bellman equations associated to the optimal feedback control of stochastic navier stokes equations in this paper we study infinite dimensional second order hamilton jaco bi bellman equations associated to the feedback synthesis of stochastic navier stokes equations forced by space time white noise uniqueness and existence of viscosity solutions are proven for these infinite dimensional partial differential equations c 2004 wiley periodicals inc

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

finite size scaling in strips antiperiodic boundary conditions the exact values of the universal amplitude a relating the correlation length xi approximately l a to the strip width l at the critical point of ising and 3 state potts models are obtained for the case of antiperiodic or twisted boundary conditions predictions are also made for the finite size scaling behaviour of the interfacial tension

crossing probabilities in one two or three directions for percolation on a cubic lattice the probabilities of crossing in exactly one two or three directions have been investigated using monte carlo simulations of percolation on cubic lattices with free boundary conditions the crossing probability in exactly one given direction shows the least influence of finite size effects and is best suited to obtain the percolation threshold pc finite size effects are most important for simultaneous crossing in three directions crossing probabilities at pc for large lattices are the same for site and bond percolation the crossing probability in a given direction regardless of crossing in other directions r1 is 0 28 exact results for small lattices agree with simulation results

percolation thresholds on elongated lattices we investigate the percolation thresholds of both random and invasion percolation in two and three dimensions on elongated lattices lattices with a geometry of l d 1 nl in d dimensions where n denotes the aspect ratio of the lattice scaling laws for the threshold and spanning cluster density for random percolation are derived and simulation confirms the behaviour a direct relationship between thresholds obtained for random percolation and invasion percolation is given and verified numerically important contributions to understanding two phase flow observations in porous media and rock have been made using percolation theory 1 3 random percolation rp is relevant to two phase displacement if the flow is very slow and the invading fluid is completely wetting invasion percolation ip is relevant when the invading fluid is completely nonwetting both variants of percolation have been used to explain the structure and the amounts of fluids in a two phase displacement at breakthrough the fractal structure of the invading fluid paths have been analysed and the properties of ip are believed to be consistent with rp in spite of this the spanning clusters are not precisely the same and no relationship between the cluster density at spanning in rp and the breakthrough threshold in ip is known estimates of ip thresholds are neglected particularly in three dimensions due to the large computational effort required when compared with rp in most studies of percolation theory a simple square or cubic geometry is considered in many applications one must consider systems with nonquadratic and noncubic geometries for example in the petroleum industry laboratory measurements e g residual saturations capillary pressure are performed on rock cores of high aspect ratio these measurements are then used as input to reservoir simulation models the crucial parameter measured is the value of the critical thresholds using scaling arguments and small scale numerical simulations monetti and albano 4 presented scaling laws for the percolation probability in an elongated geometry that depend on the aspect ratio n of the lattice in this paper we derive new scaling laws for percolation properties of elongated lattices els in both two and three dimensions and present simulation data to confirm the theoretical results we also derive relationships between thresholds observed in rp and ip for els and verify the relationships numerically using finite size scaling arguments monetti and albano assumed the expectation value of the percolation thresholdhpc n l i l 1 n 1 where is the critical exponent for the correlation length we define a lattice in d dimensions of size l d as a simple lattice sl

strong solution for a stochastic model of two dimensional second grade fluids existence uniqueness and asymptotic behavior abstract we investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of r 2 global existence and uniqueness of strong probabilistic solution is established in contrast to previous results on this model we show that the sequence of galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem we also give two results on the long time behavior of the solution mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time independent second grade fluids equations if the deterministic part of the external force does not depend on time if the deterministic forcing term explicitly depends on time then the strong probabilistic solution decays exponentially in mean square

asymptotic normality of the l_1 error of the grenander estimator groeneboom introduced a jump process that can be used among other things to study the asymptotic properties of the grenander estimator of a monotone density in this paper we derive the asymptotic normality of a suitably rescaled version of the l 1 error of the grenander estimator using properties of this jump process

a massive data framework for m estimators with cubic rate the divide and conquer method is a common strategy for handling massive data in this article we study the divide and conquer method for cubic rate estimators under the massive data framework we develop a general theory for establishing the asymptotic distribution of the aggregated m estimators using a weighted average with weights depending on the subgroup sample sizes under certain condition on the growing rate of the number of subgroups the resulting aggregated estimators are shown to have faster convergence rate and asymptotic normal distribution which are more tractable in both computation and inference than the original m estimators based on pooled data our theory applies to a wide class of m estimators with cube root convergence rate including the location estimator maximum score estimator and value search estimator empirical performance via simulations and a real data application also validate our theoretical findings

on integrated l 1 convergence rate of an isotonic regression estimator for multivariate observations we consider a general monotone regression estimation where we allow for independent and dependent regressors we propose a modification of the classical isotonic least squares estimator and establish its rate of convergence for the integrated l 1 loss function the methodology captures the shape of the data without assuming additivity or a parametric form for the regression function furthermore the degree of smoothing is chosen automatically and no auxiliary tuning is required for the theoretical analysis some simulations and two real data illustrations complement the study of the proposed estimator

universality of critical existence probability for percolation on three dimensional lattices using a histogram monte carlo simulation method we calculate the existence probability for bond percolation on simple cubic sc and body centred cubic bcc lattices and site percolation on sc lattices with free boundary conditions the spanning rule considered by reynolds stanley and klein is used to define percolating clusters we find that for such systems has very good finite size scaling behaviour and the value of at the critical point is universal and is about

estimation of a function under shape restrictions applications to reliability this paper deals with a nonparametric shape respecting estimation method for u shaped or unimodal functions a general upper bound for the nonasymptotic l 1 risk of the estimator is given the method is applied to the shape respecting estimation of several classical functions among them typical intensity functions encountered in the reliability field in each case we derive from our upper bound the spatially adaptive property of our estimator with respect to the l 1 metric it approximately behaves as the best variable binwidth histogram of the function under estimation

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

the distribution of percolating concentrations in finite systems abstract simulation of finite cubic lattices suggests that even in the limit of large systems the distribution of percolation thresholds deviates from a gaussian thus confirming ziff s two dimensional result

well posedness of stochastic second grade fluids abstract the theory of turbulent newtonian fluids shows that the choice of the boundary condition is a relevant issue because it can modify the behavior of a fluid by creating or avoiding a strong boundary layer in this study we consider stochastic second grade fluids filling a two dimensional bounded domain with the navier slip boundary condition with friction we prove the well posedness of this problem and establish a stability result our stochastic model involves a multiplicative white noise and a convective term with third order derivatives which significantly complicate the analysis

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

sharp asymptotics for isotonic regression the asymptotic behavior of the isotonic estimator of a monotone regression function that is the least squares estimator under monotonicity restriction is investigated in particular it is proved that the 1 distance between the isotonic estimator and the true function is of magnitude n 1 3 moreover it is proved that a centered version of this 1 distance converges at the n1 2 rate to a gaussian variable with fixed variance

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

finite size scaling for percolation on elongated lattices in two and three dimensions we derive scaling laws for the percolation properties of an elongated lattice i e those with dimensions of l d ensuremath 1 ifmmode times else texttimes fi mathrm nl in d dimensions where n denotes the aspect ratio of the lattice based on statistical arguments it is shown that in the direction of the extension the percolation threshold scales approximately as mathrm ln n 1 a in both two and three dimensions extensive monte carlo simulations of the site percolation model confirm this scaling behavior it is further shown that the density of the incipient infinite cluster at the percolation threshold scales differently in two and three dimensions

on the mathbb l _ p error of monotonicity constrained estimators we aim at estimating a function 0 1 r subject to the constraint that it is decreasing or increasing we provide a unified approach for studying the lp loss of an estimator defined as the slope of a concave or convex approximation of an estimator of a primitive of based on n observations our main task is to prove that the l p loss is asymptotically gaussian with explicit though unknown asymptotic mean and variance we also prove that the local lp risk at a fixed point and the global lp risk are of order n p 3 applying the results to the density and regression models we recover and generalize known results about grenander and brunk estimators also we obtain new results for the huang wellner estimator of a monotone failure rate in the random censorship model and for an estimator of the monotone intensity function of an inhomogeneous poisson process

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

the number of incipient spanning clusters in two dimensional percolation using methods of conformal field theory we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k i n the limit when k is large the study of the structure of large clusters at the percolation threshold continues to pose interesting problems whose solution sheds light on the nature of the critical state in general recently some attention has been paid to incipient spanning clusters iscs these are clusters which connect two disjoint segments of the boundary of a macroscopically large region langlands et al 1 conjectured that the probability that at least one such cluster exists that is that the segments are connected is invariant under conformal transformations this statement was placed in the context of conformal field theory in 2 where an explicit formula was given for this crossing probability for a review of the status of conformal invariance in critical percolation see 3 more recently aizenman 4 has considered among other things the probability that there exist n distinct iscs connecting the two segmentsy in the case of a rectangular region 0 kl 0 l he has proved that the probability p n k l that the strip is traversed in the direction of length kl b y nindependent clusters satisfies the bounds a e n 2 k

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

a two stage hybrid procedure for estimating an inverse regression function we consider a two stage procedure tsp for estimating an inverse regression function at a given point where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two we establish that the convergence rate of the second stage estimate can attain the parametric n 1 2 rate furthermore a bootstrapped variant of tsp btsp is introduced and its consistency properties studied this variant manages to overcome the slow speed of the convergence in distribution and the estimation of the derivative of the regression function at the unknown target quantity finally the finite sample performance of btsp is studied through simulations and the method is illustrated on a data set

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

the limit distribution of the l error of grenander type estimators let f be a nonincreasing function defined on 0 1 under standard regularity conditions we derive the asymptotic distribution of the supremum norm of the difference between f and its grenander type estimator on sub intervals of 0 1 the rate of convergence is found to be of order n log n 1 3 and the limiting distribution to be gumbel

the probability distribution of the percolation threshold in a large system we show that the distribution of the percolation threshold in a large finite system does not converge to a gaussian when the size of the system goes to infinity provided that the two widely accepted definitions of correlation length are equivalent the shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length the result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v all results are rigorously proven in the 2d case generalizations for three dimensions are also discussed

on the existence of optimal and ϵ optimal feedback controls for stochastic second grade fluids abstract this article deals with a feedback optimal control problem for the stochastic second grade fluids more precisely we establish the existence of an optimal feedback control for the two dimensional stochastic second grade fluids with navier slip boundary conditions in addition using the galerkin approximations we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs showing the existence of the so called optimal feedback control

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point