a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

infinite dimensional cohomology groups and periodic solutions of asymptotically linear hamiltonian systems abstract in this paper we study the existence of nontrivial 2 periodic solutions of asymptotically linear hamiltonian systems we consider the case of resonance both at zero and at infinity and we permit time dependent asymptotic matrices our main tools are an infinite dimensional cohomology theory and a corresponding morse theory recently constructed by w kryszewski and the first author we develop a method to compute the new critical groups

remarks on approximation methods in degree theory an approximative approach to a generalized theory of the topological mapping degree is presented some new wide classes of operators acting in banach spaces which include a proper mappings of petryshyn are introduced and studied from the viewpoint of the homotopic properties of the topological degree the results are applied in some existence aspects of abstract nonlinear equations

generalized linear models with random effects salamander mating revisited in recent years much effort has been devoted to extending regression methodology to non gaussian data where responses are not independent these methods for dependent responses are suitable for data from longitudinal studies or nested designs however use of these methods for crossed designs seems to have serious limitations due to the intensive computations involved because of the intractable nature of the joint distribution in this paper we cast the problem in a bayesian framework and use a monte carlo method the gibbs sampler to avoid current computational limitations the flexibility of this approach is illustrated by analyzing the interesting salamander mating data reported by mccullagh and nelder 1989 generalized linear models 2nd edition london chapman and hall

a note on fitting a marginal model to mixed effects log linear regression data via gee marginal generalized linear models for correlated data are often fit by generalized estimating equations liang k y and zeger s l 1986 biometrika 73 13 22 which requires the marginal expectation to be correctly specified while the covariance structure is allowed to be misspecified this note is concerned with the correct specification of the marginal mean for data from a mixed log linear regression model and the relationship between the true subject specific parameters and the true marginal parameters

a note on superlinear hamiltonian elliptic systems this paper is concerned with the superlinear periodic elliptic systems of hamiltonian type in the whole space the existence of a ground state solution as well as an infinite number of geometrically distinct solutions is obtained

on hamiltonian elliptic systems with periodic or non periodic potentials abstract this paper is concerned with the following hamiltonian elliptic system u b x u v x u h v x u v v b x v v x v h u x u v for x r n existence and multiplicity of solutions are obtained for the systems with periodic or non periodic potentials v via variational methods

periodic solutions of autonomous newtonian systems partially asymptotically linear at infinity abstract the goal of this article is to study the existence continuation and global bifurcation of autonomous newtonian systems partially asymptotically linear at infinity the main idea is to prove a kind of splitting lemma and apply the degree for s 1 equivariant gradient maps defined by rybicki 1994 in 16

infinitely many solutions for diffusion equations without symmetry abstract we consider the following diffusion system t u x u b t x x u v x u h v t x u v t v x v b t x x v v x v h u t x u v t x r r n which is an unbounded hamiltonian system in l 2 r r n r 2 m z u v r r n r m r m b c r r n r n v c r n r and h c 1 r r n r 2 m r suppose that h b and v depend periodically on t and x and that h t x z is superquadratic in z as z without a symmetry assumption on h we establish the existence of infinitely many geometrically distinct solutions via a variational approach

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

an assessment of estimation methods for generalized linear mixed models with binary outcomes two main classes of methodology have been developed for addressing the analytical intractability of generalized linear mixed models likelihood based methods and bayesian methods likelihood based methods such as the penalized quasi likelihood approach have been shown to produce biased estimates especially for binary clustered data with small clusters sizes more recent methods using adaptive gaussian quadrature perform well but can be overwhelmed by problems with large numbers of random effects and efficient algorithms to better handle these situations have not yet been integrated in standard statistical packages bayesian methods although they have good frequentist properties when the model is correct are known to be computationally intensive and also require specialized code limiting their use in practice in this article we introduce a modification of the hybrid approach of capanu and begg 2011 biometrics 67 371 380 as a bridge between the likelihood based and bayesian approaches by employing bayesian estimation for the variance components followed by laplacian estimation for the regression coefficients we investigate its performance as well as that of several likelihood based methods in the setting of generalized linear mixed models with binary outcomes we apply the methods to three datasets and conduct simulations to illustrate their properties simulation results indicate that for moderate to large numbers of observations per random effect adaptive gaussian quadrature and the laplacian approximation are very accurate with adaptive gaussian quadrature preferable as the number of observations per random effect increases the hybrid approach is overall similar to the laplace method and it can be superior for data with very sparse random effects copyright 2013 john wiley sons ltd

