generalized simulated annealing we discuss and illustrate a new stochastic algorithm generalized simulated annealing for computationally finding the global minimum of a given not necessarily convex energy cost function defined in a continuous d dimensional space this algorithm recovers as particular cases the so called classical boltzmann machine and fast cauchy machine simulated annealings and turns out to be quicker than both

sequential sequence estimation for channels with intersymbol interference of finite or infinite length a sequential sequence estimator for channels with finite or infinite intersymbol interference isi is proposed it consists of a whitened matched filter followed by a sequential algorithm a key component of a sequential estimator is a suitable metric that accounts for the unequal search depth this metric is derived for the isi channel computer simulations for three different channels show that the sequential sequence estimator has considerably less computational complexity than a maximum likelihood sequence estimator using the viterbi algorithm mlse va while still maintaining the optimum estimation performance of the mlse va a performance not achievable with any reduced state viterbi algorithm

chaotic method for generating q gaussian random variables this study proposes a pseudorandom number generator of q gaussian random variables for a range of q values q 3 based on deterministic chaotic map dynamics our method consists of chaotic maps on the unit circle and map dynamics based on the piecewise linear map we perform the q gaussian random number generator for several values of q and conduct both kolmogorov smirnov ks and anderson darling ad tests the q gaussian samples generated by our proposed method pass the ks test at more than 5 significance level for values of q ranging from 1 0 to 2 7 while they pass the ad test at more than 5 significance level for q ranging from 1 to 2 4

mother wavelet functions generalized through q exponentials we generalize some widely used mother wavelets by means of the q exponential function exq 1 1 q x 1 1 q that emerges from nonextensive statistical mechanics in particular we define extended versions of the mexican hat and the morlet wavelets we also introduce new wavelets that are q generalizations of the trigonometric functions all cases reduce to the usual ones as q 1 within nonextensive statistical mechanics departures from unity of the entropic index q are expected in the presence of long range interactions long term memory multi fractal structures among others consistently the analysis of signals associated with such features is hopefully improved by proper tuning of the value of q we exemplify with the wavelet transform modulus maxima method for mono and multi fractal self affine signals

markov partition in non hyperbolic interval dynamics we considerc2 unimodal mapsf such that all periodic points are hyperbolic the critical point is non degenerated and non recurrent and the julia set does not contain intervals we construct a markov partition for a big part of the julia set then we use it to estimate the limit capacity and hausdorff dimension of the julia set

superposition of chaotic processes with convergence to levy s stable law we construct a family of chaotic dynamical systems with explicit distributions with broad tails which always violate the central limit theorem in particular we show that the superposition of many statistically independent identically distributed random variables obeying such a chaotic process converge in density to l evy s stable laws in a full range of index parameters in a unified manner the theory related to the connection between deterministic chaos and non gaussian distributions gives us a systematic view of the purely mechanical generation of l evy s stable laws

hyperbolicity and invariant measures for general c 2 interval maps satisfying the misiurewicz condition in this paper we will show that piecewise c2 mappings on 0 1 or s1 satisfying the so called misiurewicz conditions are globally expanding in the sense defined below and have absolute continuous invariant probability measures of positive entropy we do not need assumptions on the schwarzian derivative of these maps instead we need the conditions that is piecewise c2 that all critical points of are non flat and that has no periodic attractors our proof gives an algorithm to verify this last condition our result implies the result of misiurewicz in mi where only maps with negative schwarzian derivatives are considered moreover as a byproduct the present paper implies and simplifies the proof of the results of mane in ma who considers general c2 maps without conditions on the schwarzian derivative and restricts attention to points whose forward orbit stay away from the critical points one of the main complications will be that in this paper we want to prove the existence of invariant measures and therefore have to consider points whose iterations come arbitrarily close to critical points misiurewicz deals with this problem using an assumption on the schwarzian derivative of the map this assumption implies very good control of the non linearity of even for high n in order to deal with this non linearity without an assumption on the schwarzian derivative we use the tools of m s it will turn out that the estimates we obtain are so precise that the existence of invariant measures can be proved in a very simple way in some sense much simpler than in mi the existence of these invariant measures under such general conditions was already conjectured a decade ago

general thermostatistical formalisms invariance under uniform spectrum translations and tsallis q additivity the recent proposal by tsallis of a generalized thermostatistics devised to treat systems endowed with long range interactions long range memory eects or a fractal like relevant phase space has raised interesting and profound issues concerning the properties of general thermostatistical formalisms in the present paper we identify families of thermostatistical formalisms that share some fundamental characteristics the canonical ensemble s invariance under uniform translations of the hamiltonian s energy spectrum is shown to be a universal property veried by any thermostatistical formalism based upon linear mean energy constraints we also provide multiparametric families of entropies exhibiting tsallis q additivity law c 1999 elsevier science b v all rights reserved

superstatistical fluctuations in time series applications to share price dynamics and turbulence we report a general technique to study a given experimental time series with superstatistics crucial for the applicability of the superstatistics concept is the existence of a parameter beta that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration the proposed method extracts the main superstatistical parameters out of a given data set and examines the validity of the superstatistical model assumptions we test the method thoroughly with surrogate data sets then the applicability of the superstatistical approach is illustrated using real experimental data we study two examples velocity time series measured in turbulent taylor couette flows and time series of log returns of the closing prices of some stock market indices

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

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

the non markovian property of q gaussian process abstract in this paper the q gaussian process based on the non extensive theory is discussed from a mathematical point of view which has been widely applied to many anomalous diffusion systems in physics and finance firstly the discussion of non markovian property of q gaussian process provides a numerical support for the future theoretical research secondly the martingale and self similarity of this process are obtained by tsallis distributions thirdly the long dependence is analyzed by simulations and hurst exponents are compared with those of fractional brownian motion at last the european call option price formula driven by this process is simulated by which we find that this process can better match anomalous diffusion and the volatility smile

nonextensive entropies derived from form invariance of pseudoadditivity the form invariance of pseudoadditivity is shown to determine the structure of nonextensive entropies nonextensive entropy is defined as the appropriate expectation value of nonextensive information content similar to the definition of shannon entropy information content in a nonextensive system is obtained uniquely from generalized axioms by replacing the usual additivity with pseudoadditivity the satisfaction of the form invariance of the pseudoadditivity of nonextensive entropy and its information content is found to require the normalization of nonextensive entropies the proposed principle requires the same normalization as that derived previously a k rajagopal and s abe phys rev lett 83 1711 1999 but is simpler and establishes a basis for the systematic definition of various entropies in nonextensive systems

nonextensive triangle equality and other properties of tsallis relative entropy minimization kullback leibler relative entropy has unique properties in cases involving distributions resulting from relative entropy minimization tsallis relative entropy is a one parameter generalization of kullback leibier relative entropy in the nonextensive thermostatistics in this paper we present the properties of tsallis relative entropy minimization and present some differences with the classical case in the representation of such a minimum relative entropy distribution we highlight the use of the q product an operator that has been recently introduced to derive the mathematical structure behind the tsallis statistics one of our main results is the generalization of triangle equality of relative entropy minimization to the nonextensive case

yet another induction scheme for non uniformly expanding transformations abstract we introduce a new induction scheme for non uniformly expanding maps f of compact riemannian manifolds relying upon ideas of 33 and 10 we use this induction approach to prove that the existence of a gibbs markov young structure is a necessary condition for f to preserve an absolutely continuous probability with all its lyapunov exponents positive

a model for persistent levy motion abstract we propose a model which allows us to approximate fractional levy noise and fractional levy motion our model is based on i the gnedenko limit theorem for an attraction basin of stable probability law and ii fractional noise as a result of fractional integration differentiation of a white levy noise we investigate self affine properties of the approximation and conclude that it is suitable for modeling persistent levy motion with the levy index between 1 and 2

maps on statistical manifolds exactly reduced from the perron frobenius equations for solvable chaotic maps maps on a parameter space for expressing distribution functions are exactly derived from the perron frobenius equations for a generalized boole transform family here the generalized boole transform family is a one parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the cauchy distribution with some parameters with this reduction some relations between the statistical picture and the orbital one are shown from the viewpoint of information geometry the parameter space can be identified with a statistical manifold and then it is shown that the derived maps can be characterized also with an induced symplectic structure from a statistical structure symplectic and information geometric aspects of the derived maps are discussed

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

chaotic and stochastic functions abstract we study chaotic functions that are exact solutions to nonlinear maps a generalization of these functions which cannot be expressed as a recursive procedure anymore can produce truly random sequences even if the initial conditions are known exactly the next values are in principle unpredictable we present a mathematical formulation of an elementary stochastic process several applications are discussed

algorithms associated with arithmetic geometric and harmonic means and integrable systems abstract gauss algorithm for arithmetic geometric mean agm can be regarded as a discrete time integrable dynamical system having an elliptic theta function solution and a conserved quantity in this paper we consider algorithms associated with arithmetic geometric and harmonic means from a viewpoint of integrable systems first a max plus limit and its inverse limit of the agm algorithm are discussed these mean operations are shown to be connected to each other by the max plus limit secondly continous time dynamical systems associated with the arithmetic harmonic mean ahm algorithm are found thirdly it is shown that the ahm algorithm in indefinite case has a chaotic dynamics and is a generator of numbers which obey the cauchy distribution finally an extension of the ahm algorithm to the space of positive definite symmetric matrices is considered

nonextensive foundation of levy distributions a deep connection between the ubiquity of l evy distributions in nature and the nonextensive thermal statistics introduced a decade ago has been established recently tsallis et al phys rev lett 75 3589 1995 by using unnormalized q expectation values it has just been argued on physical grounds that normalized q expectation values should be used instead we revisit within this more appropriate scheme the l evy problem and verify that the relevant analytic results become sensibly simplified whereas the basic physics remains unchanged

timing variation in an analytically solvable chaotic system abstract we present analytic solutions for a chaotic dynamical system that do not have the regular timing characteristic of recently reported solvable chaotic systems the dynamical system can be viewed as a first order filter with binary feedback the feedback state may be switched only at instants defined by an external clock signal generalizing from a period one clock we show analytic solutions for period two and higher period clocks we show that even when the clock ticks randomly the chaotic system has an analytic solution these solutions can be visualized in a stroboscopic map whose complexity increases with the complexity of the clock we provide both analytic results as well as experimental data from an electronic circuit implementation of the system our findings bridge the gap between the irregular timing of well known chaotic systems such as lorenz and rossler and the well regulated oscillations of recently reported solvable chaotic systems

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

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

sequential decoding of convolutional codes in channels with intersymbol interference sequential sequence estimation for communications channels with convolutionally coded input sequences and intersymbol interference isi is proposed the receiver consists of a whitened matched filter wmf and a vector sequential decoder the metric of the vector sequential algorithm sa is derived analytical results for error probability and computational complexity of the sequential decoder are presented the error probability is found to be dominated by the minimum euclidean distance between the possible output sequences and the computational distribution is pareto typical of a sequential decoder the computational cut off code rate r sub comp and its lower bound r sub comp are also derived numerical simulation results of the error probability and the computational complexity are presented comparing with the viterbi algorithm va reveals that the sa essentially has the same error performance as that of the va while its average computational complexity is one to two orders of magnitude less than that of the va

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

generalized box muller method for generating q gaussian random deviates the q gaussian distribution is known to be an attractor of certain correlated systems and is the distribution which under appropriate constraints maximizes a generalization of the familiar shannon entropy this generalized entropy or q entropy provides the basis of nonextensive statistical mechanics a theory which is postulated as a natural extension of the standard boltzmann gibbs statistical mechanics and which may explain the ubiquitous appearance of heavy tailed distributions in both natural and man made systems the q gaussian distribution is also used as a numerical tool for example as a visiting distribution in generalized simulated annealing a simple easy to implement numerical method for generating random deviates from a q gaussian distribution based upon a generalization of the well known box miiller method is developed and presented this method is suitable for a larger range of q values infin q 3 than has previously appeared in the literature and can generate deviates from q gaussian distributions of arbitrary width and center matlab code showing a straightforward implementation is also included

on maximum entropy principle superstatistics power law distribution and renyi parameter abstract the equilibrium distributions of probabilities are derived on the basis of the maximum entropy principle mep for the renyi and tsallis entropies new s forms for the renyi and tsallis distribution functions are found which are normalised with corresponding entropies in contrast to the usual z forms normalised with partition functions z the superstatistics based on the gibbs distribution of energy fluctuations gives rise to a distribution function of the same structure that the renyi and tsallis distributions have the long range tail of the renyi distribution is the power law distribution with the exponent s expressed in terms of the free renyi parameter q as s 1 1 q the condition s 0 gives rise to the requirement q q can be uniquely determined with the use of a further extension of mep as the condition for maximum of the difference between the renyi and boltzmann entropies for the same power law distribution dependent on q it is found that the maximum is realized for q within the range from 0 25 to 0 5 and the exponent s varies from 1 3 to 2 in dependence on parameters of stochastic systems

on the maximum entropy principle and the minimization of the fisher information in tsallis statistics we give a new proof of the theorems on the maximum entropy principle in tsallis statistics that is we show that the q canonical distribution attains the maximum value of the tsallis entropy subject to the constraint on the q expectation value and the q gaussian distribution attains the maximum value of the tsallis entropy subject to the constraint on the q variance as applications of the non negativity of the tsallis relative entropy without using the lagrange multipliers method in addition we define a q fisher information and then prove a q cramer rao inequality that the q gaussian distribution with special q variances attains the minimum value of the q fisher information

julia fatou sullivan theory for real one dimensional dynamics our aim is to show that the julia fatou sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle and of the interval under extremely mild smoothness and non flatness conditions in order to stress the similarity between real and complex one dimensional dynamics let us recall the main results from the fatou julia sullivan theory if f is a rational map then there is a dynamical decomposition of the riemann sphere into the disjoint union of two totally invariant i e both forward and backward invariant sets j f f f here f f is the domain of normality of the family of iterates of f and is called the fatou set its complement which is called the julia set off is a compact set which contains all the complications of the dynamics off the connected components of the open set f f are mapped onto each other by f hence the orbit of a component of f f is the union of some components of f f julia proved in the beginning of the century that if a component of f f is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point sullivan in the remarkable paper su showed via quasi conformal deformations that the components are eventually periodic and fall into finitely many orbits let n be either the circle s j or a compact interval of the real line and f n n be a smooth endomorphism a critical point of f is a point where the derivative vanishes a critical point is non flat if some higher derivative is non zero a critical point is an inflection point if it has a neighbourhood where f is monotone otherwise it is called a turning point assume f is not a homeomorphism i f f is a homeomorphism then these maps correspond to degree 1 rational maps of the riemann sphere and the situation is

morphology of growth by random walk deposition we simulate a deposition model with a break of symmetry induced by a point source we 1nd that tsallis anomalous distribution for random walks n x n a 1 b q 1 x 2 q q 1

hierarchy of random chaotic maps with an invariant measure hierarchy of one and many parameter families of random chaotic maps and one parameter random elliptic maps of cn type with an invariant measure have been introduced using the invariant measure sinai ruelle bowen measure the kolmogorov sinai entropy of the random chaotic maps have been calculated analytically where the numerical simulations support the results

exact invariant measures how the strength of measure settles the intensity of chaos the aim of this paper is to show how to extract dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures on the one hand we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure then we show how to identify ergodic properties by means of transitions along the phase space via exact measures on the other hand we discuss quantitatively how infinite measures imply maps having subexponential lyapunov instability weakly chaotic as opposed to finite measure ergodic maps which are fully chaotic in addition we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval a demand from this field finally we give a simple proof that infinite measure implies universal mittag leffler statistics of observables rather than narrow distributions typically observed in finite measure ergodic maps

generalization of shannon khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy tsallis entropy one parameter generalization of shannon entropy has been often discussed in statistical physics as a new information measure this new information measure has provided many satisfactory physical interpretations in nonextensive systems exhibiting chaos or fractal we present the generalized shannon khinchin axioms to nonextensive systems and prove the uniqueness theorem rigorously our results show that tsallis entropy is the simplest among all nonextensive entropies by the detailed comparisons of our axioms with the previously presented two sets of axioms we reveal the peculiarity of pseudoadditivity as an axiom in this correspondence the most fundamental basis for tsallis entropy as information measure is established in the information theoretic framework

law of error in tsallis statistics in order to theoretically explain the ubiquitous existence of power law behavior such as chaos and fractals in nature tsallis entropy has been successfully applied to the generalization of the traditional boltzmann gibbs statistics the fundamental information measure of which is shannon entropy tsallis entropy s sub q is a one parameter generalization of shannon entropy s sub 1 in the sense that lim sub q spl rarr 1 s sub q s sub 1 the generalized statistics using tsallis entropy are referred to as tsallis statistics in order to present the law of error in tsallis statistics as a generalization of gauss law of error and prove it mathematically we apply the new multiplication operation determined by q logarithm and q exponential the fundamental functions in tsallis statistics to the definition of the likelihood function in gauss law of error the present maximum likelihood principle mlp leads us to determine the so called q gaussian distribution which coincides with one of the tsallis distributions derived from the maximum entropy principle for tsallis entropy under the second moment constraint

alternative way to characterize a q gaussian distribution by a robust heavy tail measurement the q gaussians are a class of stable distributions which are present in many scientific fields and that behave as heavy tailed distributions for an especific range of q values the identification of these values which are used in the description of systems is sometimes a hard task in this work the identification of a q gaussian distribution from empirical data was done by a measure of its tail weight using robust statistics numerical methods were used to generate artificial data to find out the tail weight medcouple and also to adjust the curve between medcouple and the q value we showed that the medcouple value remains unchanged when the calculation is applied to data which have long memory a routine was made to calculate the q value and its standard deviation when applied to empirical data it is possible to identify a q gaussian by the proposed methods with higher precision than in the literature for the same data sample or as precise as found in the literature however in this case it is required a smaller sample of data we hope that this method will be able to open new ways for identifying physical phenomena that belongs to nonextensive frameworks

induced maps markov extensions and invariant measures in one dimensional dynamics a way to study ergodic and measure theoretic aspects of interval maps is by means of the markov extension this tool which ties interval maps to the theory of markov chains was introduced by hofbauer and keller more generally known are induced maps i e maps that restricted to an element of an interval partition coincide with an iterate of the original map

the schroder functional equation and its relation to the invariant measures of chaotic maps the aim of this paper is to show that the invariant measure for a class of one dimensional chaotic maps t x is an extended solution of the schroder functional equation q t x q x induced by them hence we give a unified treatment of a collection of exactly solved examples worked out in the current literature in particular we show that these examples belong to a class of functions introduced by mira see the text moreover as a new example we compute the invariant densities for a class of rational maps having the weierstrass function as an invariant one also we study the relation between that equation and the well known frobenius perron and koopman s operators

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

determinantal solutions of solvable chaotic systems it is shown that two solvable chaotic systems the arithmetic harmonic mean ahm algorithm and the ulam von neumann uvn map have determinantal solutions an additional formula for certain determinants and riccati difference equations play a key role in both cases two infinite hierarchies of solvable chaotic systems are presented which have determinantal solutions

generalized log likelihood functions and bregman divergences based on a two parameter generalization of gauss law of error a generalized log likelihood is related to a bregman divergence this relation is a two parameter generalization of the well known relation between log likelihood and kullback leibler divergence

skewness and kurtosis analysis for non gaussian distributions abstract in this paper we address a number of pitfalls regarding the use of kurtosis as a measure of deviations from the gaussian we treat kurtosis in both its standard definition and that which arises in q statistics namely q kurtosis we have recently shown that the relation proposed by cristelli et al 2012 between skewness and kurtosis can only be verified for relatively small data sets independently of the type of statistics chosen however it fails for sufficiently large data sets if the fourth moment of the distribution is finite for infinite fourth moments kurtosis is not defined as the size of the data set tends to infinity for distributions with finite fourth moments the size n of the data set for which the standard kurtosis saturates to a fixed value depends on the deviation of the original distribution from the gaussian nevertheless using kurtosis as a criterion for deciding which distribution deviates further from the gaussian can be misleading for small data sets even for finite fourth moment distributions going over to q statistics we find that although the value of q kurtosis is finite in the range of 0 q 3 this quantity is not useful for comparing different non gaussian distributed data sets unless the appropriate q value which truly characterizes the data set of interest is chosen finally we propose a method to determine the correct q value and thereby to compute the q kurtosis of q gaussian distributed data sets

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

tsallis mittag leffler distribution and its applications in gas prices abstract the theory of mittag leffler process is a fundamental concept in probability and stochastic process this paper introduces the class of tsallis mittag leffler distributions in special case our results include a new tsallis q weibull distribution one parametric mittag leffler distribution and some other well known distributions finally to illustrate the applicability of our distribution a real data set on gas prices in time series data based on 100 days moving average is analyzed

hyperbolicity sinks and measure in one dimensional dynamics letf be ac2 map of the circle or the interval and let f denote the complement of the basins of attraction of the attracting periodic orbits we prove that f is a hyperbolic expanding set if and obviously only if every periodic point is hyperbolic and f doesn t contain the critical point this is the real one dimensional version of fatou s hyperbolicity criteria for holomorphic endomorphisms of the riemann sphere we also explore other applications of the techniques used for the result above proving for instance that for everyc2 immersionf of the circle i e a map of the circle onto itself without critical points either its julia set has measure zero or it is the whole circle and thenf is ergodic i e positively invariant borel sets have zero or full measure

polar generation of random variates with the t distribution the polar method of box and muller uses two independent uniform variates in order to generate two independent normal variates it can be adapted so that two variates from student s t distribution with parameter v are generated though the two variates are now not independent an algorithm based on the polar method is exact inexpensive and valid for all v 0 box and muller s 1 polar method for generating random normal variates relies on two convenient properties of the normal distribution which we may formulate as follows i let x n o 1 then x can be regarded as the real part of a complex random variable z which has a radial distribution the contours of the density function of z form circles centered at the origin ii write z x iy reie9 then the distribution function fr 1gr of r is a simple algebraic expression so simple that it is invertible that is given g gr r we can write down a closed expression for r in term of g the aim of this article is to show that properties i and ii are shared by the student t distribution with parameter v the tv distribution defined by the density 1 ft x b v 2 1 2 i v 1 2 1 x2 v v l 2 thus we are asserting that the tv distribution like the normal has a tractable radial parent if t has the density 1 we shall write t tv many methods have been proposed for the generation of tv variates the most important ones are described in devroye 2 pp 445 450 whose masterly survey we shall not attempt to emulate the faster algorithms may require either a comparatively great programming effort or the expensive recalculation of certain quantities required by the algorithm whenever v is changed as devroye notes problems arise when v is small and the departure from normality is greatest particularly in the region 0 0 we now show how the polar method may be applied to the tv distribution to yield such an algorithm received by the editor december 23 1992 and in revised form may 12 1993 1991 mathematics subject classification primary 65c10 secondary 62e15

jacobian elliptic chebyshev rational maps we introduce the jacobian elliptic chebyshev rational maps with the the equidistributivity property or briefly edp and the semi group property like the chebyshev polynomial maps simple methods are also discussed for generating sequences of i i d p ary random variables based on the jacobian elliptic chebyshev rational maps furthermore we give conjectures on correlational properties of a real valued trajectory generated by the jacobian elliptic chebyshev rational maps

hierarchy of chaotic maps with an invariant measure and their coupling abstract hierarchy of one parameter families of chaotic maps with an invariant measure have been introduced where their appropriate coupling has led to the generation of some coupled chaotic maps with an invariant measure it is shown that these chaotic maps also the coupled maps do not undergo any period doubling or period n tupling cascade bifurcation to chaos but they have either single fixed point attractor at certain values of the parameters or they are ergodic in the complementary region using the invariant measure or sinai ruelle bowen measure the kolmogrov sinai entropy of the chaotic maps coupled maps have been calculated analytically where the numerical simulations support the results

student s t and r distributions unified derivation from an entropic variational principle abstract the optimization of the recently generalized entropy of the form s q 1 dx p x q q 1 with the constraints dx p x 1 and x 2 q dx x 2 p x q 1 yields the student s t distribution for q 1 and the r distribution for q

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

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

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

properties of the maximum q likelihood estimator for independent random variables abstract in the last two decades the nonextensive statistics proposed by tsallis have been extensively discussed in terms of the tsallis entropy which is a generalization of the boltzmann gibbs shannon entropy within the nonextensive framework many kinds of generalizations have been made suyari h suyari ieee trans on inf theory 51 2005 753 has proposed the generalized likelihood called the q likelihood in which a traditional product operator is replaced by the q product we study properties of the maximum q likelihood estimator mqle which is a generalization of the conventional maximum likelihood estimator with the use of the q likelihood we discuss mqle from the viewpoint of the minimization of the divergences in the nonextensive statistics it has been shown that optimum parameters determined by the mqle are nearly in agreement with those yielding the minimum distance of the divergence proposed by rajagopal a k rajagopal in s abe y okamoto eds nonextensive statistical mechanics and its applications springer 2000 p 99 we here show that the consistency of mqle is achieved along with the non negativity of rajagopal s divergence the asymptotic gaussianity and robustness of the mqle for independent random variables are discussed both by analytical methods and simulations

from exactly solvable chaotic maps to stochastic dynamics abstract for a class of nonlinear chaotic maps the exact solution can be written as xn p kn where p t is a periodic function is a real parameter and k is an integer number a generalization of these functions xn p zn where z is a real parameter can be proved to produce truly random sequences using different functions p t we can obtain different distributions for the random sequences similar results can be obtained with functions of type xn h f n where f n is a chaotic function and h t is a noninvertible function we show that a dynamical system consisting of a chaotic map coupled to a map with a noninvertible nonlinearity can generate random dynamics we present physical systems with this kind of behavior we report the results of real experiments with nonlinear circuits and josephson junctions we show that these dynamical systems can produce a type of complexity that cannot be observed in common chaotic systems we discuss applications of these phenomena in dynamics based computation

ultradiscretization of solvable one dimensional chaotic maps we consider the ultradiscretization of a solvable one dimensional chaotic map which arises from the duplication formula of the elliptic functions it is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously a geometric interpretation of the dynamics of the tent map is given in terms of the tropical jacobian of a certain tropical curve generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed

method of constructing exactly solvable chaos we present a new systematic method of constructing rational mappings as ergordic transformations with nonuniform invariant measures on the unit interval 0 1 as a result we obtain a two parameter family of rational mappings that have a special property in that their invariant measures can be explicitly written in terms of algebraic functions of parameters and a dynamical variable furthermore it is shown here that this family is the most generalized class of rational mappings possessing the property of exactly solvable chaos on the unit interval including the ulam neumann map y 4x 1 x based on the present method we can produce a series of rational mappings resembling the asymmetric shape of the experimentally obtained first return maps of the beloussof zhabotinski chemical reaction and we can match some rational functions with other experimentally obtained first return maps in a systematic manner

crossover between levy and gaussian regimes in first passage processes we propose an approach to the problem of the first passage time our method is applicable not only to the wiener process but also to the non gaussian l evy flights or to more complicated stochastic processes whose distributions are stable to show the usefulness of the method we particularly focus on the first passage time problems in the truncated l evy flights the so called kobol processes from koponen boyarchenko and levendorskii in which the arbitrarily large tail of the l evy distribution is cut off we find that the asymptotic scaling law of the first passage time t distribution changes from t ensuremath ensuremath alpha 1 ensuremath alpha law non gaussian l evy regime to t ensuremath 3 2 law gaussian regime at the crossover point this result means that an ultraslow convergence from the non gaussian l evy regime to the gaussian regime is observed not only in the distribution of the real time step for the truncated l evy flight but also in the first passage time distribution of the flight the nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cutoff length of the l evy distribution are discussed

bistable stochastic processes in the q exponential family abstract stochastic bistable systems whose stationary distributions belong to the q exponential family are investigated using two approaches i the langevin model subjected to additive and quadratic multiplicative noise and ii the superstatistical model previously the bistable langevin model has been analyzed under linear multiplicative noise whereas this paper reports on quadratic multiplicative noise which is more physically meaningful the stationary distribution of the langevin model under quadratic multiplicative noise which agrees with that derived by the maximum tsallis entropy method is found to be qualitatively different from its counterpart under linear multiplicative noise we also show that the stationary distribution of the superstatistical model is the same as that of the langevin model whereas their transient properties described in terms of mean first passage times mfpts are qualitatively different

q gaussian swarm quantum particle intelligence on predicting global minimum of potential energy function we present a newly developed q gaussian swarm quantum like particle optimization q gsqpo algorithm to determine the global minimum of the potential energy function swarm quantum like particle optimization sqpo algorithms have already been derived using different attractive potential fields to represent swarm particles moving in a quantum environment in this paper we propose a new sqpo algorithm using q gaussian probability density function as the attractive potential field q gsqpo rather than the gaussian one gsqpo which corresponds to harmonic potential the performance of the q gsqpo algorithm is compared with that of gsqpo algorithm the results of this study show that the new algorithm outperforms the gsqpo on most of the time in convergence to the global optimum the most obvious finding of this study is that the new algorithm increases the efficiency of sampling the phase space and avoids the premature convergence to local minima by increasing the diversity of the swarm moreover it was found that generally the computational efforts were comparable for both algorithms we tested the algorithms by determining the lowest energy positions of the particles moving in the surface landscape of 2 5 10 and 50 dimensions

least informative distributions in maximum q log likelihood estimation abstract we use the maximum q log likelihood estimation for least informative distributions lids in order to estimate the parameters in probability density functions pdfs efficiently and robustly when data include outlier s lids are derived by using convex combinations of two pdfs a convex combination of two pdfs is composed of an underlying distribution and a contamination the optimal criterion is obtained by minimizing the change of maximum q log likelihood function when the data contain small amount of contamination in this paper we make a comparison between ordinary maximum likelihood estimation maximum q log likelihood estimation mqle and lids based on mqle for parameter estimation from data with outliers we derive a new fisher information matrix based on the score function for lid from m function and use it for choice of optimal estimator in the class of mqle the model selection is done by the robust information criteria we test the methods on the real data with outliers and estimate shape and scale parameters of probability distributions as a result we show that the lids based on mqle provide the most robust and efficient estimation of the model parameters

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 most concise set of axioms and the uniqueness theorem for tsallis entropy we present the most concise set of axioms for tsallis entropy and rigorously prove the uniqueness theorem this set of axioms consists of only two distinct additivities pseudoadditivity and shannon additivity we then compare our axioms with the axioms presented by santos the peculiarity of pseudoadditivity as an axiom for tsallis entropy is also discussed

computational complexity of sequential sequence estimation for intersymbol interference channels the computational complexity of a sequential algorithm sa developed for intersymbol interference isi channels is analyzed to determine the computational complexity the finite state machine that models the channel and white matched filter system of which the sa is a part is interpreted as a special convolutional encoder followed by a binary symbol to q ary symbol mapping it follows that the computational distribution is pareto and that there exists a computational cutoff rate r sub comp for the uncoded data considered the rate is fixed and the r sub comp criterion translates into a signal to noise ratio snr criterion an upper bound on snr sub comp is found analytically by assuming a uniform input distribution iteration equations developed by s arimoto 1976 are adapted to find the true snr sub comp numerically

the q gamma and q q polygamma functions of tsallis statistics abstract an axiomatic definition is given for the q gamma function q x q r q 0 x r of tsallis non extensive statistical physics the continuous analogue of the q factorial of suyari h suyari physica a 368 1 2006 63 and the q analogue of the gamma function x of euler and gauss a working definition in closed form based on the hurwitz and riemann zeta functions including their analytic continuations is shown to satisfy this definition several relations involving the q gamma and other functions are obtained the q q polygamma functions q q m x m n defined by successive derivatives of ln q q x where ln q a 1 q 1 a 1 q 1 a 0 is the q logarithmic function are also reported the new functions are used to calculate the inferred probabilities and multipliers for tsallis systems with finite numbers of particles n

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