energy levels of a two dimensional anharmonic oscillator characteristic function approach the eigenlevels of a two dimensional quartic anharmonic oscillator are identified as the zeros of a characteristic function derived from the resolvent numerical values are obtained by means of pade approximants

a variational iterative technique applied to quantum mechanical problems a technique for solving nonlinear operator equations iteratively is applied to quantum mechanical calculations it is shown that alternative methods of applying the technique lead to another way of approaching variational calculations and generalisations of perturbation theory respectively

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

a simple iterative method for resonance calculation a recently developed procedure for solving the stationary schrodinger equation in matrix representation form is proved to be useful in calculating quasibound energy levels results for a simple quantum mechanical model with potential energy function v r 1 lambda r lambda 0 are shown to be very accurate

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

relative convergences of the wkb and swkb approximations the relative convergences of the wkb and swkb approximations are examined by calculating the eigenenergies for four potentials by one and two term wkb and swkb approximations exact eigenenergies for these four potentials are also calculated by a numerical integration of the schrodinger equation varied results are found for the four potentials it is found that in general the effect of the second term in the wkb and swkb approximations depends on the potential the parameters involved and the quantum number of the state no simple generalizations are possible

generalised meshes for quantum mechanical problems a new method to discretise schrodinger equations on a mesh is described this method is based on an accurate approximation of a variational calculation the regularly spaced mesh and meshes based on the zeros of orthogonal polynomials are studied in detail it is shown that with each type of mesh is associated a particular kinetic energy operator and an optimal formula for its discretized form the applications to some simple potential problems show that the method is very accurate as well as very simple applications to many body problems indicate that the accuracy of the results is improved by an order of magnitude with respect to conventional mesh calculations

a unified treatment of schrodinger s equation for anharmonic and double well potentials the authors exploit some exact soluble model potentials vn alpha beta v a b lambda x6 in order to extrapolate as lambda to 0 reliable eigenvalues of the one dimensional schrodinger equations with either anharmonic or symmetric double well potentials of the form v a b 1 2ax2 bx2 b 0 their procedure which corresponds to low order rayleigh schrodinger perturbation theory is found to be competitive with both high order pade summation of conventional rspt and large scale variational calculations using harmonic oscillators basis functions

numerical methods for the eigenvalue determination of second order ordinary differential equations an accurate method for the numerical solution of the eigenvalue problem of second order ordinary differential equation using the shooting method is presented the method has three steps firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method secondly the initial value problem is solved using new highly accurate formulas of the linear multistep method thirdly the eigenvalue is properly corrected at the matching point the efficiency of the proposed methods is demonstrated by their applications to bound states for the one dimensional harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential in quantum mechanics

quantum theory of anharmonic oscillators a variational and systematic general approximation method this is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1 2 omega 2q2 lambda q4 as a lowest order approximation an extremely simple formula for energy levels ei 0 i 1 2 1 4 3 alphai alphai is derived i being the quantum number of the energy level which covers any lambda i alphai is the real positive root of a cubic equation yi alphai3 alphai2 1 0 with yi 6lambda 2i2 2i 1 2i 1 this formula reproduces the exact energy levels within an error of about 1 for any lambda i the worst case is 2 for i 0 lambda to infinity systematically higher orders of this perturbation theory are developed which contains the usual perturbation theory for the limiting case of small lambda but this perturbation theory is valid for any i lambda the second order perturbation theory reduces the errors of our lowest order results by a factor of about 1 5 in general various ranges large intermediate small of i lambda are investigated and compared with the exact values obtained by the montroll group 1975 1978 for i 0 1 even the fourth order perturbation calculation can be elaborated explicitly which reduces the error to about 0 01 for any lambda for small lambda it gives correct numerical coefficients up to lambda4 terms as it should

an adaptive orthogonal search algorithm for model subset selection and non linear system identification a new adaptive orthogonal search aos algorithm is proposed for model subset selection and non linear system identification model structure detection is a key step in any system identification problem this consists of selecting significant model terms from a redundant dictionary of candidate model terms and determining the model complexity model length or model size the final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system in the new aos algorithm a modified generalized cross validation criterion called the adjustable prediction error sum of squares apress is introduced and incorporated into a forward orthogonal search procedure the main advantage of the new aos algorithm is that the mechanism is simple and the implementation is direct and easy and more importantly it can produce efficient model subsets for most non linear identification problems

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

a robust nonlinear identification algorithm using press statistic and forward regression this paper introduces a new robust nonlinear identification algorithm using the predicted residual sums of squares press statistic and forward regression the major contribution is to compute the press statistic within a framework of a forward orthogonalization process and hence construct a model with a good generalization property based on the properties of the press statistic the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation

model structure selection by cross validation two criteria for choosing between different model structures are proposed their derivation is within a natural cross validatory assessment context and is fairly assumption free in particular the two criteria can be used for discriminating between non nested model structures and more importantly the true system is not required to belong to the considered set of models should the true system belong to the model set the two proposed criteria will asymptotically reduce to some well known structure selection criteria this is believed to be a desirable feature of our proposals on the other hand it provides a nice cross validation interpretation of some well known model structure selection rules also the cross validation interpretation helps one to choose which criteria to use in a given application the paper also has a second purpose somewhat decoupled from that mentioned above it contains an extensive survey of the literature that is useful in its own right

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

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

identification of non linear additive autoregressive models summary we propose a lag selection method for non linear additive autoregressive models that is based on spline estimation and the bayes information criterion the additive structure of the autoregression function is used to overcome the curse of dimensionality whereas the spline estimators effectively take into account such a structure in estimation a stepwise procedure is suggested to implement the method proposed a comprehensive monte carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local polynomial based methods consistency of the lag selection method based on the bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing which provides the first theoretical result of this kind for spline smoothing of weakly dependent data

a new class of wavelet networks for nonlinear system identification a new class of wavelet networks wns is proposed for nonlinear system identification in the new networks the model structure for a high dimensional system is chosen to be a superimposition of a number of functions with fewer variables by expanding each function using truncated wavelet decompositions the multivariate nonlinear networks can be converted into linear in the parameter regressions which can be solved using least squares type methods an efficient model term selection approach based upon a forward orthogonal least squares ols algorithm and the error reduction ratio err is applied to solve the linear in the parameters problem in the present study the main advantage of the new wn is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks by adopting the analysis of variance anova expansion wns can now handle nonlinear identification problems in high dimensions

on the μx2 λx4 ηx6 interaction a class of exact even and odd parity solutions of the schrodinger equation for the interaction x2 x4 x6 0 is obtained in the form polynomial exponential when and satisfy some specific relations in the general case the eigenvalues of the bounded potential problem are found by the method of series solution for any value of the authors also show that the analytic continued fraction method does not always lead to the corrected results

edgeworth expansions of suitably normalized sample mean statistics for atomic markov chains this paper is devoted to the problem of estimating functionals of type f fd from observations drawn from a positive recurrent atomic markov chain open image in new window with stationary distribution the properties of different estimators are studied beyond an accurate estimation of their bias the estimation of their asymptotic variance is considered we also show that the results of malinovskii 1987 on the validity of the formal edgeworth expansion for sample mean statistics of type t n n 1 n i 1 f x i extend to their studentized versions normalized by the asymptotic variance estimates we consider

nonparametric identification of nonlinear time series selecting significant lags abstract in this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series the procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series the procedure is illustrated by simulations and sunspot data lynx data and blowfly data are analyzed it is indicated that projectors can be used in conjunction with the procedure for selecting significant lags to check the adequacy of an additive time series model

energy levels for a double well potential in three dimensional system using hill determinant approach abstract energy levels of the schrodinger equation for the double well potential v x y z z 2 1 2 z 2 x 2 y 2 z 2 1 2 a xx x 4 a yy y 4 a zz z 4 2a xy x 2 y 2 2a xz x 2 z 2 2a yz y 2 z 2 in a three dimensional system are calculated using the hill determinant approach for several eigenstates and large values of the perturbation parameters and z2 emphasis is specially placed on the larger values of z2 because the eigenvalues for different states have almost degenerate eigenvalues some of the results calculated by the hill determinant approach are compared with the results produced by other techniques

second order correctness of the blockwise bootstrap for stationary observations we show that the blockwise bootstrap approximation for the distribution of a studentized statistic computed from dependent data is second order correct provided we choose an appropriate variance estimator we also show how to adapt the bc a confidence interval of efron to the dependent case for the proofs we extend the results of gotze and hipp on the validity of the formal edgeworth expansion for a sum to the studentized mean

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

on the summation of some divergent hypergeometric series and related perturbation expansions abstract divergent hypergeometric series 2 f 0 1 occur frequently in poincare type asymptotic expansions of special functions these divergent series 2 f 0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied there is considerable evidence that levin s sequence transformation 1973 and in particular also some closely related sequence transformations weniger 1989 which were derived recently sum divergent series 2 f 0 much more efficiently than pade approximants similar summation problems occur also in the case of divergent rayleigh schrodinger perturbation expansions of elementary quantum mechanical systems a comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that levin s sequence transformation and the recently derived new sequence transformations are again more efficient than pade approximants however the superiority of levin s sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection renormalization is a method often used to approximate the eigenvalues of a hamiltonian that cannot be solved exactly it consists of splitting the hamiltonian into a solvable part and a remainder which is then minimized the inner projection technique first introduced by lowdin in the sixties was developed to bracket the eigenvalues between lower and upper bounds combining renormalization and lowdin s inner projection yielded the so called renormalized inner projection technique in this study this method will be applied to the quartic sextic and octic anharmonic oscillators lower and upper energy bounds are obtained for finite values of the coupling constant as well as for the infinite case the relation between the renormalized inner projection and perturbation theory will also be discussed another feature of this study is the importance of symbolic computation in allowing us to manipulate expressions with unevaluated parameters and to perform calculations in rational arithmetics or high decimal precision thus lowdin s rational approximants can be expressed explicitly as rational fractions in terms of the coupling constant and values for the limit constant can be obtained with amazing high accuracy namely 62 33 and 21 decimal places for the quartic sextic and octic oscillator respectively

the confined system approximation for solving non separable potentials in three dimensions the hilbert space to which the wavefunction of the three dimensional schrodinger equation belongs has been replaced by where is a bounded region the energy spectrum of the usual unbounded system is then determined by showing that the dirichlet and neumann problems in generate upper and lower bounds respectively to the eigenvalues required highly accurate numerical results for the quartic and sextic oscillators are presented for a wide range of the coupling constants

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

tracking time varying causality and directionality of information flow using an error reduction ratio test with applications to electroencephalography data this paper introduces an error reduction ratio causality err causality test that can be used to detect and track causal relationships between two signals in comparison to the traditional granger method one significant advantage of the new err causality test is that it can effectively detect the time varying direction of linear or nonlinear causality between two signals without fitting a complete model another important advantage is that the err causality test can detect both the direction of interactions and estimate the relative time shift between the two signals numerical examples are provided to illustrate the effectiveness of the new method together with the determination of the causality between electroencephalograph signals from different cortical sites for patients during an epileptic seizure

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

a characteristic function approach to the discrete spectrum of electrically charged particles the purpose of this article is to obtain a scalar function the zeros of which give the discrete energy values for a system of electrically charged particles the relation between the serial expansion of the characteristic function in powers of the system s eigenvalue and the stieltjes series has been revealed not only for the electrically charged particles but also generally for any positive or negative definite hermitian operator which has only a discrete spectrum the use of pade approximants to express the characteristic function has offered a rapidly convergent scheme to evaluate the system s eigenvalues the first few elements of the pade table for the reciprocal of the characteristic function of certain systems have been given to verify the presented idea numerically the determination of these elements needs the values of certain complicated integrals which we name zeroth order hyperspherical spectral coefficients hsc 0 the first two of these coefficients are investigated and their evaluation is realized analytically

studies on algebraic methods to solve linear eigenvalue problems generalised anharmonic oscillators a detailed presentation of the recently introduced integration free method with applications to determine the energy levels of the generalised quantum anharmonic oscillators are given numerical calculations are realised for the quartic and the sextic oscillators energy eigenvalues obtained for the ground state as well as for the first few excited states accurate to thirty digits are very impressive and demonstrate the efficiency of the method certain remarks about the selection of the basis functions and a convergence discussion on the presented simple approximation scheme are also included in this paper

a simple iterative solution of the schrodinger equation in matrix representation form a simple procedure for solving the schrodinger equation is presented it is based upon an iterative solution of the secular equation a large enough convergence rate is obtained by using a basis set of properly scaled functions the effect of the scaling parameter on the convergence rate is studied in order to improve the calculation the method is applied to simple though non trivial quantum mechanical models such as the quartic sextic and octic anharmonic oscillators a double well potential and the linear confining potential highly accurate eigenvalues for all values of the coupling parameter are obtained

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

the rayleigh ritz method still competitive abstract an extended rayleigh ritz method for computing two sided eigenvalue bounds of the one dimensional schrodinger equation is presented the method is based on the b spline approximation over the truncated domain for the low lying eigenvalues and problems with rapidly increasing spectrum the corresponding lower bounds are much more accurate than the known temple bounds numerical results for problems exhibiting different kinds of the eigenfunction behaviour are presented

failure of the hill determinant method for the sextic anharmonic oscillator an extended analysis of eigenvalues and wavefunctions resulting from the hill determinant method is carried out for the one dimensional sextic anharmonic oscillator it is shown that in spite of a seemingly natural ansatz of the method irrelevant wavefunctions may appear leading to incorrect eigenvalues the domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented

lattice dynamical wavelet neural networks implemented using particle swarm optimization for spatio temporal system identification in this brief by combining an efficient wavelet representation with a coupled map lattice model a new family of adaptive wavelet neural networks called lattice dynamical wavelet neural networks ldwnns is introduced for spatio temporal system identification a new orthogonal projection pursuit opp method coupled with a particle swarm optimization pso algorithm is proposed for augmenting the proposed network a novel two stage hybrid training scheme is developed for constructing a parsimonious network model in the first stage by applying the opp algorithm significant wavelet neurons are adaptively and successively recruited into the network where adjustable parameters of the associated wavelet neurons are optimized using a particle swarm optimizer the resultant network model obtained in the first stage however may be redundant in the second stage an orthogonal least squares algorithm is then applied to refine and improve the initially trained network by removing redundant wavelet neurons from the network an example for a real spatio temporal system identification problem is presented to demonstrate the performance of the proposed new modeling framework

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

simple modifications for signed roots of likelihood ratio statistics summary in the context of inference about a scalar parameter in the presence of nuisance parameters some simple modifications for the signed root of the log likelihood ratio statistic r are developed that reduce the order of error in the standard normal approximation to the distribution of r from o n 1 2 to o n 1 barndorff nielsen has introduced a variable u such that the error in the standard normal approximation to the distribution of r r r log u r is of order o n 3 2 but calculation of urequires the specification of an exact or approximate ancillary statistic a this paper proposes an alternative variable to u denoted by t that is available without knowledge of a and satisfies t u op n 1 in general thus the standard normal approximation to the distribution of r r log t r has error of order o n 1 and it can be used to construct approximate confidence limits having coverage error of order o n 1 in certain cases however tand uare identical the derivation of tinvolves the bayesian approach to constructing confidence limits considered by welch and peers and peers similar modifications for the signed root of the conditional likelihood ratio statistic are also developed and these modifications are seen to be useful when a large number of nuisance parameters are present several examples are presented including inference for natural parameters in exponential models and inference about location scale models with type ii censoring in each case the relationship between tand u is discussed numerical examples are also given including inference for regression models inference about the means of log normal distributions and inference for exponential lifetime models with type i censoring where barndorff nielsen s variable u is not available

is the improved likelihood ratio statistic really improved in the discrete case summary on the basis of some numerical studies we argue that bartlett adjustment of the likelihood ratio statistic for discrete data does not in general yield an improvement in the asymptotic error rate of the chi squared approximation also we argue that the small sample effect of the bartlett adjustment is unclear and that it is in no way comparable with the improvement obtained for continuous data

complementary variational principles and variational iterative principles complementary variational principles for the solution of certain linear equations are developed it is shown that these may be used iteratively for the solution of nonlinear equations examples are presented with applications in particle theory electromagnetic theory communication theory and the thomas fermi statistical theory for atoms

term and variable selection for non linear system identification the purpose of variable selection is to pre select a subset consisting of the significant variables or to eliminate the redundant variables from all the candidate variables of a system under study prior to model term detection it is required that the selected significant variables alone should sufficiently represent the system generally not all the model terms which are produced by combining different variables make an equal contribution to the system output and terms which make little contribution can be omitted a parsimonious representation which contains only the significant terms can often be obtained without the loss of representational accuracy by eliminating the redundant terms based on these observations a new variable and term selection algorithm is proposed in this paper the term detection algorithm can be applied to the general class of non linear modelling problems which can be expressed as a linear in the parameters form the variable selection procedure is based on locally linea

edgeworth expansion for student s t statistic under minimal moment conditions recent advances in the theory of edgeworth expansion allow rigorous estimation of remainder terms under explicit regularity conditions however even in the important case of student s t statistic the moment conditions required are unrealistically stringent in this note we relax those conditions to the minimum needed to define terms in the edgeworth expansion

orthogonal least squares methods and their application to non linear system identification abstract identification algorithms based on the well known linear least squares methods of gaussian elimination cholesky decomposition classical gram schmidt modified gram schmidt householder transformation givens method and singular value decomposition are reviewed the classical gram schmidt modified gram schmidt and householder transformation algorithms are then extended to combine structure determination or which terms to include in the model and parameter estimation in a very simple and efficient manner for a class of multivariate discrete time non linear stochastic systems which are linear in the parameters

variational principles and variational iterative principles for a class of problems abstract complementary variational principles are developed for linear equations in a function space and a variational iterative scheme based on these principles is introduced for non linear equations

the berry esseen theorem for strongly mixing harris recurrent markov chains let 0 1 be a stationary harris recurrent markov chain with state space e and let f e ir xi f i it is known that the sequence xi i 0 is strongly mixing i e n 0 where n are the strong or rosenblatt mixing coefficients if n decreases at a sufficiently fast rate and f is suitable then a central limit holds for the partial sums sum limits i 0 n x i the present paper gives conditions in order that the convergence rates are o n 1 2

analytical bound eigenstates and eigenvalues of a truncated coulomb potential the author obtains exact eigenfunctions and eigenvalues of the schrodinger equation with the potential v r ze2 r beta beta 0 for particular values of the cut off parameter beta each eigenfunction is a finite polynomial times an exponential and the eigenvalues are exactly those for the pure coulomb problem i e beta 0

tight bounds to the schrodinger equation eigenvalues increasing tight upper and lower bounds to the schrodinger equation eigenvalues are obtained from the riccati equation for the logarithmic derivative of the wavefunction the solution of this non linear equation is written as a pade approximant and the bounds are given by the roots of a sequence of determinants numerical results are shown for the anharmonic and quartic oscillators

conformal mappings versus other power series methods for solving ordinary differential equations illustration on anharmonic oscillators the simplicity and the efficiency of a quasi analytical method for solving nonlinear ordinary differential equations ode is illustrated on the study of anharmonic oscillators ao with a potential v x x2 x2m m 0 the method bervillier 2008 nucl phys b801 296 applies a priori to any ode with two point boundaries one being located at infinity the solution of which has fixed singularities in the complex plane of the independent variable x a conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed taylor series of the generic solution so that the boundary condition at infinity can be easily imposed in principle this constraint when applied on the logarithmic derivative of the wavefunction determines the eigenvalues to an arbitrary level of accuracy in practice for 0 or slightly negative the accuracy of the results obtained is astonishingly large with regard to the modest computing power used various aspects of the method and comparisons with some seemingly similar methods based also on expressing the solution as a taylor series are shortly reviewed presented and discussed

identification of mimo non linear systems using a forward regression orthogonal estimator abstract an orthogonal least squares estimator which was originally derived for single input single output systems is extended to multi input multi output non linear systems the estimator can provide information about the structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner multivariable non linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

converging bounds for the eigenvalues of multiminima potentials in two dimensional space the eigenvalue problem in of schrodinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls it is shown that the dirichlet and the von neumann problems in generate upper and lower bounds respectively to the eigenvalues of the unbounded system to illustrate the method rapidly converging two sided bounds for the energy levels of two and three well oscillators are presented using simple trigonometric basis functions

a new approach to the eigenvalue problem in quantum mechanics convergent perturbation theory for rising potentials a new iteration procedure for solution of the schrodinger equation with arbitrary local potentials is proposed both the eigenvalues and eigenfunctions are represented in the form of a series which is well convergent under certain conditions the solution of the k dimensional schrodinger equation within the proposed scheme reduces to a problem of the k dimensional electrostatics as an example potentials x2n n 2 3 4 and m2x2 gx4 in one dimensional space are considered

a lie algebraic study of some schrodinger equations bound state solutions of several different schrodinger equations are calculated rather efficiently using the methods of lie algebra in all cases considered either of the algebras so 3 or so 2 1 provides a suitable framework and there is no advantage to either choice however the choice of an appropriate realization of the generators is of greater significance leading to a particularly simple solution whenever the hamiltonian can be expressed as a linear function of the generators in certain cases the hamiltonian can be expressed only as a bilinear function of the generators and only part of the bound state spectrum can be calculated analytically to yield a finite set of so called quasiexact solutions these may be interrelated by means of suitable ladder operators and it is possible though by no means necessary to adopt a particular finite dimensional representation of the underlying algebra the present work emphasizes the role of similarity transformations and the connection between the lie

some inner product techniques for computing eigenvalues for three dimensional anharmonic oscillators with quartic and sextic perturbations abstract an algorithm previously used by the author for calculating the energy eigenvalues for a three dimensional oscillator with limited range of perturbation parameters eigenstates and accuracy is revisited and implemented in a more effective way to give excellent accuracy even for higher values of and nx ny nz x y z fifty coefficients of renormalized perturbation series are calculated in contrast to the previous works which were limited to 20 coefficients of the perturbation series due to computational facilities some of the results calculated by the inner product technique are compared with results produced by other techniques

fast orthogonal identification of nonlinear stochastic models and radial basis function neural networks a new fast orthogonal estimation algorithm is derived for a wide class of nonlinear stochastic models including training radial basis function neural networks the selection of significant regressors and the estimation of unknown parameters in the presence of nonlinear noise sources are considered and simulated examples are included to demonstrate the efficiency of the new procedure

pseudospectral methods for solving an equation of hypergeometric type with a perturbation almost all regular or singular sturm liouville eigenvalue problems in the schrodinger form email protected x v x j x email protected x email protected email protected email protected r j x l 2 email protected email protected for a wide class of potentials v x may be transformed into the form s x y email protected x y q x email protected email protected a b r by means of intelligent transformations on both dependent and independent variables where s x and t x are polynomials of degrees at most 2 and 1 respectively and l is a parameter the last form is closely related to the equation of the hypergeometric type eht in which q x is identically zero it will be called here the equation of hypergeometric type with a perturbation ehtp the function q x may therefore be regarded as a perturbation it is well known that the eht has polynomial solutions of degree n for specific values of the parameter l i e l email protected n 0 n t 12 n 1 s which form a basis for the hilbert space l 2 a b of square integrable functions pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of ehtp and hence the energies e of the original schrodinger equation specimen computations are performed to support the convergence numerically

solving the anharmonic oscillator tuning the boundary condition we outline a remarkably efficient method for generating solutions to quantum anharmonic oscillators with an x2m potential we solve the schrodinger equation in terms of a free parameter which is then tuned to give the correct boundary condition by generating a power series expansion of the wavefunction in x and applying a modified borel resummation technique to obtain the large x behaviour the process allows us to calculate energy eigenvalues to an arbitrary level of accuracy high degrees of precision are achieved even with modest computing power our technique extends to all levels of excitation and produces the correct solution to the double well oscillators even though they are dominated by non perturbative effects

standardized log likelihood ratio statistics for mixtures of discrete and continuous observations on considere des cas ou la statistique exhaustive minimale est partiellement discrete et partiellement continue en particulier on considere des problemes de test associes avec des durees de vie exponentielles censurees

energy levels of double well potentials in a three dimensional system the renormalized inner product perturbation technique is used to compute the energy levels for a double well potential v x y z z2 z2 x2 y2 z2 axxx4 ayyy4 azzz4 2axyx2y2 2axzx2z2 2ayzy2z2 in the three dimensional system results are produced for a wide range of parameters z2 and of state numbers nx ny nz comparison is made with results obtained by other means

validity of the formal edgeworth expansion when the underlying distribution is partly discrete validity of the formal edgeworth expansion for the distribution of the statistic ng x n n y n n is considered here x n is a continuous variate and y n is a discrete variate in general if x n y n resemble the sum of i i d variables and the partial derivative of g with respect to the first variable has full rank it is possible to establish an edgeworth expansion

quantum theory of a double well potential energy levels for symmetric and nonsymmetric double well potentials in a three dimensional system abstract energy eigenvalues are obtained for a double well potential v x y z z x 2 z y 2 z z 2 1 2 z x 2 x 2 z y 2 y 2 z z 2 z 2 a xx x 4 a yy y 4 a zz z 4 2a xy x 2 y 2 2a xz x 2 z 2 2a yz y 2 z 2 in a three dimensional system using matrix diagonalization for various values of parameters zx2 zy2 zz2 and several energy levels enx ny nz comparison is made with results obtained by other means

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case