the cohomological span of ls conley index abstract in this paper we introduce a new homotopy invariant the cohomological span of ls conley index we prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form l x k x where l is bounded linear and k is completely continuous we give examples of hamiltonian systems for which our methods give better results than the morse inequalities we also give a formula for the ls index of an isolated critical point which is an extension of the classical dancer theorem for the case of ls index

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

bias reduction of approximate maximum likelihood estimates for heritability in threshold models approximate maximum likelihood estimates such as penalized quasi likelihood and iterated reweighted reml estimates for heritability or intraclass correlation in threshold models for binary data can be seriously biased two approaches to correct for this bias are studied with emphasis on animal breeding models for binary data scope for reduction of bias of heritability estimates is found to be slim because of the commonly large number of fixed effects in animal breeding models minimal dimensions for the data are identified such that bias and root mean squared error are of modest size and useful inference on heritability or intraclass correlation is feasible

marginally specified logistic normal models for longitudinal binary data summary likelihood based inference for longitudinal binary data can be obtained using a generalized linear mixed model breslow n and clayton d g 1993 journal of the american statistical association88 9 25 wolfinger r and o connell m 1993 journal of statistical computation and simulation48 233 243 given the recent improvements in computational approaches alternatively fitzmaurice and laird 1993 biometrika80 141 151 molenberghs and lesaffre 1994 journal of the american statistical association89 633 644 and heagerty and zeger 1996 journal of the american statistical association91 1024 1036 have developed a likelihood based inference that adopts a marginal mean regression parameter and completes full specification of the joint multivariate distribution through either canonical and or marginal higher moment assumptions each of these marginal approaches is computationally intense and currently limited to small cluster sizes in this manuscript an alternative parameterization of the logistic normal random effects model is adopted and both likelihood and estimating equation approaches to parameter estimation are studied a key feature of the proposed approach is that marginal regression parameters are adopted that still permit individual level predictions or contrasts an example is presented where scientific interest is in both the mean response and the covariance among repeated measurements

infinitely many periodic solutions for the equation u tt u xx u p 1 u f x t ii on considere l existence de vibrations forcees pour l equation d onde non lineaire u tt u xx u p 1 u f x t x t 0 r u 0 t u t 0 pour t r u x t 2 u x t x t 0 r pour tout p 1 et f x t l p 1 p on demontre l existence d une infinite de solutions periodiques

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

a type of restricted maximum likelihood estimator of variance components in generalised linear mixed models summary the maximum likelihood estimator of the variance components in a linear model can be biased downwards restricted maximum likelihood reml corrects this problem by using the likelihood of a set of residual contrasts and is generally considered superior however this original restricted maximum likelihood definition does not directly extend beyond linear models we propose a reml type estimator for generalised linear mixed models by correcting the bias in the profile score function of the variance components the proposed estimator has the same consistency properties as the maximum likelihood estimator if the number of parameters in the mean and variance components models remains fixed however the estimator of the variance components has a smaller finite sample bias a simulation study with a logistic mixed model shows that the proposed estimator is effective in correcting the downward bias in the maximum likelihood estimator

a new index theory for linear self adjoint operator equations and its applications abstract we develop an index theory for linear self adjoint operator equation with the linear self adjoint operator having no compact resolvent as applications we consider the existence and multiplicity of periodic solutions of wave equations and beam equations

consistent estimators in generalized linear mixed models abstract a simple method based on simulated moments is proposed for estimating the fixed effects and variance components in a generalized linear mixed model glmm it is shown that the method is not only computationally attractive but also leads to consistent estimators on the other hand simulation shows that the method can be quite inefficient

global bifurcations of critical orbits of g invariant strongly indefinite functionals abstract let h be a separable hilbert space which is an orthogonal representation of a compact lie group g and let h r be a g invariant strongly indefinite functional of the class c 1 to study critical orbits of we have defined the degree for g invariant strongly indefinite functionals which is an element of the euler ring u g using this degree we have formulated the rabinowitz alternative for g invariant strongly indefinite functionals the abstract results are applied to the study of global bifurcations of weak solutions of non cooperative systems of elliptic differential equations

a generalized saddle point theorem abstract the well known saddle point theorem is extended to the case of functions defined on a product space x v where x is a banach space and v is a compact manifold under some linking conditions the existence of at least cuplength v 1 critical points is proved the abstract theorems are applied to the existence problems of periodic solutions of hamiltonian systems with periodic nonlinearity and or resonance

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

periodic solutions of asymptotically linear autonomous hamiltonian systems abstract in this paper we study the existence and the continuation of periodic solutions of autonomous hamiltonian systems we apply the degree for g invariant strongly indefinite functionals defined in gol biewska and rybicki 2011 2 we consider resonant as well as nonresonant case at stationary solutions

ground state solutions for a diffusion system in this paper we study the following diffusion system t u x u v x u g t x v t v x v v x v f t x u where z u v r r n r 2 v x c r n r is a general periodic function under weaker conditions on nonlinearity we establish the existence of ground state solutions via variational methods

maximum likelihood estimation for probit linear mixed models with correlated random effects the probit normal model for binary data mcculloch 1994 journal of the american statistical association 89 330 335 is extended to allow correlated random effects to obtain maximum likelihood estimates we use the em algorithm with its m step greatly simplified under the assumption of a probit link and its e step made feasible by gibbs sampling standard errors are calculated by inverting a monte carlo approximation of the information matrix rather than via the sem algorithm a method is also suggested that accounts for the monte carlo variation explicitly as an illustration we present a new analysis of the famous salamander mating data unlike previous analyses we find it necessary to introduce different variance components for different species of animals finally we consider models with correlated errors as well as correlated random effects

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

reml estimation with exact covariance in the logistic mixed model residual maximum likelihood reml estimation is adapted to certain logistic mixed models for which representation of the unconditional mean as a linear function of the fixed effects is possible only the first two moments of the unconditional distribution need be evaluated and except for the form of the covariance the maximization algorithm carries over directly from linear models the exact unconditional covariance is computed from the logistic normal mixture for input into the algorithm residual log likelihood plots provide a means of inference for the dispersion components the taylor series approximation to this covariance besides exhibiting insufficient accuracy fails to be positive definite for large values of the dispersion components as a consequence reml loglikelihood plots based on this approximate covariance attribute misleadingly high precision to the dispersion component estimates the method is presented in the context of a salamander mating experiment in which random effects corresponding to male and female animals occur in a crossed design analyses of these data by several methods are compared

comparing odds ratios measuring improvement or deterioration across repeated exposure or treatment sessions unbiased estimating equations are used to estimate and compare the odds ratios representing the odds of heightened anxiety following acute exposures relative to baseline in a cross over study examining indoor air contaminants although estimating equations are used in the creation of the estimates and standard errors inference is conditional on subject the proposed method produces estimates that are less biased and computationally simpler than pseudo likelihood in addition specification of the form of the random effects distributions is not required

robust estimation in generalised linear mixed models summary we propose robust methods for estimation of parameters of interest in an extended generalised linear mixed model in which only the conditional means of the responses given the random effects are specified a first step estimator 0 of the vector 0 of parameters is obtained by solving a system of estimating equations it is shown that 9 is consistent if furthermore the conditional variances are correctly specified a second step estimator 0 can be obtained by solving a system of optimal estimating equations whose coefficients a are estimated by 0 it is shown that 0 maintains the asymptotic optimality simulations a also indicate solid improvement of 0 over 0 two examples involving real data analysis are considered the methods are developed under a more general framework so that they may be applied to more general estimation problems

ergodipotent maps and commutativity of elements in noncommutative rings and algebras with twisted intertwining a property of algebraic dependence between two commuting elements is shown to hold in a more general setting than that in which it has previously been established key conditions are identified and some methods for establishing them are given moreover the class of algebras with a generalised weyl structure generalising the so called generalised weyl algebras gwas or hyperbolic rings is introduced and studied we also present an interesting class of algebras which are not gwas but share many of their properties by virtue of their generalised weyl structure for these classes of algebras centralisers and algebraic dependence are investigated c 2007 elsevier inc all rights reserved

computations of critical groups and applications to asymptotically linear wave equation and beam equation in this paper by using the morse index theory for strongly indefinite functionals developed in nonlinear anal tma in press we compute precisely the critical groups at the origin and at infinity respectively the abstract theorems are used to study the existence multiplicity of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero

ground state solutions of hamiltonian elliptic systems in dimension two the aim of this paper is to study hamiltonian elliptic system of the form 0 1 where 2 is a bounded domain in the second place we present existence results for the following stationary schrodinger systems defined in the whole plane 0 2 we assume that the nonlinearities f g have critical growth in the sense of trudinger moser by using a suitable variational framework based on the generalized nehari manifold method we obtain the existence of ground state solutions of both systems 0 1 and 0 2

computations of cohomology groups and nontrivial periodic solutions of hamiltonian systems by computing the e critical groups at and infinity of the corresponding functional of hamiltonian systems we proved the existence of nontrivial periodic solutions for the systems which may be resonant at and infinity under some new conditions some results in the literature are extended and some new type of theorems are proved the main tool is the e morse theory developed by kryszewski and szulkin

the existence of solutions for superquadratic hamiltonian elliptic systems on rn abstract this paper is concerned with the following hamiltonian elliptic system u v x u h u x u v in r n v v x v h v x u v in r n u x 0 and v x 0 as x where v x c r n r and h x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we obtain infinitely many solutions for the above system

existence and multiplicity of rotating periodic solutions for resonant hamiltonian systems abstract in the present paper we consider a class of resonant hamiltonian systems x j h x t x in r 2 n we will use saddle point reduction morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions i e x t t q x t for any t r with t 0 and q an symplectic orthogonal matrix in the case q k i 2 n for any positive integer k such a rotating periodic solution is just a quasi periodic solution moreover if h is even in x we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques

fence methods for mixed model selection many model search strategies involve trading off model fit with model complexity in a penalized goodness of fit measure asymptotic properties for these types of procedures in settings like linear regression and arma time series have been studied but these do not naturally extend to nonstandard situations such as mixed effects models where simple definition of the sample size is not meaningful this paper introduces a new class of strategies known as fence methods for mixed model selection which includes linear and generalized linear mixed models the idea involves a procedure to isolate a subgroup of what are known as correct models of which the optimal model is a member this is accomplished by constructing a statistical fence or barrier to carefully eliminate incorrect models once the fence is constructed the optimal model is selected from among those within the fence according to a criterion which can be made flexible in addition we propose two variations of the fence the first is a stepwise procedure to handle situations of many predictors the second is an adaptive approach for choosing a tuning constant we give sufficient conditions for consistency of fence and its variations a desirable property for a good model selection procedure the methods are illustrated through simulation studies and real data analysis

existence of solutions for nonperiodic superquadratic hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u r v x u v in r n v v x v r u x u v in r n u x 0 and v x 0 as x where r x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we prove the existence of solution for the above system

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

bias correction in generalized linear mixed models with multiple components of dispersion abstract general formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi likelihood pql in generalized linear mixed models with canonical link function and multiple sets of independent random effects easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values both first order and second order correction procedures are developed for regression coefficients estimated by pql the methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects and their properties are evaluated in a simulation study

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

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