discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

bayesian modeling of the covariance structure for irregular longitudinal data using the partial autocorrelation function in long term follow up studies irregular longitudinal data are observed when individuals are assessed repeatedly over time but at uncommon and irregularly spaced time points modeling the covariance structure for this type of data is challenging as it requires specification of a covariance function that is positive definite moreover in certain settings careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure two common settings where this occurs are studies with outcome dependent follow up and studies with ignorable missing data outcome dependent follow up occurs when individuals with a history of poor health outcomes had more follow up measurements and the intervals between the repeated measurements were shorter when the follow up time process only depends on previous outcomes likelihood based methods can still provide consistent estimates of the regression parameters given that both the mean and covariance structures of the irregular longitudinal data are correctly specified and no model for the follow up time process is required for ignorable missing data the missing data mechanism does not need to be specified but valid likelihood based inference requires correct specification of the covariance structure in both cases flexible modeling approaches for the covariance structure are essential in this paper we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function and the variance function in particular we propose semiparametric non stationary partial autocorrelation function models which do not suffer from complex positive definiteness restrictions like the autocorrelation function we describe a bayesian approach discuss computational issues and apply the proposed methods to cd4 count data from a pediatric aids clinical trial 2015 the authors statistics in medicine published by john wiley sons ltd

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

modelling of covariance structures in generalised estimating equations for longitudinal data summary when used for modelling longitudinal data generalised estimating equations specify a working structure for the within subject covariance matrices aiming to produce efficient parameter estimators however misspecification of the working covariance structure may lead to a large loss of efficiency of the estimators of the mean parameters in this paper we propose an approach for joint modelling of the mean and covariance structures of longitudinal data within the framework of generalised estimating equations the resulting estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed real data analysis and simulation studies show that the proposed approach yields efficient estimators for both the mean and covariance parameters

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

the solution of nonlinear schrodinger equations using orthogonal spline collocation abstract an orthogonal spline collocation semidiscretization previously applied in the numerical solution of the cubic schrodinger equation is extended to the solution of schrodinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives the accuracy of the method is examined as well as its ability to conserve approximations to certain invariants associated with the theoretical solution the results of numerical experiments are presented in which the integration in time is performed using a routine from a software library particular attention is paid to the value and limitations of conservation as an indicator of accuracy

discrete time orthogonal spline collocation method with application to two dimensional fractional cable equation discrete time orthogonal spline collocation osc methods are presented for the two dimensional fractional cable equation which governs the dynamics of membrane potential in thin and long cylinders such as axons or dendrites in neurons the proposed scheme is based on the osc method for space discretization and finite difference method for time which is proved to be unconditionally stable and convergent with the order o min 2 1 2 2 h r 1 in l 2 norm where h and r are the time step size space step size and polynomial degree respectively and 1 and 2 are two different exponents of fractional derivatives with 0 1 2 1 numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein and super convergence phenomena at the partition nodes is also exhibited which is a characteristic of the osc methods namely the rates of convergence in the maximum norm at the partition nodes in u x and u y are approximately h r 1 in our numerical experiment

bayesian inference for a covariance matrix a flexible class of prior distributions is proposed for the covariance matrix of a multivariate normal distribution yielding much more general hierarchical and empirical bayes smoothing and inference when compared with a conjugate analysis involving an inverted wishart distribution a likelihood approximation is obtained for the matrix logarithm of the covariance matrix via bellman s iterative solution to a volterra integral equation exact and approximate bayesian empirical and hierarchical bayesian estimation and finite sample inference techniques are developed some risk and asymptotic frequency properties are investigated a subset of the project talent american high school data is analyzed applications and extensions to multivariate analysis including a generalized linear model for covariance matrices are indicated

bayesian analysis of covariance matrices and dynamic models for longitudinal data summary parsimonious modelling of the within subject covariance structure while heeding its positive definiteness is of great importance in the analysis of longitudinal data using the cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse wishart priors our priors offer many advantages with regard to elicitation positive definiteness computations using gibbs sampling shrinking covariances toward a particular structure with considerable flexibility and modelling covariances using covariates bayesian estimation methods are developed and the results are compared using two simulation studies these simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

nonconjugate bayesian estimation of covariance matrices and its use in hierarchical models abstract the problem of estimating a covariance matrix in small samples has been considered by several authors following early work by stein this problem can be especially important in hierarchical models where the standard errors of fixed and random effects depend on estimation of the covariance matrix of the distribution of the random effects we propose a set of hierarchical priors hps for the covariance matrix that produce posterior shrinkage toward a specified structure here we examine shrinkage toward diagonality we then address the computational difficulties raised by incorporating these priors and nonconjugate priors in general into hierarchical models we apply a combination of approximation gibbs sampling possibly with a metropolis step and importance reweighting to fit the models and compare this hybrid approach to alternative markov chain monte carlo methods our investigation involves three alternative hps the first works with the spectral decomposition of the covariance matrix an

cholesky decompositions and estimation of a covariance matrix orthogonality of variance correlation parameters summary chen dunson 2003 have proposed a modified cholesky decomposition of the form e dll d for a covariance matrix where d is a diagonal matrix with entries proportional to the square roots of the diagonal entries of e and l is a unit lower triangular matrix solely determining its correlation matrix this total separation of variance and correlation is definitely a major advantage over the more traditional modified cholesky decomposition of the form ld2l pourahmadi 1999 we show that though the variance and correlation parameters of the former decomposition are separate they are not asymptotically orthogonal and that the estimation of the new parameters could be more demanding computationally we also provide statistical interpretation for the entries of l and d as certain moving average parameters and innovation variances and indicate how the existing likelihood procedures can be employed to estimate the new parameters

subfields of ample fields rational maps and definability abstract pop proved that a smooth curve c over an ample field k with c k has k many rational points we strengthen this result by showing that there are k many rational points that do not lie in a given proper subfield even after applying a rational map this has several consequences for example we gain insight into the structure of existentially definable i e diophantine subsets of ample fields

bayesian covariance selection in generalized linear mixed models repeated measures and longitudinal data are commonly collected for analysis in epidemiology clinical trials biology sociology and economic sciences in such studies a response is measured repeatedly over time for each subject under study and the number and timing of the observations often varies among subjects in contrast to cross sectional studies that collect a single measurement per subject longitudinal studies have the extra complication of within subject dependence in the repeated measures such dependence can be thought to arise due to the impact of unmeasured predictors main effects of unmeasured predictors lead to variation in the average level of response among subjects while interactions with measured predictors lead to heterogeneity in the regression coefficients this justification has motivated random effects models which allow the intercept and slopes in a regression model to be subject specific random effects models are broadly useful for modeling of dependence not only for longitudinal data but also in multicenter studies meta analysis and functional data analysis

galois theory over complete local domains complete local domains play an important role in commutative algebra and algebraic geometry and their algebraic properties were already described by cohen s structure theorem in 1946 however the galois theoretic properties of their quotient fields only recently began to unfold in 2005 harbater and stevenson considered the two dimensional case they proved that the absolute galois group of the field k x y where k is an arbitrary field is semi free in this work we settle the general case and prove that if r is a complete local domain of dimension exceeding 1 then the quotient field of r has a semi free absolute galois group

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

on spline basis selection for solving differential equations the suitability of b splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged two alternative local solution representations are considered in the context of collocating ordinary differential equations hermite type and lmonomial both are much easier and shorter to implement and somewhat more efficient than b splines a new condition number estimate for the b splines and hermite type representations is presented one choice of the hermite type representation is experimentally determined to produce roundoff errors at most as large as those for b splines the monomial representation is shown to have a much smaller condition number than the other ones and correspondingly produces smaller roundoff errors especially for extremely nonuniform meshes the operation counts for the two local representations considered are about the same the hermite type representation being slightly cheaper it is concluded that both representations are preferable

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

a class of conservative orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations abstract a class of discrete time orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations with initial and boundary conditions are considered these schemes are constructed by using piecewise cubic hermite interpolations in space combined with finite difference methods in time it is proved that the schemes have the conservation laws of discrete energy and possess second order accuracy in maximum norm and fourth order in l 2 norm for time and space respectively the conservation laws and rate of convergence are verified in numerical experiments moreover the propagations of solitary waves and collisions of two head on solitary waves are also well simulated

on optimal order error estimates for the nonlinear schro dinger equation implicit runge kutta methods in time are used in conjunction with the galerkin method in space to generate stable and accurate approximations to solutions of the nonlinear cubic schrodinger equation the temporal component of the discretization error is shown to decrease at the classical rates in some important special cases

numerical computations for n coupled nonlinear schrodinger equations by split step spectral methods in this paper various split step spectral sssp schemes are proposed for n coupled nonlinear schrodinger n cnls equations especially for the systems with n 3 for which numerical studies are few these schemes are spectrally accurate in space and sth s 1 2 4 6 8 order in time they are proved to be conservative and to admit the exact plane wave solution extensive numerical experiments are carried out for the 3 and 4 cnls systems to confirm the theoretical analysis half of the schemes are shown to possess wonderful ability of capturing high frequency waves however the eighth order schemes which seems to be optimal fail in this test accuracy and efficiency of the schemes are compared with each other and the high order schemes exhibit better since the eighth order schemes are not better than the sixth order ones it is believed that constructing too much high order schemes are unnecessary moreover interactions of two soliton solutions are also well simulated by the fourth order and the sixth order schemes so the two methods are sufficient for use

orthogonal cubic spline collocation method for the extended fisher kolmogorov equation a second order splitting combined with orthogonal cubic spline collocation method is formulated and analysed for the extended fisher kolmogorov equation with the help of lyapunov functional a bound in maximum norm is derived for the semidiscrete solution optimal error estimates are established for the semidiscrete case finally using the monomial basis functions we present the numerical results in which the integration in time is performed using radau 5 software library

discrete time orthogonal spline collocation methods for the nonlinear schrodinger equation with wave operator in this paper discrete time orthogonal spline collocation schemes are proposed for the nonlinear schrodinger equation with wave operator these schemes are constructed by using orthogonal spline collocation approaches combined with finite difference methods the conservative property the convergence and the stability of these methods are theoretically analyzed and also verified by extensive numerical experiments in addition some interesting phenomena which require further theoretical analysis are discussed numerically

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

stochastic galerkin methods for elliptic interface problems with random input in this work we consider random elliptic interface problems namely the media in elliptic equations have both randomness and interfaces a galerkin method using bi orthogonal polynomials is used to convert the random problem into an uncoupled system of deterministic interface problems a principle on how to choose the orders of the approximated polynomial spaces is given based on the sensitivity analysis in random spaces with which the total degree of freedom can be significantly reduced then immersed finite element methods are introduced to solve the resulting system convergence results are given both theoretically and numerically

patching and thickening problems in recent years much progress has been made on the structure of fundamental groups of algebraic curves by means of patching techniques in formal and rigid geometry a number of these results have concerned curves over algebraically closed fields of characteristic p e g the proofs of the abhyankar conjecture ra ha5 and of the geometric case of the shafarevich conjecture po1 ha6 and the realization of galois groups over projective curves sa1 st1 while the rigid approach to patching is often regarded as more intuitive than the formal approach its foundations are less well established but constructions involving the formal approach have tended to be technically more cumbersome the purpose of the current paper is to build on previous formal patching results in order to create a framework in which such constructions are facilitated in the process we prove a result asserting that singular curves over a field k can be thickened to curves over k t with prescribed behavior in a formal neighborhood of the singular locus and similarly for covers of curves afterwards we obtain applications to fundamental groups of curves over large fields the structure of the paper is as follows section 1 concerns patching problems for projective curves x over a power series ring r k t1 tn it is shown theorem 1 that giving a coherent projective module over x is equivalent to giving such modules compatibly on a formal neighborhood of each singular point of the closed fibre x and on the formal thickening along the complement of the singular locus s of x section 2 applies this to thickening problems in both the absolute and relative senses namely it shows theorem 3 that such an x can be constructed from its closed fibre x and from complete local thickenings near s such that x is a trivial deformation away from s moreover theorem 2 given a morphism x z and a thickening z of z such that the local thickenings near s are compatible with that of z there is a unique such thickening x of x that is compatible with the given data in section 3 these results are applied to the problem of thickening and deforming covers theorem 4 there combines the results of sections 1 and 2 to show that covers of reducible k curves can be thickened to covers of curves over k t and theorem 5 then shows how this can be used over large fields to

misspecified maximum likelihood estimates and generalised linear mixed models summary we investigate the impact of model violations on the estimate of a regression coefficient in a generalised linear mixed model specifically we evaluate the asymptotic relative bias that results from incorrect assumptions regarding the random effects we compare the impact of model violation for two parameterisations of the regression model substantial bias in the conditionally specified regression point estimators can result from using a simple random intercepts model when either the random effects distribution depends on measured covariates or there are autoregressive random effects a marginally specified regression structure that is estimated using maximum likelihood is much less susceptible to bias resulting from random effects model misspecification

the matrix logarithmic covariance model abstract a flexible method is introduced to model the structure of a covariance matrix c and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix a the matrix exponential transformation c exp a is a positive definite matrix because there is no constraint on the possible values of the upper triangular elements on a any possible structure of interest can be imposed on them the method presented here is not intended to replace the existing special models available for a covariance matrix but rather to provide a broad range of further structures that supplements existing methodology maximum likelihood estimation procedures are used to estimate the parameters and the large sample asymptotic properties are obtained a simulation study and two real life examples are given to illustrate the method introduced

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

shrinkage estimators for covariance matrices estimation of covariance matrices in small samples has been studied by many authors standard estimators like the unstructured maximum likelihood estimator ml or restricted maximum likelihood reml estimator can be very unstable with the smallest estimated eigenvalues being too small and the largest too big a standard approach to more stably estimating the matrix in small samples is to compute the ml or reml estimator under some simple structure that involves estimation of fewer parameters such as compound symmetry or independence however these estimators will not be consistent unless the hypothesized structure is correct if interest focuses on estimation of regression coefficients with correlated or longitudinal data a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure with large matrices however the inefficiency of the sandwich estimator becomes worrisome we consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients the first involves shrinking the eigenvalues of the unstructured ml or reml estimator the second involves shrinking an unstructured estimator toward a structured estimator for both cases the data determine the amount of shrinkage these estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples the final estimator chosen includes a combination of both shrinkage approaches i e shrinking the eigenvalues and then shrinking toward structure we illustrate our approach on a sleep eeg study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen we recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators

maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix the positive definiteness constraint is the most awkward stumbling block in modelling the covariance matrix pourahmadi s 1999 unconstrained parameterisation models covariance using covariates in a similar manner to mean modelling in generalised linear models the new covariance parameters have statistical interpretation as the regression coefficients and logarithms of prediction error variances corresponding to regressing a response on its predecessors in this paper the maximum likelihood estimators of the parameters of a generalised linear model for the covariance matrix their consistency and their asymptotic normality are studied when the observations are normally distributed these results along with the likelihood ratio test and penalised likelihood criteria such as bic for model and variable selection are illustrated using a real dataset

orthogonal cubic spline collocation method for the cahn hilliard equation the cahn hilliard equation plays an important role in the phase separation in a binary mixture this is a fourth order nonlinear partial differential equation in this paper we study the behaviour of the solution by using orthogonal cubic spline collocation method and derive optimal order error estimates we discuss some computational experiments by using monomial basis functions in the spatial direction and radau 5 time integrator the method we present here is better in terms of stability efficiency and conditioning of the resulting matrix since no integrals to be evaluated or approximated it behaves better than finite element method

finite element solution to the parabolic wave equation in seeking more ways to predict underwater sound fields the underwater acoustical parabolic wave equation is solved using the finite element fe method the mathematical theory for the finite element method in an n medium case is discussed when the conditions of continuity of pressure and continuity of the normal component of particle velocity are perfectly satisfied at each interface several propagation examples for horizontal and irregular interfaces are presented to demonstrate the capability of this effective fe method

modeling the random effects covariance matrix for longitudinal data with covariates measurement error longitudinal data occur frequently in practice such as medical studies and life sciences generalized linear mixed models glmms are commonly used to analyze such data it is typically assumed that the random effects covariance matrix is constant among subjects in these models in many situations however the correlation structure may differ among subjects and ignoring this heterogeneity can lead to biases in model parameters estimate recently lee et al developed a heterogeneous random effects covariance matrix for glmms for error free covariates covariates measured with error also happen frequently in the longitudinal data set up eg blood pressure and cholesterol level ignoring this issue in the data may produce bias in model parameters estimate and lead to wrong conclusions in this paper we propose an approach to properly model the random effects covariance matrix based on covariates in the class of glmms where we also have covariates measured with error the resulting parameters from the decomposition of random effects covariance matrix have a sensible interpretation and can be easily modeled without the concern of positive definiteness of the resulting estimator the performance of the proposed approach is evaluated through simulation studies which show that the proposed method performs very well in terms of bias mean squared error and coverage rate an application of the proposed method is also provided using a longitudinal data from manitoba follow up study

dynamic conditionally linear mixed models for longitudinal data we develop a new class of models dynamic conditionally linear mixed models for longitudinal data by decomposing the within subject covariance matrix using a special cholesky decomposition here dynamic means using past responses as covariates and conditional linearity means that parameters entering the model linearly may be random but nonlinear parameters are nonrandom this setup offers several advantages and is surprisingly similar to models obtained from the first order linearization method applied to nonlinear mixed models first it allows for flexible and computationally tractable models that include a wide array of covariance structures these structures may depend on covariates and hence may differ across subjects this class of models includes e g all standard linear mixed models antedependence models and vonesh carter models second it guarantees the fitted marginal covariance matrix of the data is positive definite we develop methods for bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited

orthogonal cubic spline collocation method for the nonlinear parabolic equation arising in non newtonian fluid flow using the orthogonal cubic spline collocation method solution for the nonlinear parabolic equation arising in magneto hydrodynamic unsteady poiseuille flow of the generalized newtonian fluid carreau rheological model is obtained also using the lyapunov functional a bound for the maximum norm of the semi discrete solution is derived moreover optimal error estimates are established for the semi discrete solution numerical results thus obtained are presented graphically and the salient features of the solution are discussed for various values of the parameters the results obtained reveal many interesting behaviors that warrant further study on the parabolic equations related to non newtonian fluid phenomena furthermore the analysis can be used to study the mathematical models that involve the flow of viscous fluids with shear rate dependent properties for example models dealing with polymer processing tribology and lubrication and food processing

modelling the random effects covariance matrix in longitudinal data a common class of models for longitudinal data are random effects mixed models in these models the random effects covariance matrix is typically assumed constant across subject however in many situations this matrix may differ by measured covariates in this paper we propose an approach to model the random effects covariance matrix by using a special cholesky decomposition of the matrix in particular we will allow the parameters that result from this decomposition to depend on subject specific covariates and also explore ways to parsimoniously model these parameters an advantage of this parameterization is that there is no concern about the positive definiteness of the resulting estimator of the covariance matrix in addition the parameters resulting from this decomposition have a sensible interpretation we propose fully bayesian modelling for which a simple gibbs sampler can be implemented to sample from the posterior distribution of the parameters we illustrate these models on data from depression studies and examine the impact of heterogeneity in the covariance matrix on estimation of both fixed and random effects

linear mixed models with heterogeneous within cluster variances this paper describes an extension of linear mixed models to allow for heterogeneous within cluster variances in the analysis of clustered data unbiased estimating equations based on quasilikelihood pseudolikelihood and method of moments are introduced and are shown to give consistent estimators of the regression coefficients variance components and heterogeneity parameter under regularity conditions cluster specific random effects and variances are predicted by the posterior modes the method is illustrated through an analysis of menstrual diary data and its properties are evaluated in a simulation study

an investigation into the effect of product approximation in the numerical solution of the cubic nonlinear schro dinger equation abstract we study the effect of product approximation on the galerkin solutions of the one dimensional cubic schrodinger equation a crank nicolson scheme is used to discretize in time the paper describes two numerical experiments in the first we examine the approximation obtained by the standard galerkin method and discuss the possibility of enforcing discrete analogs of the conservation laws satisfied by the exact solution in the second experiment numerical results obtained by the product approximation version of the galerkin method are compared and the effectiveness of the method for different combinations of test and trial functions is also investigated

maximum norm error bound of a linearized difference scheme for a coupled nonlinear schrodinger equations in this article a linearized conservative difference scheme for a coupled nonlinear schrodinger equations is studied the discrete energy method and an useful technique are used to analyze the difference scheme it is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm numerical experiments are presented to support the theoretical results

local galois theory in dimension two this paper proves a generalization of shafarevich s conjecture for fields of laurent series in two variables over an arbitrary field this result says that the absolute galois group gk of such a field k is quasi free of rank equal to the cardinality of k i e every non trivial finite split embedding problem for gk has exactly cardk proper solutions we also strengthen a result of pop and haran jarden on the existence of proper regular solutions to split embedding problems for curves over large fields our strengthening concerns integral models of curves which are two dimensional

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis