the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation we first consider the boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates this is the simplest boltzmann equation having the compressible euler equations as a scaling limit for it we prove a stability result for theh theorem which says that when the entropy production is small the solution of the spatially homogeneous boltzmann equation is necessarily close to equilibrium in the entropie sense and therefore strongl 1 sense we use this to prove that solutions to the spatially homogeneous boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes every initial condition with finite entropy andp th velocity moment for some p 2 belongs to such a class we then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below which covers a wide class of slightly modified physical collision kernels these results are the basis of a study of the relation between scaling limits of solutions of the boltzmann equation and hydrodynamics which will be developed in subsequent papers the program is described here

stokes fourier and acoustic limits for the boltzmann equation convergence proofs we establish a stokes fourier limit for the boltzmann equation considered over any periodic spatial domain of dimension 2 or more appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of stokes fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate l 2 initial data of the stokesfourier equations both the motion and heat equations are recovered in the limit by controlling the fluxes and the local conservation defects of the diperna lions solutions with dissipation rate estimates the scaling of the fluctuations with respect to knudsen number is essentially optimal the assumptions on the collision kernel are little more than those required for the diperna lions theory and that the viscosity and heat conduction are finite for the acoustic limit these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations both weak limits become strong when the initial fluctuations converge entropically to appropriate l 2 initial data c 2002 john wiley sons inc

quasistationary hydrodynamics for the boltzmann equation the boltzmann equation solutions are considered for small knudsen number the main attention is devoted to certain deviations from the classical navier stokes description the equations for the quasistationary slow flows are derived these equations do not contain the knudsen number and provide in this sense a limiting description of hydrodynamic variables in the isothermal case the equations reduce to incompressible navier stokes equations for bulk velocity and pressure in the stationary case they coincide with the equations of slow nonisothermal flows it is shown that the derived equations unlike the burnett equations possess all principal properties of the boltzmann equation in one dimension the equations reduce to a nonlinear diffusion equation being exactly solvable for maxwell molecules multidimensional stationary heat transfer problems are also discussed it is shown that one can expect an essential difference between the boltzmann equation solution in the limit of continuous media and the corresponding solution of navier stokes equations

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

acoustic limit for the boltzmann equation in optimal scaling based on a recent l 2 l 1 framework we establish the acoustic limit of the boltzmann equation for general collision kernels the scaling of the fluctuations with respect to knudsen number is optimal our approach is based on a new analysis of the compressible euler limit of the boltzmann equation as well as refined estimates of euler and acoustic solutions

hydrodynamics of stochastic cellular automata we investigate a stochastic version of cellular automata used for simulating hydrodynamical flows e g the hpp and fhp models the extra stochasticity consists of random exchanges between neighboring cells which conserve momentum we prove that in suitable limits these models satisfy the appropriate continuous boltzmann and hydrodynamic equations the same as those conjectured for the original models except that there is no negative viscosity contribution the results are obtained by proving a very strong form of propagation of chaos and by using hilbert chapman enskog type expansions explicit proofs are presented for the stochastic hpp model

asymptotic properties of the balanced repeated replication method for sample quantiles inference including variance estimation can be made from stratified samples by selecting a balanced set of subsamples this balanced subsampling method is generically called the balanced repeated replication method in survey data analysis which includes mccarthy s balanced half samples method and its extensions for more general stratified designs we establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of interest is the population quantile the consistency results also hold when balanced subsampling is replaced by random subsampling as a key technical prerequisite we prove a bahadurtype representation for sample quantiles in stratified random sampling

hydrodynamic limit with geometric correction of stationary boltzmann equation abstract we consider the hydrodynamic limit of a stationary boltzmann equation in a unit plate with in flow boundary the classical theory claims that the solution can be approximated by the sum of interior solution which satisfies steady incompressible navier stokes fourier system and boundary layer derived from milne problem in this paper we construct counterexamples to disprove such formulation in l both for its proof and result also we show the hydrodynamic limit with a different boundary layer expansion with geometric correction

mixed orthogonal arrays for variance estimation with unequal numbers of primary selections per stratum summary for estimating the variance of nonlinear statistics in large scale complex surveys the method of balanced repeated replications has received special attention the method at present is useful in stratified clustered designs with equal number of primary selections from each stratum this paper extends the method to general sample designs with arbitrary number of selections from each stratum it is shown that mixed orthogonal arrays of strength two or equivalently equal frequency orthogonal main effect plans for asymmetrical factorials give a set of balanced subsamples useful in variance estimation

the acoustic limit for the boltzmann equation the acoustic equations are the linearization of the compressible euler equations about a spatially homogeneous fluid state we first derive them directly from the boltzmann equation as the formal limit of moment equations for an appropriately scaled family of boltzmann solutions we then establish this limit for the boltzmann equation considered over a periodic spatial domain for bounded collision kernels appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that converge entropically and hence strongly in l 1 to a unique limit governed by a solution of the acoustic equations for all time provided that its initial fluctuations converge entropically to an appropriate limit associated to any given l 2 initial data of the acoustic equations the associated local conservation laws are recovered in the limit

diffusive expansion for solutions of the boltzmann equation in the whole space this paper is concerned with the diffusive expansion for solutions of the rescaled boltzmann equation in the whole space 0 1 tf v xf 1 q f f x rn v rn t 0 with prescribed initial data 0 2 f t x v t 0 f0 x v x rn v rn our main purpose is to justify the global validity of the diffusive expansion 0 3 f t x v f1 t x v 2f2 t x v n 1fn 1 t x v nfn t x v for a solution f t x v of the rescaled boltzmann equation 0 1 in the whole space rn for all t 0 with initial data f0 x v satisfying the initial expansion 0 4 f0 x v f1 0 x v 2f2 0 x v n 1fn 1 0 x v nfn 0 x v x v r2n here v 2 n2exp v 22 is a normalized global maxwellian under the assumption that the fluid components of the coefficients fm 0 x v 1 m n of the initial expansion f0 x v have divergence free velocity fields um0 x as well as temperature fields m0 x if we assume further that the velocity temperature fields u10 x 10 x of f1 0 x v have small amplitude in hs rn s 2 n n 2 we can determine these coefficients fm t x v 1 m n in the diffusive expansion 0 3 uniquely by an iteration method and energy method the hydrodynamic component of these coefficients fm t x v 1 m n satisfies the incompressible condition the boussinesq relations and or the navier stokes fourier system respectively while the microscopic component of these coefficients is determined by a recursive formula compared with the corresponding problem inside a periodic box studied in y guo 2006 18 the main difficulty here is due to the fact that poincare s inequality is not valid in the whole space rn and this difficulty is overcome by using the lp lq estimate on the riesz potential moreover by exploiting the energy method we can also deduce certain the space time energy estimates on these coefficients fm t x v 1 m n once the coefficients fm t x v 1 m n in the diffusive expansion 0 3 are uniquely determined and some delicate estimates have been obtained the uniform estimates with respect to on the remainders fn t x v are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion 0 3 in the large provided that 0 5 n 2n 2 notice that for m 2 um t x is no longer a divergence free vector and it is worth to pointing out that for m 3 it was in deducing certain estimates on pm t x by the lp lq estimate on the riesz potential that we need to require that n 2n 2

asymptotic analysis of a slightly rarefied gas with nonlocal boundary conditions in this paper nonlocal boundary conditions for the navier stokes equations are derived starting from the boltzmann equation in the limit for the knudsen number being vanishingly small in the same spirit of lombardo et al in j stat phys 130 69 82 2008 where a nonlocal poisson scattering kernel was introduced a gaussian scattering kernel which models nonlocal interactions between the gas molecules and the wall boundary is proposed it is proved to satisfy the global mass conservation and a generalized reciprocity relation the asymptotic expansion of the boundary value problem for the boltzmann equation provides in the continuum limit the navier stokes equations associated with a class of nonlocal boundary conditions of the type used in turbulence modeling

incompressible navier stokes and euler limits of the boltzmann equation we consider solutions of the boltzmann equation in a d dimensional torus d 2 3 for macroscopic times t n 1 t 0 when the space variations are on a macroscopic scale x n 1r n 2 x in the unit torus let u x t be for t t0 a smooth solution of the incompressible navier stokes equations ins for n 2 and of the incompressible euler equation ie for n 2 we prove that has solutions for t t0 which are close to o 2 in a suitable norm to the local maxwellian p 2 t d 2 exp v u x t 2 2t with constant density p and temperature t this is a particular case defined by the choice of initial values of the macroscopic variables of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations for n 3 these equations correspond to variable density ie while for n 2 they involve higher order derivatives of the density

from the boltzmann equation to the incompressible navier stokes equations on the torus a quantitative error estimate we investigate the boltzmann equation depending on the knudsen number in the navier stokes perturbative setting on the torus using hypocoer civity we derive a new proof of existence and exponential decay for solutions close to a global equilibrium with explicit regularity bounds and rates of convergence these results are uniform in the knudsen number and thus allow us to obtain a strong derivation of the incompressible navier stokes equations as the knudsen number tends to 0 moreover our method is also used to deal with other kinetic models finally we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time unless the initial layer conditions are satisfied

an asymptotic preserving numerical scheme for kinetic equations in the low mach number limit a numerical scheme for the nonstationary boltzmann equation in the incompressible navier stokes limit is developed the scheme is induced by the asymptotic analysis of the navier stokes limit for the boltzmann equation it works uniformly for all ranges of mean free paths in the limit the scheme reduces to the chorin projection method for the incompressible navier stokes equation numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

biometrika centenary sample surveys summary despite karl pearson s stated aim that the evolutionist has to become in the widest sense of the words a registrar general for all forms of life the papers published in biometrika on sample surveys have been mainly theoretical and do not start to appear until the 1940s there has been little work on sampling in the life sciences with the exception of capture recapture methods the major areas have been in variable probability designs the foundations of inference for finite populations analytic surveys and the role of selection mechanisms in inference in recent years the work on survey analysis has been the outcome of an interaction between sample selection mechanisms and mainstream theory with interesting consequences for both areas

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

kinetic derivation of a finite difference scheme for the incompressible navier stokes equation in the present paper the low mach number limit of kinetic equations is used to develop a discretization for the incompressible navier stokes equation the kinetic equation is discretized with a first and second order discretization in space the discretized equation is then considered in the low mach number limit using this limit a second order discretization for the convective part in the incompressible navier stokes equation is obtained numerical experiments are shown comparing different approaches

resampling inference with complex survey data abstract methods for standard errors and confidence intervals for nonlinear statistics such as ratios regression and correlation coefficients have been extensively studied for stratified multistage designs in which the clusters are sampled with replacement in particular the important special case of two sampled clusters per stratum these methods include the customary linearization or taylor method and resampling methods based on the jackknife and balanced repeated replication brr unlike the jackknife or the brr the linearization method is applicable to general sampling designs but it involves a separate variance formula for each nonlinear statistic thereby requiring additional programming efforts both the jackknife and the brr use a single variance formula for all nonlinear statistics but they are more computing intensive the resampling methods developed here retain these features of the jackknife and the brr yet permit extension to more complex designs involving sampling without replace

hydrodynamic limit of the stationary boltzmann equation in a slab we study the stationary solution of the boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature we prove that when the force is sufficiently small there exists a solution which converges in the hydrodynamic limit to a local maxwellian with parameters given by the stationary solution of the corresponding compressible navier stokes equations with no slip boundary conditions corrections to this maxwellian are obtained in powers of the knudsen number with a controlled remainder

balanced repeated replication for stratified multistage survey data under imputation abstract balanced repeated replication brr is a popular method for variance estimation in surveys the standard brr method works by first creating a set of balanced pseudoreplicated datasets from the original dataset for a survey estimator the brr variance estimator is the average of squared deviations r where r is the same as but based on the data in the rth pseudoreplicated dataset only but when there are a large number of imputed missing values nonrespondents treating the imputed values as observed data and applying the standard brr variance estimation formula does not produce valid variance estimators intuitively the variation due to imputation can be captured by the brr method if every pseudoreplicated dataset is imputed in exactly the same way as the original dataset is imputed assuming that the dataset contains flags for nonrespondents but when a random imputation method such as random hot deck imputation random ratio imputation or random regression imputation is u

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

from boltzmann s kinetic theory to euler s equations abstract the incompressible euler equations are obtained as a weak asymptotics of the boltzmann equation in the fast relaxation limit the knudsen number kn goes to zero when both the mach number ma defined as the ratio between the bulk velocity and the speed of sound and the inverse reynolds number kn ma which measures the viscosity of the fluid go to zero the entropy method used here consists in deriving some stability inequality which allows us to compare the sequence of solutions of the scaled boltzmann equation to its expected limit provided that it is sufficiently smooth it thus leads to some strong convergence result one of the main points to be understood is how to deal with the corrections to the weak limit i e the contributions converging weakly but not strongly to 0 such as the initial layer or the acoustic waves

hydrodynamic limits for the boltzmann process we study the behavior of the nonlinear markov process associated to the boltzmann equation under both hyperbolic and parabolic space time scalings in the first case the limit of the process is the solution of an o d e with vector field given by a solution of the euler equation while in the second case the limit of the process in the incompressible case turns out to be a diffusion process whose drift is a solution of the incompressible navier stokes equation

modified balanced repeated replication for complex survey data summary balanced repeated replication commonly used to estimate the variances of nonlinear estimators from stratified sampling designs can run into problems in estimating a ratio of small domain totals or in the case of poststratification or unit nonresponse adjustment involving ratio weighting within cells if some cell sizes are small this problem is mainly caused by a sharp perturbation of the weights to construct replicate estimates to avoid extreme replicate estimates robert fay proposed a modified balanced repeated replication method using gentler perturbation of weights theoretical properties of the modified method are studied here the method is extended to the case of imputation for item nonresponse both smooth and nonsmooth estimators are studied

balanced repeated replications based on mixed orthogonal arrays summary for stratified samples with arbitrary numbers of primary sampling units per stratum one can divide the units in the stratum into two groups and then apply the balanced half samples method to the two groups thus formed the efficiency loss of the grouped balanced half samples variance estimator can be substantial use of a mixed orthogonal array whose numbers of symbols correspond to the numbers of units per stratum gives a set of balanced subsamples unlike that of gupta nigam 1987 the proposed method of analysis gives a consistent variance estimator and bias reduction for nonlinear statistics to further reduce the number of replicates use of nearly orthogonal arrays is considered the balanced half samples method mccarthy 1966 1969 is commonly used for estimating the variance of nonlinear statistics from stratified samples with two primary sampling units per stratum these samples arise naturally if deep stratification is desired as in some large scale surveys the method is extended gurney jewett 1975 to the case of p primary sampling units per stratum for prime p this extension for p 2 is not adopted in common practice since it requires a much larger number of replicates also in many survey designs the numbers of primary sampling units are unequal a common practice is to divide the units in the stratum into two groups and then to apply the balanced half samples method to the two groups thus formed this has gained popularity because the software for obtaining balanced half samples is readily available and the number of replicates required is more economical as shown in 2 the resulting variance estimators can be quite inefficient and their bias based on a conditional analysis can be considerable the conditional analysis also suggests a method of grouping to reduce the bias the degree of grouping depends on the tradeoff between computational economy and statistical efficiency if the efficiency loss from grouping is not justifiable one may draw replicates by using mixed orthogonal arrays with variable numbers of symbols to accommodate the different numbers of units or groups of units in the strata as long as an orthogonal array is used for drawing replicates we call the method balanced repeated replications in 3 we show that the analysis method of gupta nigam 1987 based on mixed orthogonal arrays leads to an inconsistent variance estimator for nonlinear statistics and propose a method that is valid for variance estimation and bias reduction

asymptotic theory for the time dependent behavior of a slightly rarefied gas over a smooth solid boundary a time evolution of a slightly rarefied monoatomic gas namely a gas for small knudsen numbers which is perturbed slowly and slightly from a reference uniform equilibrium state at rest is investigated on the basis of the linearized boltzmann equation by a systematic asymptotic analysis a set of fluid dynamic type equations and its boundary conditions that describe the gas behavior up to the second order of the knudsen number are derived the developed theory covers a general intermolecular potential and a gas surface interaction it is shown that i the compressibility of the gas manifests itself from the leading order in the energy equation and from the first order in the continuity equation ii although the momentum equation is the stokes equation it contains a double laplacian of the leading order flow velocity as a source term at the second order iii a double laplacian source term also appears in the energy equation at the second order iv the slip and jump conditions are the same as those in the time independent case up to the first order and the difference occurs at the second order in the jump conditions as the terms of the divergence of flow velocity and of the laplacian of temperature numerical values of all the slip and jump coefficients are obtained for a hard sphere gas by the use of a symmetric relation developed recently

compressible navier stokes approximation to the boltzmann equation abstract even though the system of the compressible navier stokes equations is not a limiting system of the boltzmann equation when the knudsen number tends to zero it is the second order approximation by applying the chapman enskog expansion the purpose of this paper is to justify this approximation rigorously in mathematics that is if the difference between the initial data for the compressible navier stokes equations and the boltzmann equation is of the second order of the knudsen number so is the difference between two solutions for all time the analysis is based on a refined energy method for a fluid type system using the techniques for the system of viscous conservation laws

hilbert expansion from the boltzmann equation to relativistic fluids we study the local in time hydrodynamic limit of the relativistic boltzmann equation using a hilbert expansion more specifically we prove the existence of local solutions to the relativistic boltzmann equation that are nearby the local relativistic maxwellians the maxwellians are constructed from a class of solutions to the relativistic euler equations that includes a large subclass of near constant non vacuum fluid states in particular for small knudsen number these solutions to the relativistic boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic euler equations

the navier stokes limit of stationary solutions of the nonlinear boltzmann equation we consider the flow of a gas in a channel whose walls are kept at fixed different temperatures there is a constant external force parallel to the boundaries which may themselves also be moving the system is described by the stationary boltzmann equation to which are added maxwellian boundary conditions with unit accommodation coefficient we prove that when the temperature gap the relative velocity of the planes and the force are all sufficiently small there is a solution which converges in the hydrodynamic limit to a local maxwellian with parameters given by the stationary solution of the corresponding compressible navier stokes equations with no slip voundary conditions corrections to this maxwellian are obtained in powers of the knudsen number with a controlled remainder

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

balanced repeated replications based on orthogonal multi arrays summary to obtain variance estimates in stratified sampling with possibly unequal numbers of primary sampling units per stratum one can either randomly divide the units in each stratum into two groups and apply the balanced half samples method to these groups or use mixed orthogonal arrays to form a set of balanced subsamples with the first technique resulting variance estimates can be very inefficient and with the second technique it can be difficult to construct orthogonal arrays with a small number of replicates for a given stratified design the proposed extension of the orthogonal array method to orthogonal multi arrays retains all of the orthogonal array method s desirable properties but allows greater flexibility in construction and thus can handle more complex stratified designs using fewer replicates

from the boltzmann equation to an incompressible navier stokes fourier system we establish a navier stokes fourier limit for solutions of the boltzmann equation considered over any periodic spatial domain of dimension two or more we do this for a broad class of collision kernels that relaxes the grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that are compact every limit point is governed by a weak solution of a navier stokes fourier system for all time