a comparison estimate of toponogov type for ricci curvature it seems a natural question to ask to what extent the results and tools for sectional curvature remain valid for ricci curvature there is rapid progress in both positive and negative directions toponogov comparison theorem has been the most powerful tool in the study of sectional curvature underlying the proof of the soul theorem the diameter sphere theorem the uniform estimate of betti numbers and the finiteness theorems toponogov comparison theorem is also the characterizing property of lower or upper sectional curvature bounds which led to generalizations of the concept of sectional curvature bounds to non smooth space thus it could not possibly hold for ricci curvature which makes problems a lot harder for ricci curvature from a geometric point of view it also makes the ricci curvature very different from the sectional curvature as one gradually comes to realize it is interesting then that we find a comparison estimate of toponogov type for ricci curvature a hinge in a complete riemannian manifold consists of two geodesic segments vi y2 such that yi ii 2 0 9 we denote it by 1 2 where z l o we will be using the following modified version of the conjugate radius function pc p

cohomogeneity one manifolds with positive ricci curvature one of the central problems in riemannian geometry is to determine how large the classes of manifolds with positive nonnegative sectional ricci or scalar curvature are see gr for scalar curvature the situation is fairly well understood by comparison special surgery constructions as in sy wr and bundle constructions as in na have resulted in a large number of interesting manifolds with positive ricci curvature so far the only known obstructions to have positive ricci curvature come from obstructions to have positive scalar curvature see li and rs and from the classical bonnet myers theorem which implies that a closed manifold with positive ricci curvature must have finite fundamental group it is well known that among homogeneous manifolds g h this is also a sufficient condition see e g the proof of corollary 3 5 or br in this paper we prove that this is true as well when the manifold admits an action by a compact lie group g with orbits of codimension one

an extension of bonnet myers theorem we give a complementary generalization of the extensions of bonnet myers theorem obtained by calabi and also cheeger gromov taylor

relative efficiency of unequal versus equal cluster sizes in cluster randomized and multicentre trials cluster randomized and multicentre trials evaluate the effect of a treatment on persons nested within clusters for instance patients within clinics or pupils within schools optimal sample sizes at the cluster centre and person level have been derived under the restrictive assumption of equal sample sizes per cluster this paper addresses the relative efficiency of unequal versus equal cluster sizes in case of cluster randomization and person randomization within clusters starting from maximum likelihood parameter estimation the relative efficiency is investigated numerically for a range of cluster size distributions an approximate formula is presented for computing the relative efficiency as a function of the mean and variance of cluster size and the intraclass correlation which can be used for adjusting the sample size the accuracy of this formula is checked against the numerical results and found to be quite good it is concluded that the loss of efficiency due to variation of cluster sizes rarely exceeds 10 per cent and can be compensated by sampling 11 per cent more clusters

li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

bias in randomised factorial trials factorial trials are an efficient method of assessing multiple treatments in a single trial saving both time and resources however they rely on the assumption of no interaction between treatment arms ignoring the possibility of an interaction in the analysis can lead to bias and potentially misleading conclusions therefore it is often recommended that the size of the interaction be assessed during analysis this approach can be formalised as a two stage analysis if the interaction test is not significant a factorial analysis where all patients receiving treatment a are compared with all not receiving a and similarly for treatment b is performed if the interaction is significant the analysis reverts to that of a four arm trial where each treatment combination is regarded as a separate treatment arm we show that estimated treatment effects from the two stage analysis can be biased even in the absence of a true interaction this occurs because the interaction estimate is highly correlated with treatment effect estimates from a four arm analysis simulations show that bias can be severe over 100 in some cases leading to inflated type i error rates therefore the two stage analysis should not be used in factorial trials a preferable approach may be to design multi arm trials i e four separate treatment groups instead this approach leads to straightforward interpretation of results is unbiased regardless of the presence of an interaction and allows investigators to ensure adequate power by basing sample size requirements on a four arm analysis

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

generalized liouville property for schrodinger operator on riemannian manifolds in this paper we prove that the dimension of the space of positive bounded respectively mathcal l harmonic functions on a complete riemannian manifold with mathcal l regular ends is equal to the number of ends mathcal l nonparabolic ends respectively this result is a solution of an open problem of grigor yan related to the liouville property for the schrodinger operator mathcal l we also prove that if a given complete riemannian manifold is roughly isometric to a complete riemannian manifold satisfying the volume doubling condition the poincare inequality and the finite covering condition on each end then the dimension of the space of positive bounded respectively solutions for the schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends mathcal l nonparabolic ends respectively this is a partial answer of the question suggested by li related to the regularity of ends of a complete riemannian manifold especially our results directly generalize various earlier results of yau of li and tam of grigor yan and of present authors but with different techniques that the peculiarity of the schrodinger operator demands

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

calderon zygmund estimates for elliptic double phase problems with variable exponents abstract we study a nonlinear elliptic double phase problem with variable exponents to prove calderon zygmund estimates under minimal regularity requirements on the nonlinearities

analysis and geometry on manifolds with integral ricci curvature bounds ii we extend several geometrical results for manifolds with lower ricci curvature bounds to situations where one has integral lower bounds in particular we generalize colding s volume convergence results and extend the cheeger colding splitting theorem

global gradient estimates for the borderline case of double phase problems with bmo coefficients in nonsmooth domains abstract we consider a double phase problem with bmo coefficient in divergence form on a bounded nonsmooth domain the problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position which describes a feature of strongly anisotropic materials we obtain the global calderon zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small bmo and the boundary of the domain is sufficiently flat in the reifenberg sense

new volume comparison results and applications to degeneration of riemannian metrics abstract we consider a condition on the ricci curvature involving vector fields which is broader than the bakry emery ricci condition under this condition volume comparison laplacian comparison isoperimetric inequality and gradient bounds are proven on the manifold specializing to the bakry emery ricci curvature condition we initiate a short cut approach to work on the original manifold which yields under a weaker than usual assumption the results mentioned above for the original manifold these results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead applications on convergence and degeneration of riemannian metrics under this curvature condition are given to this effect in particular for the bakry emery ricci curvature condition the gradient of the potential function is allowed to have singularity of order close to 1 which is broader than those allowed by the traditional weighted method in general coupling with some new techniques in constructing model functions this approach enables us to extend some of the results in the papers 14 11 36 29 and 31 the condition also covers general ricci solitons instead of just gradient ricci solitons

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

rough isometry and energy finite solutions of elliptic equations on riemannian manifolds we prove that if the s harmonic boundary of a complete riemannian manifold consists of finitely many points then the set of bounded energy finite solutions for certain nonlinear elliptic operators on the manifold is one to one corresponding to bf r l where l is the cardinality of thes harmonic boundary we also prove that the finiteness of cardinality of s harmonic boundary is a rough isometric invariant moreover in this case the cardinality is preserved under rough isometries between complete riemannian manifolds this result generalizes those of yau of donnelly of grigor yan of li and tam of kim and the present author of holopainen and of the present author but with different techniques which are demanded by the peculiarity of nonlinearity

on complete manifolds of nonnegative kth ricci curvature in this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature key tools are comparison techniques and morse theory of busemann and distance functions

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

power and sample size requirements for gee analyses of cluster randomized crossover trials the cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel cluster randomized design which often involves a limited number of clusters in recent years the cluster randomized crossover design has been increasingly used to evaluate the effectiveness of health care policy or programs and the interest often lies in quantifying the population averaged intervention effect in this paper we consider the two treatment two period crossover design and develop sample size procedures for continuous and binary outcomes corresponding to a population averaged model estimated by generalized estimating equations accounting for both within period and interperiod correlations in particular we show that the required sample size depends on the correlation parameters through an eigenvalue of the within cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes we demonstrate that the empirical power corresponds well with the predicted power by the proposed formulae for as few as eight clusters when outcomes are analyzed using the matrix adjusted estimating equations for the correlation parameters concurrently with a suitable bias corrected sandwich variance estimator

choosing appropriate analysis methods for cluster randomised cross over trials with a binary outcome in cluster randomised cross over crxo trials clusters receive multiple treatments in a randomised sequence over time in such trials there is usual correlation between patients in the same cluster in addition within a cluster patients in the same period may be more similar to each other than to patients in other periods we demonstrate that it is necessary to account for these correlations in the analysis to obtain correct type i error rates we then use simulation to compare different methods of analysing a binary outcome from a two period crxo design our simulations demonstrated that hierarchical models without random effects for period within cluster which do not account for any extra within period correlation performed poorly with greatly inflated type i errors in many scenarios in scenarios where extra within period correlation was present a hierarchical model with random effects for cluster and period within cluster only had correct type i errors when there were large numbers of clusters with small numbers of clusters the error rate was inflated we also found that generalised estimating equations did not give correct error rates in any scenarios considered an unweighted cluster level summary regression performed best overall maintaining an error rate close to 5 for all scenarios although it lost power when extra within period correlation was present especially for small numbers of clusters results from our simulation study show that it is important to model both levels of clustering in crxo trials and that any extra within period correlation should be accounted for copyright 2016 john wiley sons ltd

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

calderon zygmund theory for nonlinear elliptic problems with irregular obstacles abstract we consider a nonhomogeneous elliptic problem with an irregular obstacle involving a discontinuous nonlinearity over an irregular domain in divergence form of p laplacian type to establish the global calderon zygmund estimate by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the nonhomogeneous term under the bmo smallness of the nonlinearity and sufficient flatness of the boundary in the reifenberg sense

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

gradient estimates for double phase problems with irregular obstacles abstract an irregular obstacle problem with non uniformly elliptic operator in divergence form of p q growth is studied we find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term in the divergence calderon zygmund type estimates are also obtained under minimal regularity requirements of the prescribed data

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

bias and inference from misspecified mixed effect models in stepped wedge trial analysis many stepped wedge trials swts are analysed by using a mixed effect model with a random intercept and fixed effects for the intervention and time periods referred to here as the standard model however it is not known whether this model is robust to misspecification we simulated swts with three groups of clusters and two time periods one group received the intervention during the first period and two groups in the second period we simulated period and intervention effects that were either common to all or varied between clusters data were analysed with the standard model or with additional random effects for period effect or intervention effect in a second simulation study we explored the weight given to within cluster comparisons by simulating a larger intervention effect in the group of the trial that experienced both the control and intervention conditions and applying the three analysis models described previously across 500 simulations we computed bias and confidence interval coverage of the estimated intervention effect we found up to 50 bias in intervention effect estimates when period or intervention effects varied between clusters and were treated as fixed effects in the analysis all misspecified models showed undercoverage of 95 confidence intervals particularly the standard model a large weight was given to within cluster comparisons in the standard model in the swts simulated here mixed effect models were highly sensitive to departures from the model assumptions which can be explained by the high dependence on within cluster comparisons trialists should consider including a random effect for time period in their swt analysis model 2017 the authors statistics in medicine published by john wiley sons ltd

embeddings of riemannian manifolds with heat kernels and eigenfunctions we show that any closed n dimensional riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points both this time and this number can be bounded in terms of the dimension a lower bound on the ricci curvature the injectivity radius and the volume it follows that the manifold can be embedded by a finite number of eigenfunctions of the laplace operator again this number only depends on the geometric bounds and the dimension in addition both maps can be made arbitrarily close to an isometry in the appendix we derive quantitative estimates of the harmonic radius so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well 2016 wiley periodicals inc

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

scalar minimizers with fractal singular sets lack of regularity of local minimizers for convex functionals with non standard growth conditions is considered it is shown that for every 0 there exists a function ac such that the functional admits a local minimizer uw1 p whose set of non lebesgue points is a closed set with dim n p and where 1 p n n q

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

discrete homotopies and the fundamental group abstract we generalize and strengthen the theorem of gromov that the fundamental group of any compact riemannian manifold of diameter at most d has a set of generators g 1 g k of length at most 2 d and relators of the form g i g m g j in particular we obtain an explicit bound for the number k of generators in terms of the number of short loops at every point and the number of balls required to cover a given semi locally simply connected geodesic space as a corollary we obtain a fundamental group finiteness theorem new even for riemannian manifolds that replaces the curvature and volume conditions of anderson and the 1 systole bound of shen wei by more general geometric hypothesis implied by these conditions this theorem in turn is a special case of a theorem for arbitrary compact geodesic spaces proved using the method of discrete homotopies introduced by the first author and v n berestovskii central to the proof is the notion of homotopy critical spectrum introduced in this paper as a natural consequence of discrete homotopy methods this spectrum is closely related to the sormani wei covering spectrum which is a subset of the classical length spectrum studied by de verdiere and duistermaat guillemin it is completely determined including multiplicity by special closed geodesics called essential circles

sample size calculation for cluster randomized cross over trials the cluster randomized cross over design has been proposed in particular because it prevents an imbalance that may bring into question the internal validity of parallel group cluster trials we derived a sample size formula for continuous outcomes that takes into account both the intraclass correlation coefficient representing the clustering effect and the interperiod correlation induced by the cross over design

an imbedding theorem for musielak sobolev spaces abstract let be a bounded domain in r n and a c 1 0 0 be a musielak orlicz function we established an imbedding theorem for the musielak sobolev space w 1 a of the form w 1 a l a where a is the sobolev conjugate function of a a musielak orlicz function defined in the paper this theorem extends the imbedding theorem established by donaldson and trudinger for orlicz sobolev spaces to the case of musielak sobolev spaces

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

universal covers for hausdorff limits of noncompact spaces we prove that if y is the gromov hausdorff limit of a sequence of complete manifolds m n i with a uniform lower bound on ricci curvature then y has a universal cover

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

examples of manifolds of positive ricci curvature on presente de nouveaux exemples de varietes de riemann simplement connexes de dimension 7 a courbure de ricci positive qui n admettent pas de metrique de courbure sectionnelle non negative dans les cas compacts et non compacts

the performance of the two stage analysis of two treatment two period crossover trials in the two treatment two period crossover trial patients are randomly allocated either to one group that receives treatment a followed by treatment b or to another group that receives the treatments in the reverse order grizzle first proposed a two stage procedure for analysing the data from such a trial this paper examines the long run sampling properties of this procedure in terms of mean square error of point estimates coverage probability of confidence intervals and actual significance level of hypothesis tests for the differences between the effects of the two treatments the advantages of incorporating baseline observations into the analysis are also explored because the preliminary test for carryover is highly correlated with the analysis of data from the first period only actual significance levels are higher than nominal levels even when there is no differential carryover when carryover is present the nominal level very seriously understates the actual level and this becomes even worse when baseline observations are ignored increasing sample size only exacerbates the problem since this adverse behaviour then occurs at smaller values of the carryover effect it is concluded that the two stage analysis is too potentially misleading to be of practical use

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below i we investigate the finiteness structure of a complete non compact n dimensional riemannian manifold m whose radial curvature at a base point of m is bounded from below by that of a non compact von mangoldt surface of revolution with its total curvature greater than we show as our main theorem that all busemann functions on m are exhaustions and that there exists a compact subset of m such that the compact set contains all critical points for any busemann function on m as corollaries by the main theorem m has finite topological type and the isometry group of m is compact

uniform convexity of musielak orlicz sobolev spaces and applications abstract this paper deals with uniform convexity of musielak orlicz sobolev spaces and its applications to variational problems some sufficient conditions and examples for uniform convexity of musielak orlicz sobolev spaces are given some special properties relative to the uniformly convex modular for uniformly convex musielak orlicz sobolev spaces are presented as an application of these abstract results the local minimizers and the mountain pass type critical point of an integral functional with more complicated growth than the p x growth are studied

complete manifolds with nonnegative ricci curvature and quadratically nonnegatively curved infinity in this paper we study complete riemannian manifolds with nonnegative ricci curvature and quadratically nonnegatively curved infinity we show that m is of finite topological type if it has either maximal volume growth or minimal volume growth

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

a volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications where dpo is the distance to po this class was first studied by u abresch abi ab2 because of the decay condition on the curvature faster than quadratic we expect the class of manifolds of asymptotically nonnegative curvature behaves similarly at infinity to manifolds of nonnegative curvature for the case of sectional curvature in abl and ab2 abresch showed that the complexity of topology of such manifolds is similar to that of nonnegative curvature namely they are all of finite topological type with uniform bound on the betti numbers in ka a kasue investigated the theory of harmonic functions for such manifolds again he demonstrated the expected similarity for ricci curvature the examples of j p sha and d g yang showed that we can not hope to control the topology of such manifolds as we do for sectional curvature very little is known for the case of asymptotically nonnegative ricci curvature except the work of abresch and gromoll ab which studied such manifolds under further conditions on the sectional curvature and diameter growth a question of recent interest stimulated by the works of p li and l f tam lt1 lt2 is whether open manifolds of asymptotically nonnegative ricci curvature has finitely many ends for the special case when a has compact support m l cai cai and z d liu liu recently gave a bound on the number of ends in lt2 under the stronger condition of f0o tn a t dt o0 li and tam also gave such a bound

large scale behavior of kernels of schrodinger operators in this paper we study the asymptotic behavior of solutions to schrodinger equations on manifolds with nonnegative ricci curvature and euclidean volume growth as a first application we get a sharp asymptotic estimate on the green s function for the laplacian which sharpens results of li and yau next we establish a definite gap in the possible rates of decay of exterior harmonic functions on these manifolds when the manifold also has quadratic curvature decay we get supremum estimates on the c 2 norm of the green s function this has function theoretic implications for these manifolds we then give a generalization to manifolds with nonnegative ricci curvature outside a compact set finally we study schrodinger operators where the potential decays quadratically at infinity in this setting we obtain a geometric version at infinity of a classical theorem of bers

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

negative ricci curvature and isometry group we show that for n dimensional manifolds with ricci curvature bounded between two negative constants the order of their isometry groups is uniformly bounded by the ricci curvature bounds the volume and the injectivity radius we also show that the degree of symmetry seex2 for denition is lower semicontinuous in the gromov hausdor topology

diagnostics for assumptions in moderate to large simple clinical trials do they really help in this article primarily we look at a case study where prior to conducting the major efficacy analysis one performs a diagnostic test for assumptions and acts upon the result if the diagnostic test rejects the assumptions specifically we show by an example that a hybrid approach of using a diagnostic test for equality of variance in a two sample t test situation can adversely affect rather than protect the operating characteristics of the study if this kind of hybrid approach fails in such a simple setting analysts should be very cautious about using hybrid approaches in more complex analyses of efficacy secondarily we present rationale as to why the classical tests or slightly modified versions can be viewed as asymptotically non parametric and can actually be more robust against failure of assumptions than rank tests readers are cautioned that this illustration is limited to efficacy analysis and is not meant as a criticism of other analyses such as modelling or exploratory ones