robustness of multipredictor adaptive regulators musmar abstract effects of multipredictor information in adaptive control are studied particularly from the standpoint of plant structural uncertainties and unmodelled dynamics by an ode analysis it is shown that for an armax plant the equilibria of a multipredictor based lq self tuning regulator namely the musmar algorithm are the extrema of a receding horizon variant of the cost further if the control horizon is large enough the musmar possible convergence points tightly approximate the local minima of the unconditional quadratic cost constrained to the chosen regulator regressor

universality classes for the fisher metric derived from relative group entropy abstract we consider the fisher metric which results from the hessian of the relative group entropy that we call group fisher metric in particular the metrics corresponding to the boltzmann gibbs tsallis kaniadakis and abe universality classes are obtained we prove that the scalar curvature derived from the group fisher metric results in a multiple of the boltzmann gibbs one with the factor of proportionality given by the local properties of the group entropy we analyze for the tsallis universality class the 2 d correlated model that presents a softening and strengthening of the scalar curvature and we illustrate with the canonical ensemble of a pair of interacting harmonic oscillators as well as a quartic harmonic oscillator

when shannon and khinchin meet shore and johnson equivalence of information theory and statistical inference axiomatics we propose a unified framework for both shannon khinchin and shore johnson axiomatic systems we do it by rephrasing shannon khinchine axioms in terms of generalized arithmetics of kolmogorov and nagumo we prove that the two axiomatic schemes yield identical classes of entropic functionals the uffink class of entropies this allows to re establish the entropic parallelism between information theory and statistical inference that has seemed to be broken by the use of non shannonian entropies

functional differential equations for the q fourier transform of q gaussians in this paper the question is the q fourier transform of a q gaussian a q gaussian with some q up to a constant factor is studied for the whole range of q fty 3 this question is connected with applicability of the q fourier transform in the study of limit processes in nonextensive statistical mechanics using the functional differential equation approach we prove that the answer is affirmative if and only if 1 q 3 excluding two particular cases of q 1 namely and complementarily we discuss some applications of the q fourier transform to nonlinear partial differential equations such as the porous medium equation

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

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

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

equivariant singularity theory with distinguished parameters two case studies of resonant hamiltonian systems we consider hamiltonian systems near equilibrium that can be formally reduced to one degree of freedom spatiotemporal symmetries play a key role the planar reduction is studied by equivariant singularity theory with distinguished parameters the method is illustrated on the conservative spring pendulum system near resonance where it leads to integrable approximations of the iso energetic poincare map the novelty of our approach is that we obtain information on the whole dynamics regarding the quasi periodic solutions the global configuration of their invariant manifolds and bifurcations of these

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

a q generalization of laplace transforms the laplace transform is generalized by using the q exponential function eqx 1 1 q x 1 1 q that emerges from tsallis non extensive statistical mechanics and some of its properties are obtained the usual transform is recovered as a limiting case q1 the use of the q laplace transform is illustrated by establishing a relation between the classical canonical q partition function and the density of states

remarks on comments on on q non extensive statistics with non tsallisian entropy physica a 466 2017 160 abstract recently in ilic and stankovic 2017 have suggested that there may be problem for the class of hybrid entropies introduced in jizba and arimitsu 2004 in this comment we point out that the problem can be traced down to the q additive entropic chain rule and to a peculiar behavior of the de finetti kolmogorov relation for escort distributions however despite this one can still safely use the proposed hybrid entropies in most of the statistical thermodynamics considerations

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

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

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

a note on the relevance of the q exponential function in the context of intertemporal choices abstract this paper shows that the q exponential function well known in the deformed algebra inspired in the tsallis s nonextensive thermodynamics may be used to model discount functions in intertemporal choices which present the phenomenon known as increasing patience moreover we show that this insight may also be used to provide a measure of the degree of dynamic inconsistency presented in such decisions

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

algebra and calculus for tsallis thermo statistics abstract we construct generalized additions and multiplications forming fields and division algebras inspired by the tsallis thermo statistics we also construct derivations and integrations in this spirit these operations do not reduce to the naively expected ones when the deformation parameter approaches zero

resonance tongues in hill s equations a geometric approach abstract the geometry of resonance tongues is considered in mainly reversible versions of hill s equation close to the classical mathieu case hill s map assigns to each value of the multiparameter the corresponding poincare matrix by an averaging method the geometry of hill s map locally can be understood in terms of cuspoid whitney singularities this adds robustness to the result the algorithmic nature of the averaging method enables a pull back to the resonance tongues of the original system

differential escort transformations and the monotonicity of the lmc renyi complexity measure abstract escort distributions have been shown to be very useful in a great variety of fields ranging from information theory nonextensive statistical mechanics till coding theory chaos and multifractals in this work we give the notion and the properties of a novel type of escort density the differential escort densities which have various advantages with respect to the standard ones we highlight the behavior of the differential shannon renyi and tsallis entropies of these distributions then we illustrate their utility to prove the monotonicity property of the lmc renyi complexity measure and to study the behavior of general distributions in the two extreme cases of minimal and very high lmc renyi complexity finally this transformation allows us to obtain the tsallis q exponential densities as the differential escort transformation of the exponential density

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

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

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

a 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

use of the geometric mean as a statistic for the scale of the coupled gaussian distributions abstract the geometric mean is shown to be an appropriate statistic for the scale of a heavy tailed coupled gaussian distribution or equivalently the student s t distribution the coupled gaussian is a member of a family of distributions parameterized by the nonlinear statistical coupling which is the reciprocal of the degree of freedom and is proportional to fluctuations in the inverse scale of the gaussian existing estimators of the scale of the coupled gaussian have relied on estimates of the full distribution and they suffer from problems related to outliers in heavy tailed distributions in this paper the scale of a coupled gaussian is proven to be equal to the product of the generalized mean and the square root of the coupling from our numerical computations of the scales of coupled gaussians using the generalized mean of random samples it is indicated that only samples from a cauchy distribution with coupling parameter one form an unbiased estimate with diminishing variance for large samples nevertheless we also prove that the scale is a function of the geometric mean the coupling term and a harmonic number numerical experiments show that this estimator is unbiased with diminishing variance for large samples for a broad range of coupling values

a note on the definition of deformed exponential and logarithm functions the recent generalizations of the boltzmann gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters in the present work we investigate whether a deformed logarithmic exponential map is a bijection from r r set of positive real numbers all real numbers to r r as their undeformed counterparts we show that their inverse map exists only in some subsets of the aforementioned co domains furthermore we present conditions which a generalized deformed function has to satisfy so that the most important properties of the ordinary functions are preserved the fulfillment of these conditions permits us to determine the validity interval of the deformation parameters we finally apply our analysis to tsallis q deformed functions and discuss the interval of concavity of the renyi entropy

arx modeling of controlled armax plants and lq adaptive controllers conditions under which controlled armax autoregressive moving average models with auxiliary inputs plants can be described by arx autoregressive models with auxiliary inputs models are studied among all possible arx models the ones that can be correctly used in long range lq linear quadratic adaptive control are characterized within this framework lq adaptive controllers embodying standard recursive least squares identifiers are singled out and shown to have equilibrium points solving related underlying lq control problems for the true armax plant

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

a definition of the coupled product for multivariate coupled exponentials abstract the coupled product and coupled exponential of the generalized calculus of nonextensive statistical mechanics are defined for multivariate functions the nonlinear statistical coupling is indexed such that d 1 d where d is the dimension of the argument of the multivariate coupled exponential the coupled gaussian distribution is defined such that the argument of the coupled exponential depends on the coupled moments but not the coupling parameter the multivariate version of the coupled product is defined such that the output dimensions are the sum of the input dimensions this enables construction of the multivariate coupled gaussian from univariate coupled gaussians the resulting construction forms a model of coupling between distributions generalizing the product of independent gaussians

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

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

stability of the inverted pendulum a topological explanation an explanation and a proof of stability of the inverted pendulum whose suspension point undergoes vertical periodic oscillations is given the main idea of the argument is topological as it turns out existence of stable regimes can be proven with little effort using only very crude qualitative information about the system more precisely let n be the number of times the pendulum becomes vertical during one forcing period if n changes by more than 4 with the change of a parameter mu then for an open interval of intermediate values of mu the pendulum will be stable

κ deformed fourier transform abstract we present a new formulation of fourier transform in the picture of the algebra derived in the framework of the generalized statistical mechanics the fourier transform is obtained from a fourier series recently introduced by scarfone 2013 the kernel of this transform that reduces to the usual exponential phase in the 0 limit is composed by a deformed phase and a damping factor that gives a wavelet like behaviour we show that the fourier transform is isomorph to the standard fourier transform through a changing of time and frequency variables nevertheless the new formalism is useful to study according to fourier analysis those functions defined in the realm of the algebra as a relevant application we discuss the central limit theorem for the sum of n iterate statistically independent random variables

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

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

binary indices of time series complexity measures and entropy plane abstract complexity of time series is an important feature of dynamical systems such as financial systems in order to bring out the complete non linear behavior of financial time series non linear tools of complexity measurement like entropy measures are indispensable sample entropy sampen and distribution entropy disten are popular methods of assessing the complexity in various fields however both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets therefore we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane in order to further improve the accuracy of the research results we take the embedding dimension m as the variable expand the entropy point into the entropy curve for further research furthermore considering the shortcomings of sample entropy and distribution entropy in practical application we generalize them by i replacing sample entropy and distribution entropy with multi scale sample entropy and multi scale distribution entropy ii generalizing shannon entropy as tsallis entropy also scale factor and entropic index q are taken respectively as variables to get the entropy curves by using the artificial data we confirm the rationality of using entropy points and entropy curves to study the complexity of time series finally we apply this method to measure the complexity of real world financial time series the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences

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

portfolio theory information theory and tsallis statistics abstract we developed a strategic of optimal portfolio based on information theory and tsallis statistics the growth rate of a stock market is defined by using q deformed functions and we find that the wealth after n days with the optimal portfolio is given by a q exponential function in this context the asymptotic optimality is investigated on causal portfolios showing advantages of the optimal portfolio over an arbitrary choice of causal portfolios finally we apply the formulation in a small number of stocks in brazilian stock market b 3 and analyzed the results

generalized entropy plane based on permutation entropy and distribution entropy analysis for complex time series abstract entropy is an accessible way to work as a measure of the irregularity and the uncertainty between the predicting knowledge and the given time series statistical complexity measure scm combining shannon entropy and the extensive jensen shannon divergence provides important additional information regarding the peculiarities of the underlying probability distribution not already detected by the entropy in this paper we extend the traditional complexity entropy causality plane which applies the diagram of scm versus normalized shannon entropy to two generalized complexity entropy plane based on permutation entropy pe and permuted distribution entropy pedisen moreover as the important extension of the shannon entropy the tsallis entropy and renyi entropy are used to construct the plane we discuss the parameter selection for the pe plane and pedisen plane respectively outlier detection is recently a heated point focusing on discovering patterns that occur infrequently in the time series in data mining however there exists few entropy plane based methods in outlier detection we apply the proposed procedure to the real world data for outlier detection it turns out that the generalized entropy plane is robust to the type of original series and is efficient for detecting outliers

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

a probabilistic choice model based on tsallis statistics abstract decision under risk and uncertainty probabilistic choice has been attracting attention in econophysics and neuroeconomics this paper proposes a probabilistic choice model based on a mathematical equivalence of delay and uncertainty in decision making and the deformed algebra developed in the tsallis non extensive thermodynamics furthermore it is shown that this model can be utilized to quantify the degree of consistency in probabilistic choice in humans and animals future directions in the application of the model to studies in econophysics neurofinance neuroeconomics and social physics are discussed

general exact solution to the problem of the probability density for sums of random variables the exact explicit expression for the probability density p n x for a sum of n random arbitrary correlated summands is obtained the expression is valid for any number n and any distribution of the random summands most attention is paid to application of the developed approach to the case of independent and identically distributed summands the obtained results reproduce all known exact solutions valid for the so called stable distributions of the summands it is also shown that if the distribution is not stable the profile of p n x may be divided into three parts namely a core small x a tail large x and a crossover from the core to the tail moderate x the quantitative description of all three parts as well as that for the entire profile is obtained a number of particular examples are considered in detail

tsallis entropy induced metrics and cat k spaces abstract generalizing the group structure of the euclidean space we construct a riemannian metric on the deformed set r q n induced by the tsallis entropy composition property we show that the tsallis entropy is a hyperbolic analogue of the euclidean boltzmann gibbs shannon entropy and find a geometric interpretation for the nonextensive parameter q we provide a geometric explanation of the uniqueness of the tsallis entropy as reflected through its composition property which is provided by the abe and the santos axioms for two or more interacting systems described by the tsallis entropy having different values of q we argue why a suitable extension of this construction is provided by the cartan alexandrov toponogov metric spaces with a uniform negative curvature upper bound

fisher information borges operators and q calculus abstract we discuss applying the increasingly popular q calculus or deformed calculus so as to suitably generalize fisher s information measure and the cramer rao inequality a q deformation can be attained in multiple ways and we show that most of them do not constitute legitimate procedures within such a context the only completely acceptable q deformation is that ensuing from using the so called borges derivative e p borges physica a 340 2004 95

tsallis renyi and nonextensive gaussian entropy derived from the respective multinomial coefficients abstract we explore the generalization of the ordinary multinomial coefficient based on the deformed q multiplication and q division aim of this study is to construct the appropriate multinomial coefficients from which one can obtain the tsallis renyi and nonextensive gaussian entropy respectively we show that for all three above entropies there are two possible ways to define the generalized multinomial coefficient its consequence is discussed

the deformed exponential functions of two variables in the context of various statistical mechanics in the recent development in various disciplines of physics it is noted the need for including the deformed versions of the exponential functions in last two decades the tsallis and kaniadakis versions have found a lot of applications in this paper we consider the deformations which have two purposes first we introduce them like beginning of a more general mathematical approach where the tsallis and kaniadakis exponential functions are the special cases then we wish to pay attention to the mathematical community that they have a lot of interesting properties from mathematical point of view and possibilities in applications really we will show the differential and difference properties of our deformations which are important for the formation and explanation of continuous and discrete models of numerous phenomena

multiscale entropy based methods for heart rate variability complexity analysis abstract physiologic complexity is an important concept to characterize time series from biological systems which associated to multiscale analysis can contribute to comprehension of many complex phenomena although multiscale entropy has been applied to physiological time series it measures irregularity as function of scale in this study we purpose and evaluate a set of three complexity metrics as function of time scales complexity metrics are derived from nonadditive entropy supported by generation of surrogate data i e sdiff q max q max and q zero in order to access accuracy of proposed complexity metrics receiver operating characteristic roc curves were built and area under the curves was computed for three physiological situations heart rate variability hrv time series in normal sinus rhythm atrial fibrillation and congestive heart failure data set were analyzed results show that proposed metric for complexity is accurate and robust when compared to classic entropic irregularity metrics furthermore sdiff q max is the most accurate for lower scales whereas q max and q zero are the most accurate when higher time scales are considered multiscale complexity analysis described here showed potential to assess complex physiological time series and deserves further investigation in wide context

the inverted pendulum a singularity theory approach the inverted pendulum with small parametric forcing is considered as an example of a wider class of parametrically forced hamiltonian systems the qualitative dynamics of the poincare map corresponding to the central periodic solution is studied via an approximating integrable normal form at bifurcation points we construct local universal models in the appropriate symmetry context using equivariant singularity theory in this context structural stability can be proved under generic conditions

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator

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

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

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

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

a reversible bifurcation analysis of the inverted pendulum the inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting parameters are given by the size of the forcing and the frequency ratio normal form theory provides an integrable approximation of the poincare map generated by a planar vector field genericity of the model is studied by a perturbation analysis where the spatial symmetry is optional here equivariant singularity theory is used

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

rescaling entropy and divergence rates based on rescaling by some suitable sequence instead of the number of time units the usual notion of divergence rate is here extended to define and determine meaningful generalized divergence rates rescaling entropy rates appears as a special case suitable rescaling is naturally induced by the asymptotic behavior of the marginal divergences closed form formulas are obtained as soon as the marginal divergences behave like powers of some analytical functions a wide class of countable markov chains is proved to satisfy this property most divergence and entropy functionals defined in the literature are concerned e g the classical shannon kullback leibler renyi and tsallis for illustration purposes ferreri or basu harris hjort jones among others are also considered

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

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

two parameter generalization of the logarithm and exponential functions and boltzmann gibbs shannon entropy the q sum x qy x y 1 q xy x 1y x y and the q product x qy x1 q y1 q 1 1 1 q x 1y xy emerge naturally within nonextensive statistical mechanics we show here how they lead to two parameter namely q and q generalizations of the logarithmic and exponential functions noted respectively lnq q x and eq q x as well as of the boltzmann gibbs shannon entropy sbgs k i 1wpilnpi noted sq q the remarkable properties of the q q generalized logarithmic function make the entropic form sq q k i 1wpilnq q 1 pi satisfy for large regions of q q important properties such as expansibility concavity and lesche stability but not necessarily composability

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

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

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

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

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

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

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

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

variational approach and deformed derivatives recently we have demonstrated that there exists a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely the tsallis framework and the kaniadakis scenario with a local form of fractional derivative operators for fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure here in this paper we present an extension of the traditional calculus of variations for systems containing deformed derivatives embedded into the lagrangian and the lagrangian densities for classical and field systems the results extend the classical euler lagrange equations and the hamiltonian formalism the resulting dynamical equations seem to be compatible with those found in the literature specially with mass dependent and with nonlinear equations for systems in classical and quantum mechanics examples are presented to illustrate applications of the formulation also the conserved noether current is worked out

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

adaptive control with the stochastic approximation algorithm geometry and convergence new geometric properties possessed by the sequence of parameter estimates are exhibited which yield valuable insight into the behavior of the stochastic approximation based algorithm as it is used in minimum variance adaptive control in particular these geometric properties together with certain probabilistic arguments prove that if the system does not have a reduced order minimum variance controller then the parameter estimates converge to a random multiple of the true parameter an explicit expression for the limiting parameter estimate is also available with strictly positive probability the limiting parameter estimate is not the true parameter and in some cases differs from the true parameter with probability one if the system possesses reduced order minimum variance controllers then convergence to a minimum variance controller in a cesaro sense is shown the geometry of the limit set is described sufficient conditions are also given for some of these results to hold for parameter estimation schemes other than stochastic approximation

implicit predictive adaptive control with stochastic gradient identifiers the replacement in an implicit predictive adaptive controller of recursive least squares rls identifiers with stochastic gradient sg identifiers is considered by an ordinary differential equation analysis local convergence properties of the new algorithm are investigated the conclusions supported by simulation results are that in contrast with the one step ahead self tuning regulator the convergence properties of the resulting adaptive controller deteriorate if rls identifiers are replaced with sg identifiers

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

paper adaptive control based on explicit criterion minimization a common approach to regulator design is to define an objective function which is minimized with respect to adjustable regulator parameters here we discuss how such objective functions can be minimized on line thus providing adaptive control such an approach to adaptive control has its roots in early contributions to learning systems and it is here further developed and discussed in the light of the recent development of the field a general algorithm is given and special attention is paid to minimization of quadratic criteria a key problem is to obtain information about the system dynamics in order to compute the derivatives of the criterion with respect to the regulator parameters it is shown that the self tuning regulator is obtained as a special case corresponding to a particular quadratic criterion and a particular way of estimating the system dynamics explicit ways using instrumental variables techniques based on extra injected noise are also discussed a specific feature of the algorithm is that it does not utilize specific knowledge about how to calculate the optimal regulator the algorithm is the same for minimum phase as well as for non minimum phase systems

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

comment on renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data in their recent paper phys rev e 99 032134 2019 2470 004510 1103 physreve 99 032134 oikonomou and bagci have argued that renyi entropy is ill suited for inference purposes because it is not consistent with the shore johnson axioms of statistical estimation theory in this comment we seek to clarify the latter statement by showing that there are several issues in oikonomou s and bagci s reasonings which lead to erroneous conclusions when all these issues are properly accounted for no violation of shore johnson axioms is found

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

probability distribution and entropy as a measure of uncertainty the relationship between three probability distributions and their maximizable entropy forms is discussed without postulating entropy property for this purpose the entropy i is defined as a measure of uncertainty of the probability distribution of a random variable x by a variational relationship a definition underlying the maximization of entropy for corresponding distribution