for the exponential function exp x 1 2 x 2 x 1 exp 0 x exp x which presents a power law asymptotic behaviour has been proposed the statistical distribution f z 1 exp e has been obtained both as stable stationary state of a proper nonlinear kinetics and as the state which maximizes a new entropic form in the present contribution starting from the algebra and after introducing the analysis we obtain the exponential exp x as the eigenstate of the derivative and study its main mathematical properties

fuzzy entropy complexity and multifractal behavior of statistical physics financial dynamics abstract the present paper introduces a novel stochastic statistical physics financial system by combining exclusion system with compound poisson process the financial system is a model concerning with price fluctuations derived from dissemination of trading attitudes of financial agents and random drastic jumps caused by exogenous economic factors respectively multifractal detrended cross correlation analysis and fuzzy entropy of simulation dataset are performed fractional fuzzy entropy based on fuzzy entropy method and fractional calculus is also developed to explore nonlinear complexity of the proposed model in the paper empirical results display that there are significant multifractality and fractal asymmetry in the proposed model and the complexity of return series increases when or increases correspondingly and same analyses for actual stock market dataset are comparatively investigated the conclusions reveal that the novel compound dynamics can reproduce various nonlinear behaviors of actual stock markets to a certain extent

sensitivity function and entropy increase rates for z logistic map family at the edge of chaos it is well known that for chaotic systems the production of relevant entropy boltzmann gibbs is always linear and the system has strong exponential sensitivity to initial conditions in recent years various numerical results indicate that basically the same type of behavior emerges at the edge of chaos if a specific generalization of the entropy and the exponential are used in this work we contribute to this scenario by numerically analyzing some generalized nonextensive entropies and their related exponential definitions using z logistic map family we also corroborate our findings by testing them at accumulation points of different cycles

generalized exponential function and discrete growth models abstract here we show that a particular one parameter generalization of the exponential function is suitable to unify most of the popular one species discrete population dynamic models into a simple formula a physical interpretation is given to this new introduced parameter in the context of the continuous richards model which remains valid for the discrete case from the discretization of the continuous richards model generalization of the gompertz and verhulst models one obtains a generalized logistic map and we briefly study its properties notice however that the physical interpretation for the introduced parameter persists valid for the discrete case next we generalize the scramble competition ricker discrete model and analytically calculate the fixed points as well as their stabilities in contrast to previous generalizations from the generalized ricker model one is able to retrieve either scramble or contest models

phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line abstract this paper provides the classification of the phase portraits in the poincare disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus the sufficient and necessary conditions for existence and uniqueness of limit cycles are also given

equivalence among different formalisms in the tsallis entropy framework abstract in a recent paper t wada a m scarfone phys lett a 335 2005 351 the authors discussed the equivalence among the various probability distribution functions of a system in equilibrium in the tsallis entropy framework in the present letter we extend these results to a system which is out of equilibrium and evolves to a stationary state according to a nonlinear fokker planck equation nfpe by means of time scale conversion it is shown that there exists a correspondence among the self similar solutions of the nfpes associated with the different tsallis formalisms the time scale conversion is related to the corresponding lyapunov functions of the respective nfpes

generalized statistics framework for rate distortion theory abstract variational principles for the rate distortion rd theory in lossy compression are formulated within the ambit of the generalized nonextensive statistics of tsallis for values of the nonextensivity parameter satisfying 0 q 1 and q 1 alternating minimization numerical schemes to evaluate the nonextensive rd function are derived numerical simulations demonstrate the efficacy of generalized statistics rd models

schrodinger dynamics and physical folia of infinite mean field quantum systems this work deals with the general i e non equilibrium schrodinger dynamics of infinite mean field quantum systems it is shown how this dynamics is related to the hamiltonian flow t q on the classical phase space e subseteqq mathbb r l recently defined by bona 10 to describe the time evolution of classical macroscopic observables of the system these connections allow us to clarify the structure of the set of all physical folia a notion introduced by sewell for the dynamical description of infinite systems in cases where this description is representation dependent they also yield a result showing that the heisenberg picture is the more general approach to such descriptions in the sense that there are more representations in which a heisenberg dynamics can be defined than ones which allow for the definition of a schrodinger dynamics finally our theory makes it possible to construct many explicit examples of physical folia in this connection it is shown that there can be overcountably many inequivalent representations with the same macroscopic dynamical structure

stability of tsallis entropy and instabilities of renyi and normalized tsallis entropies a basis for q exponential distributions the q exponential distributions which are generalizations of the zipf mandelbrot power law distribution are frequently encountered in complex systems at their stationary states from the viewpoint of the principle of maximum entropy they can apparently be derived from three different generalized entropies the renyi entropy the tsallis entropy and the normalized tsallis entropy accordingly mere fittings of observed data by the q exponential distributions do not lead to identification of the correct physical entropy here stabilities of these entropies i e their behaviors under arbitrary small deformation of a distribution are examined it is shown that among the three the tsallis entropy is stable and can provide an entropic basis for the q exponential distributions whereas the others are unstable and cannot represent any experimentally observable quantities

thermodynamic stabilities of the generalized boltzmann entropies abstract we consider the thermodynamic stability conditions tsc on the boltzmann entropies generalized by tsallis q and kaniadakis deformed logarithmic functions it is shown that the corresponding tscs are not necessarily equivalent to the concavity of the generalized boltzmann entropies with respect to internal energy nevertheless both the tscs are equivalent to the positivity of standard heat capacity

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

on the classical part of the mean field dynamics for quantum lattice systems in grand canonical representations for a class of discrete mean field models the limiting dynamics is investigated in the representations of generalized grand canonical states it is demonstrated that for a certain form of spontaneous symmetry breakdown the w automorphism dynamics exhibits a uniquely determined nontrivial classical part which is essential for the explanation of macroscopic quantum phenomena

continuous growth models in terms of generalized logarithm and exponential functions abstract consider the one parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non symmetrical hyperboles these generalizations coincide to the one obtained in the context of non extensive thermostatistics we show that these functions are suitable to describe and unify the great majority of continuous growth models which we briefly review physical interpretation to the generalization function parameter is given for the richards model which has an underlying microscopic model to justify it

a class of vectorfields on s2 that are topologically equivalent to polynomial vectorfields abstract let x be a c1 vectorfield on s2 x y z r x2 y2 z2 1 such that no open subset of s2 is the union of closed orbits of x if x has only a finite number of singular orbits and satisfies one additional condition then it is shown that x is topologically equivalent to a polynomial vectorfield a corollary is that a foliation f of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if f has only a finite number of inseparable leaves

beyond the zipf mandelbrot law in quantitative linguistics abstract in this paper the zipf mandelbrot law is revisited in the context of linguistics despite its widespread popularity the zipf mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus in particular we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation finally it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by c tsallis

annual change of pareto index dynamically deduced from the law of detailed quasi balance abstract employing data on the assessed value of land in 1983 2005 japan we investigate the dynamical behavior in the high scale region of non equilibrium systems from the detailed quasi balance and gibrat s law we derive a relation between the change of pareto index and a symmetry in the detailed quasi balance the relation is confirmed in the empirical data nicely

stabilities of generalized entropies the generalized entropic measure which is maximized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity is considered to examine if it can be of physical relevance its experimental robustness is discussed in particular lesche s criterion is analysed which states that an entropic measure is stable if its change under an arbitrary weak deformation of the distribution representing fluctuations of experimental data remains small it is essential to note the difference between this criterion and thermodynamic stability a general condition under which the generalized entropy becomes stable is derived examples known in the literature including the entropy for the stretched exponential distribution the quantum group entropy and the entropy are discussed

thermodynamic equilibrium and its stability for microcanonical systems described by the sharma taneja mittal entropy it is generally assumed that the thermodynamic stability of equilibrium states is reflected by the concavity of entropy we inquire in the microcanonical picture about the validity of this statement for systems described by the two parametric entropy s kappa r of sharma taneja and mittal we analyze the composability rule for two statistically independent systems a and b described by the entropy s kappa r with the same set of the deformation parameters it is shown that in spite of the concavity of the entropy the composability rule modifies the thermodynamic stability conditions of the equilibrium state depending on the values assumed by the deformation parameters when the relation s kappa r a union b s kappa r a s kappa r b holds superadditive systems the concavity condition does imply thermodynamics stability otherwise when the relation s kappa r a union b s kappa r a s kappa r b holds subadditive systems the concavity condition does not imply thermodynamical stability of the equilibrium state

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

a characterization of theω limit sets of planar continuous dynamical systems abstract if the family of curves g i i and the set of points s are given we find necessary and sufficient conditions for the existence of a c k map f from the plane into itself 0 k if k 0 we also assume that f is locally lipschitz such that the curves from g and the points from s are respectively nondegenerate trajectories and singular points of the system x f x and additionally s i i is the limit set of some trajectory of the system in such cases we also provide a detailed geometric description of s i i after contraction of the connected components of s

thermodynamic paths and stochastic order in open systems abstract the theorem of extremum entropy generation is related to the stochastic order of the paths inside the phase space indeed the system evolves from an indistinguishable configuration to another one on the most probable path in relation to the paths stochastic order the result is that at the stationary state the entropy generation is maximal and this maximum value is a consequence of the stochastic order of the paths in the phase space conversely the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation for the open systems at the stationary states

on the cole cole relaxation function and related mittag leffler distribution abstract in the framework of the one dimensional fractal time random walk ftrw relaxation model we rigorously show that the frequency domain response takes in both nonbiased and biased walks the only possible cole cole form the underlying reason for this is the specific form of the relaxation function the survival probability of a relaxing system determined in this model by the mittag leffler distribution we provide also analytical formulas for the propagators of the nonbiased and biased ftrws

a κ entropic approach to the analysis of the fracture problem abstract in order to study the relation between the ohmic resistance measured in a thin conducting ribbon and the length of a transversal cut we employ a one parameter deformed exponential and logarithm that were recently introduced phys rev e 66 2002 056125 in the framework of a generalized statistical mechanics the analytical results have been compared with the data that was experimentally obtained and numerically computed with the boundary element method remarkably the new deformed functions that interpolate between the standard functions and the power law functions allow the best fit of the experimental data to be obtained for a wide range of the cut length

two forms of self similarity as a fundamental feature of the power law dielectric response a new mathematical representation of the cluster model for dielectric relaxation in a bound dipole case is established by employing the extremal value theory two distinct probabilistic mechanisms which drive the dielectric response function to acquire the power law form are presented consequently two forms of self similarity one of which dominates the response at short times and the other at long times leading to a general relaxation equation are identified finally the conditions under which the derived response function takes the well known empirical forms williams watts cole cole cole davidson broadened debye and flat loss responses are recognized

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

remarks about the tsallis formalism in the present paper the conditions for the validity of the tsallis statistics are analyzed the same has been done following the analogy with the traditional case starting from the microcanonical description of the systems and taking into account their self similarity scaling properties in the thermodynamic limit it is analyzed the necessary conditions for the equivalence of microcanonical ensemble with the tsallis generalization of the canonical ensemble it is shown that the tsallis statistics is appropriate for the macroscopic description of systems with potential scaling laws of the asymptotic accessible states density of the microcanonical ensemble our analysis shows many details of the tsallis s formalism the q expectation values the generalized legendre transformations between the thermodynamic potentials as well as the conditions for its validity having a priori the possibility to estimate the value of the entropic index without the necessity of appealing to the computational simulations or the experiment on the other hand the definition of physical temperature received a modification that differs from the toral result for the case of finite systems we have generalized the microcanonical thermostatistics of gross with the generalization of the curvature tensor for this kind of description

a note on bounded entropies abstract the aim of the paper is to study the link between non additivity of some entropies and their boundedness we propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group this allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the renyi entropies including the particular case of the shannon entropy as a particular example we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity we show that tsallis havrda charvat entropy is included in the family of entropies we define finally a link is made between the approach developed in the paper and the theory of deformed logarithms

a cluster approach to the structure of imperfect materials and their relaxation spectroscopy a new theory is proposed for the explanation of observed relaxation phenomena which differs significantly from theories suggested by the authors before the theory is based on a model of structural organization of macroscopically sized samples of imperfectly structured materials both solids and liquids and is intermediate in character in terms of the model a microscopic structure is maintained over a cluster containing a number of microscopic units with an array of clusters described by a steady state distribution completing the macroscopic picture the structural regularity of each level of morphological organization is precisely defined by a coarse grained index which is given a thermodynamic interpretation in terms of binding energies and configurational entropy the limiting cases of an ideal liquid and a perfect crystal are recovered as asymptotic extremes in terms of this definition the consequences of this model for the relaxation dynamics of the structure are examined and it is shown that prepared fluctuations decay in a time power law manner as coupled zero point motions evolve either within clusters or between clusters with a power determined by the relevant regularity index as a result the origin of power law noise in materials is explained in terms of configurational entropy and its relation with gaussian and white noise which appear as asymptotic limits outlined the shape of the steady state distribution of the array of clusters is also determined without any a priori assumptions and it is shown to range from an unbounded form to a function as the regularity of the array superstructure increases experimental examples of dielectric relaxation spectroscopy have been used to illustrate these structural concepts and outline the way in which this technique can be used to deduce the structural organization of the sample finally a short description is given of some commonly observed forms of response and their structural interpretation

deformed logarithms and entropies by solving a differential functional equation inposed by the maxent principle we obtain a class of two parameter deformed logarithms and construct the corresponding two parameter generalized trace form entropies generalized distributions follow from these generalized entropies in the same fashion as the gaussian distribution follows from the shannon entropy which is a special limiting case of the family we determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm and show that important existing generalizations of the entropy are included as special cases in this two parameter class

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

a mechanism to derive multi power law functions an application in the econophysics framework abstract it is generally recognized that economical systems and more in general complex systems are characterized by power law distributions sometime these distributions show a changing of the slope in the tail so that more appropriately they show a multi power law behavior we present a method to derive analytically a two power law distribution starting from a single power law function recently obtained in the frameworks of the generalized statistical mechanics based on the sharma taneja mittal information measure in order to test the method we fit the cumulative distribution of personal income and gross domestic production of several countries obtaining a good agreement for a wide range of data

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles

inverse problems for dynamical systems in this paper we consider the problem of reconstructing a structurally stable vector field from its global phase portrait or from certain local pieces of the phase portrait we show that for arbitrary finite sets of points and of simple closed curves we can construct a structurally stable system having the elements of those sets as its critical points and limit cycles and a prescribed topological type in a neighborhood of each of them given a global structurally stable phase portrait we can construct a structurally stable polynomial system which is topologically equivalent to it some of our results were originally announced in 28 1 the general theory of dynamical systems orginated in the study of solution curves for systems of ordinary differential equations in a phase space this qualitative theory had its first major development in the work of poincare on the topology of integral curves a typical problem is to give a geometric description of the solutions of a particular system or class of systems with emphasis on the location of special features such as critical points and periodic orbits an inverse problem then is the construction of a system of equations given geometric properties of the solutions which may be in the form of a picture of the flow a schematic diagram of phase space or a list of critical orbits defined analytically a method for solving such problems is potentially useful in building models in the various fields in which the theory of dynamical systems is now being applied it could provide a basic model to which statistical identification techniques 141 could then be applied there is also an intrinsic interest in methods for constructing polynomial systems of prescribed type which is related to the lack of any progress toward a solution of hilbert s sixteenth problem the recent appearance of a counterexample 27 to the petrovsky landis conjecture for quadratic systems leaves the problem completely open one of the earliest results of this kind was obtained by forster in 1938

statistical descriptions of nonlinear systems at the onset of chaos abstract ensemble of initial conditions for nonlinear maps can be described in terms of entropy this ensemble entropy shows an asymptotic linear growth with rate k the rate k matches the logarithm of the corresponding asymptotic sensitivity to initial conditions the statistical formalism and the equality k can be extended to weakly chaotic systems by suitable and corresponding generalizations of the logarithm and of the entropy using the logistic map as a test case we consider a wide class of deformed statistical description which includes tsallis abe and kaniadakis proposals the physical criterion of finite entropy growth k strongly restricts the suitable entropies we study how large is the region in parameter space where the generalized description is useful

a conditionally exponential decay approach to scaling in finance we demonstrate how the basic ideas of the fractal and the heterogeneous market hypotheses lead to a rigorous mathematical model which can be used to solve the problem of characterizing the distribution of price changes corresponding to the empirical scaling law of volatility for high frequency data from the foreign exchange market for this purpose we adopt the conditionally exponential decay model which describes asymptotic behaviour of general complex systems we also discuss the overall rationale for why one might expect such scaling laws to hold for nancial data c 1999 elsevier science b v all rights reserved

legendre structure of the thermostatistics theory based on the sharma taneja mittal entropy abstract the statistical proprieties of complex systems can differ deeply for those of classical systems governed by boltzmann gibbs entropy in particular the probability distribution function observed in several complex systems shows a power law behavior in the tail which disagrees with the standard exponential behavior showed by gibbs distribution recently a two parameter deformed family of entropies previously introduced by sharma taneja and mittal stm has been reconsidered in the statistical mechanics framework any entropy belonging to this family admits a probability distribution function with an asymptotic power law behavior in the present work we investigate the legendre structure of the thermostatistics theory based on this family of entropies we introduce some generalized thermodynamical potentials study their relationships with the entropy and discuss their main proprieties specialization of the results to some one parameter entropies belonging to the stm family are presented

scaled bregman divergences in a tsallis scenario abstract there exist two different versions of the kullback leibler divergence k ld in tsallis statistics namely the usual generalized k ld and the generalized bregman k ld problems have been encountered in trying to reconcile them a condition for consistency between these two generalized k ld forms is derived by recourse to the additive duality of tsallis statistics it is also shown that the usual generalized k ld subjected to this additive duality known as the dual generalized k ld is a scaled bregman divergence this leads to an interesting conclusion the dual generalized mutual information is a scaled bregman information the utility and implications of these results are discussed

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

on the average uncertainty for systems with nonlinear coupling abstract the increased uncertainty and complexity of nonlinear systems have motivated investigators to consider generalized approaches to defining an entropy function new insights are achieved by defining the average uncertainty in the probability domain as a transformation of entropy functions the shannon entropy when transformed to the probability domain is the weighted geometric mean of the probabilities for the exponential and gaussian distributions we show that the weighted geometric mean of the distribution is equal to the density of the distribution at the location plus the scale i e at the width of the distribution the average uncertainty is generalized via the weighted generalized mean in which the moment is a function of the nonlinear source both the renyi and tsallis entropies transform to this definition of the generalized average uncertainty in the probability domain for the generalized pareto and student s t distributions which are the maximum entropy distributions for these generalized entropies the appropriate weighted generalized mean also equals the density of the distribution at the location plus scale a coupled entropy function is proposed which is equal to the normalized tsallis entropy divided by one plus the coupling