a priori bounds for semilinear equations and a new class of critical exponents for lipschitz domains abstract a priori bounds for positive very weak solutions of semilinear elliptic boundary value problems u f x u on a bounded domain r n with u 0 on are studied where the nonlinearity 0 f x s grows at most like s p if is a lipschitz domain we exhibit two exponents p and p which depend on the boundary behavior of the green function and on the smallest interior opening angle of we prove that for 1 p p all positive very weak solutions are a priori bounded in l for p p we construct a nonlinearity f x s a x s p together with a positive very weak solution which does not belong to l finally we exhibit a class of domains for which p p for such domains we have found a true critical exponent for very weak solutions in the case of smooth domains p p n 1 n 1 is an exponent which is well known from classical work of brezis turner h brezis r e l turner on a class of superlinear elliptic problems comm partial differential equations 2 1977 601 614 and from recent work of quittner souplet p quittner ph souplet a priori estimates and existence for elliptic systems via bootstrap in weighted lebesgue spaces arch ration mech anal 174 2004 49 81

regularized outcome weighted subgroup identification for differential treatment effects summary to facilitate comparative treatment selection when there is substantial heterogeneity of treatment effectiveness it is important to identify subgroups that exhibit differential treatment effects existing approaches model outcomes directly and then define subgroups according to interactions between treatment and covariates because outcomes are affected by both the covariate treatment interactions and covariate main effects direct modeling outcomes can be hard due to model misspecification especially in presence of many covariates alternatively one can directly work with differential treatment effect estimation we propose such a method that approximates a target function whose value directly reflects correct treatment assignment for patients the function uses patient outcomes as weights rather than modeling targets consequently our method can deal with binary continuous time to event and possibly contaminated outcomes in the same fashion we first focus on identifying only directional estimates from linear rules that characterize important subgroups we further consider estimation of comparative treatment effects for identified subgroups we demonstrate the advantages of our method in simulation studies and in analyses of two real data sets

categories with fuzzy sets and relations we define a 2 category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions we enrich this category with additional monoidal and involutive structure coming from t norms and negations on the unit interval we develop the basic properties of this category and consider its relation to other familiar categories a discussion is made of extending these results to the setting of type 2 fuzzy sets

multiplicity considerations in subgroup analysis this paper deals with the general topic of subgroup analysis in late stage clinical trials with emphasis on multiplicity considerations the discussion begins with multiplicity issues arising in the context of exploratory subgroup analysis including principled approaches to subgroup search that are applied as part of subgroup exploration exercises as well as in adaptive biomarker driven designs key considerations in confirmatory subgroup analysis based on one or more pre specified patient populations are reviewed including a survey of multiplicity adjustment methods recommended in multi population phase iii clinical trials guidelines for interpretation of significant findings in several patient populations are introduced to facilitate the decision making process and achieve consistent labeling across development programs

subgroup identification based on differential effect search a recursive partitioning method for establishing response to treatment in patient subpopulations we propose a novel recursive partitioning method for identifying subgroups of subjects with enhanced treatment effects based on a differential effect search algorithm the idea is to build a collection of subgroups by recursively partitioning a database into two subgroups at each parent group such that the treatment effect within one of the two subgroups is maximized compared with the other subgroup the process of data splitting continues until a predefined stopping condition has been satisfied the method is similar to interaction tree approaches that allow incorporation of a treatment by split interaction in the splitting criterion however unlike other tree based methods this method searches only within specific regions of the covariate space and generates multiple subgroups of potential interest we develop this method and provide guidance on key topics of interest that include generating multiple promising subgroups using different splitting criteria choosing optimal values of complexity parameters via cross validation and addressing type i error rate inflation inherent in data mining applications using a resampling based method we evaluate the operating characteristics of the procedure using a simulation study and illustrate the method with a clinical trial example copyright 2011 john wiley sons ltd

a bayesian credible subgroups approach to identifying patient subgroups with positive treatment effects many new experimental treatments benefit only a subset of the population identifying the baseline covariate profiles of patients who benefit from such a treatment rather than determining whether or not the treatment has a population level effect can substantially lessen the risk in undertaking a clinical trial and expose fewer patients to treatments that do not benefit them the standard analyses for identifying patient subgroups that benefit from an experimental treatment either do not account for multiplicity or focus on testing for the presence of treatment covariate interactions rather than the resulting individualized treatment effects we propose a bayesian credible subgroups method to identify two bounding subgroups for the benefiting subgroup one for which it is likely that all members simultaneously have a treatment effect exceeding a specified threshold and another for which it is likely that no members do we examine frequentist properties of the credible subgroups method via simulations and illustrate the approach using data from an alzheimer s disease treatment trial we conclude with a discussion of the advantages and limitations of this approach to identifying patients for whom the treatment is beneficial

lattices of convex normal functions the algebra of truth values of type 2 fuzzy sets is the set of all functions from the unit interval into itself with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min this algebra has been studied rather extensively both from a theoretical and from a practical point of view it has a number of interesting subalgebras and this paper is about the subalgebra of all convex normal functions and closely related ones these particular algebras are de morgan algebras and our concern is principally with their completeness as lattices a special feature of our treatment is a representation of these algebras as monotone functions with pointwise order making the operations more intuitive

optimal treatment assignment to maximize expected outcome with multiple treatments when there is substantial heterogeneity of treatment effectiveness it is crucial to identify individualized treatment assignment rules for comparative treatment selection traditional approaches directly model clinical outcome and define optimal treatment rule according to the interactions between treatment and covariates this approach relies on the success of separating the main effects from the covariate treatment interaction effects which may not be easy to overcome this shortcoming a recent approach called outcome weighted learning focuses on building an optimal treatment rule by maximizing the expected clinical outcome related with differential treatments however there seems to be a lack of approaches to explicitly deal with three or more treatments in this article we propose an outcome weighted learning method that extends estimating individualized treatment rules to multi treatment case by using a vector hinge loss as a target function consistency of the resulting estimator is shown in the article we demonstrate the performance of our approach in simulation studies and in a real data analysis

random forests with ensemble of feature spaces random forests receive much attention from researchers because of their excellent performance as breiman suggested the performance of random forests depends on the strength of the weak learners in the forests and the diversity among them however in the literature many researchers only considered pre processing of the data or post processing of the random forests models in this paper we propose a new method to increase the diversity of each tree in the forests and thereby improve the overall accuracy during the training process of each individual tree in the forest different rotation spaces are concatenated into a higher space at the root node then the best split is exhaustively searched within this higher space the location where the best split lies decides which rotation method to be used for all subsequent nodes the performance of the proposed method here is evaluated on 42 benchmark data sets from various research fields and compared with the standard random forests the results show that the proposed method improves the performance of the random forests in most cases

symmetry results for solutions of semilinear elliptic equations with convex nonlinearities abstract in this paper we study the symmetry properties of the solutions of the semilinear elliptic problem u f x u in u g x on where is a bounded symmetric domain in r n n 2 and f r r is a continuous function of class c1 in the second variable g is continuous and f and g are somehow symmetric in x our main result is to show that all solutions of the above problem of index one are axially symmetric when is an annulus or a ball g 0 and f is strictly convex in the second variable to do this we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of is a sufficient condition for the symmetry of the solution when f is a convex function

comparison results for pdes with a singular potential several comparison results are obtained for solutions to linear elliptic and parabolic equations with a singular potential solutions to these equations are singular in many cases and our results roughly say that they all have comparable singularities provided that they belong to an appropriate space we formulate the hypothesis on the potential in terms of an inequality which in the case of the well known inverse square potential is a consequence of an improvement of hardy s inequality due to vazquez and zuazua

survival trees by goodness of split abstract a tree based method for censored survival data is developed based on maximizing the difference in survival between groups of patients represented by nodes in a binary tree the method includes a pruning algorithm with optimal properties analogous to the classification and regression tree cart pruning algorithm uniform convergence of the estimates of the conditional cumulative hazard and survival functions is discussed and an example is given to show the utility of the algorithm for developing prognostic classifications for patients

hardyʼs inequality and curvature a hardy inequality of the form f x pdx p 1p p 1 a x f x p x pdx for all f c0 r is considered for p 1 p 1 where is a domain in rnrn n 2n 2 r r is the ridge of and x x is the distance from x x to the boundary the main emphasis is on determining the dependence of a a on the geometric properties of a hardy inequality is also established for any doubly connected domain in r2r2 in terms of a uniformization of that is any conformal univalent map of onto an annulus

recursive partitioning for heterogeneous causal effects in this paper we propose methods for estimating heterogeneity in causal effects in experimental and observational studies and for conducting hypothesis tests about the magnitude of differences in treatment effects across subsets of the population we provide a data driven approach to partition the data into subpopulations that differ in the magnitude of their treatment effects the approach enables the construction of valid confidence intervals for treatment effects even with many covariates relative to the sample size and without sparsity assumptions we propose an honest approach to estimation whereby one sample is used to construct the partition and another to estimate treatment effects for each subpopulation our approach builds on regression tree methods modified to optimize for goodness of fit in treatment effects and to account for honest estimation our model selection criterion anticipates that bias will be eliminated by honest estimation and also accounts for the effect of making additional splits on the variance of treatment effect estimates within each subpopulation we address the challenge that the ground truth for a causal effect is not observed for any individual unit so that standard approaches to cross validation must be modified through a simulation study we show that for our preferred method honest estimation results in nominal coverage for 90 confidence intervals whereas coverage ranges between 74 and 84 for nonhonest approaches honest estimation requires estimating the model with a smaller sample size the cost in terms of mean squared error of treatment effects for our preferred method ranges between 7 22

on the best constants for the brezis marcus inequalities in balls we study the best possible constants c n in the brezis marcus inequalities bn u 2dx 14 bn u 2 x x0 2dx c n 2 bn u 2dx for u h01 bn in balls bn x rn x x0 the quantity c 1 is known by our paper f g avkhadiev k j wirths unified poincare and hardy inequalities with sharp constants for convex domains zamm z angew math mech 87 8 9 26 2007 632 642 in the present paper we prove the estimate c 2 2 and the assertion limn c n n2 14 which gives that the known lower estimates in g barbatis s filippas and a tertikas in comm cont math 5 2003 no 6 869 881 for c n n 3 are asymptotically sharp as n also for the 3 dimensional ball b30 x r3 x 1 we obtain a new brezis marcus type inequality which contains two parameters m 0 0 1 m and has the following form b30 u x 2dx 14 b30 1 2m2 1 x 2 m2j 2 1 x 2 m u x 2dx where j is the first zero of the bessel function j of order and the constants 1 2m24andm2j 24 are sharp for all admissible values of parameters m and

convex normal functions revisited the lattice l u of upper semicontinuous convex normal functions with convolution ordering arises in studies of type 2 fuzzy sets in 2002 kawaguchi and miyakoshi extended t norms as logical connectives of fuzzy truth values multiple valued logic 8 1 2002 53 69 showed that this lattice is a complete heyting algebra later harding et al lattices of convex normal functions fuzzy sets and systems 159 2008 1061 1071 gave an improved description of this lattice and showed it was a continuous lattice in the sense of gierz et al a compendium of continuous lattices springer berlin 1980 in this note we show the lattice l u is isomorphic to the lattice of decreasing functions from the real unit interval 0 1 to the interval 0 2 under pointwise ordering modulo equivalence almost everywhere this allows development of further properties of l u it is shown that l u is completely distributive is a compact hausdorff topological lattice whose topology is induced by a metric and is self dual via a period two antiautomorphism we also show the lattice l u has another realization of natural interest in studies of type 2 fuzzy sets it is isomorphic to a quotient of the lattice l of all convex normal functions under the convolution ordering this quotient identifies two convex normal functions if they agree almost everywhere and their intervals of increase and decrease agree almost everywhere

multiple solutions for a semilinear boundary value problem a computational multiplicity proof abstract we prove the first genuine partial differential equation result on a conjecture concerning the number of solutions of second order elliptic boundary value problems with a nonlinearity which grows superlinearly at the proof makes massive use of computer assistance after approximate solutions have been computed by a numerical mountain pass algorithm combined with a newton iteration to improve accuracy a fixed point argument is used to show the existence of exact solutions close to the approximations

mesh independent a priori bounds for nonlinear elliptic finite difference boundary value problems abstract in this paper we prove mesh independent a priori l bounds for positive solutions of the finite difference boundary value problem h u f x u in h u 0 on h where h is the finite difference laplacian and h is a discretized n dimensional box on the one hand this completes a result of 9 on the asymptotic symmetry of solutions of finite difference boundary value problems on the other hand it is a finite difference version of a critical exponent problem studied in 10 two main results are given one for dimension n 1 and one for the higher dimensional case n 2 the methods of proof differ substantially in these two cases in the 1 dimensional case our method resembles ode techniques in the higher dimensional case the growth rate of the nonlinearity has to be bounded by an exponent p n n 1 where we believe that n n 1 plays the role of a critical exponent our method in this case is based on the use of the discrete hardy sobolev inequality as in 3 and on moser s iteration method we point out that our a priori bounds are in principal explicit

gidas ni nirenberg results for finite difference equations estimates of approximate symmetry are positive solutions of finite difference boundary value problems hu f u in h u 0 on h as symmetric as the domain to answer this question we first show by examples that almost arbitrary non symmetric solutions can be constructed this is in striking difference to the continuous case where by the famous gidas ni nirenberg theorem b gidas wei ming ni l nirenberg symmetry and related problems via the maximum principle comm math phys 68 1979 209 243 positive solutions inherit the symmetry of the underlying domain then we prove approximate symmetry theorems for solutions on equidistantly meshed n dimensional cubes explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer the quality of the estimates depends on whether or not f 0 0 the one dimensional case stands out in two ways the proofs are elementary and the estimates for the defect of symmetry are o h compared to o 1 log h in the higher dimensional case

facilitating score and causal inference trees for large observational studies assessing treatment effects in observational studies is a multifaceted problem that not only involves heterogeneous mechanisms of how the treatment or cause is exposed to subjects known as propensity but also differential causal effects across sub populations we introduce a concept termed the facilitating score to account for both the confounding and interacting impacts of covariates on the treatment effect several approaches for estimating the facilitating score are discussed in particular we put forward a machine learning method called causal inference tree cit to provide a piecewise constant approximation of the facilitating score with interpretable rules cit splits data in such a way that both the propensity and the treatment effect become more homogeneous within each resultant partition causal inference at different levels can be made on the basis of cit together with an aggregated grouping procedure cit stratifies data into strata where causal effects can be conveniently assessed within each besides a feasible way of predicting individual causal effects ice is made available by aggregating ensemble cit models both the stratified results and the estimated ice provide an assessment of heterogeneity of causal effects and can be integrated for estimating the average causal effect ace mean square consistency of cit is also established we evaluate the performance of proposed methods with simulations and illustrate their use with the nsw data in dehejia and wahba 1999 where the objective is to assess the impact of a labor training program the national supportedwork nsw demonstration on post intervention earnings

tree based methods for individualized treatment regimes individualized treatment rules recommend treatments on the basis of individual patient characteristics a high quality treatment rule can produce better patient outcomes lower costs and less treatment burden if a treatment rule learned from data is to be used to inform clinical practice or provide scientific insight it is crucial that it be interpretable clinicians may be unwilling to implement models they do not understand and black box models may not be useful for guiding future research the canonical example of an interpretable prediction model is a decision tree we propose a method for estimating an optimal individualized treatment rule within the class of rules that are representable as decision trees the class of rules we consider is interpretable but expressive a novel feature of this problem is that the learning task is unsupervised as the optimal treatment for each patient is unknown and must be estimated the proposed method applies to both categorical and continuous treatments and produces favourable marginal mean outcomes in simulation experiments we illustrate it using data from a study of major depressive disorder

a symmetry theorem and applications to nonlinear partial differential equations on donne des applications d un nouveau theoreme sur les proprietes de symetrie des solutions des equations differentielles non lineaires de la forme lu f x u dans u 0 ou u n 0 ou est une region bornee ou non bornee de r n

analytical approach of the symmetry sharp supercritical hardy sobolev inequalities and applications abstract we consider the optimal hardy sobolevinequality on smooth bounded symmetric domains of the euclidean space without any assumption concerning the shape of the boundary i e some convexity confirming that the symmetry of a domain is an intrinsic property characterizing both the domain itself and its boundary our model domain is the solid torus because of its particular interest in terms both of the geometry and of the analysis we apply the results of the above study to solve the supercritical problem p p u a x u p 1 f x u p s 1 x s u 0 on t u 0 on t 1 p 2 0 s p and p s p 2 s 2 p and some variants of it

finite difference approximations to the dirichlet problem for elliptic systems the dirichlet problem foru u 1 u n delta u f x u 0in omega u 0on gamma partial omega wheref f 1 f n is discretized in the usual way h mesh size delta h u f x u 0in omega h u 0on gamma h we consider variousmonotone convergent iterative schemes among others they can be used together with estimation theorems for upper and lower solutions to show uniqueness for solutions of 2 numerical results are given for the system delta u u a bu c upsilon 0 delta upsilon upsilon d eu f upsilon 0 from mathematical biology two competing species it is shown that there is a unique positive solution for certain values of the positive parametersa f this result is crucial for the asymptotic behavior of solutions of the corresponding parabolic system ast

quasi oracle estimation of heterogeneous treatment effects flexible estimation of heterogeneous treatment effects lies at the heart of many statistical challenges such as personalized medicine and optimal resource allocation in this paper we develop a general class of two step algorithms for heterogeneous treatment effect estimation in observational studies we first estimate marginal effects and treatment propensities in order to form an objective function that isolates the causal component of the signal then we optimize this data adaptive objective function our approach has several advantages over existing methods from a practical perspective our method is flexible and easy to use in both steps we can use any loss minimization method e g penalized regression deep neural networks or boosting moreover these methods can be fine tuned by cross validation meanwhile in the case of penalized kernel regression we show that our method has a quasi oracle property even if the pilot estimates for marginal effects and treatment propensities are not particularly accurate we achieve the same error bounds as an oracle who has a priori knowledge of these two nuisance components we implement variants of our approach based on penalized regression kernel ridge regression and boosting in a variety of simulation setups and find promising performance relative to existing baselines

hardy inequalities for robin laplacians abstract in this paper we establish a hardy inequality for laplace operators with robin boundary conditions for convex domains in particular we show explicitly how the corresponding hardy weight depends on the coefficient of the robin boundary conditions we also study several extensions to non convex and unbounded domains

the covariance inflation criterion for adaptive model selection we propose a new criterion for model selection in prediction problems the covariance inflation criterion adjusts the training error by the average covariance of the predictions and responses when the prediction rule is applied to permuted versions of the data set this criterion can be applied to general prediction problems e g regression or classification and to general prediction rules e g stepwise regression tree based models and neural nets as a by product we obtain a measure of the effective number of parameters used by an adaptive procedure we relate the covariance inflation criterion to other model selection procedures and illustrate its use in some regression and classification problems we also revisit the conditional bootstrap approach to model selection

estimation and inference of heterogeneous treatment effects using random forests many scientific and engineering challenges ranging from personalized medicine to customized marketing recommendations require an understanding of treatment effect heterogeneity in this article we

subgroup identification using covariate adjusted interaction trees we consider the problem of identifying subgroups of participants in a clinical trial that have enhanced treatment effect recursive partitioning methods that recursively partition the covariate space based on some measure of between groups treatment effect difference are popular for such subgroup identification the most commonly used recursive partitioning method the classification and regression tree algorithm first creates a large tree by recursively partitioning the covariate space using some splitting criteria and then selects the final tree from all the subtrees of the large tree in the context of subgroup identification calculation of the splitting criteria and the evaluation measure used for final tree selection rely on comparing differences in means between the treatment and control arm when covariates are prognostic for the outcome covariate adjusted estimators have the ability to improve efficiency compared to using differences in averages between the treatment and control group this manuscript develops two covariate adjusted estimators that can be used to both make splitting decisions and for final tree selection the performance of the resulting covariate adjusted recursive partitioning algorithm is evaluated using simulations and by analyzing a clinical trial that evaluates if motivational interviews improve treatment engagement for substance abusers

identification of subgroups with differential treatment effects for longitudinal and multiresponse variables we describe and evaluate a regression tree algorithm for finding subgroups with differential treatments effects in randomized trials with multivariate outcomes the data may contain missing values in the outcomes and covariates and the treatment variable is not limited to two levels simulation results show that the regression tree models have unbiased variable selection and the estimates of subgroup treatment effects are approximately unbiased a bootstrap calibration technique is proposed for constructing confidence intervals for the treatment effects the method is illustrated with data from a longitudinal study comparing two diabetes drugs and a mammography screening trial comparing two treatments and a control copyright 2016 john wiley sons ltd

hardy sobolev inequalities in unbounded domains and heat kernel estimates we deal with domains with infinite inner radius more precisely we introduce a new geometric assumption on an exterior domain rn n 3 i e complement of smooth compact domain not containing the origin under this assumption we prove the hardy inequality with optimal constant involving the distance to the boundary in addition in the case n 4 we improve this inequality by adding a critical sobolev norm furthermore we investigate the singular case n 3 and we show that under some additional geometric assumption on the hardy inequality can be improved by adding a sobolev type term with critical exponent also we prove some hardy sobolev type inequalities without any geometric assumptions on which are of independent interest finally we prove harnack inequality up to the boundary for the positive solutions of the problem ut u 14udist2 x and we prove heat kernel estimates for small times

hardy s inequality for the stokes problem abstract sharp constant in generalized hardy s inequality for the stokes problem is obtained in the case of arbitrary convex domain in rn it coincides with the same constant in the classical one dimensional case

greedy outcome weighted tree learning of optimal personalized treatment rules summary we propose a subgroup identification approach for inferring optimal and interpretable personalized treatment rules with high dimensional covariates our approach is based on a two step greedy tree algorithm to pursue signals in a high dimensional space in the first step we transform the treatment selection problem into a weighted classification problem that can utilize tree based methods in the second step we adopt a newly proposed tree based method known as reinforcement learning trees to detect features involved in the optimal treatment rules and to construct binary splitting rules the method is further extended to right censored survival data by using the accelerated failure time model and introducing double weighting to the classification trees the performance of the proposed method is demonstrated via simulation studies as well as analyses of the cancer cell line encyclopedia ccle data and the tamoxifen breast cancer data

the algebra of fuzzy truth values the purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type 2 fuzzy sets

on measuring and correcting the effects of data mining and model selection abstract in the theory of linear models the concept of degrees of freedom plays an important role this concept is often used for measurement of model complexity for obtaining an unbiased estimate of the error variance and for comparison of different models i have developed a concept of generalized degrees of freedom gdf that is applicable to complex modeling procedures the definition is based on the sum of the sensitivity of each fitted value to perturbation in the corresponding observed value the concept is nonasymptotic in nature and does not require analytic knowledge of the modeling procedures the concept of gdf offers a unified framework under which complex and highly irregular modeling procedures can be analyzed in the same way as classical linear models by using this framework many difficult problems can be solved easily for example one can now measure the number of observations used in a variable selection process different modeling procedures such as a tree based regression and a

sharp hardy inequalities on the solid torus abstract in this paper we establish the classical hardy inequality in the solid torus and some variants of it the general idea is to use the fact that sobolev embeddings can be improved in the presence of symmetries in all cases using techniques that exploit the symmetry presented by the solid torus we calculate the displayed best constants and we prove that they are the same as the standard hardy best constants which appear in convex domains although the solid torus is not convex

the finite element approximation of semilinear elliptic partial differential equations with critical exponents in the cube we consider the finite element solution of the parameterized semilinear elliptic equation delta u lambda u u 5 0 u 0 where u is defined in the unit cube and is 0 on the boundary of the cube this equation is important in analysis and it is known that there is a value lambda 0 0 such that no solutions exist for lambda lambda 0 and for the form of the spurious numerical solutions which are known to exist when lambda lambda 0 these estimates are then used to post process the numerical results to obtain a sharp estimate for lambda 0 which agrees with the conjectured value

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

rearrangement inequalities in the discrete setting and some applications abstract we prove extended hardy littlewood riesz and polya szeg inequalities in the discrete case we also establish cases of equality in these inqualities and give some applications of our result

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

a geometric characterization of a sharp hardy inequality abstract in this paper we prove that the distance function of an open connected set in r n 1 with a c 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex we then prove that hardy inequalities with a sharp constant hold on weakly mean convex c 2 domains moreover we show that the weakly mean convexity condition cannot be weakened we also prove various improved hardy inequalities on mean convex domains along the line of brezis and marcus 1997 7

a geometrical version of hardy s inequality we prove a version of hardy s type inequality in a domain omega subset of r n which involves the distance to the boundary and the volume of omega in particular we obtain a result which gives a positive answer to a question asked by h brezis and m marcus

subgroups from regression trees with adjustment for prognostic effects and postselection inference identification of subgroups with differential treatment effects in randomized trials is attracting much attention many methods use regression tree algorithms this article addresses 2 important questions arising from the subgroups how to ensure that treatment effects in subgroups are not confounded with effects of prognostic variables and how to determine the statistical significance of treatment effects in the subgroups we address the first question by selectively including linear prognostic effects in the subgroups in a regression tree model the second question is more difficult because it falls within the subject of postselection inference we use a bootstrap technique to calibrate normal theory t intervals so that their expected coverage probability averaged over all the subgroups in a fitted model approximates the desired confidence level it can also provide simultaneous confidence intervals for all subgroups the first solution is implemented in the guide algorithm and is applicable to data with missing covariate values 2 or more treatment arms and outcomes subject to right censoring bootstrap calibration is applicable to any subgroup identification method it is not restricted to regression tree models two real examples are used for illustration a diabetes trial where the outcomes are completely observed but some covariate values are missing and a breast cancer trial where the outcome is right censored

subgroup detection and sample size calculation with proportional hazards regression for survival data in this paper we propose a testing procedure for detecting and estimating the subgroup with an enhanced treatment effect in survival data analysis here we consider a new proportional hazard model that includes a nonparametric component for the covariate effect in the control group and a subgroup treatment interaction effect defined by a change plane we develop a score type test for detecting the existence of the subgroup which is doubly robust against misspecification of the baseline effect model or the propensity score but not both under mild assumptions for censoring when the null hypothesis of no subgroup is rejected the change plane parameters that define the subgroup can be estimated on the basis of supremum of the normalized score statistic the asymptotic distributions of the proposed test statistic under the null and local alternative hypotheses are established on the basis of established asymptotic distributions we further propose a sample size calculation formula for detecting a given subgroup effect and derive a numerical algorithm for implementing the sample size calculation in clinical trial designs the performance of the proposed approach is evaluated by simulation studies an application to an aids clinical trial data is also given for illustration

subgroup finding via bayesian additive regression trees we provide a bayesian decision theoretic approach to finding subgroups that have elevated treatment effects our approach separates the modeling of the response variable from the task of subgroup finding and allows a flexible modeling of the response variable irrespective of potential subgroups of interest we use bayesian additive regression trees to model the response variable and use a utility function defined in terms of a candidate subgroup and the predicted response for that subgroup subgroups are identified by maximizing the expected utility where the expectation is taken with respect to the posterior predictive distribution of the response and the maximization is carried out over an a priori specified set of candidate subgroups our approach allows subgroups based on both quantitative and categorical covariates we illustrate the approach using simulated data set study and a real data set copyright 2017 john wiley sons ltd

very weak solutions with boundary singularities for semilinear elliptic dirichlet problems in domains with conical corners let rn be a bounded lipschitz domain with a cone like corner at 0 we prove existence of at least two positive unbounded very weak solutions of the problem u up in u 0 on which have a singularity at 0 for any p slightly bigger that the generalized brezis turner exponent p on an example of a planar polygonal domain the actual size of the p interval on which the existence result holds is computed the solutions are found variationally as perturbations of explicitly constructed singular solutions in cones this approach also makes it possible to find numerical approximations of the two very weak solutions on following a gradient flow of a suitable functional and using the mountain pass algorithm two dimensional examples are presented

subgroup analysis via recursive partitioning subgroup analysis is an integral part of comparative analysis where assessing the treatment effect on a response is of central interest its goal is to determine the heterogeneity of the treatment effect across subpopulations in this paper we adapt the idea of recursive partitioning and introduce an interaction tree it procedure to conduct subgroup analysis the it procedure automatically facilitates a number of objectively defined subgroups in some of which the treatment effect is found prominent while in others the treatment has a negligible or even negative effect the standard cart breiman et al 1984 methodology is inherited to construct the tree structure also in order to extract factors that contribute to the heterogeneity of the treatment effect variable importance measure is made available via random forests of the interaction trees both simulated experiments and analysis of census wage data are presented for illustration

hardy inequalities for fractional integrals on general domains abstract we prove a sharp hardy inequality for fractional integrals for functions that are supported in a general domain the constant is the same as the one for the half space and hence our result settles a recent conjecture of bogdan and dyda

hardy inequalities for p laplacians with robin boundary conditions in this paper we study the best constant in a hardy inequality for the p laplace operator on convex domains with robin boundary conditions we show in particular that the best constant equals p

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

a regression tree approach to identifying subgroups with differential treatment effects in the fight against hard to treat diseases such as cancer it is often difficult to discover new treatments that benefit all subjects for regulatory agency approval it is more practical to identify subgroups of subjects for whom the treatment has an enhanced effect regression trees are natural for this task because they partition the data space we briefly review existing regression tree algorithms then we introduce three new ones that are practically free of selection bias and are applicable to data from randomized trials with two or more treatments censored response variables and missing values in the predictor variables the algorithms extend the guide approach by using three key ideas i treatment as a linear predictor ii chi squared tests to detect residual patterns and lack of fit and iii proportional hazards modeling via poisson regression importance scores with thresholds for identifying influential variables are obtained as by products a bootstrap technique is used to construct confidence intervals for the treatment effects in each node the methods are compared using real and simulated data

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in