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

control of chaotic dynamical systems using radial basis function network approximators abstract this paper presents a general control method based on radial basis function networks rbfns for chaotic dynamical systems for many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part under some mild conditions the rbfn can be used to well approximate the nonlinear part of the system dynamics the resulting system is then dominated by the linear part with some small or weak residual nonlinearities due to the rbfn approximation errors thus a simple linear state feedback controller can be devised to drive the system response to a desirable set point in addition to some theoretical analysis computer simulations on two representative continuous time chaotic systems the duffing and the lorenz systems are presented to demonstrate the effectiveness of the proposed method

on parameter estimation of mimo flat fading channels with frequency offsets we address the frequency offsets and channel gains estimation problem for a multi input multi output mimo flat fading channel using a training sequence the general case where the frequency offsets are possibly different for each transmit antenna is considered the cramer rao bound crb for the problem at hand is derived additionally we present a simple closed form expression for the large sample crb and show that it depends in a simple way on the channel parameters next the parameters estimation issue is investigated first the maximum likelihood estimator mle which entails solving an n dimensional maximization problem where n is the number of transmit antennas is derived then we show that the likelihood function can be written as the product of n one dimensional 1 d functions if a suitable choice of the training sequence is made based on this fact we suggest two computationally simpler methods numerical examples that illustrate the performance of the estimators and compare it with the crb are provided

nonlinear modeling and prediction by successive approximation using radial basis functions abstract in this paper a multilayer radial basis network is proposed for nonlinear system modeling and prediction it uses successive approximations first obtaining a number of coarse approximations and then optimally linearly combining the coarsely defined functions to achieve an accurate end result the more conventional approach of strict interpolation using radial basis functions does not necessarily yield high predictive accuracy is very time consuming for large data sets and can also give suboptimal accuracy due to numerical problems with nearly singular systems of equations the present approach considerably reduces computation time and can improve the predictive ability of radial basis networks while retaining good training accuracy numerical examples for chaotic time series and some classical real world time series are presented it is shown that the successive approximation radial basis network presented here is a useful tool for nonlinear modeling and prediction

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

a generalized thermodynamics for power law statistics abstract we show that there exists a natural way to define a condition of generalized thermal equilibrium between systems governed by tsallis thermostatistics under the hypotheses that i the coupling between the systems is weak ii the structure functions of the systems have a power law dependence on the energy it is found that the q values of two such systems at equilibrium must satisfy a relationship involving the respective numbers of degrees of freedom the physical properties of a tsallis distribution can be conveniently characterized by a new parameter which can vary between 0 and these limits corresponding respectively to the two opposite situations of a microcanonical distribution and of a distribution with a predominant power tail at high energies we prove that the statistical expression of the thermodynamic functions is univocally determined by the requirements that a systems at thermal equilibrium have the same temperature b the definitions of temperature and entropy are consistent with the second law of thermodynamics we find that for systems satisfying the hypotheses i and ii specified above the thermodynamic entropy is given by renyi entropy

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

joint cfo and channel estimation for multiuser mimo ofdm systems with optimal training sequences this paper addresses the problem of joint carrier frequency offset cfo and channel estimation in multiuser multiple input multiple output mimo orthogonal frequency division multiplexing ofdm systems to choose the optimal training sequences with the goal of providing the smallest estimation mean square error mse the asymptotic cramer rao bounds ascrbs are derived the optimal training sequences are designed that minimize the ascrbs for both cfo and channel estimation under the constraint that the ascrbs being channel independent a joint cfo and channel estimator is derived based on the maximum likelihood ml criterion a computationally efficient method using importance sampling technique is proposed to solve the highly demanding multidimensional exhaustive search required by the ml multi cfo estimation simulation results illustrate the merits of the proposed training sequences and also verify the effectiveness of the proposed estimation scheme

galilean invariant lattice boltzmann models with h theorem we demonstrate that the requirement of galilean invariance determines the choice of h function for a wide class of entropic lattice boltzmann models for the incompressible navier stokes equations the required h function has the form of the burg entropy for d 2 and of a tsallis entropy with q 1 2 d for d 2 where d is the number of spatial dimensions we use this observation to construct a fully explicit unconditionally stable galilean invariant lattice boltzmann model for the incompressible navier stokes equations for which attainable reynolds number is limited only by grid resolution

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

microcanonical foundation for systems with power law distributions starting from a microcanonical basis with the principle of equal a priori probability it is shown using the method of steepest descents that besides ordinary boltzmann gibbs theory with the exponential distribution a theory describing systems with power law distributions can also be derived

state space reconstruction in the presence of noise takens theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series this theorem gives little guidance however about practical considerations for reconstructing a good state space we extend takens treatment applying statistical methods to incorporate the effects of observational noise and estimation error we define the distortion matrix which is proportional to the conditional covariance of a state given a series of noisy measurements and the noise amplification which is proportional to root mean square time series prediction errors with an ideal model we derive explicit formulae for these quantities and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification we identify several different scaling regimes for distortion and noise amplification and derive asymptotic scaling laws when the dimension and lyapunov exponents are sufficiently large these scaling laws show that no matter how the state space is reconstructed there is an explosion in the noise amplification from a practical point of view determinism is lost and the time series is effectively a random process in the low noise large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension however in numerical experiments we find that estimation error complicates this issue for local approximation methods we analyze the effect of reconstruction on estimation error derive a scaling law and suggest an algorithm for reducing estimation errors

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

nonextensivity of the cyclic lattice lotka volterra model this niteness only oc curs for q 0 5 for the d 2 growth mode growing droplet and for q 0 for the d 1 one growing stripe this strongevidence of nonextensivity is consistent with the spontaneousemergence of local domains of identical particles with fractalboundaries and competing interactions such direct evidenceis for the rst time exhibited for a many body system which at the mean eld level is conservative keywords lattice lotka volterra model fractals reac tion limited processes nonextensivity pacs numbers 5 40 a 05 10 ln 05 65 b 82 40 npmany natural and arti cial systems are known today tobe hardly or not at all tractable within boltzmann gibbsstatistical mechanics hence the usual thermodynamics suchis the case of systems which include long range interactionsor long range microscopic or mesoscopic memory or othersources of multi fractality phenomena where many spa tial and or temporal scales are involved typically exhibitpower laws the celebrated boltzmann gibbs bg entropys

blind carrier frequency offset estimation for interleaved ofdma uplink in this paper we develop two novel blind carrier frequency offset cfo estimators for interleaved orthogonal frequency division multiple access ofdma uplink transmission in a multiantenna system the first estimator is the subspace based one and could blindly estimate multiple cfos from a rank reduction approach the second estimator is based on the maximum likelihood ml approach and improves the performance as compared to the first one the higher computational complexity of the ml estimator is alleviated by the alternating projection ap method both the proposed estimators support fully loaded data transmissions i e all subcarriers being occupied which provides higher bandwidth efficiency as compared to the existing schemes the numerical results are then provided to corroborate the proposed studies

universal renormalization group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics we uncover the dynamics at the chaos threshold m of the logistic map and find that it consists of trajectories made of intertwined power laws that reproduce the entire period doubling cascade that occurs for m m we corroborate this structure analytically via the feigenbaum renormalization group rg transformation and find that the sensitivity to initial conditions has precisely the form of a q exponential of which we determine the q index and the q generalized lyapunov coefficientl q our results are an unequivocal validation of the applicability of the nonextensive generalization of boltzmann gibbs statistical mechanics to critical points of nonlinear maps critical points of nonlinear maps offer a suitable playground for testing the validity of the nonextensive generalization of the boltzmann gibbs bg statistical mechanics proposed by tsallis over a decade ago 1 2 here we describe universal properties related to the dynamics of iterates at the onset of chaos in unimodal maps 3 which provide a literal confirmation of the generalized nonextensive theory to this end we employ the celebrated one dimensional logistic map f m x 512muxu 2 21 x 1 and the properties of its renormalization group rg fixed point to present evidence of previously unexposed scaling properties at the onset of chaos m5m at this state the most prominent of the map critical points the trajectories of the iterates exhibit an intricate structure that we describe and show is governed by the feigenbaum rg transformation 3 the domain of validity of bg statistical mechanics has been implicitly challenged by the proposal of its nonextensive generalization subsequent studies have offered experimental and numerical evidence that point out both the inadequacy of the standard bg statistics and the plausible competence of the generalized theory in describing various types of phenomena and systems this theoretical development represents an exceptional event in the long and trustworthy history of bg statistical mechanics however it is still in the process of being converted into a rigorously corroborated and fully understood fact the suggested circumstances under which the generalized theory is believed to be applicable at least with regards to nonlinear dynamical systems are those associated to a phase space with power law sensitivity to initial conditions to the consequent vanishing of the largest lyapunov exponent and to a fractal or multifractal geometrical structure 2 here we show that our results for the dynamics at the onset of chaos in unimodal maps constitute an unequivocal proof of the universal validity of the nonextensive statistics at such critical points in fact at the chaos threshold as well as at other critical points of the map the lyapunov exponent l 1 vanishes and the sensitivity to initial conditions j t for large iteration time t ceases to obey exponential behavior exhibiting instead a power law behavior 4 in order to describe the dynamics at such critical points the q exponential expression

critique of q entropy for thermal statistics during the past 12 years there have been numerous papers on a relation between entropy and probability which is nonadditive and has a parameter q that depends on the nature of the thermodynamic system under consideration for q 1 this relation corresponds to the boltzmann gibbs entropy but for other values of q it is claimed that it leads to a formalism that is consistent with the laws of thermodynamics however it is shown here that the joint entropy for systems having different values of q is not defined in this formalism and consequently fundamental thermodynamic concepts such as temperature and heat exchange cannot be considered for such systems moreover for q ensuremath ne 1 the probability distribution for weakly interacting systems does not factor into the product of the probability distribution for the separate systems leading to spurious correlations and other unphysical consequences e g nonextensive energy that have been ignored in various applications given in the literature

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

blind frequency synchronization for multiuser ofdm uplink with large number of receive antennas in this paper we propose a new blind frequency synchronization scheme for multiuser orthogonal frequency division multiplexing ofdm uplink transmissions the spatial multiplexing is supported in the considered model that allows the subcarriers to be simultaneously occupied by multiple users while the orthogonal frequency division multiple access ofdma can serve as a special case of our discussion we propose to assign different null subcarriers to different users and design algorithm that can perform blind carrier frequency offset cfo estimation for each individual user with the aid of large number of receive antennas which then removes the necessity of multidimensional searching after estimating cfo for one user the orthogonal complement space of the other users can be obtained which could be further utilized for multiuser interference cancelation hence the multiuser ofdm uplink model reduces to an equivalent single user signal model where the blind channel estimation with scalar ambiguity can be designed and the conventional single user data detection can be peformed to make the proposed study complete the closed form performance analysis for mean square error mse of the cfo estimation is derived based on which the discussion about optimal null subcarrier assignment scheme is further provided finally the cramer rao bound crb is derived and the numerical results are provided to corroborate the proposed studies

convergence to the critical attractor of dissipative maps log periodic oscillations fractality and nonextensivity for a family of logisticlike maps we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space we found that the phase space volume occupied by the ensemble w t depicts a power law decay with log periodic oscillations reflecting the multifractal character of the critical attractor we explore the parametric dependence of the power law exponent and the amplitude of the log periodic oscillations with the attractor s fractal dimension governed by the inflection of the map near its extremal point further we investigate the temporal evolution of w t for the circle map whose critical attractor is dense in this case we found w t to exhibit a rich pattern with a slow logarithmic decay of the lower bounds these results are discussed in the context of nonextensive tsallis entropies

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

non extensive block entropy statistics of cantor fractal sets abstract by using non extensive block entropy statistics we demonstrate analytically that the static structures of deterministic cantor sets with fractal dimension d f are characterised by a non extensive q exponent q 1 d f d for d f d where d is the embedding dimensions of the fractal set to calculate the s q entropy we use the block entropy method based on non overlapping windows and standard exact enumeration on cantor sets this result indicates that fractal structures are formed via dynamical processes which operate at the edge of chaos

prediction of noisy chaotic time series using an optimal radial basis function neural network this paper considers the problem of optimum prediction of noisy chaotic time series using a basis function neural network in particular the radial basis function rbf network in the noiseless environment predicting a chaotic time series is equivalent to approximating a nonlinear function the optimal generalization is achieved when the number of hidden units of a rbf predictor approaches infinity when noise exists it is shown that an optimal rbf predictor should use a finite number of hidden units to determine the structure of an optimal rbf predictor we propose a new technique called the cross validated subspace method to estimate the optimum number of hidden units while the subspace technique is used to identify a suitable number of hidden units by detecting the dimension of the subspace spanned by the signal eigenvectors the cross validation method is applied to prevent the problem of overfitting the effectiveness of this new method is evaluated using simulated noisy chaotic time series as well as real life oceanic radar signals results show that the proposed method can find the correct number of hidden units of an rbf network for an optimal prediction

comment on critique of q entropy for thermal statistics it was recently argued m nauenberg phys rev e 67 036114 2003 that the theory sometimes referred to as nonextensive statistical mechanics has no physical basis for a considerable variety of reasons including the impossibility of measuring the temperature out of the boltzmann gibbs bg theory we comment here on virtually all the physically and mathematically relevant issues and point out what we consider to be severe inadvertences contained in that paper in particular we factually argue through computer simulations the validity of the zeroth principle of thermodynamics and of the basic rules of thermometry for nonextensive systems this fact further supports the possible connection with the thermodynamics of nonextensive statistical mechanics which is already known to be consistent with the first second and third principles all the foundational steps e g the uniqueness of the entropy and the stationary state distribution have already been established for nonextensive thermostatistics on similar grounds than those long known for bg statistics the former corresponding to power laws expected for long range interactions when size n diverges before time t and the latter correspond to the bg exponential law expected for long range interactions when n diverges after t as well as for short range interactions in any diverging order for n and t we conclude that the invalidating arguments made by nauenberg by no means apply

generalization of shannon s theorem for tsallis entropy by using the assumptions that the entropy must i be a continuous function of the probabilities pi pi 0 1 i only ii be a monotonic increasing function of the number of states w in the case of equiprobability iii satisfy the pseudoadditivity relation sq a b k sq a k sq b k 1 q sq a sq b k2 a and b being two independent systems q r and k a positive constant and iv satisfy the relation sq pi sq pl pm pl qsq pi pl pm qsq pi pm where pl pm 1 pl i 1wlpi and pm i wl 1wpi we prove along shannon s lines that the unique function that satisfies all these properties is the generalized tsallis entropy sq k 1 i 1wpiq q 1

on the theory of bilinear time series models summary the theory of bilinear time series models is considered in this paper the sufficient conditions for asymptotic stationarity of the bilinear time series models are derived and the expressions for the variance and covariance are obtained the conditions for the invertibility of the model are also included the estimation of the parameters of the scalar bilinear time series model is considered the bilinear models are fitted to sunspot numbers and also to a p wave of a nuclear explosion the forecasting of sunspot numbers is also considered

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

nonlinear statistical coupling abstract by considering a nonlinear combination of the probabilities of a system a physical interpretation of tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed the escort probability is interpreted as the coupled probability with q 1 q defined as the degree of nonlinear coupling between the statistical states positive values of q have coupled statistical states a larger entropy metric and a maximum coupled entropy distribution of compact support coupled gaussians negative values of q have decoupled statistical states and for 2 q 0 a maximum coupled entropy distribution of heavy tail coupled gaussians the conjugate transformation between the heavy tail and compact support domains is shown to be q 2 q 2 q for coupled gaussian distributions this conjugate relationship has been used to extend the generalized fourier transform to the compact support domain and to define a scale invariant correlation structure with heavy tail limit distribution in the present paper we show that the conjugate is a mapping between the source of nonlinearity in non stationary stochastic processes and the nonlinear coupling which defines the coupled gaussian limit distribution the effects of additive and multiplicative noise are shown to be separable into the coupled variance and the coupling parameter q providing further evidence of the importance of the generalized moments