existence of weak solutions for a hyperbolic model of chemosensitive movement a hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered in contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account this qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment the model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation we assume that the speed depends on the external stimulus only and not on its gradients in that case solutions with steep gradients are expected which have the interpretation of moving swarms a notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown the proof relies on the vanishing viscosity method i e we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

bivariate categorical data analysis using normal linear conditional multinomial probability model bivariate multinomial data such as the left and right eyes retinopathy status data are analyzed either by using a joint bivariate probability model or by exploiting certain odds ratio based association models however the joint bivariate probability model yields marginal probabilities which are complicated functions of marginal and association parameters for both variables and the odds ratio based association model treats the odds ratios involved in the joint probabilities as working parameters which are consequently estimated through certain arbitrary working regression models also this later odds ratio based model does not provide any easy interpretations of the correlations between two categorical variables on the basis of pre specified marginal probabilities in this paper we develop a bivariate normal type linear conditional multinomial probability model to understand the correlations between two categorical variables the parameters involved in the model are consistently estimated using the optimal likelihood and generalized quasi likelihood approaches the proposed model and the inferences are illustrated through an intensive simulation study as well as an analysis of the well known wisconsin diabetic retinopathy status data copyright 2014 john wiley sons ltd

mixed effects models with bivariate and univariate association parameters for longitudinal bivariate binary response data summary when two binary responses are measured for each study subject across time it may be of interest to model how the bivariate associations and marginal univariate risks involving the two responses change across time to achieve such a goal marginal models with bivariate log odds ratio and univariate logit components are extended to include random effects for all components specifically separate normal random effects are specified on the log odds ratio scale for bivariate responses and on the logit scale for univariate responses assuming conditional independence given the random effects facilitates the modeling of bivariate associations across time with missing at random incomplete data we fit the model to a dataset for which such structures are feasible a longitudinal randomized trial of a cardiovascular educational program where the responses of interest are change in hypertension and hypercholestemia status the proposed model is compared to a naive bivariate model that assumes independence between time points and univariate mixed effects logit models

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

error bounds for finite difference approximations for a class of nonlinear parabolic systems in this paper we establish error bounds for a finite difference approximation to solutions of certain parabolic systems of the form t f po u we assume that the cauchy data is of class bv and we show that the sup norm of the error is bounded by o axiln axi at positive times

marginal methods for incomplete longitudinal data arising in clusters inverse probability weighted generalized estimating equations are commonly used to deal with incomplete longitudinal data arising from a missing at random mechanism when the marginal means are of primary interest in many cases however the repeated measurements themselves may arise in clusters which leads to both a cross sectional and a longitudinal correlation structure in some applications the degree of these types of correlation may become of scientific interest here we develop inverse probability weighted second order estimating equations for monotone missing data patterns which under specified assumptions facilitate consistent estimation of the marginal mean parameters and association parameters here the missing data model accommodates cross sectional clustering in the missing data indicators and the probabilities are estimated under a multivariate plackett model for computational reasons we also consider using the alternating logistic regression algorithm for estimation of the associatio

generalized estimating equations for correlated binary data using the odds ratio as a measure of association summary moment methods for analyzing repeated binary responses have been proposed by liang zeger 1986 and extended by prentice 1988 in their generalized estimating equations both liang zeger 1986 and prentice 1988 estimate the parameters associated with the expected value of an individual s vector of binary responses as well as the correlations between pairs of binary responses because the odds ratio has many desirable properties and some investigators may find the odds ratio is easier to interpret we discuss modelling the association between binary responses at pairs of times with the odds ratio we then modify the estimating equations of prentice to estimate the odds ratios in simulations the parameter estimates for the logistic regression model for the marginal probabilities appear slightly more efficient when using the odds ratio parameterization

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

a bivariate cumulative probit regression model for ordered categorical data this paper proposes a latent variable regression model for bivariate ordered categorical data and develops the necessary numerical procedure for parameter estimation the proposed model is an extension of the standard bivariate probit model for dichotomous data to ordered categorical data with more than two categories for each margin in addition the proposed model allows for different covariates for the margins which is characteristic of data from typical ophthalmological studies it utilizes the stochastic ordering implicit in the data and the correlation coefficient of the bivariate normal distribution in expressing intra subject dependency illustration of the proposed model uses data from the wisconsin epidemiologic study of diabetic retinopathy for identifying risk factors for diabetic retinopathy among younger onset diabetics the proposed regression model also applies to other clinical or epidemiological studies that involve paired organs

miscellanea on the efficiency of regression estimators in generalised linear models for longitudinal data summary liang zeger 1986 introduced a generalised estimating equations approach based on a working correlation matrix to obtain consistent and efficient estimators of regression parameters in the class of generalised linear models for repeated measures data as demonstrated by crowder 1995 because of the uncertainty of definition of the working correlation matrix the liang zeger approach may in some cases lead to a complete breakdown of the estimation of the regression parameters in this paper we show that even though the liang zeger approach in many situations yields consistent estimators for the regression parameters these estimators are usually inefficient as compared to the regression estimators obtained by using the independence estimating equations approach

stability results for a diffusion equation with functional drift approximating a chemotaxis model a hyperbolic parabolic chemotaxis system modelling aggre gation of motile cells by production of a diffusible chemoattractant is ap proximated by a scalar diffusion equation for the cell density where the drift term is an explicit functional of the current density profile we prove the unique existence and using the hopf cole transformation the local stability of an equilibrium i e a steady aggregation state we also discuss the limit ing hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps 1 introduction and development of the model in this note we shall de velop and explore the behavior of a simple two speed one dimensional model which describes the motion of chemosensitive cells that is cells whose motions are influ enced by a chemical gradient dp dx where p p x t denotes the concentration of a diffusible substance produced by each cell a well known example is the mor phogenetic aggregation of slime mold ceils 5 which previously has been modelled by keller and segel 6 using a coupled system of two equations for p and the cell density n u x t the so called chemotaxis system for a recent mathematical analysis see 4 12 in most cases as in slime molds however parameter estimations reveal that the diffusion coefficient of the chemoattractant dp 10 7cm2 sec is about ten times larger than the motility of the cells du 10 8 cm2 sec compare for example 3 if also the production rate of the chemoattractant is relatively large then the quasi stationary gradient of the rapidly diffusing substance approx imately determines the chemotactic drift of the cells this finally leads to a scalar diffusion equation for the cell density u of type

functionals of complex ornstein uhlenbeck processes abstract the exact distribution of the sufficient statistics and distribution of the maximum likelihood estimator of the drift damping parameter in a stationary complex ornstein uhlenbeck process given by 1 1 is investigated complete tables of the distribution function for different levels are given by the help of matlab the comparison with the earlier calculations are discussed the relation of the famous model of chandler wobble proposed by kolmogorov is investigated 1

analyzing bivariate ordinal data using a global odds ratio abstract a moment method is proposed for regression analysis of bivariate ordered categorical data using the global odds ratio as the measure of association for modeling the margins this method utilizes the stochastic ordering implicit in the data this method allows for covariate effects in modeling the association between responses an application of the proposed model is illustrated using the ophthalmological data from the wisconsin epidemiologic study of diabetic retinopathy for identifying risk factors among younger onset diabetics this model is also extended to a longitudinal data setting with more than two repeated measures

convergence results for an inhomogeneous system arising in various high frequency approximations this paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of hamilton jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the schrodinger equation we first state an existence and uniqueness result in the framework of viscosity and duality solutions then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence as an illustration some practical computations are provided

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

asymptotic high order mass preserving schemes for a hyperbolic model of chemotaxis we introduce a new class of finite difference schemes for approximating the solutions to an initial boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term which arises as a simple model of chemotaxis since the solutions to this problem may converge to nonconstant asymptotic states for large times standard schemes usually fail to yield a good approximation therefore we propose a new class of schemes which use an asymptotic higher order correction second and third order in our examples to balance the effects of the source term and the influence of the asymptotic solutions special care is needed to deal with boundary conditions to avoid harmful loss of mass convergence results are proved for these new schemes and several numerical tests are presented and discussed to verify the effectiveness of their behavior

hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves we consider a class of kinetic equations equipped with a single conservation law which generate l 1 contractions we discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a possibly degenerate parabolic equation the limits are obtained in the dissipative sense equivalent to the notion of entropy solutions for conservation laws which permits the use of the perturbed test function method and allows for simple proofs a general compactness framework is obtained for the diffusive scaling in l 1 the radiative transport equations satisfied by the wigner function for random acoustic waves present such a kinetic model that is endowed with conservation of energy the general theory is used to validate the diffusive approximation of the radiative transport equation

analysing ordinal longitudinal survey data generalised estimating equations approach summary longitudinal survey data may comprise ordinal polytomous repeated observations and a set of multidimensional covariates for a large number of individuals one of the main goals of the longitudinal survey is then to describe the marginal expectation of the ordinal polytomous outcome variable as a function of the covariates while accounting for the structural as well as longitudinal correlations the structural correlations come from the polytomous nature of the response variable and the longitudinal correlations from the repetition of the polytomous responses over time in this paper we develop a generalised estimating equations approach based on autocorrelation structure to deal with multivariate polytomous longitudinal survey data the univariate and bivariate analyses being special cases the regression estimators are shown to be consistent for the corresponding regression parameters the methods are illustrated by using the survey of labour and income dynamics data from statistics canada

the convergence of numerical transfer schemes in diffusive regimes i discrete ordinate method in highly diffusive regimes the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path epsilon tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis a numerical scheme for solving this problem has an epsilon 1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small epsilon thus degrading its performance in diffusive regimes in this paper we show that whenever the discrete ordinate method has the correct diffusion limit both in the interior and at the boundaries its solutions converge to the solution of the transport equation uniformly in epsilon our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete ordinate method

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

on the use of a working correlation matrix in using generalised linear models for repeated measures summary in a seminal paper liang zeger 1986 extended the use of generalised linear models to repeated measures data they based the analysis on specifications for the means and variances of the observations as usual for generalised linear models but showed how specifications for the correlations between measurements made on the same unit could be avoided by using a working correlation matrix in some cases the parameters involved in this matrix are subject to an uncertainty of definition which can lead to a breakdown of the asymptotic properties of the estimators

on the existence of a travelling wave solution for a model of actin based bacterial movement some bacteria like listeria monocytogenes shigella and rickettsia rickettsii can move inside the host cell thanks to an actin tail in this paper a one dimensional model for the motion of these bacteria introduced by b bazaliy y bazaliy and a friedman 2007 in 2 is studied in particular the model is a system of partial differential equation with two moving boundaries for which we prove the existence of a travelling wave solution

convergence of a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper a new explicit finite element method for numerically solving the drift diffusion semiconductor device equations is introduced and analyzed the method uses a mixed finite element method for the approximation of the electric field a finite element method using discontinuous finite elements is used to approximate the concentrations which may display strong gradients the use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable in this paper we carry out the analysis of the model method which employs a continuous piecewise linear approximation to the electric field and a piecewise constant approximation to the electron concentration in a model problem namely the so called unipolar case with the diffusion terms neglected the resulting system of equations is equivalent to a conservation law whose flux the electric field depends globally on the solution the concentration of electrons by exploiting the similarities of this system with classical scalar conservation laws the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme the scheme considered as a scheme for the electron concentration is shown to satisfy a maximum principle and to be total variation bounded its convergence to the unique weak solution is proven numerical experiments displaying the performance of the scheme are shown

global existence of classical solutions to a combined chemotaxis haptotaxis model with logistic source abstract this paper deals with a mathematical model of cancer invasion of tissue the model consists of a system of reaction diffusion taxis partial differential equations describing interactions between cancer cells matrix degrading enzymes and the host tissue in two space dimensions we prove global existence and uniqueness of classical solutions to this model for any 0 where is the logistic growth rate of cancer cells the crucial point of proof is to raise the regularity estimate of a solution from l 1 to l 3 0 t where r 2 is some bounded domain and t 0 is some constant this paper develops new estimate techniques and improves greatly our previous results y tao m wang global solution for a chemotactic haptotactic model of cancer invasion nonlinearity 21 2008 2221 2238 in 2 dimensions