nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

caratheodory toeplitz and nehari problems for matrix valued almost periodic functions in this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated we present nece ssary and sufficient conditions for the existence of extensions in terms of toeplit z and hankel operators on besicovitch spaces and lebesgue spaces furthermore when a solution exists a special extension the band extension is constructed which enjoys a maximum entropy property a linear fractional parameterization of the set of all extensions is also provided the techniques used xn the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method

contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

nonlinear scattering theory at low energy sequel abstract we extend the nonlinear scattering theory and apply it to schrodinger and klein gordon equations with various kinds of nonlinear terms including some nonlocal ones

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

weighted estimates for a convolution appearing in the wave equation of hartree type abstract the paper concerns the cauchy and scattering problem of the wave equation of hartree type with small initial data with fast decay we prove weighted estimates for a convolution which appears in the equation

displacement structure and maximum entropy the study of matrices with a displacement structure is mainly concerned with recursions for the so called generator matrices the recursion usually involves free parameters which can be chosen in several ways so as to simplify the resulting algorithm in this correspondence we present a choice for the parameters that is motivated by a maximum entropy formulation this choice further motivates the introduction of the so called generalized reflection coefficients which are in general different from the better known schur coefficients

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

on a semilinear wave equation the cauchy problem and the asymptotic behavior of solutions abstract the semilinear wave equation u m 2 u u p 2 u v u p 0 in r 3 is studied where denotes the d alembertian operator and means spatial convolution under mild assumptions on the real valued function v and 2 p 3 the well posedness of the cauchy problem is proved furthermore some properties of the solutions of the equation are analyzed such as the asymptotic behavior of local energy as t in the case of zero mass our results extend that of perla menzala and strauss where case p 2 was studied

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

estimating central subspaces via inverse third moments summary modern graphical tools have enhanced our ability to learn many things from data directly in recent years dimension reduction has proven to be an effective tool for generating low dimensional summary plots without appreciable loss of information some well known inverse regression methods for dimension reduction such as sliced inverse regression li 1991 and sliced average variance estimation cook weisberg 1991 have been developed to estimate summary plots for regression and discriminant analysis in this paper we suggest a new method that makes use of inverse third moments this method can find structure beyond that found by sliced inverse regression and sliced average variance estimation particularly regression mixtures illustrative examples are presented

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

approximate quasi likelihood estimation in models with surrogate predictors abstract we consider quasi likelihood estimation with estimated parameters in the variance function when some of the predictors are measured with error we review and extend four approaches to estimation in this problem all of them based on small measurement error approximations a taxonomy of the data sets likely to be available in measurement error studies is developed an asymptotic theory based on this taxonomy is obtained and includes measurement error and berkson error models as special cases

principal hessian directions for regression with measurement error summary we consider a nonlinear regression problem with predictors with measurement error we assume that the response is related to unknown linear combinations of a p dimensional predictor vector through an unknown link function instead of observing the predictors we observe a surrogate vector with the property that its expectation is linearly related to the predictor vector with constant variance we use an important linear transformation of the surrogates based on the transformed variables we develop the modified principal hessian directions method for estimating the subspace of the effective dimension reduction space we derive the asymptotic variances of the modified principal hessian directions estimators several examples are reported and comparisons are made with the sliced inverse regression method of carroll li 1992

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

a maximum entropy principle for contractive interpolants abstract in this paper a maximum entropy principle is established for a class of continuous contractive m n matrix valued functions g with preassigned fourier coefficients gj j for j 0 1 the function g in this class which achieves the maximum entropy is characterized in a number of different ways and its entropy is computed in terms of a hankel operator based on the prescribed matrices 0 1 a pair of seemingly new szego type formulas are obtained enroute and some applications to nevanlinna pick interpolation are indicated

dimension reduction for the conditional kth moment in regression the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction here we introduce the central kth moment subspace to capture information from the mean variance and so on up to the kth conditional moment of the regression new methods are studied for estimating these subspaces connections with sliced inverse regression are established and examples illustrating the theory are presented

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

small data blow up for the wave equation with a time dependent scale invariant damping and a cubic convolution for slowly decaying initial data abstract in the present paper we study the cauchy problem for the wave equation with a time dependent scale invariant damping i e 2 1 t t v and a cubic convolution x v 2 v with 0 n where v v x t is an unknown function on r n 0 t our aim of the present paper is to prove a small data blow up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data v x 0 t v x 0 such as t v x 0 o x 1 as x here belongs to the scaling supercritical case n 2 the proof of our main result is based on the combination of the arguments in the papers takamura et al 2010 and takamura 1995 especially our main new contribution is to estimate the convolution term in high spatial dimensions i e n 4 this paper is the first blow up result to treat wave equations with the cubic convolution in high spatial dimensions n 4

on surrogate dimension reduction for measurement error regression an invariance law we consider a general nonlinear regression problem where the predictors contain measurement error it has been recently discovered that several well known dimension reduction methods such as ols sir and phd can be performed on the surrogate regression problem to produce consistent estimates for the original regression problem involving the unobserved true predictor in this paper we establish a general invariance law between the surrogate and the original dimension reduction spaces which implies that at least at the population level the two dimension reduction problems are in fact equivalent consequently we can apply all existing dimension reduction methods to measurement error regression problems the equivalence holds exactly for multivariate normal predictors and approximately for arbitrary predictors we also characterize the rate of convergence for the surrogate dimension reduction estimators finally we apply several dimension reduction methods to real and simulated data sets involving measurement error to compare their performances

on a wave equation with a cubic convolution abstract we study the well posedness of the cauchy problem and the asymptotic behavior of solutions of the nonlinear wave equation u tt u m 2 u v u 2 0 in euclidean space

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

invertible completions of partial operator matrices the nonsymmetric case abstract we consider nonsymmetric partial operator matrices r whose directed graphs belong to a certain class and for which certain key fully specified principal submatrices are invertible under these circumstances we prove the existence of a unique invertible completion f of r such that f 1 ij is block zero whenever rij is unspecified in this way the existing results of johnson and lundquist are generalized to the nonsymmetric case

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

covariance analysis in generalized linear measurement error models we summarize some of the recent work on the errors in variables problem in generalized linear models the focus is on covariance analysis and in particular testing for and estimation of treatment effects there is a considerable difference between the randomized and non randomized models when testing for an effect in randomized studies simple techniques exist for testing for a treatment effect in some instances such as linear and multiplicative regression simple methods exist for estimating the treatment effect in other examples such as logistic regression estimating a treatment effect requires careful attention to measurement error in non randomized studies there is no recourse to understanding and modelling measurement error in particular ignoring measurement error can lead to the wrong conclusions for example the true but unobserved data may indicate a positive effect for treatment while the observed data indicate the opposite some of the possible methods are outlined and compared

on global solution to the klein gordon hartree equation below energy space in this paper we consider the cauchy problem for klein gordon equation with a cubic convolution nonlinearity in r3 by making use of bourgain s method in conjunction with a precise strichartz estimate of s klainerman and d tataru we establish the hs s 1 global well posedness of the cauchy problem for the cubic convolution defocusing klein gordon hartree equation before arriving at the previously discussed conclusion we obtain global solution for this non scaling equation with small initial data in hs0 hs0 1 where s0 6 but not 2 1 for this equation that we consider is a subconformal equation in some sense in doing so a number of nonlinear prior estimates are already established by using bony s decomposition flexibility of klein gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate we establish this commutator estimate by exploiting cancellation property and utilizing coifman and meyer multilinear multiplier theorem as far as we know it seems that this is the first result on low regularity for this klein gordon hartree equation

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

an extension problem for discrete time almost periodically correlated stochastic processes abstract in d alpay a chevreuil ph loubaton j time ser anal 2000 to appear an extension problem for covariance matrix of discrete time periodically correlated stochastic processes introduced by gladyshev was treated in this paper we study the same problem for discrete time almost periodically correlated stochastic processes this problem can be reformulated and solved within the framework of interpolation for upper triangular operators more precisely one can reduce the problem to an interpolation problem in the class of upper triangular operators of the schur class

an entropy formula for time varying discrete time control systems the results of this paper generalize the formula for the entropy of a transfer function to time varying systems this is done through the use of some results on spectral factorizations due to arveson and properties of the cw transform which generalizes the usual cz transform for time varying systems using the formula defined it is shown that for linear fractional transformations like those that arise in time varying f control there exists a unique bounded contraction which maximizes the entropy this generalizes known results in the time invariant case possible extensions are discussed along with state space formulae

the inverse scattering problem for schrodinger and klein gordon equations with a nonlocal nonlinearity abstract we study the inverse scattering problem for the nonlinear schrodinger equation and for the nonlinear klein gordon equation with the generalized hartree type nonlinearity we reconstruct the nonlinearity from knowledge of the scattering operator which improves the known results

slicing regression a link free regression method slicing regression a link free regression m e t h o d author s naihua duan and k e r c h a u l i s o u r c e the annals of statistics v o l 19 n o 2 t u n 1991 p p 5 0 5 5 3 0 p u b l i s h e d b y institute of mathematical statistics s t a b l e u r l http www jstor org stable 2242072 accessed 16 05 2011 your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available at http www jstor org page info about policies terms jsp jstor s terms and conditions of use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained at http www jstor org action showpublisher publishercode ims each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jstor is a not for profit service that helps scholars researchers and students discover use and build upon a wide range of content in a trusted digital archive we use information technology and tools to increase productivity and facilitate new forms of scholarship for more information about jstor please contact support jstor org http www jstor org

measurement error regression with unknown link dimension reduction and data visualization abstract a general nonlinear regression problem is considered with measurement error in the predictors we assume that the response is related to an unknown linear combination of a multidimensional predictor through an unknown link function instead of observing the predictor we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance we identify an important transformation of the surrogate variable using this transformed variable we show that if one proceeds with the usual analysis ignoring measurement error then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter up to a constant of proportionality we derive the asymptotic distribution of the estimates a simulation study is conducted applying sliced inverse regression in this context

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

the maximum distance problem and band sequences abstract we solve the following problem for 1 j k n and j k m let a jk be a given complex number with a kj jk we wish to find linearly independent vectors x 1 x n such that x k x j a jk for j k m and such that the distance from x k to the linear span of x 1 x k 1 is maximal for 2 k n we construct and characterize all such sequences of vectors our solution leads naturally to the class of m band sequences of vectors in an inner product space we study these sequences and characterize their equivalence classes under unitary transformations