dilations of symmetric operators shifted by a unitary group let q be a symmetric operator and ux a one parameter unitary group in a hilbert space such that ux dom q dom q and uxqux q x for dom q it is shown by a compactness argment that q admits a resolution q xf dx with a positive operator valued measure satisfying uxf ux f x this entails the existence of a selfadjoint extension of q to a larger space satisfying the analogus commutation relation with a suitable extended unitary group this result is generalized to n commuting symmetric operators transformed among each other by a representation of an amenable subgroup of the affine group in rn

off diagonal terms in symmetric operators in this paper we provide a quantitative comparison of two obstructions for a given symmetric operator s with dense domain in hilbert space h to be self adjoint the first one is the pair of deficiency spaces of von neumann and the second one is of more recent vintage let p be a projection in h we say that it is smooth relative to s if its range is contained in the domain of s we say that smooth projections pi i 1 diagonalize s if a i pi spi 0 for all i and b supi pi i if such projections exist then s has a self adjoint closure i e s has a spectral resolution and so our second obstruction to self adjointness is defined from smooth projections pi with i pi spi 0 we prove results both in the case of a single operator s and a system of operators

operator integrals and sesquilinear forms abstract we consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to mostly positive operator measures and positive sesquilinear form measures and investigate their relationships to each other in view of the extension theory of symmetric operators we demonstrate the associated mathematical subtleties with a physically relevant example involving moment operators of the momentum observable of a particle confined to move on a bounded interval

non commutative moment problems we are concerned with two problems called the generalized moment problem gmp and the quantum moment problem qmp both problems can be considered as non commutative versions of the classical moment problem et g be a connected lie group and 7b e g the universal enveloping algebra of the lie algebra of g suppose f is a given linear functional on 7b e g and is a fixed unitary representation of g we then ask gmp when do there exist a unitary representation u of g and a cyclic c vector for u such that f x x 7b e g qmp when is f of the form f x trdv x x 7b e g where is an operator such that dv x is of trace class for every x 7b e g

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

testing ignorable missingness in estimating equation approaches for longitudinal data summary we address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches this involves testing for whether or not the zero mean property of estimating equations holds true chen little 1999 proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data whereas we propose a more unified generalised score type test this avoids exhaustive estimation of parameters for each missing data pattern testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern specific estimating equations are zero comparisons are made for the two approaches with both simulations and real data examples

highly efficient aggregate unbiased estimating functions approach for correlated data with missing at random we develop a consistent and highly efficient marginal model for missing at random data using an estimating function approach our approach differs from inverse weighted estimating equations robins rotnitzky and zhao 1995 and the imputation method paik 1997 in that our approach does not require estimating the probability of missing or imputing the missing response based on assumed models the proposed method is based on an aggregate unbiased estimating function approach which does not require the likelihood function however it is equivalent to the score equation if the likelihood is known the aggregate unbiased approach is based on the best linear approximation of efficient scores from the full dataset we provide comparisons of the three approaches using simulated data and also a human immunodeficiency virus hiv data example

unconditional stability of explicit exponential runge kutta methods for semi linear ordinary differential equations in this paper we define unconditional stability properties of exponential kunge kutta metnoas wnen tney are apphed to semi linear systems or ordinary differential equations characterized by a stiff linear part and a non stiff non linear part these properties are related to a class of systems and to a specific norm we give sufficient conditions in order that an explicit method satisfies such properties on the basis of such conditions we analyze some of the popular methods

a test of missing completely at random for generalised estimating equations with missing data we consider inference from generalised estimating equations when data are incomplete a test for missing completely at random is proposed to help decide whether or not we should adjust estimating equations to correct the possible bias introduced by a missing data mechanism that is not missing completely at random likelihood ratio tests have been introduced to test the missing completely at random hypothesis fuchs 1982 little 1988 for the estimating equation setting following the basic idea of little 1988 we propose a wald type test based on an information decomposition and recombination procedure which also provides an alternative method for estimating parameters one application of the test is to assess the adequacy of the marginal generalised estimating equation for longitudinal data with missing values simulations are done to evaluate its performance

exponential runge kutta for the inhomogeneous boltzmann equations with high order of accuracy we consider the development of exponential methods for the robust time discretization of space inhomogeneous boltzmann equations in stiff regimes compared to the space homogeneous case or more in general to the case of splitting based methods studied in dimarco pareschi 7 a major difficulty is that the local maxwellian equilibrium state change with respect to time and thus needs a proper numerical treatment we show how to derive asymptotic preserving ap schemes of arbitrary order and in particular by using the shu osher representation of runge kutta methods we explore the monotonicity properties of such schemes like strong stability preserving ssp and positivity preserving several numerical results confirm our analysis

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

assessing the validity of weighted generalized estimating equations the inverse probability weighted generalized estimating equations approach robins et al 1994 robins et al 1995 effectively removes bias and provides valid statistical inference for regression parameter estimation in marginal models when longitudinal data contain missing values the validity of the weighted generalized estimating equations regarding consistent estimation depends on whether the underlying missing data process is properly modelled however there is little work available to examine whether or not this condition holds in this paper we propose a test constructed from two sets of estimating equations one set is known to be unbiased but the other set is not known we utilize the quadratic inference function qu et al 2000 method to assess their compatibility which is equivalent to testing for the validity of the weighted generalized estimating equations approach we conduct simulation studies to assess the performance of the proposed method the test procedure is illustrated through a real data example

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

a successive penalty based asymptotic preserving scheme for kinetic equations we propose an asymptotic preserving ap scheme for kinetic equations that is efficient also in the hydrodynamic regimes this scheme is based on the bhantnagar gross krook bgk penalty method introduced by filbet and jin j comput phys 229 2010 pp 7625 7648 but uses the penalization successively to achieve the desired asymptotic property this method possesses a stronger ap property than the original method of filbet and jin with the additional feature of being positivity preserving when applied on the boltzmann equation it is also general enough to be applicable to several important classes of kinetic equations including the boltzmann equation and the landau equation numerical experiments verify these properties

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

exponential runge kutta methods for stiff kinetic equations we introduce a class of exponential runge kutta integration methods for kinetic equations the methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of bgk type for boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes we give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times the methods also offer favorable properties such as nonnegativity of the solution and entropy inequality for this reason as we will show the methods are suitable both for deterministic as well as probabilistic numerical techniques

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

positive elements in the algebra of the quantum moment problem let mathfrak a denote the extended weyl algebra mathfrak a 0 subset mathfrak a the weyl algebra it is well known that every element of mathfrak a of the forma b k b k is positive we prove that the converse implication also holds every positive elementa in mathfrak a has a quadratic sum factorization for some finite set of elements b k in mathfrak a the corresponding result is not true for the subalgebra mathfrak a 0 we identify states on mathfrak a 0 which do not extend to states on mathfrak a it follows from a result of powers and arveson that such states on mathfrak a 0 cannot be completely positive our theorem is based on a certain regularity property for the representations which are generated by states on mathfrak a and this property is not in general shared by representations generated by states defined only on the subalgebra mathfrak a 0

classification of certain pairs of operators p q satisfying p q i id abstract in a separable hilbert space a certain class of pairs of operators p q satisfying the born heisenberg commutation relation p q i id on a dense domain is investigated this class is essentially defined by requiring q bounded and self adjoint p symmetric and p q we show that q is absolutely continuous and that p can be thought of as a first order differential operator the class considered contains the pair angle and angular momentum l z it is expected that the methods of this paper can be applied to more general classes of operators p q including the schrodinger case

uniformly stable numerical schemes for the boltzmann equation preserving the compressible navier stokes asymptotics in this paper we develop a numerical method to solve boltzmann like equations of kinetic theory which is able to capture the compressible navier stokes dynamics at small knudsen numbers our approach is based on the micro macro decomposition technique which applies to general collision operators this decomposition is performed in all the phase space and leads to an equivalent formulation of the boltzmann or bgk equation that couples a kinetic equation with macroscopic ones this new formulation is then discretized with a semi implicit time scheme combined with a staggered grid space discretization finally several numerical tests are presented in order to illustrate the efficiency of our approach incidentally we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible navier stokes asymptotics in the case of the simplified bgk model up to our knowledge this property is not known for general collision operators

a note on the bias of estimators with missing data it is well known that many standard analyses including maximum likelihood estimation and the generalized estimating equation approach liang and zeger 1986 biometrika 73 13 22 can result in biased estimation when there are missing observations in such cases it is of interest to calculate the magnitude of the bias incurred under specific assumptions about the process generating the full data and the nonresponse mechanism in this paper we give a condition that identifies the limit in probability of estimators that are solutions of estimating equations computed from the incomplete data with discrete data this condition suggests a simple algorithm to compute the asymptotic bias of these estimators that can be easily implemented with existing statistical software we illustrate our approach with asthma prevalence data in children

an autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes we are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient and many patients drop out with a reason related to the response in this paper we focus on a model that assumes a dropout process is missing at random mar in this dropout process we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified however parsimonious covariance structures for the profiles approaching asymptotes are unclear an autoregressive linear mixed effects model can express the profile with random individual asymptotes we show that this model provides a new parsimonious covariance structure the covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points in simulation studies the estimate of the asymptote is unbiased in mar dropouts but biased in non ignorable dropouts we also applied this model to actual schizophrenia trial data

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

a nonparametric test of missing completely at random for incomplete multivariate data missing data occur in many real world studies knowing the type of missing mechanisms is important for adopting appropriate statistical analysis procedure many statistical methods assume missing completely at random mcar due to its simplicity therefore it is necessary to test whether this assumption is satisfied before applying those procedures in the literature most of the procedures for testing mcar were developed under normality assumption which is sometimes difficult to justify in practice in this paper we propose a nonparametric test of mcar for incomplete multivariate data which does not require distributional assumptions the proposed test is carried out by comparing the distributions of the observed data across different missing pattern groups we prove that the proposed test is consistent against any distributional differences in the observed data simulation shows that the proposed procedure has the type i error well controlled at the nominal level for testing mcar and also has good power against a variety of non mcar alternatives

a micro macro decomposition based asymptotic preserving scheme for the multispecies boltzmann equation in this paper we extend the micro macro decomposition based asymptotic preserving scheme developed by bennoune lemou and mieussens j comput phys 227 2008 pp 3781 3803 for the single species boltzmann equation to the multispecies problems an asymptotic preserving scheme for the kinetic equation is very efficient in the fluid regime where the knudsen number is small and the collision term becomes stiff it allows a coarse independent of the knudsen number mesh size and a large time step in the fluid regime the difficulty associated with multispecies problems is that there are no local conservation laws for each species resulting in extra stiff nonlinear source terms that need to be discretized properly in order to 1 avoid newton type solvers for nonlinear algebraic systems and 2 to be asymptotic preserving we show that these extra nonlinear source terms can be solved using only linear system solvers and the scheme preserves the correct euler and navier stokes limits numerical examples are used to demonstrate the efficiency and applicability of the schemes for both euler and navier stokes regimes

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

an adaptive domain decomposition procedure for boltzmann and euler equations in this paper we present a domain decomposition approach for the coupling of boltzmann and euler equations particle methods are used for both equations this leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains adaptive time and space discretizations and a direct coupling procedure lead to considerable gains in cpu time compared to a solution of the full boltzmann equation several test cases involving a large range of knudsen numbers are numerically investigated

missing data mechanisms and homogeneity of means and variances covariances unless data are missing completely at random mcar proper methodology is crucial for the analysis of incomplete data consequently methods for effectively testing the mcar mechanism become important and procedures were developed via testing the homogeneity of means and variances covariances across the observed patterns e g kim bentler in psychometrika 67 609 624 2002 little in j am stat assoc 83 1198 1202 1988 the current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data and the normal distribution based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random although the values may not equal those of the underlying population distribution the results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the mcar mechanism even when the population distribution is multivariate normal

existence of smooth solutions to the classical moment problems let s 0 s 1 be a given sequence and define s n s n for n 0 if n q m s m n 0 holds for all finite sequences n n z then it is known that there is a positive borel measure on the circle t such that s n 75 e int d t and conversely our main theorem provides a necessary and sufficient condition on the sequence s n that the measure may be chosen to be smooth

a smooth transition model between kinetic and hydrodynamic equations this paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain the idea is to use a buffer zone in which both hydrodynamics and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a simple kinetic scheme is developed to discretize our model and numerical examples are used to validate the method

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

testing for order restricted hypotheses in longitudinal data an inherent ordering we propose a quadratic score test qst based on a quadratic inference function for detecting an order in treatment effects for correlated data the quadratic inference function is similar to the negative of a log likelihood and it provides test statistics in the spirit of a x2 test for testing nested hypotheses as well as for assessing the goodness of fit of model assumptions under the null hypothesis of no order restriction it is shown that the qst statistic has a wald type asymptotic representation and that the asymptotic distribution of the qst statistic is a weighted x2 distribution furthermore an asymptotic distribution of the qst statistic under an arbitrary convex cone alternative is provided the performance of the qst is investigated through monte carlo simulation experiments analysis of the polyposis data demonstrates that the qst outperforms the wald test when data are highly correlated with a small sample size and there is a significant amount of missing data with a small number of clusters the proposed test statistic accommodates both time dependent and time independent covariates in a model

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures