rigorous properties and uncertainty like relationships on product complexity measures application to atomic systems most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework valid for any arbitrary n dimensional distribution in the present work different product complexities constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated different results on product complexities such as rigorous bounds uncertainty like inequalities relationships among themselves and also statistical interpretations are given for the sake of completeness some of these analytical results on product complexities are numerically checked for the one particle densities of atomic systems in both conjugated spaces

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

mathematical structures derived from the q multinomial coefficient in tsallis statistics abstract we present q stirling s formula q multinomial coefficient one to one correspondence between q multinominal coefficient and tsallis entropy q pascal s triangle and a conjecture on the q central limit theorem in tsallis statistics for the generalization of the well known fundamental formulas to systems exhibiting power law behaviors the main approach is based on the q product uniquely determined by tsallis entropy which has already been successfully applied to our recent proof of the law of error in tsallis statistics

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

gamma function to beck cohen superstatistics abstract in this paper we show a mathematical construction of beck cohen superstatistics in the bayesian point of view with the help of the two representations of a gamma function furthermore it is shown how some results for superstatistics are related to each other

stock market and motion of a variable mass spring we establish an analogy between the motion of spring whose mass increases linearly with time and volatile stock market dynamics within an economic model based on simple temporal demand and supply functions e canessa j phys a 33 2000 3637 the total system energy et is shown to be proportional to a decreasing time dependent spring constant kt this model allows to derive log periodicity cos log t tc on commodity prices and oscillations surplus and shortages in the level of stocks we also made an attempt to connect these results to the tsallis statistics parameter q based on a possible force entropy correlation e canessa physica a 341 2004 165 and find that the tsallis second entropic term i 1wpiq q 1 relates to the square of the demand or supply function

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

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

age innovations and time operator of networks abstract we extend the time operator and age to network evolution models internal age formulas and the distribution of innovations are computed for erd s renyi random networks for markov networks and barabasi albert preferential attachment networks the innovation probabilities are found to be proportional to the quadratic entropy which coincides with the tsallis entropy for entropic index q 2 in all markov networks as well as in the linear growth mechanism the distribution of innovations in the barabasi albert model is a new probability distribution of the logarithmic type

thermodynamics from first principles correlations and nonextensivity the standard formulation of thermostatistics being based on the boltzmann gibbs distribution and logarithmic shannon entropy describes idealized uncorrelated systems with extensive energies and short range interactions in this rapid communication we use the fundamental principles of ergodicity via liouville s theorem the self similarity of correlations and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long range interacting systems significantly our formalism provides a justification for the well studied nonextensive thermostatistics characterized by the tsallis distribution which it includes as a special case we also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the gibbs shannon entropy the consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics

a generalized statistical complexity measure applications to quantum systems a two parameter family of complexity measures c based on the renyi entropies is introduced and characterized by a detailed study of its mathematical properties this family is the generalization of a continuous version of the lopez ruiz mancini calbet complexity which is recovered for 1 and 2 these complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system when one of the parameters or goes to infinity one of the global factors becomes a local factor for this special case the complexity is calculated on different quantum systems h atom harmonic oscillator and square well

harmonious representation of pdfs reflecting large deviations abstract the framework of multifractal analysis mfa is distilled to the most sophisticated one within this transparent framework it is shown that the harmonious representation of mfa utilizing two distinct tsallis distribution functions one for the tail part of probability density function pdf and the other for its center part explains the recently observed pdfs of turbulence in the highest accuracy superior to the analyses based on other models such as the log normal model and the p model

entropic nonextensivity as a measure of time series complexity abstract information entropy is applied to the analysis of time series generated by dynamical systems complexity of a temporal or spatio temporal signal is defined as the difference between the sum of entropies of the local linear regions of the trajectory manifold and the entropy of the globally linearized manifold when the entropies are tsallis entropies the complexity is characterized by the value of q

