a class of p q laplacian type equation with concave convex nonlinearities in bounded domain abstract in this paper our main purpose is to establish the existence of multiple solutions of a class of p q laplacian equation involving concave convex nonlinearities p u q u v x u r 2 u u p 2 u f x u x u 0 x where is a bounded domain in r n 0 1 r q p n and p n p n p is the critical sobolev exponent s u div u s 2 u is the s laplacian of u we prove that for any 0 0 is a constant there is a 0 such that for every 0 the above problem possesses infinitely many weak solutions we also obtain some results for the case 1 q p r p the existence results of solutions are obtained by variational methods

a generalized estimating equations approach for testing ordered group effects with repeated measurements in repeated measures studies we are often interested in comparing group effects in which groups are associated with a certain order relation we propose testing procedures for ordered group effects using the generalized estimating equations gee approach of liang and zeger 1986 biometrika 73 13 22 the order constrained gee estimators of group effects are approximated by the isotonic regression of the unconstrained gee estimators based on these constrained estimators we construct test statistics for detecting ordered group effects the limiting distributions of the test statistics are mixtures of chi square distributions a monte carlo experiment shows improved performances of the proposed tests over the usual chi square tests in detecting ordered group effects the proposed test procedures are illustrated by familial polyposis supplementation trial data

testing for ordered group effects with repeated measurements summary in the repeated measures experiment testing procedures for ordered group effects are proposed using the likelihood ratio principle the constrained maximum likelihood estimators of ordered group effects are approximated by the isotonic regression of the unconstrained maximum likelihood estimators based on these constrained maximum likelihood estimators test statistics are constructed and are used for detecting ordered group effects the limiting distributions of the proposed test statistics are shown to be mixtures of chi squared distributions a simulation study shows that the proposed tests have improved powers over the usual chi squared statistic for testing equality of group effects a real data set is analysed to illustrate the proposed statistics

on approximation of the level probabilities and associated distributions in order restricted inference summary the use of much of the distribution theory developed for order restricted inference has been limited by the lack of algorithms for the level probabilities an approximation for these which accounts for the pattern of large and small weights is developed this approximation and the equal weights approximation are examined both approximations appear to be reasonable for weight sets having a moderate amount of variability the quality of the equal weights approximation as a function of the amount of variability in the weights deteriorates more quickly for certain patterns of large and small weights than for others thus the approximation based upon the pattern of large and small weights is a significant improvement over the equal weights approximation finally siskind s 1976 approximation which can be applied if the number of parameters is not too large is discussed the chi bar squared 2 and e bar squared p2 distributions are fundamental to the theory of order restricted hypothesis tests these distributions have tail probabilities which are linear combinations of the tail probabilities of standard distributions and they depend upon the order restriction through the coefficients in these linear combinations the values of these coefficients are the probabilities that the order restricted maximum likelihood estimates of normal means assume specified numbers of distinct values which are called levels the estimates are based upon independent samples from each of the populations and depend upon the vector of relative precisions called weights of the sample means as estimates of the corresponding population means the use of these tests has been limited because these level probabilities can be virtually impossible to compute if the weights are not all equal in this paper we describe a technique for approximating the level probabilities for a linear order restriction and for unequal precisions this approximation is based upon an idea of chase 1974 and uses the pattern of large and small relative precisions it seems to be particularly good when the relative precisions have two distinct values however it seems to provide a satisfactory approximation so long as the variation in the relative precisions is not too large this is probably due to the lack of sensitivity of the level probabilities to changes in the weights the robustness was

existence of positive solutions for a semipositone p laplacian problem we prove the existence of positive solutions to a semipositone p laplacian problem combining mountain pass arguments comparison principles regularity principles and a priori estimates

existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball abstract we prove the existence and multiplicity of positive radial solutions to the nonlinear system u i k i x f i u j in d i u i n c i u i u i 0 on x r 0 u i x 0 as x for a certain range of 0 where i j 1 2 i j x r n x r 0 0 n 2 d i 0 k i r 0 0 c 0 0 f i 0 r are continuous with possible singularity at 0 and satisfy a combined superlinear condition at

existence of positive radial solutions for a superlinear semipositone p laplacian problem on the exterior of a ball we prove the existence of positive radial solutions to a class of semipositone p laplacian problems on the exterior of a ball subject to dirichlet and nonlinear boundary conditions using variational methods we prove the existence of a solution and then use a priori estimates to prove the positivity of the solution

analysis of positive radial solutions for singular superlinear p laplacian systems on the exterior of a ball abstract in this paper we study some well posedness properties of a positive radial solution pair u 1 u 2 to a singular boundary value problem in the form p u i k i x f i u j in r n b r 0 a i u i n c i u j u i u i 0 on b r 0 u i 0 as x where i j 1 2 i j p stands for the singular degenerate p laplacian p w div w p 2 w with 1 p n 0 a i 0 r 0 0 k i c r 0 0 and c i c 0 0 0 0 here the functions f i c 0 r i 1 2 satisfy a combined p superlinear growth at and possibly have singularities at 0 we establish an existence result for the case 0 multiplicity results of positive radial solutions for certain ranges of and nonexistence results of a positive radial solution for the case 1

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

on radial solutions for singular combined superlinear elliptic systems on annular domains abstract we prove the existence of a large positive solution to the system r n 1 1 u r n 1 f 1 v a r b r n 1 2 v r n 1 f 2 u a r b u a 0 u b v a 0 v b where a 0 is a small positive parameter f i 0 r are continuous and lim z 1 1 f 1 c 2 1 f 2 z z for all c 0

an approach to comparing treatments based on repeated measures summary a mixed effects model is assumed for comparing treatments based on repeated measurements the covariance structure of the data is interpreted as some systematic inhomogeneity of individual profiles along the time axis rather than as serial correlation then a method of comparing treatment effects is proposed as well as that of testing the homogeneity of individual profiles a follow up analysis of residuals for the resulting model is also mentioned

two sample asymptotically distribution free tests for incomplete multivariate observations abstract asymptotically distribution free tests for equality of two multivariate distributions based on censored observations are proposed by choosing appropriate test scores we obtain the multivariate versions of the gehan and log rank tests an important application of the proposed tests is in the area in which multiple observations are collected on study subjects some of whom may have missing observations the patterns of missing observations are allowed to differ for the two groups to be compared these tests are particularly useful for longitudinal studies for example the analysis of repeated measurements and growth curves real examples are also provided for illustration

existence of positive solutions for a class of p laplacian superlinear semipositone problems we consider a quasilinear elliptic problem of the form where 0 is a parameter 1 p is a strictly convex bounded domain in n n p with c 2 boundary the nonlinearity f 0 is a continuous function that is semipositone f 0 p superlinear at infinity using degree theory combined with a rescaling argument and uniform l a priori bound we establish the existence of a positive solution for small moreover we show that there exists a connected component of positive solutions bifurcating from infinity at 0 we also extend our study to systems

a complete classification of bifurcation diagrams for a class of p q laplace equations abstract we study the bifurcation problem of positive solutions for the one dimensional p q laplace equation with nonlinear term u r 1 there are five types of order relations for p q r we investigate the exact shape of the bifurcation curve in each type of the order relation we prove that there are two types of bifurcation curves that are increasing two types that are decreasing and one that is not monotone and turns exactly once moreover we study the asymptotic profile of the normalized solution u x u as u 0 or u where u denotes the l norm of u

positive solutions to classes of infinite semipositone p q laplace problems with nonlinear boundary conditions abstract we consider one dimensional p q laplace problems u h t f u t 0 1 u 0 0 a u 1 g u 1 u 1 where 0 a 0 s s p 2 s s q 2 s 1 p q h c 0 1 0 f c 0 r with lim s 0 f s 0 and g c 0 0 0 such that g r s s is nondecreasing with respect to s 0 classifying the behaviors of f near infinity we establish the existence multiplicity and nonexistence of positive solutions in particular we provide a sufficient condition on f to obtain a multiplicity result for the case when lim s f s s r 1 0 1 r q which is new even in semilinear problems p q 2 the proofs are based on a krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity

positive radial solutions to classes of singular problems on the exterior domain of a ball abstract we study positive radial solutions to singular boundary value problems of the form u k x f u u in u c u u 0 x r 0 u x 0 x where u div u is the laplacian operator of u x r n x r 0 0 n 2 0 k c r 0 0 is such that k s 1 s n for s 1 for some 1 min 1 n 2 and u is the outward normal derivative of u on x r 0 here f c 1 0 r is such that f s s 1 0 as s and c c 0 0 we analyse the cases when a f 0 0 and b f 0 0 we discuss existence non existence multiplicity and uniqueness results we prove our existence results by the method of sub and supersolutions

comparison of methods for the analysis of longitudinal interval count data longitudinal studies are often concerned with estimating the recurrence rate of a non fatal event in many cases only the total number of events occurring during successive time intervals is known we compared a mixed poisson gamma regression method proposed by thall and a quasi likelihood method proposed by zeger and liang for the analysis of such data in the case where the mean was correctly specified using simulation techniques with large samples both methods produced similar standard errors in most situations except in the case of time dependent covariates with non poisson gamma data where they were seriously underestimated by the thall method a simple method for discriminating between the variance forms of the two methods is described the findings are applied to the analyses of clinical trials of non melanoma skin cancer and familial polyposis this study extends the findings of breslow concerning variance misspecification in overdispersed poisson and quasi likelihood models to the longitudinal setting

multiplicity of positive solutions to a p q laplacian equation involving critical nonlinearity abstract in this paper our main purpose is to establish the existence of multiple solutions of a p q laplacian equation with convex and sobolev critical nonlinearity by some standard variational methods whose key is to construct homotopies between and levels of the functional i and some analytical techniques

a unified theory of commutator estimates for a class of interpolation methods abstract a general family of interpolation methods is introduced which includes as special cases the real and complex methods and also the so called or g1 and g2 methods defined by peetre and gustavsson peetre derivation operators and translation operators r are introduced for all methods of this family a theorem is proved about the boundedness of the commutators t and t r for operators t which are bounded on the spaces of the pair to which the interpolation method is applied this extends and unifies results previously known for derivation and translation operators in the contexts of the real and complex methods other results deal with higher order commutators and also include an equivalence theorem i e it is shown that as previously known only for real interpolation spaces all these interpolation spaces have two different equivalent definitions in the style of the j method and k method auxiliary results which may also be of independent interest include the equivalence of lions schechter complex interpolation spaces defined using an annulus with the same spaces defined in the usual way using a strip

on the electrostatic born infeld equation with extended charges in this paper we deal with the electrostatic born infeld equation left begin array ll operatorname div left displaystyle frac nabla phi sqrt 1 nabla phi 2 right rho quad in mathbb r n displaystyle lim x to infty phi x 0 end array right quad quad quad quad mathcal bi where rho is an assigned extended charge density we are interested in the existence and uniqueness of the potential phi and finiteness of the energy of the electrostatic field nabla phi we first relax the problem and treat it with the direct method of the calculus of variations for a broad class of charge densities assuming rho is radially distributed we recover the weak formulation of mathcal bi and the regularity of the solution of the poisson equation under the same smoothness assumptions in the case of a locally bounded charge we also recover the weak formulation without assuming any symmetry the solution is even classical if rho is smooth then we analyze the case where the density rho is a superposition of point charges and discuss the results in kiessling commun math phys 314 509 523 2012 other models are discussed as for instance a system arising from the coupling of the nonlinear klein gordon equation with the born infeld theory

semi parametric and non parametric methods for the analysis of repeated measurements with applications to clinical trials techniques applicable for the analysis of longitudinal data when the response variable is non normal are not nearly as comprehensive as for normally distributed outcomes however there have been several recent developments semi parametric and non parametric methodology for the analysis of repeated measurements is reviewed the commonly encountered design in which for each subject one assesses a univariate response variable at multiple fixed time points is considered the types of outcomes considered include binary ordered categorical and continuous but extremely non normal response variables all of the methods considered allow for incomplete data due to the occurrence of missing observations in addition discrete and or continuous covariates which may be time dependent are accommodated by some of the approaches the methods are demonstrated using data from three clinical trials

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