a one factor multivariate time series model of metropolitan wage rates abstract the paper formulates and estimates a single factor multivariate time series model the model is a dynamic generalization of the multiple indicator or factor analysis model it is shown to be a special case of the general state space model and can be estimated by maximum likelihood methods using the kalman filter algorithm the model is used to obtain estimates of the unobserved metropolitan wage rate for los angeles based on observations of sectoral wages within the standard metropolitan statistical area hypothesis tests model diagnostics and out of sample forecasts are used to evaluate the model

spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

a novel algorithm for dynamic factor analysis in this paper a dynamic factor model is proposed to extract the dynamic factors from time series data in order to deal with the problem of scaling the cross correlation matrices ccm are first employed to cluster the time series data then the dynamic factors are extracted using the revised independent component analysis ica in addition a numerical study is used to demonstrate the proposed method on the basis of the simulated results we can conclude that the proposed method can really extract the effective dynamic factors

right limits and reflectionless measures for cmv matrices we study cmv matrices by focusing on their right limit sets we prove a cmv version of a recent result of remling dealing with the implications of the existence of absolutely continuous spectrum and we study some of its consequences we further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of khrushchev to demonstrate the analogy with the jacobi case we recover corresponding previous results of simon using the same approach

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

local existence of the borel transform in euclidean φ 4 4 we bound rigorously the large order behaviour of 44 euclidean perturbative quantum field theory as the simplest example of renormalizable but non super renormalizable theory the needed methods are developed to take into account the structure of renormalization which plays a crucial role in the estimates as a main thorem it is shown that the schwinger functions at ordern are bounded byknn which implies a finite radius of convergence for the borel transform of the perturbation series

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

a true time series and its indicators an alternative approach abstract de leeuw and mckelvey proposed a method for using two imperfect indicators to predict the values of an unobserved series in this article we suggest an alternative method that uses the multiple indicator approach to model the true time series and its indicators we examine the identification status of the model estimate the parameters by maximum likelihood and use the kalman filter to obtain predictions of the unobserved series the method is applied to the u s gross private saving rate using flow of funds and national income and product accounts data as indicators

dual time frequency domain system identification in this paper we obtain the maximum likelihood estimate of the parameters of discrete time linear models by using a dual time frequency domain approach we propose a formulation that considers a reduced rank linear transformation of the available data such a transformation may correspond to different options selection of time domain data transformation to the frequency domain or selection of frequency domain data obtained from time domain samples we use the proposed approach to identify multivariate systems represented in state space form by using the expectation maximisation algorithm we illustrate the benefits of the approach via numerical examples

on isospectral sets of jacobi operators we consider the inverse spectral problem for a class of reflectionless bounded jacobi operators with empty singularly continuous spectra our spectral hypotheses admit countably many accumulation points in the set of eigenvalues as well as in the set of boundary points of intervals of absolutely continuous spectrum the corresponding isospectral set of jacobi operators is explicitly determined in terms of dirichlet type data

weighted maximum likelihood for dynamic factor analysis and forecasting with mixed frequency data for the purpose of forecasting key macroeconomic or financial variables from a panel of time series variables we adopt the dynamic factor model and propose a weighted likelihood based method for parameter estimation the loglikelihood function is split into two parts that are weighted differently the first part is associated with the key variables while the second part is associated with the related variables which may contribute to the forecasting of key variables we derive asymptotic properties including consistency and asymptotic normality of the weighted maximum likelihood estimator we show that this estimator outperforms the standard likelihood based estimator in approximating the true unknown distribution of the data as well as in out of sample forecasting accuracy we verify the new estimation method in a monte carlo study and investigate the role of different weights in different settings in the context of forecasting gross domestic product growth this key variable is typically observed at a low quarterly frequency while the supporting variables are observed at a high monthly frequency we adopt a low frequency representation of the mixed frequency dynamic factor model and discuss the computational efficiencies of this approach in our empirical study for the u s economy we present improvements in nowcasting and forecasting accuracy when the weighted likelihood based estimation procedure is adopted

effective lagrangians for scalar fields and finite size effects in field theory we first discuss the approach of effective field theory in a d dimensional euclidean space we consider a model with two interacting scalar fields lint 1 2 2 2 1 2 2 with masses m1 and m2 assuming m2 m1 we show that there is a decoupling in the effective theory describing the dynamic of the light mass field furthermore we consider the presence of two parallel hyperplanes which break translational symmetry with a natural cutoff satisfying m2 m1 then imposing dirichlet and also neumann boundary conditions we study the perturbative renormalization of the effective theory 4 in a region bounded by the two parallel hyperplanes in the one loop approximation

improving gdp measurement a measurement error perspective we provide a new measure of historical u s gdp growth obtained by applying optimal signal extraction techniques to the noisy expenditure side and income side gdp estimates the quarter by quarter values of our new measure often differ noticeably from those of the traditional measures its dynamic properties differ as well indicating that the persistence of aggregate output dynamics is stronger than previously thought

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

the strong coupling expansion and the ultra local approximation in field theory abstract we first discuss the strong coupling expansion in 4 d theory and quantum electrodynamics in a d dimensional euclidean space in a formal representation for the schwinger functional we treat the gaussian part of the action as a perturbation with respect to the remaining terms in this way we develop a perturbative expansion around the ultra local model where fields defined at different points of euclidean space are decoupled we examine the singularities of the strong coupling perturbative expansion analysing the analytic structure of the zero dimensional generating functions in the coupling constant complex planes we also discuss the ultra local generating functional in a non polynomial model in field theory defined by the following interaction lagrangian density l ii g 1 g 2 g 1 cosh g 2 x 1 finally we use the strong coupling perturbative expansion to compute the renormalized vacuum energy of the strongly coupled 4 d theory assuming that the scalar field is defined in a region bounded by two parallel hyperplanes where we are imposing dirichlet dirichlet boundary conditions

critical measures quadratic differentials and weak limits of zeros of stieltjes polynomials we investigate the asymptotic zero distribution of heine stieltjes polynomials polynomial solutions of second order differential equations with complex polynomial coefficients in the case when all zeros of the leading coefficients are all real zeros of the heine stieltjes polynomials were interpreted by stieltjes as discrete distributions minimizing an energy functional in a general complex situation one deals instead with a critical point of the energy we introduce the notion of discrete and continuous critical measures saddle points of the weighted logarithmic energy on the plane and prove that a weak limit of a sequence of discrete critical measures is a continuous critical measure thus the limit zero distributions of the heine stieltjes polynomials are given by continuous critical measures we give a detailed description of such measures showing their connections with quadratic differentials in doing that we obtain some results on the global structure of rational quadratic differentials on the riemann sphere that have an independent interest

kotani last problem and hardy spaces on surfaces of widom type this is a small theory of non almost periodic ergodic families of jacobi matrices with purely however absolutely continuous spectrum the reason why this effect may happen is that under our axioms we found an analytic condition on the resolvent set that is responsible for exactly equivalent to this effect

the transformation operator for schrodinger operators on almost periodic infinite gap backgrounds abstract we investigate the kernels of the transformation operators for one dimensional schrodinger operators with potentials which are asymptotically close to bohr almost periodic infinite gap potentials

nodal solutions of a nonlinear eigenvalue problem of the euler bernoulli equation abstract we are concerned with determining values of r for which there exist nodal solutions of the boundary value problems of the euler bernoulli equation p x u q x u r f x u u 0 x 0 l u 0 cos p u 0 sin 0 u 0 0 u l cos p u l sin 0 u l 0 where 0 2 p c 2 0 l and q c 1 0 l the proof of our main result is based upon bifurcation techniques

new trace formulae for a quadratic pencil of the schrodinger operator this work deals with the eigenvalue problem for a quadratic pencil of the schrodinger operator on a finite closed interval with the two point boundary conditions we will obtain new regularized trace formulas for this class of differential pencil

maximum likelihood estimation of the dynamic shock error model abstract in this paper an efficient estimation technique in the time domain for the dynamic shock error model is developed this approach utilizes a state space representation of the model and the kalman filtering technique to obtain the maximum likelihood estimates the resulting estimates are shown to be consistent and asymptotically normal with the inverse of the information matrix as their precision measure a consistent estimate of the information matrix is suggested to obtain the standard errors of the estimates

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

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

the continuous spin ising model g 0 φ 4 d field theory and the renormalization group we have used the method of high temperature series expansions to investigate the critical point properties of a continuous spin ising model and g0 4 d euclidean field theory we have computed through tenth order the hightemperature series expansions for the magnetization susceptibility second derivative of the susceptibility and the second moment of the spin spin correlation function on eight different lattices our analysis of these series is made using integral and pade approximants in three dimensions we find that hyperscaling fails for sufficiently ising like systems the strong coupling limit of g0 4 3 depends on how the ultraviolet cutoff is removed the level contours of the renormalized coupling constant for this model in theg0 correlation length plane exhibit a saddle point if the ultraviolet cutoff is removed beforeg0 the usual field theory results and the renormalization group fixed point with hyperscaling is obtained if the order of these limits is reversed the ising model limit where hyperscaling fails and the field theory is trivial is obtained in four dimensions we find that hyperscaling fails completely g0 4 4 is trivial for all g0 when the ultraviolet cutoff is removed

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

covariances for smoothed estimates in state space models summary ansley kohn 1982 presented a novel proof of the fixed interval smoothing algorithm for state space models in this note the ideas of their proof are used to provide a new formula for the covariance between smoothed estimates at any two points in time this equation is substantially simpler than existing special cases

vars common factors and the empirical validation of equilibrium business cycle models abstract equilibrium business cycle models have typically less shocks than variables as pointed out by altug 1989 international economic review 30 4 889 920 and sargent 1989 the journal of political economy 97 2 251 287 if variables are measured with error this characteristic implies that the model solution for measured variables has a factor structure this paper compares estimation performance for the impulse response coefficients based on a var approximation to this class of models and an estimation method that explicitly takes into account the restrictions implied by the factor structure bias and mean squared error for both factor and var based estimates of impulse response functions are quantified using as data generating process a calibrated standard equilibrium business cycle model we show that at short horizons var estimates of impulse response functions are less accurate than factor estimates while the two methods perform similarly at medium and long run horizons

consistent factor estimation in dynamic factor models with structural instability abstract this paper considers the estimation of approximate dynamic factor models when there is temporal instability in the factor loadings we characterize the type and magnitude of instabilities under which the principal components estimator of the factors is consistent and find that these instabilities can be larger than earlier theoretical calculations suggest we also discuss implications of our results for the robustness of regressions based on the estimated factors and of estimates of the number of factors in the presence of parameter instability simulations calibrated to an empirical application indicate that instability in the factor loadings has a limited impact on estimation of the factor space and diffusion index forecasting whereas estimation of the number of factors is more substantially affected

uniqueness of solutions of the kdv hierarchy via dubrovin type flows abstract we consider the cauchy problem for the kdv hierarchy a family of integrable pdes with a lax pair representation involving one dimensional schrodinger operators under a local in time boundedness assumption on the solution for reflectionless initial data we prove that the solution stays reflectionless for almost periodic initial data with absolutely continuous spectrum we prove that under craig type conditions on the spectrum dirichlet data evolve according to a lipschitz dubrovin type flow so the solution is uniquely recovered by a trace formula this applies to algebro geometric finite gap solutions more notably we prove that it applies to small quasiperiodic initial data with analytic sampling functions and diophantine frequency this also gives a uniqueness result for the cauchy problem on the line for periodic initial data even in the absence of craig type conditions

a note on the non perturbative zero dimensional λϕ4 model abstract we exhibit the partition function of the zero dimensional 4 model as a simple exact expression in terms of the macdonald function for re 0 then by analytic continuation we obtain an expression defined in the complex coupling constant plane for arg consequently the partition function understood as an analytic continuation is defined for all values of except for a branch cut along the real negative axis this shows that at least in zero dimension the partition function can be defined for negative coupling constant where the integral is formally divergent provided it has a non vanishing imaginary part we also evaluate the partition function on perturbative grounds using the borel summation technique and we found that in the common domain of validity for re 0 it coincides precisely with the exact expression furthermore a connection between the non perturbative zero dimensional solution and the ultralocal 4 model in arbitrary dimension d is presented

a note on the estimation of variances in state space models using the maximum a posteriori procedure this note examines an estimation procedure for the unknown parameters in a state space model proposed by tsang glover and bach the method is based on the maximum a posteriori map principle contrary to the claims of tsang et al it is shown that the algorithm performs very poorly compared to maximum likelihood some insight into the shortcomings of the map procedure is obtained by comparing it to the computation of maximum likelihood estimators by the em algorithm

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

em based identification of continuous time arma models from irregularly sampled data abstract in this paper we present a novel algorithm for identifying continuous time autoregressive moving average models utilizing irregularly sampled data the proposed algorithm is based on the expectation maximization algorithm and obtains maximum likelihood estimates the proposed algorithm shows a fast convergence rate good robustness to initial values and desirable estimation accuracy comparisons are made with other algorithms in the literature via numerical examples

the infinite period limit of the inverse formalism for periodic potentials soit l operateur de hill h p d 2 dx 2 v p x quand v p x est une fonction p periodique de x on montre que les formules pour la solution du probleme inverse ont une limite si on prend v p comme une repetition periodique d une fonction a support compact v x et que l on fait tendre p vers l infini

analytic standard errors for exploratory process factor analysis exploratory process factor analysis epfa is a data driven latent variable model for multivariate time series this article presents analytic standard errors for epfa unlike standard errors for exploratory factor analysis with independent data the analytic standard errors for epfa take into account the time dependency in time series data in addition factor rotation is treated as the imposition of equality constraints on model parameters properties of the analytic standard errors are demonstrated using empirical and simulated data

higher order trace formulas of the buslaev faddeev type for the half line schrodinger operator with long range potentials we deal with trace formulas for half line schrodinger operators with long range potentials we generalize the buslaev faddeev trace formulas to the case of square integrable potentials the exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given the main results are optimal

the spectral theory of the vibrating periodic beam we study the spectral theory of the fourth order eigenvalue problem left a x u x right prime prime lambda rho x u x infty x infty where the functionsa and are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described bya and we examine the structure of the spectrum establishing the fact that the periodic and antiperiodic eigenvalues are the endpoints of the spectral bands we also introduce an entire function which we denote bye connected to the spectral theory whose zeros at least the ones of odd multiplicity are shown to lie on the negative real axis where they define a collection of pseudogaps next we prove some inverse results in the spirit of two old theorems of borg for the hill s equation we finish with a determinant formula i e a multiplicative trace formula and some comments on its role in the formulation of the general inverse problem

posterior mode estimation by extended kalman filtering for multivariate dynamic generalized linear models abstract a family of multivariate dynamic generalized linear models is introduced as a general framework for the analysis of time series with observations from the exponential family besides common conditionally gaussian models this article deals with univariate models for counted and binary data and as the most interesting multivariate case models for nonstationary multicategorical time series for univariate responses a related yet different class of models has been introduced in a bayesian setting by west harrison and migon assuming conjugate prior posterior distributions for the natural parameter of the exponential family they derive an approximate filter for estimation of time varying states or parameters however their method raises some problems in particular in extending it to the multivariate case a different approach to filtering and smoothing is chosen in this article to avoid a full bayesian analysis based on numerical integration which becomes computationally critical for higher

analysis of time series subject to changes in regime abstract this paper introduces an em algorithm for obtaining maximum likelihood estimates of parameters for processes subject to discrete shifts in autoregressive parameters with the shifts themselves modeled as the outcome of a discrete valued markov process the simplicity of the em algorithm permits potential application of the approach to large vector systems

asymptotics of chebyshev polynomials i subsets of r we consider chebyshev polynomials t n z for infinite compact sets e that is the monic polynomials minimizing the sup norm t n e on e we resolve a 45 year old conjecture of widom that for finite gap subsets of r his conjectured asymptotics which we call szeg widom asymptotics holds we also prove the first upper bounds of the form t n e qc e n where c e is the logarithmic capacity of e for a class of e s with an infinite number of components explicitly for those e that obey a parreau widom condition

the em approach to the multiple indicators and multiple causes model via the estimation of the latent variable abstract this article considers both likelihood and bayesian estimation procedures for a model with multiple indicators and multiple causes of a single unobservable latent variable the model is complicated by its reduced form which displays a mixture of econometric and psychometric themes we avoid this complexity via the estimation of the unobservable latent variable through direct use of the original structural form this approach not only provides maximum likelihood estimates that are equivalent to those derived from the reduced form but it also permits a feasible bayesian approach which would be technically complex to execute otherwise

the three pass regression filter a new approach to forecasting using many predictors we forecast a single time series using many predictor variables with a new estimator called the three pass regression lter 3prf it is calculated in closed form and conveniently represented as a set of ordinary least squares regressions 3prf forecasts converge to the infeasible best forecast when both the time dimension and cross section dimension become large this requires only specifying the number of relevant factors driving the forecast target regardless of the total number of common including potentially irrelevant factors driving the cross section of predictors we derive inferential theory including limiting distributions for estimated relevant factors predictive coefcients and forecasts and provide consistent standard error estimators we explore two empirical applications forecasting macroeconomic aggregates with a large panel of economic indices and forecasting stock market aggregates with many individual assets price dividend ratios these combined with a range of monte carlo experiments demonstrate the 3prf s forecasting power

counterexamples to the kotani last conjecture for continuum schrodinger operators via character automorphic hardy spaces abstract the kotani last conjecture states that every ergodic operator in one space dimension with non empty absolutely continuous spectrum must have almost periodic coefficients this statement makes sense in a variety of settings for example discrete schrodinger operators jacobi matrices cmv matrices and continuum schrodinger operators in the main body of this paper we show how to construct counterexamples to the kotani last conjecture for continuum schrodinger operators by adapting the approach developed by volberg and yuditskii to construct counterexamples to the kotani last conjecture for jacobi matrices this approach relates the reflectionless operators associated with the prescribed spectrum to a family of character automorphic hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters the key to our approach is an explicit correspondence between the space of continuum reflectionless schrodinger operators associated with a given set and the space of reflectionless jacobi matrices associated with a derived set once this correspondence is established we can rely to a large extent on the previous work of volberg and yuditskii to produce the resulting action on the space of characters we analyze this action and identify situations where we can observe absolute continuity without almost periodicity in the appendix we show how to implement this strategy and obtain analogous results for extended cmv matrices

almost periodicity in time of solutions of the kdv equation we study the cauchy problem for the kdv equation tu 6u xu xu 0 with almost periodic initial data u x 0 v x we consider initial data v for which the associated schrodinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify and show the existence uniqueness and almost periodicity in time of solutions this establishes a conjecture of percy deift for this class of initial data the result is shown to apply to all small analytic quasiperiodic initial data with diophantine frequency vector

finite size effects in the anisotropic λ 4 φ14 φ24 d model we consider the 4 14 24 model on a d dimensional euclidean space where all but one of the coordinates are unbounded translation invariance along the bounded coordinate z which lies in the interval 0 l is broken because of the boundary conditions bcs chosen for the hyperplanes z 0 and z l dd and nn where d denotes dirichlet and n neumann respectively the renormalization procedure up to one loop order in the two point function is applied obtaining two main results the first is the fact that the renormalization program requires the introduction of counterterms which are surface interactions the second one is that the tadpole graphs for dd and nn have the same z dependent part in modulus but with opposite signs we investigate the relevance of this fact to the elimination of surface divergences

the floquet theory of the periodic euler bernoulli equation abstract we continue the study initiated in 17 of the spectral theory of the fourth order eigenvalue problem a x u x x u x where the functions a and are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described by a and the equality of the algebraic and geometric multiplicities of the periodic and antiperiodic eigenvalues is established also a spectrum like set that we called pseudospectrum or spectrum is introduced or rather discovered this spectrum is shown to lie on the negative real axis and have a band gap structure various open questions and conjectures are mentioned at the end of the paper

systems of seemingly unrelated regression equations with time varying coefficients an interplay of kalman filtering scoring em and minque method abstract this paper gives a detailed description of the implementation of ml estimation using scoring and em for the hyperparameters of a particular econometric state space we also show how the em method can be turned into an on line estimation method and how this procedure relates to an adaptive method of variance component estimation based on minque theory a small simulation study for a single equation model indicates how much can be gained in terms of computer time by using a combination of scoring and em

alternative algorithms for the estimation of dynamic factor mimic and varying coefficient regression models this paper provides a general approach to the formulation and estimation of dynamic unobserved component models after introducing the general model two methods for estimating the unknown parameters are presented both are algorithms for maximizing the likelihood function the first is based on the method of scoring the second is the em algorithm a derivative free method each iteration of em requires a kalman filter and smoother followed by straightforward regression calculations the paper suggests using the em methods to quickly locate a neighborhood of the maximum scoring can then be used to pinpoint the maximum and calculate the information matrix

an algorithm for the exact fisher information matrix of vector armax time series abstract in this paper an algorithm is developed for the exact fisher information matrix of a gaussian vector armax or varmax process the algorithm proposed in this paper is composed by chandrasekhar recursion equations at a vector matrix level and some of these recursions consist of derivatives based on appropriate differential rules applied to a state space model for a vector process the chosen representation is such that the recursions extracted from the state space model are given in terms of expectations of derivatives of innovations and not the process and observation disturbances the algorithm will be illustrated by an example on that example a comparison is made with results from e4 a toolbox for matlab and with the asymptotic information matrix

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

equality of the spectral and dynamical definitions of reflection for full line jacobi matrices schrodinger operators and cmv matrices we show that being reflectionless in the sense of the well known property of m functions is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x as t goes entirely to x as t this allows us to settle a conjecture of deift and simon from 1983 regarding ergodic jacobi matrices

the periodic euler bernoulli equation we continue the study of the floquet spectral theory of the beam equation namely the fourth order eigenvalue problem a x u x x u x x where the functions a and p are periodic and strictly positive this equation models the transverse vibrations of a thin straight periodic beam whose physical characteristics are described by a and p here we develop a theory analogous to the theory of the hill operator d dx 2 q x we first review some facts and notions from our previous works including the concept of the pseudospectrum or spectrum our new analysis begins with a detailed study of the zeros of the function f a k for any given quasimomentum k c where f k 0 is the floquet bloch variety of the beam equation the hill quantity corresponding to f k is 2 cos kb where a a is the discriminant and b the period of q we show that the multiplicity m of any zero of f a k can be one or two and m 2 for some k if and only if is also a zero of another entire function d a independent of k furthermore we show that d a has exactly one zero in each gap of the spectrum and two zeros counting multiplicities in each gap if is a double zero of f a k it may happen that there is only one floquet solution with quasimomentum k thus there are exceptional cases where the algebraic and geometric multiplicities do not agree next we show that if is an open gap of the pseudospectrum i e then the floquet matrix t a has a specific jordan anomaly at a a and a we then introduce a multipoint dirichlet type eigenvalue problem which is the analogue of the dirichlet problem for the hill equation we denote by n n z the eigenvalues of this multipoint problem and show that n n z is also characterized as the set of values of a for which there is a proper floquet solution f x a such that f 0 a 0 we also show theorem 7 that each gap of the l 2 r spectrum contains exactly one n and each gap of the pseudospectrum contains exactly two n s counting multiplicities here when we say gap or gap we also include the endpoints so that when two consecutive bands or bands touch the in between collapsed gap or gap is a point we believe that n n z can be used to formulate the associated inverse spectral problem as an application of theorem 7 we show that if is a collapsed closed gap then the floquet matrix t is diagonalizable some of the above results were conjectured in our previous works however our conjecture that if all the gaps are closed then the beam operator is the square of a second order hill type operator is still open

a standard error for the estimated state vector of a state space model abstract this paper motivates an estimate of the variance of the estimated state vector x t in a state space model when the vector of parameters characterizing system dynamics must be estimated from the data

the isospectral torus of quasi periodic schrodinger operators via periodic approximations we study the quasi periodic schrodinger operator begin aligned psi x v x psi x e psi x quad x in mathbb r end aligned in the regime of small v x sum m in mathbb z nu c m exp 2 pi i m omega x omega omega 1 dots omega nu in mathbb r nu c m le varepsilon exp kappa 0 m we show that the set of reflectionless potentials isospectral with v is homeomorphic to a torus moreover we prove that any reflectionless potential q isospectral with v has the form q x sum m in mathbb z nu d m exp 2 pi i m omega x with the same omega and with d m le sqrt 2 varepsilon exp frac kappa 0 2 m our derivation relies on the study of the approximation via hill operators with potentials tilde v x sum m in mathbb z nu c m exp 2 pi i m tilde omega x where tilde omega is a rational approximation of omega it turns out that the multi scale analysis method of damanik and goldstein publ math inst hautes etudes sci 119 217 401 2014 applies to these hill operators namely in damanik et al trans am math soc to appear arxiv 1409 2147 2016 we developed the multi scale analysis for the operators dual to the hill operators in question the main estimates obtained in damanik et al trans am math soc to appear arxiv 1409 2147 2016 allow us here to establish estimates for the gap lengths and the fourier coefficients in a form that is considerably stronger than the estimates known in the theory of hill operators with analytic potentials in the general setting due to these estimates the approximation procedure for the quasi periodic potentials is effective despite the fact that the rate of approximation omega tilde omega thicksim tilde t delta 0 delta 1 2 is slow on the scale of the period tilde t of the hill operator

disturbance smoother for state space models summary this paper develops a method to evaluate the smoothed estimator of the disturbance vector in a state space model together with its mean squared error matrix this disturbance smoother also leads to an efficient smoother for the state vector applications include a method to calculate auxiliary residuals for unobserved components time series models and an em algorithm for estimating covariance parameters in a state space model

extensions of estimation methods using the em algorithm abstract the em algorithm described by dempster laid and rubin 1977 is reviewed with the purpose of clarifying several misconceptions in the statistical and econometric literature the clarifications lead to several applications of the algorithm to models that have appeared to be less tractable the relationship between the em algorithm and the method of scoring is also explained providing estimators of the score and the information from the em algorithm the em algorithm is extended to missing data problems and an estimation method based on simulations

critical phenomena with convergent series expansions in a finite volume linked cluster expansions are generalized from an infinite to a finite volume they are performed to 20th order in the expansion parameter to approach the critical region from the symmetric phase a new criterion is proposed to distinguish first from second order transitions within a finite size scaling analysis the criterion applies also to other methods for investigating the phase structure such as monte carlo simulations our computational tools are illustrated with the example of scalar o n models with four and six point couplings forn 1 andn 4 in three dimensions it is shown how to localize the tricritical line in these models we indicate some further applications of our methods to the electroweak transition as well as to models for superconductivity

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

hidden gauss markov models for signal classification continuous state hidden markov models cs hmms are developed as a tool for signal classification analogs of the baum 1972 viterbi 1962 and baum welch algorithms are formulated for this class of models the cs hmm algorithms are then specialized to hidden gauss markov models hgmms with linear gaussian state transition and output densities a new gaussian refactorization lemma is used to show that the baum and viterbi algorithms for hgmms are implemented by two different formulations of the fixed interval kalman smoother the measurement likelihoods obtained from the forward pass of the hgmm baum algorithm and from the kalman filter innovation sequence are shown to be equal a direct link between the baum welch training algorithm and an existing expectation maximization em algorithm for gaussian models is demonstrated a new expression for the cross covariance between time adjacent states in hgmms is derived from the off diagonal block of the conditional joint covariance matrix a parameter invariance structure is noted for the hgmm likelihood function cs hmms and hgmms are extended to incorporate mixture densities for the a priori density of the initial state application of hgmms to signal classification is demonstrated with a three class test simulation

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

dynamic linear models with markov switching abstract in this paper hamilton s 1988 1989 markov switching model is extended to a general state space model this paper also complements shumway and stoffer s 1991 dynamic linear models with switching by introducing dependence in the switching process and by allowing switching in both measurement and transition equations building upon ideas in hamilton 1989 cosslett and lee 1985 and harrison and stevens 1976 a basic filtering and smoothing algorithm is presented the algorithm and the maximum likelihood estimation procedure is applied in estimating lam s 1990 generalized hamilton model with a general autoregressive component the estimation results show that the approximation employed in this paper performs an excellent job with a considerable advantage in computation time a state space representation is a very flexible form and the approach taken in this paper therefore allows a broad class of models to be estimated that could not be handled before in addition the algorithm for calculating smoothed inferences on the unobserved states is a vastly more efficient one than that in the literature

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

assessing the predictive influence of cases in a state space process an important inferential objective in state space modelling is to recover unobserved states using fixed interval smoothing thus the identification of cases which have a substantial influence on the smoothers is a relevant practical problem to facilitate this identification we propose a case deletion diagnostic which can be easily computed using the outputs of the standard filtering and smoothing algorithms our diagnostic is defined as the kullback leibler directed divergence between two versions of the conditional density which determines the smoothers one based on all the data the other based on all the data except for the case or cases in question we investigate the detection performance of the diagnostic in a practical application

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

dynamic structural systems under indirect observation identifiability and estimation aspects from a system theoretic perspective in this partly expository paper the parameter identifiability and estimation of a general dynamic structural model under indirect observation will be considered from a system theoretic perspective the general dynamic model covers dynamic factor analytic models dynamic mimic models and joreskog s linear structural model as special cases its reduced form is under a slightly different specification known in system theory and econometrics as the stochastic stationary version of the state space model by using concepts and methods from system theory such as the observability and controllability concept the steady state kalman filter and a general nonlinear ml estimation procedure known as prediction error estimation the general dynamic model will be identified it will be shown that joreskog s lisrel procedure is a special case of the prediction error estimation procedure

trace formulae for the schrodinger equation with energy dependent potential we consider the eigenvalue problems for the schrodinger equation with energy dependent potential subject to the eigenparameter boundary conditions and quasiperiodic boundary conditions respectively two new formulae of the regularized trace for this class of schrodinger equation with boundary conditions are obtained in an explicit form

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

the analytic regularization zeta function method and the cut off method in the casimir effect the zero point energy associated with a hermitian massless scalar field in the presence of perfectly reflecting plates in a 3d flat space time is discussed a new technique to unify two different methods the zeta function and a variant of the cut off method used to obtain the so called casimir energy is presented and the proof of the analytic equivalence between both methods is given

spectral properties of a class of reflectionless schrodinger operators abstract we prove that one dimensional reflectionless schrodinger operators with spectrum a homogeneous set in the sense of carleson belonging to the class introduced by sodin and yuditskii have purely absolutely continuous spectra this class includes all earlier examples of reflectionless almost periodic schrodinger operators in addition we construct examples of reflectionless schrodinger operators with more general types of spectra given by the complement of a denjoy widom type domain in c which exhibit a singular component

eigenvalue bounds in the gaps of schrodinger operators and jacobi matrices abstract we consider c a b where a is selfadjoint with a gap a b in its spectrum and b is relatively compact we prove a general result allowing b of indefinite sign and apply it to obtain a v d 2 bound for perturbations of suitable periodic schrodinger operators and a not quite lieb thirring bound for perturbations of algebro geometric almost periodic jacobi matrices