transient measures in the standard map abstract for an area preserving map each chaotic orbit appears numerically to densely cover a region an irregular component of nonzero area surprisingly the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component most prominently there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles we show that this phenomena is transient and therefore numerical ergodicity on the irregular component eventually obtains though the times involved are extremely long 1010 iterates a markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class for example a rotational circle however the density spikes do occur in a markov tree model that includes the effects of islands around islands

application of the sturm liouville theorem and shape invariance formalism to the dirac equation with hyperbolic like potential abstract we use a simple algebraic formalism i e based on the sturm liouville theorem and shape invariance formalism to study the energy spectra for dirac equation with scalar and vector hyperbolic like potentials the rosen morse and eckart potentials as typical models are performed to show the advantage of this method

dimension measurement on high dimensional systems abstract using experimental data from a variety of laboratory electrical systems the upper limits on dimension measurement are examined and it is shown that contrary to standard scaling estimates it can be possible to reliably measure correlation dimensions larger than 10 the practical implications of the constraints on such a measurement are examined including the role of statistical errors geometrical artifacts dimension algorithm metric choice and the process power spectrum these ideas are applied to electrical 1 f noise from both macroscopic and microscopic sources to show that it cannot be explained by a deterministic model with fewer than roughly 20 degrees of freedom

panel data models with multiple time varying individual effects abstract this paper considers a panel data model with time varying individual effects the data are assumed to contain a large number of cross sectional units repeatedly observed over a fixed number of time periods the model has a feature of the fixed effects model in that the effects are assumed to be correlated with the regressors the unobservable individual effects are assumed to have a factor structure for consistent estimation of the model it is important to estimate the true number of individual effects we propose a generalized methods of moments procedure by which both the number of individual effects and the regression coefficients can be consistently estimated some important identification issues are also discussed our simulation results indicate that the proposed methods produce reliable estimates

exact numerical studies of hamiltonian maps iterating without roundoff error abstract for many important hamiltonian maps e g the standard map it is possible to construct related mappings that i carry a lattice into itself ii approach the original map as the lattice spacing is decreased iii can be iterated exactly using integer arithmetic and iv are hamiltonian themselves we compare these lattice maps to maps that use floating point arithmetic to evaluate the original map we discuss the problems associated with roundoff error and we argue that lattice maps are superior to floating point maps for the study of the long term behaviour of hamiltonian dynamical systems

the complexity of finding cycles in periodic functions given a function f over a finite domain d and an arbitrary starting point x the sequence f 0 x f 1 x f 2 x cdots is ultimately periodic such sequences are typically the output of random number generators the cycle problem is to determine the first repeated element f n x in the sequence previous algorithms for this problem have required 3n o 1 operations in this paper we show that n 1 theta 1 sqrt m steps are both necessary and sufficient if m memory cells are available to store values of the function we explicitly consider the performance of the algorithm as a function of the amount of memory available and the relative cost of evaluating f and comparing sequence elements for equality

another look at the instrumental variable estimation of error components models this article develops a framework for efficient iv estimators of random effects models with information in levels which can accommodate predetermined variables our formulation clarifies the relationship between the existing estimators and the role of transformations in panel data models we characterize the valid transformations for relevant models and show that optimal estimators are invariant to the transformation used to remove individual effects we present an alternative transformation for models with predetermined instruments which preserves the orthogonality among the errors finally we consider models with predetermined variables that have constant correlation with the effects and illustrate their importance with simulations

double precision errors in the logistic map statistical study and dynamical interpretation the nature of the round off errors that occur in the usual double precision computation of the logistic map is studied in detail different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined histograms of errors in trajectories given and for the case of fully developed chaos an explicit formula is found it is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram both jumps and locations seem to present geometric convergence characterized by the two first feigenbaum constants eventually a comparison with benford s law for the distribution of the leading digit of compilation of numbers is discussed

computation of true chaotic orbits using cubic irrationals abstract we introduce a method that enables us to generate long true orbits of discrete time dynamical systems defined by one dimensional piecewise linear fractional maps with integer coefficients the method uses cubic irrationals to represent numbers and involves only integer arithmetic to compute true orbits by applying the method to the bernoulli map and a modified bernoulli map we show that it successfully generates true chaotic and intermittent orbits respectively in contrast with conventional simulation methods we demonstrate through simulations concerning invariant measures and the power spectrum that the statistical properties of the true orbits generated agree well with those of typical orbits of the two maps

an algorithmic view of pseudochaos abstract the controversial concept of quantum chaos the dynamical chaos in bounded mesoscopic quantum systems is presented as the most important and universal instance of a new generic dynamical phenomenon pseudochaos the latter characterizes the irregular behaviour of dynamical systems with discrete energy and or frequency spectrum which include classical systems with discrete phase space the question of randomness is addressed in terms of the algorithmic theory of dynamical systems using the arnold cat map for illustration the relationship between the algorithmic theory and ergodic theory is discussed

comment on iv estimation of panels with factor residuals by d robertson and v sarafidis abstract this comment discusses two issues related to the paper by robertson and sarafidis 2014 hereafter rs the first issue is the potential incidental parameters problem that may arise when data are cross sectionally heteroskedastic the second issue is an identification problem in the estimation of panel models with weakly exogenous instrumental variables

explicit wavefunctions for shape invariant potentials by operator techniques the authors obtain explicit expressions for the wavefunctions of all the known shape invariant potentials by using the operator method

improved lower bounds for the cycle detection problem abstract lower bounds for the cycle detection problem were recently investigated by fich 1981 1983 she showed that floyd s algorithm was optimal among those algorithms which have m 2 memory locations and which make a finite number of jumps a lower bound for the case where m 2 was also presented but the question of whether having more than two memory locations could actually yield a better algorithm was left open in this report we show that it cannot a lower bound was also presented by fich 1981 1983 for algorithms which have two memory locations and which make a finite number of back advances we show here that the same lower bound holds even if the restriction on back advances is dropped

effects of phase space discretization on the long time behavior of dynamical systems abstract we investigate the effects that roundoff errors have on discrete time dynamical systems computer generated orbits terminate in loops whose lengths scale with the machine precision in a universal way we present arguments to explain this fact

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

iv estimation of panels with factor residuals this paper considers panel data regression models with weakly exogenous or endogenous regressors and residuals generated by a multi factor error structure in this case the standard dynamic panel estimators fail to provide consistent estimates of the parameters we propose a new estimation approach based on instrumental variables which retains the traditional attractive features of method of moments estimators one novelty of our approach is that we introduce new parameters to represent the unobserved covariances between the instruments and the factor component of the residual these parameters are typically estimable when n is large some important estimation and identification issues are studied in detail the finite sample performance of the proposed estimators is investigated using simulated data the results show that the method produces reliable estimates of the parameters over various parametrizations and is robust to large values of the autoregressive parameter and or the variance of the factor loadings

noisy integer maps abstract the combined effects of discretization and noise in iterated maps are studied with simulations and a transfer matrix technique the resulting changes in a typical map s invariant distribution are illustrated

properties of fully developed chaos in one dimensional maps we consider single humped symmetric one dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one to calculate the probability distribution function and in turn the lyapunov exponent and the correlation function a perturbation expansion is developed for the invariant measure besides deriving some general results we treat several examples in detail and compare our theoretical results with recent numerical ones furthermore a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes

analytical approximations to the l wave solutions of the schrodinger equation with the eckart potential the bound state solutions of the schrodinger equation with the eckart potential with the centrifugal term are obtained approximately it is shown that the solutions can be expressed in terms of the generalized hypergeometric functions 2f1 a b c z the intractable normalized wavefunctions are also derived to show the accuracy of our results we calculate the eigenvalues numerically for arbitrary quantum numbers n and l it is found that the results are in good agreement with those obtained by other methods for short range potential large a two special cases for l 0 and 0 are also studied briefly

panel models with interactive effects abstract the multiplication of individual specific effects and time specific effects provides a more general formulation than the traditionally used additive form to capture the unobserved heterogeneity in panel data modeling it is also a useful approach for dimension reduction for modeling cross section dependence however and are unobservable we explore the implications for econometric modeling under various formulations of the interactive effects models and suggest a quasi likelihood approach as a common framework to study issues of estimation and statistical inference when regressors are either strictly exogenous or predetermined and under different combinations of the data size of cross sectional dimension n and time series dimensions t we also suggest some computationally simpler estimation methods in light of the quasi likelihood approach monte carlo studies are conducted to highlight the issues involved

efficient estimation of models for dynamic panel data abstract in this paper we consider a dynamic model for panel data we show that under standard assumptions there are more moment conditions than are currently exploited in the literature some of these are linear but others are quadratic so that nonlinear gmm is required we also show that exogenous regressors generate a larger number of relevant moment conditions in a dynamic model than they would in a static model

geometric integration for particle accelerators this paper is a very personal view of the field of geometric integration in accelerator physics a field where often work of the highest quality is buried in lost technical notes or even not published one has only to think of simon van der meer nobel prize work on stochastic cooling unpublished in any refereed journal so i reconstructed the relevant history of geometrical integration in accelerator physics as much as i could by talking to collaborators and using my own understanding of the field the reader should not be too surprised if this account is somewhere between history science and perhaps even fiction

rigorous chaos verification in discrete dynamical systems abstract in this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems the existence of such points guarantees that a smale horseshoe is embedded in the mapping which therefore is chaotic we take special interest in the 2 dimensional case and describe a method to obtain a computer assisted proof of chaos for these systems numerical results are presented for the standard map