interpolation operators in orlicz sobolev spaces we study classical interpolation operators for finite elements like the scott zhang operator in the context of orlicz sobolev spaces furthermore we show estimates for these operators with respect to quasi norms which appear in the study of systems of p laplace type

a class of elliptic partial differential equations with exponential nonlinearities on demontre l existence d au moins une infinite denombrable de familles a 1 parametre de solutions non triviales continues de holder pour une classe de problemes aux valeurs limites de dirichlet d ordre 2 fortement non lineaires a 2 dimensions et plus

self averaging in the statistical mechanics of some lattice models we discuss self averaging of thermodynamic properties in some random lattice models in particular we investigate when self averaging in an almost sure sense of the free energy implies self averaging of the energy and heat capacity and we discuss the connection between self averaging in the almost sure sense and self averaging in an l p sense under quite general conditions we show that the average of the finite size heat capacity converges to the second derivative of the limiting quenched average free energy we consider the application of these ideas to the problem of adsorption of a random copolymer at a surface and to some related systems

optimal orlicz domains in sobolev embeddings into marcinkiewicz spaces abstract in this paper we present a method for determining whether there exists a largest orlicz space l a satisfying the sobolev embedding w m l a y where y stands for an arbitrary so called marcinkiewicz endpoint space the tool developed in this work enables us to investigate the optimality of orlicz domain spaces in sobolev embeddings and also in sobolev trace embeddings on domains in r n with various regularity

affine orlicz polya szego principle for log concave functions abstract the affine l p polya szego principle significantly strengthens the classical polya szego principle it is an open problem whether there exists an affine orlicz polya szego principle which includes the affine l p polya szego principle as special case in this paper an affine orlicz polya szego principle for log concave functions is established by using functional steiner symmetrizations

population intervention causal effects based on stochastic interventions estimating the causal effect of an intervention on a population typically involves defining parameters in a nonparametric structural equation model pearl 2000 causality models reasoning and inference in which the treatment or exposure is deterministically assigned in a static or dynamic way we define a new causal parameter that takes into account the fact that intervention policies can result in stochastically assigned exposures the statistical parameter that identifies the causal parameter of interest is established inverse probability of treatment weighting iptw augmented iptw a iptw and targeted maximum likelihood estimators tmle are developed a simulation study is performed to demonstrate the properties of these estimators which include the double robustness of the a iptw and the tmle an application example using physical activity data is presented

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

the thermodynamic limit for long range random systems long range spin systems with random interactions are considered a simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape provided there is no external field the argument is valid for both classical and quantum spin systems and can be applied to a spins randomly distributed on a lattice and interacting via dipolar interactions and b spin systems with potentials of the formj x 1 x 2 x 1 x 2 d where thej x 1 x 2 are independent random variables with mean zero d is the dimension and 1 2 the key to the proof is a multidimensional subadditive ergodic theorem as a corollary we show that for random ferromagnets the correlation length is a nonrandom quantity

nonparametric methods for doubly robust estimation of continuous treatment effects summary continuous treatments e g doses arise often in practice but many available causal effect estimators are limited by either requiring parametric models for the effect curve or by not allowing doubly robust covariate adjustment we develop a novel kernel smoothing approach that requires only mild smoothness assumptions on the effect curve and still allows for misspecification of either the treatment density or outcome regression we derive asymptotic properties and give a procedure for data driven bandwidth selection the methods are illustrated via simulation and in a study of the effect of nurse staffing on hospital readmissions penalties

extent of self averaging in the statistical mechanics of finite random copolymers we investigate the extent of thermodynamic self averaging in a coloured self avoiding walk model of finite random copolymer adsorption we derive a bound on the extent of self averaging as a function of the length of the self avoiding walk

smooth manifolds for semilinear wave equations on r2 on the existence of almost periodic breathers abstract in this paper we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension these solutions represent time almost periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity our nonexistence results are of two kinds we first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial whenever their spectrum does not interact with the nonlinearity in a certain sense we also prove non existence results without requiring any spectral conditions but this is at the expense of having to impose more stringent conditions on the nonlinearities such as certain convexity properties in the vicinity of the equilibrium solution finally we prove an existence result for time almost periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution to accomplish this we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time we then embed that dynamical system into an appropriate banach space of time almost periodic functions so that the one parameter family of stable and unstable manifolds which we construct carry the solutions that we seek the major difficulty to overcome in our construction is a small divisor problem it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps we discuss several examples and we also stress some important qualitative differences which distinguish the almost periodic case from the purely periodic one

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

self averaging sequences in the statistical mechanics of random copolymers a constructive proof is given that certain problems in the statistical mechanics of random copolymers are thermodynamically self averaging the proof relies on the use of normal numbers and potentially gives a method for calculating expectations in the quenched averaged ensemble in particular we give a new proof that adsorbing random copolymers are self averaging

an optimal interpolation theorem of marcinkiewicz type in orlicz spaces abstract an extension of the marcinkiewicz interpolation theorem is proved yielding a necessary and sufficient condition for every quasilinear operator satisfying given endpoint estimates of weak type to be bounded from an orlicz space into another

interacting self avoiding walks and polygons in three dimensions self interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the point we consider self avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon we prove that these walks and polygons have the same limiting free energy if the interactions between nearest neighbour vertices are repulsive the attractive interaction regime is investigated using monte carlo methods and we find evidence that the limiting free energies are also equal here in particular this means that these models have the same point in the asymptotic limit the dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

self averaging in random self interacting polygons we prove that a lattice polygon model of a self interacting random ring copolymer is thermodynamically self averaging the proof is quite general and applies to any two body potential linear in the numbers of contacts of different types

population intervention models in causal inference we propose a new causal parameter which is a natural extension of existing approaches to causal inference such as marginal structural models modelling approaches are proposed for the difference between a treatment specific counterfactual population distribution and the actual population distribution of an outcome in the target population of interest relevant parameters describe the effect of a hypothetical intervention on such a population and therefore we refer to these models as population intervention models we focus on intervention models estimating the effect of an intervention in terms of a difference and ratio of means called risk difference and relative risk if the outcome is binary we provide a class of inverse probability of treatment weighted and doubly robust estimators of the causal parameters in these models the finite sample performance of these new estimators is explored in a simulation study

a self avoiding walk model of random copolymer adsorption there is an error in the proof of lemma 3 4 which can be remedied by using 3 22 and 3 13 directly the proof of lemma 3 2 can also be simplified please see pdf for details

a class of orlicz sobolev spaces with applications to variational problems involving nonlinear hill s equations necessary and sufficient conditions are proved for a b 2 young function g with independent variable t to be convex resp concave in t2 in terms of inequalities between the second derivative of g and the first derivative of its legendre transform g with independent variable s it is then proven that a young function g is convex resp concave in t2 if and only if g is concave resp convex in s2 these results along with another set of inequalities for functions g convex resp concave in t2 allow the proof of the uniform convexity and thereby of the reflexivity with respect to luxemburg s norm f g inf k 0 d g f k 1 of the orlicz space lg over an open domain rn with lebesgue measure d when applied to g t t pp and g s s p p with p 1 p 1 1 the preceding results lead to the shortest proof to date of two clarkson s inequalities and of the reflexivity of lp spaces for 1 p finally some of these results are used to solve by direct methods variational problems associated with the existence question of periodic orbits for a class of nonlinear hill s equations these variational problems are formulated on suitable orlicz sobolev spaces wmlg and thereby allow for nonlinear terms which may grow faster than any power of the variable

negative orlicz sobolev norms and strongly nonlinear systems in fluid mechanics abstract we prove a version of the negative norm theorem in orlicz sobolev spaces a study of boundedness properties of the bogovski operator between orlicz spaces is a crucial step of independent interest in our approach applications to the problem of pressure reconstruction for non newtonian fluids governed by constitutive laws which are not necessarily of power type are presented a key inequality for a numerical analysis of the underlying elliptic system is also derived

the collapse transition for the lattice trees the authors consider a lattice tree model for the collapse of dilute branched polymers in the good solvent regime in which the collapse is driven by a near neighbour contact fugacity they describe some rigorous results including bounds for the temperature dependence of the reduced limiting free energy and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature tc and the cross over exponent phi 0 in two and three dimensions they find phi 0 0 60 or 0 03 d 2 and phi 0 0 82 or 0 03 d 3 finally they speculate on a possible roughening transition which may occur at a temperature tr tc

differentiability properties of functions from higher order orlicz sobolev spaces abstract we analyze pointwise differentiability properties of functions from local orlicz sobolev spaces w loc m a on an open subset of r n n 2 a necessary and sufficient condition on the order of differentiation m and on the young function a is exhibited for any function in w loc m a to admit a differential of order m a e in an optimal approximate differentiability property in integral and norm form is established when such a condition fails

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

robust causal inference with continuous instruments using the local instrumental variable curve instrumental variables are commonly used to estimate effects of a treatment afflicted by unmeasured confounding and in practice instruments are often continuous e g measures of distance or treatment preference however available methods for continuous instruments have important limitations they either require restrictive parametric assumptions for identification or else rely on modelling both the outcome and the treatment process well and require modelling effect modification by all adjustment covariates in this work we develop the first semiparametric doubly robust estimators of the local instrumental variable effect curve i e the effect among those who would take treatment for instrument values above some threshold and not below in addition to being robust to misspecification of either the instrument or treatment or outcome processes our approach also incorporates information about the instrument mechanism and allows for flexible data adaptive estimation of effect modification we discuss asymptotic properties under weak conditions and use the methods to study infant mortality effects of neonatal intensive care units with high versus low technical capacity using travel time as an instrument

a characterization of minimal orlicz sobolev norms in the affine class abstract the orlicz sobolev norm is invariant only under rigid motions but is not invariant under volume preserving affine transformations on r n in this paper we will consider the orlicz sobolev norm in the affine class and characterize its minimum

extensions of fisher information and stam s inequality we explain how the classical notions of fisher information of a random variable and fisher information matrix of a random vector can be extended to a much broader setting we also show that stam s inequality for fisher information and shannon entropy as well as the more generalized versions proved earlier by the authors are all special cases of more general sharp inequalities satisfied by random vectors the extremal random vectors which we call generalized gaussians contain gaussians as a limiting case but are noteworthy because they are heavy tailed

estimation of the effect of interventions that modify the received treatment motivated by a study of surgical operating time and post operative outcomes for lung cancer we consider the estimation of causal effects of continuous point exposure treatments to investigate causality the standard paradigm postulates a series of treatment specific counterfactual outcomes and establishes conditions under which we may learn about them from observational study data while many choices are possible causal effects are typically defined in terms of variation of the mean of counterfactual outcomes in hypothetical worlds in which specific treatment strategies are applied to all individuals for example one might compare two worlds one where each individual receives some specific dose and a second where each individual receives some other dose for our motivating study defining causal effects in this way corresponds to hypothetical interventions that could not conceivably be implemented in the real world in this work we consider an alternative complimentary framework that investigates variation in the mean of counterfactual outcomes under hypothetical treatment strategies where each individual receives a treatment dose corresponding to that actually received but modified in some pre specified way quantification of this variation is defined in terms of contrasts for specific interventions as well as in terms of the parameters of a new class of marginal structural mean models within this framework we propose three estimators an outcome regression estimator an inverse probability of treatment weighted estimator and a doubly robust estimator we illustrate the methods with an analysis of the motivating data copyright 2013 john wiley sons ltd

sobolev embeddings in orlicz and lorentz spaces with measures abstract embedding theorems for orlicz sobolev spaces into orlicz or orlicz lorentz spaces in domains in r n endowed with a frostman measure are offered parallel embeddings for lorentz sobolev spaces into generalized lorentz spaces are also established the relevant embeddings yield an optimal target norm in the relevant classes whenever it does exist in particular various results available in the literature are improved by special choices of the function spaces and of the measure

optimal sobolev norms in the affine class abstract optimal sobolev norms under volume preserving affine transformations are considered it turns out that this minimal transform is equivalent to the p 2 fisher information matrix defined by lutwak lv yang and zhang furthermore some analytic inequalities regarding to the l p affine and l p sine energies for the optimal function are investigated

lipschitz regularity for a general class of quasilinear elliptic equations in convex domains abstract we establish a local lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with neumann boundary condition in bounded convex domains

semiparametric efficiency and its implication on the design and analysis of group sequential studies abstract authors have shown that the time sequential joint distributions of many statistics used to analyze data arising from group sequential time to event and longitudinal studies are multivariate normal with an independent increments covariance structure in theorem 1 of this article we demonstrate that this limiting distribution arises naturally when one uses an efficient test statistic to test a single parameter in a semiparametric or parametric model because we are able to think of many of the statistics in the literature in this fashion the limiting distribution under investigation is just a special case of theorem 1 using this general structure we then develop an information based design and monitoring procedure that can be applied to any type of model for any type of group sequential study provided that there is a unique parameter of interest that can be efficiently tested

parametric g formula implementations for causal survival analyses the g formula can be used to estimate the survival curve under a sustained treatment strategy two available estimators of the g formula are noniterative conditional expectation and iterative conditional expectation we propose a version of the iterative conditional expectation estimator and describe its procedures for deterministic and random treatment strategies also because little is known about the comparative performance of noniterative and iterative conditional expectation estimators we explore their relative efficiency via simulation studies our simulations show that in the absence of model misspecification and unmeasured confounding our proposed iterative conditional expectation estimator and the noniterative conditional expectation estimator are similarly efficient and that both are at least as efficient as the classical iterative conditional expectation estimator we describe an application of both noniterative and iterative conditional expectation to answer when to start treatment questions using data from the hiv causal collaboration

orlicz sobolev boundary trace embeddings we prove a sharp trace embedding theorem for orlicz sobolev spaces into orlicz spaces on the boundary an optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

vanishing exponential integrability for functions whose gradients belong to ln log e l α if the gradient of u x is nth power locally integrable on euclidean n space then the integral average over a ball b of the exponential of a constant multiple of u x ub n n 1 ub average of u over b tends to 1 as the radius of b shrinks to zero for quasi almost all center points this refines a result of n trudinger 1967 we prove here a similar result for the class of gradients in ln log e l 0 n 1 the results depend on a capacitary strong type inequality for these spaces

self averaging in models of random copolymer collapse we give a set of conditions under which a system is thermodynamically self averaging and show that several lattice models of interacting copolymers satisfy these conditions we prove this result for a general potential which is linear in the numbers of various types of contacts and show that this includes two potentials which have previously been used in models of random interacting linear copolymers

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

one parameter families of sturm liouville eigenfunctions and of periodic solutions for a class of nonlinear hill s equations i introduction in cyclic accelerators and in intersecting storage rings the orbits of the particles are periodically perturbed by strongly nonlinear electromagnetic forces whenever the two counterrotating beams undergo head on collisions at the interaction regions around the machine such perturbations in turn generate nonlinear transverse oscillations about the ideal orbits which can generally be described by nonlinear hill s equations of the form z 8 nz 8 f z d i 1 see

localization transition for a polymer near an interface consider the directed process i s i where the second component is simple random walk on mathbb z s 0 0 define a transformed path measure by weighting each n step path with a factor exp lambda sum 1 leq i leq n omega i h sign s i here omega i i geq 1 is an i i d sequence of random variables taking values pm 1 with probability 1 2 acting as a random medium while lambda in 0 infty and h in 0 1 are parameters the weight factor has a tendency to pull the path towards the horizontal because it favors the combinations s i 0 omega i 1 and s i 0 omega i 1 the transformed path measure describes a heteropolymer consisting of hydrophylic and hydrophobic monomers near an oil water interface we study the free energy of this model as n to infty and show that there is a critical curve lambda to h c lambda where a phase transition occurs between localized and delocalized behavior in the vertical direction we derive several properties of this curve in particular its behavior for lambda downarrow 0 to obtain this behavior we prove that as lambda h downarrow 0 the free energy scales to its brownian motion analogue