polish group actions dichotomies and generalized elementary embeddings the results in this paper involve two different topics in the descriptive theory of polish group actions the book becker kechris 6 is an introduction to that theory our two topics and two collections of theorems are rather unrelated but the proofs for both topics are essentially the same locally compact polish groups i e second countable locally compact groups are the traditional objects of study in the field known as topological groups more recently there has been interest in nonlocally compact polish groups such as s the symmetric group on n topologized as a subspace of nn much of this paper is concerned with a proper subclass of the class of polish groups namely those polish groups which admit a complete left invariant metric we call these cli groups the class of cli groups includes all locally compact groups proposition 3 c 2 d but also much more for example all solvable groups are cli hjorth solecki see proposition 3 c 2 f and a closed subgroup g of s is cli iff g is closed in nn see example 3 c 3 a we study continuous actions by polish groups on polish spaces and the associated orbit equivalence relation for locally compact groups all of our results are known or trivial or both but some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups one of the two topics considered in this paper is dichotomy theorems for equivalence relations these are theorems which assert that either the quotient space is small or else it contains a copy of a specific large set we prove some dichotomy theorems for the orbit space of actions by cli groups there are two types of dichotomies which we call the silver vaught dichotomy and the glimm effros dichotomy the silver vaught dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements assuming the negation of the continuum hypothesis the silver vaught dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2 0 equivalence classes the topological vaught conjecture which is open is for any continuous action by a polish group g on a polish space x the orbit equivalence relation satisfies the silver vaught dichotomy this conjecture implies and was motivated by the vaught conjecture in model theory indeed

borel subgroups of polish groups abstract we study three classes of subgroups of polish groups borel subgroups polishable subgroups and maximal divisible subgroups the membership of a subgroup in each of these classes allows one to assign to it a rank that is a countable ordinal measuring in a natural way complexity of the subgroup we prove theorems comparing these three ranks and construct subgroups with prescribed ranks in particular answering a question of mauldin we establish the existence of borel subgroups which are 0 complete 3 and 0 complete 2 in each uncountable polish group also for every 1 we construct an abelian locally compact second countable group which is densely divisible and of ulm length 1 all previously known such groups had ulm length 0 or 1

the effects of measurement error in response variables and tests of association of explanatory variables in change models biomedical studies often measure variables with error examples in the literature include investigation of the association between the change in some outcome variable blood pressure cholesterol level etc and a set of explanatory variables age smoking status etc typically one fits linear regression models to investigate such associations with the outcome variable measured with error a problem occurs when we include the baseline value of the outcome variable as a covariate in such instances one can find a relationship between the observed change in the outcome and the explanatory variables even when there is no association between these variables and the true change in the outcome variable we present a simple method of adjusting for a common measurement error bias that tends to be overlooked in the modelling of associations with change additional information for example replicates instrumental variables is needed to estimate the variance of the measurement error to perform this bias correction

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

vaught s conjecture on analytic sets conjecture 0 2 like 0 1 remains open under the assumption of the continuum hypothesis both these conjectures are trivially true and for this reason it is often customary to consider a strengthened version which has the same spirit as the original to phrase this stronger version we need to equip the collection of all models whose underlying set is n with a topology whose basic open sets consist of all m for which

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

countable sections for free actions of groups let the group g act on the set x we will call the set y a countable section if each gx n y is countable p forrest showed in 2 that if a locally compact separable metrizable group g acts freely on a standard measurable space x with quasi invariant probability measure u then there is a countable section y which is in and such that the set gy which is the image of g x y by a measurable function and therefore p measurable has pmeasure 1 in the terminology of 11 y is a complete measurable countable section we prove the following converse to forrest s result

naively haar null sets in polish groups abstract let g be a polish group we say that a set x g is haar null if there exists a universally measurable set u x and a borel probability measure such that for every g h g we have g u h 0 we call a set x naively haar null if there exists a borel probability measure such that for every g h g we have g x h 0 generalizing a result of elekes and stepr ns which answers the first part of problem fc from fremlin s list we prove that in every abelian polish group there exists a naively haar null set that is not haar null

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

covariate imbalance and random allocation in clinical trials a model is developed to estimate the effect of covariate imbalance on the size of a test of treatment efficacy in randomized clinical trials comparing two treatments when dispersion parameters are known it is concluded that tests of homogeneity on the covariates should not be performed that covariate imbalance is just as much a problem for large studies as for small ones in terms of effect on size and that the effect of correlation between covariates and measures of efficacy is more complex than has previously been suggested the best way to adjust for covariate imbalance is by an analysis of covariance

multiple imputation approaches for the analysis of dichotomized responses in longitudinal studies with missing data often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value in the event that the underlying continuous measurements are from a longitudinal study one can use the repeated measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status standard bayesian multiple imputation techniques rubin 1987 in multiple imputation for nonresponse in surveys that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within and between imputation variances are found to be conservative the frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of robins and wang 2000 biometrika 87 113 124 is shown to be more efficient we propose to apply kenward and roger 1997 biometrics 53 983 997 degrees of freedom to account for the uncertainty associated with variance covariance parameter estimates for the repeated measures model

polish group actions and the vaught conjecture we consider the topological vaught conjecture if a polish group g acts continuously on a polish space s then s has either countably many or perfectly many orbits we show 1 the conjecture is true for abelian groups 2 the conjecture is true whenever g s are recursively presented the action of g is recursive and for x e s the orbit of x is of borel multiplicative rank wx assertion 1 holds also for analytic s specializing g to a closed subgroup of co we prove that nonempty invariant borel sets not having perfectly many orbits have orbits of about the same borel rank an upper bound is derived for the borel rank of orbits when the set of orbits is finite

haar meager sets revisited abstract in the present article we investigate darji s notion of haar meager sets from several directions we consider alternative definitions and show that some of them are equivalent to the original one while others fail to produce interesting notions we define haar meager sets in nonabelian polish groups and show that many results including the facts that haar meager sets are meager and form a ideal are valid in the more general setting as well the article provides various examples distinguishing haar meager sets from haar null sets including decomposition theorems for some subclasses of polish groups as a corollary we obtain for example that z r or any banach space can be decomposed into a haar meager set and a haar null set we also establish the stability of non haar meagerness under cartesian product

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

closed form reml estimators and sample size determination for mixed effects models for repeated measures under monotone missingness we derive the closed form restricted maximum likelihood estimator and kenward roger s variance estimator for fixed effects in the mixed effects model for repeated measures mmrm when the missing data pattern is monotone as an important application of the analytic result we present the formula for calculating the power of treatment comparison using the wald t test with the kenward roger adjusted variance estimate in mmrm it allows adjustment for baseline covariates without the need to specify the covariate distribution in randomized trials a simple two step procedure is proposed to determine the sample size needed to achieve the targeted power the proposed method performs well for both normal and moderately non normal data even in small samples n 20 in simulations an antidepressant trial is analyzed for illustrative purposes copyright 2017 john wiley sons ltd

stratification for the propensity score compared with linear regression techniques to assess the effect of treatment or exposure stratifying and matching by the propensity score are increasingly popular approaches to deal with confounding in medical studies investigating effects of a treatment or exposure a more traditional alternative technique is the direct adjustment for confounding in regression models this paper discusses fundamental differences between the two approaches with a focus on linear regression and propensity score stratification and identifies points to be considered for an adequate comparison the treatment estimators are examined for unbiasedness and efficiency this is illustrated in an application to real data and supplemented by an investigation on properties of the estimators for a range of underlying linear models we demonstrate that in specific circumstances the propensity score estimator is identical to the effect estimated from a full linear model even if it is built on coarser covariate strata than the linear model as a consequence the coarsening property of the propensity score adjustment for a one dimensional confounder instead of a high dimensional covariate may be viewed as a way to implement a pre specified richly parametrized linear model we conclude that the propensity score estimator inherits the potential for overfitting and that care should be taken to restrict covariates to those relevant for outcome

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

the schrodinger equation and canonical perturbation theory lett0 ev be the schrodinger operator corresponding to the classical hamiltonianh0 ev whereh0 is thed dimensional harmonic oscillator with non resonant frequencies 1 d and the potentialv q1 qd is an entire function of order d 1 1 we prove that the algorithm of classical canonical perturbation theory can be applied to the schrodinger equation in the bargmann representation as a consequence each term of the rayleigh schrodinger series near any eigenvalue oft0 admits a convergent expansion in powers of of initial point the corresponding term of the classical birkhoff expansion moreover ifv is an even polynomial the above result and the kam theorem show that all eigenvalues n e oft0 ev such thatn coincides with a kam torus are given up to order e by a quantization formula which reduces to the bohr sommerfeld one up to first order terms in

the classification problem for von neumann factors abstract we prove that it is not possible to classify separable von neumann factors of types ii 1 ii or iii 0 1 up to isomorphism by a borel measurable assignment of countable structures as invariants in particular the isomorphism relation of type ii 1 factors is not smooth we also prove that the isomorphism relation for von neumann ii 1 factors is analytic but is not borel

non classification of cartan subalgebras for a class of von neumann algebras abstract we study the complexity of the classification problem for cartan subalgebras in von neumann algebras we construct a large family of ii1 factors whose cartan subalgebras up to unitary conjugacy are not classifiable by countable structures providing the first such examples additionally we construct examples of ii1 factors whose cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures finally we show directly that the cartan subalgebras of the hyperfinite ii1 factor up to unitary conjugacy are not classifiable by countable structures and deduce that the same holds for any mcduff ii1 factor with at least one cartan subalgebra

vaught s conjecture and the glimm effros property for polish transformation groups we extend the original glimm effros theorem for locally compact groups to a class of polish groups including the nilpotent ones and those with an invariant metric for this class we thereby obtain the topological vaught conjecture

the coset equivalence relation and topologies on subgroups the paper studies the structure of the homogeneous space g h for g a polish group and h g a borel not necessarily closed subgroup of g from the point of view of the theory of definable equivalence relations it makes a connection between the complexity of the natural it coset equivalence relation associated with g h and it polishability of h that is the possibility of introducing a polish group topology on h respecting its borel structure in particular it is proved that if h is an abelian borel subgroup of a polish group g then either h is polishable or bbb e 1 continuously embeds into the coset equivalence relation induced by h on g the same conclusion is shown to hold if h is an increasing union of a sequence of polishable subgroups of g

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

polish groupoids and functorial complexity we introduce and study the notion of functorial borel complexity for polish groupoids such a notion aims to measure the complexity of classifying the objects of a category in a constructive and functorial way in the particular case of principal groupoids such a notion coincides with the usual borel complexity of equivalence relations our main result is that on one hand for polish groupoids with an essentially treeable orbit equivalence relation functorial borel complexity coincides with the borel complexity of the associated orbit equivalence relation on the other hand for every countable equivalence relation e that is not treeable there are polish groupoids with different functorial borel complexity both having e as orbit equivalence relation in order to obtain such a conclusion we generalize some fundamental results about the descriptive set theory of polish group actions to actions of polish groupoids answering a question of arlan ramsay these include the becker kechris results on polishability of borel g spaces existence of universal borel g spaces and characterization of borel g spaces with borel orbit equivalence relations

classical limit of the quantized hyperbolic toral automorphisms the canonical quantization of any hyperbolic symplectomorphisma of the 2 torus yields a periodic unitary operator on an dimenional hilbert space n 1 h we prove that this quantum system becomes ergodic and mixing at the classical limit n n prime which can be interchanged with the time average limit the recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the lebesgue measure the wigner functions of the eigenstates supported on the classical periodic orbits are indeed proved to become uniformly speread in phase space

measure reducibility of countable borel equivalence relations we show that every basis for the countable borel equivalence relations strictly above mathbb e 0 under measure reducibility is uncountable thereby ruling out natural generalizations of the glimm effros dichotomy we also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base using arguments substantially simpler than those previously employed

borel actions of polish groups we show that a borel action of a polish group on a standard borel space is borel isomorphic to a continuous action of the group on a polish space and we apply this result to three aspects of the theory of borel actions of polish groups universal actions invariant probability measures and the topological vaught conjecture we establish the existence of universal actions for any given polish group extending a result of mackey and varadarajan for the locally compact case we prove an analogue of tarski s theorem on paradoxical decompositions by showing that the existence of an invariant borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many borel pieces we show that various natural versions of the topological vaught conjecture are equivalent to each other and in the case of the group of permutations of n with the model theoretic vaught conjecture for infinitary logic this depends on our identification of the universal action for that group

equivalence relations induced by actions of polish groups we give an algebraic characterization of those sequences hni of countable abelian groups for which the equivalence relations induced by borel or equivalently continuous actions of ho x hl x h2 x are borel in particular the equivalence relations induced by borel actions of hi h countable abelian are borel iff h p fp x z poo np where fp is a finite p group z p is the quasicyclic p group np e co and p varies over the set of all primes this answers a question of r l sami by showing that there are borel actions of polish abelian groups inducing non borel equivalence relations the theorem also shows that there exist non locally compact abelian polish groups all of whose borel actions induce only borel equivalence relations in the process of proving the theorem we generalize a result of makkai on the existence of group trees of arbitrary height

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

classical limit of the harmonic oscillator wigner functions in the bargmann representation writing the wigner functions of any pair of harmonic oscillator eigenstates in the bargmann representation a direct and detailed proof is given of their convergence in the sense of distributions to at the classical limit m n fixed

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

should baseline be a covariate or dependent variable in analyses of change from baseline in clinical trials in randomized clinical trials a pre treatment measurement is often taken at baseline and post treatment effects are measured at several time points post baseline say t 1 t at the end of the trial it is of interest to assess the treatment effect based on the mean change from baseline at the last time point t we consider statistical methods for i a point estimate and 95 per cent confidence interval for the mean change from baseline at time t for each treatment group and ii a p value and 95 per cent confidence interval for the between group difference in the mean change from baseline the manner in which the baseline responses are used in the analysis influences both the accuracy and the efficiency of items i and ii in this paper we will consider the ancova approach with change from baseline as a dependent variable and compare that with a constrained longitudinal data analysis clda model proposed by liang and zeger sankhya indian j stat ser b 2000 62 134 148 in which the baseline is modeled as a dependent variable in conjunction with the constraint of a common baseline mean across the treatment groups some drawbacks of the ancova model and potential advantages of the clda approach are discussed and illustrated using numerical simulations copyright 2009 john wiley sons ltd

methods for assessing difference between groups in change when initial measurement is subject to intra individual variation statistical analysis of difference between groups in change for some variable adjusting for initial value is complicated by the presence of intra individual variation in that variable we estimate here the asymptotic bias that results from calculating the adjusted between group difference by ordinary least squares ols from observed data we also present explicit formulae that use the ols estimates the difference between treatment groups in mean initial values and a measure of the intra individual variation to compute a corrected estimator and its variance alternatively we can use ols on transformed data to obtain unbiased estimates in which we replace initial observed values by conditional stein estimates of true values we illustrate the results with data from an observational study and a clinical trial

oscillator eigenstates concentrated on classical trajectories the highly degenerate eigenspaces of the two dimensional isotropic harmonic oscillator are proven to contain eigenstates that are optimally localized on the closed trajectories of the classical dynamics as h cross to 0 their phase space probability density converges to the unique probability density on the corresponding trajectory which is invariant under the classical flow

analysis of covariance in parallel group clinical trials with pretreatment baselines analysis of covariance ancova techniques are often employed in the analysis of clinical trials to try to account for the effects of varying pretreatment baseline values of an outcome variable on posttreatment measurements of the same variable baseline measurements of outcome variables are typically random variables which violates the usual ancova assumption that covariate values are fixed therefore the usual ancova hypothesis tests of treatment effects may be invalid and the ancova slope parameter estimator biased for this application we show however that if the pretreatment posttreatment measurements have a bivariate normal distribution then i the ancova model with residual error independent of the covariate is a valid expression of the relationship between pretreatment and posttreatment measurements ii the usual fixed covariate analysis ancova estimates of the slope parameter and treatment effect contrasts are unbiased and iii the usual ancova treatment effect contrast t tests are valid significance tests for treatment effects moreover as long as the magnitudes of the treatment effects do not depend on the true pretreatment value of the outcome variable the true slope parameter must lie in the interval 0 1 and the ancova model has a clear interpretation as an adjustment based on between and within subject variability to an analysis of variance model applied to the posttreatment pretreatment differences

turbulence and araki woods factors abstract using baire category techniques we prove that araki woods factors are not classifiable by countable structures as a result we obtain a far reaching strengthening as well as a new proof of the well known theorem of woods that the isomorphism problem for itpfi factors is not smooth we derive as a consequence that the odometer actions of z that preserve the measure class of a finite non atomic product measure are not classifiable up to orbit equivalence by countable structures