the barrier transmission problem treated by the amplitude phase method and expressed in terms of an invariant of the ermakov lewis type transmission and reflection of a quantal particle by a single hump potential barrier are analysed by means of an amplitude phase decomposition of the wavefunction on both sides of the barrier the amplitude phase analysis of the wavefunction provides a particular invariant of the ermakov lewis type which originates in the matching process the transmission and reflection coefficients turn out to be simple functions of this invariant numerical calculations of the invariant for an eckart epstein potential barrier provide very accurate results

simultaneous confidence bands in spectral density estimation we propose a method for the construction of simultaneous confidence bands for a smoothed version of the spectral density of a gaussian process based on nonparametric kernel estimators obtained by smoothing the periodogram a studentized statistic is used to determine the width of the band at each frequency and a frequency domain bootstrap approach is employed to estimate the distribution of the supremum of this statistic over all frequencies we prove by means of strong approximations that the bootstrap estimates consistently the distribution of the supremum deviation of interest and consequently that the proposed confidence bands achieve asymptotically the desired simultaneous coverage probability the behaviour of our method in finite sample situations is investigated by simulations and a real life data example demonstrates its applicability in time series analysis

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

automatic estimation of multivariate spectra via smoothing splines summary the classical method for estimating the spectral density of a multivariate time series is first to calculate the periodogram and then to smooth it to obtain a consistent estimator typically to ensure the estimate is positive definite all the elements of the periodogram are smoothed the same way there are however many situations for which different components of the spectral matrix have different degrees of smoothness we propose a bayesian approach that uses markov chain monte carlo techniques to fit smoothing splines to each component real and imaginary of the cholesky decomposition of the periodogram matrix the spectral estimator is then obtained by reconstructing the spectral estimator from the smoothed cholesky decomposition components our technique produces an automatically smoothed spectral matrix estimator along with samples from the posterior distributions of the parameters to facilitate inference

asymptotic simultaneous confidence bands for vector autoregressive spectra asymptotic simultaneous confidence bands for all in 0 0 5 are obtained for each scalar function in a spectral density matrix as well as for the logarithm of the autospectra the squared coherence and the phase spectra the bands are illustrated using a simulation study and surface elevation data

the q operator for root of unity symmetry in the six vertex model we construct the explicit q operator incorporated with the sl2 loop algebra symmetry of the six vertex model at roots of unity the functional relations involving the q operator the six vertex transfer matrix and fusion matrices are derived from the bethe equation parallel to the onsager algebra symmetry discussion in the superintegrable n state chiral potts model we show that the whole set of functional equations is valid for the q operator direct calculations in certain cases are also given here for clearer illustration about the nature of the q operator in the symmetry study of root of unity six vertex model from the functional relation aspect

data driven choice of a spectrum estimate extending the applicability of cross validation methods abstract i develop methods of objectively choosing a spectrum estimate from a general class c of available estimates c can for example simultaneously include blackman tukey and autoregressive estimates so the statistician no longer needs to choose one type or the other arbitrarily the methods work by extending the applicability of existing cross validatory techniques through the introduction of generalized leave out one spectrum estimates as special cases i obtain new objective smoothness parameter selection methods for both autoregressive and blackman tukey estimates in a monte carlo study i demonstrate the effectiveness of the particular methods that result from generalizing wahba s cvmse

a necessary and sufficient condition for asymptotic independence of discrete fourier transforms under short and long range dependence let x t be a stationary time series and let d t denote the discrete fourier transform dft of x 0 x t 1 with a data taper the main results of this paper provide a characterization of asymptotic independence of the dfts in terms of the distance between their arguments under both short and long range dependence of the process x t further asymptotic joint distributions of the dfts d t 1t and d t 2t are also established for the cases t 1t 2t o 1 as t asymptotically close ordinates and t 1t 2t as t asymptotically distant ordinates some implications of the main results on the estimation of the index of dependence are also discussed

functional relations for the order parameters of the chiral potts model low temperature expansions abstract this is the third in a series of papers in which we set up and discuss the functional relations for the split rapidity line correlation function in the n state chiral potts model the order parameters of the model can be obtained from this function here we consider the case n 3 and write the equations explicitly in terms of the hyperelliptic functions parametrization we also present four term low temperature series expansions which we hope will cast light on the analyticity properties needed to solve the relations the problem remains unsolved but we hope that this will prove to be a step in the right direction

on regge pole trajectories for a rational function approximation of thomas fermi potentials the positions of two regge poles caused by the rational function approximate thomas fermi potential a screened attractive coulomb potential are studied in detail as a function of the scattering e

generation of converging regge pole bounds for arbitrary rational fraction scattering potentials we expand upon the regge pole bounding formalism previously presented by handy and msezane 2001 j phys a math gen 34 l531 by analysing in greater detail several important singular potentials including the v r 2 r3 potential the polarization potential v r 2 r4 and the lennard jones class v r 2 rm 2 rn for m n 6 4 12 6 with respect to the polarization potential our bounds enable one to assess the accuracy of the turning point based asymptotic formula of avdonina et al avdonina n b belov s felfli z msezane a z and naboko s n 2001 cau preprint 2002 phys rev a 65 at press we also extend the method to the non singular coulomb case veff r 1 r l l 1 r2 which represents a more challenging class of problems within the present bounding formalism due to the nature of the underlying boundary conditions

the ermakov lewis invariants for coupled linear oscillators we consider n coupled linear oscillators with time dependent coefficients an exact complex amplitude real phase decomposition of the oscillatory motion is constructed this decomposition is further used to derive n exact constants of motion which generalize the so called ermakov lewis invariant of a single oscillator in the floquet problem of periodic oscillator coefficients we discuss the existence of periodic complex amplitude functions in terms of existing floquet solutions

bayesian estimation of the spectral density of a time series this article describes a bayesian approach to estimating the spectral density of a stationary time series a nonparametric prior on the spectral density is described through bernstein polynomials because the actual likelihood is very complicated a pseudoposterior distribution is obtained by updating the prior using the whittle likelihood a markov chain monte carlo algorithm for sampling from this posterior distribution is described that is used for computing the posterior mean variance and other statistics a consistency result is established for this pseudoposterior distribution that holds for a short memory gaussian time series and under some conditions on the prior to prove this asymptotic result a general consistency theorem of schwartz is extended for a triangular array of independent nonidentically distributed observations this extension is also of independent interest a simulation study is conducted to compare the proposed method with some existing methods the method is illustrated with

coupled radial schrodinger equations written as dirac type equations application to an amplitude phase approach the classical amplitude phase method due to milne wilson young and wheeler in the 1930s is known to be a powerful computational tool for determining phase shifts and energy eigenvalues in cases where a sufficiently slowly varying amplitude function can be found the key for the efficient computations is that the original single state radial schr dinger equation is transformed to a nonlinear equation the milne equation such an equation has solutions that may or may not oscillate depending on boundary conditions which then requires a robust recipe for locating the optimal almost constant solutions for its use in the method for scattering problems the solutions of the amplitude equations always approach constants as the radial distance r tends to infinity and there is no problem locating the optimal amplitude functions from a low order semiclassical approximation in the present work the amplitude phase approach is generalized to two coupled schr dinger equations similar to an earlier generalization to radial dirac equations the original scalar amplitude then becomes a vector quantity and the original milne equation is generalized accordingly numerical applications to resonant electron atom scattering are illustrated

a model based approach for estimation of two dimensional maximum entropy power spectra a stochastic model based approach is presented for estimation of the two dimensional maximum entropy power spectrum meps from given finite uniform array data the method consists of fitting an appropriate two dimensional noncausal gaussian markov random field gmrf model to the given data using the maximum likelihood ml technique for parameter estimation the nonlinear criterion function used for ml estimation is similar in structure to the function arising in the deterministic approach of lang and mcclellan the model based approach provides new insights into the two dimensional meps estimation problem for example using the asymptotic normality of ml estimates we derive simultaneous confidence bands for the estimated meps it turns out that when the true correlations are generated by a noncausal gmrf model the two dimensional meps can be obtained by solving linear equations this approach also suggests techniques for realizing two dimensional gmrf models from the given correlation data several numerical examples are given to illustrate the usefulness of the approach

scattering s matrix derived from invariants of the ermakov lewis type an amplitude phase formula for the s matrix due to a central potential is derived the derivation makes use of invariants of the ermakov lewis type

on bootstrapping kernel spectral estimates an approach to bootstrapping kernel spectral density estimates is described which is based on resampling from the periodogram of the original data we show that it is asymptotically valid under suitable conditions and we illustrate its performance for a medium sized time series sample with a small simulation study

the hybrid wild bootstrap for time series we introduce a new and simple bootstrap procedure for general linear processes called the hybrid wild bootstrap the hybrid wild bootstrap generates frequency domain replicates of the periodogram that imitate asymptotically correct the first and second order properties of the ordinary periodogram including its weak dependence structure at different frequencies as a consequence the hybrid wild bootstrapped periodogram succeeds in approximating consistently the distribution of statistics that can be expressed as functionals of the periodogram including the important class of spectral means for which all so far existing frequency domain bootstrap methods generally fail moreover by inverting the hybrid wild bootstrapped discrete fourier transform pseudo observations in the time domain are obtained the generated time domain pseudo observations can be used to approximate correctly the random behavior of statistics the distribution of which depends on the first second and to some extent on the fo

functional relations for the order parameters of the chiral potts model following the method of jimbo miwa and others we obtain functional relations for the order parameters of the chiral potts model we have not yet solved these relations here we discuss their properties and show how one should beware of spurious solutions

bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities summary we propose a general bootstrap procedure to approximate the null distribution of non parametric frequency domain tests about the spectral density matrix of a muitivariate time series under a set of easy to verify conditions we establish asymptotic validity of the bootstrap procedure proposed we apply a version of this procedure together with a new statistic to test the hypothesis that the spectral densities of not necessarily independent time series are equal the test statistic proposed is based on an 2 distance between the non parametrically estimated individual spectral densities and an overall pooled spectral density the latter being obtained by using the whole set of m time series considered the effects of the dependence between the time series on the power behaviour of the test are investigated some simulations are presented and a real life data example is discussed

weak convergence and adaptive peak estimation for spectral densities the spectral density of a stationary time series are proposed and investigated they are based on direct smoothing of the periodogram where the amount of smoothing is determined automatically in an asymptotically optimal fashion these adaptive estimators minimize the asymptotic mean squared error adaptivity is derived from the weak convergence of a two parameter stochastic process in a deviation and a bandwidth coordinate to a gaussian limit process efficient global and local bandwidth choices which lead to adaptive peak estimators and practical aspects are discussed 1 introduction in many instances of spectral analysis of a stationary time series interest focuses on the location of a peak of the spectral density the peak frequency one possibility is to derive estimates of the peak frequency from estimates within a suitable parametric model either in time or frequency domains however parametric assumptions often are not easy to justify or require difficult choices like the determination of the order of an autoregressive model therefore time honored nonparametric methods like direct smoothing of the periodogram and then reading off the desired peak frequency have kept a prominent place in the repertoire of the time series analyst these smoothing methods however depend critically on the degree of smoothing used and this paper addresses the question of how to choose the smoothing parameter for the estimation of peaks in such a way that adaptive peak estimates are obtained since the peak frequency is a local concept the desired bandwidth choice will have to be an inherently local one what happens far away from the peak in question should not unduly influence this bandwidth choice adaptive peak estimation means that employing certain datadependent bandwidths guarantees the same asymptotic distribution of the estimated peak frequency as if the optimal but unknown bandwidths which minimize the asymptotic mean squared error of the peak frequency would have been chosen we consider kernel type estimates for spectral densities which are kernelsmoothed periodograms in contrast to lag window smoothers smoothing of periodograms has been considered for a long time for the purpose of spectral

the inversion relation method for obtaining the free energy of the chiral potts model abstract we derive the free energy of the chiral potts model by the infinite lattice inversion relation method this method is non rigorous in that it always needs appropriate analyticity assumptions guided by previous calculations based on exact finite lattice functional relations we find that in addition to the usual assumption that the free energy be analytic and bounded in some principal domain of the rapidity parameter space that includes the physical regime we also need a much less obvious symmetry we can then obtain the free energy by wiener hopf factorization in the complex planes of appropriate variables together with the inversion relation this symmetry relates the values of the free energy in all neighbouring domains to those in the principal domain

a mean squared error criterion for time series data windows summary a periodogram mean integrated squared error criterion is proposed for assessing the performance of time series data windows or tapers the criterion is evaluated numerically for several known autoregressive processes to determine the optimal amount of tapering in tukey s cosine window it is found that tapering can be extremely beneficial if the spectrum contains sharp peaks also in all cases studied the optimal number of points tapered converges to a constant as the sample size increases this result is at odds with the traditional strategy of tapering a fixed percentage of the data finally a method is proposed for estimating the optimal degree of tapering on the basis of data from an unknown process some monte carlo evidence is presented to demonstrate the effectiveness of the proposed estimation technique

asymptotic spectral theory for nonlinear time series we consider asymptotic problems in spectral analysis of stationary causal processes limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given instead of the commonly used strong mixing conditions in our asymptotic spectral theory we impose conditions only involving conditional moments which are easily verifiable for a variety of nonlinear time series

two dimensional spectrum estimation using noncausal autoregressive models two dimensional 2 d spectrum estimation from raw data is of interest in signal and image processing a parametric technique for spectrum estimation using 2 d noncausal autoregressive ncar models is given the ncar models characterize the statistical dependency of the observation at location s on its neighbors in all directions this modeling assumption reduces the spectrum estimation problem to two subproblems the choice of appropriate structure of the ncar model and the estimation of parameters in ncar models by assuming that the true structure of the ncar model is known we first analyze the existence and uniqueness of gaussian maximum likelihood gml estimates of ncar model parameters due to the noncausal nature of the models the computation of gml estimates is burdensome by assuming specific boundary conditions computationally tractable expressions are obtained for the likelihood function expressions for the asymptotic covariance matrix of the gml estimates as well as the simultaneous confidence bands for the estimated spectrum using gml estimates are derived finally the usefulness of the method is illustrated by computer simulation results

some hyperelliptic function identities that occur in the chiral potts model we present a number of hyperelliptic function identities that occur in the chiral potts model some of these are relevant to the outstanding problem of solving the functional relations for the order parameter

bootstrap technology and applications bootstrap resampling methods have emerged as powerful tools for constructing inferential procedures in modern statistical data analysis although these methods depend on the availability of fast inexpensive computing they offer the potential for highly accurate methods of inference moreover they can even eliminate the need to impose a convenient statistical model that does not have a strong scientific basis in this article we review some bootstrap methods emphasizing applications through illustrations with some real data special attention is given to regression problems with dependent data and choosing tuning parameters for optimal performance

note on invariants for uncoupled ermakov systems several invariants for an ermakov system consisting of a parametric oscillator and a milne oscillator in one dimension are derived this system appears in amplitude phase analysis of parametric oscillator solutions first and second order invariants for such a system are derived using the wronskian relations of a fundamental pair of parametric oscillator solutions in this way wronskian invariants of the parametric oscillator system alone imply ermakov lewis type invariants for the combined ermakov system as well as invariants for the milne oscillator alone

a frequency domain bootstrap for ratio statistics in time series analysis the asymptotic properties of the bootstrap in the frequency domain based on studentized periodogram ordinates are studied it is proved that this bootstrap approximation is valid for ratio statistics such as autocorrelations by using edgeworth expansions it is shown that the bootstrap approximation even outperforms the normal approximation the results carry over to whittle estimates in a simulation study the behavior of the bootstrap is studied for empirical correlations and whittle estimates

empirical likelihood confidence regions in time series models summary this paper develops empirical likelihood in time series models by application of whittle s estimation method one obtains an m estimator of the parameter of a stationary time series from approximately independent observations the periodogram ordinates this estimator is used to obtain an empirical likelihood ratio which is asymptotically distributed as x2 and which can be used to construct confidence regions a procedure for the bartlett correction is also proposed finally small sample properties of the empirical likelihood confidence regions are explored through a simulation

goodness of fit for lattice processes abstract the paper discusses tests for the correct specification of a model when data is observed in a d dimensional lattice extending previous work when the data is collected in the real line as it happens with the latter type of data the asymptotic distributions of the tests are functionals of a gaussian sheet process say b 0 d since it is not easy to find a time transformation h such that b h becomes the standard brownian sheet a consequence is that the critical values are difficult if at all possible to obtain so to overcome the problem of its implementation we propose employing a bootstrap approach showing its validity in our context

the bootstrap applied to power spectral density function estimation summary a shortcoming in the use of autoregressive spectral estimation has been the lack of methods for placing confidence intervals or bands on estimated spectral density the bootstrap method is applied to obtain nonparametric confidence bands and intervals it is the aim of this paper to indicate how this can be done and to report on the outcome of some monte carlo studies of the properties of the resulting procedures some asymptotic theory concerning the properties of the bootstrap confidence band is also discussed

generation of converging regge pole bounds a new formulation of complex rotation quantization we propose an unprecedented bounding theory which generates converging bounds to the regge poles of rational fraction scattering potentials this is made possible by the recent work of handy 2001 j phys a math gen 34 l271 and handy and wang 2001 j phys a math gen 34 8297 which transforms the schrodinger equation into an equivalent fourth order linear differential equation for the probability density this new representation is better suited for numerical considerations since the rapid oscillations of the regge pole wavefunction are factored out more importantly the moments of the probability density can be constrained and thereby the underlying complex angular momentum parameter of the effective potential function through appropriate moment problem theorems as incorporated within the eigenvalue moment method of handy and bessis 1985 phys rev lett 55 931 and handy et al 1988 phys rev lett 60 253

loss of spectral peaks in autoregressive spectral estimation on montre par une analyse asymptotique comprenant les termes du second ordre que les estimateurs des moindres carres doivent etre preferes aux estimateurs de yule walker

a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

chiral potts rapidity curve descended from six vertex model and symmetry group of rapidities in this paper we present a systematical account of the descending procedure from the six vertex model to the n state chiral potts model through fusion relations of j operators following the works of bazhanov stroganov and baxter bazhanov perk a careful analysis of the descending process leads to the appearance of the high genus curve as the rapidity constraint for the chiral potts models full symmetries of the rapidity curve are identified as is its symmetry group structure by normalized transfer matrices of the chiral potts model the 2 t relation can be reduced to functional equations over a hyperelliptic curve associated with rapidities by which the degeneracy of 2 eigenvalues is revealed in the case of the superintegrable chiral potts model

the effect of order estimation on estimating the peak frequency of an autoregressive spectral density summary the effect of estimating the order of an autoregressive process on estimating the peak frequency is examined for consistent order determining methods it is shown that the asymptotic results are the same as for the case when the order is known for the order criteria which asymptotically overestimate the order the peak frequency estimator is shown to be a consistent estimator of the true peak frequency

asymptotic expansions for sums of weakly dependent random vectors it is shown that formal edgeworth expansions are valid for sums of weakly dependent random vectors the error of approximation has ordero n s 2 2 if the strong mixing coefficients in iii are decreasing at an exponential rate the above conditions can easily be checked and are often satisfied when the sequence of random vectors is a gaussian or a markov or an autoregressive process explicit formulas are given for the distribution of finite fourier transforms of a strictly stationary time series

subsampling for heteroskedastic time series abstract in this article a general theory for the construction of confidence intervals or regions in the context of heteroskedastic dependent data is presented the basic idea is to approximate the sampling distribution of a statistic based on the values of the statistic computed over smaller subsets of the data this method was first proposed by politis and romano 1994b for stationary observations we extend their results to heteroskedastic observations and prove a general asymptotic validity result under minimal conditions in contrast the usual bootstrap and moving blocks bootstrap are typically valid only for asymptotically linear statistics and their justification requires a case by case analysis our general asymptotic results are applied to a regression setting with dependent heteroskedastic errors

non oscillating solutions to uncoupled ermakov systems and the semi classical limit the amplitude phase formulation of the schrodinger equation is investigated within the context of uncoupled ermakov systems whereby the amplitude function is given by the auxiliary non linear equation the classical limit of the amplitude and phase functions is analysed by setting up a semi classical ermakov system in this limit it is shown that classical quantities such as the classical probability amplitude and the reduced action are obtained only when the semi classical amplitude and the accumulated phase are non oscillating functions respectively of the space and energy variables conversely among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction only the non oscillating ones yield classical quantities in the limit 0

simultaneous confidence bands for autoregressive spectra summary a shortcoming in the use of autoregressive spectral estimation has been the lack of methods for placing confidence intervals or bands on the estimated spectral density we propose and illustrate a method for determining asymptotic simultaneous confidence bands using scheffe projections

analytic approximation to the harmonic oscillator equation with a sub period time dependent parameter analytic solutions are presented for the time dependent harmonic oscillator equation as well as for the corresponding amplitude ermakov and phase equations the solutions are adequate approximations for an initially constant amplitude and a time dependent parameter that varies continuously in a time much shorter than the characteristic period the amplitude and phase representation of the solution is shown not to be unique the relationship between the phase independent amplitudes is derived from the orthogonal functions invariant

effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density tion and so bias must be corrected by other means when the bootstrap is used to construct confidence intervals for a probability density bias may either be estimated explicitly or allowed for by undersmoothing the curve estimator which of these two approaches is to be preferred in the present paper we address this question from the viewpoint of coverage accuracy assuming a given number of derivatives of the unknown density we conclude that the simpler undersmoothing method is more efficacious undersmoothing also has advantages from the standpoint of minimizing interval width we derive formulae for bandwidths which are optimal in terms of coverage accuracy and also give formulae for the coverage error which results from using those bandwidths 1 introduction the bootstrap can be a notoriously poor estimator of bias in problems of nonparametric curve estimation e g 5 and this fact should be taken into account when using bootstrap methods to construct confidence intervals bias is usually significant when a curve estimator is constructed so as to minimize mean squared error but decreases as the amount of statistical smoothing is reduced therefore one way of alleviating bias is to smooth the curve estimator less than would be appropriate for point estimation as opposed to interval estimation a statistician must either estimate bias explicitly or allow for bias by undersmoothing the curve estimator which of these two methods is to be preferred in this paper we address this problem in the case of nonparametric density estimation and from the viewpoint of coverage error we use the percentile t approach to confidence interval construction and show that in a sense which we shall make precise two paragraphs below the undersmoothing method produces confidence intervals with greater coverage accuracy than those obtained by explicit bias correction l l f x nh h for the density estimator based on kernel k and bandwidth h let b and o2

semi classical analysis and pseudo spectra abstract we prove an approximate spectral theorem for non self adjoint operators and investigate its applications to second order differential operators in the semi classical limit this leads to the construction of a twisted fbi transform we also investigate the connections between pseudo spectra and boundary conditions in the semi classical limit

a bootstrap causality test for covariance stationary processes abstract this paper examines a nonparametric test for granger causality for a vector covariance stationary linear process under possibly the presence of long range dependence we show that the test converges to a nondistribution free multivariate gaussian process say vec b indexed by 0 1 because contrary to the scalar situation it is not possible except in very specific cases to find a time transformation g such that vec b g is a vector with independent brownian motion components it implies that inferences based on vec b will be difficult to implement to circumvent this problem we propose to bootstrapping the test by two alternative although similar algorithms showing their validity and consistency

stability and periodicity in coupled pinney equations abstract we show that the coupled pinney equations with periodic coefficient constitute a class of nonlinear coupled hill s equations for which the problems of stability and periodicity of the solutions reduce to those for a pair of uncoupled linear hill s equations one of which is of ince type

an amplitude phase approach to calculating regge pole positions and residues new amplitude phase formulae for regge pole positions and residues are derived the derivation makes use of certain invariants of the ermakov lewis type the formulas allow calculation to be made on the real r axis with an additional flexibility to optimize its numerical aspects

the dependent wild bootstrap we propose a new resampling procedure the dependent wild bootstrap for stationary time series as a natural extension of the traditional wild bootstrap to time series setting the dependent wild bootstrap offers a viable alternative to the existing block based bootstrap methods whose properties have been extensively studied over the last two decades unlike all of the block based bootstrap methods the dependent wild bootstrap can be easily extended to irregularly spaced time series with no implementational difficulty furthermore it preserves the favorable bias and mean squared error property of the tapered block bootstrap which is the state of the art block based method in terms of asymptotic accuracy of variance estimation and distribution approximation the consistency of the dependent wild bootstrap in distribution approximation is established under the framework of the smooth function model in addition we obtain the bias and variance expansions of the dependent wild bootstrap variance estimat

regge poles as complex eigenvalues we report a new direct method for calculating regge pole positions and residues upon introduction of an absorbing optical potential sufficiently far in the asymptotic region the problem reduces to several iterative diagonalizations of a symmetric complex matrix the method is tested on well studied cases of potential scattering

small sample effects in time series analysis a new asymptotic theory and a new estimate on introduit un modele mathematique qui couvre les effets des petits echantillons dans l approximation de whittle 1953 on montre que l estimation du maximum de vraisemblance exacte est toujours optimale dans ce modele

the six and eight vertex models revisited elliott lieb s ice type models opened up the whole field of solvable models in statistical mechanics here we discuss the commuting transfer matrix t qequations for these models writing them in a more explicit and transparent notation that we believe offers new insights the approach manifests the relationship between the six vertex and chiral potts models and between the eight vertex and kashiwara miwa models

confidence intervals for spectral mean and ratio statistics summary we propose a new method to construct confldence intervals for spectral mean and related ratio statistics of a stationary process that avoids direct estimation of their asymptotic variances by introducing a bandwidth a self normalization procedure is adopted and the distribution of the new statistic is asymptotically nuisance parameter free the bandwidth is chosen using information criteria and a moving average sieve approximation through a simulation study we demonstrate good flnite sample performance of our method when the sample size is moderate while a comparison with empirical likelihood based method for ratio statistics is made conflrming wider applicability of our method

choice of bandwidth for kernel regression when residuals are correlated summary standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated in this paper we propose a modified version of an existing plug in bandwidth selector the method is generalized to stationary error variables by estimating a functional of the residual covariance function the proposed bandwidth selector shows good properties in asymptotic theory and in simulations without assuming a parametric model for the error process

nonparametric spectral density estimation using penalized whittle likelihood abstract the penalized likelihood approach is not well developed in time series analysis even though it has been applied successfully in a number of nonparametric function estimation problems chow and grenander proposed a penalized likelihood type approach to the nonparametric estimation of the spectral density of gaussian processes in this article this estimator is extended to more general stationary processes its practical implementation is developed in some detail and some asymptotic rates of convergence are established its performance is also compared to more widely used alternatives in the field a computational algorithm involving an iterative least squares initialized by the log periodogram is first developed then motivated by an asymptotic linearization an estimator of the integrated squared error between the estimated and true log spectral densities is proposed from this a data dependent procedure for selection of the amount of smoothing is constructed the methodology is illustrated

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

the algebraic structure of generalized ermakov systems in three dimensions the characteristic algebra of generalized ermakov systems is sl 2 r the structure of these systems in three dimensions is obtained a subset in the form of an equation of motion with the additional requirement of so 3 symmetry is studied it includes the classical equation of the magnetic monopole the existence of three vectors of poincare type is established consideration is given to weak generalized ermakov systems in which the symmetry breaking occurs in the radial equation

fusion operators in the generalized τ 2 model and root of unity symmetry of the xxz spin chain of higher spin we construct the fusion operators in the generalized 2 model using the fused l operators and verify the fusion relations with the truncation identity the algebraic bethe ansatz discussion is conducted on two special classes of 2 which include the superintegrable chiral potts model we then perform the parallel discussion on the xxz spin chain at roots of unity and demonstrate that the sl2 loop algebra symmetry exists for the root of unity xxz spin chain with a higher spin where the evaluation parameters for the symmetry algebra are identified by the explicit fabricius mccoy current for the bethe states parallels are also drawn to the comparison with the superintegrable chiral potts model

automatic estimation of the cross spectrum of a bivariate time series summary the penalised whittle likelihood has recently been shown to have good properties in nonparametric estimation of spectral density functions this paper extends the approach to the estimation of the cross spectrum of a bivariate time series one major difference from the univariate case is that the cross spectrum estimate is not constrained to be positive but must result in a positive definite spectral density matrix an efficient computational method based on iterative reweighted least squares is described and an estimate of the integrated squared error loss is derived and used as an objective criterion to allow automatic selection of the smoothing parameters numerical experiments indicate that the proposed estimate improves on the standard estimate based on kernel smoothing of the cross periodograms an analysis of respiration and heart rate time series is given as an illustrative example

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

numerical solution of non self adjoint sturm liouville problems and related systems this paper gives the analysis and numerics underlying a shooting method for approximating the eigenvalues of non self adjoint sturm liouville problems we consider even order problems with equally divided separated boundary conditions the method can find the eigenvalues in a rectangle and in a left half plane it combines the argument principle with the compound matrix method using the magnus expansion in some cases the computational cost of compound matrices can be reduced by transforming to a second order vector sturm liouville problem we study the asymptotics of the solutions of the ode for large absolute values of the eigenvalue parameter in order to calculate the eigenvalues in a left half plane the method is applied to the orr sommerfeld equation and other examples

can milne s method work well for the coulomb like potentials we extend the improved milne s milne spline method for obtaining eigenvalues and eigenfunctions to the cases of long range and singular potentials for which we have conjectured that it is difficult to apply the method contrary to our conjecture it turned out that the method is valid also for coulomb potential and repulsive 1 xn n 2 3 type potential further we applied the method for two cases for which the solutions are not known in order to investigate the stability of the multi dimensional universe it has been shown that the extra dimensional internal space of our universe is not stable in classical einstein gravity as well as canonically quantized one two possibilities for stabilization were investigated i noncanonically quantized einstein gravity and ii canonically quantized higher curvature gravity it has been suggested that the space is stable by qualitative and approximate methods exact analytical treatment is very difficult so that numerical investigation is highly desirable numerical investigation shows that the space is stable with sufficient reliability

a frequency domain selection criterion for regression with autocorrelated errors abstract we consider the regression model yt t e t t 0 1 n 1 where y t are scalar observations t is the unknown regression function and e t are unobservable errors generated by a zero mean weakly stationary process independent of t and with completely unknown autocorrelation structure we propose a data driven method for selecting a parametric or nonparametric estimator of t the method is based on cross validation in the frequency domain and requires no assumptions about the form of the estimator or the error correlations it does however require the discrete fourier transform dft of the signal t to be a smooth complex function of frequency as is the case for example with transient signals or growth and decay curves after giving some general motivations for the method we focus on the special case of linear estimators of a nonparametric regression function including both kernel and spline estimators for these estimators we develop efficient methods of evaluating t

a method for determining periods in time series abstract the periods corresponding to peaks in the autoregressive and window spectral density estimators are shown to be consistent and asymptotically normal estimators of peak periods in the true spectral density under assumptions of known autoregressive order and under slight modifications of the assumptions often made to show the asymptotic normality of window estimators respectively the asymptotic variances are obtained and the use of the theory is illustrated by obtaining an asymptotic confidence interval for a peak period using a set of hormone data

the maximum of the periodogram of a non gaussian sequence it is a well known fact that the periodogram ordinates of an lid mean zero gaussian sequence at the fourier frequencies constitute an lid exponential vector hence the maximum of these periodogram ordinates has a limiting gumbel distribution we show for a non gaussian lid mean zero finite variance sequence that this statement remains valid we also prove that the point process constructed from the periodogram ordinates converges to a poisson process this implies the joint weak convergence of the upper order statistics of the periodogram ordinates these results are in agreement with the empirically observed phenomenon that various functionals of the periodogram ordinates of an lid finite variance sequence have very much the same asymptotic behavior as the same functionals applied to an lid exponential sample

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

improved amplitude phase method for complex angular momentum analysis an amplitude phase formula for the s matrix using two milne solutions and the regular schrodinger solution is derived the formula is particularly useful in the analysis of regge poles located far

generating converging eigenenergy bounds for the discrete states of the ix3 non hermitian potential recent investigations by bender and boettcher and by mezincescu have argued that the discrete spectrum of the non hermitian potential v x ix3 should be real we give further evidence for this through a novel formulation which transforms the general one dimensional schrodinger equation with complex potential into a fourth order linear differential equation for x 2 this permits the application of the eigenvalue moment method developed by handy bessis and coworkers yielding rapidly converging lower and upper bounds to the low lying discrete state energies we adapt this formalism to the pure imaginary cubic potential generating tight bounds for the first five discrete state energy levels

the q operator and functional relations of the eight vertex model at root of unity for odd n following baxter s method of producing q 72 operator we construct the q operator of the root of unity eight vertex model for the crossing parameter 2mk n with odd n where q 72 does not exist we use this new q operator to study the functional relations in the fabricius mccoy comparison between the root of unity eight vertex model and the superintegrable n state chiral potts model by the compatibility of the constructed q operator with the structure of baxter s eight vertex solid on solid sos model we verify the set of functional relations of the root of unity eight vertex model using the explicit form of the q operator and fusion weights of the sos model

transfer matrix functional relations for the generalized τ 2 t q model the n state chiral potts model in lattice statistical mechanics can be obtained as a descendant of the six vertex model via an intermediate q or 2 t q model here we generalize this to obtain a column inhomogeneous 2 t q model and derive the functional relations satisfied by its row to row transfer matrix we do not need the usual chiral potts relations between the nth powers of the rapidity parameters a p b p c p d p of each column this enables us to readily consider the case of fixed spin boundary conditions on the left and right most columns we thereby re derive the simple direct product structure of the transfer matrix eigenvalues of this model which is closely related to the superintegrable chiral potts model with fixed spin boundary conditions

a simple span selector for periodogram smoothing summary one approach to estimating the spectral density of a stationary time series is to smooth the periodogram and one important component of this approach is the choice of the span for smoothing this note proposes a new span selector which is based on unbiased risk estimation the proposed span selector is simple and does not impose strong conditions on the unknown spectrum for example it does not require the unknown spectrum to possess a second derivative which is a typical requirement of most plug in type or spline based methods the finite sample performance of the proposed span selector is illustrated via a small simulation

is the north atlantic oscillation just a pink noise inthis paper the authors address the problem of predictability for the nao in dex series the spectral analysis completed with a bootstrap procedure shows a rather featureless structure of the index in other words the actual time series could be a realisation of many di6erent stochastic processes an analysis of the hurst exponent does suggest a slightly red noise as a model for the index which is interpreted as the nao being driven by meteorological noise a nonlinear study of the series embedding dimension fractal correlation dimension and leading lyapunov exponent shows little predictive performance as well c 2003 published by elsevier science b v

autoregressive aided periodogram bootstrap for time series a bootstrap methodology for the periodogram of a stationary process is proposed which is based on a combination of a time domain parametric and a frequency domain nonparametric bootstrap the parametric fit is used to generate periodogram ordinates that imitate the essential features of the data and the weak dependence structure of the periodogram while a nonparametric kernel based correction is applied in order to catch features not represented by the parametric fit the asymptotic theory developed shows validity of the proposed bootstrap procedure for a large class of periodogram statistics for important classes of stochastic processes validity of the new procedure is also established for periodogram statistics not captured by existing frequency domain bootstrap methods based on independent periodogram replicates

a characterization of rapidity curve in the chiral potts model we study the geometry of high genus curves of rapidity variables in chiral potts model in terms of symmetries we characterize these rieman surfaces and derive their period matrices by the theory of prime forms the temperature like parameter is expressed by hyperelliptic theta functions

multivariate spectral analysis using cholesky decomposition summary we propose to smooth the cholesky decomposition of a raw estimate of a multivariate spectrum allowing different degrees of smoothness for different elements the final spectral estimate is reconstructed from the smoothed cholesky elements and is consistent and positive definite more importantly the cholesky decomposition matrix of the spectrum can be used as a transfer function in generating time series whose spectrum is identical to the given spectrum at the fourier frequencies this not only provides us with much flexibility in simulations but also allows us to construct bootstrap confidence intervals for the multivariate spectrum by generating bootstrap samples using the cholesky decomposition of the spectral estimate a numerical example and an application to electroencephalogram data are used as illustrations

generalization of the amplitude phase s matrix formula for coupled scattering states the amplitude phase method is generalized to coupled schr dinger scattering states with a common angular momentum quantum number a pair of exponential type amplitude phase solutions u j r exp ij r for each channel is obtained containing a common complex scalar phase function j r and two column vector amplitudes u j r the amplitude functions satisfy certain nonlinear generalized milne equations and the scalar product of the two amplitudes determines the derivative of the common phase function fundamental amplitude phase matrix solutions that are proportional to jost like schr dinger matrix solutions are constructed it is shown how a generalized amplitude phase s matrix formula can be derived from wronskian relations involving the two amplitude phase matrix solutions and a regular matrix solution

multi state complex angular momentum residues a relation between a multi state complex angular momentum cam pole residue and the corresponding cam state wavefunction is derived for a real symmetric potential matrix the result generalizes a residue formula available for single channel atomical collision systems and it is based on a diagonalization of the s matrix together with the use of exact wronskian relations

attainable conditions and exact invariant for the time dependent harmonic oscillator the time dependent oscillator equation is solved numerically for various trajectories in amplitude and phase variables the solutions exhibit a finite time dependent parameter whenever the squared amplitude times the derivative of the phase is invariant if the invariant relationship does not hold the time dependent parameter has divergent singularities these observations lead to the proposition that the harmonic oscillator equation with finite time dependent parameter must have amplitude and phase solutions fulfilling the invariant relationship since the time dependent parameter or the potential must be finite for any real oscillator implementation the invariant must hold for any such physically realizable system