lie structures in parasupersymmetric quantum mechanics i the standard supersymmetrization procedure parasupersymmetric quantum mechanics leads to the construction of new lie structures that are called lie parasuperalgebras they are characterized by double commutation relations between even and odd generators and include as particular substructures the usual lie superalgebras oscillatorlike systems in one spatial dimension are especially exploited in order to put such structures in evidence but n dimensional extensions are easily determined these developments in the standard supersymmetrization procedure will be considered first and the spin orbit coupling one will be considered in a subsequent article j beckers and n debergh j math phys 1991

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

finite dimensional lie algebras of order f f lie algebras are natural generalizations of lie algebras f 1 and lie superalgebras f 2 when f 2 not many finite dimensional examples are known in this article we construct finite dimensional f lie algebras f 2 by an inductive process starting from lie algebras and lie superalgebras matrix realizations of f lie algebras constructed in this way from su n sp 2n so n and sl n m osp 2 m are given we obtain nontrivial extensions of the poincare algebra by inonu wigner contraction of certain f lie algebras with f 2

analysis of subspace fitting and ml techniques for parameter estimation from sensor array data it is shown that the multidimensional signal subspace method termed weighted subspace fitting wsf is asymptotically efficient this results in a novel compact matrix expression for the cramer rao bound crb on the estimation error variance the asymptotic analysis of the maximum likelihood ml and wsf methods is extended to deterministic emitter signals the asymptotic properties of the estimates for this case are shown to be identical to the gaussian emitter signal case i e independent of the actual signal waveforms conclusions concerning the modeling aspect of the sensor array problem are drawn

concentrated cramer rao bound expressions we present a method to simplify the analytical computation of the cramer rao bound the method circumvents bound calculations for so called nuisance parameters under mild regularity conditions the technique which replaces expectations with almost sure limits can significantly lower the analytical complexity as compared to traditional methods the dimension of a matrix that requires computation and inversion is reduced to the length of the parameter vector of interest we give applications to random variables having densities in the exponential family for normal distributions the resulting expressions take on particularly simple closed forms

reversible jump mcmc for joint detection and estimation of sources in colored noise this paper presents a novel bayesian solution to the difficult problem of joint detection and estimation of sources impinging on a single array of sensors in spatially colored noise with arbitrary covariance structure robustness to the noise covariance structure is achieved by integrating out the unknown covariance matrix in an appropriate posterior distribution the proposed procedure uses the reversible jump markov chain monte carlo mcmc method to extract the desired model order and direction of arrival parameters we show that the determination of model order is consistent provided a particular hyperparameter is within a specified range simulation results support the effectiveness of the method

turning bosons into fermions exclusion statistics fractional statistics and the simple harmonic oscillator motivated by haldane s exclusion statistics we construct creation and annihilation operators for g ons using a bosonic algebra we find that g ons appear due to the breaking of a discrete symmetry of the original bosonic system this symmetry is intimately related to the braid group and we demonstrate a link between exclusion statistics and fractional statistics

positive representations of general commutation relations allowing wick ordering we consider the problem of representing in hilbert space commutation relations of the form graphics where the tklij are essentially arbitrary scalar coefficients examples comprise the q canonical commutation relations introduced by greenberg bozejko and speicher and the twisted canonical anti commutation relations studied by pusz and woronowicz as well as the quantum group s u 2 using these relations any polynomial in the generators ai and their adjoints can uniquely be written in wick ordered form in which all starred generators are to the left of all unstarred ones in this general framework we define the fock representation as well as coherent representations we develop criteria for the natural scalar product in the associated representation spaces to be positive definite and for the relations to have representations by bounded operators in a hilbert space we characterize the relations between the generators ai not involving a i which are compatible with the basic relations the relations may also be interpreted as defining a non commutative differential calculus for generic coefficients tklij however all differential forms of degree 2 and higher vanish we exhibit conditions for this not to be the case and relate them to the ideal structure of the wick algebra and conditions of positivity we show that the differential calculus is compatible with the involution iff the coefficients t define a representation of the braid group this condition is also shown to imply improved bounds for the positivity of the fock representation finally we study the kms states of the group of gauge transformations defined by aj exp it aj

maximum likelihood direction of arrival estimation in unknown noise fields using sparse sensor arrays we address the problem of maximum likelihood ml direction of arrival doa estimation in unknown spatially correlated noise fields using sparse sensor arrays composed of multiple widely separated subarrays in such arrays intersubarray spacings are substantially larger than the signal wavelength and therefore sensor noises can be assumed to be uncorrelated between different subarrays this leads to a block diagonal structure of the noise covariance matrix which enables a substantial reduction of the number of nuisance noise parameters and ensures the identifiability of the underlying doa estimation problem a new deterministic ml doa estimator is derived for this class of sparse sensor arrays the proposed approach concentrates the ml estimation problem with respect to all nuisance parameters in contrast to the analytic concentration used in conventional ml techniques the implementation of the proposed estimator is based on an iterative procedure which includes a stepwise concentration of the log likelihood ll function the proposed algorithm is shown to have a straightforward extension to the case of uncalibrated arrays with unknown sensor gains and phases it is free of any further structural constraints or parametric model restrictions that are usually imposed on the noise covariance matrix and received signals in most existing ml based approaches to doa estimation in spatially correlated noise

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

wigner quantum oscillators we present three groups of non canonical quantum oscillators the position and momentum operators of each group generate basic lie superalgebras namely sl 1 3 osp 1 6 and osp 3 2 the sl 1 3 oscillators have finite energy spectrum and finite dimensions the osp 1 6 oscillators are related to the para bose statistics the internal angular momentum s of the osp 3 2 oscillators takes no more than three half integer values in a particular representation s 1 2

doa estimation and detection in colored noise using additional noise only data in a typical array processing scenario noise acting on the array can not be assumed spatially white it is in many cases necessary to use quiet periods when only noise is received to estimate the noise covariance if estimation of the signal parameters and noise covariance is performed jointly performance can be improved this is especially true when stationarity considerations limit the amount of available valid noise only data an asymptotically valid approximative maximum likelihood method aml for the estimation problem is derived in this work the resulting criterion is when concentrated with respect to the signal parameters relatively simple in numerical experiments aml shows promising small sample performance compared to earlier methods the criterion function is also well suited for numerical optimization the new criterion function allows for the development of a novel mode like non iterative estimation procedure if the array belongs to the important class of uniform linear arrays the resulting procedure retains the asymptotic properties of maximum likelihood and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting wsf formulation of mode for the detection problem no method has been presented that takes the unknown noise covariance into account here a well known detection scheme for wsf is extended to work in this scenario as well the derivations of this scheme further stress the importance of using the correct weighting in wsf when the noise covariance is unknown it is also shown that the minimum value of the criterion function associated with aml can be used for the detection purpose numerical experiments indicate very promising performance for the aml detection scheme

irreducible representations of the extended poincare parasuperalgebra we classify and explicitly construct all the irreducible hermitian representations of the extended poincar parasuperalgebra these representations which include the representations of the usual extended poincar superalgebra as a particular case can serve as a group theoretical foundation of parasupersymmetric quantum field theory i e as a general viewpoint to reformulate the quantum field theory and quantum mechanics of identical particles on the general basis of paraquantization and supersymmetry

realizability of a model in infinite statistics following greenberg and others we study a space with a collection of operatorsa k satisfying the q mutator relations a l a k a l k l corresponding forq 1 to classical bose and fermi statistics we show that then n matrixa n q representing the scalar products ofn particle states is positive definite for alln ifq lies between 1 and 1 so that the commutator relations have a hilbert space representation in this case this has also been proved by fivel and by bozejko and speicher we also give an explicit factorization ofa n q as a product of matrices of the form 1 q jt 1 with 1 j n andt a permutation matrix in particular a n q is singular if and only ifq m 1 for some integerm of the formk 2 k 2 k n

gaussian cramer rao bound for direction estimation of noncircular signals in unknown noise fields this paper focuses on the stochastic cramer rao bound crb on direction of arrival doa estimation accuracy for noncircular gaussian sources in the general case of an arbitrary unknown gaussian noise field parameterized by a vector of unknowns explicit closed form expressions of the stochastic crb for doa parameters alone are obtained directly from the slepian bangs formula for general noncircular complex gaussian distributions as a special case the crb under the nonuniform white noise assumption is derived our expressions can be viewed as extensions of the well known results by stoica and nehorai ottersten et al weiss and friedlander pesavento and gershman and gershman et al some properties of these crbs are proved and finally these bounds are numerically compared with the conventional crbs under the circular complex gaussian distribution for different unknown noise field models

a new kind of deformed calculus and parabosonic coordinate representation the general expression for the momentum operator p of parabose quantization theory in the coordinate x diagonal representation is derived from the heisenberg equations of motion based on this a new kind of deformed calculus is developed by its application the eigenequation for a free parabosonic hamiltonian in coordinate representation is solved and the results correspond exactly to the well known ones in number representation

relationships between adaptive minimum variance beamforming and optimal source localization for many years the popular minimum variance mv adaptive beamformer has been well known for not having been derived as a maximum likelihood ml estimator this paper demonstrates that by use of a judicious decomposition of the signal and noise the log likelihood function of source location is in fact directly proportional to the adaptive mv beamformer output power in the proposed model the measurement consists of an unknown temporal signal whose spatial wavefront is known as a function of its unknown location which is embedded in complex gaussian noise with unknown but positive definite covariance further in cases where the available observation time is insufficient a constrained ml estimator is derived here that is closely related to mv beamforming with a diagonally loaded data covariance matrix estimate the performance of the constrained ml estimator compares favorably with robust mv techniques giving slightly better root mean square error rmse angle of arrival estimation of a plane wave signal in interference more importantly however the fact that such optimal ml techniques are closely related to conventional robust mv methods such as diagonal loading lends theoretical justification to the use of these practical approaches

on supersymmetries in nonrelativistic quantum mechanics one dimensional nonrelativistic systems are studied when time independent potential interactions are involved their supersymmetries are determined and their closed subsets generating kinematical invariance lie superalgebras are pointed out the study of even supersymmetries is particularly enlightened through the already known symmetries of the corresponding schrodinger equation three tables collect the even odd and total supersymmetries as well as the invariance super algebras

optimum array design to maximize fisher information for bearing estimation source bearing estimation is a common application of linear sensor arrays the cramer rao bound crb sets a lower bound on the achievable mean square error mse of any unbiased bearing estimate in the spatially white noise case the crb is minimized by placing half of the sensors at each end of the array however many realistic ocean environments have a mixture of both white noise and spatially correlated noise in shallow water environments the correlated ambient noise can be modeled as cylindrically isotropic this research designs a fixed aperture linear array to maximize the bearing fisher information fi under these noise conditions the fi is the inverse of the crb so maximizing the fi minimizes the crb the elements of the optimum array are located closer to the array ends than uniform spacing but are not as extreme as in the white noise case the optimum array results from a trade off between maximizing the array bearing sensitivity and minimizing output noise power variation over the bear

maximum likelihood direction finding in spatially colored noise fields using sparse sensor arrays we consider the problem of maximum likelihood ml direction of arrival doa estimation of narrowband signals using sparse sensor arrays which consist of widely separated subarrays such that the unknown spatially colored noise field is uncorrelated between different subarrays we develop ml doa estimators under the assumptions of zero mean and non zero mean gaussian signals based on an expectation maximization em framework for doa estimation of non zero mean gaussian signals we derive the cramer rao bound crb as well as the asymptotic error covariance matrix of the ml estimator that improperly assumes zero mean gaussian signals we provide analytical and numerical performance comparisons for the existing deterministic and the proposed stochastic ml estimators the results show that the proposed estimators normally provide better accuracy than the existing deterministic estimator and that the nonzero means in the signals improve the accuracy of doa estimation

maximum likelihood doa estimation and asymptotic cramer rao bounds for additive unknown colored noise this paper is devoted to the maximum likelihood estimation of multiple sources in the presence of unknown noise with the spatial noise covariance modeled as a function of certain unknown parameters e g an autoregressive ar model a direct and systematic way is developed to find the exact maximum likelihood ml estimates of all parameters associated with the direction finding problem including the direction of arrival doa angles spl theta the noise parameters spl alpha the signal covariance spl phi sub s and the noise power spl sigma sup 2 we show that the estimates of the linear part of the parameter set spl phi sub s and spl sigma sup 2 can be separated from the nonlinear parts spl theta and spl alpha thus the estimates of spl phi sub s and spl sigma sup 2 become explicit functions of spl theta and spl alpha this results in a significant reduction in the dimensionality of the nonlinear optimization problem asymptotic analysis is performed on the estimates of spl theta and spl alpha and compact formulas are obtained for the cramer rao bounds crb s finally a newton type algorithm is designed to solve the nonlinear optimization problem and simulations show that the asymptotic crb agrees well with the results from monte carlo trials even for small numbers of snapshots

poschl teller paradoxes in the well known poschl teller trigonometric potential well a symmetric regularization x x ie of the impenetrable end point barriers is performed this leads to the four different solvable generalizations of the model as a byproduct the scheme clarifies certain apparent paradoxes encountered in the classically forbidden coupling regime

universal domination and stochastic domination estimation simultaneously under a broad class of loss functions several theoretical questions are resolved in this paper in particular the criterion of universal domination is shown to be equivalent to the criterion of stochastic domination that compares the estimators by the stochastic ordering of their euclidean distances from the estimators to the true parameter concrete results about universal domination relating to the usual estimator are also established in particular when x 0 has a p variate t distribution and p 1 2 there exists no estimator for 0 that universally dominates x however for p 2 3 estimators of the type of james stein positive part estimators that universally dominate x are specified when x has a p variate normal distribution with mean 0 and identity covariance matrix we show that for any dimension p no james stein positive part estimators universally dominate x however under slightly smaller classes of losses some jamesstein positive part estimators are shown to simultaneously dominate x these hitherto unstudied losses are bounded and fairly practical

maximum likelihood localization of multiple sources by alternating projection an algorithm referred to as apm for computing the maximum likelihood estimator of the locations of simple sources in passive sensor arrays is presented the algorithm is equally applicable to the case of coherent signals and to the case of a single snapshot the algorithm is iterative the maximum of the likelihood function is computed by successive approximations the convergence of the algorithm to the global maximum is demonstrated for a variety of scenarios the key to this global convergence is the initialization scheme

optimal utilization of signal free samples for array processing in unknown colored noise fields many algorithms for direction of arrival doa estimation require the noise covariance matrix to be known or to possess a known structure in many cases the noise covariance is in fact estimated from separate measurements this paper addresses the combined effects of finite sample sizes both in the estimated noise covariance matrix and in the data with signals present it is assumed that a batch of signal free samples is available in addition to the signal containing samples no assumption is made on the structure of the noise covariance in this paper the asymptotic covariance of the weighted subspace fitting wsf algorithm is derived for the case in which the data are whitened using an estimated noise covariance the expression obtained suggests an optimal weighting that improves performance compared to the standard choice in addition a new method based on covariance matching is proposed both methods are asymptotically statistically efficient the cramer rao lower bound crb on the covariance of the estimate for the data model is also derived monte carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results

bounds for maximum likelihood regular and non regular doa estimation in k distributed noise we consider the problem of estimating the direction of arrival of a signal embedded in k distributed noise when secondary data which contains noise only are assumed to be available based upon a recent formula of the fisher information matrix fim for complex elliptically distributed data we provide a simple expression of the fim with the two data sets framework in the specific case of k distributed noise we show that under certain conditions the fim for the deterministic part of the model can be unbounded while the fim for the covariance part of the model is always bounded in the general case of elliptical distributions we provide a sufficient condition for unboundedness of the fim accurate approximations of the fim for k distributed noise are also derived when it is bounded additionally the maximum likelihood estimator of the signal doa and an approximated version are derived assuming known covariance matrix the latter is then estimated from secondary data using a conventional regularization technique when the fim is unbounded an analysis of the estimators reveals a rate of convergence much faster than the usual t 1 simulations illustrate the different behaviors of the estimators depending on the fim being bounded or not

a nonlinear version of the gauss markov theorem abstract this article gives a nonlinear version of the gauss markov theorem it is shown that in a linear regression model y x u the lower bound for the risk matrix of a nonlinear estimator of belonging to a certain class is the covariance matrix of the gauss markov estimator provided the distribution of error term u belongs to the class of elliptically symmetric distributions with second moments

eigenstates of paraparticle creation operators eigenstates of the parabose and parafermi creation operators are constructed in the dirac contour representation the parabose eigenstates correspond to the dual vectors of the parabose coherent states in order p 2 conserved charge parabose creation operator eigenstates are also constructed the contour forms of the associated resolutions of unity are obtained

extended poincare parasuperalgebra with central charges and invariant wave equations we describe irreducible representations of the extended poincare parasuperalgebra ppsa which includes an arbitrary number n of parasupercharges n n n 2 central charges and an internal symmetry group we also discuss wave equations which are invariant with respect to the ppsa and propose a parasupersymmetric generalization of the wess zumino model for arbitrary p and n

parafermions for higher order extensions of the poincare algebra and their associated superspace parafermions of orders 2 and 3 are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the poincare algebra the corresponding superfields are constructed and some of their main properties are analyzed in detail in this context the existence problem of operators acting like covariant derivatives is analyzed and the associated operators are explicitly constructed

exclusion statistics operator algebras and fock space representations we study exclusion statistics within the second quantized approach we consider operator algebras with positive definite fock space and restrict them in a such a way that certain state vectors in fock space are forbidden ab initio we describe three characteristic examples of such exclusion namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space e g the calogero sutherland type of exclusion statistics exclusion in the single oscillator fock space where some states in single oscillator fock space are forbidden e g the gentile realization of exclusion statistics and a combination of these two exclusions e g green s realization of para fermi statistics for these types of exclusions we discuss the extended haldane statistics parameters g recently introduced by two of us in 1996 mod phys lett a 11 3081 and associated counting rules within these three types of exclusions in fock space the original haldane exclusion statistics cannot be realized

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

volumetric array prewhitening direction finders such as music experience loss of resolution and increased bias in the presence of nonwhite noise this paper presents two versions of a steepest descent gradient algorithm that prewhiten the signal received by an arbitrarily oriented volumetric sensor array minimizing these undesirable effects the algorithms optimize a whiteness functional over a surface with desirable properties including low dimensionality unimodality and concavity two ambient noise models facilitate algorithm development through succinct parameterization the first which is a novel linear matricial ambient noise model based on a spherical harmonic expansion places no constraints on array geometry the second model requires a spatially uniformly sampled sensor array and reduces problem dimensionality associated with exact multidimensional autoregressive modeling the algorithms estimate a stacked vector arrangement of the model parameters application with music demonstrates enhanced performance in terms of angular resolution and detection of low signal to noise ratio snr sources

a note on the unbiasedness of feasible gls quasi maximum likelihood robust adaptive and spectral estimators of the linear model on presente un resultat general qui etablit la symetrie autour du parametre reel des distributions d une large classe d estimateurs des parametres de la regression dans le modele lineaire

symmetry based inference in an instrumental variable setting we describe exact inference based on group invariance assumptions that specify various forms of symmetry in the distribution of a disturbance vector in a general nonlinear model it is shown that such mild assumptions can be equivalently formulated in terms of exact confidence sets for the parameters of the functional form when applied to the linear model this exact inference provides a unified approach to a variety of parametric and distribution free tests in particular we consider exact instrumental variable inference based on symmetry assumptions the unboundedness of exact confidence sets is related to the power to reject a hypothesis of underidentification in a multivariate instrumental variables context generalizations of anderson rubin confidence sets are considered

polynomial families and schrodinger equation one example for nonhypergeometric type of correspondence we discuss the explicit construction of the schrodinger equations admitting representation through some family of general nonorthogonal polynomials the specific choice of the third order polynomial coefficient functions that lead to quasi solvable families of schrodinger potentials is considered in detail

example of infinite statistics i give a second quantized example of infinite statistics in which any representation of the symmetric group can occur for identical particles and discuss the properties of the example

bayesian estimation and prediction using asymmetric loss functions abstract estimators and predictors that are optimal relative to varian s asymmetric linex loss function are derived for a number of well known models their risk functions and bayes risks are derived and compared with those of usual estimators and predictors it is shown that some usual estimators for example a scalar sample mean or a scalar least squares regression coefficient estimator are inadmissible relative to asymmetric linex loss by providing alternative estimators that dominate them uniformly in terms of risk

best median unbiased estimation in linear regression with bounded asymmetric loss functions abstract this article considers optimal median unbiased estimation in a linear regression model with the distribution of the errors lying in a subclass of the elliptically symmetric distributions the generalized least squares gls estimator is shown to be best for any monotone loss function that is any loss function that is nondecreasing as the magnitude of underestimation or overestimation increases this includes bounded asymmetric loss functions for the same loss functions a restricted gls estimator is shown to be best when the estimand is known to lie in an interval for the case of normal errors a best median unbiased estimator of the error variance 2 is given for restricted and unrestricted parameter spaces this estimator differs from the sample variance s 2 in comparison with best mean unbiased estimators of regression and variance parameters the best median unbiased estimators considered here take advantage of restrictions on the parameter space and are optimal with respect to a much wi

irreducible representations of the poincare parasuperalgebra we explicitly describe all the irreducible unitary representations of the poincare parasuperalgebra i e the parasupersymmetric extension of the lie algebra of the poincare group this parasuperalgebra includes as a particular case the usual poincare superalgebra and can serve as the group theoretical foundation of parasupersymmetric quantum field theory

lie structures in parasupersymmetric quantum mechanics ii the spin orbit coupling supersymmetrization procedure the spin orbit coupling supersymmetrization procedure is considered in parasupersymmetric quantum mechanics describing oscillatorlike systems the superposition of bosonic and parafermionic degrees of freedom is mainly studied in the cases of two and three spatial dimensions but the n dimensional case is also determined new lie superalgebras and lie parasuperalgebras are pointed out they differ from those already obtained in connection with the standard procedure of supersymmetrization studied in part i of this series j beckers and n debergh j math phys 1991

on the deterministic crb for doa estimation in unknown noise fields using sparse sensor arrays the cramer rao bound crb plays an important role in direction of arrival doa estimation because it is always used as a benchmark for comparison of the different proposed estimation algorithms in this correspondence using well known techniques of global analysis and differential geometry four necessary conditions for the maximum of the log likelihood function are derived two of which seem to be new the crb is derived for the general class of sensor arrays composed of multiple arbitrary widely separated subarrays in a concise way via a coordinate free form of the fisher information the result derived in 1 is confirmed