global dynamics in a commodity market model a b s t r a c t we study the global behavior of the price dynamics in a commodity market governed by a balance between demand and supply while the dependence of demand on price is considered instantaneous the supply term contains a delay leading to a delay differential equation a discrete model is naturally defined as a limit case of this equation we provide a thorough study of the discrete case and use these results to get new sufficient conditions for the global convergence of the solutions to the positive equilibrium in the continuous case for when the equilibrium is unstable we provide some bounds for the amplitude of the oscillations that are quite sharp when the delay is large

discrete halanay type inequalities and applications in this paper we derive new discrete halanay type inequalities and show some applications in the investigation of the asymptotic behavior of nonlinear di4erence equations in particular di4erence equations involving the maximum functional are considered 2003 elsevier ltd all rights reserved

benford s law a poisson perspective abstract benford s law is a counterintuitive statistical law asserting that the distribution of leading digits taken from a large ensemble of positive numerical values that range over many orders of scale is logarithmic rather than uniform as intuition suggests in this paper we explore benford s law from a poisson perspective considering ensembles of positive numerical values governed by poisson process statistics we show that this poisson setting naturally accommodates benford s law and i establish a poisson characterization and a poisson multidigit extension of benford s law ii study a system invariant leading digit distribution which generalizes benford s law and establish a poisson characterization and a poisson multidigit extension of this distribution iii explore the universal emergence of the system invariant leading digit distribution couple this universal emergence to the universal emergence of the weibull and frechet extreme value distributions and distinguish the special role of benford s law in this universal emergence iv study the continued fractions counterpart of the system invariant leading digit distribution and establish a poisson characterization of this distribution and v unveil the elemental connection between the system invariant leading digit distribution and its continued fractions counterpart this paper presents a panoramic poisson approach to benford s law to its system invariant generalization and to its continued fractions counterpart

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 significant digit law in statistical physics abstract the occurrence of the nonzero leftmost digit i e 1 2 9 of numbers from many real world sources is not uniformly distributed as one might naively expect but instead the nature favors smaller ones according to a logarithmic distribution named benford s law we investigate three kinds of widely used physical statistics i e the boltzmann gibbs bg distribution the fermi dirac fd distribution and the bose einstein be distribution and find that the bg and fd distributions both fluctuate slightly in a periodic manner around benford s distribution with respect to the temperature of the system while the be distribution conforms to it exactly whatever the temperature is thus benford s law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature furthermore various elegant properties of benford s law especially the mantissa distribution of data sets are discussed

the essential histogram the histogram is widely used as a simple exploratory display of data but it is usually not clear how to choose the number and size of bins we construct a confidence set of distribution functions that optimally address the two main tasks of the histogram estimating probabilities and detecting features such as increases and modes in the distribution we define the essential histogram as the histogram in the confidence set with the fewest bins thus the essential histogram is the simplest visualization of the data that optimally achieves the main tasks of the histogram the only assumption we make is that the data are independent and identically distributed we provide a fast algorithm for the essential histogram and illustrate our methodology with examples an r package is available on cran

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

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

benford s law and the fsd distribution of economic behavioral micro data abstract in this paper we focus on the first significant digit fsd distribution of european micro income data and use information theoretic entropy based methods to investigate the degree to which benford s fsd law is consistent with the nature of these economic behavioral systems we demonstrate that benford s law is not an empirical phenomenon that occurs only in important distributions in physical statistics but that it also arises in self organizing dynamic economic behavioral systems the empirical likelihood member of the minimum divergence entropy family is used to recover country based income fsd probability density functions and to demonstrate the implications of using a benford prior reference distribution in economic behavioral system information recovery

on stochastic complexity and nonparametric density estimation summary we use the concepts of stochastic complexity description length and model selection to develop data based methods for choosing smoothing parameters in nonparametric density estimation in the case of histogram estimators we derive a simple exact formula for stochastic complexity when the prior distribution of cell probabilities is uniform over the class of all possible choices the formula depends only on the data and the smoothing parameter which is readily chosen according to the criterion of minimum stochastic complexity approaches based on stochastic complexity and description length are shown to be asymptotically equivalent in certain circumstances they produce a degree of smoothing which is almost optimal from the viewpoint of minimizing l or supremum distance but which smooths a little more than is optimal in the sense of minimizing l distance for any finite value of r

on some algebraic difference equations un 2un ψ un 1 in r related to families of conics or cubics generalization of the lyness sequences abstract in this paper and in a forthcoming one we study difference equations in r of the types 2 u n 2 u n a b u n 1 u n 1 2 4 u n 2 u n a b u n 1 c u n 1 2 c u n 1 6 u n 2 u n a b u n 1 c u n 1 2 c d u n 1 u n 1 2 which are linked to families of conics cubics and quartics respectively these equations generalize lyness one u n 2 u n a u n 1 studied in several papers whose behavior was completely elucidated in g bastien m rogalski in press through methods which are transposed in the present paper for the study of 1 and 2 and in the forthcoming one for 3 in particular we prove in the present paper a form of chaotic behavior for solutions of difference equations 1 and 2 and find all the possible periods for these solutions

benford s law and complex atomic spectra we found that in transition arrays of complex atomic spectra the strengths of electric dipolar lines obey benford s law which means that their significant digits follow a logarithmic distribution favoring the smallest values this indicates that atomic processes result from the superposition of uncorrelated probability laws and that the occurrence of digits reflects the constraints induced by the selection rules furthermore benford law can be a useful test of theoretical spectroscopic models its applicability to the statistics of electric dipolar lines can be understood in the framework of random matrix theory and is consistent with the porter thomas law

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

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

global stability of periodic orbits of non autonomous difference equations and population biology elaydi and yakubu showed that a globally asymptotically stable gas periodic orbit in an autonomous dierence equation must in fact be a fixed point whenever the phase space is connected in this paper we extend this result to periodic nonautonomous dierence equations via the concept of skew product dynamical systems we show that for a k periodic dierence equation if a periodic orbit of period r is gas then r must be a divisor of k in particular sub harmonic or long periodic oscillations cannot occur moreover if r divides k we construct a non autonomous dynamical system having minimum period k and which has a gas periodic orbit with minimum period r our methods are then applied to prove a conjecture by j cushing and s henson concerning a non autonomous beverton holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates

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

global asymptotic stability for two recursive difference equations two sufficient conditions are obtained for the global asymptotic stability of the following two recursive difference equationsx n 1 x nx n 1 x n 2 ax n x n 1x n 2 a n 0 1 2 andx n 1 x n 1 x nx n 2 ax nx n 1 x n 2 a n 0 1 2 where a 0 and the initial values x 2 x 1 x 0 0 some known results are included

akaike s information criterion and the histogram summary by interpreting the histogram as a step function we explore the use of akaike s information criterion in an automatic procedure to determine the histogram class width we obtain an asymptotic relationship and present some results from a small simulation study

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

on the validation of the newcomb benford law and the weibull distribution in neuromuscular transmission abstract the neuromuscular junction represents a relevant substrate for revealing important biophysical mechanisms of synaptic transmission in this context calcium ions are important in the synapse machinery providing the nerve impulse transmission to the muscle fiber in this work we carefully investigated whether intervals of spontaneous electrical activity recorded in seven different calcium concentrations conform to the newcomb benford law our analysis revealed that electrical discharge of neuromuscular junction yields the expected values for newcomb benford law for first and second digits for different calcium concentrations on the other hand first two digits archived conformity especially for concentrations much above the physiological level we next examined previous theoretical studies establishing a relation between the law and lognormal and weibull distributions we showed that the weibull distribution is more appropriate to fit the intervals as compared to the lognormal distribution altogether the present findings strongly suggest that the electrical discharge dynamics is characterized by base invariance and scale invariance being the time intervals also well described by weibull statistics

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

dichotomy of a perturbed lyness difference equation abstract we investigate in this paper the perturbed lyness difference equation bx n 2 x n x n 1 x n 2 n 0 1 2 where b are arbitrary positive real numbers and 0 and the initial values x 1 x 0 0 which is a generalization of the lyness difference equation x n 2 x n a x n 1 extensively studied it is known that for the lyness difference equation i e the perturbed lyness difference equation with 0 all its solutions are periodic or strictly oscillatory however one here finds that this perturbed lyness difference equation possesses the following dichotomy for 0 b all of its solutions are globally asymptotically stable for b all the sequences generated by it converge to we hence find that there exists the essential difference for the properties of solutions between the unperturbed lyness difference equation and the perturbed lyness difference equation

zeeman s monotonicity conjecture abstract in this paper we prove a conjecture of zeeman 1 about the monotonicity of the rotation number of a family of diffeomorphisms of the first quadrant q of r 2

on global asymptotic stability of nonlinear higher order difference equations in this paper we generalize the main theorem of liz and ferreiro e liz j b ferreiro a note on the global stability of generalized difference equations appl math lett 15 2002 655 659 and some other global stability results for nonautonomous higher order difference equations to the case when contraction type steps are incorporated together with the steps when the difference sequence can increase the relation to the q method for discretization of delay equations is discussed and some sufficient stability conditions for the numerical scheme are deduced several examples are presented to demonstrate sharpness and applications of the results

akaike s information criterion and kullback leibler loss for histogram density estimation we take a detailed look at akaike s information criterion aic and kullback leibler cross validation klcv in the problem of histogram density estimation two different definitions of number of unknown parameters inaic are considered a careful description is given of the influence of density tail properties on performance of both types ofaic and onklcv a number of practical conclusions emerge in particular we find thataic will often give problems when used with heavy tailed unbounded densities but can perform quite well with compactly supported densities in the latter case both types ofaic produce similar results and those results will sometimes be asymptotically equivalent to the ones obtained fromklcv however depending on the shape of the true density theklcv method can fail to balance bias and variance components of loss with the result thatklcv andaic may produce very different results

first digit law in nonextensive statistics nonextensive statistics characterized by a nonextensive parameter q is a promising and practically useful generalization of the boltzmann statistics to describe power law behaviors from physical and social observations we here explore the unevenness of the first digit distribution of nonextensive statistics analytically and numerically we find that the first digit distribution follows benford s law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature the fluctuation decreases when q increases and the result converges to benford s law exactly as q approaches 2 the relevant regularities between nonextensive statistics and benford s law are also presented and discussed