numerical solution of compartmental models by meshless and finite difference methods abstract in this paper an operator splitting method based on meshless and finite difference procedures is being considered for numerical solution of compartmental epidemiological population models with and without diffusion a one step explicit meshless procedure is also applied for the numerical solution of the nonlinear model the compartmental model contains susceptible vaccinated exposed infected and recovered sveir classes of the population effects of the diffusion on the simulation results of the model are being studied stability of endemic equilibrium point along with bifurcation analysis has also been investigated due to non availability of the exact solution the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well

on the l p solvability of higher order parabolic and elliptic systems with bmo coefficients we prove the solvability in sobolev spaces for both divergence and non divergence form higher order parabolic and elliptic systems in the whole space on a half space and on a bounded domain the leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders for the proof we develop a set of new techniques to produce mean oscillation estimates for systems on a half space

uniformly nondegenerate elliptic equations with partially bmo coefficients in nonsmooth domains abstract a global estimate in weighted lorentz sobolev spaces is obtained for the weak solutions to divergence form uniformly nondegenerate elliptic equations over a bounded nonsmooth domain here the leading coefficients are assumed to be merely measurable in one variable and have small bmo semi norms in the remaining variables under the assumption that one variable direction is perpendicular to the boundary points which is close to the boundary while a geometric assumption on the boundary is a locally bounded reifenberg flatness in addition we also investigate regularities in lorentz morrey morrey and holder spaces for elliptic equations under the same assumptions on the leading coefficients and the boundary of domain

testing for autoregressive disturbances in a time series regression with missing observations abstract this paper constructs tests for autoregressive disturbances in a time series regression with missing observations where the disturbance terms are generated by i an ar 1 process and ii an ar p process with a possible seasonal component a point optimal invariant poi test is constructed for each problem in addition we show how to compute exact small sample p values for the tests in o n operations and give a computationally efficient procedure for choosing the specific alternative against which to make poi test most powerful invariant

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

elliptic equations with measurable coefficients in reifenberg domains abstract we prove w 1 p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in n 1 directions we assume the boundary to be reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary

global weighted estimates for quasilinear elliptic equations with non standard growth abstract in this paper we obtain a new global gradient estimates in weighted lorentz spaces for weak solutions of p x laplacian type equation with small bmo coefficients in a reifenberg flat domain the modified vitali covering lemma the maximal function technique and the appropriate localization method are the main analytical tools our results improve the known results for such equations

testing for serial correlation in regression with missing observations summary in order to test for serial correlation in residuals for static time series regression in the presence of missing data the score principle is applied both to the likelihood conditional on the observation times and to an unconditional form of likelihood asymptotic distributions of the test statistics are established under both the null hypothesis of no serial correlation and sequences of local correlated alternatives enabling analytic comparison of efficiency

pattern formation of a spatial predator prey system abstract there are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors to investigate how these migration directed movement and diffusion random movement affect predator prey systems we have studied the spatiotemporal complexity in a predator prey system with holling tanner form a theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated at the same time we present the properties of pattern solutions the results of numerical simulations show that migration has prominent effect on the pattern formation of the population i e changing turing pattern into traveling pattern this study suggests that modelling by migration and diffusion in predator prey systems can account for the dynamical complexity of ecosystems

numerical study of an influenza epidemic model with diffusion abstract a diffusive epidemic model is investigated with a view to describe the transmission of influenza as an epidemic the equations are solved numerically using the splitting method under different initial distribution of population density it is shown that the initial population distribution and diffusion play an important role for spread of disease it is also shown that interventions medical and nonmedical significantly slow down the spread of disease stability of equilibria of the numerical solutions are also established

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

computing p values for the generalized durbin watson and other invariant test statistics abstract shively ansley and kohn 1990 give an o n algorithm for computing the p values of the durbin watson and other invariant test statistics in time series regression they do so by evaluating the characteristic function of a quadratic form in standard normal random variables and then numerically inverting it in this paper we obtain a new expression for the characteristic function which simplifies the handling of the independent regressors and so is easier to evaluate we also obtain general easily computable bounds on the integration and truncation errors which arise in the numerical inversion of the characteristic function empirical results are presented on the speed and accuracy of our algorithm

calderon zygmund theory for nonlinear elliptic problems with irregular obstacles abstract we consider a nonhomogeneous elliptic problem with an irregular obstacle involving a discontinuous nonlinearity over an irregular domain in divergence form of p laplacian type to establish the global calderon zygmund estimate by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the nonhomogeneous term under the bmo smallness of the nonlinearity and sufficient flatness of the boundary in the reifenberg sense

lorentz estimates for degenerate and singular evolutionary systems we prove estimates of calderon zygmund type for evolutionary p laplacian systems in the setting of lorentz spaces we suppose the coefficients of the system to satisfy only a vmo condition with respect to the space variable our results hold true mutatis mutandis also for stationary p laplacian systems

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

marcinkiewicz estimates for degenerate parabolic equations with measure data we consider solutions to degenerate parabolic equations with measurable coefficients having on the right hand side a measure satisfying a suitable density condition we prove integrability results for the gradient in the marcinkiewicz scale

testing for fourth order autocorrelation in regression disturbances when first order autocorrrelation is present abstract this paper considers the problem of testing the null hypothesis of first order autoregressive disturbances in the linear regression model against the alternative that the disturbances follow a joint first order simple fourth order autoregressive process the class of approximate point optimal invariant tests are discussed and rules are given for choosing an appropriate member from this class of tests the beneficial nature of these rules is illustrated by a limited empirical power comparison which shows the recommended test has good small sample power properties

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

variable lorentz estimate for nonlinear elliptic equations with partially regular nonlinearities abstract we prove global calderon zygmund type estimate in lorentz spaces for variable power of the gradients to weak solution of nonlinear elliptic equations in a non smooth domain we mainly assume that the nonlinearities are merely measurable in one of the spatial variables and have sufficiently small bmo semi norm in the other variables the boundary of domain belongs to reifenberg flatness and the variable exponents p x satisfy log holder continuity

elliptic equations with measurable nonlinearities in nonsmooth domains abstract we study elliptic equations with measurable nonlinearities in nonsmooth domains we establish an optimal global w 1 p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small bmo semi norm in the other variables while the underlying domain is sufficiently flat in the reifenberg sense that the boundary of the domain is locally trapped between two narrow strips

elliptic equations in divergence form with partially bmo coefficients the solvability in sobolev spaces is proved for divergence form second order elliptic equations in the whole space a half space and a bounded lipschitz domain for equations in the whole space or a half space the leading coefficients aij are assumed to be only measurable in one direction and have locally small bmo semi norms in the other directions for equations in a bounded domain additionally we assume that aij have small bmo semi norms in a neighborhood of the boundary of the domain we give a unified approach of both the dirichlet boundary problem and the conormal derivative problem we also investigate elliptic equations in sobolev spaces with mixed norms under the same assumptions on the coefficients

global regularity in lorentz spaces for nonlinear elliptic equations with lp logl growth abstract we prove a global calderon zygmund type estimate in the framework of lorentz spaces for the gradients of weak solutions of nonlinear elliptic equations with l p log l growth more precisely by considering the zero dirichlet problem of the elliptic equation div a x d u div h x f x with a x d a x p x log e for any 0 a x 0 0 and h x f p x 1 log e f we obtain x f l t q x d u l t q for t 1 and q 0 where x p x log e under the assumptions that the function a has a small bounded mean oscillation seminorm p x satisfies strong log holder continuity and the underlying domain is reifenberg flat

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

reproduction numbers and the stability of equilibria of si models for heterogeneous populations a major project in deterministic epidemiological modeling of heterogeneous populations is to find conditions for the local and global stability of the equilibria and to work out the relations among these stability conditions the thresholds for epidemic take off and endemicity and the basic reproduction number s most of the work to date has been on models of homogeneous populations of constant size motivated by their analysis of models of the dynamics of human immunodeficiency virus acquired immunodeficiency syndrome hiv aids the authors carry out this project for si models of diseases that have multiple stages which can lead to death and which infect heterogeneous populations with intricate mixing patterns and varying sizes in such models it is even difficult to find the analytical expression for the stability threshold and the reproduction number the authors show how the number of disease transmitting contacts by infectives can be used as a lyapunov function to carry out a systematic stabili

the alternative durbin watson test an assessment of durbin and watson s choice of test statistic abstract this paper examines durbin and watson s 1950 choice of test statistic for their test of first 0order autoregressive regression disturbances attention is focused on an alternative statistic d theoretical and empirical power properties of the d test are compared with those of the durbin watson test the former is found to be locally best invariant while the latter is approximately locally best invariant the d test is also found to be more powerful than its counterpart against negative autocorrelation and for small values of the autocorrelation coefficient against positive autocorrelation selected bounds for significance points of d are tabulated

lorentz estimates for fully nonlinear parabolic and elliptic equations abstract we prove an interior lorentz estimate of the hessian of strong solutions to fully nonlinear parabolic equations u t f d 2 u x t f x t and elliptic equations f d 2 u x f x respectively here we assume that the associated nonlinearities satisfy uniformly parabolic condition or ellipticity certain growth condition and the r vanishing condition we establish lorentz estimates for such fully nonlinear equations based on the approach of the large m inequality principle introduced by acerbi mingione

a point optimal test for autoregressive disturbances abstract this paper is concerned with testing for first order autoregressive disturbances in the linear regression model and recommends an alternative test to the durbin watson test the new test is most powerful invariant in a given neighbourhood of the alternative hypothesis parameter space an empirical power comparison indicates that the test is generally more powerful than the durbin watson test the comparison also suggests that for many economic applications the difference in power will be small although circumstances do exist in which the power advantage of the new test is very real selected bounds for the test s significance points are tabulated

diffusion has no influence on the global asymptotical stability of the lotka volterra prey predator model incorporating a constant number of prey refuges by constructing a suitable lyapunov function we obtain the global asymptotical stability of the lotka volterra prey predator model incorporating a constant number of prey refuges and diffusion it is shown that diffusion has no influence on the global asymptotical stability of the system with homogeneous neumann boundary condition

existence of traveling wave solutions for influenza model with treatment abstract to investigate the spreading speed of influenza and the influence of treatment on the spreading speed a reaction diffusion influenza model with treatment is established the existence of traveling wave solutions is shown by introducing an auxiliary system and applying the schauder fixed point theorem the non existence of traveling wave solutions is proved by a two sided laplace transform which needs a new approach for the prior estimate of exponential decay of traveling wave solutions

w 1 p regularity for elliptic equations with measurable coefficients in nonsmooth domains we establish global w 1 p estimates for second order elliptic equations in divergence form under the natural assumption that p is log holder continuous to this end we assume that the coefficients are measurable in one variable and have small bmo semi norms in the other variables and the boundary of the domain is reifenberg flat our work is an optimal and natural extension of w 1 p regularity for such equations with merely measurable coefficients beyond lipschitz domains

global estimates for quasilinear elliptic equations on reifenberg flat domains new global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems the coefficients are assumed to have small bmo seminorms and the boundary of the domain is sufficiently flat in the sense of reifenberg the main regularity estimates obtained are in weighted lorentz spaces other regularity results in lorentz morrey morrey and holder spaces are shown to follow from the main estimates

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

an analysis of tests for regression coefficient stability abstract an exact small sample point optimal test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a first order autoregressive process in addition it is shown how watson and engle s 1985 asymptotic test can be modified so that its small sample level can be computed and its small sample power properties investigated simulation results show that the point optimal test developed in this paperoutperforms the small sample watson engle test as well as other tests given in the literature a point optimal test is also constructed for a stochastic coefficient generated by an arima 1 1 0 process