metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

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

enriched stick breaking processes for functional data in many applications involving functional data prior information is available about the proportion of curves having different attributes it is not straightforward to include such information in existing procedures for functional data analysis generalizing the functional dirichlet process fdp we propose a class of stick breaking priors for distributions of functions these priors incorporate functional atoms drawn from constrained stochastic processes the stick breaking weights are specified to allow user specified prior probabilities for curve attributes with hyperpriors accommodating uncertainty compared with the fdp the random distribution is enriched for curves having attributes known to be common theoretical properties are considered methods are developed for posterior computation and the approach is illustrated using data on temperature curves in menstrual cycles

functional clustering by bayesian wavelet methods summary we propose a nonparametric bayes wavelet model for clustering of functional data the wavelet based methodology is aimed at the resolution of generic global and local features during clustering and is suitable for clustering high dimensional data based on the dirichlet process the nonparametric bayes model extends the scope of traditional bayes wavelet methods to functional clustering and allows the elicitation of prior belief about the regularity of the functions and the number of clusters by suitably mixing the dirichlet processes posterior inference is carried out by gibbs sampling with conjugate priors which makes the computation straightforward we use simulated as well as real data sets to illustrate the suitability of the approach over other alternatives

on computing structural changes in evolving surfaces and their appearance as a surface undergoes a one parameter family of deformations its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur this paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph an algorithm based on homotopy continuation and curve tracing is proposed in the case of polynomial density functions whose zero set is an algebraic surface it has been implemented and examples are presented

a finite element approximation for a class of degenerate elliptic equations in this paper we exhibit a finite element method fitting a suitable geometry naturally associated with a class of degenerate elliptic equations usually called grushin type equations in a plane region and we discuss the related error estimates

dealing with label switching in mixture models summary in a bayesian analysis of finite mixture models parameter estimation and clustering are sometimes less straightforward than might be expected in particular the common practice of estimating parameters by their posterior mean and summarizing joint posterior distributions by marginal distributions often leads to nonsensical answers this is due to the so called label switching problem which is caused by symmetry in the likelihood of the model parameters a frequent response to this problem is to remove the symmetry by using artificial identifiability constraints we demonstrate that this fails in general to solve the problem and we describe an alternative class of approaches relabelling algorithms which arise from attempting to minimize the posterior expected loss under a class of loss functions we describe in detail one particularly simple and general relabelling algorithm and illustrate its success in dealing with the label switching problem on two examples

inegalites de sobolev produit sur les groupes de lie nilpotents on generalise dans le cadre des groupes de lie nilpotents reels et connexes l inegalite classique due a gagliardo et nirenberg

metric differentiation monotonicity and maps to l1 this is one of a series of papers on lipschitz maps from metric spaces to l1 here we present the details of results which were announced in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 sect 1 8 a new approach to the infinitesimal structure of lipschitz maps into l1 and as a first application an alternative proof of the main theorem of cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 that the heisenberg group does not admit a bi lipschitz embedding in l1 the proof uses the metric differentiation theorem of pauls commun anal geom 9 5 951 982 2001 and the cut metric description in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 to reduce the nonembedding argument to a classification of monotone subsets of the heisenberg group a quantitative version of this classification argument is used in our forthcoming joint paper with assaf naor cheeger et al in arxiv 0910 2026 2009

isoperimetric inequality under kahler ricci flow let bf m g t be a k ahler ricci flow with positive first chern class first we prove a uniform isoperimetric inequality for all time in the process we also prove a cheng yau type log gradient bound for positive harmonic functions on bf m g t without assuming the ricci curvature is bounded from below

