a generalized computational approach to stability of static equilibria of nonlinearly elastic rods in the presence of constraints abstract we present a generalized approach to stability of static equilibria of nonlinearly elastic rods subjected to general loading boundary conditions and constraints of both point wise and integral type based upon the linearized dynamics stability criterion discretization of the governing equations leads to a non standard singular generalized eigenvalue problem a new efficient sparse matrix friendly algorithm is presented to determine its few left most eigenvalues which in turn yield stability instability information for conservative problems the eigenvalue problem arising from the linearized dynamics stability criterion is also shown to be equivalent to that arising in the determination of constrained local minima of the potential energy we illustrate the method with several examples a novel variational formulation for extensible and unshearable rods is also proposed within the context of one of the example problems

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

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

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

an asymptotic numerical method for continuation of spatial equilibria of special cosserat rods abstract we present an efficient numerical scheme based on asymptotic numerical method for continuation of spatial equilibria of special cosserat rods using quaternions to represent rotation the equations of static equilibria of special cosserat rods are posed as a system of thirteen first order ordinary differential equations having cubic nonlinearity the derivatives in these equations are further discretized to yield a system of cubic polynomial equations as asymptotic numerical methods are typically applied to polynomial systems having quadratic nonlinearity a modified version of this method is presented in order to apply it directly to our cubic nonlinear system we then use our method for continuation of equilibria of the follower load problem and demonstrate our method to be highly efficient when compared to conventional solvers based on the finite element method finally we demonstrate how our method can be used for computing the buckling load as well as for continuation of postbuckled equilibria of hemitropic rods

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

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

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

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

isogeometric collocation methods for cosserat rods and rod structures abstract we present a novel method for the mechanical simulation of slender elastic spatial rods and rod structures subject to large deformation and rotation we develop an isogeometric collocation method for the geometrically exact nonlinear cosserat rod theory the rod centerlines are represented as spatial nurbs curves and cross section orientations are parameterized in terms of unit quaternions as 4 dimensional nurbs curves within the isogeometric framework the strong forms of the equilibrium equations of forces and moments of the discretized cosserat model are collocated leading to an efficient method for higher order discretizations for rod structures consisting of multiple connected rods we introduce a formulation with rigid quasi g 1 coupling it is based on the strong enforcement of continuity of displacement and change of cross section orientation at interfaces we also develop a mixed isogeometric formulation which is based on an independent discretization of internal forces and moments and alleviates shear locking for thin rods the novel rod simulation methods are verified by numerical convergence studies further computational examples include realistic applications with large deformations and rotations as well as a large scale rod structure with several hundreds of coupled rods and complex buckling behavior

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

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

stability analysis of mean field type nonlinear fokker planck equations associated with a generalized entropy and its application to the self gravitating system multidimensional nonlinear fokker planck equations of mean field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions two types of eigenvalue equations are studied the nonlinear fokker planck equations are shown to exhibit an h theorem with a liapunov functional that takes the form of a free energy involving generalized entropies of tsallis the second order variation of the liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean field coupling potential assuming quasiequilibrium for the velocity distribution the reduced eigenvalue equation with space coordinates alone is also obtained the alternative type of eigenvalue equation based on the linearization of the nonlinear fokker planck equations is presented taking the mean field coupling potential to be the gravitational one the nonlinear fokker planck equation in terms of three dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean field model of self gravitating system by solving the eigenvalue equation for the eigenfunction with 0 eigenvalue the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed

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

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

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

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

an algebraic procedure for the spectral corrections using the miss distance functions in regular and singular sturm liouville problems a general method based on the evaluation of the zeros of a suitable polynomial is suggested in order to have an estimation of the spectral error in the numerical treatment of sturm liouville problems the method is strictly concerned with the miss distance function arising in the shooting algorithm for eigenvalues the error correcting procedure derived from the method is particularly helpful when difficulties arise in the numerical integration two kinds of sturm liouville problems are considered the standard regular problems on a closed interval and the problems where an eigenvalue is nonlinearly involved and embedded in an essential spectrum giving origin to an inner singularity numerical experiments clearly highlight the efficaciousness of the proposed method both in the regular and singular case

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

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

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

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

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

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

conditional correlation as a measure of mediated interactivity in fmri and meg eeg many measures have been proposed so far to extract brain functional interactivity from functional magnetic resonance imaging fmri and magnetoencephalography electroencephalography meg eeg data sets unfortunately none has been able to provide a relevant self contained and common definition of brain interaction in this paper we propose a first step in this direction we first introduce a common terminology together with a cross modal definition of interaction in this setting we investigate the commonalities shared by some measures of interaction proposed in the literature we show that temporal correlation nonlinear correlation mutual information generalized synchronization phase synchronization coherence and phase locking value plv actually measure the same quantity namely correlation when one is investigating linear interactions between independently and identically distributed gaussian variables we also demonstrate that these data driven measures can only partly account for the interaction patterns that can be expressed by the effective connectivity of structural equation modeling sem to bridge this gap we suggest the use of conditional correlation which is shown to be related to mediated interaction

high performance computations of steady state bifurcations in 3d incompressible fluid flows by asymptotic numerical method this paper presents a powerful numerical model that implements the asymptotic numerical method to compute 3d steady state incompressible fluid flow solutions this continuation algorithm enables to explore branches of steady state solutions stable or unstable to accurately determine any simple steady state bifurcation points and their emanating bifurcated branches the powerfulness of the model stands on an optimal step length continuation thanks to the combination of power series analysis in the framework of anm along with an efficient parallel implementation of the resulting algorithm on high performance computers the outcome of this approach is demonstrated throughout 3d incompressible fluid flows inside a sudden expansion channel expansion ratio e 3 cross section aspect ratio 10 b 20 we have computed for the first time up to four steady symmetry breaking pitchfork bifurcations together with their associated bifurcated branches the main characteristic of this 3d symmetric expansion configuration is that for a given cross section aspect ratio the first bifurcation mode induces a top bottom asymmetry as in the 2d case whereas the subsequent ones modulate the former in the span wise direction with increasing wave numbers the asymptotic numerical method is a very efficient continuation technique judicious usage of power series analysis increases computational efficiency an hpc implementation of the resulting algorithm is presented in this paper its numerical efficiency enables to investigate problems out of reach until now span wise symmetry breaking bifurcations in a symmetric sudden expansion channel

generative model for feedback networks we propose a model for network formation and study some of its statistical properties the motivation for the model comes from the growth of several kinds of real networks i e kinship and trading networks networks of corporate alliances networks of autocatalytic chemical reactions these networks grow either by establishing closer connections by adding links in the existing network or by adding new nodes a node in these networks lacks the information of the entire network in order to establish a closer connection to other nodes it starts a search in the neighboring part of the network and waits for a possible feedback from a distant node that received the searching signal our model imitates this behavior by growing the network via the addition of a link that creates a cycle in the network or via the addition of a new node with a link to the network the forming of a cycle creates feedback between the two ending nodes after choosing a starting node a search is made for another node at a suitable distance if such a node is found a link is established between this and the starting node otherwise such a node cannot be found a new node is added and is linked to the starting node we simulate this algorithm and find that we cannot reject the hypothesis that the empirical degree distribution is a q exponential function which has been used to model long range processes in nonequilibrium statistical mechanics

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

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

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

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

computing finite rotations of shells by an asymptotic numerical method abstract we present an asymptotic numerical algorithm for the computation of elastic shells with large rotations the theoretical formulation involves a three field hu washizu functional which allows us to put the problem into a quadratic framework the spatial discretization is based on geometrically exact element recently presented by buchter et al buchter et al three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept int j numer methods engrg 37 1994 2551 1568 several classical benchmarks are discussed to define the best strategy and to assess the validity and the efficiency of the present method as compared to more classical iterative algorithms

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

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

generalized mutual information fmri analysis a study of the tsallis q parameter abstract the main objective of analyzing functional magnetic resonance imaging fmri data sets is to screen the physiologically induced signals from noise or artifacts resulting either from involuntary patient motion or mri detection techniques one problem that is not yet solved is related to the dependency of the results in regard to the threshold applied within each analysis recently an alternative fmri time series analysis method has been proposed to decrease the dependency of the results with the applied threshold this method is based on the calculation of the generalized mutual information gmi between the blood oxygenation level dependence time series and the paradigm slope since the analysis relies heavily on the tsallis q parameter a broad spectrum of results can be produced so simulated data analysis was performed in order to construct the receiver operating characteristic curves the parameters that evaluate the quality of the curves were determined thus allowing for an optimization of the q values results for simulated and real fmri data were compared for both gmi and classical deterministic analysis

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

a polynomial approach to the spectral corrections for sturm liouville problems a technique based on the evaluation of the zeros of a polynomial is proposed to estimate the spectral errors and set up a correcting procedure in sturm liouville problems the method suggested shows its effectiveness both in the regular and nonregular case and can be successfully applied also to those problems containing the eigenvalue parameter rationally some numerical experiments clearly confirm the theoretical results in the case of an eigenvalue embedded in the essential spectrum the correcting procedure seems to be particularly helpful because the inner singularity gives rise to possible decay of the performance of some classical methods for the numerical integration

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

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

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

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

spectral corrections for sturm liouville problems abstract the numerical solution of the sturm liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction in this paper we use the shooting method both for eigenvalues and eigenfunctions in integrating the corresponding initial value problems we resort to the boundary value method the technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems a technique to estimate the eigenvalue errors is also suggested and seems to be particularly effective for the higher index eigenvalues numerical experiments on some classical sturm liouville problems are presented

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

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

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

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

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

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

locking free isogeometric collocation formulation for three dimensional geometrically exact shear deformable beams with arbitrary initial curvature abstract this paper presents a displacement based and a mixed isogeometric collocation formulations for three dimensional geometrically exact shear deformable beams with arbitrarily curved initial geometry strong forms of both formulations are consistently linearized with respect to the underlying configuration manifold high efficiency is achieved by using a geometrically consistent kinematic model in which incremental rotations are parametrized through the spatial rotation vector several numerical tests demonstrate that the proposed mixed formulation is totally locking free for any choice of approximation degree mixed formulation also exhibits higher robustness in the incremental iterative algorithm elimination of shear locking and low computational cost assured by the combination of isogeometric collocation with a consistent kinematic model result in a very efficient accurate and geometrically powerful method able to handle geometrically complex beam problems

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

generalized relative entropy in functional magnetic resonance imaging abstract the generalized kullback leibler distance d q q is the tsallis parameter is shown to be an useful measure for analysis of functional magnetic resonance imaging fmri data series this generalized form of entropy is used to evaluate the distance between the probability functions p 1 and p 2 of the signal levels related to periods of stimulus and non stimulus in event related fmri experiments the probability densities of the mean distance d q averaged over the n epochs of the entire experiment are obtained through numerical simulations for different values of signal to noise ratio snr and found to be fitted very well by gamma distributions 2 0 0008 for small values of n n 30 these distributions allow us to determine the sensitivity and specificity of the method by construction of the receiver operating characteristic roc curves the performance of the method is also investigated in terms of the parameters q and l number of signal levels and our results indicate that the optimum choice is q 0 8 and l 3 the entropic index q is found to exert control on both sensitivity and specificity of the method as q q 0 is raised sensitivity increases but specificity decreases finally the method is applied in the analysis of a real event related fmri motor stimulus experiment and the resulting maps show activation in primary and secondary motor brain areas

