maps completely preserving commutativity and maps completely preserving jordan zero product abstract let x y be real or complex banach spaces with infinite dimension and let a b be standard operator algebras on x and y respectively in this paper we show that every map completely preserving commutativity from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism every map completely preserving jordan zero product from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism

maps on idempotent matrices over division rings abstract let d be an arbitrary division ring and p n d the set of all n n idempotent matrices over d under some mild conditions we give a complete description of maps on p n d that preserve either commutativity or order or orthogonality we give examples showing that our assumptions cannot be relaxed much further as an application we will prove a quaternionic analogue of ovchinnikov s result that is important in quantum mechanics other applications of our theorems include results on automorphisms of operator and matrix semigroups local automorphisms linear preserver problems and geometry of matrices and grassmannians

assessment of stratum covariate interactions in cox s proportional hazards regression model we assess stratum e g treatment interactions with covariates and with the baseline hazard function in the proportional hazards ph regression model for lifetime data we consider models incorporating stratum interactions both with and without stratification of the risk sets in the likelihood function and describe likelihood ratio statistics for tests of the presence of these interactions we also present step down methods for building reduced models which include stratum specific parameters corresponding to covariates which interact with treatment we apply ph models with such interactions to a clinical trial of des in the treatment of prostate cancer to determine optimal treatment conditional on each patient s covariates

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

nonlinear maps preserving numerical radius of indefinite skew products of operators let h be a complex hilbert space of dimension greater than 2 and j b h be an invertible self adjoint operator denote by a j 1a j the indefinite conjugate of a b h with respect to j and denote by w a the numerical radius of a let w and v be subsets of b h which contain all rank one operators and let w v be a surjective map we show that satisfies w ab w a b and w a b w a b for all a b w if and only if there exist scalars i 1 1 i 1 2 unitary or conjugate unitary operators u v on h satisfying u u 1i v v 2i and a functional w c with a 1 such that a a uav for all a w satisfies w ab a w a b a for all a b w if and only if either there exist 1 1 a unitary or conjugate unitary operator u on h satisfying u u i and a functional w c with a 1 such that a a uau for all a w or there exist a nonzero real number b a unitary or conjugate unitary operator u on h satisfying u ju bj 1 and a functional w c with a 1 such that a a ua u for all a w

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

isometries of grassmann spaces abstract botelho jamison and molnar have recently described the general form of surjective isometries of grassmann spaces on complex hilbert spaces under certain dimensionality assumptions in this paper we provide a new approach to this problem which enables us first to give a shorter proof and second to remove dimensionality constraints completely in one of the low dimensional cases which was not covered by botelho jamison and molnar an exceptional possibility occurs as a byproduct we are able to handle the real case as well furthermore in finite dimensions we remove the surjectivity assumption a variety of tools is used in order to achieve our goal such as topological geometrical and linear algebra techniques the famous two projections theorem for two finite rank projections will be re proven using linear algebraic methods a theorem of gyory and the second author on orthogonality preservers on grassmann spaces will be strengthened as well this latter result will be obtained by using chow s fundamental theorem of geometry of grassmannians

well posedness and stability results for a quasilinear periodic muskat problem abstract we study the muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two dimensional geometry we first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown secondly we show that this evolution equation has a quasilinear structure which is at a formal level not obvious and we also disclose the parabolic character of the equation exploiting these aspects we establish the local well posedness of the problem for arbitrary initial data in h s s with s 3 2 2 determine a new criterion for the global existence of solutions and uncover a parabolic smoothing property besides we prove that the zero steady state solution is exponentially stable

a center manifold analysis for the mullins sekerka model abstract the mullins sekerka model is a nonlocal evolution model for hypersurfaces which arises as a singular limit for the cahn hilliard equation we show that classical solutions exist globally and tend to spheres exponentially fast provided that they are close to a sphere initially our analysis is based on center manifold theory and on maximal regularity

linear regression analysis for highly stratified failure time data abstract in this article we consider cases where data consist of many small and independent groups of correlated failure time observations for each failure time which may be censored some important covariates are also recorded our goal is to examine the covariate effects on the individual failure time observations we assume that the logarithm of each failure time is linearly related to its covariates we then take a population averaged model approach to obtain inference procedures for the regression parameters without specifying the joint distribution of the observations within the group the new proposals do not need complicated and unstable nonparametric estimates for the hazard function of the error term their properties are extensively examined for practical sample sizes comparisons among various procedures are also performed all the methods studied in this article are illustrated with examples

automorphisms of the poset of skew projections abstract any automorphism of the poset b of all idempotents of the algebra b h of all bounded linear operators in a hilbert space h dim h 3 is shown to be of the form p apa 1 or p ap a 1 with a c where c is the set of all semi linear bijections a h h if h is finite dimensional and the set of all continuous invertible linear or conjugate linear operators in h if h is infinite dimensional further applications of the method are outlined

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

transition probability preserving maps on a grassmann space in a semifinite factor abstract in this paper the surjective transition probability preserving transformations on the grassmann space of infinite projections in an infinite semifinite factor are characterized as in the classical wigner s theorem all these maps are induced by isomorphisms or anti isomorphisms of the semifinite factor

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

transformations on the sets of states and density operators in this paper we consider two physically relevant numerical quantities namely the concurrence and fidelity and an operation called product defined on the set of all quantum states or density operators our main aim is to present the complete descriptions of all bijective transformations which preserve any of these quantities or operation

characterizing sequential isomorphisms on hilbert space effect algebras let be any sequential product on the hilbert space effect algebra with and be a bijective map we show that if satisfies ab a b for then there is either a unitary or an anti unitary operator u such that a uau for every let be a borel function satisfying g 0 0 g 1 1 and let us define a binary operation g on by a gb a1 2 g a bg a a1 2 where t denotes the conjugate of the operator t we also show that a bijective map satisfies a gb a g b for if and only if there is either a unitary or an anti unitary operator u such that a uau for every

on operators preserving james orthogonality abstract we characterise isometries and coisometries in b h in terms of james orthogonality as a consequence we obtain a characterisation of surjective linear or conjugate linear mappings b h b h that preserve james orthogonality in both directions

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

phase isometries on real normed spaces abstract we say that a map f from a normed space x to another normed space y is a phase isometry if the equality f x f y f x f y x y x y holds for all x y x a normed space x is said to have the wigner property if for any normed space y and every surjective phase isometry f x y there exists a phase function e x 1 1 such that e f is a linear isometry we show that all smooth normed spaces l type spaces and l 1 spaces enjoy this property

a stable manifold theorem for the curve shortening equation on presente une famille de solutions homothetiques de l equation pour une courbe planaire x kn et on demontre l existence de varietes non lineaires stables et instables autour de telles solutions

entropy preserving maps on quantum states abstract in the paper we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations let s h be the set of all quantum states positive operators with the unit trace on a complex hilbert space h with dim h n and s the quantum entropy of the quantum state for arbitrary a surjective map s h s h we claim that preserves quantum entropy of convex combinations of quantum states i e satisfies s t 1 t s t 1 t for arbitrary s h and t 0 1 if and only if there exists a unitary or anti unitary operator u on h such that u u for all s h

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

endomorphisms of the poset of idempotent matrices abstract let d be a division ring n 3 an integer and p n d the poset of all n n idempotent matrices over d with the partial order defined by p q if p q q p p let t m n d be an invertible matrix and d d an endomorphism d d an anti endomorphism for any p p n d we denote by p p the idempotent matrix obtained from p by applying entrywise the map p n d p n d defined by p t p t 1 p p n d p t t p t 1 p p n d is an injective map preserving order in both directions every such map is called a standard map it has been known before that if d is an eas division ring then every injective order preserving map p n d p n d is either standard or of a very special degenerate form in this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the eas assumption is indispensable then we define generalized standard maps and using them we describe the general form of injective order preservers on p n d for an arbitrary division ring d our proof is shorter than the original one for the special case of eas division rings under somewhat stronger assumptions we get two characterizations of standard maps all the results are optimal as shown by counterexamples

twistor space minkowski space and the conformal group it is shown that the conformal group of compactified minkowski space is isomorphic to a group of rays of semilinear transformations of twistor space the action of the conformal group on twistor space is given by an explicit realisation of this isomorphism in this way we determine the transformation of twistor space under space inversion and time inversion

maps on states preserving generalized entropy of convex combinations abstract let s h be the set of all linear positive semidefinite self adjoint trace one operators states on h where h is an at least two dimensional finite dimensional real or complex hilbert space or at least three dimensional left quaternionic hilbert space of dimension n given a strictly convex function f 0 1 r for any s h we define f i f i where 1 2 n are the eigenvalues of counted with multiplicities in this note we completely describe maps s h s h having the property f t 1 t f t 1 t for all t 0 1 and every s h it turns out that u u s h where u is a real linear isometry of h note that there is no surjectivity assumption and that our result in particular improves the description of maps preserving the von neumann entropy of convex combinations of states in the complex hilbert space it can as well be applied to preserving schatten or some other strictly convex norms of convex combinations of states

a general framework for random effects survival analysis in the cox proportional hazards setting summary the use of random effects modeling in statistics has increased greatly in recent years the introduction of such modeling into event time analysis has proceeded more slowly however previously random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit in this paper we develop a method of random effect analysis of survival data the hierarchical cox model that is an extension of cox s original formulation in that the baseline hazard function remains unspecified this method also allows an arbitrary distribution for the random effects we accomplish this using markov chain monte carlo methods in a bayesian setting the method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control this analysis is more satisfying than standard approaches such as studying the first event for each subject which does not fully use the data or assuming independence which in this case would overestimate the precision

transition probabilities of normal states determine the jordan structure of a quantum system let x1d516 m1 x1d516 m2 be a bijection not assumed affine nor continuous between the sets of normal states of two quantum systems modelled on the self adjoint parts of von neumann algebras m1 and m2 respectively this paper concerns with the situation when preserves or partially preserves one of the following three notions of transition probability on the normal state spaces the transition probability pu introduced by uhlmann rep math phys 9 273 279 1976 the transition probability pr introduced by raggio lett math phys 6 233 236 1982 and an asymmetric transition probability p0 as introduced in this article it is shown that the two systems are isomorphic i e m1 and m2 are jordan isomorphic if preserves all pairs with zero uhlmann respectively raggio or asymmetric transition probability in the sense that for any normal states and we have p 0 if and only if p 0 where p stands for pu respectively pr or p0 furthermore as an extension of wi

generalized symmetry transformations on quaternionic indefinite inner product spaces an extension of quaternionic version of wigner s theorem uhlhorn s extension of wigner s unitary antiunitary theorem has been recently generalized by molnar to indefinite inner product spaces we present the quaternionic version of this result

regularity of solutions to a free boundary problem modeling tumor growth by stokes equation in this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid like tissues the model equations include a quasi stationary diffusion equation for the nutrient concentration and a stokes equation with a source representing the proliferation density of the tumor cells subject to a boundary condition with stress tensor effected by surface tension this problem is a fully nonlinear problem involving nonlocal terms based on the employment of the functional analytic method and the theory of maximal regularity we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

maps leaving functional values of operator products invariant abstract let a be a standard operator algebra on a complex hilbert space h of dimension greater than 2 by invariants of certain functional values of operator products we characterize some surjective maps on a furthermore several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as for example k numerical radius k 1 operator norm ky fan k norm schatten p norm 1 p and so on

parametric spatial cure rate models for interval censored time to relapse data summary several recent papers e g chen ibrahim and sinha 1999 journal of the american statistical association94 909 919 ibrahim chen and sinha 2001a biometrics57 383 388 have described statistical methods for use with time to event data featuring a surviving fraction i e a proportion of the population that never experiences the event such cure rate models and their multivariate generalizations are quite useful in studies of multiple diseases to which an individual may never succumb or from which an individual may reasonably be expected to recover following treatment e g various types of cancer in this article we extend these models to allow for spatial correlation estimable via zip code identifiers for the subjects as well as interval censoring our approach is bayesian where posterior summaries are obtained via a hybrid markov chain monte carlo algorithm we compare across a broad collection of rather high dimensional hierarchical models using the deviance information criterion a tool recently developed for just this purpose we apply our approach to the analysis of a smoking cessation study where the subjects reside in 53 southeastern minnesota zip codes in addition to the usual posterior estimates our approach yields smoothed zip code level maps of model parameters related to the relapse rates over time and the ultimate proportion of quitters the cure rates

wigner type theorem on transition probability preserving maps in semifinite factors abstract the wigner s theorem which is one of the cornerstones of the mathematical formulation of quantum mechanics asserts that every symmetry of quantum system is unitary or anti unitary this classical result was first given by wigner in 1931 thereafter it has been proved and generalized in various ways by many authors recently g p geher extended wigner s and molnar s theorems and characterized the transformations on the grassmann space of all rank n projections which preserve the transition probability the aim of this paper is to provide a new approach to describe the general form of the transition probability preserving not necessarily bijective maps between grassmann spaces as a byproduct we are able to generalize the results of molnar and g p geher

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

symmetry transformations in indefinite metric spaces a generalization of wigner s theorem we formulate and prove a generalization to indefinite metric spaces of uhlhorn s version of wigner s theorem

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

bayesian analysis of proportional hazards models built from monotone functions we consider the usual proportional hazards model in the case where the baseline hazard the covariate link and the covariate coefficients are all unknown both the baseline hazard and the covariate link are monotone functions and thus are characterized using a dense class of such functions which arises upon transformation as a mixture of beta distribution functions we take a bayesian approach for fitting such a model since interest focuses more upon the likelihood we consider vague prior specifications including jeffreys s prior computations are carried out using sampling based methods model criticism is also discussed finally a data set studying survival of a sample of lung cancer patients is analyzed

parabolic equations for curves on surfaces part i curves with p integrable curvature this is the first of a two part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so called curve shortening of flow by mean curvature problem as well as to a number of models for phase transitions in two dimensions we introduce a class of equations for which the initial value problem is solvable for initial data with p integrable curvature and we also give estimates for the rate at which the p norms of the curvature must blow up if the curve becomes singular in finite time a detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part

order preserving maps on the poset of upper triangular idempotent matrices abstract we obtain the general form of bijective order and orthogonality preserving maps on the poset of all n n upper triangular idempotent matrices over an arbitrary field f we also show that the assumption of surjectivity can be removed if every nonzero homomorphism g f f is surjective

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained

hierarchical proportional hazards regression models for highly stratified data summary in clinical trials conducted over several data collection centers the most common statistically defensible analytic method a stratified cox model analysis suffers from two important defects first identification of units that are outlying with respect to the baseline hazard is awkward since this hazard is implicit rather than explicit in the cox partial likelihood second and more seriously identification of modest treatment effects is often difficult since the model fails to acknowledge any similarity across the strata we consider a number of hierarchical modeling approaches that preserve the integrity of the stratified design while offering a middle ground between traditional stratified and unstratified analyses we investigate both fully parametric weibull and semiparametric models the latter based not on the cox model but on an extension of an idea by gelfand and mallick 1995 biometrics51 843 852 which models the integrated baseline hazard as a mixture of monotone functions we illustrate the methods using data from a recent multicenter aids clinical trial comparing their ease of use interpretation and degree of robustness with respect to estimates of both the unit specific baseline hazards and the treatment effect

a generalization of wigner s unitary antiunitary theorem to hilbert modules let h be a hilbert c module over a matrix algebra a it is proved that any function t h h which preserves the absolute value of the generalized inner product is of the form tf f uf f h where is a phase function and u is an a linear isometry the result gives a natural extension of wigner s classical unitary antiunitary theorem for hilbert modules

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

wigner s theorem on grassmann spaces wigner s celebrated theorem which is particularly important in the mathematical foundations of quantum mechanics states that every bijective transformation on the set of all rank one projections of a complex hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator this vital theorem has been generalised in various ways by several scientists in 2001 molnar provided a natural generalisation namely he provided a characterisation of not necessarily bijective maps which act on the grassmann space of all rank n projections and leave the system of jordan principal angles invariant see 17 and 20 in this paper we give a very natural joint generalisation of wigner s and molnar s theorems namely we prove a characterisation of all not necessarily bijective transformations on the grassmann space which fix the quantity trpq i e the sum of the squares of cosines of principal angles for every pair of rank n projections p and q

proportional hazards model with random effects we propose a general proportional hazards model with random effects for handling clustered survival data this generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk in a way similar to the modelling of random effects in linear generalized linear and non linear mixed models the distribution of the random effects is generally assumed to be multivariate normal but other preferably symmetrical distributions are also possible maximum likelihood estimates of the regression parameters the variance components and the baseline hazard function are obtained via the em algorithm the e step of the algorithm involves computation of the conditional expectations of functions of the random effects for which we use markov chain monte carlo mcmc methods approximate variances of the estimates are computed by louis formula and posterior expectations and variances of the individual random effects can be obtained as a by product of the estimation the inference procedure is exemplified on two data sets

a modified log rank test for highly stratified data summary clinical trials often employ a stratified randomization so that treatment arms are balanced within prognostic factors in these circumstances the use of the ordinary log rank test for comparing treatment groups with respect to survival is invalid the stratified log rank test although valid loses efficiency as the number of strata increases this paper proposes a modified log rank test that is fully efficient when there is no strata effect and is more efficient than the stratified log rank test when the number of strata is large furthermore the test has been designed so as to remain valid when the censoring pattern depends on the treatment group as it would in a historically controlled study

generalization of wigner s unitary antiunitary theorem for indefinite inner product spaces we present a generalization of wigner s unitary antiunitary theorem for pairs of ray transformations as a particular case we get a new wigner type theorem for non hermitian indefinite inner product spaces

transformations on the set of all n dimensional subspaces of a hilbert space preserving principal angles wigner s classical theorem on symmetry transformations plays a fundamental role in quantum mechanics it can be formulated for example in the following way every bijective transformation on the set of all 1 dimensional subspaces of a hilbert space h which preserves the angle between the elements of is induced by either a unitary or an antiunitary operator on h the aim of this paper is to extend wigner s result from the 1 dimensional case to the case of n dimensional subspaces of h with n fixed