universal positive mass theorems in this paper we develop a general study of contributions at infinity of bochner weitzenbock type formulas on asymptotically flat manifolds inspired by witten s proof of the positive mass theorem as an application we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied

rescaling entropy and divergence rates based on rescaling by some suitable sequence instead of the number of time units the usual notion of divergence rate is here extended to define and determine meaningful generalized divergence rates rescaling entropy rates appears as a special case suitable rescaling is naturally induced by the asymptotic behavior of the marginal divergences closed form formulas are obtained as soon as the marginal divergences behave like powers of some analytical functions a wide class of countable markov chains is proved to satisfy this property most divergence and entropy functionals defined in the literature are concerned e g the classical shannon kullback leibler renyi and tsallis for illustration purposes ferreri or basu harris hjort jones among others are also considered

un theoreme de la masse positive pour le probleme de yamabe en dimension paire soit m g une variete compacte conformement plate de dimension n geq4 et de courbure scalaire strictement positive selon une theoreme de la masse positive du a schoen et yau le terme constant dans le developpement de la fonction de green du laplacien conforme est strictement positif quand m g n est pas conforme a la sphere ronde sur les varietes spin ammann et humbert en ont donne une demonstration elementaire basee sur une preuve de witten en utilisant les formes differentielles au lieu des spineurs nous en donnons une demonstration elementaire sur les varietes de dimension paire sans autre hypothese sur la topologie

comparing entropy rates on finite and infinite rooted trees we consider denumerable stochastic processes with or without memory their evolution is encoded by a finite or infinite rooted tree the main goal is to compare the entropy rates of a given base process and a second one to be considered as a perturbation of the former the processes are described by probability measures on the boundary of the given tree and by corresponding forward transition probabilities at the inner nodes the comparison is in terms of kullback leibler divergence we elaborate and extend ideas and results of bocherer and amjad our extensions involve length functions on the edges of the tree as well as nodes with countably many successors in particular in section v we consider trees with infinite nonbacktracking paths and random perturbations of a given process

computation and estimation of generalized entropy rates for denumerable markov chains we study entropy rates of random sequences for general entropy functionals including the classical shannon and renyi entropies and the more recent tsallis and sharma mittal ones in the first part we obtain an explicit formula for the entropy rate for a large class of entropy functionals as soon as the process satisfies a regularity property known in dynamical systems theory as the quasi power property independent and identically distributed sequence of random variables naturally satisfy this property markov chains are proven to satisfy it too under simple explicit conditions on their transition probabilities all the entropy rates under study are thus shown to be either infinite or zero except at a threshold where they are equal to shannon or renyi entropy rates up to a multiplicative constant in the second part we focus on the estimation of the marginal generalized entropy and entropy rate for parametric markov chains estimators with good asymptotic properties are built through a plug in procedure using a maximum likelihood estimation of the parameter

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

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

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

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

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

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

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

on computing the 2 d fft a a new implementation of the two dimensional fft 2 d fft is proposed compared with the usual separable solution the new realization of the 2 d fft has reduced arithmetic complexity computational savings are achieved because the 2 d case enables after some modifications of the basic separable algorithm scaling and inverse scaling of butterfly operators the new improvement is also applied to other 2 d transforms dct iv dct and lapped transforms

interpretations of renyi entropies and divergences abstract in this paper a new operational definition of renyi divergence and renyi entropy is presented other operational definitions are mentioned

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

a note on bounded entropies abstract the aim of the paper is to study the link between non additivity of some entropies and their boundedness we propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group this allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the renyi entropies including the particular case of the shannon entropy as a particular example we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity we show that tsallis havrda charvat entropy is included in the family of entropies we define finally a link is made between the approach developed in the paper and the theory of deformed logarithms

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

fast unified computation of the multi dimensional discrete sinusoidal transforms based on a one to one index mapping approach conversions of two types of m dimensional discrete w transforms into the multiple sums involving a number of one dimensional discrete w transforms 1 d dwts are first presented a unified fast m 1 dimensional polynomial transform m 1 d pt and 1 d dwts algorithm for computing these two types of m d dwts are then proposed by revealing the relationships among the m dimensional discrete w transforms m d dwts the m dimensional generalized discrete fourier transforms m d gdfts the m dimensional discrete cosine transforms m d dcts and the m dimensional discrete sine transforms m d dsts the generalized fast algorithms for 14 types of the m d discrete sinusoidal transforms including four types of m d dwts and the same number of types of m d gdfts and three types of m d dcts and the same number of types of m d dsts are presented the number of multiplications for all 14 types of the m d discrete sinusoidal transforms needed by the proposed algorithm is only 1 m times that of the widely used corresponding row column methods the number of additions required by the proposed algorithm is also reduced considerably finally with the help of the software platform visual c 6 0 for windows 98 the computation time comparisons between the proposed and the reported algorithms are given the numerical experiments show that the proposed algorithms are not only highly efficient but are also very simple in computational structure

information theoretical properties of tsallis entropies a chain rule and a subadditivity for the entropy of type which is one of the nonadditive entropies were derived by daroczy in this paper we study the further relations among tsallis type entropies which are typical nonadditive entropies the chain rule is generalized by showing it for tsallis relative entropy and the nonadditive entropy we show some inequalities related to tsallis entropies especially the strong subadditivity for tsallis type entropies and the subadditivity for the nonadditive entropies the subadditivity and the strong subadditivity naturally lead to define tsallis mutual entropy and tsallis conditional mutual entropy respectively and then we show again chain rules for tsallis mutual entropies we give properties of entropic distances in terms of tsallis entropies finally we show parametrically extended results based on information theory