long time large scale limit of the wigner transform for a system of linear oscillators in one dimension we consider the long time large scale behavior of the wigner transform w t x k of the wave function corresponding to a discrete wave equation on a 1 d integer lattice with a weak multiplicative noise this model has been introduced in basile et al in phys rev lett 96 2006 to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum the kinetic limit for the wigner transform has been shown in basile et al in arch rat mech 195 1 171 203 2009 in the present paper we prove that in the unpinned case there exists 0 0 such that for any 0 0 the weak limit of w t 3 2 x k as 1 satisfies a one dimensional fractional heat equation partial t w t x hat c partial x 2 3 4 w t x with hat c 0 in the pinned case an analogous result can be claimed for w t 2 x k but the limit satisfies then the usual heat equation

stability of a novel stochastic epidemic model with double epidemic hypothesis abstract in this paper a novel sir disease transmission model is formulated under double epidemic hypothesis and stochastic perturbation we present the stability conditions of the disease free equilibrium of the sir model without stochastic perturbation and with stochastic perturbation we obtain the deterministic stability threshold 1 and 2 of the disease free equilibrium under which the disease free equilibrium is stochastically stable such that the disease will disappear finally leaving all the population susceptible the results show that double epidemic hypothesis and stochastic perturbation have significant effects on the dynamics behaviors of the model

stationary distribution of a stochastic delayed sveir epidemic model with vaccination and saturation incidence abstract in this paper we develop and analyze a stochastic delayed sveir epidemic model with vaccination and saturation incidence by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model the existence of a stationary distribution implies stochastic weak stability numerical simulations are introduced to illustrate the analytical result

extinction and stationary distribution of a stochastic sirs epidemic model with standard incidence rate and partial immunity abstract in this paper a stochastic sirs model with standard incidence rate and partial temporary immunity is proposed the sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established our results reveal that random perturbations in the environment can restrain the spread of disease that is the deterministic model neglecting random perturbations overestimates the ability of the disease to spread the numerical simulations are carried out to demonstrate the analytical results

analysis of a deterministic and a stochastic epidemic model with two distinct epidemics hypothesis abstract in this paper we discuss an epidemic model with double hypothesis combined two different transmission mechanisms first we prove the local asymptotic stability of the equilibrium solutions of deterministic system moreover we establish the global stability for the free equilibrium second we show for stochastic system the extinction and persistence in mean of the disease under some conditions on parameters of the model finally numerical simulations are used to confirm our analytical results

macroscopic behavior of microscopic oscillations in harmonic lattices via wigner husimi transforms we consider the dynamics of infinite harmonic lattices in the limit of the lattice distance tending to 0 we allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by fourier transform and linear algebra methods

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

energy transport in stochastically perturbed lattice dynamics we consider lattice dynamics with a small stochastic perturbation of order varepsilon and prove that for a space time scale of order varepsilon 1 the local spectral density wigner function evolves according to a linear transport equation describing inelastic collisions for an energy and momentum conserving chain the transport equation predicts a slow decay as 1 sqrt t for the energy current correlation in equilibrium this is in agreement with previous studies using a different method

asymptotics of the solutions of the random schrodinger equation we consider solutions of the schrodinger equation with a weak time dependent random potential it is shown that when the two point correlation function of the potential is rapidly decaying then the fourier transform hat zeta epsilon t xi of the appropriately scaled solution converges point wise in to a stochastic complex gaussian limit on the other hand when the two point correlation function decays slowly we show that the limit of hat zeta epsilon t xi has the form hat zeta 0 xi rm exp ib kappa t xi where b t is a fractional brownian motion

analysis of the deterministic and stochastic sirs epidemic models with nonlinear incidence abstract in this paper the deterministic and stochastic sirs epidemic models with nonlinear incidence are introduced and investigated for deterministic system the basic reproductive number r 0 is obtained furthermore if r 0 1 then the disease free equilibrium is globally asymptotically stable and if r 0 1 then there is a unique endemic equilibrium which is globally asymptotically stable for stochastic system to begin with we verify that there is a unique global positive solution starting from the positive initial value then when r 0 1 we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent when r 0 1 a result on fluctuation of the solution around the disease free equilibrium of deterministic model is obtained under appropriate conditions at last if the intensity of the white noise is sufficiently small and r 0 1 then there is a unique stationary distribution to stochastic system

two sample asymptotically distribution free tests for incomplete multivariate observations abstract asymptotically distribution free tests for equality of two multivariate distributions based on censored observations are proposed by choosing appropriate test scores we obtain the multivariate versions of the gehan and log rank tests an important application of the proposed tests is in the area in which multiple observations are collected on study subjects some of whom may have missing observations the patterns of missing observations are allowed to differ for the two groups to be compared these tests are particularly useful for longitudinal studies for example the analysis of repeated measurements and growth curves real examples are also provided for illustration

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

the extinction and persistence of the stochastic sis epidemic model with vaccination abstract in this paper we discuss the dynamics of a stochastic sis epidemic model with vaccination when the noise is large the infective decays exponentially to zero regardless of the magnitude of r 0 when the noise is small sufficient conditions for extinction exponentially and persistence in the mean are established the results are illustrated by computer simulations

asymptotics of the solutions of the stochastic lattice wave equation we consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing the multiplicative noise conserves energy and in the unpinned case also conserves momentum we obtain a time inhomogeneous ornstein uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data the limit is understood in the point wise sense in the former case and in the weak sense in the latter on the other hand the weak limit for square integrable initial data is deterministic

sequential comparison of changes with repeated measurements data abstract there are many clinical trials in which the patients enter sequentially and a response variable is measured repeatedly over time for each patient a group sequential procedure is proposed for comparing the rates of change between two treatment groups some existing procedures for testing the equality of means between two treatment groups with repeated measurements data such as those proposed by armitage stratton and worthington and by geary can be interpreted as special cases of the proposed procedure under a linear mixed effects model the asymptotic joint distribution of the sequentially computed statistics is derived construction of the group sequential boundaries is based on this distribution theory the extension to nonnormal distributions of errors is also discussed the proposed procedure allows for using either the sequential method of slud and wei or that of lan and demets this procedure is then applied to repeated bone density measurements on a sample of 74 middle aged women and

a stochastic chemostat model with an inhibitor and noise independent of population sizes abstract in this paper a stochastic chemostat model with an inhibitor is considered here the inhibitor is input from an external source and two organisms in chemostat compete for a nutrient firstly we show that the system has a unique global positive solution secondly by constructing some suitable lyapunov functions we investigate that the average in time of the second moment of the solutions of the stochastic model is bounded for a relatively small noise that is the asymptotic behaviors of the stochastic system around the equilibrium points of the deterministic system are studied however the sufficient large noise can make the microorganisms become extinct with probability one although the solutions to the original deterministic model may be persistent finally the obtained analytical results are illustrated by computer simulations

analysis of a novel stochastic sirs epidemic model with two different saturated incidence rates abstract this paper presents a stochastic sirs epidemic model with two different nonlinear incidence rates and double epidemic asymmetrical hypothesis and we devote to develop a mathematical method to obtain the threshold of the stochastic epidemic model we firstly investigate the boundness and extinction of the stochastic system furthermore we use ito s formula the comparison theorem and some new inequalities techniques of stochastic differential systems to discuss persistence in mean of two diseases on three cases the results indicate that stochastic fluctuations can suppress the disease outbreak finally numerical simulations about different noise disturbance coefficients are carried out to illustrate the obtained theoretical results

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

group sequential large sample t2 like chi2 tests for multivariate observations in many studies a k degree of freedom large sample 2 test is used to assess the effect of treatment on a multivariate response such as an omnibus t2 like test of a difference between two treatment groups in any of k repeated measures alternately a k df 2 test may be used to test the equality of k 1 groups in a single outcome measure jennison and turnbull biometrika 1991 78 133 141 describe group sequential 2 and f tests for normal errors linear models and proschan follmann and geller statist med 1994 13 1441 1452 describe group sequential tests for k 1 group comparisons these methods apply to sequences of statistics that can be characterized as having an independent increments variance covariance structure thus simplifying the computation of the sequential variance covariance matrix and the resulting sequential test boundaries however many commonly used statistics do not share this structure including a liang zeger biometrika 1986 73 13 22 gee longitudinal analysis with an independence working correlation structure and a wei lachin j amer statist assoc 1984 79 653 661 multivariate wilcoxon rank test among others for such analyses this paper describes the computation of group sequential boundaries for the interim analysis of emerging results using k df tests that are expressed as quadratic forms in a statistics vector that is distributed as multivariate normal at least asymptotically we derive the elements of the covariance matrix of multiple successive k df 2 statistics based on established theorems on the distribution of quadratic forms this covariance matrix is estimated by augmenting the data from the successive interim analyses into a single analysis from which the component sequential tests and their variance covariance matrix can then be extracted boundary values for the sequential statistics can then be computed using the method of slud and wei j amer statist assoc 1982 77 862 868 or using the spending function of lan and demets biometrika 1983 70 659 663 with a surrogate measure of information an example is presented using the analysis of repeated cholesterol measurements in a clinical trial copyright 2003 john wiley sons ltd

exact calculations for sequential t x2 and f tests summary sequential and group sequential procedures are proposed for monitoring repeated t x2or f statistics these can be used to construct sequential hypothesis tests or repeated confidence intervals when the parameter of interest is a normal mean with unknown variance or a multivariate normal mean with variance matrix known or known up to a scale factor exact methods for calculating error probabilities and sample size distributions are described and tables of critical values needed to implement the procedures are provided

interim analyses with repeated measurements in a sequential clinical trial summary in comparing two treatments suppose that patients are available for the study sequentially repeated measurements of the same characteristic are collected serially over time for each study patient sequential procedures are proposed to monitor the trial for early evidence of treatment difference or harmful side effects based on such repeated measurements all the methods are illustrated with a real life example

a class of stochastic delayed sir epidemic models with generalized nonlinear incidence rate and temporary immunity abstract a class of sir epidemic model with generalized nonlinear incidence rate is presented in this paper temporary immunity and stochastic perturbation are also considered the existence and uniqueness of the global positive solution is achieved sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established moreover the threshold behavior is discussed and the threshold value r 0 is obtained we show that if r 0 1 the disease eventually becomes extinct with probability one whereas if r 0 1 then the system remains permanent in the mean

the threshold of a stochastic delayed sir epidemic model with temporary immunity abstract this paper is concerned with the asymptotic properties of a stochastic delayed sir epidemic model with temporary immunity sufficient conditions for extinction and persistence in the mean of the epidemic are established the threshold between persistence in the mean and extinction of the epidemic is obtained compared with the corresponding deterministic model the threshold affected by the white noise is smaller than the basic reproduction number r 0 of the deterministic system

local and global bifurcations in an sirs epidemic model this paper studies the existence and stability of the disease free equilibrium and endemic equilibria for the sirs epidemic model with the saturated incidence rate considering the factor of population dynamics such as the disease related the natural mortality and the constant recruitment of population analytical techniques are used to show for some parameter values the periodic solutions can arise through the hopf bifurcation which is important to carry different strategies for the controlling disease then the codimension two bifurcation i e bt bifurcation is investigated by using a global qualitative method and the curves of saddle node bifurcation hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium moreover several numerical simulations are given to support the theoretical analysis

dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis abstract in this paper we propose new mathematical models with nonlinear incidence rate and double epidemic hypothesis then we dedicate to develop a method to obtain the threshold of the stochastic sis epidemic model to this end first we investigate the stability of the equilibria of the deterministic system and obtain the conditions for the extinction and the permanence of two epidemic diseases second we explore and obtain the threshold of a stochastic sis system for the extinction and the permanence in mean of two epidemic diseases the results show that a large stochastic disturbance can cause infectious diseases to go to extinction in other words the persistent infectious disease of a deterministic system can become extinct due to the white noise stochastic disturbance this implies that the stochastic disturbance is conducive to epidemic diseases control to illustrate the performance of the theoretical results we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients

a delayed vaccinated epidemic model with nonlinear incidence rate and levy jumps abstract a stochastic susceptible infectious recovered epidemic model with nonlinear incidence rate is formulated to discuss the effects of temporary immunity vaccination and le vy jumps on the transmission of diseases we first determine the existence of a unique global positive solution and a positively invariant set for the stochastic system sufficient conditions for extinction and persistence in the mean of the disease are then achieved by constructing suitable lyapunov functions based on the analysis we conclude that noise intensity and the validity period of vaccination greatly influence the transmission dynamics of the system

the phonon boltzmann equation properties and link to weakly anharmonic lattice dynamics for low density gases the validity of the boltzmann transport equation is well established the central object is the one particle distribution function f which in the boltzmann grad limit satisfies the boltzmann equation grad and much refined cercignani argue for the existence of this limit on the basis of the bbgky hierarchy for hard spheres at least for a short kinetic time span the argument can be made mathematically precise following the seminal work of lanford in this article a corresponding program is undertaken for weakly nonlinear both discrete and continuum wave equations our working example is the harmonic lattice with a weakly nonquadratic on site potential we argue that the role of the boltzmann f function is taken over by the wigner function which is a very convenient device to filter the slow degrees of freedom the wigner function so to speak labels locally the covariances of dynamically almost stationary measures one route to the phonon boltzmann equation is a gaussian decoupling which is based on the fact that the purely harmonic dynamics has very good mixing properties as a further approach the expansion in terms of feynman diagrams is outlined both methods are extended to the quantized version of the weakly nonlinear wave equation the resulting phonon boltzmann equation has been hardly studied on a rigorous level as one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal wigner functions for three phonon processes such a result requires extra conditions on the dispersion law we also outline the reasoning leading to fourier s law for heat conduction

group sequential distribution free methods for the analysis of multivariate observations many studies involve the collection of multivariate observations such as repeated measures on two groups of subjects who are recruited over time i e with staggered entry of subjects various marginal distribution free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random using the multivariate u statistic of wei and johnson 1985 biometrika 72 359 364 we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially both within and among subjects we describe a multivariate generalization of the hodges and lehmann 1963 annals of mathematical statistics 34 598 611 estimator of a location shift that can be obtained via the multivariate u statistic with the mann whitney wilcoxon kernel we then describe large sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures we also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed these methods are applied to a real life example

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

threshold behavior of a stochastic sis model with levy jumps in this paper the dynamics of a stochastic sis model with levy jumps are investigated we first prove that this model has a unique global positive solution starting from the positive initial value then taking the accumulated jump size into account we find a threshold of the model denoted by r 0 which completely determines the extinction and prevalence of the disease if r 0 1 the disease dies out exponentially with probability one if r 0 1 the solution of the model tends to a point in time average which leads to the stochastical persistence of the disease from the view of epidemiology the existence of threshold is useful in determining treatment strategies and forecasting epidemic dynamics moreover we find that levy noise can suppress disease outbreak finally we introduce some numerical simulations to support the main results obtained