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

rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

almost reducibility and non perturbative reducibility of quasi periodic linear systems in this paper we prove that a quasi periodic linear differential equation in sl 2 with two frequencies 1 is almost reducible provided that the coefficients are analytic and close to a constant in the case that is diophantine we get the non perturbative reducibility we also obtain the reducibility and the rotations reducibility for an arbitrary irrational under some assumption on the rotation number and give some applications for schrodinger operators our proof is a generalized kam type iteration adapted to all irrational

rotation numbers for random dynamical systems on the circle in this paper we study rotation numbers of random dynamical systems on the circle we prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems as an application we prove a theorem on analytic conjugacy to a circle rotation

unified lasso estimation by least squares approximation we propose a method of least squares approximation lsa for unified yet simple lasso estimation our general theoretical framework includes ordinary least squares generalized linear models quantile regression and many others as special cases specifically lsa can transfer many different types of lasso objective functions into their asymptotically equivalent least squares problems thereafter the standard asymptotic theory can be established and the lars algorithm can be applied in particular if the adaptive lasso penalty and a bayes information criterion type tuning parameter selector are used the resulting lsa estimator can be as efficient as the oracle extensive numerical studies confirm our theory

dispersive estimate for quasi periodic schrodinger operators on 1 d lattices abstract consider the one dimensional discrete schrodinger operator h h q n q n 1 q n 1 v n q n n z with r d diophantine and v a real analytic function on t d r 2 z d for v sufficiently small we prove the dispersive estimate for every l 1 z 1 e i t h l k 0 ln e 0 a ln ln 2 t 2 d t 1 3 l 1 t 1 t 2 with a and k 0 two absolute constants and e 0 an analytic norm of v the estimate holds for every t d

the rotation number of the linear schrodinger equation with discontinuous almost periodic potentials abstract in this paper we study a linear schrodinger equation with a stepanov almost periodic potential where the potential is discontinuous from r to r we first show the existence of rotation numbers for the equation we then give the optimal estimate of rotation numbers

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

second phase transition line we study the phase transion line of the almost mathieu operator that separates arithmetic regions corresponding to singular continuous and a e pure point regimes and prove that both purely singular continuous and a e pure point spectrum occur for dense sets of frequencies

poisson lie structures on poincare and euclidean groups in three dimensions the complete list of poisson lie structures on poincare and euclidean groups in three dimensions is presented some new solutions for inhomogenous are given

variable selection for cox s proportional hazards model and frailty model a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in fan and li 2001a it has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance such a property is called an oracle property the proposed procedures were illustrated in the context of linear regression robust linear regression and generalized linear models in this paper the nonconcave penalized likelihood approach is extended further to the cox proportional hazards model and the cox proportional hazards frailty model two commonly used semi parametric models in survival analysis as a result new variable selection procedures for these two commonly used models are proposed it is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function further with a proper choice of the regularization parameter and the penalty function the proposed estimators possess an oracle property standard error formulae are derived and their accuracies are empirically tested simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection and they reduce model complexity as effectively as the best subset variable selection compared with the lasso which is the penalized likelihood method with the l 1 penalty proposed by tibshirani the newly proposed approaches have better theoretic properties and finite sample performance

the lasso method for variable selection in the cox model summary i propose a new method for variable selection and shrinkage in cox s proportional hazards model my proposal minimizes the log partial likelihood subject to the sum of the absolute values of the parameters being bounded by a constant because of the nature of this constraint it shrinks coe cients and produces some coe cients that are exactly zero as a result it reduces the estimation variance while providing an interpretable final model the method is a variation of the lasso proposal of tibshirani designed for the linear regression context simulations indicate that the lasso can be more accurate than stepwise selection in this setting

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

case control current status data summary in this paper we show that the distribution function of survival times is identified up to a one parameter family of distribution functions based on information from case control current status data with supplementary information on the population frequency of cases relative to controls a simple weighted version of the nonparametric maximum likelihood estimator for prospective current status data provides a natural estimator for case control samples following the parametric results of scott wild 1997 we show that this estimator is in fact the nonparametric maximum likelihood estimator

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

spectral behavior of quasi periodic potentials spectral properties of schrodinger operators of the typeh v where is the laplacian v a quasiperiodic potential and a coupling constant are developed v is taken to be finite sum of exponentials with generic frequencies for small a strong stability is shown on the other hand examples in the finite diffeence case are given for which a transition in the type of spectrum occurs as is increased

linear relations in the calkin algebra for composition operators we consider this and related questions when is a finite linear combination of composition operators acting on the hardy space or the standard weighted bergman spaces on the unit disk a compact operator

rotation numbers for quasiperiodically forced circle maps mode locking vs strict monotonicity rotation numbers for quasiperiodically forced circle maps mode locking vs strict monotonicity

estimates for the hill operator ii consider the hill operator ty y q t y in l2 r where the real potential q is 1 periodic and q q l2 0 1 the spectrum of t consists of spectral bands separated by gaps n n 1 with length n 0 we obtain two sided estimates of the gap lengths n2 n 2 in terms of 01q t 2dt moreover we obtain the similar two sided estimates for spectral data the height of the corresponding slit on the quasimomentum domain action variables for the kdv equation and so on in order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the hill operator that makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory then the proof is based on the analysis of the conformal mapping the embedding theorems and the identities furthermore we obtain the similar two sided estimates for potentials which have p 2 derivatives

the schrodinger equation with a quasi periodic potential we consider the schrodinger equation dx2 v e cosx cos cix 0 v eqi where e is small and a satisfies the diophantine inequality lp qcl c q2 forp qez q 0 we look for solutions of the form ikx ikx in x im ax 0 m x e q x e 2m ne inxne e if we try to solve for v v mn we are led to the schrodinger equation on the lattice z2 h k q ba v k q en where a is the discrete laplacian without diagonal terms and v k is some 2 potential on z we have two main results 4 for e sufficiently small h k has pure point spectrum for almost every k 2 for e sufficiently small the operator d dx 2 e cosx cos ax has no point spectrum to prove our results we must get decay estimates on the green s function e h 1 the decay of the eigenfunction follows from this in general we must keep track of small divisors which can make the green s function large this is accomplished by a kam kolmogorov arnold moser type of multiscale perturbation analysis

applications of the exponential ordering in the study of almost periodic delayed hopfield neural networks abstract this paper studies almost periodic neural networks of hopfield type described by delayed differential equations the authors introduce an exponential ordering to analyze the long term behavior of the solutions they prove some theorems of global convergence and deduce the stabilization role of the fast inhibitory self connections the proof which in the case of the neural networks considered in this paper requires uniform stability uses arguments of comparison of differential equations and methods of the non autonomous monotone theory of dynamical systems when the connections between different neurons are excitatory improved conditions of convergence are obtained and the stabilization role of strongly positive inputs is also shown the applicability of the results is illustrated with several numerical experiments based on two different families of neural networks

adaptive lasso for cox s proportional hazards model summary we investigate the variable selection problem for cox s proportional hazards model and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience the new method is based on a penalized log partial likelihood with the adaptively weighted l1 penalty on regression coefficients providing what we call the adaptive lasso estimator the method incorporates different penalties for different coefficients unimportant variables receive larger penalties than important ones so that important variables tend to be retained in the selection process whereas unimportant variables are more likely to be dropped theoretical properties such as consistency and rate of convergence of the estimator are studied we also show that with proper choice of regularization parameters the proposed estimator has the oracle properties the convex optimization nature of the method leads to an efficient algorithm both simulated and real examples show that the method performs competitively

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

isospectral deformations of random jacobi operators we show the integrability of infinite dimensional hamiltonian systems obtained by making isospectral deformations of random jacobi operators over an abstract dynamical system the time 1 map of these so called random toda flows can be expressed by aqr decomposition

sieve maximum likelihood regression analysis of dependent current status data current status data occur in contexts including demographic studies and tumorigenicity experiments in such cases each subject is observed only once and the failure time of interest is either left or right censored kalbfleisch prentice 2002 many methods have been developed for the analysis of such data huang 1996 sun 2006 most of which assume that the failure time and the observation time are independent completely or given covariates in this paper we present a sieve maximum likelihood approach for current status data when independence does not hold a copula model and monotone i splines are used and the asymptotic properties of the resulting estimators are established in particular the estimated regression parameters are shown to be semiparametrically efficient an illustrative example is provided

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

simultaneous estimation and variable selection for interval censored data with broken adaptive ridge regression the simultaneous estimation and variable selection for cox model has been discussed by several authors when one observes right censored failure time data however there does not seem to exist an e

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

a sieve semiparametric maximum likelihood approach for regression analysis of bivariate interval censored failure time data abstractinterval censored failure time data arise in a number of fields and many authors have discussed various issues related to their analysis however most of the existing methods are for univariate data and there exists only limited research on bivariate data especially on regression analysis of bivariate interval censored data we present a class of semiparametric transformation models for the problem and for inference a sieve maximum likelihood approach is developed the model provides a great flexibility in particular including the commonly used proportional hazards model as a special case and in the approach bernstein polynomials are employed the strong consistency and asymptotic normality of the resulting estimators of regression parameters are established and furthermore the estimators are shown to be asymptotically efficient extensive simulation studies are conducted and indicate that the proposed method works well for practical situations supplementary materials for this article are

holder continuity of absolutely continuous spectral measure for multi frequency schrodinger operators abstract we establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi frequency schrodinger operators with diphantine frequencies and small analytic potentials

abundance of mode locking for quasiperiodically forced circle maps we study the phenomenon of mode locking in the context of quasiperiodically forced non linear circle maps as a main result we show that under certain mathcal c 1 open condition on the geometry of twist parameter families of such systems the closure of the union of mode locking plateaus has positive measure in particular this implies the existence of infinitely many mode locking plateaus open arnold tongues the proof builds on multiscale analysis and parameter exclusion methods in the spirit of benedicks and carleson which were previously developed for quasiperiodic rm sl 2 mathbb r cocycles by young and bjerklov the methods apply to a variety of examples including a forced version of the classical arnold circle map