the zakharov shabat inverse spectral problem for operators many soliton equations their solutions and their properties can be deduced from simpler essentially linear equations involving more complicated objects e g integral operators the deduction process from the operator valued equation to the scalar one amounts to some rather simple procedure like taking a trace or a determinant this is described for the class of soliton equations solvable with the zakharov shabat spectral transform

extension of noncommutative soliton hierarchies a linear system which generates a moyal deformed two dimensional soliton equation as an integrability condition can be extended to a three dimensional linear system treating the deformation parameter as an additional coordinate the supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter the flow of which commutes with the flow of the deformed soliton equation in this way a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations we prove the extended hierarchy properties for the deformed akns hierarchy and specialize to the cases of deformed nls kdv and mkdv hierarchies corresponding results are also obtained for the deformed kp hierarchy a deformation equation determines a kind of seiberg witten map from classical solutions to solutions of the respective noncommutative deformed equation

the noncommutative akns system projection to matrix systems countable superposition and soliton like solutions starting from the recent work on noncommutative akns systems for functions with values in the bounded operators on a banach space it is shown how their formal 1 soliton solution depending on operator parameters can be mapped to solutions of matrix akns systems the main result is rather general solution formulas for matrix akns systems the most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton like solutions of the classical akns system

existence and uniqueness of weak solutions for a non uniformly parabolic equation in this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial boundary value problem of a non uniformly parabolic equation some well known parabolic equations are the special cases of this equation

on realizations of solutions of the kdv equation by determinants on operator ideals using new developments in the theory of traces determinants and elementary operators on quasi banach operator ideals we clarify and extend marchenko s method for realizing solutions of the kdv equation moreover we point out why abstract traces and determinants on quasi banach operator ideals are appropriate tools for obtaining solutions of the kdv equation the method we present can also be applied to other nonlinear equations in soliton physics

gradient estimates for problems with orlicz growth abstract we study a general nonlinear elliptic equation in the orlicz setting with data not belonging to the dual of the energy space we provide several lorentz type and morrey type estimates for the gradients of solutions under various conditions on the data

new classes of toda soliton solutions we provide a detailed investigation of limits of n soliton solutions of the toda lattice as n tends to infinity our principal results yield new classes of toda solutions including in particular new kinds of soliton like i e reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional jacobi operators and explicitly construct tri diagonal matrices that yield a purely absolutely continuous spectrum in 1 1 and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of mr bs 1 1

cauchy type determinants and integrable systems it is well known that the sylvester matrix equation ax xb c has a unique solution x if and only if 0 spec a spec b the main result of the present article are explicit formulas for the determinant of x in the case that c is one dimensional for diagonal matrices a b we reobtain a classical result by cauchy as a special case the formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine gordon equation and the toda lattice we will provide a concise introduction to the background from soliton theory an operator theoretic approach originating from work of marchenko and carl and discuss examples for the application of the main results

generalized multilevel structural equation modeling a unifying framework for generalized multilevel structural equation modeling is introduced the models in the framework called generalized linear latent and mixed models gllamm combine features of generalized linear mixed models glmm and structural equation models sem and consist of a response model and a structural model for the latent variables the response model generalizes glmms to incorporate factor structures in addition to random intercepts and coefficients as in glmms the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower level units in the higher level units and missing data a wide range of response processes can be modeled including ordered and unordered categorical responses counts and responses of mixed types the structural model is similar to the structural part of a sem except that it may include latent and observed variables varying at different levels for example unit level latent variables factors or random coefficients can be regressed on cluster level latent variables special cases of this framework are explored and data from the british social attitudes survey are used for illustration maximum likelihood estimation and empirical bayes latent score prediction within the gllamm framework can be performed using adaptive quadrature in gllamm a freely available program running in stata

higher integrability for minimizers of integral functionals with p q growth abstract we prove higher integrability for minimizers u r n of integral functionals f du a x u dx where f satisfies a non standard growth condition of p q type z p f z l 1 z q p

an estimate of the covariance between variables which are not jointly observed situations sometimes arise in which variables collected in a study are not jointly observed this typically occurs because of study design an example is an equating study where distinct groups of subjects are administered different sections of a test in the normal maximum likelihood function to estimate the covariance matrix among all variables elements corresponding to those that are not jointly observed are unidentified if a factor analysis model holds for the variables however then all sections of the matrix can be accurately estimated using the fact that the covariances are a function of the factor loadings standard errors of the estimated covariances can be obtained by the delta method in addition to estimating the covariance matrix in this design the method can be applied to other problems such as regression factor analysis two examples are presented to illustrate the method

noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods here noncommutative hierarchies of nonlinear equations are studied they represent a generalization to the operator level of corresponding hierarchies of scalar equations which can be obtained from the operator ones via a suitable projection a key tool is the application of backlund transformations to relate different operator valued hierarchies indeed in the case when hierarchies in 1 1 dimensions are considered a backlund chart depicts links relating in particular the korteweg de vries kdv to the modified kdv mkdv hierarchy notably analogous links connect the hierarchies of operator equations the main result is the construction of an operator soliton solution depending on an infinite dimensional parameter to start with the potential kdv hierarchy is considered then backlund transformations are exploited to derive solution formulas in the case of kdv and mkdv hierarchies it is remarked that hierarchies of matrix equations of any dimension are also incorporated in the present framework

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

estimation for structural equation models with missing data a direct method in handling incomplete data in general covariance structural models is investigated asymptotic statistical properties of the generalized least squares method are developed it is shown that this approach has very close relationships with the maximum likelihood approach iterative procedures for obtaining the generalized least squares estimates the maximum likelihood estimates as well as their standard error estimates are derived computer programs for the confirmatory factor analysis model are implemented a longitudinal type data set is used as an example to illustrate the results

pseudo maximum likelihood estimation of mean and covariance structures with missing data abstract a nonlinear mean and covariance structure model for one or more groups is constructed the model subsumes the usual linear model considered in the literature it is then shown how to estimate the parameters of the model and the asymptotic covariance matrix of the parameter estimates using pseudo maximum likelihood pml estimation the resulting estimates are strongly consistent under general regularity conditions provided only that the model for the first two moments is correctly specified nevertheless because the data are not necessarily drawn from a multivariate normal distribution the usual likelihood ratio tests for model comparisons in mean and covariance structure models do not apply wald tests and lagrange multiplier tests may be used to implement such comparisons next the standard results on ml estimation with missing data are extended to the case of pml estimation with missing data and the results are applied to the model the approach to the missing data problem adopted whic

integrable evolution equations on associative algebras this paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra which includes matrix clifford and group algebra valued systems a variety of new examples of integrable systems possessing higher order symmetries are presented symmetry reductions lead to an associative algebra valued version of the painleve transcendent equations the basic theory of hamiltonian structures for associative algebra valued systems is developed and the bihamiltonian structures for several examples are found

some existence results on periodic solutions of euler lagrange equations in an orlicz sobolev space setting abstract in this paper we consider the problem of finding periodic solutions of certain euler lagrange equations we employ the direct method of the calculus of variations i e we obtain solutions minimizing certain functional i we give conditions which ensure that i is finitely defined and differentiable on certain subsets of orlicz sobolev spaces w 1 l associated to an n function we show that in some sense it is necessary for the coercivity that the complementary function of satisfy the 2 condition we conclude by discussing conditions for the existence of minima of i

quasideterminant solutions of a non abelian toda lattice and kink solutions of a matrix sine gordon equation two families of solutions of a generalized non abelian toda lattice are considered these solutions are expressed in terms of quasideterminants constructed by means of darboux and binary darboux transformations as an example of the application of these solutions we consider the 2 periodic reduction to a matrix sine gordon equation in particular we investigate the interaction properties of polarized kink solutions

stochastic production frontiers and panel data a latent variable framework abstract this paper deals with the issue of estimating production frontier and measuring efficiency from a panel data set first it proposes an alternate method for the estimation of a production frontier on a short panel data set the method is based on the so called mean and covariance structure analysis which is closely related to the generalized method of moments one advantage of the method is that it allows us to investigate the presence of correlations between individual effects and exogenous variables without the requirement of some available instruments uncorrelated with the individual effects as in instrumental variable estimation another advantage is that the method is well suited to a panel data set with a short number of periods second the paper considers the question of recovering individual efficiency levels from the estimates obtained from the mean and covariance structure analysis since individual effects are here viewed as latent variables they can be estimated as factor scores i e weighted sums of the observed variables we illustrate the proposed methods with the estimation of a stochastic production frontier on a short panel data of french fruit growers

least squares theory based on general distributional assumptions with an application to the incomplete observations problem the linear regression modely x e is reanalyzed taking the modest position that x is an approximation of the best predictor ofy we derive the asymptotic distribution ofb andr 2 under mild assumptions the method of derivation yields an easy answer to the estimation of from a data set which contains incomplete observations where the incompleteness is random

the time dependent schrodinger equation of dimension k 1 explicit and rational solutions via gbdt and multinodes a version of the binary darboux transformation is constructed for a non stationary schrodinger equation of dimension k 1 where k 1 is the number of space variables this is an iterated generalized backlund darboux transformation version new families of non singular and rational potentials and solutions are obtained some results are also new for the case that k 1 a certain generalization of a colligation introduced by m s livsic and a generalization of the s node introduced by l a sakhnovich have been successfully used in our construction

robust structural equation modeling with missing data and auxiliary variables the paper develops a two stage robust procedure for structural equation modeling sem and an r package rsem to facilitate the use of the procedure by applied researchers in the first stage m estimates of the saturated mean vector and covariance matrix of all variables are obtained those corresponding to the substantive variables are then fitted to the structural model in the second stage a sandwich type covariance matrix is used to obtain consistent standard errors se of the structural parameter estimates rescaled adjusted as well as corrected and f statistics are proposed for overall model evaluation using r and eqs the r package rsem combines the two stages and generates all the test statistics and consistent ses following the robust analysis multiple model fit indices and standardized solutions are provided in the corresponding output of eqs an example with open closed book examination data illustrates the proper use of the package the method is further applied to the analysis of a data set from the national longitudinal survey of youth 1997 cohort and results show that the developed procedure not only gives a better endorsement of the substantive models but also yields estimates with uniformly smaller standard errors than the normal distribution based maximum likelihood

on the asymptotic relative efficiency of planned missingness designs in planned missingness pm designs certain data are set a priori to be missing pm designs can increase validity and reduce cost however little is known about the loss of efficiency that accompanies these designs the present paper compares pm designs to reduced sample rn designs that have the same total number of data points concentrated in fewer participants in 4 studies we consider models for both observed and latent variables designs that do or do not include an x set of variables with complete data and a full range of between and within set correlation values all results are obtained using asymptotic relative efficiency formulas and thus no data are generated this novel approach allows us to examine whether pm designs have theoretical advantages over rn designs removing the impact of sampling error our primary findings are that a in manifest variable regression models estimates of regression coefficients have much lower relative efficiency in pm designs as compared to rn designs b relative efficiency of factor correlation or latent regression coefficient estimates is maximized when the indicators of each latent variable come from different sets and c the addition of an x set improves efficiency in manifest variable regression models only for the parameters that directly involve the x set variables but it substantially improves efficiency of most parameters in latent variable models we conclude that pm designs can be beneficial when the model of interest is a latent variable model recommendations are made for how to optimize such a design

global lorentz estimates for nonuniformly nonlinear elliptic equations via fractional maximal operators abstract this article is a contribution to the study of regularity theory for nonlinear elliptic equations the aim of this article is to establish some global estimates for nonuniformly elliptic equations in divergence form as div u p 2 u a x u q 2 u div f p 2 f a x f q 2 f which arises from double phase functional problems in particular the main results provide the regularity estimates for the distributional solutions in terms of maximal and fractional maximal operators this work extends that of colombo and mingione 2016 23 and byun and oh 2017 9 by dealing with the global estimates in lorentz spaces it also extends our recent result tran and nguyen 2020 66 concerning new estimates of divergence elliptic equations using cutoff fractional maximal operators for future research the approach developed in this article will allow global estimates of distributional solutions to nonuniformly nonlinear elliptic equations to be obtained in the framework of other spaces

pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models using the theory of pseudo maximum likelihood estimation the asymptotic covariance matrix of maximum likelihood estimates for mean and covariance structure models is given for the case where the variables are not multivariate normal this asymptotic covariance matrix is consistently estimated without the computation of the empirical fourth order moment matrix using quasi maximum likelihood theory a hausman misspecification test is developed this test is sensitive to misspecification caused by errors that are correlated with the independent variables this misspecification cannot be detected by the test statistics currently used in covariance structure analysis

a unified approach to exploratory factor analysis with missing data nonnormal data and in the presence of outliers factor analysis is regularly used for analyzing survey data missing data data with outliers and consequently nonnormal data are very common for data obtained through questionnaires based on covariance matrix estimates for such nonstandard samples a unified approach for factor analysis is developed by generalizing the approach of maximum likelihood under constraints statistical properties of the estimates for factor loadings and error variances are obtained a rescaled bartlett corrected statistic is proposed for evaluating the number of factors equivariance and invariance of parameter estimates and their standard errors for canonical varimax and normalized varimax rotations are discussed numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures

from akns to derivative nls hierarchies via deformations of associative products using deformations of associative products derivative nonlinear schrodinger dnls hierarchies are recovered as akns type hierarchies since the latter can also be formulated as gelfand dickey type lax hierarchies a recently developed method to obtain functional representations can be applied we actually consider hierarchies with dependent variables in any possibly noncommutative associative algebra e g an algebra of matrices of functions this also covers the case of hierarchies of coupled derivative nls equations

on the lipschitz regularity of minimizers of anisotropic functionals abstract we prove a lipschitz regularity result for minimizers of functionals of the calculus of variations of the form f du x dx where f is a continuous convex function from r n into 0 not necessarily depending on the modulus of du

using follow up data to avoid omitted variable bias an application to cardiovascular epidemiology omitted variable bias is discussed in the context of linear models it is shown that the effect of omitted variables can be controlled in linear models for metric dependent variables by using data from follow up studies two different models for analysing such data are proposed in the first model the omitted variables are assumed to be uncorrelated with the explanatory variables in the model and to be constant over time these assumptions lead to a special structure of the covariance matrix of the errors over time efficient estimation of the parameters in the linear model has to take this special covariance matrix of the errors into account by using appropriate generalized least squares or maximum likelihood methods in the second model the omitted variables are assumed to be time constant additionally they are allowed to be correlated with the explanatory variables that is these variables are omitted confounders in the usual epidemiological sense it is shown that even in this case the parameters of the linear model can be estimated consistently with ordinary least squares if a follow up study is available the differences between the parameter estimates under the first and the second model may be used to construct a hausman test for misspecification the models the estimation methods and the hausman test are illustrated by the example that explores the determinants of serum cholesterol in german adoloscents of both sexes

on structural equation modeling with data that are not missing completely at random a general latent variable model is given which includes the specification of a missing data mechanism this framework allows for an elucidating discussion of existing general multivariate theory bearing on maximum likelihood estimation with missing data here missing completely at random is not a prerequisite for unbiased estimation in large samples as when using the traditional listwise or pairwise present data approaches the theory is connected with old and new results in the area of selection and factorial invariance it is pointed out that in many applications maximum likelihood estimation with missing data may be carried out by existing structural equation modeling software such as lisrel and liscomp several sets of artifical data are generated within the general model framework the proposed estimator is compared to the two traditional ones and found superior

interior behaviour of minimizers for certain functionals with nonstandard growth for all t e 0 00 for example we see that f t2 atp btq a o b o l p 1 satisfy the ellipticity condition 2 recently a number of authors have considered regularity questions of minimizers for z under nonstandard growth conditions onf see l 7 indeed in the scalar cases choe 3 has already obtained c regularity result for minimizers of z when f t2 tp log 1 t p 1 using lewis method 8 in this paper we prove that a locally bounded minimizer u is c f under the assumption that l prq p l 3

obstacle problem for a class of parabolic equations of generalized p laplacian type abstract we study nonlinear parabolic pdes with orlicz type growth conditions the main result gives the existence of a unique solution to the obstacle problem related to these equations to achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution moreover we prove that if the obstacle is continuous so is the solution