tensor notation and cumulants of polynomials a modified and extended tensor notation is introduced that is sufficient to cover multivariate moments and cumulants as special cases using this notation two basic identities are given the first of these expresses generalized cumulants in terms of ordinary cumulants the second gives the joint cumulant generating function of any polynomial transformation in terms of the cumulants of the original variables three applications of the basic identities are given the first application is concerned with sample cumulants or k statistics the second to edgeworth series and the third to exponential family models

lp gradient estimate for elliptic equations with high contrast conductivities in rn abstract uniform estimate for the solutions of elliptic equations with high contrast conductivities in r n is concerned the space domain consists of a periodic connected sub region and a periodic disconnected matrix block subset the elliptic equations have fast diffusion in the connected sub region and slow diffusion in the disconnected subset suppose 0 1 is the diameter of each matrix block and 2 0 1 is the conductivity ratio of the disconnected matrix block subset to the connected sub region it is proved that the w 1 p norm of the elliptic solutions in the connected sub region is bounded uniformly in when the l p norm of the elliptic solutions in the whole space is bounded uniformly in the w 1 p norm of the elliptic solutions in perforated domains is bounded uniformly in however the l p norm of the second order derivatives of the solutions in the connected sub region may not be bounded uniformly in

on a class of preconditioning methods for dense linear systems from boundary elements in this paper we discuss several sparse matrix based preconditioners suitable for preconditioning dense linear systems of boundary element equations all preconditioners involve only o n nonzeros we provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners for singular integral equations a more efficient preconditioner is proposed that has a band 2 structure

application of adaptive basis functions for a diagonal moment matrix solution of arbitrarily shaped three dimensional conducting body problems we present the application of specially constructed adaptive basis functions that generate a diagonal matrix in the method of moments solution procedure for the calculation of scattered electromagnetic fields from arbitrarily shaped conducting bodies excited by a plane electromagnetic wave the arbitrary body is modeled using planar triangular patches the crucial step in the solution procedure is the construction of the adaptive basis functions to generate the diagonal matrix this task is accomplished with the help of well known rwg basis functions the solution thus obtained is very efficient accurate and applicable to truly arbitrary bodies several numerical examples are presented to validate the new method

non uniform elliptic equations in convex lipschitz domains abstract non uniform elliptic equations in convex lipschitz domains are concerned the non smooth domains consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected matrix block subset to the connected sub region the w 1 p norm for p 1 of the elliptic solutions in the high permeability sub region is shown to be bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in roughly speaking if the sources in the low permeability subset are small enough the solutions in that subset are bounded uniformly in otherwise the solutions cannot be bounded uniformly in relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

a holder estimate for non uniform elliptic equations in a random medium abstract uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned the medium is separated by a random ensemble of simply closed interfaces into a connected sub region with high conductivity and a disconnected subset with low conductivity the elliptic equations whose diffusion coefficients depend on the conductivity have fast diffusion in the connected sub region and slow diffusion in the disconnected subset without a stationary ergodic assumption a uniform holder estimate in for the elliptic solutions is derived where is a realization of the random ensemble 0 1 is the length scale of the interfaces and 2 0 1 is the conductivity ratio of the disconnected subset to the connected sub region results show that if external sources are small enough in the disconnected subset the uniform holder estimate in holds in the whole domain if not it holds only in the connected sub region meanwhile the elliptic solutions change rapidly in the disconnected subset

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

uniform estimates in two periodic homogenization problems we analyze two partial differential equations that are posed on perforated domains we provide a priori estimates that do not depend on the size of the perforation a sequence of solutions is uniformly bounded in a sobolev space of regular functions the first homogenization problem concerns the laplace and the mean curvature operator with neumann boundary conditions we derive uniform lipschitz estimates for the solutions the result is used in the analysis of a free boundary system of fluid mechanics a contractive iteration yields the existence of solutions and uniform estimates the key is the use of function spaces that are different from the usual lp spaces 2000 john wiley sons inc

homogenization of multiparameter integrals where is a bounded domain in rn u rm 0 and w rn rmn 0 is supposed to be 0 1 periodic with respect to the 7rst variable x rn the distinguishing feature of 1 here is that the integrand is permitted to depend on a vector of parameters 1 k r with k 1 we are interested in the asymptotic behavior of 1 as 0 and 0 integral functionals of this type appear in the mathematical modeling of cellular composite materials when n m 3 1 can be interpreted as the stored strain energy of an elastic and heterogeneous material u being a deformation or displacement 7eld when m 1 u may be a di erence of potential in a condenser in any case the medium under consideration is composed of several materials which are periodically distributed at the microscopic scale given by in the applications we often deal with

residuals in generalized linear models abstract generalized linear models are regression type models for data not normally distributed appropriately fitted by maximum likelihood rather than least squares typical examples are models for binomial or poisson data with a linear regression model for a given ordinarily nonlinear function of the expected values of the observations use of such models has become very common in recent years and there is a clear need to study the issue of appropriate residuals to be used for diagnostic purposes several definitions of residuals are possible for generalized linear models the statistical package glim baker and nelder 1978 routinely prints out residuals where v is the function relating the variance to the mean of y and is the maximum likelihood estimate of the ith mean as fitted to the regression model these residuals are the signed square roots of the contributions to the pearson goodness of fit statistic another choice of residual is the signed square root of the contribution to the devia

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

a new form of the hypersingular boundary integral equation for 3 d acoustics and its implementation with c0 boundary elements abstract an improved weakly singular form of the hypersingular boundary integral equation hbie for 3 d acoustic wave problems is presented in this paper compared with the weakly singular form of the hbie published earlier y j liu and f j rizzo a weakly singular form of the hypersingular boundary integral equation applied to 3 d acoustic wave problems comput methods appl mech engrg 96 1992 271 287 this new form involves only tangential derivatives of the density function and thus its discretization using the boundary element method bem is easier to perform instead of using nonconforming and c1 continuous boundary elements advocated earlier c0 continuous conforming quadratic elements are employed in the discretization of this weakly singular form of the hbie some justifications on using c0 elements for hbies are provided to reflect the current views on this crucial issue for hbies it is postulated that the original c1 continuity requirement for the density function in the analytical hbie formulation can be relaxed to piecewise c1 continuity in the numerical implementation of the weakly singular forms of the hbie numerical examples of both scattering and radiation problems clearly demonstrate the accuracy and versatility of the new weakly singular form of the hbie for 3 d acoustics

regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity we prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form the equation is required to be strictly elliptic but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite d th moment where d is the dimension in the general stationary ergodic framework we show that the equation homogenizes to a deterministic uniformly elliptic equation and we obtain an explicit estimate of the effective ellipticity which is new even in the uniformly elliptic context showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory we prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity which are controlled in the macroscopic limit by the ergodic theorem we show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite p th moment for every p less than d but for which regularity and homogenization break down while these results are new even for linear equations we prove them in the fully nonlinear context in probabilistic terms our homogenization results correspond to quenched invariance principles for controlled diffusion processes in random media including linear diffusions as well as diffusions controlled by one controller or two competing players

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

homogenization of the linearized ionic transport equations in rigid periodic porous media in this paper we undertake the rigorous homogenization of a system of partial differential equations describing the transport of a n component electrolyte in a dilute newtonian solvent through a rigid porous medium the motion is governed by a small static electric field and a small hydrodynamic force which allows us to use o brien s linearized equations as the starting model we establish convergence of the homogenization procedure and discuss the homogenized equations even if the symmetry of the effective tensor is known from the literature j r looker and s l carnie transp porous media 65 107 2006 its positive definiteness does not seem to be known based on the rigorous study of the underlying equations we prove that the effective tensor satisfies onsager properties namely is symmetric positive definite this result justifies the approach of many authors who use onsager theory as starting point

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

homogenization of periodic linear degenerate pdes it is well known under the name of periodic homogenization that under a centering condition of the drift a periodic diffusion process on r d converges under diffusive rescaling to a d dimensional brownian motion existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion in this paper we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set as a consequence it is no longer the case that the effective diffusivity matrix is necessarily non degenerate it turns out that provided that some very weak regularity conditions are met the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion in particular this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank we also discuss the application of our results to the homogenization of a class of elliptic and parabolic pdes c 2008 elsevier inc all rights reserved

an iterative method to solve acoustic scattering problems using a boundary integral equation in this work a simple iterative method to solve the acoustic scattering radiation problems using the boundary integral equation bie formulation is presented the operator equation obtained in the bie formulation is converted into a matrix equation using the well known method of moments solution procedure the present method requires much fewer mathematical operations per iteration when compared to other available iterative methods further the present iterative method can easily handle multiple incident fields a highly desirable feature not available in any other iterative method much the same way as direct solution techniques several numerical examples are presented to illustrate the efficiency and accuracy of the method

linear elliptic equations in composite media with anisotropic fibres abstract linear elliptic equations in composite media with anisotropic fibres are concerned the media consist of a periodic set of anisotropic fibres with low conductivity included in a connected matrix with high conductivity inside the anisotropic fibres the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions the coefficients of the elliptic equations depend on the conductivity this work is to derive the holder and the gradient l p estimates uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix for the solutions of the elliptic equations furthermore it is shown that inside the fibres the solutions have higher regularity along the fibres than in the transverse directions

application of the method of moments to acoustic scattering from multiple bodies of arbitrary shape the method proposed in this paper provides a computational technique for solving acoustic scattering problems associated with arbitrarily shaped three dimensional bodies in particular using the free space green s function along with the associated boundary conditions on the surface of the body integral equations to calculate the velocity potential and the exterior field of the scattering body are derived the equations are then solved by approximating the scatterers by planar triangular patches and using the method of moments test cases are shown involving single and multiple bodies of spherical cylindrical and cubical geometry the method presented in this paper is novel simple and efficient and yet general enough to obtain the sonic fields scattered by multiple arbitrarily shaped bodies oriented in random fashion

convergence for elliptic equations in periodic perforated domains abstract convergence for the solutions of elliptic equations in periodic perforated domains is concerned let denote the size ratio of the holes of a periodic perforated domain to the whole domain it is known that by energy method the gradient of the solutions of elliptic equations is bounded uniformly in in l 2 space also when approaches 0 the elliptic solutions converge to a solution of some simple homogenized elliptic equation in this work above results are extended to general w 1 p space for p 1 more precisely a uniform w 1 p estimate in for p 1 and a w 1 p convergence result for p n n 2 for the elliptic solutions in periodic perforated domains are derived here n is the dimension of the space domain one also notes that the l p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in

deviance residuals and normal scores plots summary we discuss the use of normal order statistics plots based on deviance residuals to check distributional assumptions in regression models continuous and discrete error distributions are considered as are censored data misspecified error distributions and discrimination between competing models are discussed with an example residual plots to detect inadequacies in normal linear regression models have a long history an example is the normal scores or rankit plot a plot of ordered residuals against normal order statistics which is used to detect outliers and to check distributional assumptions under or overdispersion in such a plot may also indicate a misspecified systematic component of the model the purpose of this paper is to discuss such plots for regressions with nonnormal errors such as generalized linear models for which deviance residuals are commonly used deviance residuals are known to be approximately normal in many cases in this paper we briefly describe the properties of rankit plots based on them for continuous distributions outline analogous results for discrete and censored data and describe a method for discriminating between competing models with different error distributions most roads lead to rome for the normal distribution in the sense that many definitions of residuals are functions of y jt o not so for other distributions for which various

a new method to generate an almost diagonal matrix in the boundary integral equation formulation in this work a new numerical procedure is developed to generate an almost diagonal matrix for the solution of boundary integral equation formulation dealing with acoustic scattering problems the major drawback of the traditional boundary integral equation procedure resulting in a dense system matrix is eliminated in this new procedure by grouping the basis functions into a cluster the geometry of the structure is modeled by planar triangles and the basis functions are defined on the nodes by doing so one can get a benefit of a smaller size matrix to begin with furthermore by grouping these node based basis functions into a cluster an almost diagonal matrix is generated thus the solution procedure developed in this work may be utilized for very large scattering problems since the required computer resources are very low the solution procedure developed in this work is validated for the scattering cross section of the simple shapes with the closed form solutions wherever available and with the other numerical solution procedures