conformal iterated function systems with applications to the geometry of continued fractions in this paper we obtain some results about general conformal iterated function systems we obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior we then apply these results to the set of values of real continued fractions with restricted entries we pay special attention to the hausdorff and packing measures of these sets we also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries

when should epidemiologic regressions use random coefficients summary regression models with random coefficients arise naturally in both frequentist and bayesian approaches to estimation problems they are becoming widely available in standard computer packages under the headings of generalized linear mixed models hierarchical models and multilevel models i here argue that such models offer a more scientifically defensible framework for epidemiologic analysis than the fixed effects models now prevalent in epidemiology the argument invokes an antiparsimony principle attributed to l j savage which is that models should be rich enough to reflect the complexity of the relations under study it also invokes the countervailing principle that you cannot estimate anything if you try to estimate everything often used to justify parsimony regression with random coefficients offers a rational compromise between these principles as well as an alternative to analyses based on standard variable selection algorithms and their attendant distortion of uncertainty assessments these points are illustrated with an analysis of data on diet nutrition and breast cancer

blowup for fractional nls abstract we consider fractional nls with focusing power type nonlinearity i t u s u u 2 u t x r r n where 1 2 s 1 and 0 for s n 2 and 0 2 s n 2 s for s n 2 we prove a general criterion for blowup of radial solutions in r n with n 2 for l 2 supercritical and l 2 critical powers 2 s n in addition we study the case of fractional nls posed on a bounded star shaped domain r n in any dimension n 1 and subject to exterior dirichlet conditions in this setting we prove a general blowup result without imposing any symmetry assumption on u t x for the blowup proof in r n we derive a localized virial estimate for fractional nls in r n which uses balakrishnan s formula for the fractional laplacian s from semigroup theory in the setting of bounded domains we use a pohozaev type estimate for the fractional laplacian to prove blowup

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

the performance of random coefficient regression in accounting for residual confounding greenland 2000 biometrics 56 915 921 describes the use of random coefficient regression to adjust for residual confounding in a particular setting we examine this setting further giving theoretical and empirical results concerning the frequentist and bayesian performance of random coefficient regression particularly we compare estimators based on this adjustment for residual confounding to estimators based on the assumption of no residual confounding this devolves to comparing an estimator from a nonidentified but more realistic model to an estimator from a less realistic but identified model the approach described by gustafson 2005 statistical science 20 111 140 is used to quantify the performance of a bayesian estimator arising from a nonidentified model from both theoretical calculations and simulations we find support for the idea that superior performance can be obtained by replacing unrealistic identifying constraints with priors that allow modest departures from those constraints in terms of point estimator bias this superiority arises when the extent of residual confounding is substantial but the advantage is much broader in terms of interval estimation the benefit from modeling residual confounding is maintained when the prior distributions employed only roughly correspond to reality for the standard identifying constraints are equivalent to priors that typically correspond much worse

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

doubly nonlocal system with hardy littlewood sobolev critical nonlinearity abstract this article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of hardy littlewood sobolev inequality s u u q 2 u v y 2 x y d y u 2 2 u in s v v q 2 v u y 2 x y d y v 2 2 v in u v 0 in r n where is a smooth bounded domain in r n n 2 s s 0 1 s is the well known fractional laplacian 0 n 2 2 n n 2 s is the upper critical exponent in the hardy littlewood sobolev inequality 1 q 2 and 0 are real parameters we study the fibering maps corresponding to the functional associated with p and show that minimization over suitable subsets of nehari manifold renders the existence of at least two non trivial solutions of p for suitable range of and

penalization bias reduction and default priors in logistic and related categorical and survival regressions penalization is a very general method of stabilizing or regularizing estimates which has both frequentist and bayesian rationales we consider some questions that arise when considering alternative penalties for logistic regression and related models the most widely programmed penalty appears to be the firth small sample bias reduction method albeit with small differences among implementations and the results they provide which corresponds to using the log density of the jeffreys invariant prior distribution as a penalty function the latter representation raises some serious contextual objections to the firth reduction which also apply to alternative penalties based on t distributions including cauchy priors taking simplicity of implementation and interpretation as our chief criteria we propose that the log f 1 1 prior provides a better default penalty than other proposals penalization based on more general log f priors is trivial to implement and facilitates mean squared error reduction and sensitivity analyses of penalty strength by varying the number of prior degrees of freedom we caution however against penalization of intercepts which are unduly sensitive to covariate coding and design idiosyncrasies copyright 2015 john wiley sons ltd

nonexistence results for a class of fractional elliptic boundary value problems abstract in this paper we study a class of fractional elliptic problems of the form s u f x u in u 0 in r n where s 0 1 we prove nonexistence of positive solutions when is star shaped and f is supercritical we also derive a nonexistence result for subcritical f in some unbounded domains the argument relies on the method of moving spheres applied to a reformulated problem using the caffarelli silvestre extension caffarelli and silvestre 2007 11 of a solution of the above problem

a global compactness type result for palais smale sequences in fractional sobolev spaces abstract we extend the global compactness result by struwe 1984 to any fractional sobolev spaces h s for 0 s n 2 and r n a bounded domain with smooth boundary the proof is a simple direct consequence of the so called profile decomposition of gerard 1998

the pohozaev identity for the fractional laplacian in this paper we prove the pohozaev identity for the semilinear dirichlet problem delta s u f u in omega u equiv0 in mathbb r n backslash omega here s in 0 1 s is the fractional laplacian in mathbb r n and is a bounded c1 1 domain to establish the identity we use among other things that if u is a bounded solution then u delta s omega is c up to the boundary where x dist x in the fractional pohozaev identity the function u delta s partial omega plays the role that u plays in the classical one surprisingly from a nonlocal problem we obtain an identity with a boundary term an integral over which is completely local as an application of our identity we deduce the nonexistence of nontrivial solutions in star shaped domains for supercritical nonlinearities

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

on critical systems involving fractional laplacian abstract consider the following non local critical system 0 1 s u 1 u 1 u 2 2 u 2 u 2 u v in s v 2 v 2 v 2 2 v 2 u v 2 v in u 0 v 0 in r n where s is fractional laplacian 0 s 1 and all 1 2 1 2 0 2 2 n n 2 s is a fractional sobolev critical exponent n 2 s 1 2 and is an open bounded domain in r n with lipschitz boundary under proper conditions we establish the existence result of the ground state solution to system 0 1

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

large gaps between consecutive prime numbers containing perfect k th powers of prime numbers abstract let k 2 be a fixed natural number we establish the existence of infinitely many pairs of consecutive primes p n p n 1 satisfying p n 1 p n c log p n log 2 p n log 4 p n log 3 p n with c being a fixed positive constant for which the interval p n p n 1 contains the k th power of a prime number

heat kernel estimates for the fractional laplacian with dirichlet conditions we give sharp estimates for the heat kernel of the fractional laplacian with dirichlet condition for a general class of domains including lipschitz domains

p values for composite null models abstract the problem of investigating compatibility of an assumed model with the data is investigated in the situation when the assumed model has unknown parameters the most frequently used measures of compatibility are p values based on statistics t for which large values are deemed to indicate incompatibility of the data and the model when the null model has unknown parameters p values are not uniquely defined the proposals for computing a p value in such a situation include the plug in and similar p values on the frequentist side and the predictive and posterior predictive p values on the bayesian side we propose two alternatives the conditional predictive p value and the partial posterior predictive p value and indicate their advantages from both bayesian and frequentist perspectives

concentration compactness principle for nonlocal scalar field equations with critical growth abstract the aim of this paper is to study a concentration compactness principle for homogeneous fractional sobolev space d s 2 r n for 0 s min 1 n 2 as an application we establish palais smale compactness for the lagrangian associated to the fractional scalar field equation s u f x u for 0 s 1 moreover using an analytic framework based on d s 2 r n we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range

multiple imputation of industry and occupation codes in census public use samples using bayesian logistic regression abstract we describe methods used to create a new census data base that can be used to study comparability of industry and occupation classification systems this project represents the most extensive application of multiple imputation to date and the modeling effort was considerable as well hundreds of logistic regressions were estimated one goal of this article is to summarize the strategies used in the project so that researchers can better understand how the new data bases were created another goal is to show how modifications of maximum likelihood methods were made for the modeling and imputation phases of the project to multiply impute 1980 census comparable codes for industries and occupations in two 1970 census public use samples logistic regression models were estimated with flattening constants for many of the regression models considered the data were too sparse to support conventional maximum likelihood analysis so some alternative had to be employed these methods solve existence and

hypothesis testing in case control studies summary prentice pyke 1979 have shown that valid estimators of the odds ratio parameters in a logistic regression model may be obtained from case control data by fitting the model as if the data had been obtained in a prospective study the resulting estimates of standard errors are also valid asymptotically we extend the prentice pyke result to likelihood ratio tests obtained from fitting a prospective model these tests are asymptotically exact if the hypothesis involves only the odds ratio parameters and are asymptotically conservative if stratum effects are involved

a new perspective on priors for generalized linear models abstract this article deals with specifications of informative prior distributions for generalized linear models our emphasis is on specifying distributions for selected points on the regression surface the prior distribution on regression coefficients is induced from this specification we believe that it is inherently easier to think about conditional means of observables given the regression variables than it is to think about model dependent regression coefficients previous use of conditional means priors seems to be restricted to logistic regression with one predictor variable and to normal theory regression we expand on the idea of conditional means priors and extend these to arbitrary generalized linear models we also consider data augmentation priors where the prior is of the same form as the likelihood we show that data augmentation priors are special cases of conditional means priors with current monte carlo methodology such as importance sampling and gibbs sampling our priors result in

on the structure of the nodal set and asymptotics of least energy sign changing radial solutions of the fractional brezis nirenberg problem abstract in this paper we study the asymptotic and qualitative properties of least energy radial sign changing solutions of the fractional brezis nirenberg problem ruled by the s laplacian in a ball of r n when s 0 1 and n 6 s as usual is the positive parameter in the linear part in u we prove that for sufficiently small such solutions cannot vanish at the origin we show that they change sign at most twice and their zeros coincide with the sign changes moreover when s is close to 1 such solutions change sign exactly once finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a tower of bubbles as 0 i e the positive and negative parts concentrate at the same point with different concentration speeds

chains of large gaps between consecutive primes where log n log n log n log log i n this result has so far not been improved it implies in particular that lim sup d log n co since by the prime number theorem the average size of d is log n this means that the largest gaps exceed the average by a ratio approaching infinity westzynthius lo in 1949 erdgs 4 proposed the problem for a fixed k and n co can k consecutive gaps be simultaneously much larger than log n or more precisely does lim sup min d d log n co erdos proved this for the case k 2 4 in this paper we prove erdos conjecture for all k moreover we obtain the same lower bound for k consecutive gaps as that given by rankin s theorem for single gaps the only difference is that as k increases the constant implied by o decreases

generalized conjugate priors for bayesian analysis of risk and survival regressions le choix pour les analyses bayesiennes portant sur les risques relatifs de distributions a priori conjuguees peut etre tres limitant du fait que la forme de cette distribution depend du parametre de valeur centrale cependant en introduisant separement ce parametre on generalise ces distributions a priori ce qui permet de modeliser une tres large palette d opinions a priori tout en conservant les proprietes de simplicite des calculs propres aux analyses conjuguees cet article illustre les resultats fournis par ces analyses avec des conjuguees generalisees a partir d exemples sur des enquetes cas x temoins dont le but etait d etudier le lien des amenagements en c bles electriques wire codes en zone residentielle et des champs magnetiques avec les cas de leucemie chez l enfant

putting background information about relative risks into conjugate prior distributions in bayesian and empirical bayes analyses of epidemiologic data the most easily implemented prior specifications use a multivariate normal distribution for the log relative risks or a conjugate distribution for the discrete response vector this article describes problems in translating background information about relative risks into conjugate priors and a solution traditionally conjugate priors have been specified through flattening constants an approach that leads to conflicts with the true prior covariance structure for the log relative risks one can however derive a conjugate prior consistent with that structure by using a data augmentation approximation to the true log relative risk prior although a rescaling step is needed to ensure the accuracy of the approximation these points are illustrated with a logistic regression analysis of neonatal death risk

asymptotic distribution of p values in composite null models abstract we investigate the compatibility of a null model h 0 with the data by calculating a p value that is the probability under h 0 that a given test statistic t exceeds its observed value when the null model consists of a single distribution the p value is readily obtained and it has a uniform distribution under h 0 on the other hand when the null model depends on an unknown nuisance parameter one must somehow get rid of e g by estimating it to calculate a p value various proposals have been suggested to remove each yielding a different candidate p value but unlike the simple case these p values typically are not uniformly distributed under the null model in this article we investigate their asymptotic distribution under h 0 we show that when the asymptotic mean of the test statistic t depends on the posterior predictive p value of guttman and rubin and the plug in p value are conservative i e their asymptotic distributions are more concentrated around 1 2 than a uni

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

existence results and bifurcation for nonlocal fractional problems with critical sobolev exponent abstract this paper is concerned with existence and bifurcation of nontrivial solutions for the following critical nonlocal problem l k u f x u u 2 2 u in u 0 in r n where 0 1 is an open bounded subset of r n n 2 with continuous boundary is a positive real parameter 2 2 n n 2 is the fractional critical sobolev exponent and f satisfies suitable growth condition l k is the integrodifferential operator defined as l k u x r n u x y u x y 2 u x k y d y x r n the single solution results extend the main results of servadei et al 2015 3 18 and include theorem 1 1 of servadei 2013 the bifurcation result extends those got by sang 1994 for classical elliptic equations to the case of nonlocal fractional operators and also generalizes the result got by fiscella et al 2016

data augmentation priors for bayesian and semi bayes analyses of conditional logistic and proportional hazards regression data augmentation priors have a long history in bayesian data analysis formulae for such priors have been derived for generalized linear models but their accuracy depends on two approximation steps this note presents a method for using offsets as well as scaling factors to improve the accuracy of the approximations in logistic regression this method produces an exceptionally simple form of data augmentation that allows it to be used with any standard package for conditional logistic or proportional hazards regression to perform bayesian and semi bayes analyses of matched and survival data the method is illustrated with an analysis of a matched case control study of diet and breast cancer

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

prospective analysis of case control data under general multiplicative intercept risk models summary we show that under a general multiplicative intercept model for risk case control data can be analyzed by maximum likelihood as if they had arisen prospectively up to an unidentifiable multiplicative constant which depends on the relative sampling fractions this generalizes earlier work of anderson 1972 1979 by showing that not only the point estimates but also the standard errors based on the observed information matrix are correct when the prospective likelihood is maximized likelihood ratio testing is also valid under this broad class of risk models data on disease status from a much larger cohort are shown to add no information to the estimation of the covariate related parameters

a semi bayes approach to the analysis of correlated multiple associations with an application to an occupational cancer mortality study thomas et al presented the application of empirical bayes methods to the problem of multiple inference in epidemiologic studies one limitation of their approach which they noted was the need to assume exchangeable log relative risk parameters and independent relative risk estimates numerical integration was also required here i generalize their approach to allow for non exchangeable parameters and non independent estimates the resulting method is bayesian in so far as some feature of the prior distribution are specified from prior information but is empirical bayes in so far as some explicit parameters in the prior distribution are estimated from the data estimation is based on approximations to the posterior distribution this allows one to implement the approach with standard software packages for matrix algebra the method is illustrated in an occupational mortality study of 84 exposure cancer associations

existence of entire solutions for schrodinger hardy systems involving two fractional operators abstract this paper deals with the existence of nontrivial nonnegative solutions of schrodinger hardy systems driven by two possibly different fractional laplacian operators via various variational methods the main features of the paper are the presence of the hardy terms and the fact that the nonlinearities do not necessarily satisfy the ambrosetti rabinowitz condition moreover we consider systems including critical nonlinear terms as treated very recently in literature and present radial versions of the main theorems finally we briefly show how to extend the previous results when the fractional laplacian operators are replaced by more general elliptic nonlocal integro differential operators

neumann conditions for the higher order s fractional laplacian δ su with s 1 abstract in this paper we study a variational neumann problem for the higher order fractional laplacian s s 1 in the process we introduce some nonlocal neumann boundary conditions that appear in a natural way from a gauss like integration formula

integration by parts and pohozaev identities for space dependent fractional order operators abstract consider a classical elliptic pseudodifferential operator p on r n of order 2a 0 a 1 with even symbol for example p a x d a where a x d is a second order strongly elliptic differential operator the fractional laplacian a is a particular case for solutions u of the dirichlet problem on a bounded smooth subset r n we show an integration by parts formula with a boundary integral involving d a u where d x dist x this extends recent results of ros oton serra and valdinoci to operators that are x dependent nonsymmetric and have lower order parts we also generalize their formula of pohozaev type that can be used to prove unique continuation properties and nonexistence of nontrivial solutions of semilinear problems an illustration is given with p m 2 a the basic step in our analysis is a factorization of p p p p where we set up a calculus for the generalized pseudodifferential operators p that come out of the construction

on higher dimensional singularities for the fractional yamabe problem a nonlocal mazzeo pacard program we consider the problem of constructing solutions to the fractional yamabe problem which are singular at a given smooth submanifold for which we establish the classical gluing method of mazzeo and pacard j differential geom 1996 for the scalar curvature in the fractional setting this proof is based on the analysis of the model linearized operator which amounts to the study of a fractional order ordinary differential equation ode thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal odes note however that no traditional phase plane analysis is available here instead we first provide a rigorous construction of radial fast decaying solutions by a blowup argument and a bifurcation method then second we use conformal geometry to rewrite this nonlocal ode giving a hint of what a nonlocal phase plane analysis should be third for the linear theory we use complex analysis and some non euclidean harmonic analysis to examine a fractional schrodinger equation with a hardy type critical potential we construct its green s function deduce fredholm properties and analyze its asymptotics at the singular points in the spirit of frobenius method surprisingly enough a fractional linear ode may still have a 2 dimensional kernel as in the second order case