schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

the schur algorithm applied to the one dimensional continuous inverse scattering problem the one dimensional continuous inverse scattering problem can be solved by the schur algorithm in the discrete time domain using sampled scattering data the sampling rate of the scattering data should be increased to reduce the discretization error but the complexity of the schur algorithm is proportional to the square of the sampling rate to improve this tradeoff between the complexity and the accuracy we propose a schur algorithm with the richardson extrapolation sare the asymptotic expansion of the schur algorithm necessary for the richardson extrapolation is derived in powers of the discretization step which shows that the accuracy order with respect to the discretization step of the schur algorithm is 1 the accuracy order of the sare with the n step richardson extrapolation is increased to n 1 with comparable complexity to the schur algorithm therefore the discretization error of the schur algorithm can be decreased in a computationally efficient manner by the sare

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

layer stripping for the helmholtz equation we develop a new layer stripping technique for the inverse scattering problem for the one dimensional helmholtz equation on the half line the technique eliminates the use of trace formulas relying instead on a nonlinear plancherel equality that provides a simple and precise characterization of the reflection data we prove both convergence of the algorithm and wellposedness of the forward and inverse scattering problems

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

linear and nonlinear inverse scattering in this paper we discuss one dimensional scattering and inverse scattering for the helmholtz equation on the half line from the point of view of the layer stripping by full or nonlinear scattering we mean the transformation between the index of refraction actually half of its logarithmic derivative and the reflection coefficient we refer to this mapping as nonlinear scattering because the mapping itself is nonlinear another appropriate name is multiple scattering as this model includes the effects of multiple reflections by linear scattering we mean the born or single scattering approximation this is the frechet derivative of the full scattering transform at the constant index of refraction which can be calculated to be exactly the fourier transform in j sylvester d p winebrenner and f gylys colwell siam j appl math 56 1996 pp 736 754 we introduced a variant of layer stripping based on causality and the riesz transform rather than on trace formulas see a brickstein and

quasi projective quotients by compact equivalence relations in 9 iii we tried to construct quasi projective moduli schemes m h for certain moduli functors 4 h of polarized compact complex manifolds x with hilbert polynomial h however being unable to handle the compact part of the equivalence relation we could achieve our aim only after changing the definition of the moduli functor 9 iii 1 10 instead of considering the functor of polarized manifolds up to q numerical equivalence as it was done in 6 and 7 we allowed only isomorphisms of the invertible sheaves giving the polarization in this article we want to show that our cheating was unnecessary and we will prove

uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions in this paper we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

direct and inverse scattering in the time domain via invariant imbedding equations this paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain which is applied here to the case of a one dimensional lossless medium the method is based on an integro differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques an example of an exact solution to this equation is given numerical schemes for solving the direct and inverse problems are derived by discretizing this equation the behavior of the resulting algorithms is then tested on several examples the results suggest that the inversion method is stable in the presence of additive noise

some matrix factorization identities for discrete inverse scattering this note analyzes the relationships between several inverse scattering methods and points out that all classical approaches implicitly construct a lower upper triangular factorization of a given positive definite toeplitz matrix it is shown that the various inversion methods implicitly obtain the factorization by solving different nested sets of linear equations and expressing the factors in terms of the solutions obtained the fact that the triangular factorization of a matrix with nonzero leading minors is unique immediately yields various identities e g involving the so called central mass sequence that were derived in the literature with considerably more manipulation in fact our basic factorization results are somewhat more general since they do not require the toeplitz assumption

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

differential and integral methods for multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this procedure operates by recursively reconstructing the radon transform of the potential from the jump in the radon transform of the scattered field at the wave front this reconstructed potential is then used to propagate the wave front and scattered field differentially further into the support of the potential the connections between this differential procedure and integral equation procedures are then illustrated by the derivations of two well known exact integral equation procedures using the radon transform and a generalized radon transform these procedures as well as the layer stripping procedure are then reduced to the familiar born approximation result for this problem by neglecting multiple scattering events this illustrates the central role of the radon transform in both exact and approximate inversion procedures

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

on the comparison principle for unbounded solutions of elliptic equations with first order terms abstract we prove a comparison principle for unbounded weak sub super solutions of the equation u div a x d u h x d u in where a x is a bounded coercive matrix with measurable ingredients 0 and h x has a super linear growth and is convex at infinity we improve earlier results where the convexity of h x was required to hold globally

generalized schur parameters and the effects of perturbations abstract let r be a matrix of the form r uu t vv t where u and v are real lower triangular toeplitz matrices and the main diagonal of v consists of zeros lozenge algorithms relating u v and schur parameters for r are presented using two known descriptions of the cholesky factors of r an explicit formula for the schur parameters generalizing the levinson durbin formula is proved statistical estimates of the effects of variations in u and v on the schur parameters are derived

reconstructing the transition rate function of a broadwell random walk from exit times in this paper we utilize the layer stripping method a method used in seismology by geophysicists to study a stochastic inverse problem arising from a one dimensional broadwell process the broadwell process can be described as a random walk of a particle that transitions with a spatially dependent flip rate between two states associated with the same speed but with different signs of velocities our goal is to reconstruct the spatially dependent flip rate function of a one dimensional broadwell process from exit time distributions out of a finite interval in principle we are able to reconstruct flip rate functions with error within o 10 2 from o 10 5 exit times when the particle speed is unity for smaller particle speeds the noise increases for a fixed number of exit times this method is less time consuming compared with traditional projection methods which involve optimization

eigenvector computation for almost unitary hessenberg matrices and inversion of szegő vandermonde matrices via discrete transmission lines abstract in this paper we use a discrete transmission line model known to geophysicists as a layered earth model to derive several computationally efficient solutions for the following three problems i as is well known a hessenberg matrix capturing recurrence relations for szeg polynomials differs from unitary only by its last column hence the first problem is how to rapidly evaluate the eigenvectors of this almost unitary hessenberg matrix ii the second problem is to design a fast o n 2 algorithm for inversion of szeg vandermonde matrices generalizing the well known traub algorithm for inversion of the usual vandermonde matrices iii finally the third problem is to extend the well known horner rule to evaluate a polynomial represented in the basis of szeg polynomials as we shall see all three problems are closely related and their solutions can be computed by the same family of fast algorithms although all the results can be derived algebraically here we reveal a connection to system theory to deduce these algorithms via elementary operations on signal flow graphs for digital filter structures including the celebrated markel gray filter widely used in speech processing and certain other filter structures this choice not only clarifies the derivation and suggests a variety of possible computational schemes but it also makes an interesting connection to many other results related to szeg polynomials which have already been interpreted via signal flow graphs for generalized lattice filter structures including the formulas of the gohberg semencul type for inversion of toeplitz like matrices schur type and levinson type algorithms etc for example this connection allows us to show that moment matrices corresponding to what we called horner szeg polynomials though not toeplitz are quasi toeplitz i e they have a certain shift invariance property

maximum likelihood estimation with side information of a 1 d discrete layered medium from its noisy impulse reflection response we consider the problem of computing the maximum likelihood estimates of the reflection coefficients of a discrete 1 d layered medium from noisy observations of its impulse reflection response we have side information in that a known subset of the reflection coefficients are known to be zero this knowledge could come from either a priori knowledge of a homogeneous subregion inside the scattering medium or from a thresholding operation in which noisy reconstructed reflection coefficients with absolute values below a threshold are known to be zero our procedure converges in one or two iterations each of which requires only setting up and solving a small system of linear equations and running the levinson algorithm numerical examples are provided that demonstrate not only the operation of the algorithm but also that the side information improves the reconstruction of unconstrained reflection coefficients as well as constrained ones due to the nonlinearity of the inverse scattering problem

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

a layer stripping solution of the inverse problem for a one dimensional elastic medium a fast algorithm for recovering profiles of density and lame parameters as functions of depth for the inverse seismic problem in an elastic medium is obtained the medium is probed with planar impulsive p and sv waves at oblique incidence and the medium velocity components are measured at the surface the interconversion of p and sv waves defines reflection coefficients from which the medium parameter profiles are obtained recursively the algorithm works on a layer stripping principle and it is specified in both differential and recursive forms a physical interpretation of this procedure is given in terms of a lattice filter where the first reflections of the downgoing waves in each layer yield the various reflection coefficients for that layer a computer run of the algorithm on the synthetic impulsive plane wave responses of a twenty layer medium shows that the algorithm works satisfactorily

a kalman filter solution of the inverse scattering problem with a rational reflection coefficient abstract this paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two component wave equations or the potential of a schrodinger equation when the reflection coefficient is rational this method relies on the so called chandrasekhar equations which implement the kalman filter associated to a stationary state space model these equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium and by specializing these recursions to the case when the probing waves have a state space model

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

an inverse problem for toeplitz matrices and the synthesis of discrete transmission lines abstract let a m be a positive definite m x m toeplitz matrix let a k be its k x k principal minor for any k m which is also positive definite and toeplitz define the central mass sequence 1 m by k sup a k k 0 in which k is the k x k matrix of all 1 s we show how knowledge of the sequence k is equivalent to knowledge of the matrix a m this result has application to the direct and inverse problems for a transmission line which consists of piecewise constant components knowing the impulse response of the transmission line we can calculate the capacitance taper of the line and vice versa

application of the schur algorithm to the inverse problem for a layered acoustic medium the schur algorithm is a signal processing algorithm which works on a layer stripping principle its time domain version the fast cholesky recursion is a fast and efficient algorithm well suited for high speed data processing in this paper these algorithms are applied to the inverse problem for a continuous layered acoustic medium three different excitations of the medium are considered impulsive plane waves at normal incidence impulsive plane waves at oblique incidence and spherical waves emanating from an impulsive point source the fast algorithms obtained for each of these problems seem to be computationally superior to past work done on these problems that employed gelfand levitan theory to reconstruct the potential of a schrodinger equation

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

levinson like and schur like fast algorithms for solving block slanted toeplitz systems of equations arising in wavelet based solution of integral equations the wiener hopf integral equation of linear least squares estimation of a wide sense stationary random process and the krein integral equation of one dimensional 1 d inverse scattering are fredholm equations with symmetric toeplitz kernels they are transformed using a wavelet based galerkin method into a symmetric block slanted toeplitz bst system of equations levinson like and schur like fast algorithms are developed for solving the symmetric bst system of equations the significance of these algorithms is as follows if the kernel of the integral equation is not a calderon zygmund operator the wavelet transform may not sparsify it the kernel of the krein and wiener hopf integral equations does not in general satisfy the calderon zygmund conditions as a result application of the wavelet transform to the integral equation does not yield a sparse system matrix there is therefore a need for fast algorithms that directly exploit the symmetric block slanted toeplitz structure of the system matrix and do not rely on sparsity the first such o n sup 2 algorithms viz a levinson like algorithm and a schur 1917 like algorithm are presented these algorithms are also applied to the factorization of the bst system matrix the levinson like algorithm also yields a test for positive definiteness of the bst system matrix the results obtained are directly applicable to the problem of constrained deconvolution of a nonstationary signal where the locations of the smooth regions of the signal being deconvolved are known a priori

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

state space approach to factorization of lossless transfer functions and structured matrices abstract the concept of j lossless triangular state space models is reviewed and its relation to lyapunov equations and to matrices with a displacement structure is characterized in detail a new recursive procedure for cascade synthesis of such state space models is introduced in contrast to previous state space based techniques for cascade decomposition factorization of j lossless transfer functions which require conversion of a given state space representation into an equivalent balanced form our new procedure recursively determines a sequence of unbalancedj lossless state space models the cascade decomposition of a given j lossless transfer function obtained by the new procedure is the same as the one obtained by the previous techniques but the attendant computational requirements are significantly reduced furthermore the final computational formulation of our procedure subsumes many previous methods for efficient triangular factorization of structured matrices the so called generalized schur or generalized fast cholesky algorithms

geometric aspects of the caratheodory extension problem abstract both euclidean geometric and non euclidean geometric descriptions for the regions of the solution set to the caratheodory extension problem are discussed in particular it is shown that all the solution functions are located inside a euclidean disk the weyl disk and a hyperbolic disk the apollonian disk the radius of each disk depends on the norm of the parametrizing schur functions moreover the so called maximum entropy solution is exactly the hyperbolic center of the apollonian disk while the associated power spectral density function is the geometric mean center of the real region where the weyl disk is projected to sensitivity analysis on the weyl radius and apollonian center due to the perturbation of the given finite moments is presented the applications of these results to the spectrum estimation and inverse scattering problems are discussed

inverse scattering for a layered acoustic medium using the first order equations of motion the inverse scattering problem for a layered acoustic medium is considered from the first order differential equations of motion resulting in a vector formulation of the problem and using a vector form of the schrodinger inverse scattering methods the result is a vector marchenko equation the differentiability constraints on the acoustic impedance are somewhat relaxed compared to the more standard approach of beginning with the wave equation the solution for plane waves at normal incidence is given along with a good approximate solution which is easily obtainable and takes into account transmission losses not included in the normal wkbj born approximation a new solution for extracting separately the velocity and density of the medium using the reflection response for two different angles of incidence is given which involves a nonlinear integral equation to relate the apparent traveltimes to depth

one dimensional profile inversion of a half space over a two part impedance ground a method which permits one to reveal the one dimensional 1 d electromagnetic profile of a half space bounded by a two part impedance ground is established the method reduces the problem to the solution of two functional equations one of these equations is solved exactly by reducing it to a riemann hilbert problem while the other is reduced under the born approximation to a fredholm equation of the first kind whose kernel involves the solution to the first equation since this latter constitutes an ill posed problem its regularized solution in the sense of tikhonov is given an illustrative application shows the applicability and the accuracy of the theory the functional equations are valid also when the impedance of the ground varies in an arbitrary manner the theory can be applied to a quick determination of the constitutive electromagnetic parameters of the atmosphere over a nonhomogeneous impedance ground

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

pluri canonical divisors on kahler manifolds soit p x m une application reguliere propre de varietes complexes a fibres connexes de dimension k o est un point de m on suppose que p 1 0 x 0 est dans la classe 7b c soit m un entier positif et on suppose le diviseur d une section non nulle dans h 0 x 0 xd 0 k m est lisse alors il y a un voisinage analytique u de o dans m tel que p l p 1 z p l x 0 pour l 1 m et pour tout z dans u

analogues of split levinson schur and lattice algorithms for three dimensional random field estimation problems fast algorithms for computing the linear least squares estimate of a three dimensional random field from noisy observations inside a sphere are proposed the algorithms can be viewed as three dimensional analogues of the split levinson schur and lattice algorithms of linear prediction since they exploit an assumed toeplitz plus hankel structure of the double radon transform of the random field covariance therefore these algorithms require fewer computations than would the solution of the three dimensional wiener hopf integral equation unlike previous generalized levinson algorithms no quarter plane or asymmetric half plane support assumptions for the filter are necessary nor is the three dimensional filtering problem treated as a multichannel vector filtering problem the algorithms work in three stages first the three dimensional split schur algorithm computes a potential from the covariance of the random field this potential is a three dimensional analogue of the parameter appearing in the

wave splitting and lattice dynamics a wave splitting approach used elsewhere to solve a black hole scattering problem is adapted to the formal solution of arbitrary self adjoint wave equations in 1 1 dimensions and yields potentially useful results in this more general case it is then shown that the self adjoint wave equation being solved and the non self adjoint linear wave equations satisfied by the one way component waves are naturally related to a pair of motions of the 1 1 dimensional toda lattice that together comprise a motion of the kac van moerbeke lattice this provides a partial explanation for the peculiar fact that every motion of the kac van moerbeke lattice can be viewed as two interpolated motions of the toda lattice

frequency content of randomly scattered signals the statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it the asymptotic analy

completion problems and scattering problems for dirac type differential equations with singularities abstract a completion problem to recover a rational matrix function which is j unitary on the line is treated a dirac type system with singularities on the semi axis is recovered explicitly from its left reflection coefficient the close relation between these two problems is discussed

signal flow graph approach to inversion of h m quasiseparable vandermonde matrices and new filter structures abstract we use the language of signal flow graph representation of digital filter structures to solve three purely mathematical problems including fast inversion of certain polynomial vandermonde matrices deriving an analogue of the horner and clenshaw rules for polynomial evaluation in a h m quasiseparable basis and computation of eigenvectors of h m quasiseparable classes of matrices while algebraic derivations are possible using elementary operations specifically flow reversal on signal flow graphs provides a unified derivation reveals connections with systems theory etc

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

inverse scattering experiments structured matrix inequalities and tensor algebra this paper deals with a general matrix extension problem with structural constraints and provides a recursive solution in terms of an inverse scattering experiment both the stationary and nonstationary cases are considered in addition to connections to tensor algebra

rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle abstract the techniques of topological dynamics and differential dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle it is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process in particular the techniques mentioned above are used to prove a gap labelling result

green s matrices abstract our goal is to identify and understand matrices a that share essential properties of the unitary hessenberg matrices m that are fundamental for szego s orthogonal polynomials those properties include i recurrence relations connect characteristic polynomials rk x of principal minors of a ii a is determined by generators parameters generalizing reflection coefficients of unitary hessenberg theory iii polynomials rk x correspond not only to a but also to a certain cmv like five diagonal matrix iv the five diagonal matrix factors into a product bc of block diagonal matrices with 2 2 blocks v submatrices above and below the main diagonal of a have rank 1 vi a is a multiplication operator in the appropriate basis of laurent polynomials vii eigenvectors of a can be expressed in terms of those polynomials conditions v connects our analysis to the study of quasi separable matrices but the factorization requirement iv narrows it to the subclass of green s matrices that share properties i vii the key tool is twist transformations that provide 2n matrices all sharing characteristic polynomials of principal minors with a one such twist transformation connects unitary hessenberg to cmv another twist transformation explains findings of fiedler who noticed that companion matrices give examples outside the unitary hessenberg framework we mention briefly the further example of a daubechies wavelet matrix infinite matrices are included

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

application of the multilevel single linkage method to one dimensional electromagnetic inverse scattering problem an inverse scattering method for the reconstruction of the permittivity and conductivity profiles of a multilayered medium and for that of the impedance profile of a nonuniform transmission line is proposed the inversion is based on the global minimization of an objective function by the multilevel single linkage method the objective function is defined as the mean square error between the measured data and the data obtained from the solution of the forward problem an exact formulation for the gradient of the objective function in closed form is derived the necessary condition for the unique solution of the inverse problem of a nonuniform transmission line is discussed reconstruction examples using both experimental and noisy synthetic data are presented

uniqueness results for nonlinear elliptic equations with a lower order term abstract we prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is u w 0 1 p div a x 1 u 2 p 2 2 u b x 1 u 2 1 2 f in d where is a bounded open subset of r n n 2 p satisfies p 2 n n 1 a is a function belonging to l such that a x 0 f belongs to the dual space w 1 p b belongs to some lebesgue space l r with r r n p and belongs to the interval 0 n p r with n p r and r n p functions which are specified below

lattice filter parameterization and modeling of nonstationary processes a general theory of constant parameter modular lattice models for discrete time nonstationary second order processes is presented a complete parametrization of such processes in terms of schur and congruence coefficients is derived by developing a natural connection between the displacement structure of a covariance matrix and schur s test for positive definiteness of matrices schur coeffieients provide a simple solution to problems of covariance extension and rational spectral approximation for nonstationary covariances and they coincide with the well known reflection or parcor coefficients when the covariance is stationary the congruence coefficients provide the time varying gains of a tapped delay line realization of the whitening filter for the process a constant parameter realization of the same filter is derived by combining a lattice filter structure with a tapped delay line both with time invariant gains this configuration also provides a recursive relation for the congruence coefficients namely a generalized levinson szego recursion the tapped delay line part of the realization can be eliminated by introducing the concept of admissibility admissibility also reduces the parametrization of a nonstationary process to schur coefficients alone in analogy to stationary processes which are completely characterized by their parcor coefficients

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

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

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

layer stripping solutions of multidimensional inverse scattering problems a layer stripping procedure for solving three dimensional schrodinger equation inverse scattering problems is developed this method operates by recursively reconstructing the potential from the jump in the scattered field at the wave front and then using the reconstructed potential to propagate the wave front and the scattered field further into the inhomogeneous region it is thus a generalization of algorithms that have been developed for one dimensional inverse scattering problems although the procedure has not yet been numerically tested the corresponding one dimensional algorithms have performed well on synthetic data the procedure is applied to a two dimensional inverse seismic problem connections between simplifications of this method and born approximation inverse scattering methods are also noted

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

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

on the feasibility of impulse reflection response data for the two dimensional inverse scattering problem layer stripping algorithms for inverse scattering problems are very fast but have the reputation of being numerically unstable especially when applied to noisy data the goal of this paper is to provide an explicitly discrete framework for layer stripping algorithms for the two dimensional 2 d schrodinger equation inverse scattering problem we determine when 2 d layer stripping algorithms are numerically stable explain why they are stable and specify exactly the discrete problem they solve when they are stable we reformulate the 2 d schrodinger equation as a multichannel two component wave system by fourier transforming the schrodinger equation in the lateral spatial variable discretization results in new 2 d layer stripping algorithms which incorporate multichannel transmission effects this leads to an important new feasibility condition on impulse reflection response data for stability of these algorithms a 2 d discrete schrodinger equation is defined and analogous results are obtained numerical examples illustrate the new results especially how rendering noisy data feasible stabilizes layer stripping algorithms

elementary backlund transformations for a discrete ablowitz ladik eigenvalue problem elementary backlund transformations bts are described for a discretization of the zakharov shabat eigenvalue problem a special case of the ablowitz ladik eigenvalue problem elementary bts allow the process of adding bound states to a system i e the add one soliton bt to be factorized to solving two simpler sub problems they are used to determine the effect on the scattering data when bound states are added they are shown to provide a method of calculating discrete solitons this is achieved by constructing a lattice of intermediate potentials with the parameters used in the calculation of the lattice simply related to the soliton scattering data when the potentials s n t n in the system are related by s n t n they enable simple derivations to be obtained of the add one soliton bt and the nonlinear superposition formula

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

inverse problem in nonstationary multidimensional medium the problem of a scalar wave propagation from the point impulsive source in the layer of a nonstationary multidimensional medium is considered the boundary problem for the wave equation is reformulated in the problem with the initial condition using the invariant imbedding method the integral differential inverse procedures of the various orders were obtained from the imbedding equations using the singularities method the order of inverse procedure is defined by the degree of a polynomial in the analytical representation of the medium characteristic near the layer boundary it was shown that the coefficients of the polynomial are calculated with the help of the differential characteristics of the point impulsive source in the inhomogeneous medium the cause and character of the multidimensional inverse problem overdefiniteness are considered the application of the proposed procedure for a statistical problem is discussed

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

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

fast triangular factorization of the sum of quasi toeplitz and quasi hankel matrices abstract the literature contains several recent fast algorithms for the triangular factorization of strongly regular toeplitz plus hankel matrices in this paper we study the rather more general sum of quasi toeplitz and quasi hankel matrices both hermitian and non hermitian quasi toeplitz and quasi hankel matrices are those that are congruent to toeplitz and hankel matrices in a special sense the derivation is based on the concept of displacement structure and its intimate relation to the schur reduction procedure for triangular factorization various special cases covering displacement ranks from two to eight are considered several other problems e g factorization of the inverse matrix solution of exact or overdetermined linear systems can be reduced to the direct factorization problem

on the brody hyperbolicity of moduli spaces for canonically polarized manifolds we show that the moduli stack mh of canonically polarized complex manifolds with hilbert polynomial h is brody hyperbolic hence if mh denotes the corresponding coarse moduli scheme and if u mh is a quasi finite morphism induced by a family then there are no nonconstant holomorphic maps c u

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

iterative trace deconvolution and noncausal transform for processing band limited data in 1983 carrion and patton showed that if recorded seismic data do not have low frequencies in their spectrum inversion for acoustic impedance is unstable this concept is commonly accepted here we infer acoustic impedance from band limited data without low frequencies using iterative trace deconvolution with the so called noncausal projection which moves initial data samples corresponding to small eigenvalues to negative times we show that this procedure can solve ill posed problems by migrating large residuals uncertainties in the trend of the recovered impedance to the bottom of the model an important property of the proposed method is that it converges to the true solution independently of the chosen initial model another advantage of the proposed algorithm is that unlike conventional dynamic deconvolution characteristic integration schemes it increases neither errors nor the condition number with depth the method is illustrated on synthetic and real data

multidimensional inverse scattering an orthogonalization formulation the three dimensional schrodinger equation inverse scattering problem is solved using an orthogonalization approach the plane waves propagating in free space are orthogonalized with respect to an inner product defined in terms of a jost operator the resulting integral equation is identical to the generalized gel fand levitan equation of newton although the present derivation is simpler and more physical than that of newton newton s generalized marchenko equation is derived from the defining integral equation for the jost operator these integral equations are shown to be solved by fast algorithms derived directly from the properties of their solutions this paper thus presents a simple interpretation of newton s two integral equations two fast algorithms for solving these integral equations and relations between the various approaches this is a generalization of previously obtained results which are also reviewed here for the one dimensional inverse scattering problem

some remarks on elliptic problems with critical growth in the gradient abstract in this work we analyze existence nonexistence multiplicity and regularity of solution to problem 1 u u u 2 f x in u 0 on where is a continuous nondecreasing positive function and f belongs to some suitable lebesgue spaces

a transcendental approach to kollar s injectivity theorem ii we treat kollar s injectivity theorem from the analytic or differential geometric viewpoint more precisely we give a curvature condition which implies kollar type cohomology injectivity theorems our main theorem is formulated for a compact kahler manifold but the proof uses the space of harmonic forms on a zariski open set with a suitable complete kahler metric we need neither covering tricks desingularizations nor leray s spectral sequence

inverse scattering in multimode structures we consider the inverse scattering problem associated with any number of interacting modes in one dimensional structures the coupling between the modes is contradirectional in addition to codirectional and may be distributed continuously or in discrete points the local coupling as a function of position is obtained from reflection data using a layer stripping type method and the separate identification of the contradirectional and codirectional coupling is obtained using matrix factorization ambiguities are discussed in detail and different a priori information that can resolve the ambiguities is suggested the method is exemplified by applications to multimode optical waveguides with quasi periodical perturbations

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