weighted sobolev poincare inequalities and pointwise estimates for a class of degenerate elliptic equations in this paper we prove a sobolev poincare inequality for a class of function spaces associated with some degenerate elliptic equations these estimates provide us with the basic tool to prove an invariant harnack inequality for weak positive solutions in addition holder regularity of the weak solutions follows in a standard way let y z 7 ai aij9 a be a second order degenerate elliptic operator in divergence form with measurable coefficients in this paper we shall obtain pointwise estimates for the weak solutions of su 0 h61der continuity of the weak solutions and harnack inequality for nonnegative solutions let us recall that the original results for elliptic operators were obtained by de giorgi nash and moser an extensive bibliography about the degenerate case can be found in fli fl2 fs to introduce the results of the present paper let us recall some recent results in fl1 fl2 a suitable metric d is associated with the differential operator y in such a way that we obtain a new geometry which is natural for the degenerate operator as the euclidean geometry is natural for the laplace operator or more precisely as a suitable riemannian geometry is natural for a secondorder elliptic operator in the smooth case this idea is contained in many papers we refer to fp nsw the basic results in fl 1 fl2 are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the hormander hypoellipticity condition for sum of squares operators we note that the same idea is used in nsw s j v to obtain pointwise estimates for sum of squares operators on the other hand a different class of degenerate elliptic operators is considered in fks instead of a geometrical degeneracy a measure degeneracy is allowed a typical example of this class is given by yu div wo x vu where cl is a weight function belonging to the a2 class of muckenhoupt unified results for a class containing both the operators in fl 1 and in fks have received by the editors august 1 1989 1980 mathematics subject classification 1985 revision primary 46e35 35j70 partially supported by g n a f a of c n r and m u r s t italy 1991 american mathematical society 0002 9947 91 1 00 25 per page

the gauss green theorem in stratified groups abstract we lay the foundations for a theory of divergence measure fields in noncommutative stratified nilpotent lie groups such vector fields form a new family of function spaces which generalize in a sense the bv fields they provide the most general setting to establish gauss green formulas for vector fields of low regularity on sets of finite perimeter we show several properties of divergence measure fields in stratified groups ultimately achieving the related gauss green theorem

non absolutely convergent integrals in metric spaces abstract we develop the theory of henstock kurzweil type integral of functions with respect to metric distributions in the framework of metric spaces in the setting of metric currents as originated by e de giorgi l ambrosio and b kirchheim we apply the new integral to study a generalization of the stokes theorem

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

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

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

comparisons of relative bv capacities and sobolev capacity in metric spaces abstract we study relations between the variational sobolev 1 capacity and versions of variational bv capacity in a complete metric space equipped with a doubling measure and supporting a weak 1 1 poincare inequality we prove the equality of 1 modulus and the continuous 1 capacity extending the known results for 1 p to also cover the more geometric case p 1 then we give alternative definitions for variational bv capacities and obtain equivalence results between them finally we study relations between total 1 capacity and versions of bv capacity

bayesian inference on changes in response densities over predictor clusters in epidemiology it often is of interest to assess how individuals with different trajectories over time in an environmental exposure or biomarker differ with respect to a continuous response for ease in interpretation and presentation of results epidemiologists typically categorize predictors before analysis to extend this approach to time varying predictors individuals can be clustered by their predictor trajectory with the cluster index included as a predictor in a regression model for the response this article develops a semiparametric bayes approach that avoids assuming a prespecified number of clusters and allows the response to vary nonparametrically over predictor clusters this methodology is motivated by interest in relating trajectories in weight gain during pregnancy to the distribution of birth weight adjusted for gestational age at delivery in this setting the proposed approach allows the tails of the birth weight density to vary flexibly over weight gain clusters

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

constant mean curvature surfaces in sub riemannian geometry we investigate the minimal and isoperimetric surface problems in a large class of sub riemannian manifolds the so called vertically rigid spaces we construct an adapted connection for such spaces and using the variational tools of bryant griffiths and grossman derive succinct forms of the euler lagrange equations for critical points for the associated variational problems using the euler lagrange equations we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points moreover we use the formalism to construct a horizontal second fundamental form ii0 for vertically rigid spaces and as a first application use ii0 to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in carnot groups cannot be locally strictly horizontally geometrically convex we note that the convexity condition is distinct from others currently in the literature

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

global bifurcation sets and stable projections of nonsingular algebraic surfaces the view graph of a surface is a planar graph whose nodes are the stable views projections of the surface and whose edges represent transitional views of codimension one the space of all directions of orthogonal projection can be identified with the projective plane the set of bad projection directions associated with the degenerate views of positive codimension forms a graph in the projective plane the view bifurcation set this graph is dual to the view graph and divides the projective plane into a certain number of connected regions whose representatives are the nodes of the view graph we assume that the projected surface is nonsingular and parameterized by polynomials of degree d we present an estimate for the number of nodes in the view graph in terms of d and describe symbolic algorithms for computing the bifurcation set and the view graph of a surface from a parametrization

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

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

on gradient bounds for the heat kernel on the heisenberg group it is known that the couple formed by the two dimensional brownian motion and its levy area leads to the heat kernel on the heisenberg group which is one of the simplest sub riemannian space the associated diffusion operator is hypoelliptic but not elliptic which makes difficult the derivation of functional inequalities for the heat kernel however driver and melcher and more recently h q li have obtained useful gradient bounds for the heat kernel on the heisenberg group we provide in this paper simple proofs of these bounds and explore their consequences in terms of functional inequalities including cheeger and bobkov type isoperimetric inequalities for the heat kernel

a method for combining inference across related nonparametric bayesian models summary we consider the problem of combining inference in related nonparametric bayes models analogous to parametric hierarchical models the hierarchical extension formalizes borrowing strength across the related submodels in the nonparametric context modelling is complicated by the fact that the random quantities over which we define the hierarchy are infinite dimensional we discuss a formal definition of such a hierarchical model the approach includes a regression at the level of the nonparametric model for the special case of dirichlet process mixtures we develop a markov chain monte carlo scheme to allow efficient implementation of full posterior inference in the given model

a note on the extension of bv functions in metric measure spaces in this note a classical extension result for bv functions due to yu d burago and v g maz ja yu d burago v g maz ja potential theory and function theory for irregular regions seminars in math vol 3 v a steklov math inst leningrad 1969 translated from russian is generalized to the abstract setting of doubling metric measure spaces endowed with a differentiation structure called d structure

compression bounds for lipschitz maps from the heisenberg group to l 1 we prove a quantitative bi lipschitz non embedding theorem for the heisenberg group with its carnot caratheodory metric and apply it to give a lower bound on the integrality gap of the goemans linial semidefinite relaxation of the sparsest cut problem

lq l holder continuity for quasilinear parabolic equations associated to sobolev derivations abstract let x be a locally compact metrizable space endowed with a couple of equivalent finite radon measures m and and let e be a hilbert c monomodule over c0 x we consider a class of abstract nonlinear parabolic equations defined as follows let be a closed derivation from l2 x m to l2 e and tt be the strongly continuous nonlinear semigroup naturally associated in the sense of brezis 1973 to the convex l s c functional e u x u x p d x where is the natural modulus function associated to e the generator of the semigroup considered is a natural generalization of the usual p laplacian operator we suppose that a suitable sobolev like inequality of the form u l2d d 2 x m c u l2 e holds true for some d 2 with p 2 d then tt is a nonlinear markov semigroup in the sense that it is order preserving and nonexpansive on each lq x m for any q 2 and moreover it satisfies ttu ttv l x m cm x t u v lq x m for all q 2 and suitable constants depending only on p q d examples include the semigroup generated by the p laplacian on finite measure manifolds with boundary and homogeneous dirichlet boundary conditions as well as by p laplacian like operators associated both to regular sub riemannian structures and to systems of possibly singular or degenerate vector fields satisfying the appropriate sobolev inequalities

l1 poincare inequalities for differential forms on euclidean spaces and heisenberg groups abstract in this paper we prove interior poincare and sobolev inequalities in euclidean spaces and in heisenberg groups in the limiting case where the exterior resp rumin differential of a differential form is measured in l 1 norm unlike for l p p 1 the estimates are doomed to fail in top degree the singular integral estimates are replaced with inequalities which go back to bourgain brezis in euclidean spaces and to chanillo van schaftingen in heisenberg groups

computing exact aspect graphs of curved objects algebraic surfaces this article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface orthographic projection is assumed the algorithm is based on a catalog of visual events available from singularity theory it uses curve tracing cell decomposition homotopy continuation and ray tracing to construct the regions of the view sphere delineated by visual event curves the algorithm has been fully implemented and examples are presented

multilevel quantile function modeling with application to birth outcomes summary infants born preterm or small for gestational age have elevated rates of morbidity and mortality using birth certicate records in texas from 2002 2004 and environmental protection agency air pollution estimates we relate the quantile functions of birth weight and gestational age to ozone exposure and multiple predictors including parental age race and education level we introduce a semi parametric bayesian quantile approach that models the full quantile function rather than just a few quantile levels our multilevel quantile function model establishes relationships between birth weight and the predictors separately for each week of gestational age and between gestational age and the predictors separately across texas public health regions we permit these relationships to vary nonlinearly across gestational age spatial domain and quantile level and we unite them in a hierarchical model via a basis expansion on the regression coefficients that preserves interpretability very low birth weight is a primary concern so we leverage extreme value theory to supplement our model in the tail of the distribution gestational ages are rounded into weekly values so we present methodology for modeling quantile functions of discrete response data in a simulation study we show that pooling information across gestational age and quantile level substantially reduces mse of predictor effects relative to standard frequentist quantile regression wend that ozone is negatively associated with the lower tail of gestational age in south texas and across the distribution of birth weight for high gestational ages our methods are available in the r package bsquare

gagliardo nirenberg inequalities for differential forms in heisenberg groups the l 1 sobolev inequality states that for compactly supported functions u on the euclidean n space the l n n 1 norm of a compactly supported function is controlled by the l 1 norm of its gradient the generalization to differential forms due to lanzani and stein and bourgain and brezis is recent and states that a the l n n 1 norm of a compactly supported differential h form is controlled by the l 1 norm of its exterior differential du and its exterior codifferential delta u in special cases the l 1 norm must be replaced by the mathcal h 1 hardy norm we shall extend this result to heisenberg groups in the framework of an appropriate complex of differential forms

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

self improving properties of john nirenberg and poincare inequalities on spaces of homogeneous type abstract we give a condition which ensures that if one inequality of sobolev poincare type is valid then other stronger inequalities of a similar type also hold including weighted versions our main result includes many previously known results as special cases we carry out the analysis in the context of spaces of homogeneous type but the main result is new even in the usual euclidean setting

isoperimetric inequality from the poisson equation via curvature in this paper we establish an isoperimetric inequality in a metric measure space via the poisson equation let x d be a complete pathwise connected metric space with locally ahlfors q regular measure where q 1 that supports a local l2 poincare inequality we show that for the poisson equation u g if the local l norm of the gradient du can be bounded by the lorentz norm lq 1 of g then we obtain an isoperimetric inequality and a sobolev inequality in x d with optimal exponents by assuming a suitable curvature lower bound we establish such optimal bounds on 2011 wiley periodicals inc

clustering for sparsely sampled functional data we develop a flexible model based procedure for clustering functional data the technique can be applied to all types of curve data but is particularly useful when individuals are observed at a sparse set of time points in addition to producing final cluster assignments the procedure generates predictions and confidence intervals for missing portions of curves our approach also provides many useful tools for evaluating the resulting models clustering can be assessed visually via low dimensional representations of the curves and the regions of greatest separation between clusters can be determined using a discriminant function finally we extend the model to handle multiple functional and finite dimensional covariates and show how it can be applied to standard finite dimensional clustering problems involving missing data

the scale space aspect graph currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape the viewpoint and the projected image this means that the aspect graph may include details that an observer could never see in practice introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is large enough to merit explicit representation this effectively allows control over the number of nodes retained in the aspect graph this paper introduces the concept of the scale space aspect graph defines three different interpretations of the scale dimension and presents a detailed example for a simple class of objects with scale defined in terms of the spatial extent of features in the image

visibility occlusion and the aspect graph this chapter studies the ways in which the topology of the image of a polyhedron changes with changing viewpoint we catalog the ways that the topological appearance or aspect can change this enables us to find maximal regions of viewpoints of the same aspect we use these techniques to construct the viewpoint space partition vsp a partition of viewpoint space into maximal regions of constant aspect and its dual the aspect graph here we present tight bounds on the maximum size of the vsp and the aspect graph and give algorithms for their construction first in the convex case and then in the general case in particular we give bounds on the maximum size of n 2 and n 6 under an orthographic projection viewing model and of n 3 and n 9 under a perspective viewing model the algorithms make use of a new representation of the appearance of polyhedra from all viewpoints called the aspect representation or asp we believe that this representation is one of the significant contributions of this paper

rectifiability of flat chains in banach spaces with coefficients in zp in this paper we study the behaviour of a class of currents with coefficients in zp in metric spaces obtained by taking a suitable quotient of currents with coefficients in z we prove under fairly general assumption on the ambient space the rectifiability of the so called slice mass and under more restrictive assumptions the rectifiability of the mass measure

existence of isoperimetric regions in contact sub riemannian manifolds abstract we prove existence of regions minimizing perimeter under a volume constraint in contact sub riemannian manifolds whose quotient by the group of contact transformations preserving the sub riemannian metric is compact

equivalent definitions of bv space and of total variation on metric measure spaces abstract in this paper we introduce a new definition of bv based on measure upper gradients and prove the equivalence of this definition and the coincidence of the corresponding notions of total variation with the definitions based on relaxation of l 1 norm of the slope of lipschitz functions or upper gradients as in the previous work by the first author with gigli and savare in the sobolev case the proof requires neither local compactness nor doubling and poincare

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

differentiating maps into l1 and the geometry of bv functions this is one of a series of papers examining the interplay between differentiation theory for lipschitz maps x v and bi lipschitz nonembeddability where x is a metric measure space and v is a banach space here we consider the case v l 1 where differentiability fails we establish another kind of differentiability for certain x including n and h the heisenberg group with its carnot caratheodory metric it follows that does not bi lipschitz embed into l 1 as conjectured by j lee and a naor when combined with their work this provides a natural counterexample to the goemans linial conjecture in theoretical computer science the first such counterexample was found by khot vishnoi kv05 a key ingredient in the proof of our main theorem is a new connection between lipschitz maps to l 1 and functions of bounded variation which permits us to exploit results on the structure of bv functions on the heisenberg group fssc01

quantitative isoperimetry a la levy gromov on a riemannian manifold with a positive lower bound on the ricci tensor the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius the deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise these results are actually obtained in the more general context of possibly non smooth metric measure spaces with curvature dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method

online learning of a dirichlet process mixture of beta liouville distributions via variational inference a large class of problems can be formulated in terms of the clustering process mixture models are an increasingly important tool in statistical pattern recognition and for analyzing and clustering complex data two challenging aspects that should be addressed when considering mixture models are how to choose between a set of plausible models and how to estimate the model s parameters in this paper we address both problems simultaneously within a unified online nonparametric bayesian framework that we develop to learn a dirichlet process mixture of beta liouville distributions i e an infinite beta liouville mixture model the proposed infinite model is used for the online modeling and clustering of proportional data for which the beta liouville mixture has been shown to be effective we propose a principled approach for approximating the intractable model s posterior distribution by a tractable one which we develop such that all the involved mixture s parameters can be estimated simultaneously and effectively in a closed form this is done through variational inference that enjoys important advantages such as handling of unobserved attributes and preventing under or overfitting we explain that in detail the effectiveness of the proposed work is evaluated on three challenging real applications namely facial expression recognition behavior modeling and recognition and dynamic textures clustering

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

bayesian variable selection for latent class models in this article we develop a latent class model with class probabilities that depend on subject specific covariates one of our major goals is to identify important predictors of latent classes we consider methodology that allows estimation of latent classes while allowing for variable selection uncertainty we propose a bayesian variable selection approach and implement a stochastic search gibbs sampler for posterior computation to obtain model averaged estimates of quantities of interest such as marginal inclusion probabilities of predictors our methods are illustrated through simulation studies and application to data on weight gain during pregnancy where it is of interest to identify important predictors of latent weight gain classes

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

perimeter as relaxed minkowski content in metric measure spaces abstract in this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the minkowski contents w r t convergence in measure

parabolic curves of evolving surfaces in this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1 parameter family we concentrate on the parabolic set and its image under the gauss map but the same techniques also classify the changes in the dual of the surface all these have significance for computer vision for example through their connection with specularities and apparent contours with the aid of our complete classification which includes all the phenomena associated with multi contact tangent planes as well as those associated with parabolic sets we re examine examples given by j koenderink in his book 1990 under the title of morphological scripts we also explain some of the connections between parabolic sets and ridges of a surface where principal curvatures achieve turning values along lines of curvature the point of view taken is the analysis of the contact between surfaces and their tangent planes a systematic investigation of this yields the results using singularity theory the mathematical details are suppressed here and appear in bruce et al 1993

the chain rule and a compactness theorem for bv functions in the heisenberg group hn abstract at first in the setting of the heisenberg group we show the chain rule for a function u bv h when composed with a lipschitz function f r r and prove that v f u belongs to bv h and dhv dhu more precisely the following result is shown d h v f u h u l 2n 1 2 2n 1 2n 1 f u f u u s d q 1 j u f u d h c u secondly using the chain rule above we prove a compactness theorem for sbvh functions

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

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

a notable family of entire intrinsic minimal graphs in the heisenberg group which are not perimeter minimizing one of the main objectives of this paper is to unravel a new interesting phenomenon of the sub riemannian bernstein problem with respect to its euclidean ancestor with the purpose of also indicating a possible line of attack toward its solution we show that the global intrinsic graphs 1 2 are unstable critical points of the horizontal perimeter as a consequence of this fact the study of the stability acquires a central position in the problem itself

efficiently computing and representing aspect graphs of polyhedral objects an efficient algorithm and a data structure for computing and representing the aspect graph of polyhedral objects under orthographic projection are presented the aspect graph is an approach to representing 3 d objects by a set of 2 d views for the purpose of object recognition in this approach the viewpoint space is partitioned into regions such that in each region the qualitative structure of the line drawing does not change the viewing data of an object is the partition of the viewpoint space together with a representative view in each region the algorithm computes the viewing data for line drawings of polyhedral objects under orthographic projection

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

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

computing exact aspect graphs of curved objects solid of revolution this paper introduces a new approach to computing the exact orthographic aspect graph of curved objects curves corresponding to various visual events partition the view sphere into regions where the image structure is stable a catalog of these events for piecewise smooth objects is available from singularity theory for a solid of revolution whose generator is an algebraic curve each visual event is characterized by a system of polynomial equations whose roots can be computed by continuation methods within each region the stable image structure is characterized by a variation of cylindrical algebraic decomposition and ray tracing this approach has been implemented and several examples are presented

bayesian clustering and product partition models summary we present a decision theoretic formulation of product partition models ppms that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously a key observation in our construction is the fact that ppms can be formulated in the context of model selection the underlying partition structure in these models is closely related to that arising in connection with dirichlet processes this allows a straightforward adaptation of some computational strategies originally devised for nonparametric bayesian problems to our framework the resulting algorithms are more flexible than other competing alternatives that are used for problems involving ppms we propose an algorithm that yields bayes estimates of the quantities of interest and the groups of experimental units we explore the application of our methods to the detection of outliers in normal and student t regression models with clustering structure equivalent to that induced by a dirichlet process prior we also discuss the sensitivity of the results considering different prior distributions for the partitions

sub riemannian calculus and monotonicity of the perimeter for graphical strips we consider the class of minimal surfaces given by the graphical strips mathcal s in the heisenberg group mathbb h 1 and we prove that for points p along the center of mathbb h 1 the quantity frac sigma h mathcal s cap b p r r q 1 is monotone increasing here q is the homogeneous dimension of mathbb h 1 we also prove that these minimal surfaces have maximum volume growth at infinity

product partition models for change point problems product partition models assume that observations in different components of a random partition of the data are independent if the probability distribution of random partitions is in a certain product form prior to making the observations it is also in product form given the observations the product model thus provides a convenient machinery for allowing the data to weight the partitions likely to hold and inference about particular future observations may then be made by first conditioning on the partition and then averaging over all partitions these models apply with special computational simplicity to change point problems where the partitions divide the sequence of observations into components within which different regimes hold we show with appropriate selection of prior product models that the observations can eventually determine approximately the true partition