on the well posedness for nls in hs abstract in this article we prove the local well posedness of the cauchy problem for the nonlinear schrodinger equation i t u u c u u in h s r n with 0 4 n 2 s and 1 s min n 2 2

strichartz type estimates in mixed besov spaces with application to critical nonlinear schrodinger equations abstract we study the cauchy problem for the nonlinear schrodinger equation with power nonlinearity in the fractional order sobolev space h s r n we consider the case where the nonlinear term is h s critical and the differentiability thereof is less than s we prove the existence of time global solutions of the cauchy problem for small initial data to this end we prove strichartz type estimates for linear inhomogeneous schrodinger equations in mixed besov spaces

global strong solutions for nonlinear schro dinger equations on considere l existence globale unique des solutions fortes pour les equations de schrodinger non lineaire a interaction en loi de puissance i u t u u p 1 u t r x r n u t 0 x u 0 x x r n ou t 0 r

multivariate calibration with more variables than observations multivariate calibration involves the use of an estimated linear relationship between a q variate response vector y and a p dimensional explanatory vector x to estimate or predict future unknown x from further observed y in some applications for example when y represents automatic intensity measurements at q different wavelengths on n chemical standard samples the feasible sample size n may be restricted whereas the dimension q can be chosen quite large in this article the singular cases appearing when n q the traditional solutions to the estimation and prediction problems that is the generalized least squares estimator and the estimated best linear predictor are both unique whereas for smaller n q n 1 dimensional hyperplanes of solutions are obtained the same in both problems the properties of the predictor are also empirically studied in an example with p 1 q 6 and varying n

maximum likelihood regression on relevant components summary most regression methods for multicollinear data are only based on assumptions about the conditional distribution of the dependent variable given the explanatory variables here we propose a new prediction method taking a joint multinormal distribution for all the variables as the starting point a hypothesis is formulated stating that only a fixed small number of components constructed from the explanatory variables is relevant for prediction and maximum likelihood type estimates of the parameters in the model are developed under this hypothesis

generalized linear regression on sampled signals and curves a p spline approach we consider generalized linear regression with many highly correlated regressors for instance digitized points of a curve on a spatial or temporal domain we refer to this setting as signal regression which requires severe regularization because the number of regressors is large often exceeding the number of observations we solve collinearity by forcing the coefficient vector to be smooth on the same domain dimension reduction is achieved by projecting the signal coefficient vector onto a moderate number of b splines a difference penalty between the b spline coefficients further increases smoothness the p spline framework of eilers and marx the procedure is regulated by a penalty parameter chosen using information criteria or cross validation

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

additive risk models for survival data with high dimensional covariates as a useful alternative to cox s proportional hazard model the additive risk model assumes that the hazard function is the sum of the baseline hazard function and the regression function of covariates this article is concerned with estimation and prediction for the additive risk models with right censored survival data especially when the dimension of the covariates is comparable to or larger than the sample size principal component regression is proposed to give unique and numerically stable estimators asymptotic properties of the proposed estimators component selection based on the weighted bootstrap and model evaluation techniques are discussed this approach is illustrated with analysis of the primary biliary cirrhosis clinical data and the diffuse large b cell lymphoma genomic data it is shown that this methodology is numerically stable and effective in dimension reduction while still being able to provide satisfactory prediction and classification results

continuum regression is not always continuous in regression with near collinear explanatory variables the least squares predictor has large variance ordinary least squares regression olsr often leads to unrealistic regression coefficients several regularized regression methods have been proposed as alternatives well known are principal components regression pcr ridge regression rr and continuum regression cr the latter two involve a continuous metaparameter offering additional flexibility for a univariate response variable cr incorporates olsr plsr and pcr as special cases for special values of the metaparameter cr is also closely related to rr however cr can in fact yield regressors that vary discontinuously with the metaparameter thus the relation between cr and rr is not always one to one we develop a new class of regression methods lsrr essentially the same as cr but without discontinuities and prove that any optimization principle will yield a regressor proportional to a rr provided only that the principle implies maximizing some function of the regressor s sample correlation coefficient and its sample variance for a multivariate response vector we demonstrate that a number of well established regression methods are related in that they are special cases of basically one general procedure we try a more general method based on this procedure with two meta parameters in a simulation study we compare this method to ridge regression multivariate plsr and repeated univariate plsr for most types of data studied all methods do approximately equally well there are cases where rr and lsrr yield larger errors than the other methods and we conclude that one factor methods are not adequate for situations where more than one latent variable are needed to describe the data among those based on latent variables none of the methods tried is superior to the others in any obvious way

continuum regression and ridge regression we demonstrate the close relationship between first factor continuum regression and standard ridge regression the difference is that continuum regression inserts a scalar compensation factor for that part of the shrinkage in ridge regression that has no connection with tendencies towards collinearity we interpret this to mean that first factor continuum regression is preferable in principle to ridge regression if we want protection against near colliearity but do not admit shrinkage as a general principle furthermore our experience indicates that with first factor continuum regression we can obtain predictors that are at least as mean squared error efficient as with ridge regression but with less sensitivity to the choice of ridge constant

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

local existence of solutions of self gravitating relativistic perfect fluids this paper deals with the evolution of the einstein gravitational fields which are coupled to a perfect fluid we consider the einstein euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity and that the pressure is a fractional power of the energy density in this setting we prove local in time existence uniqueness and well posedness of classical solutions the zero order term of our system contains an expression which might not be a c function and therefore causes an additional technical difficulty in order to achieve our goals we use a certain type of weighted sobolev space of fractional order in brauer and karp j diff eqs 251 1428 1446 2011 we constructed an initial data set for these of systems in the same type of weighted sobolev spaces

principal components regression with data chosen components and related methods multiple regression with correlated explanatory variables is relevant to a broad range of problems in the physical chemical and engineering sciences chemometricians in particular have made heavy use of principal components regression and related procedures for predicting a response variable from a large number of highly correlated variables in this article we develop a general theory for selecting principal components that yield estimates of regression coefficients with low mean squared error our numerical results suggest that the theory also can be used to improve partial least squares regression estimators and regression estimators based on rotated principal components although our work has been motivated by the statistical genetics problem of mapping quantitative trait loci the results are applicable to any problem in which estimation of regression coefficients for correlated explanatory variables is of interest

downdating the moore penrose generalized inverse for cross validation of centred least squares prediction summary given ap x p symmetric matrix b and ap vector c outside the range of b a simple expression is formulated for the action of the moore penrose generalized inverse b on a vector b in the range of b in terms of b c and b cc a special case of this formula gives a downdating formula for the vector of least squares regression coefficients based on the moore penrose inverse of the centred sum of squares and products matrix the formula is analogous to that given by a albert for uncentred least squares fitting an explicit downdating formula for the moore penrose inverse b itself is derived and then more general downdating expressions for the inverse and the vector b b when the p vector c is replaced by a matrix array c of such vectors outside the range of b

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

when the data are functions a datum is often a continuous functionx t of a variable such as time observed over some interval one or more such functions are observed for each subject or unit of observation the extension of classical data analytic techniques designed forp variate observations to such data is discussed the essential step is the expression of the classical problem in the language of functional analysis after which the extension to functions is a straightforward matter a schematic device called the duality diagram is a very useful tool for describing an analysis and for suggesting new possibilities least squares approximation descriptive statistics principal components analysis and canonical correlation analysis are discussed within this broader framework

multivariate calibration when the error covariance matrix is structured this article concerns multivariate calibration in linear models the error covariance matrix is assumed to have linear factor structure a new approach to calibration incorporating distributional properties of the observations of the calibration set is proposed the proposed predictor is based on estimating the parameters in the best linear predictor in the assumed model the predictor is tested on two data sets meat data and fish data

some insights into continuum regression and its asymptotic properties continuum regression encompasses ordinary least squares regression partial least squares regression and principal component regression under the same umbrella using a nonnegative parameter however there seems to be no literature discussing the asymptotic properties for arbitrary continuum regression parameter this article establishes a relation between continuum regression and sufficient dimension reduction and studies the asymptotic properties of continuum regression for arbitrary under inverse regression models theoretical and simulation results show that the continuum seems unnecessary when the conditional distribution of the predictors given the response follows the multivariate normal distribution

continuum regression cross validated sequentially constructed prediction embracing ordinary least squares partial least squares and principal components regression read before the royal statistical society at a meeting organized by the research section on wednesday october 25th 1989 professor d v hinkley in the chair summary the paper addresses the evergreen problem of construction of regressors for use in least squares multiple regression in the context of a general sequential procedure for doing this it is shown that with a particular objective criterion for the construction the procedures of ordinary least squares and principal components regression occupy the opposite ends of a continuous spectrum with partial least squares lying in between there are two adjustable parameters controlling the procedure alpha in the continuum 0 11 and omega the number of regressors finally accepted these control parameters are chosen by crossvalidation the method is illustrated by a range of examples of its application

modified strichartz estimates with an application to the critical nonlinear schrodinger equation abstract the cauchy problem for the critical nonlinear schrodinger equation is considered in the sobolev space of fractional order some modified strichartz estimates are constructed and applied to the problem to obtain small global solutions with less regularity assumption for the nonlinear term

the peculiar shrinkage properties of partial least squares regression partial least squares regression has been widely adopted within some areas as a useful alternative to ordinary least squares regression in the manner of other shrinkage methods such as principal components regression and ridge regression in this paper we examine the nature of this shrinkage and demonstrate that partial least squares regression exhibits some undesirable properties

linear dimensionality reduction survey insights and generalizations linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data due to their simple geometric interpretations and typically attractive computational properties these methods capture many data features of interest such as covariance dynamical structure correlation between data sets input output relationships and margin between data classes methods have been developed with a variety of names and motivations in many fields and perhaps as a result the connections between all these methods have not been highlighted here we survey methods from this disparate literature as optimization programs over matrix manifolds we discuss principal component analysis factor analysis linear multidimensional scaling fisher s linear discriminant analysis canonical correlations analysis maximum autocorrelation factors slow feature analysis sufficient dimensionality reduction undercomplete independent component analysis linear regression distance metric learning and more this optimization framework gives insight to some rarely discussed shortcomings of well known methods such as the suboptimality of certain eigenvector solutions modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver which accepts as input data and an objective to be optimized and returns as output an optimal low dimensional projection of the data this simple optimization framework further allows straightforward generalizations and novel variants of classical methods which we demonstrate here by creating an orthogonal projection canonical correlations analysis more broadly this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox objective agnostic numerical technology

local well posedness for the h 2 critical nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation iu t delta u lambda u frac 4 n 4 u in r n n ge 5 with lambda in c we prove local well posedness local existence unconditional uniqueness continuous dependence in the critical space dot h 2 r n

bootstrap and cross validation estimates of the prediction error for linear regression models different estimates of the mean squared error of prediction for linear regression models are derived by the bootstrap and cross validation approaches a comparison is made under normal error distributions especially by the biases and the mean square errors the results indicate that the bias corrected bootstrap estimator is best unbiased and should be the first choice while its simulated variant has approximately the same behaviour on the other hand if only a comparison between uncorrected estimators is made with implications for nonlinear regression models in mind then other variants of bootstrap estimates are preferable for a large or a small dimension of the model parameter for a small dimension the cross validation estimate and sometimes grouped variants of it seem also to be acceptable if the model error is known to be small

on well posedness for nonlinear schrodinger equations with power nonlinearity in fractional order sobolev spaces abstract we study the well posedness for the nonlinear schrodinger equation nls i t u 1 2 u u p 1 u in r 1 n where p 1 c and prove that nls is locally well posed in h s if 2 s 4 and s 2 p 1 4 n 2 s to obtain a good lower bound for p we systematically use strichartz type estimates in fractional order besov spaces for the time variable

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3