why tsallis statistics abstract it is a remarkable fact that the traditional teaching of thermodynamics as reflected in the textbooks and including the long developments about ensembles and thermodynamic functions is almost entirely about systems in equilibrium the time variable does not enter there is one exception however the single most important item the flagship of the thermodynamic navy the second law is about the irreversibility of the time evolution of systems out of equilibrium this is a bizarre situation to say the least a glaring case of the drunk man looking for his key under the lamp post when he knows that he lost it in the dark part of the street the moment has come for us to go looking in the dark part the behavior of systems as a function of time we have been given a powerful new flashlight chaos theory we should use it there on the formerly dark pavement we can find tsallis statistics

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

incidence of nonextensive thermodynamics in temporal scaling at feigenbaum points abstract recently in phys rev lett 95 2005 140601 grassberger addresses the interesting issue of the applicability of q statistics to the renowned feigenbaum attractor he concludes there is no genuine connection between the dynamics at the critical attractor and the generalized statistics and argues against its usefulness and correctness yet several points are not in line with our current knowledge nor are his interpretations we refer here only to the dynamics on the attractor to point out that a correct reading of recent developments invalidates his basic claim

on the definition of temperature using time averages abstract this paper is a natural continuation of a previous one by the author which was concerned with the foundations of statistical thermodynamics far from equilibrium one of the problems left open in that paper was the correct definition of temperature in the literature temperature is in general defined through the mean kinetic energy of the particles of a given system in this paper instead temperature is defined a la caratheodory the system being coupled to a heat bath and temperature being singled out as the right integrating factor of the exchanged heat as a byproduct the right expression for the entropy is also obtained in particular in the case of a q distribution the entropy turns out to be that of tsallis

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

a counterexample against the lesche stability of a generic entropy functional region des pays de la loire of france 2007 6088 2007 11953 huaqiao university people s republic of china 07bs105 ismans

temporal scaling at feigenbaum points and nonextensive thermodynamics we show that recent claims for the nonstationary behavior of the logistic map at the feigenbaum point based on nonextensive thermodynamics are either incorrect or can be easily deduced from well known properties of the feigenbaum attractor in particular there is no generalized pesin identity for this system the existing attempts at proofs being based on misconceptions about basic notions of ergodic theory in deriving several new scaling laws of the feigenbaum attractor thorough use is made of its detailed structure but there is no obvious connection to nonextensive thermodynamics

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

thermodynamics and time averages abstract for a dynamical system far from equilibrium one has to deal with empirical probabilities defined through time averages and the main problem then is how to formulate an appropriate statistical thermodynamics the common answer is that the standard functional expression of boltzmann gibbs for the entropy should be used the empirical probabilities being substituted for the gibbs measure other functional expressions have been suggested but apparently with no clear mechanical foundation here it is shown how a natural extension of the original procedure employed by gibbs and khinchin in defining entropy with the only proviso of using the empirical probabilities leads to a functional expression for the entropy which is in general different from that of boltzmann gibbs in particular the gibbs entropy is recovered for empirical probabilities of the poisson type while the tsallis entropies are recovered for a deformation of the poisson distribution

separation of trajectories and its relation to entropy for intermittent systems with a zero lyapunov exponent i t where is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points the mean of is determined by the infinite invariant density using semianalytical arguments we calculate the infinite invariant density for the pomeaumanneville map and with it we obtain excellent agreement between numerical simulation and theory we show that is equal to krengel s entropy and to the complexity calculated by the lempel ziv compression algorithm this generalized pesin s identity shows that and krengel s entropy are the natural generalizations of usual lyapunov exponent and entropy for these systems

possible canonical distributions for finite systems with nonadditive energy region des pays de la loire of france 2007 6088 science research fund 07bs105 huaqiao university people s republic of china

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

on the meromorphic solutions of non homogeneous linear differential equations with polynomial coefficients on considere un operateur differentiel lineaire d ordre k l k de la forme l k w w k a k 1 w k 1 a 0 w ou k 2 et a 0 a k 1 sont polynomiaux on etudie les fonctions f entieres ou meromorphes sans zero pour lesquelles on peut avoir une fonction meromorphe a l equation l k w f qui n a pas de zeros

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

local regularity of the navier stokes equations near the curved boundary abstract we present some regularity conditions for suitable weak solutions of the navier stokes equations near the curved boundary of a sufficiently smooth domain our extend the work that was results established near a flat boundary by gustafson kang and tsai 2006 6

on the growth of solutions of a class of higher order linear differential equations with coefficients having the same order abstract in this paper the authors investigate the growth of solutions of a class of higher order linear differential equations f k a k 1 f k 1 a 0 f 0 when most coefficients in the above equations have the same order with each other and obtain some results which improve previous results due to k h kwon k h kwon nonexistence of finite order solutions of certain second order linear differential equations kodai math j 19 1996 378 387 and z x chen z x chen the growth of solutions of the differential equation f e z f q z f 0 sci china ser a 31 2001 775 784 in chinese z x chen on the hyper order of solutions of higher order differential equations chinese ann math ser b 24 2003 501 508 in chinese z x chen on the growth of solutions of a class of higher order differential equations acta math sci ser b 24 2004 52 60 in chinese z x chen c c yang quantitative estimations on the zeros and growth of entire solutions of linear differential equations complex var 42 2000 119 133

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

the possible orders of solutions of linear differential equations with polynomial coefficients we find specific inforrnation about the possible orders of transeendental solutions of equations of the form f n pn l z f n l po z f0 where po z pl z apnl z are polynomials with po z g 0 several examples are given

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

a collocation method to find solutions of linear complex differential equations in circular domains in this study we introduce a collocation approach for solving high order linear complex differential equations in circular domain by using collocation points defined in a circular domain and bessel functions of the first kind this method transforms the linear complex differential equations into a matrix equation the matrix equation corresponds to a system of linear equations with the unknown bessel coefficients proposed method gives the analytic solution when the exact solutions are polynomials numerical examples are given to demonstrate the validity and applicability of the technique and the comparisons are made with existing results the results obtained from the examples demonstrate the efficiency and accuracy of the present work all of the numerical computations have been computed on computer using a code written in matlab

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

q deformed statistical mechanical property in the dynamics of trajectories en route to the feigenbaum attractor we demonstrate that the dynamics toward and within the feigenbaum attractor combine to form a q deformed statistical mechanical construction the rate at which ensemble trajectories converge to the attractor and to the repellor is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor the values of the q indices involved are given by the unimodal map universal constants while the thermodynamic structure is closely related to that formerly developed for multifractals as an essential component in our demonstration we expose in great detail the features of the dynamics of trajectories that either evolve toward the feigenbaum attractor or are captured by its matching repellor the dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period doubling transition to chaos elements in our analysis are the following i the preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops ii the limiting form of this rank structure results in an infinite number of families of well defined phase space gaps in the positions of the feigenbaum attractor or of its repellor iii the gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions iv the power law with log periodic modulation associated with the rate of approach of trajectories toward the attractor and to the repellor is explained in terms of the progression of gap formation v the relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated vi a mean field evaluation of the atypical partition function a thermodynamic interpretation of the time evolution process and a crossover to ordinary exponential statistics are given we make clear the dynamical origin of the anomalous thermodynamic framework existing at the feigenbaum attractor

boundary regularity for the navier stokes equations in a half space weak solutions to the nonstationary navier stokes equations in a half space are locally bounded at the boundary except for a closed set with finite one dimensional hausdorff measure

the production of tsallis entropy in the limit of weak chaos and a new indicator of chaoticity abstract we study the connection between the appearance of a metastable behaviour of weakly chaotic orbits characterized by a constant rate of increase of the tsallis q entropy c tsallis possible generalization of boltzmann gibbs statistics journal of statistical physics 52 1988 479 487 and the solutions of the variational equations of motion for the same orbits we demonstrate that the variational equations yield transient solutions lasting for long time intervals during which the length of deviation vectors of nearby orbits grows in time almost as a power law the associated power exponent can be simply related to the entropic exponent for which the q entropy exhibits a constant rate of increase this analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits that we call average power law exponent aple we compare the aple with other established indicators of the literature in particular we give examples of application of the aple in a a thin separatrix layer of the standard map b the stickiness region around an island of stability in the same map and c the web of resonances of a 4d symplectic map in all these cases we identify weakly chaotic orbits exhibiting the metastable behaviour associated with the tsallis q entropy

a non extensive approach to the entropy of symbolic sequences symbolic sequences with long range correlations are expected to result in a slow regression to a steady state of entropy increase however we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained provided that the extensive entropy of tsallis with entropic index q is adopted thereby resulting in a new form of entropy that we shall refer to as kolmogorov sinai tsallis kst entropy we assume that the same symbols either 1 or 1 are repeated in strings of length l with the probability distribution p l 1 l the numerical evaluation of the kst entropy suggests that at the value 2 a sort of abrupt transition might occur for the values of in the range 1 3 it is argued that this phase transition like property signals the onset of the thermodynamical regime at 2

regularity criteria for suitable weak solutions of the navier stokes equations near the boundary abstract we present some new regularity criteria for suitable weak solutions of the navier stokes equations near the boundary in dimension three we prove that suitable weak solutions are holder continuous up to the boundary provided that the scaled mixed norm l x t p q with 3 p 2 q 2 2 q p q 3 2 is small near the boundary our methods yield new results in the interior case as well partial regularity of weak solutions is also analyzed under some additional integral conditions