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

power series analysis as a major breakthrough to improve the efficiency of asymptotic numerical method in the vicinity of bifurcations this paper presents the outcome of power series analysis in the framework of the asymptotic numerical method we theoretically demonstrate and numerically evidence that the emergence of geometric power series in the vicinity of simple bifurcation points is a generic behavior so we propose to use this hallmark as a bifurcation indicator to locate and compute very efficiently any simple bifurcation point finally a power series that recovers an optimal step length is build in the neighborhood of bifurcation points the reliability and robustness of this powerful approach is then demonstrated on two application examples from structural mechanics and hydrodynamics

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

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

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

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

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

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

a high order purely frequency based harmonic balance formulation for continuation of periodic solutions the case of non polynomial nonlinearities abstract in this paper we extend the method proposed by cochelin and vergez a high order purely frequency based harmonic balance formulation for continuation of periodic solutions journal of sound and vibration 324 2009 243 262 to the case of non polynomial nonlinearities this extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems the principle remains to transform the original ode system into an extended polynomial quadratic system for an easy application of the harmonic balance method hbm the transformation of non polynomial terms is based on the differentiation of state variables with respect to the time variable shifting the nonlinear non polynomial nonlinearity to a time independent initial condition equation not concerned with the hbm the continuation of the resulting algebraic system is here performed by the asymptotic numerical method high order taylor series representation of the solution branch using a further differentiation of the non polynomial algebraic equation with respect to the path parameter a one dof vibro impact system is used to illustrate how an exponential nonlinearity is handled showing that the method works at very high order 1000 in that case various kinds of nonlinear functions are also treated and finally the nonlinear free pendulum is addressed showing that very accurate periodic solutions can be computed with the proposed method

nonlinear dynamics of filaments i dynamical instabilities abstract the kirchhoff model provides a well established mathematical framework to study both computationaly and theoretically the dynamics of thin filaments within the approximations of linear elasticity theory the study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time dependent version of the kirchhoff model in the euler angle frame here we study the linear stability of the full time independent equations by introducing a new are length preserving perturbation scheme as an application we consider the instabilities of various stationary solutions such as the planar ring and straight rod subjected to twisting perturbations this scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments

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

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

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

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

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

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

maximum entropy approach to stretched exponential probability distributions we introduce a nonextensive entropy functionals whose optimization under simple constraints mean values of some standard quantities yields stretched exponential probability distributions which occur in many complex systems the new entropy functional is characterized by a parameter the stretching exponent such that for d 1 the standard logarithmic entropy is recovered we study its mathematical properties showing that the basic requirements for a well behaved entropy functional are verified i e s possesses the usual properties of positivity equiprobability concavity and irreversibility and verifies khinchin axioms except the one related to additivity since s is nonextensive the entropys is shown to be superadditive for 1

a finite element estimate with trigonometric hat functions for sturm liouville eigenvalues abstract the computation of eigenvalues of regular sturm liouville problems is considered numerical results show that the error of the k th eigenvalue obtained by the finite element method with trigonometric hat functions and mesh length h is of the same order as the one obtained with the linear hat functions together with a simple asymptotic correction technique of paine de hoog and anderssen 1981

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

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

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

analysis of return distributions in the coherent noise model return distributions of the coherent noise model are studied for the system size independent case it is shown that in this case these distributions are in the shape of q gaussians which are the standard distributions obtained in nonextensive statistical mechanics moreover an exact relation connecting the exponent of avalanche size distribution and the q value of appropriate q gaussian has been obtained as q 2 making use of this relation one can easily determine q parameter values of the appropriate q gaussians a priori from one of the well known exponents of the system since the coherent noise model has the advantage of producing different t values by varying a model parameter clear numerical evidences on the validity of the proposed relation have been achieved for various cases finally the effect of the system size has also been analyzed and an analytical expression has been proposed which is corroborated by the numerical results

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

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

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