power and sample size estimation for the clustered wilcoxon test the wilcoxon rank sum test is widely used for two group comparisons of nonnormal data an assumption of this test is independence of sampling units both within and between groups which will be violated in the clustered data setting such as in ophthalmological clinical trials where the unit of randomization is the subject but the unit of analysis is the individual eye for this purpose we have proposed the clustered wilcoxon test to account for clustering among multiple subunits within the same cluster rosner glynn and lee 2003 biometrics 59 1089 1098 2006 biometrics 62 1251 1259 however power estimation is needed to plan studies that use this analytic approach we have recently published methods for estimating power and sample size for the ordinary wilcoxon rank sum test rosner and glynn 2009 biometrics 65 188 197 in this article we present extensions of this approach to estimate power for the clustered wilcoxon test simulation studies show a good agreement between estimated and empirical power these methods are illustrated with examples from randomized trials in ophthalmology enhanced power is achieved with use of the subunit as the unit of analysis instead of the cluster using the ordinary wilcoxon rank sum test

empirical regularities of opening call auction in chinese stock market abstract we study the statistical regularities of an opening call auction using the ultra high frequency data of 22 liquid stocks traded on the shenzhen stock exchange in 2003 the distribution of the relative price defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price the detrended fluctuation analysis dfa method is adopted to investigate the long term memory of relative order prices we further study the statistical regularities of order sizes in the opening call auction and observe a phenomenon of number preference known as order size clustering the probability density function pdf of order sizes could be well fitted by a q gamma function and the long term memory also exists in order sizes in addition both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit order book lob established immediately after the opening call auction and a price clustering phenomenon is observed

a defect relation for holomorphic curves intersecting hypersurfaces in 1933 h cartan proved a defect relation q j 1 f h j n 1 for a linearly nondegenerate holomorphic curve f inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i and hyperplanes h j 1 j q in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i in general position this paper extends it to holomorphic curves intersecting hypersurfaces in 1979 b shiffman conjectured that if f inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i is an algebraically non degenerate holomorphic map and d 1 d q are hypersurfaces in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i in general position then q j 1 f d j n 1 this paper proves this conjecture

steering in spin tomographic probability representation abstract the steering property known for two qubit state in terms of specific inequalities for the correlation function is translated for the state of qudit with the spin j 3 2 since most steering detection inequalities are based on the correlation functions we introduce analogs of such functions for the single qudit systems the tomographic probability representation for the qudit states is applied the connection between the correlation function in the two qubit system and the single qudit is presented in an integral form with an intertwining kernel calculated explicitly in tomographic probability terms

power and sample size estimation for the wilcoxon rank sum test with application to comparisons of c statistics from alternative prediction models the wilcoxon mann whitney wmw u test is commonly used in nonparametric two group comparisons when the normality of the underlying distribution is questionable there has been some previous work on estimating power based on this procedure lehmann 1998 nonparametrics in this article we present an approach for estimating type ii error which is applicable to any continuous distribution and also extend the approach to handle grouped continuous data allowing for ties we apply these results to obtaining standard errors of the area under the receiver operating characteristic curve auroc for risk prediction rules under h 1 and for comparing auroc between competing risk prediction rules applied to the same data set these results are based on sas callable functions to evaluate the bivariate normal integral and are thus easily implemented with standard software

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

a new class of entropic information measures formal group theory and information geometry in this work we study generalized entropies and information geometry in a group theoretical framework we explore the conditions that ensure the existence of some natural properties and at the same time of a group theoretical structure for a large class of entropies in addition a method for defining new entropies using previously known ones with some desired group theoretical properties is proposed in the second part of this work the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied finally a method of constructing new divergences from known ones is discussed in particular some results concerning the riemannian structure associated with the class of divergences under investigation are formulated

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

generalized quantum entropy the generalized quantum entropies are introduced in this paper some important properties such as nonnegativity continuity and concavity are proved but different from the von neumann entropy the subadditivity and additivity fail for the quantum unified r s entropy in general

the renormalization group and optimization of non extensive entropy criticality in non linear one dimensional maps abstract we examine the pitchfork and tangent bifurcations in unimodal maps to illustrate a connection between renormalization group rg fixed points and entropy extremal properties we observe that the exact rg solution for the tangent bifurcation is also applicable to the period doubling cascade and assess its physical meaning since the expression for the fixed point map can be put into the form of the non extensive expressions for the temporal evolution of phase space volume and sensitivity of initial conditions we conclude that the map critical points possess the properties of this formalism the universality of the rg solution makes this interpretation inclusive to all one dimensional maps of non linearity z 1

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

recent advances of stretched gaussian distribution underlying hausdorff fractal distance and its applications in fitting stretched gaussian noise abstract this paper summarizes the latest advances of the third author s research group on stretched gaussian distribution underlying the hausdorff fractal theory and its applications in fitting stretched gaussian noise firstly the hausdorff fractal metrics are introduced as an extension of non euclidean distance based on the fractal scaling the hausdorff derivative is derived which can describe the problems of interest to construct non integer differential equations secondly we introduce the hausdorff derivatives in tackling partial differential equations and the fundamental solution of hausdorff derivative diffusion equation is stretched gaussian distribution thirdly we analyze the feasibility of the least square method for stretched gaussian noise which obeys stretched gaussian distribution the least square method is inapplicable when the noise level is larger than 5 finally by using the hausdorff fractal distance we introduce the stretched least square method which improves the traditional least square method to fit stretched gaussian noise

power and sample size calculations for the wilcoxon mann whitney test in the presence of death censored observations we consider a clinical trial of a potentially lethal disease in which patients are randomly assigned to two treatment groups and are followed for a fixed period of time a continuous endpoint is measured at the end of follow up for some patients however death or severe disease progression may preclude measurement of the endpoint a statistical analysis that includes only patients with endpoint measurements may be biased an alternative analysis includes all randomized patients with rank scores assigned to the patients who are available for the endpoint measurement on the basis of the magnitude of their responses and with worst rank scores assigned to those patients whose death precluded the measurement of the continuous endpoint the worst rank scores are worse than all observed rank scores the treatment effect is then evaluated using the wilcoxon mann whitney test in this paper we derive closed form formulae for the power and sample size of the wilcoxon mann whitney test when missing measurements of the continuous endpoints because of death are replaced by worst rank scores we distinguish two approaches for assigning the worst rank scores in the tied worst rank approach all deaths are weighted equally and the worst rank scores are set to a single value that is worse than all measured responses in the untied worst rank approach the worst rank scores further rank patients according to their time of death so that an earlier death is considered worse than a later death which in turn is worse than all measured responses in addition we propose four methods for the implementation of the sample size formulae for a trial with expected early death we conduct monte carlo simulation studies to evaluate the accuracy of our power and sample size formulae and to compare the four sample size estimation methods

cartan s conjecture for moving hypersurfaces let f be a holomorphic curve in mathbb p n mathbb c and let mathcal d d 1 ldots d q be a family of moving hypersurfaces defined by a set of homogeneous polynomials mathcal q q 1 ldots q q for j 1 ldots q denote by q j sum nolimits i 0 cdots i n d j a j i z x 0 i 0 cdots x n i n where i i 0 ldots i n in mathbb z ge 0 n 1 and a j i z are entire functions on mathbb c without common zeros let mathcal k mathcal q be the smallest subfield of meromorphic function field mathcal m which contains mathbb c and all frac a j i z a j i z with a j i z not equiv 0 1 le j le q in previous known second main theorems for f and mathcal d f is usually assumed to be algebraically nondegenerate over mathcal k mathcal q in this paper we prove a second main theorem in which f is only assumed to be nonconstant this result can be regarded as a generalization of cartan s conjecture for moving hypersurfaces

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

fractal surfaces from simple arithmetic operations fractal surfaces patchwork quilts are shown to arise under most general circumstances involving simple bitwise operations between real numbers a theory is presented for all deterministic bitwise operations on a finite alphabet it is shown that these models give rise to a roughness exponent h that shapes the resulting spatial patterns larger values of the exponent leading to coarser surfaces

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

statistical analysis of geopotential height gh timeseries based on tsallis non extensive statistical mechanics abstract in this paper we perform statistical analysis of time series deriving from earth s climate the time series are concerned with geopotential height gh and correspond to temporal and spatial components of the global distribution of month average values during the period 1948 2012 the analysis is based on tsallis non extensive statistical mechanics and in particular on the estimation of tsallis q triplet namely q stat q sens q rel the reconstructed phase space and the estimation of correlation dimension and the hurst exponent of rescaled range analysis r s the deviation of tsallis q triplet from unity indicates non gaussian tsallis q gaussian non extensive character with heavy tails probability density functions pdfs multifractal behavior and long range dependences for all timeseries considered also noticeable differences of the q triplet estimation found in the timeseries at distinct local or temporal regions moreover in the reconstructive phase space revealed a lower dimensional fractal set in the gh dynamical phase space strong self organization and the estimation of hurst exponent indicated multifractality non gaussianity and persistence the analysis is giving significant information identifying and characterizing the dynamical characteristics of the earth s climate

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

on the putative essential discreteness of q generalized entropies abstract it has been argued in abe 2010 entitled essential discreteness in generalized thermostatistics with non logarithmic entropy that continuous hamiltonian systems with long range interactions and the so called q gaussian momentum distributions are seen to be outside the scope of non extensive statistical mechanics the arguments are clever and appealing we show here that however some mathematical subtleties render them unconvincing

disequilibrium complexity the schottky effect and q entropies in paramagnetism abstract we investigate connections between statistical quantifiers and paramagnetism more concretely we apply the notions of i disequilibrium and ii statistical complexity to a paramagnetic system of non coupled dipoles interesting insights are thereby obtained in particular we encounter a kind of criticality not associated to the temperature but to the disequilibrium

on the stability of generalized entropies by the use of a probabilistic coupling technique and generalizing a recent result of zhang 2007 lett math phys 80 171 we derive several sharp inequalities concerning the uniform continuity of some generalized entropies these inequalities capture certain aspects of the interplay between the entropies and the uniform distance variational distance and establish the stability of the involved entropies

sample size determination for diagnostic accuracy studies involving binormal roc curve indices receiver operating characteristic roc curves and their associated indices are valuable tools for the assessment of the accuracy of diagnostic tests the area under the roc curve is a popular summary measure of the accuracy of a test the full area under the roc curve however has been criticized because it gives equal weight to all false positive error rates alternative indices include the area under the roc curve in a particular range of false positive rates partial area and the sensitivity of the test for a single fixed false positive rate fpr we present a unified approach for computing sample size for binormal roc curves and their indices our method uses taylor series expansions to derive approximate large sample estimates of the variance and covariance of binormal roc curve parameters several examples from diagnostic radiology illustrate the proposed method

on a comparative study between dependence scales determined by linear and non linear measures in this manuscript we present a comparative study about the determination of the relaxation i e independence time scales obtained from the correlation function the mutual information and a criterion based on the evaluation of a nonextensive generalisation of mutual entropy our results show that for systems with a small degree of complexity standard mutual information and the criterion based on its nonextensive generalisation provide the same scale whereas for systems with a higher complex dynamics the standard mutual information presents a time scale consistently smaller

extremum entrop y based heisenberg like uncertainty relations in this work we use the extremization method of various information theoretic measures fisher information shannon entropy tsallis entropy for d dimensional quantum systems which complementarily describe the spreading of the quantum states of natural systems under some given constraints usually one or two radial expectation values this variational method allows us to determine an extremum entropy distribution which is the least biased one to characterize the state among all those compatible with the known data then we use it together with the spin dependent uncertainty like relations of daubechies thakkar type as a tool to obtain relationships between the position and momentum radial expectation values of the type for d dimensional systems of n fermions with spin s the resulting uncertainty like products which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system are shown to often improve the best corresponding relationships existing in the literature

indices for testing neural codes one of the most critical challenges in systems neuroscience is determining the neural code a principled framework for addressing this can be found in information theory with this approach one can determine whether a proposed code can account for the stimulus response relationship specifically one can compare the transmitted information between the stimulus and the hypothesized neural code with the transmitted information between the stimulus and the behavioral response if the former is smaller than the latter i e if the code cannot account for the behavior the code can be ruled out the information theoretic index most widely used in this context is shannon s mutual information the shannon test however is not ideal for this purpose while the codes it will rule out are truly nonviable there will be some nonviable codes that it will fail to rule out here we describe a wide range of alternative indices that can be used for ruling codes out the range includes a continuum from shannon information to measures of the performance of a bayesian decoder we analyze the relationship of these indices to each other and their complementary strengths and weaknesses for addressing this problem

tsallis based variational method for finding approximate wave functions abstract we introduce trial functions of a form that maximizes a tsallis information measure and use them in conjunction with the variational approach of cooper dawson and shepard cds that is based upon supersymmetry ideas

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

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

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

diffusion on fractal phase spaces and entropy production abstract complex systems may show strongly non exponential relaxation the origins of these behavior seem to be related to the fractal structure of the phase space in this article we consider the diffusion on a dilute hypercube this stochastic process is a coarse grained model for the time evolution of a glassy system where both the dynamics and the structure of the phase space was considered as simple as possible we characterize this process using two quantities a memory function and the tsallis entropy the first one is a measure of mixing while the second is related to non extensivity we characterize quantitatively their relationship finally we discuss the implications for other glassy and complex systems

a possible deformed algebra and calculus inspired in nonextensive thermostatistics we present a deformed algebra related to the q exponential and the q logarithm functions that emerge from nonextensive statistical mechanics we also develop a q derivative and consistently a q integral for which the q exponential is an eigenfunction the q derivative and the q integral have a dual nature that is also presented

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

on a general thue s equation we analyze the integral points on varieties defined by one equation of the form f 1 f r g where the f i g are polynomials in n variables with algebraic coefficients and g has small degree we shall use a method that we recently introduced in the context of siegel s theorem for integral points on curves classical very particular instances of our equations arise e g in a well known corollary of roth s theorem the case n 2 f i linear forms deg g r 2 and with the norm form equations treated by w m schmidt here we shall prove thm 1 that the integral points are not zariski dense provided deg f i n max deg f i deg g and provided the f i g satisfy certain mild assumptions which are generically verified our conclusions also cover certain complete intersection subvarieties of our hypersurface thm 2 finally we shall prove thm 3 an analogue of the schmidt s subspace theorem for arbitrary polynomials in place of linear forms

troublesome aspects of the renyi maxent treatment we study in great detail the possible existence of a renyi associated thermodynamics with negative results in particular we uncover a hidden relation in renyi s variational problem maxent this relation connects the two associated lagrange multipliers canonical ensemble with the mean energy u and the renyi parameter as a consequence of such relation we obtain anomalous renyi maxent thermodynamic results

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

solution of fractional kinetic equation by a class of integral transform of pathway type solutions of fractional kinetic equations are obtained through an integral transform named p transform introduced in this paper the p transform is a binomial type transform containing many class of transforms including the well known laplace transform the paper is motivated by the idea of pathway model introduced by mathai linear algebra appl 396 317 328 2005 10 1016 j laa 2004 09 022 the composition of the transform with differential and integral operators are proved along with convolution theorem as an illustration of applications to the general theory of differential equations a simple differential equation is solved by the new transform being a new transform the p transform of some elementary functions as well as some generalized special functions such as h function g function wright generalized hypergeometric function generalized hypergeometric function and mittag leffler function are also obtained the results for the classical laplace transform is retrieved by letting 1

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

zeroth law compatibility of nonadditive thermodynamics nonextensive thermodynamics is criticized by the statement that the zeroth law cannot be satisfied with nonadditive composition rules corrected in this paper we determine the general functional form of those nonadditive composition rules that are compatible with the zeroth law of thermodynamics we find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these our result offers a possible solution to the long standing questions about equilibrium between extensive and nonextensive systems or systems with different nonextensivity parameters

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

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

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

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

irreversibility entropy and incomplete information abstract the open system has been proved to be a system with perfect accessibility represented as a probability space in which is defined a p a measure but the p a measure is not yet known consequently it is difficult to develop the statistical thermodynamics for an irreversible system here its integral expression is obtained in order to its use in the statistical thermodynamic analysis of the complex and irreversible systems

third law of thermodynamics as a key test of generalized entropies the laws of thermodynamics constrain the formulation of statistical mechanics at the microscopic level the third law of thermodynamics states that the entropy must vanish at absolute zero temperature for systems with nondegenerate ground states in equilibrium conversely the entropy can vanish only at absolute zero temperature here we ask whether or not generalized entropies satisfy this fundamental property we propose a direct analytical procedure to test if a generalized entropy satisfies the third law assuming only very general assumptions for the entropy s and energy u of an arbitrary n level classical system mathematically the method relies on exact calculation of ds du in terms of the microstate probabilities pi to illustrate this approach we present exact results for the two best known generalizations of statistical mechanics specifically we study the kaniadakis entropy s which is additive and the tsallis entropy sq which is nonadditive we show that the kaniadakis entropy correctly satisfies the third law only for 1 1 thereby shedding light on why is conventionally restricted to this interval surprisingly however the tsallis entropy violates the third law for q 1 finally we give a concrete example of the power of our proposed method by applying it to a paradigmatic system the one dimensional ferromagnetic ising model with nearest neighbor interactions

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

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

tsallis like information measures and the analysis of complex signals abstract a tsallis like information measure that elaborates upon those of the tsallis type is shown to be a rather powerful tool for the analysis of complex non stationary signals as exemplified by the henon map and eegs the ensuing results could be of importance concerning the automatic detection of epileptic spikes

information gain within nonextensive thermostatistics we discuss the information theoretical foundations of the kullback information gain recently generalized within a nonextensive thermostatistical formalism general properties are studied and in particular a consistent test for measuring the degree of correlation between random variables is proposed in addition minimum entropy distributions are discussed and the h theorem is proved within the generalized context

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

extracting features of gaussian self similar stochastic processes via the bandt pompe approach by recourse to appropriate information theory quantifiers normalized shannon entropy and martin plastino rosso intensive statistical complexity measure we revisit the characterization of gaussian self similar stochastic processes from a bandt pompe viewpoint we show that the ensuing approach exhibits considerable advantages with respect to other treatments in particular clear quantifiers gaps are found in the transition between the continuous processes and their associated noises

on the derivation of power law distributions within classical statistical mechanics far from the thermodynamic limit we show that within classical statistical mechanics without taking the thermodynamic limit the most general boltzmann factor for the canonical ensemble is a q exponential function the only assumption here is that microcanonical distributions have to be separated from the total system energy which is the prerequisite for any sensible measurement we derive that all separable distributions are parametrized by a mathematical separation constant q which can be related to the non extensivity q parameter in tsallis distributions we further demonstrate that nature fixes the separation constant q to 1 for large dimensionality of gibbs phase space our results will be relevant for systems with a low dimensional space for example nanosystems comprised of a small number of particles or for systems with a dimensionally collapsed phase space which might be the case for a large class of complex systems

on sample size of the kruskal wallis test with application to a mouse peritoneal cavity study summary as the nonparametric generalization of the one way analysis of variance model the kruskal wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown although the kruskal wallis test has been widely used for data analysis power and sample size methods for this test have been investigated to a much lesser extent this article proposes new power and sample size calculation methods for the kruskal wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model no assumption is made on the shape of the underlying population distributions simulation results show that in terms of sample size calculation for the kruskal wallis test the proposed methods are more reliable and preferable to some more traditional methods a mouse peritoneal cavity study is used to demonstrate the application of the methods

roundoff induced attractors and reversibility in conservative two dimensional maps we numerically study two conservative two dimensional maps namely the baker map whose lyapunov exponent is known to be positive and a typical one exhibiting a vanishing lyapunov exponent chosen from the generalized shift family of maps introduced by c moore phys rev lett 64 1990 2354 in the context of undecidability we calculate the time evolution of the entropy sq 1 i 1wpiq q 1 s1 sbg i 1wpilnpi we exhibit the dramatic effect introduced by numerical precision indeed in spite of being area preserving maps they present well after the initially concentrated ensemble has spread virtually all over the phase space unexpected pseudo attractors fixed point like for the baker map and more complex structures for the moore map these pseudo attractors and the apparent time partial reversibility they provoke gradually disappear for increasingly large precision in the case of the moore map they are related to zero lebesgue measure effects associated with the frontiers existing in the definition of the map in addition to the above and consistent with the results by v latora and m baranger phys rev lett 82 1999 520 we find that the rate of the far from equilibrium entropy production of baker map numerically coincides with the standard kolmogorov sinai entropy of this strongly chaotic system

optimal sample size planning for the wilcoxon mann whitney test there are many different proposed procedures for sample size planning for the wilcoxon mann whitney test at given type i and type ii error rates and respectively most methods assume very specific models or types of data to simplify calculations eg ordered categorical or metric data location shift alternatives etc we present a unified approach that covers metric data with and without ties count data ordered categorical data and even dichotomous data for that we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following synthetic data approach we evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities then well known relations for the ranks of the data are used for the calculations in addition to computing the necessary sample size n for a fixed allocation proportion t n1 n where n1 is the sample size in the first group and n n1 n2 is the total sample size we provide an interval for the optimal allocation rate t which minimizes the total sample size n it turns out that for certain distributions a balanced design is optimal we give a characterization of such distributions furthermore we show that the optimal choice of t depends on the ratio of the two variances which determine the variance of the wilcoxon mann whitney statistic under the alternative this is different from an optimal sample size allocation in case of the normal distribution model

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