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

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

some more universal scaling laws for critical mappings we study such nonlinear mappingsxn 1 f xn bcr of an intervali into itself for which the feigenbaum scaling laws hold i e for which bcr is an accumulation point of bifurcation points letx0 be a random variable with some absolutely continuous distribution ini we show in particular that i the geometric average distance ofxn from the nearest point of the attractor decreases liken 1 93387 ii the geometric average of xn x0 increases liken0 60 iii the geometric mean distance xn yn between the iterates of two close by pointsx0 y0 asymptotically tends towards a value x0 y0 0 77 these and other properties are also borne out from a simple probabilistic model which depicts the evolution as a random walklike process

self similar sequences and universal scaling of dynamical entropies symbol sequences play a prominent role in the context of symbolic dynamics important features of a dynamical system are reflected by related statistics of subsequences a dynamical behavior giving rise to a self similar attractor and universal scaling relations expressed by critical exponents will lead to self similar statistics of subsequences in the present paper we show how self similar distributions of subsequences i e temporal self similarity can be connected with a scaling relation for dynamical entropies moreover the effect of slightly perturbing perfectly self similar sequences by contaminating them with noise is investigated the achieved results are of importance for physical processes marking the borderline between order and chaos textcopyright 1996 the american physical society

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

weak insensitivity to initial conditions at the edge of chaos in the logistic map abstract we extend existing studies of weakly sensitive points within the framework of tsallis non extensive thermodynamics to include weakly insensitive points at the edge of chaos analyzing tangent points of the logistic map we have verified that the generalized entropy with suitable entropic index q correctly describes the approach to the attractor

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

linear differential equations with entire coefficients of p q order in the complex plane in this paper we firstly investigate the complex high order linear differential equations in which the coefficients are entire functions of p q order and obtain some results which improve and generalize some previous results in cao 2009 3 chen and yang 2000 4 heittokangas et al 2006 12 kinnunen 1998 17 tu and yi 2008 20

thermodynamic paths and stochastic order in open systems abstract the theorem of extremum entropy generation is related to the stochastic order of the paths inside the phase space indeed the system evolves from an indistinguishable configuration to another one on the most probable path in relation to the paths stochastic order the result is that at the stationary state the entropy generation is maximal and this maximum value is a consequence of the stochastic order of the paths in the phase space conversely the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation for the open systems at the stationary states

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

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

routes to chaos universality and glass formation abstract we review recent results obtained for the dynamics of incipient chaos these results suggest a common picture underlying the three universal routes to chaos displayed by the prototypical logistic and circle maps namely the period doubling intermittency and quasiperiodicity routes in these situations the dynamical behavior is exactly describable through infinite families of tsallis q exponential functions furthermore the addition of a noise perturbation to the dynamics at the onset of chaos of the logistic map allows to establish parallels with the behavior of supercooled liquids close to glass formation specifically the occurrence of two step relaxation aging with its characteristic scaling property and subdiffusion and arrest is corroborated for such a system

random texts exhibit zipf s law like word frequency distribution it is shown that the distribution of word frequencies for randomly generated texts is very similar to zipf s law observed in natural languages such as english the facts that the frequency of occurrence of a word is almost an inverse power law function of its rank and the exponent of this inverse power law is very close to 1 are largely due to the transformation from the word s length to its rank which stretches an exponential function to a power law function

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

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

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

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

the growth oscillation and fixed points of solutions of complex linear differential equations in the unit disc abstract we consider the complex differential equations of the form a k z f k a k 1 z f k 1 a 1 z f a 0 z f f z where a 0 0 a 1 a k and f are analytic functions in the unit disc d z c z 1 some results on the finite iterated order and the finite iterated convergence exponent of zero points in d of meromorphic analytic solutions are obtained the fixed points of solutions of differential equations are also investigated in this paper

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

stationary open systems a brief review on contemporary theories on irreversibility abstract open systems are very important in science and engineering for their applications and the analysis of the real word at their steady state two apparently opposed principles for their rate of entropy production have been proposed the minimum entropy production rate and the maximum entropy production useful in the analysis of dissipation and irreversibility of different processes in physics chemistry biology and engineering both principles involve an extremum of the rate of the entropy production at the steady state under non equilibrium conditions on the other hand in engineering thermodynamics dissipation and irreversibility are analyzed using the entropy generation for which there exist two principle of extrema too the minimum and the maximum principle finally oppositions to the extrema principle have been developed too in this paper all these extrema principles will be analyzed in order to point out the relations among them and a synthesis useful in engineering applications in physical and chemical process analysis and in biology and biotechnology will be proposed

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