algebraic shape invariant potentials in two steps the simplified algebraic structure of the shape invariance condition in two steps is developed by imposing an extra relation on the two superpotentials of the corresponding two step potentials this simplified version of potential algebra is found to be similar to that of the shape invariance condition in one step the solvable potentials of shape invariance in two steps with a translation change of parameters a1 a0 are shown to possess such a simplified version of potential algebra the condition under which the potential algebra of shape invariance in more than two steps becomes simplified is also discussed

deformed shape invariance and exactly solvable hamiltonians with position dependent effective mass known shape invariant potentials for the constant mass schrodinger equation are taken as effective potentials in a position dependent effective mass pdem one the corresponding shape invariance condition turns out to be deformed its solvability imposes the form of both the deformed superpotential and the pdem a lot of new exactly solvable potentials associated with a pdem background are generated in this way a novel and important condition restricting the existence of bound states whenever the pdem vanishes at an end point of the interval is identified in some cases the bound state spectrum results from a smooth deformation of that of the conventional shape invariant potential used in the construction in others one observes a generation or suppression of bound states depending on the mass parameter values the corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials

on line bayesian estimation of signals in symmetric spl alpha stable noise in this paper we describe on line bayesian filtering methods for time series models with heavy tailed spl alpha stable noise the filtering methodology is based on a scale mixtures of normals smin representation of the spl alpha stable distribution which reexpresses the intractable stable distribution in a conditionally gaussian form we describe how the method can be used for estimation of time varying autoregressive signals buried in symmetric spl alpha stable noise efficiently implemented using an adaptation to an existing rao blackwellized particle filter the methodology is shown to work well with both simulated and real corrupted audio data for which the spl alpha stable noise distribution is found to fit the noise data better than other more standard heavy tailed distributions

algebraic solution for the natanzon hypergeometric potentials an algebraic method based on a strategy that makes use of a realization of the algebra so 2 1 in terms of differential operators is used to solve the bound state problem for the most general natanzon potentials for which the schrodinger equation can be reduced to hypergeometric form hence hypergeometric potentials

an algebraic approach to shape invariance with reflections we consider a realization of supersymmetric shape invariant quantum systems where generalized ladder operators are constructed through the association of the supersymmetric partner operators the parameter potential translation operator and the reflection operator the basic algebraic relations as well as the system eigenstates and energy eigenvalues are obtained as an illustration we apply the generalized formalism for two important shape invariant potential systems the scarf hyperbolic and the morse potentials

generalized coherent states for associated hypergeometric type functions we introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric type operators this makes possible the definition of generalized coherent states for the associated eigenfunctions these generalized coherent states satisfy the relevant properties i e label continuity overcompleteness temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system

cancellation of polarized impulsive noise using an azimuth dependent conditional mean estimator the separation of signals from noisy vector measurements is obtained by taking advantage of the middleton class a model of noise amplitude and the correlation of the components of the noise process due to their polarization the signal is assumed to be white gaussian noise is a superposition of m non gaussian processes each with a fixed azimuth of polarization neither the number of processes m nor their azimuths are known the separation of signal from noise is based on the conditional mean estimators in addition to the optimum estimator which can be derived from a knowledge of the bivariate density functions two suboptimum solutions for polarized noise are discussed the circularly symmetric estimator and the azimuth dependent one circular symmetry is suitable for the nonpolarized noise vector whereas the azimuth dependent estimator is tailored to polarized noise the azimuth dependent approach consists of two steps first the data vector process is discretized into azimuth sectors and then in those classified as noisy the signal is separated from the noise statistical model parameters of random processes are estimated by using the optimum classification based on the likelihood ratio test decision directed method iterative whitening methods are also discussed for correlated vector signals numerical examples show the effectiveness of the above technique in canceling polarized noise

volterra series arising from the discrete schrodinger wave equation in l2 a b space the wave solution of the stationary schrodinger wave equation has been studied extensively however after removing restriction on potential function the problem of obtaining the wave solution becomes considerably difficult methods available in the literature for the closed form wave solution are applicable to specific potential for this reason it is important to strive for the wave solution with a general class of potential functions this paper introduces the volterra series framework to address this limitation the volterra system is a natural extension of linear systems such systems take into account quadratic cubic and more generally polynomial kind of non linearities in the systems the volterra system is a non linear system with the structure of the volterra series here we decompose first a wave function in an appropriate function space and subsequently the wave equation is discretized as a result of this we obtain a bilinear system treating potential as input and wave function as output such an analysis is based on inherent bilinear character of the schrodinger equation in which potential is bilinearly coupled to the wave function this paper discusses state space methodology to recover the volterra series from the bilinear systems the volterra series is attractive from the system theoretic point of view since it describes the output of a class of non linear systems explicitly in terms of the input rather than involving coupling terms and allows substantial simplifications for the numerical simulation a simulation result is introduced to demonstrate the efficacy of the proposed analysis

identification of nonstationary stochastic systems using parallel estimation schemes the parallel multiple model schemes for identification of nonstationary stochastic systems are considered first the form of the optimal local bayesian predictor is derived under the assumptions that system coefficients vary according to the random walk model and that the kalman filter based algorithms are used for identification purposes a rational extension of this strategy which can be applied to identification algorithms of any form is discussed specific suggestions are made concerning the possible choice of adaptation gains of the competitive adaptive filters computer simulation results confirming the good estimation robustness properties of the parallel identification schemes are presented it is shown that the proposed scheme can significantly decrease sensitivity of the identification algorithm to the rate of nonstationarity of the analyzed system or alternatively to the choice of design parameters such as adaptation gains and forgetting factors

a smoothness priors time varying ar coefficient modeling of nonstationary covariance time series a smoothness priors time varying ar coefficient model approach for the modeling of nonstationary in the covariance time series is shown smoothness priors in the form of a difference equation constraint excited by an independent white noise are imposed on each ar coefficient the unknown white noise variances are hyperparameters of the ar coefficient distribution the critical computation is of the likelihood of the hyperparameters of the bayesian model this computation is facilitated by a state space representation kalman filter implementation the best difference equation order best ar model order best hyperparameter model locally in time is selected using the minimum aic method also an instantaneous spectral density is defined in terms of the instantaneous ar model coefficients and a smoothed estimate of the instantaneous time series variance an earthquake record is analyzed the changing spectral analysis of the original data and of simulations from a time varying ar coefficient model of that data are shown

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

algebraic solution for the natanzon confluent potentials an algebraic method is used to solve the bound state problem for the most general natanzon potential for which the schrodinger equation can be reduced to the confluent hypergeometric form hence confluent potentials the solution is obtained straightforwardly a simple argument is given to sustain that the natanzon potential is the most general confluent potential

a second class of solvable potentials related to the jacobi polynomials levai 1991 has used a simple method for generating solvable wave equations to construct potentials whose solutions include as factors jacobi polynomials of functions g x which solve specific differential equations application of this method to a previously unconsidered differential equation involving jacobi polynomials results in a new class of solvable potentials missing from earlier compilations an interesting property of these new potentials is that when two of their defining parameters are fixed they all possess the same energy eigenvalue spectrum independent of any change in a third parameter leading to an indefinite number of different potentials with the same energies

an approximate κ state solutions of the dirac equation for the generalized morse potential under spin and pseudospin symmetry by using an improved approximation scheme to deal with the centrifugal pseudo centrifugal term we solve the dirac equation for the generalized morse potential with arbitrary spin orbit quantum number in the presence of spin and pseudospin symmetry the analytic bound state energy eigenvalues and the associated upper and lower spinor components of two dirac particles are found by using the basic concepts of the nikiforov uvarov method we study the special cases when 1 l l 0 s wave the non relativistic limit and the limit when becomes zero kratzer potential model the present solutions are compared with those obtained by other methods

a unified treatment of exactly solvable and quasi exactly solvable quantum potentials by exploiting the hidden algebraic structure of the schrodinger hamiltonian namely the sl 2 we propose a unified approach of generating both exactly solvable and quasi exactly solvable potentials we obtain in this way two new classes of quasi exactly solvable systems one of which is of periodic type while the other hyperbolic

factorization of sturm liouville operators solvable potentials and underlying algebraic structure in this paper a general method of factorization of sturm liouville sl operators is provided a method to solve sl eigenvalue problems is presented new classes of exactly solvable potentials are obtained the supersymmetry and shape invariance approaches are generalized to the sl operators it is shown that the sl shape invariance potentials have an underlying algebraic structure this algebra is in general infinite dimensional the condition of finite algebra is obtained

mixture based extension of the ar model and its recursive bayesian identification an extension of the autoregressive ar model is studied which allows transformations and distortions on the regressor to be handled many important signal processing problems are amenable to this extended ar i e ear model it is shown that bayesian identification and prediction of the ear model can be performed recursively in common with the ar model itself the ear model does however require that the transformation be known when it is unknown the associated transformation space is represented by a finite set of candidates what follows is a mixture based ear model i e the mear model an approximate identification algorithm for mear is developed using a restricted variational bayes vb method this restores the elegant recursive update of sufficient statistics the mear model is applied to the robust identification of ar processes corrupted by outliers and burst noise respectively and to click removal for speech

transformation properties of the schrodinger equation with the momentum dependent potential x 2ord using suitable coordinate transformations enables us to convert the schrodinger equation with the momentum dependent potential x 2ord into a pair of coupled s wave poschl teller systems with zero energy an equivalent zero energy system with a related singular potential has also been derived this time by using an appropriate wavefunction transformation

a connection between multi linear and volterra systems the volterra system is a non linear system with the structure of the volterra series the volterra series is attractive from the system theoretic point of view since it enables to obtain the output of a class of non linear systems in terms of the input explicitly rather than involving input output coupling terms and allows substantial simplifications for the numerical simulation the volterra system allows to derive the stability condition as well i e obtain a bound on the output for a given bound on the input function especially for the bilinear system the bilinear system possesses the volterra series this paper derives the volterra series formalism from the multi linear system involving the coupling term attributed to k 1 th order non linearity and output function where 1

adaptive interpolation of discrete time signals that can be modeled as autoregressive processes this paper presents an adaptive algorithm for the restoration of lost sample values in discrete time signals that can locally be described by means of autoregressive processes the only restrictions are that the positions of the unknown samples should be known and that they should be embedded in a sufficiently large neighborhood of known samples the estimates of the unknown samples are obtained by minimizing the sum of squares of the residual errors that involve estimates of the autoregressive parameters a statistical analysis shows that for a burst of lost samples the expected quadratic interpolation error per sample converges to the signal variance when the burst length tends to infinity the method is in fact the first step of an iterative algorithm in which in each iteration step the current estimates of the missing samples are used to compute the new estimates furthermore the feasibility of implementation in hardware for real time use is established the method has been tested on artificially generated auto regressive processes as well as on digitized music and speech signals

learning from normalized local and global discriminative information for semi supervised regression and dimensionality reduction semi supervised dimensionality reduction is one of the important topics in pattern recognition and machine learning during the past decade laplacian regularized least square laprls and semi supervised discriminant analysis sda are the two widely used semi supervised dimensionality reduction methods in this paper we show that sda and laprls can be unified into a constrained manifold regularized least square framework the manifold term however cannot fully utilize the underlying discriminative information we thus introduce a new and effective semi supervised dimensionality reduction method called learning from local and global information llgdi to solve the problem the proposed llgdi method adopts a set of local classification functions to preserve both local geometrical and discriminative information of dataset it also adopts a global classification function to preserve the global discriminative information and an uncorrelated constraint to calculate the projection matrix for simultaneously solving regression and dimensionality reduction problem as a result the llgdi method is able to preserve local discriminative manifold information as well as the global discriminative information theoretical analysis and extensive simulations presented in the paper show the effectiveness of the llgdi algorithm the results also demonstrate llgdi can achieve superior performance compared with other existing methods

adaptive scheme for elimination of broadband noise and impulsive disturbances from ar and arma signals the problem of elimination of measurement noise and impulsive disturbances from autoregressive and autoregressive moving average signals is considered it is shown that the task of simultaneous detection tracking restoration can be stated as a nonlinear filtering problem and solved using the theory of extended kalman filter numerical tests carried out for real audio signals corrupted by both real and artificially generated disturbances confirm very good properties of the proposed algorithm

solvable potentials of shape invariance in two steps within the framework of supersymmetric quantum mechanics we obtain a class of solvable potentials of shape invariance in two steps where the parameters a1 and a2 of partner potentials are related to each other by translation a2 a1 it is found that discontinuity at some x points is a characteristic of the two step superpotentials therefore giving rise to dirac delta function singularities to the corresponding potentials

finite dimensional smoothers for map state estimation of bilinear systems in this paper we present two finite dimensional iterative algorithms for maximum a posteriori map state sequence estimation of bilinear systems bilinear models are appealing in their ability to represent or approximate a broad class of nonlinear systems our iterative algorithms for state estimation are based on the expectation maximization em algorithm and outperform the widely used extended kalman smoother eks unlike the eks these em algorithms are optimal in the map sense finite dimensional solutions to the state sequence estimation problem for bilinear models we also present recursive on line versions of the two algorithms and show that they outperform the extended kalman filter ekf our main conclusion is that the em based algorithms presented in this paper are novel nonlinear filtering methods that perform better than traditional methods such as the ekf