f murtagh 2005 correspondence analysis and data coding with java and r 230 pp us 76 00 isbn 1584885289

a stochastic model for microtubule length dynamics abstract in this paper we use random walk theory to describe the length dynamics of microtubules one of the principal components of the cytoskeleton we present a simple two state model growing and shrinking of microtubule length evolution that incorporates a variable rate of switching between the states using the generating function technique we calculate the mean length of microtubule its variance and diffusion coefficient we also report analytical and computer simulation results for the mean number of positive monomers in microtubule and find good qualitative agreement with experimental data

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

a quantitative central limit theorem for the effective conductance on the discrete torus we study a random conductance problem on a d dimensional discrete torus of size l 0 the conductances are independent identically distributed random variables uniformly bounded from above and below by positive constants the effective conductance al of the network is a random variable depending on l that converges almost surely to the homogenized conductance ahom our main result is a quantitative central limit theorem for this quantity as l in particular we prove there exists some 0 such that dk ld 2 a ahom g l d 2 logd l where dk is the kolmogorov distance and gis a standard normal variable the main achievement of this contribution is the precise asymptotic description of the variance of al 2015 wiley periodicals inc

normal approximation for a random elliptic equation we consider solutions of an elliptic partial differential equation in mathbb r d with a stationary random conductivity coefficient that is also periodic with period l boundary conditions on a square domain of width l are arranged so that the solution has a macroscopic unit gradient we then consider the average flux that results from this imposed boundary condition it is known that in the limit l rightarrow infty this quantity converges to a deterministic constant almost surely our main result is that the law of this random variable is very close to that of a normal random variable if the domain size l is large we quantify this approximation by an error estimate in total variation the error estimate relies on a second order poincare inequality developed recently by chatterjee

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

nonlinear stochastic homogenization and ergodic theory on etudie l homogeneisation stochastique non lineaire d une classe de fonctionnelles integrales on relie le probleme a la theorie ergodique

noise stability and central limit theorems for effective resistance of random electric networks we investigate the generalized walsh decomposition of point to point effective resistances on countable random electric networks with i i d resistances we show that it is concentrated on low levels and thus point to point effective resistances are uniformly stable to noise for graphs that satisfy some homogeneity property we show in addition that it is concentrated on sets of small diameter as a consequence we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length n in zd when n goes to infinity

an analysis and synthesis of multiple correspondence analysis optimal scaling dual scaling homogeneity analysis and other methods for quantifying categorical multivariate data we discuss a variety of methods for quantifying categorical multivariate data these methods have been proposed in many different countries by many different authors under many different names in the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data in the second major section of the paper we introduce the notion of a duality diagram and use this diagram to synthesize the many superficially different methods into a single method

multivariate analysis with linearizable regressions we study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables this class seems more realistic than the multivariate normal or the elliptical distributions and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis in particular a two stage procedure which first scales the variables optimally and then fits a simultaneous equations model is studied in detail and is shown to have some desirable properties

the additive structure of elliptic homogenization one of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions since the latter are nonlocal functions of the former in this paper we address this problem in a new way in the context of linear elliptic equations in divergence form by showing that certain quantities associated to the energy density of solutions are essentially additive as a result we are able to prove quantitative estimates on the weak convergence of the gradients fluxes and energy densities of the first order correctors under blow down which are optimal in both scaling and stochastic integrability the proof of the additivity is a bootstrap argument completing the program initiated in armstrong et al commun math phys 347 2 315 361 2016 using the regularity theory recently developed for stochastic homogenization we reduce the error in additivity as we pass to larger and larger length scales in the second part of the paper we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables in particular we prove that the first order correctors converge in the large scale limit to a variant of the gaussian free field

correspondence analysis used complementary to loglinear analysis loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables in this paper we will show that there are cases in which these two techniques can be used complementary to each other more specifically we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices each following a specific loglinear model therefore in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models a generalization of correspondence analysis recently proposed by escofier will also be discussed with this decomposition which includes classical correspondence analysis as a special case it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis in this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models

a central limit theorem for the effective conductance linear boundary data and small ellipticity contrasts given a resistor network on mathbb z d with nearest neighbor conductances the effective conductance in a finite set with a given boundary condition is the minimum of the dirichlet energy over functions with the prescribed boundary values for shift ergodic conductances linear dirichlet boundary conditions and square boxes the effective conductance scaled by the volume of the box converges to a deterministic limit as the box size tends to infinity here we prove that for i i d conductances with a small ellipticity contrast also a non degenerate central limit theorem holds the proof is based on the corrector method and the martingale central limit theorem a key integrability condition is furnished by the meyers estimate more general domains boundary conditions and ellipticity contrasts will be addressed in a subsequent paper

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

mesoscopic higher regularity and subadditivity in elliptic homogenization we introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form unlike previous works on the topic our method does not use concentration inequalities such as poincare or logarithmic sobolev inequalities in the probability space and relies instead on a higher ck k 1 regularity theory for solutions of the heterogeneous equation which is valid on length scales larger than a certain specified mesoscopic scale this regularity theory which is of independent interest allows us to in effect localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument the fluctuations of the energy are then tightly controlled using subadditivity the convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes that is it quantifies the weak convergence of these quantities which yields by a new multiscale poincare inequality quantitative estimates on the sublinearity of the corrector

annealed estimates on the green function we consider a random uniformly elliptic coefficient field a x on the d dimensional integer lattice mathbb z d we are interested in the spatial decay of the quenched elliptic green function g a x y next to stationarity we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic sobolev inequality holds for the ensemble langle cdot rangle we prove that all stochastic moments of the first and second mixed derivatives of the green function that is langle nabla x g x y p rangle and langle nabla x nabla y g x y p rangle have the same decay rates in x y gg 1 as for the constant coefficient green function respectively this result relies on and substantially extends the one by delmotte and deuschel probab theory relat fields 133 358 390 2005 which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of g that is langle nabla x g x y 2 rangle and langle nabla x nabla y g x y rangle as an application we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast

comment on correspondence analysis used complementary to loglinear analysis in van der heijden and de leeuw 1985 it was proposed to use loglinear analysis to detect interactions in a multiway contingency table and to explore the form of these interactions with correspondence analysis after performing the exploratory phase of the analysis we will show here how the results found in this phase can be used for confirmation

correlation structure of the corrector in stochastic homogenization recently the quantification of errors in the stochastic homogenization of divergence form operators has witnessed important progress our aim now is to go beyond error bounds and give precise descriptions of the effect of the randomness in the large scale limit this paper is a first step in this direction our main result is to identify the correlation structure of the corrector in dimension 33 and higher this correlation structure is similar to but different from that of a gaussian free field

correspondence analysis of incomplete contingency tables correspondence analysis can be described as a technique which decomposes the departure from independence in a two way contingency table in this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi independence model this form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended for example in case of missing data or structural zeros it is shown that nora s reconstitution of order zero a procedure well known in the french literature is formally identical to our correspondence analysis of incomplete tables therefore reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi independence model furthermore correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis

a new method of normal approximation we introduce a new version of stein s method that reduces a large class of normal approximation problems to variance bounding exercises thus making a connection between central limit theorems and concentration of measure unlike skorokhod embeddings the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily as an application we derive a general clt for functions that are obtained as combinations of many local contributions where the definition of local itself depends on the data several examples are given including the solution to a nearest neighbor clt problem posed by p bickel

bounded correctors in almost periodic homogenization we show that certain linear elliptic equations and systems in divergence form with almost periodic coefficients have bounded almost periodic correctors this is proved under a new condition we introduce which quantifies the almost periodic assumption and includes but is not restricted to the class of smooth quasiperiodic coefficient fields which satisfy a diophantine type condition previously considered by kozlov mat sb n s 107 149 199 217 1978 the proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by armstrong and shen pure appl math 2016 for equations with almost periodic coefficients this yields control on spatial averages of the gradient of the corrector which is converted into estimates on the size of the corrector itself via a multiscale poincare type inequality

lipschitz regularity for elliptic equations with random coefficients we develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients the main result is a large scale l type estimate for the gradient of a solution the estimate is proved with optimal stochastic integrability under a one parameter family of mixing assumptions allowing for very weak mixing with non integrable correlations to very strong mixing for example finite range of dependence we also prove a quenched l2 estimate for the error in homogenization of dirichlet problems the approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence form equations