subelliptic p harmonic maps into spheres and the ghost of hardy spaces the story begins with the paper of muller 59 who for the sake of an application to nonlinear elasticity proved that if the jacobian determinant ju of a sobolev map u w 1 n loc rn rn is nonnegative then it belongs locally to l log l the result is quite intriguing since a priori holder inequality implies only that ju l1 and one does not suspect any higher integrability if one does not assume that ju is nonnegative then as coifman lions meyer and semmes 14 have proved ju belongs to the local hardy space h 1 loc since a nonnegative function belongs to the local hardy space if and only if it belongs locally to l log l the result of coifman lions meyer and semmes generalizes that of muller in fact coifman lions meyer and semmes proved more namely for 1 p the scalar product of a divergence free vector field e lp rn rn and a curl free vector field b lp p 1 rn rn belongs to the hardy space h 1 a priori holder inequality implies integrability of this expression only note that the jacobian is of this form the discovery that many algebraic expressions involving partial derivatives belong to the hardy space turned out to be important and widely applicable in nonlinear partial differential equations this was first shown by helein 40 41 42 let us describe briefly his result consider maps u bn s m from the n dimensional ball to the mdimensional sphere such that the p energy of u given by the functional

estimationslpdes operateurs de schrodinger sur les groupes nilpotents resume dans cet article on se propose d etudier la continuite dans les espaces l p des groupes nilpotents des operateurs 2 w 1 w 1 2 nabla w 1 et w 1 2 ou est un sous laplacien et w un potentiel qui verifie d assez bonnes conditions

parameter expansion to accelerate em the px em algorithm summary the em algorithm and its extensions are popular tools for modal estimation but ar e often criticised for their slow convergence we propose a new method that can often make em much faster the intuitive idea is to use a covariance adjustment to correct the analysis of the m step capitalising on extra information captured in the imputed complete data the way we accomplish this is by parameter expansion we expand the complete data model while preserving the observed data model and use the expanded complete data model to generate em this parameter expanded ei m px em algorithm shares the simplicity and stability of ordinary em but has a faster rate of convergence since its m step performs a more efficient analysis the px em algorithm is illustrated for the multivariate t distribution a random effects model factor analysis probit regression and a poisson imaging model

small time heat kernel asymptotics at the sub riemannian cut locus for a sub riemannian manifold provided with a smooth volume we relate the small time asymptotics of the heat kernel at a point y of the cut locus from x with roughly how much y is conjugate to x this is done under the hypothesis that all minimizers connecting x to y are strongly normal i e all pieces of the trajectory are not abnormal our result is a renement of the one of leandre 4t logpt x y d 2 x y for t 0 in which only the leading exponential term is detected our results are obtained by extending an idea of molchanov from the riemannian to the sub riemannian case and some details we get appear to be new even in the riemannian context these results permit us to obtain properties of the sub riemannian distance starting from those of the heat kernel and vice versa for the grushin plane endowed with the euclidean volume we get the expansion pt x y t 5 4 exp d 2 x y 4t where y is reached from a riemannian point x by a minimizing geodesic which is conjugate at y

a unified framework for the comparison of treatments with ordinal responses different latent variable models have been used to analyze ordinal categorical data which can be conceptualized as manifestations of an unobserved continuous variable in this paper we propose a unified framework based on a general latent variable model for the comparison of treatments with ordinal responses the latent variable model is built upon the location scale family and is rich enough to include many important existing models for analyzing ordinal categorical variables including the proportional odds model the ordered probit type model and the proportional hazards model a flexible estimation procedure is proposed for the identification and estimation of the general latent variable model which allows for the location and scale parameters to be freely estimated the framework advances the existing methods by enabling many other popular models for analyzing continuous variables to be used to analyze ordinal categorical data thus allowing for important statistical inferences such as location and or dispersion comparisons among treatments to be conveniently drawn analysis on real data sets is used to illustrate the proposed methods

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

isoperimetric inequalities and the homology of groups an isoperimetric function for a finitely presented group g bounds the number of defining relations needed to prove that a word w g 1 in terms of the length of w suppose g f r is a finitely presented group where f is a finitely generated free group and r is the normal closure of a finite set of relators for each w r we define the area w to be the least number of uses of the given relators needed to prove that w g 1 the isoperimetric function g n is defined to be the maximum of the areas of all the words in r of length n one compares the size of such functions using a suitable order 1 on functions different presentations give rise to equivalent isoperimetric functions where is the equivalence relation determined by 1 interest in such functions was stimulated by gromov 13 who characterized word hyperbolic groups as those with linear isoperimetric function that is g n n further information was obtained by cannon et al 8 who showed that automatic groups satisfy g 1 n and that asynchronously automatic groups satisfy g 1 2 they also showed the free nilpotent group of class 2 and rank 2 has g n n gersten 10 used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups he exploited the fact that for a torsion free one relator groupg the usual 2 dimensional complex having fundamental group isomorphic to g is aspherical the general technique for giving upper bounds for g n is to describe a solution to the word problem for g and analyze its use of defining relations it seems quite difficult to establish lower bounds for g n in this manner in this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for g instead of trying to analyze r directly we consider some of its quotient groups for which additional machinery is available for instance we consider for each w r its image w r f in r r f and define a suitable centralized isoperimetric function g n such that g n 1 g n using the hopf formula it can be shown that r r f is a direct sum of the second homology group h2 g z with a free abelian

bayesian analysis of multivariate probit models with surrogate outcome data a new class of parametric models that generalize the multivariate probit model and the errors in variables model is developed to model and analyze ordinal data a general model structure is assumed to accommodate the information that is obtained via surrogate variables a hybrid gibbs sampler is developed to estimate the model parameters to obtain a rapidly converged algorithm the parameter expansion technique is applied to the correlation structure of the multivariate probit models the proposed model and method of analysis are demonstrated with real data examples and simulation studies

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

the intrinsic hypoelliptic laplacian and its heat kernel on unimodular lie groups we present an invariant definition of the hypoelliptic laplacian on sub riemannian structures with constant growth vector using the popp s volume form introduced by montgomery this definition generalizes the one of the laplace beltrami operator in riemannian geometry in the case of left invariant problems on unimodular lie groups we prove that it coincides with the usual sum of squares we then extend a method first used by hulanicki on the heisenberg group to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular lie group of type i the main tool is the noncommutative fourier transform we then study some relevant cases su 2 so 3 sl 2 with the metrics inherited by the killing form and the group se 2 of rototranslations of the plane our study is motivated by some recent results about the cut and conjugate loci on these sub riemannian manifolds the perspective is to understand how singularities of the sub riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

bayesian inference for multivariate ordinal data using parameter expansion multivariate ordinal data arise in many applications this article proposes a new efficient method for bayesian inference for multivariate probit models using markov chain monte carlo techniques the key idea is the novel use of parameter expansion to sample correlation matrices a nice feature of the approach is that inference is performed using straightforward gibbs sampling bayesian methods for model selection are also discussed our approach is motivated by a study of how women make decisions on taking medication to reduce the risk of breast cancer furthermore we compare and contrast the performance of our approach with other methods

rounding error and perturbation bounds for the indefinite qr factorization abstract indefinite qr factorization is a generalization of the well known qr factorization where q is a unitary matrix with respect to the given indefinite inner product matrix j this factorization can be used for accurate computation of eigenvalues of the hermitian matrix a g jg where g and j are initially given or naturally formed from initial data the classical example of such a matrix is a b b c c with given b and c in this paper we present the rounding error and perturbation bounds for the so called triangular case of the indefinite qr factorization these bounds fit well into the relative perturbation theory for hermitian matrices given in factorized form

fine properties of functions with bounded variation in carnot caratheodory spaces abstract we study properties of functions with bounded variation in carnot caratheodory spaces we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives under an additional assumption on the space called property r we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative

double sampling for exact values in some multivariate measurement error problems abstract increasing attention is being given to measurement error models in which the dimension of the proxy or surrogate values is different from that of the missing true values even if they are of the same dimension the error model may not be the simple additive one of observed true error where the error has mean 0 the use of broader models relating true and observed values requires the use of external or internal data containing some true values to calibrate validate the measurement error model this article considers a multivariate normal framework in which measurement error is allowed in any subset of the variables with a broad class of multivariate regression models relating true and observed values the classical additive model is a special case multiple regression with random regressors the structural case is treated within this framework correcting for measurement error is made possible through double sampling in which true values are obtained for a randomly chosen subset of the main

a monte carlo implementation of the em algorithm and the poor man s data augmentation algorithms abstract the first part of this article presents the monte carlo implementation of the e step of the em algorithm given the current guess to the maximizer of the posterior distribution latent data patterns are generated from the conditional predictive distribution the expected value of the augmented log posterior is then updated as a mixture of augmented log posteriors mixed over the generated latent data patterns multiple imputations in the m step of the algorithm this mixture is maximized to obtain the update to the maximizer of the observed posterior the gradient and hessian of the observed log posterior are also expressed as mixtures mixed over the multiple imputations the relation between the monte carlo em mcem algorithm and the data augmentation algorithm is noted two modifications to the mcem algorithm the poor man s data augmentation algorithms which allow for the calculation of the entire posterior are then presented these approximations serve as diagnostics for the validity o

regular submanifolds graphs and area formula in heisenberg groups we describe intrinsically regular submanifolds in heisenberg groups h low dimensional and low codimensional submanifolds turn out to be of a very different nature the first ones are legendrian surfaces while low codimensional ones are more general objects possibly non euclidean rectifiable nevertheless we prove that they are graphs in a natural group way as well as that an area formula holds for the intrinsic hausdorff measure finally they can be seen as federer fleming currents given a natural complex of differential forms on h

sub riemannian geometry of the coefficients of univalent functions abstract we consider coefficient bodies m n for univalent functions based on the lowner kufarev parametric representation we get a partially integrable hamiltonian system in which the first integrals are kirillov s operators for a representation of the virasoro algebra then m n are defined as sub riemannian manifolds given a lie poisson bracket they form a grading of subspaces with the first subspace as a bracket generating distribution of complex dimension two with this sub riemannian structure we construct a new hamiltonian system to calculate regular geodesics which turn to be horizontal lagrangian formulation is also given in the particular case m 3

seeking efficient data augmentation schemes via conditional and marginal augmentation data augmentation sometimes known as the method of auxiliary variables is a powerful tool for constructing optimisation and simulation algorithms in the context of optimisation meng van dyk 1997 1998 reported several successes of the working parameter approach for constructing efficient data augmentation schemes for fast and simple em type algorithms this paper investigates the use of working parameters in the context of markov chain monte carlo in particular in the context of tanner wong s 1987 data augmentation algorithm via a theoretical study of two working parameter approaches the conditional augmentation approach and the marginal augmentation approach posterior sampling under the univariate t model is used as a running example which particularly illustrates how the marginal augmentation approach obtains a fast mixing positive recurrent markov chain by first constructing a nonpositive recurrent markov chain in a larger space

hausdorff volume in non equiregular sub riemannian manifolds in this paper we study the hausdorff volume in a non equiregular sub riemannian manifold and we compare it with a smooth volume we first give the lebesgue decomposition of the hausdorff volume then we study the regular part show that it is not commensurable with the smooth volume and give conditions under which it is a radon measure we finally give a complete characterization of the singular part we illustrate our results and techniques on numerous examples and cases e g to generic sub riemannian structures

fractional isoperimetric inequalities and subgroup distortion isoperimetric inequalities measure the complexity of the word problem in finitely presented groups by giving a bound on the number of relators that one must apply in order to show that a word w in the given generators represents the identity such bounds are given in terms of the length of w and the function describing the optimal bound is known as the dehn function of the group modulo a standard equivalence relation the dehn function is an invariant of the group not just the given finite presentation about seven years ago as the result of efforts by a number of authors 9 11 14 it was established that for every positive integer d one can construct finitely presented groups whose dehn function is polynomial of degree d the question of whether or not there exist groups whose dehn functions are of fractional degree has attracted a good deal of interest e g 18 25 according to a theorem of m gromov one does not get fractional exponents less than 2 because if a group satisfies a sub quadratic isoperimetric inequality then it actually satisfies a linear isoperimetric inequality see 13 10 19 22 the main purpose of this article is to prove the following

sub riemannian interpolation inequalities we prove that ideal sub riemannian manifolds i e admitting no non trivial abnormal minimizers support interpolation inequalities for optimal transport a key role is played by sub riemannian jacobi fields and distortion coefficients whose properties are remarkably different with respect to the riemannian case as a byproduct we characterize the cut locus as the set of points where the squared sub riemannian distance fails to be semiconvex answering to a question raised by figalli and rifford geom funct anal 20 1 124 159 2010 as an application we deduce sharp and intrinsic borell brascamp lieb and geodesic brunn minkowski inequalities in the aforementioned setting for the case of the heisenberg group we recover in an intrinsic way the results recently obtained by balogh et al calc var part differ equ 57 2 61 2018 and we extend them to the class of generalized h type carnot groups our results do not require the distribution to have constant rank yielding for the particular case of the grushin plane a sharp measure contraction property and a sharp brunn minkowski inequality

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

empirical likelihood based inference in linear errors in covariables models with validation data summary linear errors in covariables models are considered assuming the availability of independent validation data on the covariables in addition to primary data on the response variable and surrogate covariables we first develop an estimated empirical loglikelihood with the help of validation data and prove that its asymptotic distribution is that of a weighted sum of independent standard z2 random variables with unknown weights by estimating the unknown weights consistently we construct an estimated empirical likelihood confidence region for the regression parameter vector we also suggest an adjusted empirical loglikelihood and prove that its asymptotic distribution is a standard x2 to avoid estimating the unknown weights or the adjustment factor we propose a partially smoothed bootstrap empirical loglikelihood for constructing a confidence region which has asymptotically correct coverage probability a simulation study is conducted to compare the proposed methods with a method based on a normal approximation in terms of coverage accuracy and average length of the confidence interval

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

filling invariants of stratified nilpotent lie groups filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities in this article we examine filling functions and higher divergence functions we prove for a class of stratified nilpotent lie groups that in the low dimensions the filling functions grow as fast as the ones of the euclidean space and in the high dimensions slower than the filling functions of the euclidean space we do this by developing a purely algebraic condition on the lie algebra of a stratified nilpotent lie group further we find a sufficient criterion for such groups to have a filling function in a special dimension with faster growth as the appropriate filling function of the euclidean space further we bound the higher divergence functions of stratified nilpotent lie groups

analysis of multivariate probit models summary this paper provides a practical simulation based bayesian and non bayesian analysis of correlated binary data using the multivariate probit model the posterior distribution is simulated by markov chain monte carlo methods and maximum likelihood estimates are obtained by a monte carlo version of the em algorithm a practical approach for the computation of bayes factors from the simulation output is also developed the methods are applied to a dataset with a bivariate binary response to a four year longitudinal dataset from the six cities study of the health effects of air pollution and to a sevenvariate binary response dataset on the labour supply of married women from the panel survey of income dynamics

sub riemannian vs euclidean dimension comparison and fractal geometry on carnot groups we solve gromov s dimension comparison problem for hausdorff and box counting dimension on carnot groups equipped with a carnot caratheodory metric and an adapted euclidean metric the proofs use sharp covering theorems relating optimal mutual coverings of euclidean and carnot caratheodory balls and elements of sub riemannian fractal geometry associated to horizontal self similar iterated function systems on carnot groups inspired by falconer s work on almost sure dimensions of euclidean self affine fractals we show that carnot caratheodory self similar fractals are almost surely horizontal as a consequence we obtain explicit dimension formulae for invariant sets of euclidean iterated function systems of polynomial type jet space carnot groups provide a rich source of examples

limit sets of discrete groups of isometries of exotic hyperbolic spaces let be a geometrically finite discrete group of isometries of hyperbolic space hf where f r c h or o in which case n 2 we prove that the critical exponent of equals the hausdorff dimension of the limit sets and that the smallest eigenvalue of the laplacian acting on square integrable functions is a quadratic function of either of them when they are sufficiently large a generalization of hopf ergodicity theorem for the geodesic flow with respect to the bowen margulis measure is also proven

block reflectors theory and computation a block reflector is an orthogonal symmetric matrix that reverses a subspace whose dimension may be greater than one we shall develop the properties of block reflectors and give some algorithms for computing a block reflector that introduces a block of zeros into a matrix we consider the compact representation of block reflectors some applications and their use in parallel computers

computable algebra and group embeddings this work grew out of our investigations of the infinitely generated subgroups of finitely presented groups a useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in l this technique was then applied in 1 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group it became apparent from 2 that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups in this paper these extensions are carried out with the consequent application that every countable locally abelian by nilpotent group can be embedded in a finitely presented group much of the effective commutative algebra we require seems to be known see for instance 6 9 14 171 unfortunately a systematic account is not available in particular we need a proof of the hilbert basis theorem which

the dehn function of richard thompson s group f is quadratic we prove that the dehn function that is the smallest isoperimetric function of the r thompson s group f is quadratic

modeling discrete choice with response error food stamp participation abstract validation of food stamp program participation by marquis and moore revealed net bias of 13 in mean estimates of participation in the 1984 survey of income and program participation we extend the analysis incorporating demographic and economic covariates in models for underreporting and overreporting we use the resulting models of response error to refine estimates of participation models using pseudo maximum likelihood estimation methods we find that the probability of underreporting rises with increased family income estimates of the participation model indicate the presence of substantial bias in income and asset related parameters compared to values obtained without incorporating response error

orthosymmetric block reflectors we develop a general theory of reflectors and block reflectors in a class of non euclidean scalar product spaces generated by orthosymmetric scalar product matrices j these j reflectors are generalizations of ordinary householder transformations and we show that they can always be expressed in a householder like representation reflection and mapping properties of j reflectors are completely described block j reflectors can be used for block annihilation in qr like factorizations where q is j unitary and r is block upper triangular

the conjugacy and isomorphism problems for combable groups there exist combable groups in which the conjugacy problem is unsolvable the isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups

random effects logistic regression analysis with auxiliary covariates summary we study a semiparametric estimation method for the random effects logistic regression when there is auxiliary covariate information about the main exposure variable we extend the semiparametric estimator of pepe and fleming 1991 journal of the american statistical association86 108 113 to the random effects model using the best linear unbiased prediction approach of henderson 1975 biometrics31 423 448 the method can be used to handle the missing covariate or mismeasured covariate data problems in a variety of real applications simulation study results show that the proposed method outperforms the existing methods we analyzed a data set from the collaborative perinatal project using the proposed method and found that the use of ddt increases the risk of preterm births among us children

large deviation on a covering graph with group of polynomial growth we discuss a large deviation principle of a periodic random walk on a covering graph with its transformation group of polynomial volume growth in view of geometry as we shall observe the behavior of a random walk at infinity is closely related to the gromov hausdorff limit of an infinite graph and in the case where the graph admits an action of a group of polynomial volume growth the carnot caratheodory metric shows up in its limit space

g reflectors analogues of householder transformations in scalar product spaces we characterize the analogues of householder transformations in matrix groups associated with scalar products and precisely delimit their mapping capabilities given a matrix group image and vectors x y necessary and sufficient conditions are derived for the existence of a householder like analogue image such that gx y when g exists we show how it can be constructed from x and y examples of matrix groups to which these results apply include the symplectic and pseudo unitary groups

a structure preserving qr factorization for centrosymmetric real matrices we construct a qr factorization of a given centrosymmetric real matrix a into centrosymmetric real matrices q and r we describe in detail a householder type algorithm based on perplectic orthogonal block reflectors to obtain such a factorization and demonstrate an application of this result to solving centrosymmetric linear systems of full rank

parameter expansion for data augmentation abstract viewing the observed data of a statistical model as incomplete and augmenting its missing parts are useful for clarifying concepts and central to the invention of two well known statistical algorithms expectation maximization em and data augmentation recently liu rubin and wu demonstrated that expanding the parameter space along with augmenting the missing data is useful for accelerating iterative computation in an em algorithm the main purpose of this article is to rigorously define a parameter expanded data augmentation px da algorithm and to study its theoretical properties the px da is a special way of using auxiliary variables to accelerate gibbs sampling algorithms and is closely related to reparameterization techniques we obtain theoretical results concerning the convergence rate of the px da algorithm and the choice of prior for the expansion parameter to understand the role of the expansion parameter we establish a new theory for iterative conditional sampling under the tra

solvable groups with polynomial dehn functions given a finitely presented group h finitely generated subgroup b of h and a monomorphism b h we obtain an upper bound of the dehn function of the corresponding hnn extension g hh t j t i1 bt b i in terms of the dehn function of h and the distortion of b in g using such a bound we construct first examples of non polycyclic solvable groups with polynomial dehn functions the constructed groups are metabelian and contain the solvable baumslag solitar groups in particular this answers a question posed by birget ol shanskii rips and sapir

isoperimetric functions of groups and computational complexity of the word problem we prove that the word problem of a finitely generated group g is in np solvable in polynomial time by a nondeterministic turing machine if and only if this group is a subgroup of a finitely presented group h with polynomial isoperimetric function the embedding can be chosen in such a way that g has bounded distortion in h this completes the work started in 6 and 25

rounding error and perturbation bounds for the symplectic qr factorization to compute the eigenvalues of a skew symmetric matrix a we can use a one sided jacobi like algorithm to enhance accuracy this algorithm begins by a suitable cholesky like factorization of a a gtjg in some applications a is given implicitly in that form and its natural cholesky like factor g is immediately available but tall i e not of full row rank this factor g is unsuitable for the jacobi like process to avoid explicit computation of a and possible loss of accuracy the factor has to be preprocessed by a qr like factorization in this paper we present the symplectic qr algorithm to achieve such a factorization together with the corresponding rounding error and perturbation bounds these bounds fit well into the relative perturbation theory for skew symmetric matrices given in factorized form

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

pauls rectifiable and purely pauls unrectifiable smooth hypersurfaces abstract this paper is related to the problem of finding a good notion of rectifiability in sub riemannian geometry in particular we study which kind of results can be expected for smooth hypersurfaces in carnot groups our main contribution will be a consequence of the following result there exists a c hypersurface s without characteristic points that has uncountably many pairwise non isomorphic tangent groups on every positive measure subset the example is found in a carnot group of topological dimension 8 it has hausdorff dimension 12 and so we use on it the hausdorff measure h 12 as a consequence we show that any lipschitz map defined on a subset of a carnot group of hausdorff dimension 12 with values in s has negligible image with respect to the hausdorff measure h 12 in particular we deduce that s cannot be lipschitz parametrizable by countably many maps each defined on some subset of some carnot group of hausdorff dimension 12 as main consequence we have that a notion of rectifiability proposed by s pauls is not equivalent to one proposed by b franchi r serapioni and f serra cassano at least for arbitrary carnot groups in addition we show that given a subset u of a homogeneous subgroup of hausdorff dimension 12 of a carnot group every bi lipschitz map f u s satisfies h 12 f u 0 finally we prove that such an example does not exist in heisenberg groups we prove that all c hypersurfaces in h n with n 2 are countably h n 1 r rectifiable according to pauls definition even with bi lipschitz maps

algorithms and geometry for graph products of groups recent work of gromov epstein cannon thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups see 16 17 and 14 many of these structures are preserved by taking graph products the graph product of groups not to be confused with the fundamental group of a graph of groups is a product mixing direct and free products whether the product between two groups in the graph product is free or direct is determined by a simplicial graph given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge

isoperimetric and isodiametric functions of groups this is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity in particular we show how to construct a finitely presented group with np complete word problem one of the main results of this paper states that if a real number cx 4 is computable in time 0 then na is equivalent big o to the dehn function of a finitely presented group the smallest isodiametric function of this group is n3 8 4 on the other hand if na is equivalent to the dehn function of a finitely presented group then oe is computable in time 4 are equivalent to dehn functions of some finitely presented groups and that n1r and na for all rational numbers oe 3 are equivalent to the smallest isodiametric functions of finitely presented groups moreover we describe all dehn functions of finitely presented groups f n4 as time functions of turing machines modulo two conjectures

on 2 step corank 2 nilpotent sub riemannian metrics in this paper we study the nilpotent 2 step corank 2 sub riemannian metrics that are nilpotent approximations of general sub riemannian metrics we exhibit optimal syntheses for these problems it turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression as a by product of this study we get some smoothness properties of the spherical hausdorff measure in the case of a generic 6 dimensional 2 step corank 2 sub riemannian metric

miscellanea a robust imputation method for surrogate outcome data summary we consider estimation for regression analysis with surrogate or auxiliary outcome data assume that the regression model for the conditional mean of the outcome is a known function of a linear combination of the covariates with unknown coefficients which are the regression parameters of interest such a class of models includes the generalised linear models as special cases suppose further that the outcome variable of interest is only observed in a validation subset which is a simple random subsample from the whole sample and that data on covariates as well as on one or more easily measured but less accurate surrogate outcome variables is collected from the whole sample we propose a robust imputation approach which replaces the unobserved value of the outcome by its predicted value generated from a specified working parametric model estimation of the regression parameters is conducted as if the outcome data were completely observed the resulting estimator of the regression parameter is consistent even if the working model is misspecified large and finite sample properties for the proposed estimator are investigated

inference using surrogate outcome data and a validation sample summary in the context of estimating 3 from the regression model p yi x relating response y to covariates x suppose that only a surrogate response s is available for most study subjects suppose that for a random subsample of the study cohort termed the validation sample the true outcome y is available in addition to s we consider maximum likelihood estimation of 3 from such data and show that it is nonrobust to misspecification of the distribution relating the surrogate to the true outcome p s i y x an alternative semiparametric method is also considered which is nonparametric with respect to p s i y x large sample distribution theory for maximum estimated likelihood estimates is developed an illustrative example is presented

intrinsic random walks and sub laplacians in sub riemannian geometry on a sub riemannian manifold we define two type of laplacians the macroscopic laplacian as the divergence of the horizontal gradient once a volume is fixed and the microscopic laplacian as the operator associated with a geodesic random walk we consider a general class of random walks where all sub riemannian geodesics are taken in account this operator depends only on the choice of a complement c to the sub riemannian distribution and is denoted l c we address the problem of equivalence of the two operators this problem is interesting since on equiregular sub riemannian manifolds there is always an intrinsic volume e g popp s one p but not a canonical choice of complement the result depends heavily on the type of structure under investigation on contact structures for every volume there exists a unique complement c such that l c on carnot groups if h is the haar volume then there always exists a complement c such that h l c however this complement is not unique in general for quasi contact structures in general p l c for any choice of c in particular l c is not symmetric w r t popp s measure this is surprising especially in dimension 4 where in a suitable sense p is the unique intrinsic macroscopic laplacian a crucial notion that we introduce here is the n intrinsic volume i e a volume that depends only on the set of parameters of the nilpotent approximation when the nilpotent approximation does not depend on the point a n intrinsic volume is unique up to a scaling by a constant and the corresponding n intrinsic sub laplacian is unique this is what happens for dimension smaller or equal than 4 and in particular in the 4 dimensional quasi contact structure mentioned above

on quadratic dehn functions we confirm with new examples that solvable groups of high rank are expected to satisfy a polynomial isoperimetric inequality gro93 5a9 to that end we study invariant quasi geodesic foliations in simply connected solvable lie groups endowed with left invariant riemannian metrics whose leaves are isometric to closed subgroups we establish a decomposition theorem which implies upper bounds on the dehn or filling function of loops by disks of the solvable group in terms of the dehn functions of the leaves we obtain examples of metabelian polycyclic groups with exponential growth and quadratic dehn functions we also deduce that the horospheres in sl 4 so 4 which bound an invariant core for sl 4 and that the horospheres which bound an invariant core for hilbert modular groups in open image in new window have quadratic filling functions the main theorem also applies to some solvable lie groups which are not quasi isometric to horospheres in symmetric spaces

the weighted isoperimetric type and sobolev type inequalities for hypersurfaces in carnot groups abstract this paper mainly studies hypersurfaces immersed in a sub riemannian step k carnot group g with a density e f where f is a smooth function on g we give the first variation formula about the weighted horizontal area of hypersurfaces furthermore on the base of some work we derive the weighted isoperimetric type and sobolev type inequalities for hypersurfaces

on the geometry of normal forms in discrete groups we study normal forms in finitely generated groups from the geometric viewpoint of combings we introduce notions of combability considerably weaker than those commonly in use we prove that groups which satisfy these conditions are finitely presented and satisfy isoperimetric and isodiametric inequalities of a controlled nature

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

likelihood inference in a correlated probit regression model correlated binary observations arise in a variety of applications for example in animal studies the term litter effect is used to describe the greater alikeness of responses within a litter as compared to that between litters at a given set of experimental conditions in a setting that motivated this work cells on individual atomic bomb survivors were scored as to the presence or absence of chromosomal aberrations the number of aberrant cells varied among subjects at a specified radiation exposure estimate age at exposure city of exposure and sex in a manner that substantially exceeds that consistent with a binomial error structure otake prentice 1984 this extra binomial variation or overdispersion reflects the fact that the binary response of cells from a given survivor tend to be more alike than are the responses of cells from distinct survivors having the same estimated radiation exposure level and other covariate values such overdispersion may result from individual differences in susceptibility to radiation damage from omitted covariates or most plausibly in this setting from substantial random errors in the estimated radiation exposure levels failure to acknowledge overdispersion may lead to serious underestimation of the standard errors associated with regression parameter estimates and to unduly precise inferences

precision of double sampling estimators for comparing two probabilities summary it is assumed that both error prone and error free observations can be made on a binary response the second type being more expensive in double sampling error prone observations are made on all sample units and for a random subsample error free observations are made as well tenenbein 1970 studied allocation of units in double sampling for estimating a single probability we consider double sampling for comparing two probabilities for different cost ratios for error prone versus error free observations and for fixed total cost we assess how the large sample variance of the log odds ratio estimator varies with the proportion of units subject to both error prone and error free observation we determine the optimal double sampling proportion and the possible gain from using double sampling versus single sampling in which only error free observations are made

log linear models for doubly sampled categorical data fitted by the em algorithm abstract double sampling experiments can be expressed as incomplete multiway contingency tables and analyzed by using techniques appropriate for fitting log linear models the framework described can be applied to experiments with any number of factors and measurement devices maximum likelihood estimation via the em algorithm leads to straightforward expressions for covariances of estimates of the parameters and functions of the parameters

high dimensional multivariate probit analysis a computationally practical form of probit analysis for multiple response variables based on an assumed common factor model for the latent tolerances is proposed numerical integration over the factor space provides maximum likelihood estimation of the probit regression parameters and of the probabilities of response combinations under the model the procedure is applied to five variables from the pneumoconiosis field trial two variables of which were previously analyzed by ashford and sowden 1970 biometrics 26 535 546

the subelliptic heat kernel on su 2 representations asymptotics and gradient bounds the lie group su 2 endowed with its canonical subriemannian structure appears as a three dimensional model of a positively curved subelliptic space the goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities

cheeger constants of surfaces and isoperimetric inequalities we show that if the isoperimetric profile of a bounded genus non compact surface grows faster than vt then it grows at least as fast as a linear function this generalizes a result of gromov for simply connected surfaces we study the isoperimetric problem in dimension 3 we show that if the filling volume function in dimension 2 is euclidean while in dimension 3 it is sub euclidean and there is a g such that minimizers in dimension 3 have genus at most g then the filling function in dimension 3 is almost linear

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea