universal mechanism for the intermittent route to strange nonchaotic attractors in quasiperiodically forced systems to examine the universality for the intermittent route to strange nonchaotic attractors snas we investigate the quasiperiodically forced henon map ring map and toda oscillator which are high dimensional invertible systems in these invertible systems dynamical transition to an intermittent sna occurs via a phase dependent saddle node bifurcation when a smooth torus collides with a ring shaped unstable set we note that this bifurcation mechanism for the appearance of intermittent snas is the same as that found in a simple system of the quasiperiodically forced noninvertible logistic map hence the intermittent route to snas seems to be universal in the sense that it occurs through the same mechanism in typical quasiperiodically forced systems of different nature

on the rediscovery of halley s iterative method for computing the zero of an analytic function abstract we show that halley s basic sequence resulting from accelerating the order of convergence of newton s method is the most efficient way of doing so in terms of usage of certain derivatives this fact could explain why this process of accelerating the convergence of newton s method is so frequently rediscovered then we present an algorithmic way of recognizing halley s family and we apply this algorithm to examples of rediscoveries

on rediscovered methods for solving equations abstract in the recent papers of gerlach siam rev 36 1994 272 276 and ford and pennline siam rev 38 1996 658 659 a class of iterative methods for solving a single equation f x 0 with arbitrary rate of convergence has been presented in this paper we show that this class is equivalent to five other classes of iterative methods derived earlier in various ways and expressed in different forms the proofs of equivalence of all considered iteration formulas are given

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

a basic family of iteration functions for polynomial root finding and its characterizations abstract let p x be a polynomial of degree n 2 with coefficients in a subfield k of the complex numbers for each natural number m 2 let lm x be the m m lower triangular matrix whose diagonal entries are p x and for each j 1 m 1 its jth subdiagonal entries are p j x j for i 1 2 let lmi x be the matrix obtained from lm x by deleting its first i rows and its last i columns l1 1 x 1 then the function bm x x p x det l m 1 1 x det l m 1 x defines a fixed point iteration function having mth order convergence rate for simple roots of p x for m 2 and 3 bm x coincides with newton s and halley s respectively the function bm x is a member of s m m n 2 where for any m m s m m is the set of all rational iteration functions g x k x such that for all roots of p x then g x i mm i x x i with i x k x and well defined at any simple root given g s m m and a simple root of p x gi 0 i 1 m 1 and the asymptotic constant of convergence of the corresponding fixed point iteration is m 1 g m m for bm x we obtain m 1 m det l m 1 2 det l m 1 if all roots of p x are simple bm x is the unique member of s m m n 2 by making use of the identity 0 i 0 n p i x i x i we arrive at two recursive formulas for constructing iteration functions within the s m m family in particular the family of bm x can be generated using one of these formulas moreover the other formula gives a simple scheme for constructing a family of iteration functions credited to euler as well as schroder whose mth order member belongs to s m mn m 2 the iteration functions within s m m can be extended to any arbitrary smooth function f with the uniform replacement of p j with f j in g as well as in m

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

maxent second variation and generalized statistics abstract there are two kinds of tsallis probability distributions heavy tail ones and compact support distributions we show here by appeal to functional analysis tools that for lower bound hamiltonians the second variation s analysis of the entropic functional guarantees that the heavy tail q distribution constitutes a maximum of tsallis entropy on the other hand in the compact support instance a case by case analysis is necessary in order to tackle the issue

convergence in reaction diffusion systems an information theory approach abstract we have applied an information theory approach in order to study the problem of convergence to point like or extended attractors in reaction diffusion systems a distance between two states based on the kullback leibler relative information was defined different forms of the probability distribution some of them based on the knowledge of the nonequilibrium potential when accesible give the possibility to look for a faster and or more accurate convergence this approach offers the chance to estimate the attraction basins of the different attractors as well as detecting limit circles together with an easy evaluation of their periods

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

bifurcations and transitions in the quasiperiodically driven logistic map abstract we discuss several bifurcation phenomena that occur in the quasiperiodically driven logistic map this system can have strange nonchaotic attractors snas in addition to chaotic and regular attractors on snas the dynamics is aperiodic but the largest lyapunov exponent is nonpositive there are a number of different transitions that occur here from periodic attractors to snas from snas to chaotic attractors etc we describe some of these transitions by examining the behavior of the largest lyapunov exponent distributions of finite time lyapunov exponents and the invariant densities in the phase space

distributivity and deformation of the reals from tsallis entropy we propose a one parameter family rq of deformations of the reals which is motivated by the generalized additivity of the tsallis entropy we introduce a generalized multiplication which is distributive with respect to the generalized addition of the tsallis entropy these operations establish a one parameter family of field isomorphisms q between r and rq through which an absolute value on rq is introduced this turns out to be a quasisymmetric map whose metric and measure theoretical implications are pointed out

inconsistency of the hydrogen atom with β β transformation in tsallis statistics abstract in this article the transformation method is applied to a hydrogen atom in the canonical ensemble to obtain the re normalized real temperatures associated with the normalized q expectation values formalism in the tsallis method of approach the results indicate that the re normalized real temperature t which is a function of the intermediate temperature t is not a monotonically increasing function for the allowable range of q of course this behavior of the t is not desirable

on the closed form solutions for non extensive value at risk abstract we examine possible closed form solutions for the cumulative distribution function for systems where the probability density function can be adequately described by the generalized non extensive statistics framework application to financial time series as a possible value at risk technique indicates reasonable agreement with the data under consideration including all possible extremes and asymmetries of the returns numerical results to illustrate the efficiency of the method are presented

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

on a q generalization of circular and hyperbolic functions a generalization of the circular and hyperbolic functions is proposed based on the tsallis statistics and on a consistent generalization of the euler formula some properties of the presently proposed q trigonometry are then investigated the generalized functions are exact solutions of a nonlinear oscillator original circular and hyperbolic relations are recovered as the limiting case

a tsallis statistics based neural network model for novel word learning abstract we invoke the tsallis entropy formalism a nonextensive entropy measure to include some degree of non locality in a neural network that is used for simulation of novel word learning in adults a generalization of the gradient descent dynamics realized via nonextensive cost functions is used as a learning rule in a simple perceptron the model is first investigated for general properties and then tested against the empirical data gathered from simple memorization experiments involving two populations of linguistically different subjects numerical solutions of the model equations corresponded to the measured performance states of human learners in particular we found that the memorization tasks were executed with rather small but population specific amounts of nonextensivity quantified by the entropic index q our findings raise the possibility of using entropic nonextensivity as a means of characterizing the degree of complexity of learning in both natural and artificial systems

classical and quantum conformational analysis using generalized genetic algorithm abstract recently the generalized genetic algorithm has been proposed based on genetic algorithms and tsallis statistics here we study molecular preferences of the simple molecules and polypeptides by coupling the generalized genetic algorithm and both a classical force field and a semi empirical quantum mechanics procedure the main characteristics of this stochastic algorithm is the capability to find the greatest number of minima for a given energy in an n dimensional hyper surface supplying information about the visiting rates for each one of the conformational structures the main advantage of this method over other global optimization procedures is that this approach provides additional information about visiting frequency of all local and global minima these characteristics may provide the populations of each conformation without using the quantum or classical monte carlo procedure

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

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

equivalence between maximum entropy principle and enforcing du tds we study here a hitherto unexplored microscopic connection between the well known thermodynamical relation du tds and jaynes maximum entropy principle maxent for determining probability distributions for the canonical ensemble

quantum corrections for general partition functions abstract the classical limit for quite general partition functions is obtained using coherent states in this framework a general procedure is presented to obtain all the quantum corrections to the classical limit in particular the first and second order corrections are worked out explicitly and the classical limit for the tsallis thermostatistics is determined the results of this work generalize the ones obtained by wigner phys rev 40 1932 749 for usual statistical mechanics

growing 1d and quasi 2d unstable manifolds of maps we present a new 1d algorithm for computing the global one dimensional unstable manifold of a saddle point of a map our method can be generalized to compute two dimensional unstable manifolds of maps with three dimensional state spaces this is shown here with a quasi 2d q2d algorithm for the special case of a quasiperiodically forced map which allows for a substantial simplification of the general case described in krauskopf and osinga 1998 the key idea is to grow the manifold in steps which consist of finding a new point on the manifold at a prescribed distance from the last point the speed of growth is determined only by the curvature of the manifold and not by the dynamics the performance of the 1d algorithm is demonstrated with a constructed test example and it is then used to compute one dimensional manifolds of a map modeling mixing in a stirring tank with the q2d algorithm we compute two dimensional unstable manifolds in the quasiperiodically forced henon map

generalized kolmogorov entropy in the dynamics of multifractal generation abstract we point out that applying a maximization principle on a tsallis like generalized form of the kolmogorov entropy for iterated function systems naturally provides a canonical statistical frame for the description of the multifractal measures generated by such dynamical processes multifractal spectra can then be characterized by usual statistical parameters in particular the temperature we show that in the limit of zero temperature the multifractal measure collapses to a homogeneous distribution over a fractal support for finite temperatures multifractal spectra are studied numerically in an illustrative example

the pade iterations for the matrix sign function and their reciprocals are optimal abstract it is proved that among the rational iterations locally converging with order s 1 to the sign function the pade iterations and their reciprocals are the unique with the lowest sum of the degrees of numerator and denominator

anisotropic fractional diffusion equation abstract we analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence the presence of external forces and time fractional derivatives we obtain new exact classes of solutions for a linear anisotropic fractional diffusion equation and investigate the time scaling behavior and an asymptotic solution for a nonlinear anisotropic fractional diffusion equation we connect the asymptotic solution obtained with the distribution that emerges from the nonextensive statistics to the nonlinear case we also verify different diffusive behavior for instance subdiffusion and superdiffusion in each direction

asymmetric unimodal maps some results from q generalized bit cumulants in this study using q generalized bit cumulants q is the nonextensivity parameter of the recently introduced tsallis statistics we investigate the asymmetric unimodal maps x t 1 1 ensuremath a x t z i i 1 2 correspond to x t g0 and x t l0 respectively z i g1 0lal 2 t 0 1 2 dots the study of the q generalized second cumulant c 2 q of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs z 1 z 2 the slope of the c 2 q versus c 2 1 plot where c 2 1 is the standard second cumulant provides the necessary tool to accomplish this task the slope behaves exactly the same as the proper q values say q that were obtained for logisticlike maps z 1 z 2 z by costa et al phys rev e 56 245 1997 it appears that as z 2 ensuremath z 1 ensuremath rightarrow ifmmode pm else textpm fi ensuremath infty this slope approaches unity this behavior is very similar to the behavior of q as a function of the inflection parameter for some z dependent maps namely as stackrel ensuremath rightarrow z ensuremath infty q approaches 1

on general convergence in extracting radicals via a fundamental family of iteration functions newton s method is well known to be generally convergent for solving x n c 0 in this paper we first extend this result to the next two members of an infinite family of high order methods referred to here as the basic family which starts with newton s method while computing roots of unity numerically is a trivial task studying the general convergence of the basic family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family with the aid of polynomiography techniques for the visualization of polynomial root finding we further conjecture the general convergence of all members of the basic family when extracting radicals using the computer algebra system maple we obtain some partial results toward the proof of our conjecture

generalization of taylor s theorem and newton s method via a new family of determinantal interpolation formulas and its applications abstract the general form of taylor s theorem for a function f k k where k is the real line or the complex plane gives the formula f pn rn where pn is the newton interpolating polynomial computed with respect to a confluent vector of nodes and rn is the remainder whenever f 0 for each m 2 n 1 we describe a determinantal interpolation formula f pm n rm n where pm n is a rational function in x and f itself these formulas play a dual role in the approximation of f or its inverse for m 2 the formula is taylor s and for m 3 is a new expansion formula and a pade approximant by applying the formulas to pn for each m 2 p m m 1 p m m n 2 is a set of n rational approximations that includes pn and may provide a better approximation to f than pn hence each taylor polynomial unfolds into an infinite spectrum of rational approximations the formulas also give an infinite spectrum of rational inverse approximations as well as a fundamental k point iteration function bm k for each k m defined as the ratio of two determinants that depend on the first m k derivatives application of our formulas have motivated several new results obtained in sequel papers i theoretical analysis of the order of b m k k 1 m proving that it ranges from m to the limiting ratio of generalized fibonacci numbers of order m ii computational results with the first few members of bm k indicating that they outperform traditional root finding methods e g newton s iii a novel polynomial rootfinding method requiring only a single input and the evaluation of the sequence of iteration functions bm 1 at that input this amounts to the evaluation of special toeplitz determinants that are also computable via a recursive formula iv a new strategy for general root finding v new formulas for approximation of e and other special numbers

formulation of the hellmann feynman theorem for the second choice version of tsallis thermostatistics abstract an approach to formulating the hellmann feynman theorem within the second choice formalism of non extensive statistical mechanics is considered for the state of thermal equilibrium we derive a relation of hellmann feynman type between the derivative of the non extensive free energy with respect to the external parameter and the quantum statistical q average of the derivative of the hamilton operator we also give a proper extension for an arbitrary observable commuting with the hamiltonian some reasons for the usefulness of new formulas are discussed

q exponential weibull and q weibull distributions an empirical analysis abstract in a comparative study the q exponential and weibull distributions are employed to investigate frequency distributions of basketball baskets cyclone victims brand name drugs by retail sales and highway length in order to analyze the intermediate cases a distribution the q weibull one which interpolates the q exponential and weibull ones is introduced it is verified that the basketball baskets distribution is well described by a q exponential whereas the cyclone victims and brand name drugs by retail sales ones are better adjusted by a weibull distribution on the other hand for highway length the q exponential and weibull distributions do not give satisfactory adjustment being necessary to employ the q weibull distribution furthermore the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law q exponential with q 1 controversy

generalized parallel sampling abstract we develop a generalized version of the parallel tempering algorithm based upon the non extensive thermostatistics of tsallis and coworkers the effectiveness of the method is demonstrated on a simple one dimensional problem and on a lennard jones cluster

scaling transformation of random walk and generalized statistics abstract we use a decimation procedure in order to obtain the dynamical renormalization group transformation rgt properties of random walk distribution in a 1 1 lattice we obtain an equation similar to the chapman kolmogorov equation first we show the existence of invariants of the rgt and that the tsallis distribution r q x 1 b q 1 x 2 1 1 q q 1 is a quasi invariant of the rgt we obtain the map q f q from the rgt and show that this map has two fixed points q 1 attractor and q 2 repellor which are the gaussian and the lorentzian respectively finally we use those concepts to show that the nonadditivity of the tsallis entropy needs to be reviewed

boundary crisis in quasiperiodically forced systems abstract we study boundary crisis in quasiperiodically forced dissipative systems using the henon map as a characteristic example the crisis is due to a homoclinic tangency of the stable and unstable manifolds of an accessible invariant circle of saddle type on the basin boundary of the attractor numerical evidence shows that the type of boundary crisis can change as the saddle circle loses its accessibility due to the quasiperiodic forcing this implies the existence of codimension two double crisis vertices where a curve of boundary crisis and a curve of interior crisis meet we argue that bifurcation points of higher codimension must exist in the full parameter space

stability analysis of the entropies for superstatistics abstract it seems reasonable to consider concavity with regard to all probability distributions and stability under arbitrarily small deformations of any given probability distribution as necessary for an entropic form to be a physical one in the thermostatistical sense most known entropic forms e g renyi entropy s q r ln i p i q 1 q violate one and or the other of these conditions in contrast the boltzmann gibbs entropy s bg i p i ln p i and the nonextensive one s q 1 i p i q q 1 q 1 satisfy both sbg and sq belong in fact to a larger class of entropies s satisfying both namely those which through appropriate optimization yield the beck cohen superstatistics we briefly review here the proof of stability of s and illustrate for an important particular case namely the log normal superstatistical entropy the satisfaction of both concavity and stability appears to be very helpful to identify physically admissible entropic forms

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

entropy and bifurcations in a chaotic laser we compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection we consider the poincare return map to a fixed plane and are able to compute the stable and unstable manifolds of periodic points globally even though it is impossible to find a plane on which the poincare map is globally smoothly defined in this way we obtain the information that forms the input of the entropy calculations and characterize the boundary crisis in which the chaotic attractor is destroyed this boundary crisis involves a periodic point with negative eigenvalues and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation

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

the gradually truncated levy flight for systems with power law distributions abstract power law distributions have been observed in various economical and physical systems levy flights have infinite variance which discourage a physical approach we introduce a class of stochastic processes the gradually truncated levy flight in which large steps of a levy flight are gradually eliminated it has finite variance and the system can be analyzed in a closed form we applied the present method to explain the distribution of a particular economical index the present method can be applied to describe time series in a variety of fields i e turbulent flow anomalous diffusion polymers etc

a computational comparison of the first nine members of a determinantal family of root finding methods abstract for each natural number m greater than one and each natural number k less than or equal to m there exists a root finding iteration function bm k defined as the ratio of two determinants that depend on the first m k derivatives of the given function this infinite family is derived in kalantari j comput appl math 126 2000 287 318 and its order of convergence is analyzed in kalantari bit 39 1999 96 109 in this paper we give a computational study of the first nine root finding methods these include newton secant and halley methods our computational results with polynomials of degree up to 30 reveal that for small degree polynomials bm k 1 is more efficient than bm k but as the degree increases bm k becomes more efficient than bm k 1 the most efficient of the nine methods is b4 4 having theoretical order of convergence equal to 1 927 newton s method which is often viewed as the method of choice is in fact the least efficient method

computation of energy exchanges by combining information theory and a key thermodynamic relation physical applications abstract by considering a simple thermodynamic system in thermal equilibrium at a temperature t and in the presence of an external parameter a we focus our attention on the particular thermodynamic macroscopic relation d u t d s w using standard axioms from information theory and the fact that the microscopic energy levels depend upon the external parameter a we show that all usual results of statistical mechanics for reversible processes follow straightforwardly without invoking the maximum entropy principle for the simple system considered herein two distinct forms of heat contributions appear naturally in the clausius definition of entropy t d s q t q a c a t d t c t a d a we give a special attention to the amount of heat q a c t a d a associated with an infinitesimal variation d a at fixed temperature for which a generalized heat capacity c t a t s a t may be defined the usefulness of these results is illustrated by considering some simple thermodynamic cycles

generalization of the single site callen identity within tsallis statistics abstract within the framework of a generalized thermostatistics introduced by tsallis we establish the generalization of the single site callen identity it is used to calculate the critical temperature of the ising ferromagnet within a consistently generalized mean field approximation

double crises in two parameter dynamical systems a crisis is a sudden discontinuous change in a chaotic attractor as a system parameter is varied we investigate phenomena observed when two parameters of a dissipative system are varied simultaneously following a crisis along a curve in the parameter plane two such curves intersect at a point we call a double crisis vertex the phenomena we study include the double crisis vertex at which an interior and a boundary crisis coincide and related forms of double crisis we show how an experimenter can infer a crisis from observations of other related crises at a vertex

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

metamorphoses sudden jumps in basin boundaries in some invertible maps of the plane that depend on a parameter boundaries of basins of attraction are extremely sensitive to small changes in the parameter a basin boundary can jump suddenly and as it does change from being smooth to fractal such changes are calledbasin boundary metamorphoses we prove under certain non degeneracy assumptions that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency

generalized ensemble theory with non extensive statistics abstract the non extensive canonical ensemble theory is reconsidered with the method of lagrange multipliers by maximizing tsallis entropy with the constraint that the normalized term of tsallis q average of physical quantities the sum p j q is independent of the probability p i for tsallis parameter q the self referential problem in the deduced probability and thermal quantities in non extensive statistics is thus avoided and thermodynamical relationships are obtained in a consistent and natural way we also extend the study to the non extensive grand canonical ensemble theory and obtain the q deformed bose einstein distribution as well as the q deformed fermi dirac distribution the theory is further applied to the generalized planck law to demonstrate the distinct behaviors of the various generalized q distribution functions discussed in literature

symmetries of the julia sets of konig s methods for polynomials abstract let r be the group of symmetries of the julia set of a rational map r and k f n be the konig s method for polynomial f of order n 2 for any given integer n 2 we prove that if f is in normal form then f is a subgroup of k f n we also obtain a necessary and sufficient condition for the julia set of k f n to be a line

newton s method and generation of a determinantal family of iteration functions abstract it is well known that halley s method can be obtained by applying newton s method to the function f f gerlach siam rev 36 1994 272 276 gives a generalization of this approach and for each m 2 recursively defines an iteration function gm x having order m kalantari et al j comput appl math 80 1997 209 226 and kalantari technical report dcs tr 328 department of computer science rutgers university new brunswick nj 1997 derive and characterize a determinantal family of iteration functions called the basic family bm x m 2 in this paper we prove gm x bm x on the one hand this implies that gm x enjoys the previously derived properties of bm x i e the closed formula its efficient computation an expansion formula which gives its precise asymptotic constant as well as its multipoint versions on the other this gives a new insight on the basic family and newton s method

an infinite family of bounds on zeros of analytic functions and relationship to smale s bound smale s analysis of newton s iteration function induce a lower bound on the gap between two distinct zeros of a given complex valued analytic function f z in this paper we make use of a fundamental family of iteration functions b m z m 2 to derive an infinite family of lower bounds on the above gap however even for m 2 where b 2 z coincides with newton s our lower bound is more than twice as good as smale s bound or its improved version given by blum cucker shub and smale when f z is a complex polynomial of degree n for small m the corresponding bound is computable in o n log n arithmetic operations for quadratic polynomials as m increases the lower bounds converge to the actual gap we show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of f z in particular using the latter result we show that given a complex polynomial f z a n z n a 0 a n a o 0 for each m 2 we can compute upper and lower bounds u m and l m such that the roots of f z lie in the annulus z l m z u m in particular l 2 1 2 max a k a 0 1 k k 1 n u 2 2 max a n k a n 1 k k 1 n and l3 5 1 2 max a 1 a k 1 a 0 a k a 2 0 1 k k 2 n 1 u 3 5 1 2 max a n 1 a n k 1 a n o n k a 2 n 1 k k 2 n 1 where a 1 a n 1 0 an application of the latter bounds is within weyl s classical quad tree algorithm for computing all roots of a given complex polynomial

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

birth of strange nonchaotic attractors due to interior crisis abstract we study the interior crisis in the period 3 window of the quasiperiodically forced logistic map two routes from quasiperiodicity to chaos involving strange nonchaotic attractors sna are discovered along one route we observe a sudden widening of the sna this is similar to the interior crisis in chaotic systems along the other route we find a direct transition from an invariant curve to a strange nonchaotic attractor exactly at the interior crisis point this is a new mechanism of the appearance of strange nonchaotic attractors beyond the interior crisis the temporal behavior can be described as a crisis induced intermittency whose scaling behavior is discussed

the effects of the nonextensivity on the dimerization process and nematic ordering abstract the influence of the dimerization process on the nematic ordering is investigated by using a nonextensive thermostatistics namely tsallis thermostatistics tt a theoretical model taking into account the dimerization influence on the nematic scalar order parameter has been recently presented phys lett a 256 1999 309 our study considers this model and the theoretical predictions for the nematic order parameter are improved by using tt

majorization relations and disorder in generalized statistics abstract the theory of majorization is applied to examine the disorder properties of generalized thermal distributions arising in non extensive statistics we show that they share with the boltzmann gibbs thermal state the property of becoming more mixed as the temperature increases implying the increase of any associated disorder measure we also show that power law distributions exhibit a second mixing parameter associated with the non extensivity index as application we examine the thermal response of quantum entanglement in a spin system for different statistics

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

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

the constrained entropy and cross entropy functions abstract this study introduces the constrained forms of the shannon information entropy and kullback leibler cross entropy functions applicable to a closed system these functions incorporate the constraints on the system in a generic fashion irrespective of the form or even the number of the constraints since they are fully constrained the constrained functions may be pulled apart to give their partial or local constrained forms providing the means to examine the probability of each outcome or state relative to its local maximum entropy or minimum cross entropy position independently of the other states of the system the shannon entropy and kullback leibler cross entropy functions do not possess this functionality the features of each function are examined in detail and the similarities between the constrained and tsallis entropy functions are examined the continuous constrained entropy and cross entropy functions are also introduced the integrands of which may be used to independently examine each infinitesimal state physical element of the system

relations between a typical scale and averages in the breaking of fractal distribution abstract we study distributions which have both fractal and non fractal scale regions by introducing a typical scale into a scale invariant system as one of models in which distributions follow power law in the large scale region and deviate further from the power law in the smaller scale region we employ 2 dim quantum gravity modified by the r 2 term as examples of distributions in the real world which have similar property to this model we consider those of personal income in japan over latest twenty fiscal years we find relations between the typical scale and several kinds of averages in this model and observe that these relations are also valid in recent personal income distributions in japan with sufficient accuracy we show the existence of the fiscal years so called bubble term in which the gap has arisen in power law by observing that the data are away from one of these relations we confirm therefore that the distribution of this model has close similarity to those of personal income in addition we can estimate the value of pareto index and whether a big gap exists in power law by using only these relations as a result we point out that the typical scale is an useful concept different from average value and that the distribution function derived in this model is an effective tool to investigate these kinds of distributions

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

generalized statistics variational perturbation approximation using q deformed calculus abstract a principled framework to generalize variational perturbation approximations vpas formulated within the ambit of the nonadditive statistics of tsallis statistics is introduced this is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy gfe with a variational procedure formulated using q deformed calculus a candidate q deformed generalized vpa gvpa is derived with the aid of the hellmann feynman theorem the generalized bogoliubov inequality for the approximate gfe are derived for the case of canonical probability densities that maximize the tsallis entropy numerical examples demonstrating the application of the q deformed gvpa are presented the qualitative distinctions between the q deformed gvpa model vis a vis prior gvpa models are highlighted

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

probability distribution of schwammle and tsallis two parameter entropies and the lambert w function abstract we investigate a two parameter entropy introduced by schwammle and tsallis and obtain its probability distribution in the canonical ensemble the probability distribution is given in terms of the lambert w function which has been used in many branches of physics especially in fractal structures also extensivity of s q q is discussed and a relationship is found to exist between the probabilities of a composite system and its subsystems so that the two parameter entropy s q q is extensive

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

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

enhancement of stochastic resonance the role of non gaussian noises abstract we have analyzed the phenomenon of stochastic resonance in a double well potential driven by a colored non gaussian noise using a path integral approach we have obtained a consistent markovian approximation that enables us to get through the two state theory analytical expressions for the signal to noise ratio finding an enhancement of this quantity when the system departs from gaussian behavior this finding is supported by extensive numerical simulations we discuss the relation of these results to some experiments in sensory systems

what should a statistical mechanics satisfy to reflect nature there is no compelling reason imposing that the methods of statistical mechanics should be restricted to the dynamical systems which follow the usual boltzmann gibbs prescriptions more specifically ubiquitous natural and artificial systems exhibit generic stationary states which are not ergodic nor close to it in any geometrically simple subset of the a priori allowed phase space in any even extended trivial sense a vast class of such systems appears nevertheless to be tractable within thermostatistical methods completely analogous to the usual ones the question posed in the title arises then naturally some answer to this complex question is advanced in the present review of nonextensive statistical mechanics and its recent connections

generalized fisher information matrix in nonextensive systems with spatial correlation by using the q gaussian distribution derived by the maximum entropy method for spatially correlated n unit nonextensive systems we have calculated the generalized fisher information matrix of g ensuremath theta n ensuremath theta m for ensuremath theta 1 ensuremath theta 2 ensuremath theta 3 ensuremath mu q ensuremath sigma q 2 s where ensuremath mu q ensuremath sigma q 2 and s denote the mean variance and degree of spatial correlation respectively for a given entropic index q it has been shown from the cram er rao theorem that 1 an accuracy of an unbiased estimate of ensuremath mu q is improved degraded by a negative positive correlation s 2 that of ensuremath sigma q 2 is worsen with increasing s and 3 that of s is much improved for s ensuremath simeq ensuremath 1 n ensuremath 1 or s ensuremath simeq 1 0 though it is worst at s n ensuremath 2 2 n ensuremath 1 our calculation provides a clear insight to the long standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles we discuss also a calculation of the q gaussian distribution applying the superstatistics to the langevin model subjected to spatially correlated inputs

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

fixed point and newton s methods for solving a nonlinear equation from linear to high order convergence in this paper we revisit the necessary and sufficient conditions for linear and high order convergence of fixed point and newton s methods based on these conditions we extend schroder s process of the first kind to increase the order of convergence of the fixed point method we also obtain two processes to increase the order of convergence of newton s method one of which is schroder s process of the second kind for which several forms are also presented a link between schroder s two processes is given examples and numerical experiments are included

generalized mutual information tests applied to fmri analysis abstract to analyze functional magnetic resonance imaging fmri data sets the physiologically induced signals have to be separated from noise or from artifacts as a result of involuntary patient motion or mri detection techniques in the past years various methods of analysis of fmri data were proposed however one of the unsolved problems is related to dependence of the results with the threshold applied within each analysis an alternative fmri time series analysis method based on information theory and tsallis entropy is proposed to decrease this dependency the method applies the generalized mutual information gmi concept within the fmri framework the gmi between the hemodynamic response function of a given voxel and the paradigm slope is evaluated as a measure of the amount of information that the paradigm could be encoded by a given brain region or voxel this value represented the activation of the voxels however the analysis relies heavily on the tsallis q parameter allowing for a broad spectrum of results as a test of applicability of the method real fmri data were obtained in a simple motor paradigm and subsequently analyzed with gmi for several values of a q as well as with classical methods cross correlation and student s t test the simulated data were used to compare quantitatively the presented method with the classical ones by means of the construction of receiver operating characteristic roc curve the results showed the applicability of the gmi method as well as the range of optimum q values

on two sequences of algorithms for approximating square roots abstract for approximating n where n is positive real two sequences of iterative methods derived from a continued fraction expansion of n are presented we prove the convergence of these algorithms and determine their order of convergence as we show algorithms of arbitrary order for approximating n can be constructed moreover a connection between these algorithms and those proposed by johnson 1989 is proved and some other results concerning specific algorithms are given

solutions for a fractional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces we analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our investigation by considering the linear case and the nonlinear case afterward the linear case is discussed taking fractional time and spatial derivatives into account in a unified approach we also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by d x t d t x for the nonlinear case we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation

irreversible processes the generalized affinities within tsallis statistics abstract within the generalized statistical mechanics introduced recently by tsallis the generalized form of the affinities the quantities which drive a process in the theory of irreversible thermodynamics was derived the affinities were obtained by considering changes in the generalized entropy s q aub of a system composed by two subsystems a and b the non extensive character of the tsallis entropy is taken into account at the end the equilibrium condition is discussed and the zeroth law in the framework of the generalized thermodynamics is consistently recovered as in the usual case two systems which are in equilibrium with a third one are necessarily in equilibrium among them and share the same temperature

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

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 schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way

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