cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

the limit behavior of solutions for the cauchy problem of the complex ginzburg landau equation this paper is devoted to the study of the limit behavior as e 0 or e 0 and a 0 for the solutions of the cauchy problem of the complex ginzburg landau equation ut e u i u a i u u 0 u 0 x u0 x 4 n 4 n 2 0 0 s 0 1 and s n 2 2 moreover the sharp convergence rate in such spaces is also given 2002 john wiley sons inc

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

analysis of factorial survival experiments several new methodological issues that arise within two way factorial designs for survival experiments are discussed within the framework of asymptotic theory for the proportional hazards model with two binary treatment covariates these issues include the proper formulation of null hypotheses and alternatives the choice among log rank and adjusted or stratified log rank statistics the asymptotic correlation between test statistics for the separate main effects the asymptotic power under the various possible methods of analysis of tests to detect main effects and interactions the comparison of power to detect main effects within a 2 x 2 factorial design with power in a three group trial where no patients are randomized simultaneously to both treatments and the problems of analysis arising when accrual or exposure to one of the treatments is terminated early for ethical reasons

sequential linear rank tests for two sample censored survival data under extremely general patterns of patient arrival allocation to treatment and loss to follow up in randomized clinical trial settings the sequentially computed logrank statistic mantel 1966 is shown under the null hypothesis of identically distributed lifetimes to have exactly uncorrelated increments and is shown via rebolledo s 1980 martingale invariance principle to satisfy a functional central limit theorem justifying sequential logrank tests of jones and whitehead 1979 generalizations are made to other twosample rank tests for censored survival data and practical applicability to real randomized clinical trials is discussed

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

random attractors for the stochastic coupled fractional ginzburg landau equation with additive noise this paper is concerned with the stochastic coupled fractional ginzburg landau equation with additive noise we first transform the stochastic coupled fractional ginzburg landau equation into random equations whose solutions generate a random dynamical system then we prove the existence of random attractor for random dynamical system

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

simultaneous inference on treatment effects in survival studies with factorial designs a clinical trial with a 2 2 factorial design involves randomization of subjects to treatment a or a and within each group further randomization to treatment b or b under this design one can assess the effects of treatments a and b on a clinical endpoint using all patients one may additionally compare treatment a treatment b or combination therapy ab to a b with multiple comparisons however it may be desirable to control the overall type i error especially for regulatory purposes because the subjects overlap in the comparisons the test statistics are generally correlated by accounting for the correlations one can achieve higher statistical power compared to the conventional bonferroni correction herein we derive the correlation between any two stratified or unstratified log rank statistics for a 2 2 factorial design with a survival time endpoint such that the overall type i error for multiple treatment comparisons can be properly controlled in addition we allow for adjustment of prognostic factors in the treatment comparisons and conduct simultaneous inference on the effect sizes we use simulation studies to show that the proposed methods perform well in realistic situations we then provide an application to a recently completed randomized controlled clinical trial on alcohol dependence finally we discuss extensions of our approach to other factorial designs and multiple endpoints

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

the principal spectrum for linear nonautonomous parabolic pdes of second order basic properties abstract we put forward a theory extending the notion of principal eigenfunction and principal eigenvalue to the case of linear nonautonomous parabolic pdes of second order u t i j 1 n a ij t x 2 u x i x j i 1 n a i t x u x i a 0 t x u x on a bounded domain r n with dirichlet or robin boundary conditions a canonically defined one dimensional subbundle s corresponding to the solutions that are globally defined and of the same sign serves as an analog of principal eigenfunction the principal spectrum is defined to be the dynamical sacker sell spectrum of the linear skew product flow on s characterizations of principal spectrum in terms of logarithmic growth rates of positive solutions are given finally monotonicity of the principal spectrum with respect to zero order terms is proved

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

globally positive solutions of linear parabolic pdes of second order with robin boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with robin boundary conditions are unique up to multiplication by a positive constant

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

nonparametric inference in factorial designs with censored data a method is proposed for testing the hypotheses of no main effects and no interaction in factorial designs with several observations per cell the method uses the fact that these hypotheses can be expressed in terms of a vector of contrasts it is based on the observation that nonparametric estimation of these contrasts is no more difficult than estimation of the location difference in the two sample problem to implement the method with censored data a new extension of the hodges lehmann estimator is proposed the estimator is simple to compute and its variance is easily evaluated a simulation study examines the performance of the proposed estimation and testing method in the context of a two by two design and a real data set from a three way layout with heavy censoring is analyzed

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

well posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials abstract the current paper is devoted to the well posedness and dynamics of the stochastic coupled fractional ginzburg landau equation which describes a class of nonlinear optical fiber materials with active and passive coupled cores by the commutation estimates and fourier galerkin approximation the global existence of weak solutions and the uniqueness criterion are established moreover the existence of a global attractor is shown finally we consider the long time behavior of the stochastic coupled fractional ginzburg landau equation scfgl with multiplicative noise and prove the existence of a random attractor for the random dynamical system generated by the scfgl equation

sequential analysis of the proportional hazards model abstract for the proportional hazards model of survival analysis an appropriate large sample theory is developed for cases of staggered entry and sequential analysis the principal techniques involve an approximation of the score process by a suitable martingale and a random rescaling of time based on the observed fisher information as a result we show that the maximum partial likelihood estimator behaves asymptotically like brownian motion author

an overview of periodic elliptic operators the article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas close attention is paid to studying analytic properties of bloch and fermi varieties which influence significantly most properties of such operators the approaches described are applicable not only to the standard model example of schrodinger operator with periodic electric potential v x but to a wide variety of elliptic periodic equations and systems equations on graphs operator and other operators on abelian coverings of compact bases many important applications are mentioned however due to the size restrictions they are not dealt with in details

asymptotic behavior for non autonomous fractional stochastic ginzburg landau equations on unbounded domains in this paper we first prove the existence and uniqueness of tempered pullback random attractors for a non autonomous stochastic fractional ginzburg landau equation driven by multiplicative noise with 0 1 in l2r3 then we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero due to the lack of the compactness of sobolev embeddings on unbounded domains we establish the pullback asymptotic compactness of solutions in l2 r3 by the tail estimates of solutions

repeated confidence intervals for the median survival time summary we describe methods for the construction of a confidence interval for median survival time based on right censored data these methods are extended to the construction of repeated confidence intervals for the median based on accumulating data here the overall probability that all intervals contain the true median is guaranteed at a fixed level the use of repeated confidence intervals for median survival time in post marketing surveillance is discussed a confidence interval for median survival time provides a useful summary of the survival experience of a group of patients if confidence intervals are calculated repeatedly as data accumulates the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly jennison turnbull 1984 have proposed methods for calculating repeated confidence intervals appropriate to such situations similar ideas have also been discussed by lai 1984 in 2 we propose a new form of single sample nonparametric confidence interval this interval has asymptotically correct coverage probability and monte carlo simulations suggest it is superior to its competitors for small sample sizes repeated confidence intervals for the median are presented in 3 and their stnall sample size performance is assessed by monte carlo simulation an example of their use is given in 4 all the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

solutions for the fractional landau lifshitz equation abstract this article considers the dynamic equation of a reduced model for thin film micromagnetics deduced by a desimone r v kohn and f otto in a desimone r v kohn f otto a reduced theory for thin film micromagnetics comm pure appl math 55 2002 1 53 to derive the existence of weak solutions under periodical boundary condition the authors first prove the existence of smooth solutions for the approximating equation then prove the convergence of the viscosity solution when the viscosity term vanishes which implies the existence of solutions for the original equation

nonparametric methods for factorial designs with censored data abstract nonparametric hypotheses of no main effects no simple effects and no interaction effects are considered in the context of factorial designs with censored observations to test these hypotheses the asymptotic distribution of quadratic forms of rank statistics is derived using the methodology of martingales for counting processes the weights used reduce to the usual logistic scores with uncensored data but are different from the weights commonly used for testing the equality of k samples the formulation of all results includes tied observations approximations to the small sample distributions of the test statistics are given the performance of the tests is examined via simulation studies the procedures are illustrated on a real dataset

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

semiparametric two sample tests in clinical trials with a post randomisation response indicator in many clinical trials involving survival endpoints one has additional data on some binary indicator of response such as initial tumour shrinkage in cancer trials this paper studies the case of randomised clinical trials where the response indicator is available shortly after randomisation and where one can assume that within each stratum defined by the response indicator a two treatment group proportional hazards model holds the same model may also describe some incompletely randomised or observational studies asymptotic relative efficiencies for kaplan meier based estimators versus maximum partial likelihood estimators are examined under this model for estimating either the difference in survival probabilities at a specified time or the parameter estimated by the logrank numerator it is shown that the efficiency gains using the model are more promising when estimating the difference in survival probabilities an example is given comparing the long term survival experience of two groups of patients with advanced hodgkin s disease

global weak solutions of the fractional landau lifshitz maxwell equation abstract the fractional landau lifshitz maxwell equation is considered in this paper we show the global existence of suitable weak solutions by galerkin method and the vanishing viscosity method the main difficulties in this study are due to the loss of compactness of this system and the fact that the nonlinear term is nonlocal and of the same order of the equation to overcome these difficulties we introduce the commutator estimates and some cancellation properties to this equation which prove to be proper tools in the study of landau lifshitz type equations

relative efficiency of the log rank test within a multiplicative intensity model summary for large sample clinical trials with independent individuals randomly allocated to two treatment groups in which survival times follow a log linear multiplicative intensity model with treatment group as one covariate this paper calculates the asymptotic relative efficiency of the log rank test for treatment effect as compared with the optimal score test the method is to exhibit the failure hazard intensity not of proportional hazards form obtained by ignoring all covariates other than treatment group the efficiency formulae are illustrated in two examples and estimation from data of the loss of efficiency is illustrated for two clinical trial datasets

globally positive solutions of linear parabolic partial differential equations of second order with dirichlet boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with dirichlet boundary conditions are unique up to multiplication by a positive constant

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

the cauchy problem in local spaces for the complex ginzburg landau equation ii contraction methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional