a test for a specific principal component of a correlation matrix abstract in the application of principal components analysis it is common to replace an observed sample principal component vector by another vector closely resembling the sample vector but which is easier to use or interpret a useful test of hypothesis in this case is one that specifies the true ith principal component in this article we obtain an asymptotically chi squared procedure suitable for testing such a hypothesis when the principal components analysis is performed on a correlation matrix the procedure easily extends to a principal components analysis based on m estimates of scatter

the heat equation with rough boundary conditions and holomorphic functional calculus abstract in this paper we consider the laplace operator with dirichlet boundary conditions on a smooth domain we prove that it has a bounded h calculus on weighted l p spaces for power weights which fall outside the classical class of a p weights furthermore we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity in particular we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data

partial common principal component subspaces summary we consider the principal components analysis of g groups of m variables for those situations in which for each group the first k principal components account for most of the total variability of observations in that group if the set of these gk principal component vectors spans a space of dimension r where r is less than m then it will be possible simultaneously to reduce the dimensionality for all groups from m to r while retaining most of the within group variability methods are already available for determining if r k in which case the g groups have a common principal component subspace in this paper we develop a general procedure for determining if r s for arbitrary s this can then be used repeatedly to determine r when r k

almost sure exponential behavior of a directed polymer in a fractional brownian environment this paper studies the asymptotic behavior of a one dimensional directed polymer in a random medium the latter is represented by a gaussian field bh on r r with fractional brownian behavior in time hurst parameter h and arbitrary function valued behavior in space the partition function of such a polymer is u t eb exp 0tbh dr br here b is a continuous time nearest neighbor random walk on z with fixed intensity 2 defined on a complete probability space pb independent of bh the spatial covariance structure of bh is assumed to be homogeneous and periodic with period 2 for h 12 we prove that the upper and lower almost sure limits lim supt t 2hlogu t and lim inft t 2hlogt logu t are non trivial in the sense that they are bounded respectively above and below by finite strictly positive constants thus as h passes through 12 the exponential behavior of u t changes abruptly this can be considered as a phase transition phenomenon novel tools used in this paper include sub gaussian concentration theory via the malliavin calculus detailed analyses of the long range memory of fractional brownian motion and an almost superadditivity property

covariance reducing models an alternative to spectral modelling of covariance matrices we introduce covariance reducing models for studying the sample covariance matrices of a random vector observed in different populations the models are based on reducing the sample covariance matrices to an informational core that is sufficient to characterize the variance heterogeneity among the populations they possess useful equivariance properties and provide a clear alternative to spectral models for covariance matrices

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

sharp estimation of the almost sure lyapunov exponent for the anderson model in continuous space in this article we study the exponential behavior of the continuous stochastic anderson model i e the solution of the stochastic partial differential equation u t x 1 0 t x u s x ds 0 t w ds x u s x when the spatial parameter x is continuous specifically x r and w is a gaussian field on r r that is brownian in time but whose spatial distribution is widely unrestricted we give a partial existence result of the lyapunov exponent defined as lim t t 1 log u t x furthermore we find upper and lower bounds for lim sup t t 1 log u t x and lim inf t t 1 log u t x respectively as functions of the diffusion constant which depend on the regularity of w in x our bounds are sharper work for a wider range of regularity scales and are significantly easier to prove than all previously known results when the uniform modulus of continuity of the process w is in the logarithmic scale our bounds are optimal

two generalizations of the common principal component model summary under the common principal component model the covariance matrices ti of k populations are assumed to have identical eigenvectors that is the same orthogonal matrix diagonalizes all ti simultaneously this paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all ti while the remaining p q eigenvectors are specific in each group this is called a partial common principal component model a related modification assumes that q eigenvectors of each matrix span the same subspace a problem that was first considered by krzanowski 1979 for both modifications this paper derives the normal theory maximum likelihood estimators it is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available the methods are illustrated by numerical examples

lyapunov exponent for the parabolic anderson model in rd abstract we consider the asymptotic almost sure behavior of the solution of the equation u t x u 0 x 2 0 t u s x d s 0 t u s x w x s where w x x r d is a field of brownian motions in fact we establish existence of the lyapunov exponent lim t 1 t log u t x we also show that c 1 1 3 c 2 1 5 as 0 under the assumption that the correlation function of the background field w x x r d is c for 1 2

moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions we study the nonlinear stochastic heat equation in the spatial domain r driven by space time white noise a central special case is the parabolic anderson model the initial condition is taken to be a measure on r such as the dirac delta function but this measure may also have noncompact support and even be nontempered e g with exponentially growing tails existence and uniqueness of a random field solution is proved without appealing to gronwall s lemma by keeping tight control over moments in the picard iteration scheme upper bounds on all pth moments p 2 are obtained as well as a lower bound on second moments these bounds become equalities for the parabolic anderson model when p 2 we determine the growth indices introduced by conus and khoshnevisan probab theory related fields 152 2012 681 701

intermittency in a catalytic random medium in this paper we study intermittency for the parabolic anderson equation u t u u where u d 0 is the diffusion constant is the discrete laplacian and d 0 is a space time random medium we focus on the case where is times the random medium that is obtained by running independent simple random walks with diffusion constant starting from a poisson random field with intensity throughout the paper we assume that 0 the solution of the equation describes the evolution of a reactant u under the influence of a catalyst we consider the annealed lyapunov exponents that is the exponential growth rates of the successive moments of u and show that they display an interesting dependence on the dimension d and on the parameters with qualitatively different intermittency behavior in d 1 2 in d 3 and in d 4 special attention is given to the asymptotics of these lyapunov exponents for 0 and

a t 1 theorem for non integral operators let x be a space of homogeneous type and let l be a sectorial operator with bounded holomorphic functional calculus on l 2 x we assume that the semigroup e tl t 0 satisfies davies gaffney estimates associated with l are certain approximations of the identity we call an operator t a non integral operator if compositions involving t and these approximations satisfy certain weighted norm estimates the davies gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on t in calderon zygmund theory in this paper we show under the additional assumption that a vertical littlewood paley stein square function associated with l is bounded on l 2 x that a non integral operator t is bounded on l 2 x if and only if t 1 in bmo l x and t 1 in bmo l x here bmo l x and bmo l x denote the recently defined bmo x spaces associated with l that generalize the space bmo x of john and nirenberg generalizing a recent result due to f bernicot we show a second version of a t 1 theorem under weaker off diagonal estimates which gives a positive answer to a question raised by him as an application we prove l 2 x boundedness of a paraproduct operator associated with l we moreover study criterions for a t b theorem to be valid

quadratic estimates and functional calculi of perturbed dirac operators we prove quadratic estimates for complex perturbations of dirac type operators and thereby show that such operators have a bounded functional calculus as an application we show that spectral projections of the hodge dirac operator on compact manifolds depend analytically on l changes in the metric we also recover a unified proof of many results in the calderon program including the kato square root problem and the boundedness of the cauchy operator on lipschitz curves and surfaces

riesz continuity of the atiyah singer dirac operator under perturbations of the metric we prove that the atiyah singer dirac operator in mathrm l 2 mathrm depends riesz continuously on mathrm l infty mathrm perturbations of complete metrics text g on a smooth manifold the lipschitz bound for the map depends on bounds on ricci curvature and its first derivatives as well as a lower bound on injectivity radius our proof uses harmonic analysis techniques related to calderon s first commutator and the kato square root problem we also show perturbation results for more general functions of general dirac type operators on vector bundles

phase transitions for the long time behavior of interacting diffusions let xi t i zd t 0 be the system of interacting diffusions on 0 defined by the following collection of coupled stochastic differential equations dxi t j zd a i j xj t xi t dt bxi t dwi t i z t 0 here a is an irreducible random walk transition kernel on z z b 0 is a diffusion parameter and wi t i zd t 0 is a collection of independent standard brownian motions on r the initial condition is chosen such that xi 0 i zd is a shift invariant and shift ergodic random field on 0 with mean 0 the evolution preserves the mean we show that the long time behaviour of this system is the result of a delicate interplay between a and b in contrast to systems where the diffusion function is subquadratic in particular let i j 12 a i j a j i i j z denote the symmetrised transition kernel we show that a if is recurrent then for any b 0 the system locally dies out b if is transient then there exist b b2 0 such that b1 the system converges to an equilibrium with mean if 0 b b3 has a finite 2 nd moment if and only if 0 b2 the equilibrium is shown to be associated and mixing for all 0 b2 we further conjecture that the system locally dies out at b b for the case where a is symmetric and transient we further show that c there exists a sequence b2 b3 b4 0 such that c1 has a finite m th moment if and only if 0 bm c3 b2 m 1 bm 2 c4 limm m 1 bm c supm 2 m 1 bm mathematisches institut universitat erlangen nurnberg bismarckstrasse 1 1 2 d 91504 erlangen germany greven mi uni erlangen de mathematical institute leiden university p o box 9512 2300 ra leiden the netherlands denholla math leidenuniv nl eurandom p o box 513 5600 mb eindhoven the netherlands

spectral models for covariance matrices summary a new model for the simultaneous eigenstructure of multiple covariance matrices is proposed the model is much more flexible than existing models and subsumes most of them as special cases a fisher scoring algorithm for computing maximum likelihood estimates of the parameters under normality is given asymptotic distributions of the estimators are derived under normality as well as under arbitrary distributions having finite fourth order cumulants special attention is given to elliptically contoured distributions likelihood ratio tests are described and sufficient conditions are given under which the test statistics are asymptotically distributed as chi squared random variables procedures are derived for evaluating bartlett corrections under normality some conjectures made by flury 1988 are verified others are refuted a small simulation study of the adequacy of the bartlett correction is described and the new procedures are illustrated on two datasets

dissipation in parabolic spdes the study of intermittency for the parabolic anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space in this paper we set up the equation on a finite spatial interval and study the other part of intermittency i e the part of the probability space on which the solution is close to zero this set has probability very close to one and we show that on this set the supremum of the solution over space is close to 0 as a consequence we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases we also show that if the noise term is very large then the probability of the set on which the supremum of the solution is very small has a very high probability

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

multilinear littlewood paley estimates with applications to partial differential equations abstract we obtain a collection of multilinear littlewood paley estimates which we then apply to two problems in partial differential equations the first problem is the estimation of the square root of an elliptic operator in divergence form and the second is the estimation of solutions to the cauchy problem for nondivergence form parabolic equations

common principal component subspaces in two groups summary one important practical application of principal component analysis is to reduce a large number of variables say p to a smaller number m by making use of the first m principal components this technique can easily be extended to two or more groups if the subspaces spanned by the first m principal components are the same for all groups in this paper we develop an approximate procedure for testing such a hypothesis of common subspaces when two groups are involved the adequacy of the approximation is investigated by a simulation and the method is illustrated by a numerical example

convolution singular integrals on lipschitz surfaces we prove the lp boundedness of convolution singular integral op erators on a lipschitz surface g x eo x e r 1 x e rn where g is a lipschitz function which satisfies ilvgli we have embedded r i in the clifford algebra r n with identity eo and are considering convolution with right monogenic functions 0 which satisfy 10 x i 0 so x xo x e r lxol where p a w provided there exists an l function q satisfying 0o r 0 r xern then the related convolution singular integral operator t u x lim o f yei x y n y u y dsy 0 en x u x e o jl is bounded on lp for 1 tronics macquarie university nsw 2109 australia e mail address chun li chun macadam mpce mq edu au e mail address alan mcintosh alan macadam mpce mq edu au stephen semmes department of mathematics rice university houston texas 77251 e mail address semmes rice edu this content downloaded from 157 55 39 143 on wed 15 jun 2016 04 40 33 utc all use subject to http about jstor org terms

asymptotic theory for common principal component analysis on donne la distribution asymptotique des estimations du maximum de vraisemblance et on utilise les resultats pour obtenir des tests d egalite des vecteurs propres et de redondance de p q composantes principales

the dirichlet problem for higher order equations in composition form the present paper commences the study of higher order differential equations in composition form specifically we consider the equation lu divb adiva u 0 where a and b are elliptic matrices with complex valued bounded measurable coefficients and a is an accretive function elliptic operators of this type naturally arise for instance via a pull back of the bilaplacian 2 from a lipschitz domain to the upper half space more generally this form is preserved under a lipschitz change of variables contrary to the case of divergence form fourth order differential equations we establish well posedness of the dirichlet problem for the equation lu 0 with boundary data in l2 and with optimal estimates in terms of nontangential maximal functions and square functions

limiting results for the free energy of directed polymers in random environment with unbounded jumps we study asymptotic behaviors of the free energy for the directed polymer in random environment the polymer is allowed to make unbounded jumps and the environment is given by the bernoulli variables we first establish the existence and continuity including the negative infinity value of the coupling constant our proof of existence at differs from existing ones in that it avoids the direct use of subadditivity secondly we identify the asymptotics of the free energy at in the limit of the success probability of the bernoulli variables tending to one it is described by using the so called time constant of a certain directed first passage percolation our proof relies on a certain continuity property of the time constant which is of independent interest

square roots of elliptic operators consider an elliptic sesquilinear form defined on v v by j u v ajk u xk t6v xj ak u xkv ju t6v xj auvdx where v is a closed subspace of h1 which contains c0 is a bounded lipschitz domain in rn ajk ak j a l and re ajk k j 0 for all cn with 1 let l be the operator with largest domain satisfying j u v lu v for all v then l i is a maximal accretive operator in l2 for a sufficiently large real number it is proved that l i 12 is a bounded operator from v to l2 provided mild regularity of the coefficients is assumed in addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense then the corresponding square root operators also depend differentiably on t the latter result is new even when the forms j are hermitian

gibbs measures on mutually interacting brownian paths under singularities we are interested in the analysis of gibbs measures defined on two independent brownian paths in d interacting through a mutual self attraction this is expressed by the hamiltonian 2dv x y dx dy with two probability measures and representing the occupation measures of two independent brownian motions we will be interested in a class of potentials v that are singular e g dirac or coulomb type interactions in 3 or the correlation function of the parabolic anderson problem with white noise potential the mutual interaction of the brownian paths inspires a compactification of the quotient space of orbits of product measures which is structurally different from the self interacting case introduced in 27 owing to the lack of shift invariant structure in the mutual interaction we prove a strong large deviation principle for the product measures of two brownian occupation measures in such a compactification and derive asymptotic path behavior under gibbs measures on wiener paths arising from mutually attracting singular interactions for the spatially smoothened parabolic anderson model with white noise potential our analysis allows a direct computation of the annealed lyapunov exponents and a strict ordering of them implies the intermittency effect present in the smoothened model 2017 wiley periodicals inc

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

common principal components in k groups abstract this article generalizes the method of principal components to so called common principal components as follows consider the hypothesis that the covariance matrices i for k populations are simultaneously diagonalizable that is there is an orthogonal matrix such that i is diagonal for i 1 k i derive the normal theory maximum likelihood estimates of the common component i matrices and the log likelihood ratio statistics for testing this hypothesis the solution has some favorable properties that do not depend on normality assumptions numerical examples illustrate the method applications to data reduction multiple regression and nonlinear discriminant analysis are sketched

weighted chi squared tests for partial common principal component subspaces summary we consider tests of the null hypothesis that g covariance matrices have a partial common principal component subspace of dimension s our approach uses a dimensionality matrix which has its rank equal to s when the hypothesis holds the test can then be based on a statistic computed from the eigenvalues of an estimate of this dimensionality matrix the asymptotic distribution of this statistic is that of a linear combination of independent one degree of freedom chi squared random variables simulation results indicate that this test yields significance levels that come closer to the nominal level than do those of a previously proposed method the procedure is also extended to a test that g correlation matrices have a partial common principal component subspace

conical stochastic maximal l p regularity for 1 leqslant p infty let a mathrm div a cdot nabla be a second order divergence form elliptic operator on mathbb r n with bounded measurable real valued coefficients and let w be a cylindrical brownian motion in a hilbert space h our main result implies that the stochastic convolution process begin aligned u t int 0 t e t s a g s dw s quad t geqslant 0 end aligned satisfies for all 1 leqslant p infty a conical maximal l p regularity estimate begin aligned mathbb e vert nabla u vert t 2 p 2 mathbb r times mathbb r n p leqslant c p p mathbb e vert g vert t 2 p 2 mathbb r times mathbb r n h p end aligned here t 2 p 2 mathbb r times mathbb r n and t 2 p 2 mathbb r times mathbb r n h are the parabolic tent spaces of real valued and h valued functions respectively this contrasts with krylov s maximal l p regularity estimate begin aligned mathbb e vert nabla u vert l p mathbb r l 2 mathbb r n mathbb r n p leqslant c p mathbb e vert g vert l p mathbb r l 2 mathbb r n h p end aligned which is known to hold only for 2 leqslant p infty even when a delta and h mathbb r the proof is based on an l 2 estimate and extrapolation arguments which use the fact that a satisfies suitable off diagonal bounds our results are applied to obtain conical stochastic maximal l p regularity for a class of nonlinear spdes with rough initial data

semiparametric partial common principal component analysis for covariance matrices we consider the problem of jointly modeling multiple covariance matrices by partial common principal component analysis pcpca which assumes a proportion of eigenvectors to be shared across covariance matrices and the rest to be individual specific this paper proposes consistent estimators of the shared eigenvectors in the pcpca as the number of matrices or the number of samples to estimate each matrix goes to infinity we prove such asymptotic results without making any assumptions on the ranks of eigenvalues that are associated with the shared eigenvectors when the number of samples goes to infinity our results do not require the data to be gaussian distributed furthermore this paper introduces a sequential testing procedure to identify the number of shared eigenvectors in the pcpca in simulation studies our method shows higher accuracy in estimating the shared eigenvectors than competing methods applied to a motor task functional magnetic resonance imaging data set our estimator identifies meaningful brain networks that are consistent with current scientific understandings of motor networks during a motor paradigm

on the brownian directed polymer in a gaussian random environment in this paper we introduce a model of brownian polymer in a continuous random environment the asymptotic behavior of the partition function associated to this polymer measure is studied and we are able to separate a weak and strong disorder regime under some reasonable assumptions on the spatial covariance of the environment some further developments concerning some concentration inequalities for the partition function are given for the weak disorder regime

principal component models for correlation matrices summary distributional theory regarding principal components is less well developed for correlation matrices than it is for covariance matrices the intent of this paper is to reduce this disparity methods are proposed that enable investigators to fit and to make inferences about flexible principal components models for correlation matrices the models allow arbitrary eigenvalue multiplicities and allow the distinct eigenvalues to be modelled parametrically or nonparametrically local parameterisations and implicit functions are used to construct full rank unconstrained parameterisations first order asymptotic distributions are obtained directly from the theory of estimating functions second order accurate distributions for making inferences under normality are obtained directly from likelihood theory simulation studies show that the bartlett correction is effective in controlling the size of the tests and that first order approximations to nonnull distributions are reasonably accurate the methods are illustrated on a dataset

estimation of common linear functional relationships in k data sets summary this paper studies a functional type linear measurement error model in which k data sets of observations are assumed to have in common a set of linear restrictions on their respective unobservable constant vectors each data set consists of replicated measurements of each one of several generally unequal constant vectors with normal error assumed independent for each measurement made the error covariance matrices are unspecified and are equal for measurements within a data set but in general unequal among the data sets this work generalises the results of anderson 1951 1984 who derived maximum likelihood estimators for the constant vectors and error covariance matrix of a single such data set the model recalls the common space analysis described by flury 1987 1988 pp 131 4 but differs by the presence of a fixed effects subspace maximum likelihood estimation of the model parameters requires special numerical methods an example from electroencephalography is used to introduce and illustrate the method

the kato square root problem on locally uniform domains abstract we obtain the kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in r d under two simple geometric conditions the dirichlet boundary part is ahlfors david regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the neumann boundary part this improves upon all existing results even in the case of pure dirichlet or neumann boundary conditions we also treat elliptic systems with lower order terms as a side product we establish new regularity results for the fractional powers of the laplacian with boundary conditions in our geometric setup

lyapunov exponent for the parabolic anderson model with levy noise we consider the asymptotic almost sure behavior of the solution of the equation where y x x z d is a field of independent levy processes and is the discrete laplacian

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

comparison of partition functions in a space time random environment let z 1 and z 2 be partition functions in the random polymer model in the same environment but driven by different underlying random walks we give a comparison in concave stochastic order between z 1 and z 2 if one of the random walks has more randomness than the other we also treat some related models the parabolic anderson model with space time levy noise brownian motion among space time obstacles and branching random walks in space time random environments we also obtain a necessary and sufficient criterion for z 1 preceq cv z 2 if the lattice is replaced by a regular tree

kato s square root problem in banach spaces abstract let l be an elliptic differential operator with bounded measurable coefficients acting in bochner spaces l p r n x of x valued functions on r n we characterize kato s square root estimates l u p u p and the h functional calculus of l in terms of r boundedness properties of the resolvent of l when x is a banach function lattice with the umd property or a noncommutative l p space to do so we develop various vector valued analogues of classical objects in harmonic analysis including a maximal function for bochner spaces in the special case x c we get a new approach to the l p theory of square roots of elliptic operators as well as an l p version of carleson s inequality

on large deviations for the parabolic anderson model the focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic anderson model above the mean versus large deviations below the mean the functionals we treat are the solution u x t to the spatially discrete parabolic anderson model and a functional an which is used in analyzing the a s lyapunov exponent for u x t both satisfy a law of large numbers with lim t to infty frac 1 t log u x t lambda kappa and lim n to infty frac a n n alpha we then think of n and t as being the mean of the respective quantities an and log u t x typically the large deviations for such functionals exhibits a strong asymmetry large deviations above the mean take on a different order of magnitude from large deviations below the mean we develop robust techniques to quantify and explain the differences

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

dissipation and high disorder given a fieldfb x gx2zd of independent standard brownian motions indexed by z d the generator of a suitable markov process on z d g and suciently nice function 0 1 0 1 we consider the influence of the parameter on the behavior of the system dut x gut x dt ut x dbt x t 0 x2 z d u0 x c0 0 x we show that for any

rough metrics on manifolds and quadratic estimates we study the persistence of quadratic estimates related to the kato square root problem across a change of metric on smooth manifolds by defining a class of rough riemannian like metrics that are permitted to be of low regularity and degenerate on sets of measure zero we also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi lipschitz as a consequence we demonstrate the invariance of the kato square root problem under lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric furthermore we show that a lower bound on the injectivity radius is not a necessary condition to solve the kato square root problem

some tests for common principal component subspaces in several groups summary an approximate test based on sample eigenprojections is obtained for testing the hypothesis that the subspaces spanned by the first m principal components of several different covariance matrices are identical this procedure can fairly easily be extended to an analogous test of hypothesis concerning correlation matrices extensions to a robust principal components analysis are also considered in recent years principal components analysis has been generalized from the typical single population setting to that of several populations this paper focuses on an extension that may be useful in those situations in which one needs a variance preserving simultaneous reduction of dimensionality in several groups this sort of reduction will be possible when the subspace spanned by the most important principal component vectors of a particular group is the same as that of any other group formally suppose the same p variables are being measured on objects in g different groups with the covariance

raw data maximum likelihood estimation for common principal component models a state space approach the specifications of state space model for some principal component related models are described including the independent group common principal component cpc model the dependent group cpc model and principal component based multivariate analysis of variance some derivations are provided to show the equivalence of the state space approach and the existing wishart likelihood approach for each model a numeric example is used to illustrate the state space approach in addition a simulation study is conducted to evaluate the standard error estimates under the normality and nonnormality conditions in order to cope with the nonnormality conditions the robust standard errors are also computed finally other possible applications of the state space approach are discussed at the end

regularized generalized canonical correlation analysis for multiblock or multigroup data analysis this paper presents an overview of methods for the analysis of data structured in blocks of variables or in groups of individuals more specifically regularized generalized canonical correlation analysis rgcca which is a unifying approach for multiblock data analysis is extended to be also a unifying tool for multigroup data analysis the versatility and usefulness of our approach is illustrated on two real datasets

the cauchy integral calderon commutators and conjugations of singular integrals in ⁿ we consider the cauchy integral and hilbert transform for lipschitz domains in the clifford algebra based on r 1 the hilbert transform is shown to be the generating function for the calderon commutators in r 1 we make use of an intrinsic characterization of these commutators to obtain l2 estimates these estimates are used to show the analyticity of the hilbert transform and of the conjugation of singular integral operators by bi lipschitz changes of variable in r 1

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon