maximum likelihood estimation for multivariate mixture observations of markov chains corresp to use probabilistic functions of a markov chain to model certain parameterizations of the speech signal we extend an estimation technique of liporace to the eases of multivariate mixtures such as gaussian sums and products of mixtures we also show how these problems relate to liporace s original framework

a general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems this paper introduces a general technique for the construction of multistep methods capable of integrating without local truncation error homogeneous linear odes with constant coefficients including those in particular that result in oscillatory solutions moreover these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies even confluent ones that occur from nonhomogeneous terms in the differential equation our procedure allows the derivation of many of the existing codes with similar properties as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency the properties of the methods are studied within a general framework and numerical examples are presented these demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes particularly when integrating the class of equations with perturbed oscillatory solutions the methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known as occurs in satellite orbit propagation the underlying ideas have already been applied to the improvement of some chebyshev methods that are not multistep

a two step explicit p stable method of high phase lag order for linear periodic ivps in this paper we present a nonlinear two step explicit p stable method of fourth algebraic order and 12th phase lag order for solving one dimensional second order linear periodic initial value problems ivps of ordinary differential equations based on a special vector arithmetic with respect to an analytic function the method can be extended to be vector applicable for multi dimensional problems directly some numerical results are reported to illustrate the efficiency of the method

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

a family of high order multistep methods with vanished phase lag and its derivatives for the numerical solution of the schrodinger equation many simulation algorithms chemical reaction systems differential systems arising from the modelling of transient behaviour in the process industries etc contain the numerical solution of systems of differential equations for the efficient solution of the above mentioned problems linear multistep methods or runge kutta single step methods are used for the simulation of chemical procedures the radial schrodinger equation is used frequently in the present paper we will study a class of linear multistep methods more specifically the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial schrodinger equation and related problems this algorithm belongs in the category of the multistep methods in order to produce an efficient multistep method the phase lag property and its derivatives are used hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions the reason of their efficiency as the analysis proved is that the phase lag and its derivatives are eliminated another reason of the efficiency of the new obtained methods is that they have high algebraic order

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

a two step explicit p stable method of high phase lag order for second order ivps this paper presents a two step explicit p stable method of high phase lag order for solving second order ordinary differential equations basing on a special vector operation with respect to a smooth function the method can be extended to be vector applicable in multi dimensional space

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

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

fourier spectral approximation to a dissipative system modeling the flow of liquid crystals we study a fourier spectral method for a dissipative system modeling the flow of liquid crystals we first prove its convergence in a suitable sense and establish the existence of a global weak solution of the original problem and its uniqueness in the two dimensional case then we derive error estimates which exhibit the spectral accuracy of the fourier spectral method we also construct a fully discrete scheme and carry out a complete stability and error analysis for it finally we present some illustrative numerical results

a multiple cause mixture model for unsupervised learning this paper presents a formulation for unsupervised learning of clusters reflecting multiple causal structure in binary data unlike the hard k means clustering algorithm and the soft mixture model each of which assumes that a single hidden event generates each data point a multiple cause model accounts for observed data by combining assertions from many hidden causes each of which can pertain to varying degree to any subset of the observable dimensions we employ an objective function and iterative gradient descent learning algorithm resembling the conventional mixture model a crucial issue is the mixing function for combining beliefs from different cluster centers in order to generate data predictions whose errors are minimized both during recognition and learning the mixing function constitutes a prior assumption about underlying structural regularities of the data domain we demonstrate a weakness inherent to the popular weighted sum followed by sigmoid squashing and offer alternative forms of the nonlinearity for two types of data domain results are presented demonstrating the algorithm s ability successfully to discover coherent multiple causal representations in several experimental data sets

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

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

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

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

new finite difference formulas for numerical differentiation conventional numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series and closed form expressions of these finite difference formulas have already been presented in this paper we present new finite difference formulas which are more accurate than the available ones especially for the oscillating functions having frequency components near the nyquist frequency closed form expressions of the new formulas are given for arbitrary order a comparison of the previously available three types of approximations is given with the presented formulas a computer program written in mathematica based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point

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

four step methods for y f x y abstract a systematic investigation is undertaken here on the possible practical consequences of the fact that the linear four step methods for y f x y form a family with a certain structure the mentioned property is found to be surprisingly rich in consequences and three of them are of particular importance i drastic reduction in the number of iterations at each step ii increased flexibility with respect to the start of integration and to the modification of the step size during the integration process iii possibility of direct and simple appraisal of the local truncation error

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

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

a p stable exponentially fitted method for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the resonance problem of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

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

a new method for the numerical solution of fourth order bvp s with oscillating solutions abstract a new finite difference scheme with minimal phase lag for the numerical solution of fourth order differential equations is developed in this paper numerical and theoretical results show that this new approach is more efficient compared with previously derived methods

corrigendum fourier spectral approximation to a dissipative system modeling the flow of liquid crystals dm 2 2 2 2 e received by the editors october 20 2002 accepted for publication november 26 2002 published electronically may 12 2003 http www siam org journals sinum 41 2 41653 html department of mathematics penn state university 307 mcallister bldg state college pa 16802 qdu math psu edu department of mathematics shanghai normal university 100 guilin road shanghai 200234 people s republic of china byguo guomai sh cn department of mathematics purdue university west lafayette in 47907 shen math purdue edu

explicit exponentially fitted methods for the numerical solution of the schrodinger equation we present here some accurate explicit methods for the numerical integration of the radial schrodinger equation the formulae considered contain free parameters which are defined in order to integrate exponential functions numerical results also indicate that the new methods are much more accurate than other explicit numerov type wellknown methods

a family of four step exponential fitted methods for the numerical integration of the radial schrodinger equation abstract a family of exponential four step methods is developed for the numerical integration of the one dimensional schrodinger equation the formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions the new methods integrate more exponential functions and are very simple compared with the well known sixth algebraic order runge kutta type methods numerical results indicate that the new methods are much more accurate than other exponentially fitted methods

p stable exponentially fitted methods for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the bound states problem we solve the radial schrodinger equation in order to find eigenvalues for which the wavefunction and its derivative are continuous and the boundary conditions are satisfied and the resonance problem the point of a resonance is that phase changes rapidly through of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind the method can be applied to any problem of physics and chemistry which can be expressed as system of coupled second order differential equations which have oscillatory or periodic solutions this is because it has the property of the p stability i e the interval of periodic stability of the proposed method is equal to 0 which allow is to integrate successful problems with high oscillatory or periodic solution

a two step explicit p stable method for solving second order initial value problems the conceptions of periodicity interval and p stability were introduced by lambert and watson j inst math appl 18 1976 189 in connection with multistep methods for second order initial value problems this paper presents a new two step explicit p stable method of order two for solving initial value problems of second order ordinary differential equations based on a special vector operation the method can be extended to be vector applicable

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

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

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

new variable step procedure for the numerical integration of the one dimensional schro dinger equation abstract a new variable step procedure is developed for the numerical integration of the one dimensional schrodinger equation the new variable step method is based on two p stable methods of order eight and ten numerical results indicate that the new procedure is more efficient than similar variable step procedures

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

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

hidden markov models with factored gaussian mixtures densities we present a factorial representation of gaussian mixture models for observation densities in hidden markov models hmms which uses the factorial learning in the hmm framework we derive the reestimation formulas for estimating the factorized parameters by the expectation maximization em algorithm and propose a novel method for initializing them to compare the performances of the proposed models with that of the factorial hidden markov models and hmms we have carried out extensive experiments which show that this modelling approach is effective and robust

a saddle point approach to the computation of harmonic maps in this paper we consider numerical approximations of a constraint minimization problem where the object function is a quadratic dirichlet functional for vector fields and the interior constraint is given by a convex function the solutions of this problem are usually referred to as harmonic maps the solution is characterized by a nonlinear saddle point problem and the corresponding linearized problem is well posed near strict local minima the main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions iterative schemes of newton type for the discrete nonlinear saddle point problems are investigated and mesh independent preconditioners for the iterative methods are proposed

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

a class of two step explicit methods for periodic ivps in this paper we present a class of non linear explicit second order methods for solving one dimensional periodic initial value problems ivps which are p stable in the sense of lambert and watson and of high order phase lag error for multi dimensional problems we introduce a special vector operation such that our methods can be extended to be vector applicable directly however it is not sufficient to investigate the stability behavior of the vector form of the methods based on the scalar test equation y l 2y after introducing a new test system and extending the definitions of the interval of periodicity and the phase lag order we discuss the stability property and the phase lag of the vector applicable methods the methods are p stable and of high phase lag for one frequency problems for multi frequency problems methods are of only second phase lag order and may have not a primary interval of periodicity some numerical results illustrate the conclusions

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

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

a family of four step exponentially fitted predictor corrector methods for the numerical integration of the schro dinger equation a family of predictor corrector exponential four step methods is developed for the numerical integration of the one dimensional schrodinger equation the formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions the new methods integrate more exponential functions and are very simple compared with the well known sixth algebraic order runge kutta type methods numerical results indicate that the new method is much more accurate than other exponentially fitted methods

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

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

an explicit sixth order method with phase lag of order eight for y f t y abstract recently thomas 6 gave a two step sixth order method with phase lag 1 101478 67 h8 for the numerical integration of periodic initial value problems y t y y t 0 y 0 y t 0 y 0 however thomas method is implicit it possesses an interval of periodicity of size 2 77 and for nonlinear problems her method requires 6i 1 function evaluations for i modified newton iterations for the solution of the resulting implicit equations at each step in the present paper we present a new two step sixth order method which also has phase lag of order eight but with a smaller constant given by 1 3 628 800 h8 in contrast with thomas method our method is explicit possesses a larger interval of periodicity of size 4 63 and it is decidedly more economical since it involves only six function evaluations per step numerical experiments confirm the superiority of our present method over thomas method