the use of mixed logit models to reflect heterogeneity in capture recapture studies summary we examine issues in estimating population size n with capture recapture models when there is variable catchability among subjects we focus on a logistic normal mixed model for which the logit of the probability of capture is an additive function of a random subject and a fixed sampling occasion parameter when the probability of capture is small or the degree of heterogeneity is large the log likelihood surface is relatively flat and it is difficult to obtain much information about n we also discuss a latent class model and a log linear model that account for heterogeneity and show that the log linear model has greater scope models assuming homogeneity provide much narrower intervals for n but are usually highly overly optimistic the actual coverage probability being much lower than the nominal level

analysis of capture recapture data with a rasch type model allowing for conditional dependence and multidimensionality summary in this article we show that if subjects are assumed to be homogeneous within a finite set of latent classes the basic restrictions of the rasch model conditional independence and unidimensionality can be relaxed in a flexible way by simply adding appropriate columns to a basic design matrix when discrete covariates are available so that subjects may be classified into strata we show how a joint modeling approach can achieve greater parsimony parameter estimates may be obtained by maximizing the conditional likelihood given the total number of captures with a combined use of the em and fisher scoring algorithms we also discuss a technique for obtaining confidence intervals for the size of the population under study based on the profile likelihood

braid group representations arising from quantum supergroups with arbitrary q and link polynomials the theory of quantum supergroups is applied to the construction of braid group representations and link invariants a method developed in an earlier publication is reviewed which produces link polynomials from braid group representations generated by quantum supergroup invariant r matrices when q is a generic complex parameter this method applied to the universal r matrices yields hierarchies of link polynomials associated with different irreps of quantum supergroups when q is a root of unity suitable constraints are imposed on quantum supergroups to turn them into finite dimensional quasitriangular z2 graded hopf algebras the corresponding universal r matrices are obtained which in turn lead to series of link polynomials upon using the above mentioned method as concrete examples the link polynomials related to the vector irreps of the quantum supergroups uq osp m 2n all the irreps of uq osp 1 2 with generic q and a class of doubly atypical irreps of uq gl 2 1 with q a root of unity

the presentation and q deformation of special linear lie superalgebras it is shown that the naive serre presentation corresponding to the simplest cartan matrix of the special linear lie superalgebra sl m n yields for m n 2 a larger lie superalgebra s m n of which sl m n is a nontrivial quotient the missing relations are found and the q deformation of the universal enveloping algebra of sl m n is constructed some auxiliary results related to associative algebras and e lie algebras defined in terms of generators and relations are also included

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

displacement structure for hankel vandermonde and related derived matrices abstract we introduce some generalized concepts of displacement structure for structured matrices obtained as products and inverses of toeplitz hankel and vandermonde matrices the toeplitz case has already been studied at some length and the corresponding matrices have been called near toeplitz or toeplitz like or toeplitz derived we focus on hankel and vandermonde like matrices and in particular show how the appropriately defined displacement structure yields fast triangular and orthogonal factorization algorithms for such matrices the main contribution of this paper is presenting a unified framework rather than obtaining the fastest algorithm for each special matrix

extended chandrasekhar recursions we extend the discrete time chandrasekhar recursions for least squares estimation in constant parameter state space models to a class of structured time variant state space models special cases of which often arise in adaptive filtering it can be shown that the much studied exponentially weighted recursive least squares filtering problem can be reformulated as an estimation problem for a state space model having this special time variant structure other applications arise in the multichannel and multidimensional adaptive filtering context

statistical inference for poisson and multinomial models for capture recapture experiments summary the classical multinomial model used for estimating the size of a closed population is compared to the highly flexible poisson models introduced by cormack 1981 the multinomial model and generalizations of it which allow for dependence between samples may be obtained from that of cormack by conditioning on the population size the maximum likelihood estimators for n the population size and 0 the vector of parameters describing the capture process are the same in both models completely general formulae for the asymptotic variances of the maximum likelihood estimates of n for both models are given the substantial differences between the variances under the two models are discussed hypotheses concerning 0 may be tested using the log likelihood ratio the procedures which result from both models are asymptotically equivalent under the null hypothesis but differ in power under the alternative

the spectral approximation of multiplication operators via asymptotic structured linear algebra abstract a multiplication operator on a hilbert space may be approximated with finite sections by choosing an orthonormal basis of the hilbert space multiplication operators with nonzero symbols defined on l 2 spaces of functions are never compact and then such approximations cannot converge in the norm topology instead we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol i e the multiplier we discuss the case of real orthogonal polynomial bases and the relations with the classical fourier basis whose choice leads to the well studied toeplitz case indeed the asymptotic approximation of the spectrum by the spectra of the associated toeplitz sections is possible only under precise geometric assumptions on the range of the symbol conversely the use of circulant approximations leads to constructive algorithms with o n log n complexity n number of sections working in general and generalizable to the separable multivariate and matrix valued cases as well

structured matrices and unconstrained rational interpolation problems we describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept we use the interpolation data to implicitly define a convenient non hermitian structured matrix and then apply a computationally efficient procedure for its triangular factorization this leads to a transmission line interpretation that makes evident the interpolation properties we further discuss connections with the lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

eigenvalues of casimir invariants for unitary irreps of a fully explicit formula for the eigenvalues of casimir invariants for u q gl m n is given which applies to all unitary irreps this is achieved by making some interesting observations on atypicality indices for irreps occurring in the tensor product of unitary irreps of the same type these results have applications in the determination of link polynomials arising from unitary irreps of u q gl m n

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

classification of all star and grade star irreps of gl n 1 a method for inducing nondegenerate forms on irreducible gl m n modules that implies some general results on star and grade star representations is investigated these results are applied to obtain a complete classification in terms of highest weights of the irreducible star and grade star representations of gl n 1 it is demonstrated that while gl n 1 admits a large class of star representations the irreducible grade star representations are comparatively rare moreover for n 2 all grade star irreducible representations are also star representations and for n 2 are atypical the superalgebra gl 2 1 proves to be a special case and admits a two parameter family of four dimensional typical grade star irreducible representations that are not star representations in particular typical grade star irreducible representations of gl n 1 exist only for n 1 2

inadmissibility of the maximum likelihood estimator in the sequential estimation of the size of a population summary for the sequential estimation of the size of a finite population the maximum likelihood estimator is a commonly used estimator but has no explicit expression in this paper it is shown that the maximum likelihood estimator is inadmissible an estimator which is simple in form and dominates the maximum likelihood estimator asymptotically is proposed to study the optimality property of the estimator for the stopping rule sb from a decision theoretical approach we investigate the admissibility of the estimator under a general weighted squared error loss function although the maximum likelihood estimator and its risk have no closed form expression the estimator will be shown to be inadmissible by means of a general theorem that we establish later as theorem 2 1 in this paper we only consider the simple capture capture model discussed by samuel 1968 more complex and realistic capture recapture models have been developed subsequently pollack otto 1983 pickands raghavachari 1987 bonett woodward bentler 1987 cormack 1989 huggins 1989 we conjecture that many of the

differential methods in inverse scattering this paper discusses a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium by writing the medium equations in the form of a two component system describing the interaction of rightward and leftward propagating waves the causality of the propagation phenomena is exploited in order to identify the medium layer by layer the recursive procedure that we obtain constitutes a continuous version of an algorithm first derived by schur in order to test for the boundedness of functions analytic inside the unit circle it recovers the local reflectivity function of the medium using similar ideas some other differential methods can also be derived to reconstruct alternative parametrizations of the layered medium in terms of the local impedance or of the potential function the differential inverse scattering methods turn out to be very efficient since in some sense they let the medium perform the inversion by itself and thus

fast recursive identification of state space models via exploitation of displacement structure abstract the seemingly computational burden of state space model identification has limited its real time application though it offers some important advantages over methods based on input output transfer functions it has been shown recently that ideas from the theory of displacement structure can be used in state space identification to reduce the computational burden of batch processing from o mn 2 to o mn flops when the data matrix is of size m n where n m however in many on line identification scenarios with slowly time varying systems it is desirable to update the model as time goes on with the minimal computational burden in this paper we extend our results of the batch processing algorithm to allow updating of the identified state space model with o m 2 flops again the theories of displacement structure and of the fast subspace decomposition fsd technique play crucial roles in the realization of the fast updating algorithm some computer simulation results are also presented

multilist population estimation with incomplete and partial stratification multilist capture recapture methods are commonly used to estimate the size of elusive populations in many situations lists are stratified by distinguishing features such as age or sex stratification has often been used to reduce biases caused by heterogeneity in the probability of list membership among members of the population however it is increasingly common to find lists that are structurally not active in all strata we develop a general method to deal with cases when not all lists are active in all strata using an expectation maximization em algorithm we use a flexible log linear modeling framework that allows for list dependencies and differential probabilities of ascertainment in each list finally we apply our method of estimating population size to two examples

inference for poisson and multinomial models for capture recapture experiments summary capture recapture models have been formulated both as poisson and as multinomial distributions maximum likelihood estimates of parameters under the two models are compared for parameters which do not involve the population size the asymptotic covariances are shown to be the same for certain classes of capture recapture experiments both poisson and multinomial models can be used in the poisson models cormack 1979 the numbers of observed individuals with different total capture histories w are regarded as independent poisson variates with means npf o n being some measure of the unknown size of the population and p 0 being specified functions of parameters 0 representing the demography of the population and the behavioural interaction of individuals with the sampling process in the multinomial models darroch 1958 n individuals are regarded as being multinomially distributed into a number of capture histories the observable ones with probabilities ph and the unobservable category with probability l p where p xp s 1 sandland cormack 1984 explored the relationships between the poisson and multinomial models and showed that the asymptotic variance of the maximum likelihood estimate of n under the poisson model is related to that obtained fienberg 1972 under the multinomial model by

logistic regression in capture recapture models the effect of population heterogeneity in capture recapture or dual registration models is discussed an estimator of the unknown population size based on a logistic regression model is introduced the model allows different capture probabilities across individuals and across capture times the probabilities are estimated from the observed data using conditional maximum likelihood the resulting population estimator is shown to be consistent and asymptotically normal a variance estimator under population heterogeneity is derived the finite sample properties of the estimators are studied via simulation an application to finnish occupational disease registration data is presented

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

fast matrix factorizations via discrete transmission lines abstract fast algorithms associated with the names of schur and levinson are known for the triangular factorization of symmetric positive definite toeplitz matrices and their inverses in this paper we show that these algorithms can be derived from simple arguments involving causality symmetry and energy conservation in discrete lossless transmission lines the results not only provide a nice interpretation of the classical schur and levinson algorithms and a certain toeplitz inversion formula of gohberg and semencul but they also show immediately that the same fast algorithms apply not only to toeplitz matrices but to all matrices with so called displacement inertia 1 1 the results have been helpful in suggesting new digital filter structures and in the study of nonstationary second order processes

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

stabilized hyperbolic householder transformations a modification of the hyperbolic householder scheme is introduced which is demonstrably stable theoretically according to an established stability criterion and which exhibits superior numerical behavior in simulations the modified transform scheme effects downdating by applying conventional orthonormal rather than hyperbolic householder transformations to the data the latter have preferable numerical properties however the construction of these orthonormal operators itself requires hyperbolic computations thus the proposed method is in some sense half hyperbolic and half orthonormal there is no computational penalty incurred with these stabilized hyperbolic householder transforms they enjoy an operation count identical to their conventional counterparts

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

a general theory for modeling capture recapture data from a closed population a general theory for estimating the size of a closed population from multiple recapture data is presented this theory is easily extended to open population models for multiple recapture data estimation is based on a log linear model developed for modeling dependent capture recapture data when capture probabilities vary temporally with behavioral response and are heterogeneous among animals models for complete capture history data are developed along with the first log linear models for removal data the maximum likelihood estimator of the model parameters along with the estimated covariance matrix are presented

remarks on the origin of the displacement rank concept an account is given of how a relationship was found between some equations of chandrasekhar for solving the wiener hopf integral equations and the nonlinear riccati differential equations encountered in linear least squares estimation problems for systems described in state space form a paper by casti kalaba and murthy 1972 provided the inspiration for seeking such a relationship and then extending it and applying it in many other fields using the concept of displacement structure

column reduced rational matrix functions with given null pole data in the complex plane abstract explicit formulas are given for rational matrix functions which have a prescribed null pole structure in the complex plane and are column reduced at infinity a full parametrization of such functions is obtained the results are specified and developed further for matrix polynomials

fast algorithms for generalized displacement structures and lossless systems abstract we derive an efficient recursive procedure for the triangular factorization of strongly regular matrices with generalized displacement structure that includes as special cases a variety of previously studied classes such as toeplitz like and hankel like matrices the derivation is based on combining a simple gaussian elimination procedure with displacement structure and leads to a transmission like interpretation in terms of two cascades of first order sections we further derive state space realizations for each section and for the entire cascades and show that these realizations satisfy a generalized embedding result and a generalized notion of j losslessness the cascades turn out to have intrinsic blocking properties which can be shown to be equivalent to interpolation constrains

on drazin inverse of singular toeplitz matrix in this paper we show that the drazin inverse of singular toeplitz matrix can be represented as a sum of products of lower and upper triangular toeplitz matrices and give the explicit expression which generalizes the well known result by gohberg and semencul regarding the inverse of toeplitz matrix

coupling operators wedderburn forney spaces and generalized inverses abstract the pole zero structure of a transfer function has been analyzed by two distinct approaches module theoretic coordinate free and analytic basis dependent the former the wyman sain conte perdon school emphasizes pole modules zero modules exact sequences of maps and abstract realization theory applications developed include the model matching problem zero structure of one sided inverses and connections with geometric systems theory the latter the gohberg school works with pole chains zero chains matrix equations and concrete factorization theory minimal wiener hopf inner outer and interpolation zero pole lagrange sylvester nevanlinna pick in this paper we make explicit the connection between the two approaches in particular we describe the zero pole exact sequence of a transfer function without a pole or zero at infinity in the coordinate system provided by a global left null pole triple an important tool for us and an object of interest in its own right is a generalized inverse g of a transfer function g we relate the poles of g with the zeros of g we also show that if the pole module of g is isomorphic to the zero module of g then left right pole pairs of g are left right null pairs of g

an estimator of number of species from quadrat sampling summary we consider the problem of estimating the number of distinct species s in a study area from the recorded presence or absence of species in each of a sample of quadrats a generalized jackknife estimator of s is derived along with an estimate of its variance it is compared with the jackknife estimator for s proposed by heltshe and forrester 1983 biometrics39 1 12 and the empirical bayes estimator of mingoti and meeden 1992 biometrics48 863 875 we show that the empirical bayes estimator has the form of a generalized jackknife estimator under a specific model for species distribution we compare the new estimators of s to the empirical bayes estimator via simulation we characterize circumstances under which each is superior

applications and extensions of chao s moment estimator for the size of a closed population summary this article revisits chao s 1989 biometrics 45 427 438 lower bound estimator for the size of a closed population in a mark recapture experiment where the capture probabilities vary between animals model mh first an extension of the lower bound to models featuring a time effect and heterogeneity in capture probabilities mth is proposed the biases of these lower bounds are shown to be a function of the heterogeneity parameter for several loglinear models for mth small sample bias reduction techniques for chao s lower bound estimator are also derived the application of the loglinear model underlying chao s estimator when heterogeneity has been detected in the primary periods of a robust design is then investigated a test for the null hypothesis that chao s loglinear model provides unbiased abundance estimators is provided the strategy of systematically using chao s loglinear model in the primary periods of a robust design where heterogeneity has been detected is investigated in a monte carlo experiment its impact on the estimation of the population sizes and of the survival rates is evaluated in a monte carlo experiment

generalized t j model by parametrizing the t j model we present a new electron correlation model with one free parameter in one dimension this model is of suq 1 mod 2 symmetry the energy spectra are shown to be modulated by the free parameter in the model the solution and symmetric structures of the hilbert space as well as the bethe ansatz approach are discussed for special cases

mark recapture models with parameters constant in time the jolly seber method which allows for both death and immigration is easy to apply but often requires a larger number of parameters to be estimated tha would otherwise be necessary if i survival rate phi or ii probability of capture p or iii both phi and p can be assumed constant over the experimental period models with a reduced number of parameters are desirable in the present paper maximum likelihood ml solutions for these three situations are derived from the general ml equations of jolly 1979 in sampling biological populations r m cormack g p patil and d s robson eds 277 282 a test is proposed for heterogeneity arising from a breakdown of assumptions in the general jolly seber model tests for constancy of phi and p are provided an example is given in which these models are fitted to data from a local butterfly population

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

population size estimation in a two list surveillance system with a discrete covariate capture recapture methods are widely adopted to estimate sizes of populations of public health interest using information from surveillance systems for a two list surveillance system with a discrete covariate a population is divided into several subpopulations a unified framework is proposed in which the logits of presence probabilities are decomposed into case effects and list effects the estimators for the whole population and subpopulation sizes their adjusted versions and asymptotic standard errors admit closed form expressions asymptotic and bootstrap individual and simultaneous confidence intervals are easily constructed conditional likelihood ratio tests are used to select one from three possible models real examples are investigated

a schur algorithm and linearly connected processor array for toeplitz plus hankel matrices a levinson durbin type algorithm for solving toeplitz plus hankel t h linear systems of equations is used to induce a schur type algorithm for such systems a schur type algorithm is defined as one which efficiently computes the ldu decomposition of the matrix on the other hand levinson durbin type algorithms are defined as those algorithms which efficiently compute the udl decomposition of the inverse of a matrix it is shown that the schur algorithm so obtained is amenable to efficient implementation on a linearly connected array of processors in a manner which generalizes the results of s y kung and y h ku 1983 for symmetric toeplitz matrices specifically if t h is of order n then the schur algorithm runs on o n processors in o n time

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

a look ahead bareiss algorithm for general toeplitz matrices the bareiss algorithm is one of the classical fast solvers for systems of linear equations with toeplitz coefficient matrices the method takes advantage of the special structure and it computes the solution of a toeplitz system of order n with only o n 2 arithmetic operations instead of o n 3 operations however the original bareiss algorithm requires that all leading principal submatrices be nonsingular and the algorithm is numerically unstable if singular or ill conditioned submatrices occur in this paper an extension of the bareiss algorithm to general toeplitz systems is presented using look ahead techniques the proposed algorithm can skip over arbitrary blocks of singular or ill conditioned submatrices and at the same time it still fully exploits the toeplitz structure implementation details and operations counts are given and numerical experiments are reported we also discuss special versions of the proposed look ahead bareiss algorithm for hermitian indefinite toeplitz systems and banded toeplitz systems

the hyberbolic singular value decomposition and applications a new generalization of the singular value decomposition svd the hyperbolic svd is advanced and its existence is established under mild restrictions the hyperbolic svd accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products signal processing applications where this task arises include the covariance differencing algorithm for bearing estimation in sensor arrays sliding rectangular windowing and array calibration two algorithms for effecting this decomposition are detailed one is sequential and follows a similar pattern to the sequential bidiagonal based svd algorithm the other is for parallel implementation and mimics hestenes svd algorithm 1958 numerical examples demonstrate that like its conventional counterpart the hyperbolic svd exhibits superior numerical behavior relative to explicit formation and solution of the normal equations furthermore the hyperbolic svd applies in problems where the conventional svd cannot be employed

displacement structure approach to q adic polynomial vandermonde and related matrices abstract in the present paper a new class of the so called q adic polynomial vandermonde like matrices over an arbitrary non algebraically closed field is introduced this class generalizes both the simple and the confluent polynomial vandermonde like matrices over the complex field and the q adic vandermonde and the q adic chebyshev vandermonde like matrices studied earlier by different authors three kinds of displacement structures and two kinds of fast inversion formulas are obtained for this class of matrices by using displacement structure matrix method which generalize the corresponding results of the polynomial vandermonde like and the q adic vandermonde like matrices

a general methodology for the analysis of capture recapture experiments in open populations we trace the development of a likelihood function representation for the open population capture recapture jolly seber experiment we find that the modelling of the birth process in the general model is not consistent with the reduced death only model and that all formulations to date lead to difficulties in imposing constraints upon the parameters of the birth process we propose a generalisation to the usual jolly seber representation that models births using a multinomial distribution from a super population we show how this leads to simplifications in the numerical optimization of the likelihood and how constraints upon the parameters of the model can now be easily imposed we show how covariate models using auxiliary variables such as sampling effort or weather conditions to explain capture or survival rates can also be easily added we also show how this model can be generalised to more than one group of animals finally a numerical example is provided which fits a class of models where the capture probabilities survival probabilities and birth probabilities can each vary over time or among groups or both this permits sequential model fitting within a comprehensive model framework an approach akin to that of lebreton et al ecological monographs 62 67 118

finite dimensional irreducible representations of the quantum supergroup uq gl m n finite dimensional irreducible representations of the quantum supergroup uq gl m n at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction the representation theory is rather similar to that of gl m n at generic q but drastically different when q is a root of unity in the latter case atypicality conditions of highest weight irreducible representations irreps are substantially altered and such finite dimensional irreps arise that do not have highest weight and or lowest weight vectors as concrete examples the irreps of uq gl 2 1 are classified

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

log linear models for capture recapture summary log linear models are developed for capture recapture experiments and their advantages and disadvantages discussed ways in which they can be extended sometimes with only partial success to open populations subpopulations trap dependence and long chains of recapture periods are presented the use of residual patterns and analysis of subsets of data to identify behavioural patterns and acceptable models is emphasised and illustrated with two examples 1 capture recapture the biological problem with which we are concerned is that of estimating the parameters of a free living population of unknown size from a sequence of s samples in each of which any individual either is or is not observed the models in this paper require that each animal be marked naturally or artificially in such a way that we can observe the numbers of animals which have any particular pattern of being or not being observed through the sequence of samples there are t 2s 1 different observable patterns which we shall refer to as capture histories although there is no requirement for physical capture and denote each pattern by a sequence of l s and 2 s representing respectively seen or not seen in a particular sample a general pattern of capture history will be denoted a with ra the number of individuals observed to have this pattern thus for example with s 3 samples and a 121 r 2 is the number of animals seen in the first and third samples but not in the second the pattern with all 2 s is by definition unobservable in general the population will be subject to demographic change by birth or immigration and by death or emigration the simplest assumption for the response of an animal to the observer one which underlies the classical petersen schnabel or jolly seber estimators is that every animal alive in the population at a sampling time has the same probability of being observed this probability may vary from one sample to another or be constant over all or a number of samples if this assumption fails it may do so for either or both of two reasons the first is that of trap dependence the probability of an individual being observed in a sample depending on its past history of capture the second is that of heterogeneity between individuals who because of differences in their behaviour or in where they live inherently have different probabilities of being observed in any sample the basic idea of heterogeneity refers to properties of individuals which remain constant over time but individual animals may also have time varying behaviour the aspect most commonly considered being temporary departure of an individual from the population various excellent texts describe theoretical and practical aspects of capture recapture notably the comprehensive one by seber 1982 the introduction by begon 1979 the

universal r matrices and invariants of quantum supergroups a general method is developed for constructing invariants of quantum supergroups using universal r matrices applied to uq gl 2 1 this method yields the invariants of this quantum supergroup in explicit form the eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed

inverse scattering and minimal partial realizations we present an inverse scattering interpretation of the classical minimal partial realization problem posed in a slightly generalized context our approach starts by considering a canonical cascade form structure for the realization of arbitrary transfer functions where the cascade structure can be interpreted as the description of a layered wave scattering medium in this context the partial realization problem calls for a recursive process of layer identification from a given input response pair the scattering data the realization algorithm uses a causality principle to progressively determine the parameters of cascaded linear 2 ports that model the successive wave interaction layers this method for approaching the realization problem turns out to fit nicely into a framework that was also used to obtain fast structured linear estimation algorithms and cascade realizations for digital filters

modified martingale estimation for recapture experiments with hetrogeneous capture probabilities summary an estimation method of lloyd yip 1991 based on martingales and the assumption of beta distributed capture probabilities is modified to improve its performance for all but highly heterogeneous populations this estimator is compared to the jackknife estimator by simulation under both beta and non beta conditions some expressions for standard error are developed but found to behave poorly for realistic sample sizes bootstrap estimates of standard error appear superior but do not lead to accurate confidence intervals because of the difficulties in bias estimation

a recursive method for solving unconstrained tangential interpolation problems an efficient recursive solution is presented for the one sided unconstrained tangential interpolation problem the method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data the recursive procedure admits a physical interpretation in terms of discretized transmission lines in this framework the generating system is constructed as a cascade of first order sections singular steps occur only when the input data is contradictory i e only when the interpolation problem does not have a solution various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants the algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general hermite matrix interpolation problem a recursive method is proposed to compute a column reduced generating system that can be used to solve the minimal tangential interpolation problem

simple capture recapture models permitting unequal catchability and variable sampling effort we consider two capture recapture models that imply that the logit of the probability of capture is an additive function of an animal catchability parameter and a parameter reflecting the sampling effort the models are special cases of the rasch model and satisfy the property of quasi symmetry one model is log linear and the other is a latent class model for the log linear model point and interval estimates of the population size are easily obtained using standard software such as glim

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

efficiency of martingale methods in recapture studies summary the asymptotic variances of various estimators of the population size from a recapture study are calculated the estimation methods compared are maximum likelihood and a class of natural martingale estimators with both time variation and behavioural response it is found that i the martingale methods appropriate for time variation or behavioural response are fully efficient ii there is no loss of efficiency at all in allowing for time variation iii there is large loss of efficiency in allowing for behavioural response iv there is a natural martingale method that allows for both time variation and behavioural response

on discrete schrodinger equations and their two component wave equation equivalents an approach to inverse scattering problems for discrete schrodinger equations which are discrete three term recursions is presented by systematically transforming them into discrete two component wave propagation equations the wave propagation equations permit the immediate application of certain computationally efficient and physically insightful layer peeling algorithms for inverse scattering the mapping of three term recursions to two component evolution equations is one to many because the relation between the potential sequence parametrizing schrodinger equations and the reflection coefficient sequence determining local wave interaction is a nonlinear difference equation this mapping is examined in some detail and it is used to study both direct and inverse scattering problems associated with discrete schrodinger equations

variances of mark recapture estimates when mark recapture data are analysed by log linear models as described by cormack 1989 biometrics 45 395 413 the analysis is in terms of parameters that are related nonlinearly to the parameters of biological interest for several models formulae are given expressing the variance of the interesting parameter estimates in terms of those provided by a computer log linear analysis a device is presented that allows such expressions to be computed simply

quantum group invariants and link polynomials a general method is developed for constructing quantum group invariants and determining their eigenvalues applied to the universal matrix this method leads to the construction of a closed formula for link polynomials to illustrate the application of this formula the quantum groups uq es uq so 2m 1 and uq gl m are considered as examples and corresponding link polynomials are obtained

classification of finite dimensional unitary irreps for uq gl m n all finite dimensional irreducible unitary representations of the quantum supergroup uq gl m n are classified in terms of their highest weights for generic q 0

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

low risk population size estimates in the presence of capture heterogeneity summary population estimation methods are used for estimating the size of a population from samples of individuals in many applications the probability of being observed in the sample varies across individuals resulting in sampling bias we show that in this setting estimators of the population size have high and sometimes infinite risk leading to large uncertainty in the population size as an alternative we propose estimating the population of individuals with observation probability exceeding a small threshold we show that estimators of this quantity have lower risk than estimators of the total population size the proposed approach is shown empirically to result in large reductions in mean squared error in a common model for capture recapture population estimation with heterogeneous capture probabilities

computational aspects of n mixture models the n mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication royle 2004 biometrics 60 105 115 we explain and exploit the equivalence of n mixture and multivariate poisson and negative binomial models which provides powerful new approaches for fitting these models we show that particularly when detection probability and the number of sampling occasions are small infinite estimates of abundance can arise we propose a sample covariance as a diagnostic for this event and demonstrate its good performance in the poisson case infinite estimates may be missed in practice due to numerical optimization procedures terminating at arbitrarily large values it is shown that the use of a bound k for an infinite summation in the n mixture likelihood can result in underestimation of abundance so that default values of k in computer packages should be avoided instead we propose a simple automatic way to choose k the methods are illustrated by analysis of data on hermann s tortoise testudo hermanni

unified maximum likelihood estimates for closed capture recapture models using mixtures summary agresti 1994 biometrics50 494 500 and norris and pollock 1996a biometrics52 639 649 suggested using methods of finite mixtures to partition the animals in a closed capture recapture experiment into two or more groups with relatively homogeneous capture probabilities this enabled them to fit the models mh mbh norris and pollock and mth agresti of otis et al 1978 wildlife monographs62 1 135 in this article finite mixture partitions of animals and or samples are used to give a unified linear logistic framework for fitting all eight models of otis et al by maximum likelihood likelihood ratio tests are available for model comparisons for many data sets a simple dichotomy of animals is enough to substantially correct for heterogeneity induced bias in the estimation of population size although there is the option of fitting more than two groups if the data warrant it

a q superdimension formula for irreps of type i quantum superalgebras we derive a closed formula for the q superdimensions of a wide class of irreps including all unitary irreps of the type i quantum superalgebras

bias reduction for multiple recapture estimators of closed population size a bias correction for multiple recapture sampling is proposed the correction is similar to chapman s 1951 university of california publication in statistics 1 131 160 correction of the lincoln petersen estimator of population size small sample properties of the modified population size estimator are examined using monte carlo simulation substantial reductions in estimator bias and variance are obtained with the modified estimator

classification of unitary and grade star irreps for uq osp 2 2n all the unitary and grade star irreps of the quantum supergroup uq osp 2 2n are classified in terms of their highest weight labels

an improved nonparametric lower bound of species richness via a modified good turing frequency formula summary it is difficult to accurately estimate species richness if there are many almost undetectable species in a hyperdiverse community practically an accurate lower bound for species richness is preferable to an inaccurate point estimator the traditional nonparametric lower bound developed by chao 1984 scandinavian journal of statistics 11 265 270 for individual based abundance data uses only the information on the rarest species the numbers of singletons and doubletons to estimate the number of undetected species in samples applying a modified good turing frequency formula we derive an approximate formula for the first order bias of this traditional lower bound the approximate bias is estimated by using additional information namely the numbers of tripletons and quadrupletons this approximate bias can be corrected and an improved lower bound is thus obtained the proposed lower bound is nonparametric in the sense that it is universally valid for any species abundance distribution a similar type of improved lower bound can be derived for incidence data we test our proposed lower bounds on simulated data sets generated from various species abundance models simulation results show that the proposed lower bounds always reduce bias over the traditional lower bounds and improve accuracy as measured by mean squared error when the heterogeneity of species abundances is relatively high we also apply the proposed new lower bounds to real data for illustration and for comparisons with previously developed estimators

generalized chandrasekhar recursions from the generalized schur algorithm presents a new approach to the chandrasekhar recursions and some generalizations thereof the derivation uses the generalized schur recursions which are o n sup 2 recursions for the triangular factorization of n spl times n matrices having a certain toeplitz like displacement structure it is shown that when the extra structure provided by an underlying state space model is properly incorporated into the generalized schur algorithm it reduces to the chandrasekhar recursions which are o nn sup 2 recursions for estimating the n dimensional state of a time invariant or constant parameter system from n measured outputs it is further noted that the generalized schur algorithm factors more general structured matrices and this fact is readily used to extend the chandrasekhar recursions to a class of time variant state space models special cases of which often arise in adaptive filtering

capture recapture estimation with samples of size one using frequency data summary one way in which a capture recapture experiment can be designed is to count each capture occasion as a separate sample and tally the number of individuals caught once twice and so on this paper discusses the estimators currently used for this type of experiment and suggests some new ones each of the estimators is categorized by its derivation following this their properties are discussed particularly their bias and variance with the aim of choosing one or two of the estimators which are the best to use especially in handling a heterogeneous population it is found that the bias adjusted estimator of chao 1989 is the best to use when the number of captures is relatively small and that the estimator of darroch ratcliff 1980 should be used otherwise