singular behavior of vacuum states for compressible fluids in this survey paper we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics that is euler equations euler poisson equations and navier stokes equations the main concern is the well posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas furthermore the relation of the solutions for the gas dynamics with vacuum to those of the boltzmann equation will also be discussed in fact the results obtained so far for vacuum states are far from being complete and satisfactory therefore this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems moreover the problems considered here are limited to the author s interest and knowledge in this area

the boltzmann grad limit of a hard sphere system analysis of the correlation error we present a quantitative analysis of the boltzmann grad low density limit of a hard sphere system we introduce and study a set of functions the correlation errors measuring the deviations in time from the statistical independence of particles propagation of chaos in the context of the bbgky hierarchy a correlation error of order k measures the event where k particles are connected by a chain of interactions preventing the factorization we show that provided k varepsilon alpha such an error flows to zero with the average density varepsilon for short times as varepsilon gamma k for some positive alpha gamma in 0 1 this provides an information on the size of chaos namely j different particles behave as dictated by the boltzmann equation even when j diverges as a negative power of varepsilon the result requires a rearrangement of lanford perturbative series into a cumulant type expansion and an analysis of many recollision events

fluid dynamic description of flocking via the povzner boltzmann equation we introduce and discuss the possible dynamics of groups of indistinguishable agents which are interacting according to their relative positions with the aim of deriving hydrodynamic equations these models are developed to mimic the collective motion of groups of species such as bird flocks fish schools herds of quadrupeds or bacteria colonies our starting model for these interactions is the povzner equation 21 which describes a dilute gas in which binary collisions of elastic spheres depend on their relative positions following the cucker and smale model 9 we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance under the assumption of weak dissipation it is shown that the povzner equation is modified through a correction in the form of a nonlinear friction type operator using this correction we formally obtain from the povzner equation in a direct way a fluid dynamic description of a system of agents with weak dissipative interactions with a coefficient of restitution that depends on their relative distance

general population systems macroscopic limit of a class of stochastic semigroups relationships between three possible descriptions microscopic mesoscopic and macroscopic that are related to a large population composed of several subpopulations are formulated the systems describing the evolution of a large number of individuals undergoing stochastic interactions a description at microscopic level in terms of stochastic semigroups are considered the solutions of the general lotka volterra type equations without or with weak diffusion descriptions at macroscopic level are approximated by solutions of stochastic systems when the number of individuals n tends to the rate of approximation is controlled

kinetic hierarchy and propagation of chaos in biological swarm models we consider two models of biological swarm behavior in these models pairs of particles interact to adjust their velocities one to each other in the first process called bdg they join their average velocity up to some noise in the second process called cl one of the two particles tries to join the other one s velocity this paper establishes the master equations and bbgky hierarchies of these two processes it investigates the infinite particle limit of the hierarchies at large time scales it shows that the resulting kinetic hierarchy for the cl process does not satisfy propagation of chaos numerical simulations indicate that the bdg process has similar behavior to the cl process c 2012 elsevier b v all rights reserved

longitudinal and repeated cross sectional cluster randomization designs using mixed effects regression for binary outcomes bias and coverage of frequentist and bayesian methods as medical applications for cluster randomization designs become more common investigators look for guidance on optimal methods for estimating the effect of group based interventions over time this study examines two distinct cluster randomization designs 1 the repeated cross sectional design in which centres are followed over time but patients change and 2 the longitudinal design in which individual patients are followed over time within treatment clusters simulations of each study design stipulated a multiplicative treatment effect on the log odds scale between 5 and 15 clusters in each of two treatment arms and followed over two time periods estimation options included linear mixed effects models using restricted maximum likelihood reml generalized estimating equations gee mixed effects logistic regression using both penalized quasi likelihood pql and numerical integration and bayesian monte carlo analysis for the repeated cross sectional designs most methods performed well in terms of bias and coverage when clusters were numerous 30 and variability across clusters of baseline risk and treatment effect was modest with few clusters two groups of five and higher variability only the bayesian methods maintained coverage in the longitudinal designs the common methods of reml gee or pql performed poorly when compared to numerical integration while bayesian methods demonstrated less bias and better coverage for estimates of both log odds ratios and risk differences the performance of common statistical tools for the analysis of cluster randomization designs depends heavily on the precise design the number of clusters and the variability of baseline outcomes and treatment effects across centres

a comparison of statistical methods for clustered data analysis with gaussian error we investigate by simulation the properties of four different estimation procedures under a linear model for correlated data with gaussian error maximum likelihood based on the normal mixed linear model generalized estimating equations a four stage method and a bootstrap method that resamples clusters rather than individuals we pay special attention to the group randomized trials where the number of independent clusters is small cluster sizes are big and the correlation within the cluster is weak we show that for balanced and near balanced data when the number of independent clusters is small 10 the bootstrap is superior if analysts do not want to impose strong distribution and covariance structure assumptions otherwise ml and four stage methods are slightly better all four methods perform well when the number of independent clusters reaches 50

an evaluation of constrained randomization for the design and analysis of group randomized trials with binary outcomes group randomized trials are randomized studies that allocate intact groups of individuals to different comparison arms a frequent practical limitation to adopting such research designs is that only a limited number of groups may be available and therefore simple randomization is unable to adequately balance multiple group level covariates between arms therefore covariate based constrained randomization was proposed as an allocation technique to achieve balance constrained randomization involves generating a large number of possible allocation schemes calculating a balance score that assesses covariate imbalance limiting the randomization space to a prespecified percentage of candidate allocations and randomly selecting one scheme to implement when the outcome is binary a number of statistical issues arise regarding the potential advantages of such designs in making inference in particular properties found for continuous outcomes may not directly apply and additional variations on statistical tests are available motivated by two recent trials we conduct a series of monte carlo simulations to evaluate the statistical properties of model based and randomization based tests under both simple and constrained randomization designs with varying degrees of analysis based covariate adjustment our results indicate that constrained randomization improves the power of the linearization f test the kc corrected gee t test kauermann and carroll 2001 journal of the american statistical association 96 1387 1396 and two permutation tests when the prognostic group level variables are controlled for in the analysis and the size of randomization space is reasonably small we also demonstrate that constrained randomization reduces power loss from redundant analysis based adjustment for non prognostic covariates design considerations such as the choice of the balance metric and the size of randomization space are discussed

a completeness relation for the q analogue coherent states by q integration q integration is defined for the q oscillator realization of quantum groups this is used to prove a completeness relation for the q analogue of the usual coherent states these states are overcomplete

on the size of chaos in the mean field dynamics we consider the error arising from the approximation of an n particle dynamics with its description in terms of a one particle kinetic equation we estimate the distance between the j marginal of the system and the factorized state obtained in a mean field limit as n our analysis relies on the evolution equation for the correlation error rather than on the usual bbgky hierarchy the rate of convergence is shown to be o j 2 n in any bounded interval of time size of chaos as expected from heuristic arguments our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data covering a stochastic jump processes converging to the homogeneous boltzmann and the povzner equation and b quantum systems giving rise to the hartree equation

small sample performance of bias corrected sandwich estimators for cluster randomized trials with binary outcomes the sandwich estimator in generalized estimating equations gee approach underestimates the true variance in small samples and consequently results in inflated type i error rates in hypothesis testing this fact limits the application of the gee in cluster randomized trials crts with few clusters under various crt scenarios with correlated binary outcomes we evaluate the small sample properties of the gee wald tests using bias corrected sandwich estimators our results suggest that the gee wald z test should be avoided in the analyses of crts with few clusters even when bias corrected sandwich estimators are used with t distribution approximation the kauermann and carroll kc correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes however in cases with large variations in cluster sizes the fay and graubard fg correction should be used instead furthermore we derive a formula to calculate the power and minimum total number of clusters one needs using the t test and kc correction for the crts with binary outcomes the power levels as predicted by the proposed formula agree well with the empirical powers from the simulations the proposed methods are illustrated using real crt data we conclude that with appropriate control of type i error rates under small sample sizes we recommend the use of gee approach in crts with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure copyright 2014 john wiley sons ltd

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

on the boltzmann grad limit for the broadwell model of the boltzmann equation an example is given of a model dynamics for which the broadwell model of the boltzmann equation seems to appear in the formal stage of the boltzmann grad limit but actually does not

global validity of the boltzmann equation for two and three dimensional rare gas in vacuum erratum and improved result we point out an error in our earlier papers 1 and 2 and present a more direct and natural proof which although based on the same physical ideas of the previous ones saves and actually improves the validity results for the boltzmann equation given in 1 and 2

analysis of data from group randomized trials with repeat observations on the same groups this study used monte carlo simulations to evaluate the performance of alternative models for the analysis of group randomized trials having more than two time intervals for data collection the major distinction among the models tested was the sampling variance of the intervention effect in the mixed model anova the sampling variance of the intervention effect is based on the variance among group x time interval means in the random coefficients model the sampling variance of the intervention effect is based on the variance among the group specific slopes these models are equivalent when the design includes only two time intervals but not when there are more than two time intervals the results indicate that the mixed model anova yields unbiased estimates of sampling variation and nominal type i error rates when the group specific time trends are homogenous however when the group specific time trends are heterogeneous the mixed model anova yields downwardly biased estimates of sampling variance and inflated type i error rates in contrast the random coefficients model yields unbiased estimates of sampling variance and the nominal type i error rate regardless of the pattern among the groups we discuss implications for the analysis of group randomized trials with more than two time intervals

correlational kinetics ii boltzmann collision terms abstract a derivation of generalized boltzmann collision terms in correlational kinetics is given such a kinetic theory allows the treatment of correlations for systems of particles even with long ranged and retarded interactions the derivation of the collision terms is based upon a number of assumptions which have already been used in connection with the boltzmann equation

the boltzmann grad limit for a one dimensional boltzmann equation in a stationary state in this paper we consider a one dimensional model of interacting particles in a bounded interval with possibly not homogeneous diffusive boundary conditions we prove that when the number of particlesn goes to infinity and the interaction is suitably rescaled the boltzmann grad limit the one particle distribution function of the unique invariant measure for the particle system converges to the unique solution of the boltzmann equation of the model provided that the mean free path is sufficiently large

the propagation of chaos for a rarefied gas of hard spheres in the whole space we discuss old and new results on the mathematical justification of boltzmann s equation the classical result along these lines is a theorem which was proven by lanford in the 1970s this paper is naturally divided into three parts

from the n body schrodinger equation to the quantum boltzmann equation a term by term convergence result in the weak coupling regime in this paper we analyze the asymptotic dynamics of a system of n quantum particles in a weak coupling regime particles are assumed statistically independent at the initial time

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

maximum principles for a fourth order equation from thin plate theory abstract this paper focuses on a nonlinear equation from thin plate theory of the form d x w 1 d w c x f w 0 we obtain maximum principles for certain functions defined on the solution of this equation using p functions or auxiliary functions of the types used by payne l e payne some remarks on maximum principles j anal math 30 1976 421 433 and schaefer p w schaefer solution gradient and laplacian bounds in some nonlinear fourth order elliptic equations siam j math anal 18 1987 430 434

maximum principles and bounds for a class of fourth order nonlinear elliptic equations we deduce maximum principles for a class of fourth order nonlinear elliptic equations by using auxiliary functions containing the square of the second gradient of the solution of such equations a priori bounds on various quantities of interest are obtained

analysis of a cluster randomized trial with binary outcome data using a multi level model the use of multi level logistic regression models was explored for the analysis of data from a cluster randomized trial investigating whether a training programme for general practitioners reception staff could improve women s attendance at breast screening twenty six general practices were randomized with women nested within them requiring a two level model which allowed for between practice variability comparisons were made with fixed effect fe and random effects re cluster summary statistic methods ordinary logistic regression and a marginal model based on generalized estimating equations with robust variance estimates an fe summary statistic method and ordinary logistic regression considerably understated the variance of the intervention effect thus overstating its statistical significance the marginal model produced a higher statistical significance for the intervention effect compared to that obtained from the re summary statistic method and the multi level model because there was only a moderate number of practices and these had unbalanced cluster sizes reliable asymptotic properties for the robust standard errors used in the marginal model may not have been achieved while the re summary statistic method cannot handle multiple covariates easily marginal and multi level models can do so in contrast to multi level models however marginal models do not provide direct estimates of variance components but treat these as nuisance parameters estimates of the variance components were of particular interest in this example additionally parametric bootstrap methods within the multi level model framework provide confidence intervals for these variance components as well as a confidence interval for the effect of intervention which allows for the imprecision in the estimated variance components the assumption of normality of the random effects can be checked and the models extended to investigate multiple sources of variability

multiscale analysis for interacting particles relaxation systems and scalar conservation laws we investigate the derivation of semilinear relaxation systems and scalar conservation laws from a class of stochastic interacting particle systems these systems are markov jump processes set on a lattice they satisfy detailed mass balance but not detailed balance of momentum and are equipped with multiple scalings using a combination of correlation function methods with compactness and convergence properties of semidiscrete relaxation schemes we prove that at a mesoscopic scale the interacting particle system gives rise to a semilinear hyperbolic system of relaxation type while at a macroscopic scale it yields a scalar conservation law rates of convergence are obtained in both scalings

a stochastic system of particles modelling the euler equations we consider a system of n spheres interacting through elastic collisions at a stochastic distance in the limit n for a suitable rescaling of the interaction parameters we prove that the one particle distribution function converges to a local maxwellian whose gross density velocity and temperature satisfy the euler equation

uniform approximation and coherent state path integrals one possible formulation of mean field approximations is in terms of coherent state path integrals in this paper the construction of uniform semiclassical approximation to coherent state integrals is investigated as a specific example the method is applied to the forced harmonic oscillator

global validity of the boltzmann equation for a three dimensional rare gas in vacuum we consider a system ofn hard disks in 2 in the boltzmann grad limit i e n d 0 n d 1 0 whered is the diameter of the disks if is sufficiently small and if the joint distribution densities factorize at time zero we prove that the time evolved one particle distribution converges for all times to the solution of the boltzmann equation with the same initial datum

the effects of mixture distribution misspecification when fitting mixed effects logistic models summary mixed effects logistic models are often used to analyze binary response data which have been gathered in clusters or groups responses are assumed to follow a logistic model within clusters with an intercept which varies across clusters according to a specified probability distribution g in this paper we examine the performance of mixed effects logistic regression analysis when a main component of the model the mixture distribution is misspecified we show that when the mixture distribution is misspecified estimates of model parameters including the effects of covariates typically are asymptotically biased i e inconsistent however we present some approximations which suggest that the magnitude of the bias in the estimated covariate effects is typically small these findings are corroborated by a set of simulations which also suggest that valid variance estimates of estimated covariate effects can be obtained when the mixture distribution is misspecified

intra cluster correlation selection for cluster randomized trials in this paper we give focus to cluster randomized trials also known as group randomized trials which randomize clusters or groups of subjects to different trial arms such as intervention or control outcomes from subjects within the same cluster tend to exhibit an exchangeable correlation measured by the intra cluster correlation coefficient icc our primary interest is to test if the intervention has an impact on the marginal mean of an outcome using recently developed methods we propose how to select a working icc structure with the goal of choosing the structure that results in the smallest standard errors for regression parameter estimates and thus the greatest power for this test specifically we utilize small sample corrections for the estimation of the covariance matrix of regression parameter estimates this matrix is incorporated within correlation selection criteria proposed in the generalized estimating equations literature to choose one of multiple working icc structures under consideration we demonstrate the potential power and utility of this approach when used in cluster randomized trial settings via a simulation study and application example and we discuss practical considerations for its use in practice copyright 2016 john wiley sons ltd

bayesian methods of analysis for cluster randomized trials with binary outcome data we explore the potential of bayesian hierarchical modelling for the analysis of cluster randomized trials with binary outcome data and apply the methods to a trial randomized by general practice an approximate relationship is derived between the intracluster correlation coefficient icc and the between cluster variance used in a hierarchical logistic regression model by constructing an informative prior for the icc on the basis of available information we are thus able implicitly to specify an informative prior for the between cluster variance the approach also provides us with a credible interval for the icc for binary outcome data several approaches to constructing informative priors from empirical icc values are described we investigate the sensitivity of results to the prior specified and find that the estimate of intervention effect changes very little in this data set while its interval estimate is more sensitive the bayesian approach allows us to assume distributions other than normality for the random effects used to model the clustering this enables us to gain insight into the robustness of our parameter estimates to the classical normality assumption in a model with a more complex variance structure bayesian methods can provide credible intervals for a difference between two variance components in order for example to investigate whether the effect of intervention varies across clusters we compare our results with those obtained from classical estimation discuss the relative merits of the bayesian framework and conclude that the flexibility of the bayesian approach offers some substantial advantages although selection of prior distributions is not straightforward

kinetic limits for a class of interacting particle systems we study one dimensional particle systems in which particles travel as independent random walks and collide stochastically the collision rates are chosen so that each particle experiences finitely many collisions per unit time we establish the kinetic limit and derive the discrete boltzmann equation for the macroscopic particle density

small data existence for the enskog equation in l 1 an existence theorem for the enskog equation with small initial data is proved in anl1 setting this type of result is not available for the boltzmann equation

maximum principles and bounds in a class of fourth order uniformly elliptic equations this paper is devoted to a class of fourth order uniformly elliptic equations posed by schaefer in 1987 we obtain maximum principles for certain functions which are defined on solutions of the elliptic equations the principles are then used to deduce some bounds on important quantities in the physical problems of interest

a random effects regression model for meta analysis many meta analyses use a random effects model to account for heterogeneity among study results beyond the variation associated with fixed effects a random effects regression approach for the synthesis of 2 x 2 tables allows the inclusion of covariates that may explain heterogeneity a simulation study found that the random effects regression method performs well in the context of a meta analysis of the efficacy of a vaccine for the prevention of tuberculosis where certain factors are thought to modify vaccine efficacy a smoothed estimator of the within study variances produced less bias in the estimated regression coefficients the method provided very good power for detecting a non zero intercept term representing overall treatment efficacy but low power for detecting a weak covariate in a meta analysis of 10 studies we illustrate the model by exploring the relationship between vaccine efficacy and one factor thought to modify efficacy the model also applies to the meta analysis of continuous outcomes when covariates are present

a new approach to quantitative propagation of chaos for drift diffusion and jump processes this paper is devoted the study of the mean field limit for many particle systems undergoing jump drift or diffusion processes as well as combinations of them the main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles as the number of particles goes to infinity to this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators consistency estimate together with fine stability estimates on the flow of the limiting nonlinear equation stability estimates then we apply this method to a boltzmann collision jump process for maxwell molecules to a mckean vlasov drift diffusion process and to an inelastic boltzmann collision jump process with stochastic thermal bath to our knowledge our approach yields the first such quantitative results for a combination of jump and diffusion processes

using second order generalized estimating equations to model heterogeneous intraclass correlation in cluster randomized trials in cluster randomized trials it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters however the correlation may in fact be heterogeneous and depend on cluster characteristics accurate modeling of the correlation has the potential to improve inference we use second order generalized estimating equations to model heterogeneous correlation in cluster randomized trials using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size we apply the methods to a cluster randomized trial of an intervention to promote breast cancer screening

explaining heterogeneity in meta analysis a comparison of methods exploring the possible reasons for heterogeneity between studies is an important aspect of conducting a meta analysis this paper compares a number of methods which can be used to investigate whether a particular covariate with a value defined for each study in the meta analysis explains any heterogeneity the main example is from a meta analysis of randomized trials of serum cholesterol reduction in which the log odds ratio for coronary events is related to the average extent of cholesterol reduction achieved in each trial different forms of weighted normal errors regression and random effects logistic regression are compared these analyses quantify the extent to which heterogeneity is explained as well as the effect of cholesterol reduction on the risk of coronary events in a second example the relationship between treatment effect estimates and their precision is examined in order to assess the evidence for publication bias we conclude that methods which allow for an additive component of residual heterogeneity should be used in weighted regression a restricted maximum likelihood estimator is appropriate although a number of other estimators are also available methods which use the original form of the data explicitly for example the binomial model for observed proportions rather than assuming normality of the log odds ratios are now computationally feasible although such methods are preferable in principle they often give similar results in practice

the grad limit for a system of soft spheres a model describing the dynamics of a system of soft spheres is introduced the model contains a random mechanism which mollifies the deterministic model of hard sphere molecules it allows a rigorous deduction of the corresponding boltzmann equation for a wide class of initial data and arbitrarily long times

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

the homogeneous boltzmann hierarchy and statistical solutions to the homogeneous boltzmann equation an existence and uniqueness result for the homogeneous boltzmann hierarchy is proven by exploiting the statistical solutions to the homogeneous boltzmann equation

a stochastic model associated with enskog equation and its kinetic limit we examine a system of particles in which the particles travel deterministically in between stochastic collisions the collisions are elastic and occur with probability d when two particles are at a distance when the number of particles n goes to infinity and n d goes to a nonzero constant we show that the particle density converges to a solution of the enskog equation

baseline adjustments for binary data in repeated cross sectional cluster randomized trials analysis of covariance models which adjust for a baseline covariate are often used to compare treatment groups in a controlled trial in which individuals are randomized such analysis adjusts for any baseline imbalance and usually increases the precision of the treatment effect estimate we assess the value of such adjustments in the context of a cluster randomized trial with repeated cross sectional design and a binary outcome in such a design a new sample of individuals is taken from the clusters at each measurement occasion so that baseline adjustment has to be at the cluster level logistic regression models are used to analyse the data with cluster level random effects to allow for different outcome probabilities in each cluster we compare the estimated treatment effect and its precision in models that incorporate a covariate measuring the cluster level probabilities at baseline and those that do not in two data sets taken from a cluster randomized trial in the treatment of menorrhagia the value of baseline adjustment is only evident when the number of subjects per cluster is large we assess the generalizability of these findings by undertaking a simulation study and find that increased precision of the treatment effect requires both large cluster sizes and substantial heterogeneity between clusters at baseline but baseline imbalance arising by chance in a randomized study can always be effectively adjusted for

quantitative boltzmann gibbs principles via orthogonal polynomial duality we study fluctuation fields of orthogonal polynomials in the context of particle systems with duality we thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions where the order of every term can be quantified this implies a quantitative generalization of the boltzmann gibbs principle in the context of independent random walkers we complete this program including also fluctuation fields in non stationary context local equilibrium for other interacting particle systems with duality such as the symmetric exclusion process similar results can be obtained under precise conditions on the n particle dynamics

a simulation study of odds ratio estimation for binary outcomes from cluster randomized trials we used simulation to compare accuracy of estimation and confidence interval coverage of several methods for analysing binary outcomes from cluster randomized trials the following methods were used to estimate the population averaged intervention effect on the log odds scale marginal logistic regression models using generalized estimating equations with information sandwich estimates of standard error gee unweighted cluster level mean difference cl u weighted cluster level mean difference cl w and cluster level random effects linear regression cl re methods were compared across trials simulated with different numbers of clusters per trial arm numbers of subjects per cluster intraclass correlation coefficients rho and intervention versus control arm proportions two thousand data sets were generated for each combination of design parameter values the results showed that the gee method has generally acceptable properties including close to nominal levels of confidence interval coverage when a simple adjustment is made for data with relatively few clusters cl u and cl w have good properties for trials where the number of subjects per cluster is sufficiently large and rho is sufficiently small cl re also has good properties in this situation provided a t distribution multiplier is used for confidence interval calculation in studies with small numbers of clusters for studies where the number of subjects per cluster is small and rho is large all cluster level methods may perform poorly for studies with between 10 and 50 clusters per trial arm

some considerations on the derivation of the nonlinear quantum boltzmann equation ii the low density regime in this paper we analyse the asymptotic dynamics of a system of n identical quantum particles in a low density regime our approach follows the strategy introduced by the authors in a previous work 2 to treat the simpler weak coupling regime the time evolution of the wigner transform of the one particle reduced density matrix is represented by means of a perturbative series the expansion is obtained upon iterating the duhamel formula in the spirit of the paper by lanford 32 for short times and small interaction potential we rigorously prove that a subseries of the complete perturbative series converges to the solution of the nonlinear boltzmann equation that is physically relevant in the context an important point is that we completely identify the cross section entering the limiting boltzmann equation as being the born series expansion of quantum scattering

coherent states for the noncompact supergroups osp 2 2n r an analysis of the coherent states for the noncompact supergroup osp 2 2n r is presented in contrast to osp 1 2n r both typical and atypical representations have to be considered the measure of integration in general for osp 2 2n r coherent states is calculated it is then used to construct the decomposition of unity for the special case of osp 2 2 r it is found however that the typical and atypical representations of osp 2 2 r have to be treated separately it is verified that the coherent states for osp 2 2 r are closest to classical in the sense of perelomov

a cluster expansion approach to a one dimensional boltzmann equation a validity result we consider a stochastic particle system on the line and prove that when the number of particles diverges and the probability of a collision is suitably rescaled the system is well described by a one dimensional boltzmann equation the result holds globally in time without any smallness assumption

boltzmann grad limits for stochastic hard sphere models we study a d dimensional stochastic particle system in which the particles travel deterministically in between stochastic collisions the collisions are elastic and occur with a probability of order when two particles are at a distance less than when the number of particles n goes to infinity and n d 1 goes to a nonzero constant we show that the particle density converges to a solution of the boltzmann equation provided that d 1

on a subharmonic functional of some even order elliptic problems abstract a procedure for finding a subharmonic functional of a solution for the problem lu n i 0 1 i c i i u 0 in the domain e which is either in a plane or on a line is presented the functional obtained is used to derive the maximum principles and uniqueness theorems for some higher order elliptic problems the procedure proposed is also applicable to nonlinear differential equations

a derivation of the broadwell equation we consider a stochastic system of particles in a two dimensional lattice and prove that under a suitable limit i e n e 0 ne2 const wheren is the number of particles and e is the mesh of the lattice the one particle distribution function converges to a solution of the two dimensional broadwell equation for all times for which the solution of this equation exists propagation of chaos is also proven

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

a q analogue of bargmann space and its scalar product a q analogue of bargmann space is defined using the properties of coherent states associated with a pair of q deformed bosons the space consists of a class of entire functions of a complex variable z and has a reproducing kernel on this space the q boson creation and annihilation operators are represented as multiplication by z and q differentiation with respect to z respectively a q integral analogue of bargmann s scalar product is defined involving the q exponential as a weight function associated with this is a completeness relation for the q coherent states

diffusion approximation and hyperbolic automorphisms of the torus abstract in this article a diffusion equation is obtained as a limit of a reversible kinetic equation scaled appropriately this limiting diffusion is produced by the collisions of the particles with the boundary indeed these particles follow a reversible reflection law having convenient mixing properties this model based on arnold s cat map can be handled with fourier series instead of the symbolic dynamics associated to a markov partition as a consequence optimal convergence results can be obtained by elementary means and illustrate the apparition of irreversibility in macroscopic limits

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

large deviations from a kinetic limit we study a one dimensional particle system in which particles travel deterministically in between stochastic collisions as the total number of particles tends to infinity the empirical density converges to a solution of a discrete boltzmann equation we establish the large deviation principle for the convergence with a rate function that is given by a variational formula some of the properties of the rate function are discussed and a nonvariational expression for the rate function is given

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

on the limit of the nonlinear enskog equation corresponding with fluid dynamics we prove that if a scale of the diameter of particles is not greater than a scale of the mean free path to a high enough power if the scale of the mean free path is sufficiently small and if the non hydrodynamic part of the initial data satisfies an assumption of smallness then a solution of the initial value problem exists for the enskog equation on a macroscopic time interval as long as a smooth solution of the euler equations for compressible fluids exists this enskog solution is approximated both by the corresponding solution of the boltzmann equation and by the solution of the euler equations

slq 2 realizations for kepler and oscillator potentials and q canonical transformations the realizations of the lie algebra corresponding to the dynamical symmetry group so 2 1 of the kepler and oscillator potentials are q deformed the q canonical transformation connecting two realizations is given and a general definition for the q canonical transformation is deduced a q schrodinger equation for a kepler like potential is obtained from the q oscillator schrodinger equation the energy spectrum and the ground state wavefunction are calculated

algebraic nature of shape invariant and self similar potentials self similar potentials generalize the concept of shape invariance which was originally introduced to explore exactly solvable potentials in quantum mechanics in this paper it is shown that the previously introduced algebraic approach to the latter can be generalized to the former the infinite lie algebras introduced in this context are shown to be closely related to the q algebras the associated coherent states are investigated

coherent states for the harmonic oscillator representations of the orthosymplectic supergroup osp 1 2n r coherent states for the harmonic oscillator representations of the noncompact supergroup osp 1 2n r are introduced and the invariant integration measure is calculated by studying transformation properties of supercoset variables the generalized bogoliubov transformation for mixed systems of bosons and fermions is also obtained an example for the simple harmonic oscillator is given

on a one parameter family of q exponential functions we examine the properties of a family of q exponential functions which depend on an extra parameter these functions have a well defined meaning for both the 0 1 cases if only it is shown that any two members of this family with different values of the parameter are related to each other by a fourier gauss transformation

kinetic limits of the hpp cellular automaton we study the boltzmann grad limit in various versions of the two dimensional hpp cellular automaton in the completely deterministic case we prove convergence to an evolution that is not of kinetic type a well known phenomenon after uchyiama s paper on the broadwell gas whereas the limiting equation becomes of kinetic type in the model with random collisions the main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting between any two successive hpp updatings stirring updatings 1 being any fixed positive number and a parameter which tends to 0 the initial measure is a product measure with average occupation numbers of the order of low density limit and varying on distances of the order of 1 the limit as 0 of the system evolved for times of the order of 1 corresponds to the boltzmann grad limit we prove propagation of chaos and that the renormalized average occupation numbers i e divided by converge to the solution of the broadwell equation convergence is proven at all times for which the solution of the broadwell equation is bounded

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures