isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

models for n tuples of noncommuting operators abstract this paper proves versions of the rota model theorem the de branges rovnyak model theorem and the coisometric extension theorem for n tuples of not necessarily commuting operators this generalizes the work of a e frazho j funct anal 48 1982 1 11 for pairs of operators the methods involve applying the single operator results to matrices of operators

complements to models for noncommuting operators abstract this paper develops a dilation theory for two noncommutative operators many of the results and techniques in dilation theory for one operator are extended to this setting this is done by extending the schaffer construction for one operator to the fock space setting a new wold decomposition for two isometrics with orthogonal range is obtained this wold decomposition and the schaffer construction is used to extend the sz nagy foias lifting theorem and the characteristic function theory to the noncommutative setting the characteristic function is operator valued and defined in a fock space

a hierarchical bayesian model for combining multiple 2 x 2 tables using conditional likelihoods summary this paper introduces a hierarchical bayesian model for combining multiple 2 2 tables that allows the flexibility of different odds ratio estimates for different tables and at the same time allows the tables to borrow information from each other the proposed model however is different from a full bayesian model in that the nuisance parameters are eliminated by conditioning instead of integration the motivation is a more robust model and a faster and more stable gibbs algorithm we work out a gibbs scheme using the adaptive rejection sampling for log concave density and an algorithm for the mean and variance of the noncentral hypergeometric distribution the model is applied to a multicenter ulcer clinical trial

fast computation of exact confidence limits for the common odds ratio in a series of 2 2 tables abstract the odds ratio is widely used as a measure of association in epidemiologic studies and clinical trials we consider calculation of exact confidence limits for the common odds ratio in a series of independent 2 2 tables and propose three modifications of the network algorithm of mehta patel and gray 1 formulating and dealing with the problem in algebraic instead of graph theoretic terms 2 performing convolutions on the natural scale instead of the logarithmic scale and 3 using the secant method instead of binary search to compute roots of polynomial equations enhancement of computational efficiency exceeding an order of magnitude afforded by these modifications is empirically demonstrated we also compare the modified method with one based on the fast fourier transform fft further we show that the fft method can also result in considerable loss of numerical accuracy the modifications proposed in this article yield an algorithm that is not only fast and accurate but that combine

conservative linear systems unitary colligations and lax phillips scattering multidimensional generalizations connections between conservative linear systems lax phillips scattering and operator model theory are well known a common thread in all the theories is a contractive analytic operator valued function on the unit disc t z having a representation of the form t z d z c i za 1 b known as the transfer or frequency response function in the system theory community the scattering function in the mathematical physics community and the characteristic operator function in the operator theory community here we consider analogues of this circle of ideas in the more general setting of multidimensional systems multi evolution scattering systems multivariable function theoretic operator theory three particular extensions are discussed from the point of view of system theory these involve 1 a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex euclidean space 2 a non commutative multidimensional linear system with evoluti

empirical bayes methods for combining likelihoods comment suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 x 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for he priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empirical bayes relationship most of the discussion is presented in terms of a hierarchical bayes model and concerns how such a model can be implemented without requiring large amounts of bayesian input frequentist approaches such as bias correction and robustness play a central role in the methodology

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

inner functions on the bidisk and associated hilbert spaces abstract matrix valued inner functions on the bidisk have a number of natural subspaces of the hardy space on the torus associated to them we study their relationship to agler decompositions regularity up to the boundary and restriction maps into one variable spaces we give a complete description of the important spaces associated to matrix rational inner functions the dimension of some of these spaces can be computed in a straightforward way and this ends up having an application to the study of three variable rational inner functions examples are included to highlight the differences between the scalar and matrix cases

functional models of γn contractions and characterization of γn isometries abstract we show that a weighted bergman space on the polydisc splits up into the orthogonal direct sum of subspaces each corresponding to an irreducible representation of the symmetric group the multiplication by the n tuple of elementary symmetric functions is a n contraction on each of these subspaces a necessary condition for a commuting n tuple to be a n contraction is obtained in terms of pencil of operators we provide a model theory for n isometries a beurling lax halmos type representation for the invariant subspaces of n isometries is given

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

unitary colligations reproducing kernel hilbert spaces and nevanlinna pick interpolation in several variables abstract recently j agler studied the class s d of scalar valued analytic functions of d complex variables f for which f t 1 t d has norm at most 1 for any collection of d commuting contractions t 1 t d on a hilbert space h among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d variable linear system as well as a nevanlinna pick interpolation theorem for this class of functions in this note we examine the system theory aspects and uniqueness of the transfer function representation and give a simpler proof of the nevanlinna pick interpolation theorem for the class s d and obtain a d variable version of the toeplitz corona theorem by using ideas of arov and grossman introduced for 1 variable problems as a bonus we obtain a collection of linear fractional maps which parametrize the set of all s d solutions of an interpolation problem

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

classes of tuples of commuting contractions satisfying the multivariable von neumann inequality we obtain a decomposition for multivariable schur class functions on the unit polydisk which to a certain extent is analogous to agler s decomposition for functions from the schur agler class as a consequence we show that d tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d variable von neumann inequality for an arbitrary operator valued bounded analytic function on the polydisk also this decomposition yields a necessary condition for solvability of the finite data nevanlinna pick interpolation problem in the schur class on the unit polydisk 2008 elsevier inc all rights reserved

nevanlinna representations in several variables abstract we generalize to several variables the classical theorem of nevanlinna that characterizes the cauchy transforms of positive measures on the real line we show that for the loewner class a large class of analytic functions that have non negative imaginary part on the upper polyhalf plane there are representation formulae in terms of densely defined self adjoint operators on a hilbert space we find four different representation formulae and we show that every function in the loewner class has one of the four representations corresponding precisely to four different growth conditions at infinity

computing an exact confidence interval for the common odds ratio in several 2 2 contingency tables abstract a quadratic time network algorithm is provided for computing an exact confidence interval for the common odds ratio in several 2 2 independent contingency tables the algorithm is shown to be a considerable improvement on an existing algorithm developed by thomas 1975 which relies on exhaustive enumeration problems that would formerly have consumed several cpu hours can now be solved in a few cpu seconds the algorithm can easily handle sparse data sets where asymptotic results are suspect the network approach on which the algorithm is based is also a powerful tool for exact statistical inference in other settings

meta analysis for 2 2 tables a bayesian approach this paper develops and implements a fully bayesian approach to meta analysis in which uncertainty about effects in distinct but comparable studies is represented by an exchangeable prior distribution specifically hierarchical normal models are used along with a parametrization that allows a unified approach to deal easily with both clinical trial and case control study data monte carlo methods are used to obtain posterior distributions for parameters of interest integrating out the unknown parameters of the exchangeable prior or random effects distribution the approach is illustrated with two examples the first involving a data set on the effect of beta blockers after myocardial infarction and the second based on a classic data set comprising 14 case control studies on the effects of smoking on lung cancer in both examples rather different conclusions from those previously published are obtained in particular it is claimed that widely used methods for meta analysis which involve complete pooling of o e values lead to understatement of uncertainty in the estimation of overall or typical effect size

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

minimal sets in monotone and concave skew product semiflows ii two dimensional systems of differential equations abstract recurrent nonautonomous two dimensional systems of differential equations of ordinary finite delay and reaction diffusion types given by cooperative and concave vector fields define monotone and concave skew product semiflows whose dynamics is analyzed in this paper a complete description of all the minimal sets in a very interesting dynamical situation is provided and some criteria to determine its possible existence and the area on which they lie are given suitable examples prove the optimality of the results

equivalence between conditional and mixture approaches to the rasch model and matched case control studies with applications analyses of data using rasch models including the special case of matched case control studies are common applications of conditional likelihood in which the usual inferential procedures are applied only after conditioning on an approximately ancillary statistic another common approach to the analysis of rasch models is to integrate the nuisance parameters over a mixing distribution using the marginal likelihood obtained as the basis for inference we show that the full conditional likelihood can always be obtained exactly via the marginal approach given a particular choice of mixing distribution and derive necessary and sufficient conditions for the two approaches to agree previous work has shown that with sufficient flexibility in the mixing distribution the maxima of the marginal and conditional likelihoods will be equivalent under concordance criteria our argument requires no such criteria and for any dataset guarantees the equivalence of the whole of the two likelihoods not just their maxi

an algorithm for the mean and variance of the noncentral hypergeometric distribution the mean and variance of the noncentral hypergeometric distribution are essential for obtaining maximum likelihood estimates for conditional logistic models for 2 x 2 tables the numerical difficulty in directly computing them using the definition is discussed in detail by satten and kupper 1990 they also give an algorithm based on a continued fraction representation for the mean we present here another algorithm based on two equations in harkness 1965

favard theory for the adjoint equation and fredholm alternative abstract fredholm alternative is a classical tool of periodic linear equations allowing to describe the existence of periodic solutions of an inhomogeneous equation in terms of the adjoint equation a few partial extensions have been proposed in the literature for recurrent equations our aim is to point out that they have a common root and discuss whether such a root gives rise to a general fredholm type alternative sacker sell spectral theory and favard theory are main ingredients in this discussion a considerable effort is devoted to understand how favard theory is affected by adjunction at least for planar equations

analysis of ulcer data using hierarchical generalized linear models in multi centre clinical trials heterogeneities in individual hospital treatment effects can be modelled as random effects estimates of the individual hospital treatment effects and estimate of the mean treatment effect allowing for the presence of overall hospital differences are required together with some measure of their uncertainty systematic inferences from the hierarchical likelihood are now possible using hierarchical generalized linear models we show how to construct profile likelihoods for the treatment effects of individual hospitals

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

multivariable bohr inequalities operator valued multivariable bohr type inequalities are obtained for i a class of noncommutative holomorphic functions on the open unit ball of generalizing the analytic functions on the open unit disc ii the noncommutative disc algebra and the noncommutative analytic toeplitz algebra iii a class of noncommutative selfadjoint harmonic functions on the open unit ball of generalizing the real valued harmonic functions on the open unit disc iv the cuntz toeplitz algebra the reduced resp full group algebra resp of the free group with generators v a class of analytic functions on the open unit ball of the classical bohr inequality is shown to be a consequence of fejer s inequality for the coefficients of positive trigonometric polynomials and haager up de la harpe inequality for nilpotent operators moreover we provide an inequality which for analytic polynomials on the open unit disc is sharper than bohr s inequality

bayesian approaches to random effects meta analysis a comparative study current methods for meta analysis still leave a number of unresolved issues such as the choice between fixed and random effects models the choice of population distribution in a random effects analysis the treatment of small studies and extreme results and incorporation of study specific covariates we describe how a full bayesian analysis can deal with these and other issues in a natural way illustrated by a recent published example that displays a number of problems such analyses are now generally available using the bugs implementation of markov chain monte carlo numerical integration techniques appropriate proper prior distributions are derived and sensitivity analysis to a variety of prior assumptions carried out current methods are briefly summarized and compared to the full bayes analysis

an operator valued poisson kernel abstract versions of the cauchy and poisson formulas in the unit ball of c n are defined by means of appropriate operator valued kernels connections with some problems of dilation theory are then made

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

empirical bayes methods for combining likelihoods abstract suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for the priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empiri

minimal sets in monotone and concave skew product semiflows i a general theory abstract the long term dynamics of a general monotone and concave skew product semiflow is analyzed paying special attention to the region delimited from below by the graph of a semicontinuous subequilibrium or by a minimal set admitting a flow extension different possibilities arise depending on the existence and number or absence of minimal sets strongly above the initially fixed one as well as on the coexistence or not of bounded and unbounded semiorbits on the region previous results are unified and extended and significative differences with the sublinear case are established some scenarios which are impossible in the autonomous or periodic cases may occur in this setting

canonical agler decompositions and transfer function realizations a seminal result of agler proves that the natural de branges rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift invariant pieces agler s decomposition is non constructive a problem remedied by work of ball sadosky vinnikov which uses scattering systems to produce agler decompositions through concrete hilbert space geometry this method while constructive so far has not revealed the rich structure shown to be present for special classes of functions inner and rational inner functions in this paper we show that most of the important structure present in these special cases extends to general bounded analytic functions we give characterizations of all agler decompositions we prove the existence of coisometric transfer function realizations with natural state spaces and we characterize when schur functions on the bidisk possess analytic extensions past the boundary in terms of associated hilbert spaces

positive extensions fejer riesz factorization and autoregressive filters in two variables in this paper we treat the two variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial this problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes we show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition as a corollary of the main result a necessary and sufficient condition for the existence of a spectral fejerriesz factorization of a strictly positive two variable trigonometric polynomial is given in terms of the fourier coefficients of its reciprocal tools in the proofs include a specific two variable kronecker theorem based on certain elements from algebraic geometry as well as a two variable christoffel darboux like formula the key ingredient is a matrix valued polynomial that appears in a parametrized version of the schur cohn test for stability the results also have consequences in the theory of two variable orthogonal polynomials where a spectral matching result is obtained as well as in the study of inverse formulas for doubly indexed toeplitz matrices finally numerical results are presented for both the autoregressive filter problem and the factorization problem