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 system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

diagonal matrix solutions of a discrete time lyapunov inequality diagonal solutions of a lyapunov inequality for companion matrices are studied such solutions are required if states of a discrete time linear system are computed with a finite precision arithmetic

the group inverse of the transformation j x ax xb abstract the group inverse j of the sylvester transformation j x ax xb is provided that it exists represented in polynomial form j y ij c ij a i yb j in terms of the minimal polynomials of a and b in one of the representations the coefficient matrix c ij is expressed with the help of a new class of derived hankel matrices

nonextensive foundation of levy distributions a deep connection between the ubiquity of l evy distributions in nature and the nonextensive thermal statistics introduced a decade ago has been established recently tsallis et al phys rev lett 75 3589 1995 by using unnormalized q expectation values it has just been argued on physical grounds that normalized q expectation values should be used instead we revisit within this more appropriate scheme the l evy problem and verify that the relevant analytic results become sensibly simplified whereas the basic physics remains unchanged

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

computing a non maxwellian velocity distribution from first principles we investigate a family of single particle anomalous velocity distribution by solving a particular class of stochastic liouville equations the stationary state is obtained analytically and the maxwell boltzmann distribution is reobtained in a particular limit we discuss the comparison with other different methods to obtain the stationary state extensions when the models cannot be solved in an exact way are also pointed out in connection with the one ficton approximation

generalized lyapunov equations matrices with displacement structure and generalized bezoutians abstract matrix equations of the form i 0 r j 0 s f ij a i xb j c are consider ed it is shown that solving such equations can be reduced to finding the solutions of certain well chosen auxiliary equations the latter equations are essentially simpler than the original equation and their solutions are matrices with a displacement structure or connected with generalized bezoutians in the case r s 1 the approaches given here along with some results on structured matrices and generalized bezoutians lead to explicit formulas for the solution

accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes abstract the von neumann halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces we propose acceleration schemes making use of two ideas firstly each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection and secondly it is easy to project onto a finite intersection of such hyperplanes we give conditions for which our accelerations converge strongly finally we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem

toward solution of matrix equation x af x b c abstract this paper studies the solvability existence of unique solution closed form solution and numerical solution of matrix equation x af x b c with f x x t f x x and f x x h where x is the unknown it is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard stein equations in the form of w a w b c where the dimensions of the coefficient matrices a b and c are the same as those of the original equation closed form solutions of equation x af x b c can then be obtained by utilizing standard results on the standard stein equation on the other hand some generalized stein iterations and accelerated stein iterations are proposed to obtain numerical solutions of equation x af x b c necessary and sufficient conditions are established to guarantee the convergence of the iterations

tsallis entropy production for diffusion on the diluted hypercube abstract we investigate the production of tsallis entropy for the probability distribution of random walkers on a hypercube with a fraction p of allowed sites we consider two regimes p 1 and from p 1 2 until the percolation limit pc in the first regime we show that the entropy rate can be obtained from the boltzmann gibbs shannon entropy time evolution on the second regime the same numerical experiments indicate a trivial limit for this quantity while by measuring the production of tsallis entropy for different q values there exists a particular value for the entropic index q that adequately describe the relaxation these results are compared to similar numerical experiments performed with non linear maps and indicate an analogy between the phase space of both systems

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

the matrix equation axb gxd e over the quaternion field abstract using the method of complex representation this paper obtains necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation axb gxd e over the quaternion field extends w e roth s theorem and a jameson s theorem to the quaternion field and gives the representation of the solution

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

constructing a statistical mechanics for beck cohen superstatistics the basic aspects of both boltzmann gibbs bg and nonextensive statistical mechanics can be seen through three different stages first the proposal of an entropic functional s bg k sigma i p i ln p i for the bg formalism with the appropriate constraints sigma i p i 1 and sigma i p i e i u for the bg canonical ensemble second through optimization the equilibrium or stationary state distribution p i e betae i z bg with z bg sigma j e betae j for bg third the connection to thermodynamics e g f bg 1 beta ln z bg and u bg partial differential partial differential beta ln z bg assuming temperature fluctuations beck and cohen recently proposed a generalized boltzmann factor b e integral infinity 0 dbetaf beta e betae this corresponds to the second stage described above in this paper we solve the corresponding first stage i e we present an entropic functional and its associated constraints which lead precisely to b e we illustrate with all six admissible examples given by beck and cohen

on the jordan form of a family of linear mappings abstract let a and b be square matrices over a field f having their eigenvalues and in f and let g x y r sarsxrys be a polynomial over f assuming the jordan forms of a and b to be known the jordan form of r sarsar bs is determined when the partial derivatives g x and g y are nonzero at thus generalizing the classical cases g x y x y and g x y xy the particular cases g x y xk yl and g x y xkyl are also generalized by using properties of the partial hasse derivatives of g at the case where a and b are 2 2 or 3 3 matrices g being arbitrary is discussed exhaustively

two generalizations of dykstra s cyclic projections algorithm dykstra s cyclic projections algorithm allows one to compute best approximations to any pointx in a hilbert space from the intersectionc lrci of a finite number of closed convex setsci by reducing it to a sequence of best approximation problems from theindividual setsci here we present two generalizations of this algorithm first we allow the number of setsci to beinfinite rather than finite secondly we allow arandom rather than cyclic ordering of the setsci

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

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

two dimensional maps at the edge of chaos numerical results for the henon map the mixing properties or sensitivity to initial conditions of the two dimensional henon map have been explored numerically at the edge of chaos three independent methods which have been developed and used so far for one dimensional maps have been used to accomplish this task these methods are i the measure of the divergence of initially nearby orbits ii analysis of the multifractal spectrum and iii computation of nonextensive entropy increase rates the results obtained closely agree with those of the one dimensional cases and constitute a verification of this scenario in two dimensional maps this obviously makes the idea of weak chaos even more robust

rates of convergence of non extensive statistical distributions to levy distributions in full and half spaces the levy type distributions are derived using the principle of maximum tsallis non extensive entropy both in the full and half spaces the rates of convergence to the exact levy stable distributions are determined by taking the n fold convolutions of these distributions the marked difference between the problems in the full and half spaces is elucidated analytically it is found that the rates of convergence depend on the ranges of the levy indices an important result emerging from the present analysis is deduced if interpreted in terms of random walks implying the dependence of the asymptotic long time behaviours of the walks on the ranges of the levy indices if n is identified with the total time of the walks

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

algebraic methods for the solution of some linear matrix equations abstract an algebraic viewpoint permits the formulation of necessary and sufficient conditions for the existence of a unique solution to some linear matrix equations the theory developed is then used to express the solution in a finite series form and to prove a stability theorem

geometry of escort distributions given an original distribution its statistical and probabilistic attributes may be scanned using the associated escort distribution introduced by beck and schlogl and employed in the formulation of nonextensive statistical mechanics here the geometric structure of the one parameter family of the escort distributions is studied based on the kullback leibler divergence and the relevant fisher metric it is shown that the fisher metric is given in terms of the generalized bit variance which measures fluctuations of the crowding index of a multifractal the cramer rao inequality leads to a fundamental limit for the precision of the statistical estimate of the order of the escort distribution we also show quantitatively that it is inappropriate to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics

an algorithm for restricted least squares regression abstract a commonly occurring problem in statistics is that of minimizing a least squares expression subject to side constraints here a simple iterative algorithm is presented and shown to converge to the desired solution several examples are presented including finding the closest concave convex function to a set of points and other general quadratic programming problems the dual problem to the basic problem is also discussed and a solution for it is given in terms of the algorithm finally extensions to expressions other than least squares are given

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

accelerating the convergence of the method of alternating projections the powerful von neumann halperin method of alternating projections map is an algorithm for determining the best approximation to any given point in a hilbert space from the intersection of a finite number of subspaces it achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the it individual subspaces which make up the intersection the main practical drawback of this algorithm at least for some applications is that the method is slowly convergent in this paper we consider a general class of iterative methods which includes the map as a special case for such methods we study an accelerated version of this algorithm which was considered earlier by gubin polyak and raik 14 and by gearhart and koshy 13 we show that the accelerated algorithm converges faster than the map in the case of two subspaces but is in general it not faster than the map for more than two subspaces however for a symmetric version of the map the accelerated algorithm always converges faster for any number of subspaces our proof seems to require the use of the spectral theorem for self adjoint mappings

acceleration schemes for the method of alternating projections abstract in this paper we develop and analyze schemes for accelerating the convergence of the alternating projections method for finding the projection of a point onto the intersection of a finite number of subspaces of a hilbert space a special method for the case of two subspaces is developed and two general schemes are determined for the case of an arbitrary number of subspaces proofs of convergence and acceleration are given for each method finally techniques for constructing test problems are outlined and the results of computational experiments are reported

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

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

linear matrix equations the module theoretic approach abstract the matrix equation fikaixbk c is discussed based on the algebraic approach of djaferis and mitter

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

set intersection problems supporting hyperplanes and quadratic programming we study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem for the problem of finding the closest point in the intersection of closed convex sets we propose an algorithm that like dykstra s algorithm converges strongly in a hilbert space moreover this algorithm converges in finitely many iterations when the closed convex sets are cones in mathbb r n r n satisfying an alignment condition next we propose modifications of the alternating projection algorithm and prove its convergence the algorithm converges superlinearly in mathbb r n r n under some nice conditions under a conical condition the convergence can be finite lastly we discuss the case where the intersection of the sets is empty