a proportional hazards model for interval censored failure time data this paper develops a method for fitting the proportional hazards regression model when the data contain left right or interval censored observations results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log rank test for comparison of several survival curves the method is used to analyze data from an animal tumorigenicity study and also a clinical trial

nearly inviscid faraday waves in annular containers of moderately large aspect ratio nearly inviscid parametrically excited surface gravity capillary waves in two dimensional domains of finite depth and large aspect ratio are considered coupled equations describing the evolution of the amplitudes of resonant left and right traveling waves and their interaction with a mean flow in the bulk are derived and the conditions for their validity established under suitable conditions the mean flow consists of an inviscid part together with a viscous mean flow driven by a tangential stress due to an oscillatory viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer these forcing mechanisms are important even in the limit of vanishing viscosity and provide boundary conditions for the navier stokes equation satisfied by the mean flow in the bulk for moderately large aspect ratio domains the amplitude equations are nonlocal but decouple from the equations describing the interaction of the slow spatial phase and the viscous mean flow two cases are considered in detail gravity capillary waves and capillary waves in a microgravity environment

valuation theory of exponential hardy fields i we describe the valuation theoretic properties of the hardy fields associated to models of t exp where t is the theory of a polynomially bounded o minimal expansion of the reals and exp is the real exponential function we deduce that t exp has levels with parameters and is exponentially bounded we establish a maximality property of h mathbb r rm an powers the hardy field of the expansion by the restricted analytic functions and power functions

tropical totally positive matrices abstract we investigate the tropical analogues of totally positive and totally nonnegative matrices these arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field like the field of real puiseux series we show that the nonarchimedean valuation sends the totally positive matrices precisely to the monge matrices this leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices we also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices we finally determine the eigenvalues of tropical totally nonnegative matrices and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields

tangent cones and c 1 regularity of definable sets abstract let x r n be a connected locally closed set which is definable in an o minimal structure we prove that the following three statements are equivalent i x is a c 1 manifold ii the tangent cone and the paratangent cone of x coincide at every point in x iii for every x x the tangent cone of x at the point x is a k dimensional linear subspace of r n k does not depend on x varies continuously in x and the density x x 3 2

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

global bifurcations to strange attractors in a model for skew varicose instability in thermal convection abstract we present a global bifurcation study of a four dimensional system of differential equations proposed by f h busse and coworkers modeling instabilities of convection rolls in the rayleigh benard experiment the rayleigh and prandtl numbers are two natural parameters on which the system depends we focus on a detailed mathematical study combining numerical pathfollowing and bifurcation analysis of chaotic dynamics and transitions to chaotic dynamics numerical continuation makes clear how homoclinic and heteroclinic bifurcations organize the bifurcation diagram in the parameter plane combined with a theoretical bifurcation analysis this explains the development of patterns and the creation of chaotic spatio temporal dynamics in the model the organizing centers such as heteroclinic cycles with resonance conditions among eigenvalues and homoclinic loops with geometric degeneracies inclination flips are identified and their unfoldings are analyzed this ties the creation of strange attractors of various geometric structures to codimension two global bifurcations

heteroclinic dynamics in a model of faraday waves in a square container we study periodic orbits associated with heteroclinic bifurcations in a model of the faraday system for containers with square cross section and single frequency forcing these periodic orbits correspond to quasiperiodic surface waves in the physical system the heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable hamiltonian limit some of which persist in the presence of small damping the dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate poincare maps with predictions that agree with numerical computations the results suggest a great richness of possible dynamics of faraday waves even in simple geometries and with single frequency forcing

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

orbits homoclinic to resonances the hamiltonian case abstract in this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits hyperbolic fixed points or combinations of hyperbolic fixed points and or periodic orbits in a class of two degree of freedom integrable hamiltonian systems subject to arbitrary hamiltonian perturbations our methods differ from previous methods in that the invariant sets periodic orbits fixed points are created and become hyperbolic as a result of the interaction of the perturbation with a resonance in the unperturbed system this results in a very degenerate situation that requires a combination of geometric singular perturbation theory higher dimensional melnikov type methods and transversality theory we establish a simple energy phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance we apply our methods to a two mode truncation of the driven nonlinear schrodinger equation first studied by bishop et al in this example we show that as the energy is increased at resonance orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle node bifurcation of homoclinic orbits

smooth functions in o minimal structures abstract fix an o minimal expansion of the real exponential field that admits smooth cell decomposition we study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the whitney topology this implies that abstract definable smooth manifolds are affine moreover abstract definable smooth manifolds are definably c diffeomorphic if and only if they are definably c 1 diffeomorphic

ilyashenko algebras based on transserial asymptotic expansions abstract we construct a hardy field that contains ilyashenko s class of germs at of almost regular functions found in 12 as well as all log exp analytic germs this implies non oscillatory behaviour of almost regular germs with respect to all log exp analytic germs in addition each germ in this hardy field is uniquely characterized by an asymptotic expansion that is an le series as defined by van den dries et al 7 as these series generally have support of order type larger than the notion of asymptotic expansion itself needs to be generalized

maximum likelihood estimation for interval censored data using a weibull based accelerated failure time model the accelerated failure time regression model is most commonly used with right censored survival data this report studies the use of a weibull based accelerated failure time regression model when left and interval censored data are also observed two alternative methods of analysis are considered first the maximum likelihood estimates mles for the observed censoring pattern are computed these are compared with estimates where midpoints are substituted for left and interval censored data midpoint estimator or mde simulation studies indicate that for relatively large samples there are many instances when the mle is superior to the mde for samples where the hazard rate is flat or nearly so or where the percentage of interval censored data is small the mde is adequate an example using framingham heart study data is discussed

comparison theorems for o minimal singular co homology fct grant sfrh bpd 6015 2001 european research and training network hprn ct 2001 00271 raag

a multiple imputation approach for clustered interval censored survival data multivariate interval censored failure time data arise commonly in many studies of epidemiology and biomedicine analysis of these type of data is more challenging than the right censored data we propose a simple multiple imputation strategy to recover the order of occurrences based on the interval censored event times using a conditional predictive distribution function derived from a parametric gamma random effects model by imputing the interval censored failure times the estimation of the regression and dependence parameters in the context of a gamma frailty proportional hazards model using the well developed em algorithm is made possible a robust estimator for the covariance matrix is suggested to adjust for the possible misspecification of the parametric baseline hazard function the finite sample properties of the proposed method are investigated via simulation the performance of the proposed method is highly satisfactory whereas the computation burden is minimal the proposed method is also applied to the diabetic retinopathy study drs data for illustration purpose and the estimates are compared with those based on other existing methods for bivariate grouped survival data

the real field with convergent generalized power series we construct a model complete and o minimal expansion of the field of real numbers in which each real function given on 0 1 by a series cnxn with 0 n and cn r n 1 is definable this expansion is polynomially bounded

multi pulse jumping orbits and homoclinic trees in a modal truncation of the damped forced nonlinear schro dinger equation abstract in this paper we prove the existence of multi pulse orbits homoclinic to a slow manifold in a two mode truncation of the damped forced nonlinear schrodinger equation first suggested by bishop et al these orbits are jumping i e the corresponding solutions keep switching in time between neighborhoods of the two charateristic breathers of the integrable limit in the case of no damping we find multi pulse smale horseshoes in the two mode model while in the dissipative case we establish the existence of structurally stable multi pulse heterolinic connections between two unstable equilibria the orbits we construct are not amenable to melnikov type perturbation methods in both the hamiltonian and the dissipative case we find homoclinic trees which describe the repeated bifurcations of multi pulse solutions to illustrate the theoretical predictions we also present visualizations of these complicated structures

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

two mechanisms of the transition to chaos in finite dimensional models of convection abstract the onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation a connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point this connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem two routes are considered a well known mechanism suggested by lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle point the asymptotical properties of the latter route prove to be entirely determined by the above ratio

estimating survival curves with left truncated and interval censored data under monotone hazards we show that the nonparametric maximum likelihood estimator npmle of a survival function may severely underestimate the survival probabilities at very early times for left truncated and interval censored data as an alternative we propose to compute the nonparametric mle under a nondecreasing hazard assumption the monotone mle by a gradient projection algorithm when the assumption holds the projection step is accomplished via an isotonic regression algorithm the pool adjacent violators algorithm this gradient projection algorithm is computationally efficient and converges globally monte carlo simulations show superior performance of the monotone mle over that of the npmle in terms of either bias or variance even for large samples the methodology is illustrated with the application to the wisconsin epidemiological study of diabetic retinopathy data to estimate the probability of incidence of retinopathy

interval censored parametric regression survival models and the analysis of longitudinal trials this paper develops interval censoring likelihood methods in the context of parametric proportional hazard ph and non ph regression models in the longitudinal study setting to reanalyze the medical research council s randomized controlled trial of teletherapy in age related macular degeneration we compare the performance of the interval censoring likelihood with proxy likelihoods that were used to analyze the original data it is shown analytically that the use of such proxy likelihoods in selected ph models leads to biased estimators such estimators are artificially precise further the magnitude of their percentage bias is quantified in a data directed simulation study for non ph models we demonstrate that these results obtained from ph models do not hold uniformly and explain the implications of this finding for the reanalysis of proxy likelihood trial data our final analysis of the age related macular degeneration trial data based on fitting ph and non ph models reassuringly confirms the published findings from the original trial copyright 2013 john wiley sons ltd

effects of mid point imputation on the analysis of doubly censored data doubly censored data arise in some cohort studies of the aids incubation period because the time of infection may be known only up to an interval defined by two successive screening tests for hiv antibody a simple analytic approach is to impute the infection time by the mid point of the interval and then apply standard survival techniques for right censored data the objective of this paper is to investigate the statistical properties of such a mid point imputation approach we investigated the asymptotic bias of the kaplan meier estimate coverage probabilities of associated confidence intervals bias in hazard ratio and the size of the logrank test we show that the statistical properties of mid point imputation depend strongly on the underlying distributions of infection times and the incubation periods and the width of the interval between screening tests in the absence of treatment the median incubation period of hiv infection is approximately 10 years and we conclude that for this situation mid point imputation is a reasonable procedure for interval widths of 2 years or less

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

remez type inequality on sets with cusps abstract a remez type inequality is proved for a large family of sets with cusps in r n including compact fat and semialgebraic or subanalytic sets

definable zero sum stochastic games definable zero sum stochastic games involve a finite number of states and action sets and reward and transition functions that are definable in an o minimal structure prominent examples of such games are finite semi algebraic or globally subanalytic stochastic games we prove that the shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure definability in the same structure does not hold systematically we provide a counterexample of a stochastic game with semi algebraic data yielding a non semi algebraic but globally subanalytic shapley operator our definability results on shapley operators are used to prove that any separable definable game has a uniform value in the case of polynomially bounded structures we also provide convergence rates using an approximation procedure we actually establish that general zero sum games with separable definable transition functions have a uniform value these results highlight the key role played by the tame structure of transition functions as particular cases of our main results we obtain that stochastic games with polynomial transitions definable games with finite actions on one side and definable games with perfect information or switching controls have a uniform value applications to nonlinear maps arising in risk sensitive control and perron frobenius theory are also given

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

temporal forcing of small amplitude waves in anisotropic systems we investigate the effect of resonant temporal forcing on an anisotropic system that exhibits a hopf bifurcation to obliquely traveling waves in the absence of this forcing we find that the forcing can excite various phase locked standing wave structures rolls rectangles and cross rolls at onset at most one of the two char22 rolls or rectangles char22 is stable the cross rolls can arise in a secondary bifurcation and can be stable experimentally they would appear as a periodic switching between a structure in which the zig component dominates and one with a dominating zag structure since there are two symmetry related states of this kind one may expect disordered structures to arise due to the breakup of the pattern into domains the results are consistent with recent experiments on electroconvection in nematic liquid crystals by de la torre ju arez and rehberg phys rev a 42 2096 1990 we also apply the general analysis to a model of the behavior near a lifshitz point where the angle of obliqueness vanishes this analysis indicates that phase locked standing rectangles are always unstable in this parameter regime

an energy equation for the weakly damped driven nonlinear schro dinger equations and its application to their attractors abstract in this note we present an energy equation concerning the h 2 norm of a forced one dimensional nonlinear schrodinger equation as an application we are able to corroborate recent results on attractors of weakly damped forced nonlinear schrodinger equation by proving that the attractor is an attractor in the strong topology of h 2 and its hausdorff and fractal dimensions are finite in h 2 the energy equation for the h 1 norm can be derived in the same fashion and it implies corresponding results for the attractor in h 1

n pulse homoclinic orbits in perturbations of resonant hamiltonian systems in this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable two degree of freedom resonant hamiltonian systems our energy phase method applies to both hamiltonian and dissipative perturbations and reveals families of multi pulse solutions which are not amenable to melnikov type methods as an example we study a two mode approximation of the nonlinear nonplanar oscillations of a parametrically forced inextensional beam in this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically

estimation in the cox proportional hazards model with left truncated and interval censored data nous montrons que l estimateur non parametrique du maximum de vraisemblance npmle du coefficient de regression obtenu a partir de la vraisemblance jointe du coefficient de regression et de la survie de base est correct pour le modele de cox a hasards proportionnels avec des donnees tronquees a gauche et censurees par intervalle mais le npmle pent sous estimer la survie de base deux alternatives sont egalement etudiees premierement la vraisemblance marginale en etendant satten 1996 aux donnees tronquees la survie de base etant eliminee comme un parametre nuisible deuxiemement l estimateur monotone du maximum de vraisemblance qui maximise la vraisemblance jointe en supposant que la fonction de hasard est non decroissante pour la distribution de base ceci a ete initialement propose pour resoudre la sous estimation de la survie par le npmle dans le cas de donnees tronquees a gauche sans covariate tsai 1988 le boostrap est utilise a des fins d inference et des simulations sont realisees pour evaluer leur performance les methodes sont appliquees pour evaluer leur performance les methodes sont appliquees aux donnees de l etude du massachusetts health care panel afin de comparer chez les seniors en fonction du sexe les probabilites de perte d independance fonctionnelle

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

models for interval censoring and simulation based inference for lifetime distributions summary interval censored lifetime data arise when individuals in a study are inspected inter mittently so that a lifetime is observed to lie between two successive times in settings where only these two times are available methods exist for nonparametric or parametric estimation of lifetime distributions however there has been virtually no discussion of how inspection processes may be estimated or identified such estimates are needed if one is to generate interval censored data by simulation this paper identifies which aspects of an independent inspection process are estimable from interval censored data and shows how to obtain nonparametric estimates the results allow interval censored data from any specified distribution to be generated and give new simulation procedures for estimation or testing a new omnibus goodness of fit test is introduced

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field