a note on differentiability of the cluster density for independent percolation in high dimensions the cluster density function of independent percolation in ad dimensional lattice is considered for eachn it is shown that p has finitenth leftderivative at critical probabilityp c ifd is sufficiently large this result agrees with the bethe lattice approximation where thenth one sided derivative of p is bounded atp c for alln

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

the survival probability and r point functions in high dimensions in this paper we investigate the survival probability n in high dimensional statistical physical models where n denotes the probability that the model survives up to time n we prove that if the r point functions scale to those of the canonical measure of super brownian motion and if certain self repellence and total population tail bound conditions are satised then n n 2 av where a is the asymptotic expected number of particles alive at timen andv is the vertex factor of the model our results apply to spread out lattice trees above 8 dimensions spread out oriented percolation above 4 1 dimensions and the spread out contact process above 4 1 dimensions in the case of oriented percolation this reproves a result by the rst author den hollander and slade which was proved using heavy lace expansion arguments at the cost of losing explicit error estimates we further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n our proofs are based on simple weak convergence arguments

the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

mean field behavior for the survival probability and the percolation point to surface connectivity we consider the critical survival probability up to time t for oriented percolation and the contact process and the point to surface of the ball of radius t connectivity for critical percolation let t denote both quantities we prove in a unifled fashion that if t exhibits a power law and both the two point function and its certain restricted version exhibit the same mean fleld behavior then t t i1 for the time oriented models with d 4 and t t i2 for percolation with d 7

non negative solutions of the evolution p laplacian equation initial traces and cauchy problem when 1 p 2 from the introduction we continue here the investigation initiated earlier by us trans amer math soc 314 1989 no 1 187 224 concerning the solvability of the cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation 1 ut div du p 2 du 0 in st rn 0 t 0 2 both in terms of results and techniques the main difference stems from the fact that unlike solutions in the case p 2 solutions of 1 are not in general locally bounded

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

the regularity for nonlinear parabolic systems of p laplacian type with critical growth abstract we study the holder regularity of weak solutions to the evolutionary p laplacian system with critical growth on the gradient we establish a natural criterion for proving that a small solution and its gradient are locally holder continuous almost everywhere actually our regularity result recovers the classical result in the case p 2 16 and can be applied to study the regularity of the heat flow for m dimensional h systems as well as the m harmonic flow

existence of solutions for nonlinear parabolic problem with p x growth abstract in this paper we study the existence of solutions for nonlinear parabolic initial boundary value problem with p x growth conditions in the space w 1 x l p x q l 0 t l 2 and give an existence theorem of weak solutions for the following equation u t a u f where a u div a x t u u a 0 x t u u a x t u u and a 0 x t u u satisfy p x growth conditions with respect to u and u and f w 1 x l q x q

estimation of reliability in field performance studies likelihood based methods are developed for the analysis of field performance studies with particular attention centered on the estimation of regression coefficients in parametric models failure record data are those in which the time to failure and the regressor variables are observed only for those items that fail in some prespecified follow up or warranty period 0 t it is noted that for satisfactory inference about baseline failure rates or regression effects it is usually necessary to supplement the failure record data either by incorporating specific prior information about x or by taking a supplementary sample of items that survive t general methods are outlined and specific formulas for various likelihood based methods are obtained when the failure time model is exponential or weibull in these models the methods are compared with respect to asymptotic efftciency of estimation several extensions to more complicated sampling plans are considered

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

global regularity in nikolskij spaces for elliptic equations with p structure on polyhedral domains abstract nonlinear elliptic equations with p structure on non convex polyhedral domains under homogeneous dirichlet boundary values are considered global regularity in fractional order nikolskij and sobolev spaces is proved

incipient infinite percolation clusters in 2d we study several kinds of large critical percolation clusters in two dimensions we show that from the microscopic lattice scale perspective these clusters can be described by kesten s incipient infinite cluster iic as was conjectured by aizenman more specifically we establish this for incipient spanning clusters large clusters in a finite box and the inhomogeneous model of chayes chayes and durrett our results prove the equivalence of several natural definitions of the iic we also show that for any k 1 the difference in size between the kth and k 1 st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity in addition the distribution of the chayes chayes durrett cluster is shown to be singular with respect to the iic

random graph asymptotics on high dimensional tori ii volume diameter and mixing time for critical bond percolation on high dimensional torus this paper proves sharp lower bounds on the size of the largest cluster removing a logarithmic correction in the lower bound in heydenreich and van der hofstad comm math phys 270 2 335 358 2007 this improvement finally settles a conjecture by aizenman nuclear phys b 485 3 551 582 1997 about the role of boundary conditions in critical high dimensional percolation and it is a key step in deriving further properties of critical percolation on the torus indeed a criterion of nachmias and peres ann probab 36 4 1267 1286 2008 implies appropriate bounds on diameter and mixing time of the largest clusters we further prove that the volume bounds apply also to any finite number of the largest clusters finally we show that any weak limit of the largest connected component is non degenerate which can be viewed as a significant sign of critical behavior the main conclusion of the paper is that the behavior of critical percolation on the high dimensional torus is the same as for critical erd s renyi random graphs

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

fitting prospective regression models to case control data summary we consider fitting prospective regression models to data obtained by case control or response selective sampling from a finite population with known population totals in each response category maximum likelihood estimation is developed and compared with two pseudo likelihood approaches the relative efficiencies of the methods are explored in the special case of estimating the parameters of the proportional odds model for ordinal responses for such applications the method hsieh manski mcfadden 1985 call conditional maximum likelihood is shown to be essentially as efficient as maximum likelihood the latter is considerably more difficult to implement in contrast the use of a weighted estimate of the prospective likelihood can lead to a substantial loss of efficiency

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

parametric models for response biased sampling summary suppose that subjects in a population follow the model f y lx 6 where y denotes a response x denotes a vector of covariates and 6 is the parameter to be estimated we consider response biased sampling in which a subject is observed with a probability which is a function of its response such response biased sampling frequently occurs in econometrics epidemiology and survey sampling the semiparametric maximum likelihood estimate of 6 is derived along with its asymptotic normality efficiency and variance estimates the estimate proposed can be used as a maximum partial likelihood estimate in stratified response selective sampling some computation algorithms are also provided

random walk on barely supercritical branching random walk let mathcal t be a supercritical galton watson tree with a bounded offspring distribution that has mean mu 1 conditioned to survive let varphi mathcal t be a random embedding of mathcal t into mathbb z d according to a simple random walk step distribution let mathcal t p be percolation on mathcal t with parameter p and let p c mu 1 be the critical percolation parameter we consider a random walk x n n ge 1 on mathcal t p and investigate the behavior of the embedded process varphi mathcal t p x n as n rightarrow infty and simultaneously mathcal t p becomes critical that is p p n searrow p c we show that when we scale time by n p n p c 3 and space by sqrt p n p c n the process varphi mathcal t p x n n ge 1 converges to a d dimensional brownian motion we argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments

estimation of rate and mean functions from truncated recurrent event data abstract this article investigates data on recurrent events that arise from sources such as warranty claims where the observation period for a unit is unknown until it experiences at least one event this creates a type of truncation in the data we consider nonparametric estimation of means and rates of the event occurrences with such zero truncated data and examine the case where the population size and the distribution of observation times across units are at least approximately known we study the behaviors of the proposed estimators by simulation we examine some car warranty data by applying the methodology developed here and considering the underlying assumptions

multiple phase transitions in long range first passage percolation on square lattices we consider a model of long range first passage percolation on the d dimensional square lattice in which any two distinct vertices are connected by an edge having exponentially distributed passage time with mean where 0 is a fixed parameter and is the l1 norm on we analyze the asymptotic growth rate of the set t which consists of all such that the first passage time between the origin 0 and x is at most t as t we show that depending on the values of there are four growth regimes i instantaneous growth for 2d 1 like the nearest neighbor first passage percolation model corresponding to 2015 wiley periodicals inc

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

l q estimates of gradients for evolutional p laplacian systems abstract we study l q estimates of gradients for evolutional p laplacian systems with discontinuous coefficients and external force given by the divergence of bmo functions the result obtained here may be regarded as a partial generalization of the calderon zygmund theory in l q estimate for the parabolic operator of second order to the evolutional p laplace operator

nonparametric estimation of a lifetime distribution when censoring times are missing we consider datasets for which lifetimes associated with the units in a population are observed if they occur within certain time intervals but for which lengths of the time intervals or censoring times of unfailed units are missing we consider nonparametric estimation of the lifetime distribution for the population from such data a maximum likelihood estimator and a simple moment estimator are obtained an example involving automobile warranty data is discussed at some length

semiparametric methods for response selective and missing data problems in regression summary suppose that data are generated according to the model f ylx 0 g x where y is a response and x are covariates we derive and compare semiparametric likelihood and pseudolikelihood methods for estimating 0 for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution the probability that a unit is fully observed may depend on y and there may be a subset of covariates which is observed only for a subsample of individuals our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that y x belongs to and that the stratum membership is observed for every unit applications include case control studies in epidemiology field reliability studies and broad classes of missing data and measurement error problems our results make fully efficient estimation of 0 feasible and they generalize and provide insight into a variety of methods that have been proposed for specific problems

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

a holder estimate for nonlinear parabolic systems of p laplacian type abstract we extend the holder regularity results for weak solutions of the p laplacian systems originally obtained by dibenedetto to a larger class of right hand side terms by appropriately arranging two scale parameters on the sizes of a local parabolic cylinder and of the gradient of the solutions we get the same holder exponent as in the scalar case our method can also recover the classical parabolic case p 2

doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions in this paper we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions we first prove a local existence result by the regularization method next we construct a weak comparison principle then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions necessary and sufficient conditions on the global existence of all positive weak solutions are obtained

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

boundary estimates for solutions of nonlinear degenerate parabolic systems on etablit la regularite partout jusqu a la frontiere pour des solutions faibles de systemes paraboliques du type u l t div du p 2 du l f l x t u du dans t l 1 2 m u t r m m n p max 1 2n n 2 est un ouvert borne de r n n 2 0 t t 0 t

regularity in sobolev spaces for doubly nonlinear parabolic equations abstract the doubly nonlinear parabolic equation u t div u m 1 u p 2 u m 1 u m 1 m p 1 1 is considered in several dimensions and regularity results in fractional order sobolev spaces are obtained the main tools in the proof are a difference quotient technique and the imbedding theorem of nikolskii spaces into sobolev spaces

least squares regression with data arising from stratified samples of the dependent variable regression analysis is being widely used in the analysis of survey data in situations where complex sampling designs are employed often the selection procedure is based on values of the independent variables in which case standard regression methods apply however the literature abounds with examples where the selection procedure is based on values of the dependent variable for example demets halperin 1977 consider the relationship between serum cholesterol the dependent variable and dietary cholesterol the independent variable from an initial large random sample a smaller subsample was selected with the procedure that only persons with very high or very low initial serum cholesterol were chosen for subsequent measurement of the regression variables in a study of the relationship of blood pressure the dependent variable and certain anthropometric factors in adolescents kotchen et al 1980 performed follow up on a subsample of a previous survey in which individuals in the original highest and lowest blood pressure groups were deliberately oversampled a regression analysis was subsequently performed using standard least square methods finally hausman wise 1982 discuss income maintenance experiments in econometrics where high income families were undersampled from the population in a situation where income is the dependent variable of a subsequent regression analysis there are many reasons why complex sampling designs may be used on the basis of the dependent variable y for example an initial goal of the study may be to estimate the mean of y with high precision for which stratified sampling on the basis of y values may be useful subsequently regression analysis with y as the dependent variable may be desired secondly oversampling extreme values of y is a method of reducing the variances of estimated regression coefficients in cases where it may not be feasible to use traditional efficient designs based on the independent variables for such sampling situations it is now well known that in general ordinary least squares regression estimates will be biased even asymptotically see for example holt smith winter 1980 and nathan holt 1980 various adjusted estimators have been considered which we discuss briefly below

estimation from truncated lifetime data with supplementary information on covariates and censoring times summary in epidemiology reliability and other areas one encounters situations where response times and covariates for individuals in a population are observed only if the response time does not exceed an associated censoring time methods for obtaining supplementary information about covariates and censoring times for nonresponding individuals are considered estimation procedures are developed for the resulting combination of truncated data and supplementary data theoretical and simulation results are given and the methodology illustrated on some automobile failure data collected under a warranty plan

on the scaling of the chemical distance in long range percolation models we consider the unoriented long range percolation on z d in dimensions d 1 where distinct sites x y z d get connected with probability p xy 0 1 assuming p xy x y s o 1 as x y where s 0 and is a norm distance on z d and supposing that the resulting random graph contains an infinite connected component c we let d x y be the graph distance between x and y measured on c our main result is that for s d 2d d x y log x y o 1 x y c x y where 1 is the binary logarithm of 2d s and o 1 is a quantity tending to zero in probability as x y besides its interest for general percolation theory this result sheds some light on a question that has recently surfaced in the context of small world phenomena as part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

the scaling limit of lattice trees in high dimensions we prove that above eight dimensions the scaling limit of sufficiently spread out lattice trees is the variant of super brownian motion known as integrated super brownian excursion ise as conjectured by aldous the same is true for nearest neighbour lattice trees in sufficiently high dimensions the proof uses the lace expansion

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

on sign changing and multiple solutions of the p laplacian abstract in this paper we construct the pseudo gradient vector field in w 0 1 p by which we obtain the positive and negative cones of w 0 1 p are both invariant sets of the descent flow of the corresponding functional then we use differential equations theory in banach spaces and dynamics theory to study p laplacian boundary value problems with jumping nonlinearities at zero or infinity and get new multiple solutions and sign changing solutions theorems of p laplacian

semiparametric estimation in logistic measurement error models summary we describe semiparametric estimation and inference in a logistic regression model with measurement error in the predictors the particular measurement error model consists of a primary data set in which only the response y and a fallible surrogate w of the true predictor x are observed plus a smaller validation data set for which y x w are observed except for the underlying assumption of a logistic model in the true predictor no parametric distributional assumption is made about the true predictor or its surrogate we develop a semiparametric parameter estimate of the logistic regression parameter which is asymptotically normally distributed and computationally feasible the estimate relies on kernel regression techniques for scalar predictors by a detailed analysis of the mean squared error of the parameter estimate we obtain a representation for an optimal bandwidth

estimating a distribution function with truncated data let mathscr p be a finite population with n geq 1 elements for each e in mathscr p let x e and y e be independent positive random variables with unknown distribution functions f and g and suppose that the pairs x e y e are i i d we consider the problem of estimating f g and n when the data consist of those pairs x e y e for which e in mathscr p and y e leq x e the nonparametric maximum likelihood estimators mles of f and g are described and their asymptotic properties as n rightarrow infty are derived it is shown that the mles are consistent against pairs f g for which f and g are continuous g 1 0 leq f 1 0 and g 1 1 leq f 1 1 sqrt n times estimation error for f converges in distribution to a gaussian process if int infty 0 1 g df infty but may fail to converge if this integral is infinite

a power counting theorem for feynman integrals on the lattice a convergence theorem is proved which states sufficient conditions for the existence of the continuum limit for a wide class of feynman integrals on a space time lattice a new kind of a uv divergence degree is introcduced which allows the formulation of the theorem in terms of power counting conditions

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

local and global existence for quasilinear parabolic systems with a strongly nonlinear source abstract we study the cauchy problem for quasilinear parabolic systems with a strongly nonlinear source on the assumptions that the initial values are taken in the sense of l l o c 1 r n we obtain the local existence global existence and nonexistence of solutions particularly in the case of global solvability some exact estimates of a solution are obtained

regularity in sobolev spaces for the fast diffusion and the porous medium equation abstract the degenerate parabolic equation u t u m 1 u m 0 is considered in a cylinder 0 t under homogeneous dirichlet boundary values the regularity of weak solutions for the fast diffusion equation m 1 and the porous medium equation m 1 are investigated regularity of u and u m 1 u in weighted sobolev spaces and in fractional order nikolskii spaces are proved

inference using surrogate outcome data and a validation sample summary in the context of estimating 3 from the regression model p yi x relating response y to covariates x suppose that only a surrogate response s is available for most study subjects suppose that for a random subsample of the study cohort termed the validation sample the true outcome y is available in addition to s we consider maximum likelihood estimation of 3 from such data and show that it is nonrobust to misspecification of the distribution relating the surrogate to the true outcome p s i y x an alternative semiparametric method is also considered which is nonparametric with respect to p s i y x large sample distribution theory for maximum estimated likelihood estimates is developed an illustrative example is presented

the scaling limit of the incipient infinite cluster in high dimensional percolation i critical exponents this is the first of two papers on the critical behavior of bond percolation models in high dimensions in this paper we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d 6 and for sufficiently spread out models in all dimensions d 6 the exponent describes the low frequency behavior of the fourier transform of the critical two point connectivity function while describes the behavior of the magnetization at the critical point our main result is an asymptotic relation showing that in a joint sense 0 and 2 the proof uses a major extension of our earlier expansion method for percolation this result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on d known as integrated super brownian excursion ise in dimensions above 6 in the sequel to this paper we extend our methods to prove that the scaling limits of the incipient infinite cluster s two point and three point functions are those of ise for the nearest neighbor model in dimensions d 6

a convergence theorem for lattice feynman integrals with massless propagators it is shown that for non vanishing lattice spacing conventional infrared power counting conditions are sufficient for convergence of lattice feynman integrals with zero mass propagators if these conditions are supplemented by ultraviolet convergence conditions the continuum limit of such a diagram exists and is universal

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

percolation critical exponents under the triangle condition for independent percolation models it is shown that if the diagrammatic triangle condition is satisfied then the critical exponents delta and hat beta exist and take their mean field values generalizing the criterion introduced in 1984 by aizenman and newman for the mean field value of gamma in nonoriented percolation the results apply to a broad class of nonoriented as well as oriented weakly homogeneous models in which the range of the connecting bonds need not be bounded for the nonoriented case the condition reduces to the finiteness at the critical point of nabla equiv sum x y tau 0 x tau x y tau y 0 with tau u nu the probability that the site u is connected to nu which was recently established by hara and slade for models with sufficiently spread out connections in d 6 dimensions our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter m beta h whose value at h 0 yields the percolation density with beta parametrizing the bond or site occupation probabilities and with h h geq 0 a ghost field the conclusion is that under the triangle condition in the vicinity of the critical point beta c 0 m beta 0 approx beta beta c hat beta and m beta c h approx h 1 delta with hat beta 1 and delta 2

semiparametric estimation exploiting covariate independence in two phase randomized trials recent results for case control sampling suggest when the covariate distribution is constrained by gene environment independence semiparametric estimation exploiting such independence yields a great deal of efficiency gain we consider the efficient estimation of the treatment biomarker interaction in two phase sampling nested within randomized clinical trials incorporating the independence between a randomized treatment and the baseline markers we develop a newton raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate spmle our algorithm accommodates both continuous phase one outcomes and continuous phase two biomarkers the profile information matrix is computed explicitly via numerical differentiation in certain situations where computing the spmle is slow we propose a maximum estimated likelihood estimator mele which is also capable of incorporating the covariate independence this estimated likelihood approach uses a one step empirical covariate distribution thus is straightforward to maximize it offers a closed form variance estimate with limited increase in variance relative to the fully efficient spmle our results suggest exploiting the covariate independence in two phase sampling increases the efficiency substantially particularly for estimating treatment biomarker interactions

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results