pairs of analytical eigenfunctions for the x2 λ x2 1 gx2 interaction a general method is discussed for deriving pairs of exact analytical eigenfunctions of the x2 x2 1 gx2 interaction recently discussed by blecher and leach and commented on by gallas it is shown that those pairs always consist of an odd and an even parity solution in each case and g are related by the conditions 0 and g as a result of their investigations the authors conjecture the possible existence of an infinite number of solution pairs having and g connected by several different relations

generalized monte carlo significance tests summary simple monte carlo significance testing has many applications particularly in the preliminary analysis of spatial data the method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis however there are situations in which a sample of values can only be conveniently generated using a markov chain initiated by the observed data so that independence is violated this paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out the methods are applied to the rasch model to the finite lattice ising model and to the testing of association between spatial processes power is discussed in a simple case

finite difference order doubling in two dimensions an order doubling process previously used to obtain eighth order eigenvalues from the fourth order numerov method is applied to the perturbed oscillator in two dimensions a simple method of obtaining high order finite difference operators is reported and an odd parity boundary condition is found to be effective in facilitating the smooth operation of the order doubling process

high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

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

a quick perturbative method for schrodinger equations within the framework of perturbation theory we propose firstly an iterative method which may serve as a source of optimal unperturbed solutions in both one and more dimensions it combines the runge kutta and newton algorithm and its efficiency is illustrated on a few quartic oscillators secondly admitting also an arbitrary perturbation of potentials we generalize the existing runge kutta one dimensional rayleigh schrodinger constructions of energies and wavefunctions to more dimensions

backlund transformation of painleve iii d 8 τ function we study the explicit formula suggested by gamayun iorgov and lisovyy for the painleve iii d 8 function in terms of virasoro conformal blocks with a central charge of 1 the painleve equation has two types of bilinear forms which we call toda like and okamoto like we obtain these equations from the representation theory using an embedding of the direct sum of two virasoro algebras in a certain superalgebra these two types of bilinear forms correspond to the neveu schwarz sector and the ramond sector of this algebra we also obtain the functions of the algebraic solutions of the painleve iii d 8 from the special representations of the virasoro algebra of the highest weight n 1 4 2

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

on the schrodinger equation for the interaction x2 λx2 1 gx2 by forming the 6 6 pade approximants to the energy perturbation series for the interaction x2 lambda x2 1 gx2 as obtained from the hypervirial relations the ground and the first three excited state energy levels are calculated for g or 2 the results of the present calculation are in good agreement with those of the variational calculations the authors also present another set of exact solutions to the schrodinger equation for the interaction x2 lambda x2 1 gx2

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

definite integrals as solutions for the x2 λx2 1 gx2 potential presents exact solutions and eigenvalues for the schrodinger equation with the x2 x2 1 gx2 interaction on the half axis x or 0 the solutions are given in the form of definite integrals and the eigenvalues by means of a well defined limiting procedure

representation theory of neveu schwarz and ramond algebras i verma modules abstract in this article we study the structure of verma modules of n 1 super virasoro algebras as applications we construct bernstein gel fand gel fand type resolutions this article is the detailed and expanded version of iohara and koga c r acad sci paris ser i 328 1999 381

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

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

simple restricted modules for neveu schwarz algebra abstract in this paper we give a construction of simple modules generalizing both highest weight and whittaker modules for the neveu schwarz algebra in the spirit of the work of mazorchuk and zhao on simple virasoro modules we establish a 1 1 correspondence between simple restricted neveu schwarz modules and simple modules of a family of finite dimensional solvable lie superalgebras associated to the neveu schwarz algebra moreover for two of these superalgebras all simple modules are classified

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

on the numerical integration of the schrodinger equation for symmetrical potentials a highly stable shooting method with the numerov integrator the problem of the stability of the numerical solution of the one dimensional schrodinger equation with symmetrical potential v x is considered this problem is illustrated by the example y e x2 y 0 with y 0 1 and y x to 0 when x to infinity for which the conventional shooting method using the numerov integrator fails for e 1 to find y x beyond x 5 it is shown that these values are reached by using a different procedure for shooting with the same numerov integrator this procedure starts the integration at any large value l of x and steps backward towards x 0 the method is applied to another value of e for which an exact value of y x is available this tests shows that the accuracy of the computed values of y x is independent of the choice of l thus the method does not improve the eigenvalue computation but it allows the determination of the solution y x for large x when such values are needed

on the convergence of monte carlo maximum likelihood calculations summary monte carlo maximum likelihood for normalized families of distributions can be used for an extremely broad class of models given any family he 0 e 0 of non negative integrable functions maximum likelihood estimates in the family obtained by normalizing the functions to integrate to 1 can be approximated by monte carlo simulation the only regularity conditions being a compactification of the parameter space such that the evaluation maps 0 h0 x remain continuous then with probability 1 the monte carlo approximant to the log likelihood hypoconverges to the exact log likelihood its maximizer converges to the exact maximum likelihood estimate approximations to profile likelihoods hypoconverge to the exact profile and level sets of the approximate likelihood support regions converge to the exact sets in painleve kuratowski set convergence the same results hold when there are missing data if a wald type integrability condition is satisfied asymptotic normality of the monte carlo error and convergence of the monte carlo approximation to the observed fisher information are also shown

an extended numerov type method for the numerical solution of the schrodinger equation abstract a new method which is based on the new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases 1 i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x 2 ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new method is more efficient compared with previously derived methods

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

two sided eigenvalue bounds for the spherically symmetric states of the schro dinger equation the eigenvalues of the radial schr6dinger equation are calculated very accurately by obtaining exact upper and lower bounds by truncating the usual unbounded domain 0 cx of the system to a finite interval of the form 0 g two auxiliary eigenvalue problems are defined it is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required moreover each auxiliary eigenvalue problem gives rise to an orthonormal set involving bessel functions the matrix representation of the hamiltonian is therefore derived by expanding the wave function into a fourier bessel series numerical results for single and doublewell polynomial oscillators as well as gaussian type non polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy whenever the boundary parameter g is in the neighborhood of some critical value denoted by ecr 1998 elsevier science b v all rights reserved

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

a finite difference approach for the calculation of perturbed oscillator energies a simple numerical method for calculating eigenvalues and corresponding eigenvectors of the schrodinger equation for a perturbed oscillator is described the derived results are compared with previously derived numerical data and with available exact values

on the existence of maximum likelihood estimates in logistic regression models summary the problems of existence uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature haberman 1974 chapter 2 wedderburn 1976 silvapulle 1981 for multinomial logistic regression models we prove existence theorems by considering the possible patterns of data points which fall into three mutually exclusive and exhaustive categories complete separation quasicomplete separation and overlap our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables

intrinsic autoregressions and related models on the two dimensional lattice summary stationary autoregressions on a two dimensional lattice are generalized to intrinsic models where only increments are assumed to be stationary prediction formulae and the asymptotic behaviour of the semivariogram are derived for parameter estimation we propose an approximate maximum likelihood estimator a generalization of whittle s estimator it is derived also for general intrinsic models

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

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

eigenenergies of the r2 lambda r2 1 gr2 and 1 2 r2 or gr4 1 g alpha r2 potentials obtained by the hypervirial method the energy levels of the schrodinger equation involving the potentials v r r2 lambda r2 1 gr2 and v or r 1 2 r2 or gr4 1 g alpha r2 are calculated by using hypervirial and pade approximant methods

an accurate finite difference method for the numerical solution of the schro dinger equation abstract an accurate finite difference approach for computing eigenvalues of schrodinger equations is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x we investigate the well known potential of the onedimensional anharmonic oscillator the symmetric double well potential the razavy potential and the doubly anharmonic oscillator potential ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient than previously derived methods

recursion formulae for the characteristic polynomial of symmetric banded matrices abstract in this article we treat the algebraic eigenvalue problem for real symmetric and banded matrices of size n n say for symmetric tridiagonal matrices there is a well known two term recursion to evaluate the characteristic polynomials of its principal submatrices this recursion is superfast i e it requires o n additions and multiplications moreover it is used as the basis for a numerical algorithm to compute particular eigenvalues of the matrix via bisection we derive similar recursion formulae also with o n numerical operations for symmetric matrices with arbitrary bandwidth containing divisions the main results are divisionfree recursions for penta and heptadiagonal symmetric matrices these recursions yield similarly as in the tridiagonal case stable and superfast algorithms to compute any particular eigenvalue

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

some finite difference methods for computing eigenvalues and eigenvectors of special two point boundary value problems abstract direct numerical integration methods will be discussed for calculating eigenvalues and eigenvectors of two point boundary value problems involving the differential equation y a p x y 0 with p x p x the derived results are compared with previously derived numerical data and with available exact values

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

energy levels of a nonpolynomial oscillator using finite difference technique abstract the energy levels of a one dimensional system are calculated for the rational potentials v 1 x n g x 2 x 2n 1 gx 2 and v 2 x i n m g x 2i gx 2n 1 g x 2m using the finite difference approach over wide range of values of the perturbation parameters g i n m and for various state numbers n the numerical results are compared with previous numerical data and with available exact values

on the x2 λx2 1 gx2 interaction presents an infinite set of exact solutions of odd parity for the one dimensional schrodinger equation with the interaction x2 lambda x2 1 gx2 these complement the exact solutions of even parity that have been obtained recently by flessas phys lett a vol 84 p 121 1981

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

representation theory of n 2 super virasoro algebra twisted sector abstract in this article we study the structure of verma modules and fock modules over the twisted sector of the n 2 super virasoro algebras

a new finite difference scheme with minimal phase lag for the numerical solution of the schrodinger equation a new finite difference approach for the numerical solution of the eigenvalue problem of the schrodinger equations is developed in this paper the new approach is based on the maximization of the algebraic order and the minimization of the phase lag we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient when compared with previously derived methods

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

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

on the structure of verma modules over virasoro and neveu schwarz algebras in the paper we present a different proof of the theorem of b l feigin and d b fuchs about the structure of verma modules over virasoro algebra we state some new results about the structure of verma modules over neveu schwarz the proof has two advantages first it is simpler in the most interesting cases for example in the so called minimal models second it can be generalized for neveu schwarz algebra for some class of verma modules

sequential monte carlo p values summary the assessment of statistical significance by monte carlo simulation may be costly in computer time this paper looks at a number of ways of calculating exact monte carlo p values by sequential sampling such p values are shown to have properties similar to those obtained by sampling with a fixed sample size both standard and generalized monte carlo procedures are discussed and in particular a sequential method is proposed for dealing with situations in which values can only be conveniently generated using a markov chain conditioned to pass through the observed data

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed