linearly degenerate hierarchies of quasiclassical sdym type we demonstrate that sdym self dual yang mills equations for the lie algebra of one dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy we define the reduction in terms of wave functions and introduce a generating relation lax sato equations and the dressing scheme for the reduced hierarchy a multidimensional case is also discussed

the monge ampere equation and its link to optimal transportation we survey old and new regularity theory for the monge ampere equation show its connection to optimal transportation and describe the regularity properties of a general class of monge ampere type equations arising in that context

on the validity of the maximum principle and of the euler lagrange equation for a minimum problem depending on the gradient we consider the limiting case alpha infty of the problem of minimizing int omega nabla u x alpha g u dx hbox on u in u 0 w 1 alpha 0 omega where g is differentiable and strictly monotone if this infimum is finite it is evidently attained we show that any minimizing function u satisfies the appropriate form of the euler lagrange equation i e for some function p div p x g u x quad rm for quad p x in partial j b nabla u x where jb is the indicator function of the closed unit ball in the euclidean norm of bbb r n and partial is the subdifferential of the convex function jb

nearly round spheres look convex we prove that a riemannian manifold m g close enough to the round sphere in the c 4 topology has uniformly convex injectivity domains so m appears uniformly convex in any exponential chart the proof is based on the ma trudinger wang nonlocal curvature tensor which originates from the regularity theory of optimal transport

sandpiles and river networks extended systems with nonlocal interactions a proposed continuum model of sandpile growth is able to demonstrate instantaneous long range interactions and sudden collapses typical of real sandpiles the model is shown to be equivalent to an evolutionary quasivariational inequality in the limiting case of zero angle of repose the inequality describes water transport in a river network

equivalent formulations for monge kantorovich equation abstract in this paper we study some equivalent formulations in divergence form for the optimization problem max d w 0 1 1 s t x k x a e x where k c and k 0 in this is the so called dual equation of monge kantorovich problem

holder regularity of the normal distance with an application to a pde model for growing sandpiles given a bounded domain omega in mathbb r 2 with smooth boundary the cut locus overline sigma is the closure of the set of nondifferentiability points of the distance d from the boundary of omega the normal distance to the cut locus tau x is the map which measures the length of the line segment joining x to the cut locus along the normal direction dd x whenever x notin overline sigma recent results show that this map restricted to boundary points is lipschitz continuous as long as the boundary of omega is of class c 2 1 our main result is the global holder regularity of tau in the case of a domain omega with analytic boundary we will also show that the regularity obtained is optimal as soon as the set of the so called regular conjugate points is nonempty in all the other cases lipschitz continuity can be extended to the whole domain omega the above regularity result for tau is also applied to derive the holder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer

randomized algorithms for the low rank approximation of matrices we describe two recently proposed randomized algorithms for the construction of low rank approximations to matrices and demonstrate their application inter alia to the evaluation of the singular value decompositions of numerically low rank matrices being probabilistic the schemes described here have a finite probability of failure in most cases this probability is rather negligible 10 17 is a typical value in many situations the new procedures are considerably more efficient and reliable than the classical deterministic ones they also parallelize naturally we present several numerical examples to illustrate the performance of the schemes

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

the distance function from the boundary of a domain with corners abstract we study the regularity of the distance function to the boundary of a domain in r 2 with respect to some asymmetric norms we allow the boundary of the domain to have corners we obtain an explicit formula for the second derivative of these distance functions furthermore we study a generalized notion of the ridge of a domain which is the set of singularities of a distance function to the boundary of the domain we completely characterize the ridge by using a generalized notion of curvature

low rank matrix completion by riemannian optimization the matrix completion problem consists of finding or approximating a low rank matrix based on a few samples of this matrix we propose a new algorithm for matrix completion that minimizes the least square distance on the sampling set over the riemannian manifold of fixed rank matrices the algorithm is an adaptation of classical nonlinear conjugate gradients developed within the framework of retraction based optimization on manifolds we describe all the necessary objects from differential geometry necessary to perform optimization over this low rank matrix manifold seen as a submanifold embedded in the space of matrices in particular we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions finally we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations the numerical experiments indicate that our approach

propagation of singularities along broken characteristics abstract this paper contributes to the analysis of propagation of singularities for semiconcave solutions of hamilton jacobi equations under certain conditions we establish the existence and uniqueness of certain broken characteristics termed strong characteristics if u is a viscosity solution of a hamilton jacobi equation then strong characteristics carry the singularities of u

singular gradient flow of the distance function and homotopy equivalence let m be a riemannian manifold and let varomega be a bounded open subset of m it is well known that significant information about the geometry of varomega is encoded into the properties of the distance d partial varomega from the boundary of varomega here we show that the generalized gradient flow associated with the distance preserves singularities that is if x 0 is a singular point of d partial varomega then the generalized characteristic starting at x 0 stays singular for all times as an application we deduce that the singular set of d partial varomega has the same homotopy type as varomega

a counterexample to uniqueness of generalized characteristics in hamilton jacobi theory abstract the notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for hamilton jacobi equations this note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

the free boundary of variational inequalities with gradient constraints abstract in this paper we prove that the free boundary of the minimizer of i v u 1 2 d v 2 v d x subject to the pointwise gradient constraint d v p 1 is as regular as the tangent bundle of the boundary of the domain to this end we study a generalized notion of ridge of a domain in the plane which is the set of singularity of the distance function in the p norm to the boundary of the domain

propagation of singularities for solutions of hamilton jacobi equations abstract we consider the viscosity solution of the cauchy problem for a class of hamilton jacobi equations and we show that the points of the c 1 singular support of such a function propagate along the generalized characteristics for all the times

generation of singularities from the initial datum for hamilton jacobi equations abstract we study the generation of singularities from the initial datum for a solution of the cauchy problem for a class of hamilton jacobi equations of evolution for such equations we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point

dynamic and asymptotic behavior of singularities of certain weak kam solutions on the torus abstract for mechanical hamiltonian systems on the torus we study the dynamical properties of the generalized characteristic semiflows associated with the hamilton jacobi equations and build the relation between the limit sets of the semiflows and the projected aubry sets

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

bilinear generalized approximate message passing part ii applications in this paper we extend the generalized approximate message passing g amp approach originally proposed for high dimensional generalized linear regression in the context of compressive sensing to the generalized bilinear case in part i of this two part paper we derived our bilinear g amp big amp algorithm as an approximation of the sum product belief propagation algorithm in the high dimensional limit and proposed an adaptive damping mechanism that aids convergence under finite problem sizes an expectation maximization em based method to automatically tune the parameters of the assumed priors and two rank selection strategies here in part ii we discuss the specializations of big amp to the problems of matrix completion robust pca and dictionary learning and present the results of an extensive empirical study comparing big amp to state of the art algorithms on each problem our numerical results using both synthetic and real world datasets demonstrate that em big amp yields excellent reconstruction accuracy often best in class while maintaining competitive runtimes

hardyʼs inequality and curvature a hardy inequality of the form f x pdx p 1p p 1 a x f x p x pdx for all f c0 r is considered for p 1 p 1 where is a domain in rnrn n 2n 2 r r is the ridge of and x x is the distance from x x to the boundary the main emphasis is on determining the dependence of a a on the geometric properties of a hardy inequality is also established for any doubly connected domain in r2r2 in terms of a uniformization of that is any conformal univalent map of onto an annulus

some properties of semiconcave functions with general modulus abstract the class of semiconcave functions represents a useful generalization of the one of concave functions such an extension can be achieved requiring that a function satisfies a suitable one sided estimate in this paper the structure of the set of points at which a semiconcave function fails to be differentiable the singular set is studied first we prove some results on the existence of arcs contained on the singular set then we show how these abstract results apply to semiconcave solutions of hamilton jacobi equations

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

propagation of singularities for solutions of nonlinear first order partial differential equations few results are available in the mathematical literature for studying the structure of the singular set of a weak solution u of f x u du 0 this paper provides new techniques to analyse such a set when u is semiconcave and f is a nonlinear convex function with respect to p the main objective achieved here is a classification of the singularities of u that propagate along lipschitz arcs such a propagation phenomenon is also described by means of a generalized characteristics inclusion

doubrov ferapontov general heavenly equation and the hyper kahler hierarchy we give a description of a recently introduced doubrov ferapontov general heavenly equation in terms of a closed differential plucker two form rationally depending on the spectral parameter we demonstrate that the general heavenly equation is an important generating equation in the context of takasaki hyper kahler hierarchy and it is also directly connected to hyper kahler geometry through the gindikin construction we develop a dressing scheme and introduce a formula for the potential satisfying of the general heavenly equation multidimensional generalization is also outlined

fast slow diffusion and collapsing sandpiles we regard the limit as p of the flow governed by the p laplacian as providing a simplistic model for the collapse of an initially unstable sandpile upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases involving nonlocal geometric curvature motion in particular we note that such models entail instantaneous mass transfer governed by monge kantorovich theory

a variational approach to the macroscopic electrodynamics of anisotropic hard superconductors we consider bean s critical state model for anisotropic superconductors a variational problem solved by the quasi static evolution of the internal magnetic field is obtained as the limit of functionals arising from maxwell s equations combined with a power law for the dissipation moreover the quasi static approximation of the internal electric field is recovered using a first order necessary condition if the sample is a long cylinder subjected to an axial uniform external field the macroscopic electrodynamics is explicitly determined

dunajski generalization of the second heavenly equation dressing method and the hierarchy dunajski generalization of the second heavenly equation is studied a dressing scheme applicable to the dunajski equation is developed an example of constructing solutions in terms of implicit functions is considered a dunajski equation hierarchy is described and its lax sato form is presented the dunajski equation hierarchy is characterized by the conservation of a three dimensional volume form in which a spectral variable is taken into account

a stochastic model for growing sandpiles and its continuum limit we study a stochastic model for the sandpile growth the height of the sandpile above every site of a d dimensional lattice is denoted by an integer the height changes randomly and after a suitable scaling of the time and space we show that the rescaled height converges to a deterministic function the limit is governed by a nonlinear evolution equation

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

hardy inequalities for robin laplacians abstract in this paper we establish a hardy inequality for laplace operators with robin boundary conditions for convex domains in particular we show explicitly how the corresponding hardy weight depends on the coefficient of the robin boundary conditions we also study several extensions to non convex and unbounded domains

the distance function from the boundary in a minkowski space let the space be endowed with a minkowski structure that is is the gauge function of a compact convex set having the origin as an interior point and with boundary of class and let be the asymmetric distance associated to given an open domain of class let be the minkowski distance of a point from the boundary of we prove that a suitable extension of to which plays the role of a signed minkowski distance to is of class in a tubular neighborhood of and that is of class outside the cut locus of that is the closure of the set of points of nondifferentiability of in in addition we prove that the cut locus of has lebesgue measure zero and that can be decomposed up to this set of vanishing measure into geodesics starting from and going into along the normal direction with respect to the minkowski distance we compute explicitly the jacobian determinant of the change of variables that associates to every point outside the cut locus the pair where denotes the unique projection of on and we apply these techniques to the analysis of pdes of monge kantorovich type arising from problems in optimal transportation theory and shape optimization

taut submanifolds and foliations we give an equivalent description of taut submanifolds of complete riemannian manifolds as exactly those submanifolds whose normal exponential map has integrable fibers it turns out that every taut submanifold is also mathbb z 2 taut so that tautness is essentially the same as mathbb z 2 tautness in the case where the normal exponential map of a submanifold has integrable fibers we explicitely construct generalized bott samelson cycles for the critical points of the energy functionals on the path spaces which generically represent a basis for the mathbb z 2 cohomology we also consider singular riemannian foliations all of whose leaves are taut and discuss some of their main features using our characterization of taut submanifolds we are able to show that tautness of a singular riemannian foliation is actually a property of the quotient

evolution monge kantorovich equation abstract the paper deals with monge kantorovich equation mk for short in an open bounded domain with dirichlet boundary condition we study existence and uniqueness of a solution to the associated evolution problem emk for short and we prove the convergence to a solution of mk when time goes to a solution is a couple u where u is the potential and is the transportation flux we study the problem for a given radon measure source term and we show how to use the numerical method of dumont and igbida 2009 22 to provide numerical approximation of the solution of mk

near optimal matrix recovery from random linear measurements in matrix recovery from random linear measurements one is interested in recovering an unknown m by n matrix x 0 from n m n measurements y i t r a i x 0 where each a i is an m by n measurement matrix with i i d random entries i 1 n we present a matrix recovery algorithm based on approximate message passing which iteratively applies an optimal singular value shrinker a nonconvex nonlinearity tailored specifically for matrix estimation our algorithm typically converges exponentially fast offering a significant speedup over previously suggested matrix recovery algorithms such as iterative solvers for nuclear norm minimization nnm it is well known that there is a recovery tradeoff between the information content of the object x 0 to be recovered specifically its matrix rank r and the number of linear measurements n from which recovery is to be attempted the precise tradeoff between r and n beyond which recovery by a given algorithm becomes possible traces the so called phase transition curve of that algorithm in the r n plane the phase transition curve of our algorithm is noticeably better than that of nnm interestingly it is close to the information theoretic lower bound for the minimal number of measurements needed for matrix recovery making it not only state of the art in terms of convergence rate but also near optimal in terms of the matrices it successfully recovers

parallel translation of curvature along geodesics soient m et m des varietes de riemann connexes completes de dimension 2 et certaines conditions si m est simplement connexe et si chaque sous ensemble compact du lieu de coupure de p a une mesure de hansdorff fini a 1 dimension alors il y a une immersion isometrique f m m avec f p p et df i en p

optimal shrinkage of singular values we consider the recovery of low rank matrices from noisy data by shrinkage of singular values in which a single univariate nonlinearity is applied to each of the empirical singular values we adopt an asymptotic framework in which the matrix size is much larger than the rank of the signal matrix to be recovered and the signal to noise ratio of the low rank piece stays constant for a variety of loss functions including mean square error mse square frobenius norm the nuclear norm loss and the operator norm loss we show that in this framework there is a well defined asymptotic loss that we evaluate precisely in each case in fact each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities we provide a general method for evaluating these optimal nonlinearities and demonstrate our framework by working out simple explicit formulas for the optimal nonlinearities in the frobenius nuclear and operator norm cases for example for a square low rank n by n matrix observed in white noise with level sigma the optimal nonlinearity for mse loss simply shrinks each data singular value y to sqrt y 2 4n sigma 2 or to 0 if y this optimal nonlinearity guarantees an asymptotic mse of 2nr sigma 2 which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding providing guarantees of 3nr sigma 2 and 6nr sigma 2 respectively our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision as an example we compute optimal shrinkers for the schatten p norm loss for any p 0

the phase transition of matrix recovery from gaussian measurements matches the minimax mse of matrix denoising let be an unknown m by n matrix in matrix recovery one takes linear measurements of where and each is an m by n matrix a popular approach for matrix recovery is nuclear norm minimization nnm solving the convex optimization problem for all where denotes the nuclear norm namely the sum of singular values empirical work reveals a phase transition curve stated in terms of the undersampling fraction rank fraction and aspect ratio specifically when the measurement matrices ai have independent standard gaussian random entries a curve exists such that if nnm typically succeeds for large m n whereas if it typically fails an apparently quite different problem is matrix denoising in gaussian noise in which an unknown m by n matrix is to be estimated based on direct noisy measurements where the matrix z has independent and identically distributed gaussian entries a popular matrix denoising scheme solves the unconstrained optimization problem when optimally tuned this scheme achieves the asymptotic minimax mean squared error where we report extensive experiments showing that the phase transition in the first problem matrix recovery from gaussian measurements coincides with the minimax risk curve in the second problem matrix denoising in gaussian noise for any rank fraction at each common aspect ratio our experiments considered matrices belonging to two constraint classes real m by n matrices of various ranks and aspect ratios and real symmetric positive semidefinite n by n matrices of various ranks

the geodesic x ray transform with matrix weights author s paternain gp salo m uhlmann g zhou h abstract 2019 by johns hopkins university press consider a compact riemannian manifold of dimension 3 with strictly convex boundary such that the manifold admits a strictly convex function we show that the attenuated ray transform in the presence of an arbitrary connection and higgs field is injective modulo the natural obstruction for functions and one forms we also show that the connection and the higgs field are uniquely determined by the scattering relation modulo gauge transformations the proofs involve a reduction to a local result showing that the geodesic x ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions as a somewhat striking corollary we show that these integral geometry problems can be solved on strictly convex manifolds of dimension 3 having nonnegative sectional curvature similar results were known earlier in negative sectional curvature we also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

excess action and broken characteristics for hamilton jacobi equations abstract we study propagation of singularities for hamilton jacobi equations s t h t x s 0 t x 0 r n by means of the excess lagrangian action and a related class of characteristics in a sense the excess action gauges how far a curve x t is from being action minimizing for a given viscosity solution s t x of the hamilton jacobi equation broken characteristics are defined as curves along which the excess action grows at the slowest pace possible in particular we demonstrate that broken characteristics carry the singularities of the viscosity solution

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

semigroups of relations and subdifferentials of the minimal time function and of the geodesic distance given a closed subset s of a banach space x we study the minimal time function to reach a point x of x starting from s we relate this problem to the study of the geodesic distance from x to s associated with a riemannian metric or a finsler metric it appears that both cases can be treated simultaneously and yield solutions to a hamilton jacobi equation we detect simple links between the general framework we propose and multivalued semigroups new concepts of infinitesimal generators of such semigroups are considered

the regularity of the distance function propagates along minimizing geodesics abstract we consider the distance function from the boundary of an open bounded set r n associated to a riemannian metric with c 1 1 coefficients we show that the c 1 1 regularity propagates towards the boundary along the distance minimizing geodesics hence we show that the cut locus is invariant with respect to the generalized gradient flow associated to the distance function and that it has the same homotopy type as

stability analysis for semilinear parabolic problems in general unbounded domains abstract we introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain under boundary condition of the oblique derivative type we employ these notions in the stability analysis of semilinear problems some of the properties we derive are new even in the dirichlet or in the whole space cases as an application we show the validity of the hair trigger effect for the fisher kpp equation on general uniformly smooth domains

an infinite number of hidden variables in hyper kahler metrics three types of hidden variables both independent and dependent are shown to underlie the hyper kahler geometry in a complexified setting the variables satisfy an infinite set of differential equations hierarchy just as in the case of most nonlinear integrable systems the notion of the plebanski key functions j math phys 16 2395 1975 is extended to this hierarchy to give an analog of the notion of the function two examples of special solutions which are reminiscent of several solution techniques in the theory of nonlinear integrable systems are presented for illustration

a geometric characterization of a sharp hardy inequality abstract in this paper we prove that the distance function of an open connected set in r n 1 with a c 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex we then prove that hardy inequalities with a sharp constant hold on weakly mean convex c 2 domains moreover we show that the weakly mean convexity condition cannot be weakened we also prove various improved hardy inequalities on mean convex domains along the line of brezis and marcus 1997 7

global structure and regularity of solutions to the eikonal equation the evolutionary eikonal equation is a hamilton jacobi equation with hamiltonian h p p which is not strictly convex nor smooth the regularizing effect of hamiltonian for the eikonal equation is much weaker than that of strictly convex hamiltonians therefore leading to new phenomena in this paper we study the set of singularity points of solutions in the upper half space for c1 or c2 initial data with emphasis on the countability of connected components of the set the regularity of solutions in the complement of the set of singularity points is also obtained

a sard theorem for the distance function in this article we prove that the set of all critical values of the distance function from a submanifold of a complete riemannian manifold whose dimension is less than 5 is of lebesgue measure zero

closing aubry sets ii given a tonelli hamiltonian h t m of class ck with k 2 we prove the following results 1 assume there exist a recurrent point of the projected aubry set and a critical viscosity subsolution u such that u is a c1 critical solution in an open neighborhood of the positive orbit of suppose further that u is c2 at then there exists a ck potential v m small in c2 topology for which the aubry set of the new hamiltonian h v is either an equilibrium point or a periodic orbit 2 if m is two dimensional 1 holds replacing c1 critical solution c2 at by cm3 critical subsolution these results can be considered as a first step through the attempt of proving the mane s conjecture in c2 topology in a second paper 27 we will generalize 2 to arbitrary dimension moreover such an extension will need the introduction of some new techniques which will allow us to prove in 27 the mane s density conjecture in c1 topology our proofs are based on a combination of techniques coming from finite dimensional control theory and hamilton jacobi theory together with some of the ideas that were used to prove c1 closing lemmas for dynamical systems 2014 wiley periodicals inc

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below i we investigate the finiteness structure of a complete non compact n dimensional riemannian manifold m whose radial curvature at a base point of m is bounded from below by that of a non compact von mangoldt surface of revolution with its total curvature greater than we show as our main theorem that all busemann functions on m are exhaustions and that there exists a compact subset of m such that the compact set contains all critical points for any busemann function on m as corollaries by the main theorem m has finite topological type and the isometry group of m is compact

on the local semiconcavity of the solutions of the eikonal equation abstract we show that if the hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave furthermore in the 1d case we show that every viscosity solution of the eikonal equation is semiconcave if and only if the hamiltonian is lipschitz continuous with respect to the state variable

generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains using three different notions of the generalized principal eigenvalue of linear second order elliptic operators in unbounded domains we derive necessary and sufficient conditions for the validity of the maximum principle as well as for the existence of positive eigenfunctions for the dirichlet problem relations between these principal eigenvalues their simplicity and several other properties are further discussed 2015 wiley periodicals inc

minimax risk of matrix denoising by singular value thresholding an unknown m by n matrix x 0 is to be estimated from noisy measurements y x 0 z where the noise matrix z has i i d gaussian entries a popular matrix denoising scheme solves the nuclear norm penalization problem operatorname min x y x f 2 2 lambda x where x denotes the nuclear norm sum of singular values this is the analog for matrices of ell 1 penalization in the vector case it has been empirically observed that if x 0 has low rank it may be recovered quite accurately from the noisy measurement y in a proportional growth framework where the rank r n number of rows m n and number of columns n all tend to infty proportionally to each other r n m n rightarrow rho m n n rightarrow beta we evaluate the asymptotic minimax mse mathcal m rho beta lim m n n rightarrow infty inf lambda sup operatorname rank x leq r n operatorname mse x 0 hat x lambda our formulas involve incomplete moments of the quarter and semi circle laws beta 1 square case and the marcenko pastur law beta 1 nonsquare case for finite m and n we show that mse increases as the nonzero singular values of x 0 grow larger as a result the finite n worst case mse a quantity which can be evaluated numerically is achieved when the signal x 0 is infinitely strong the nuclear norm penalization problem is solved by applying soft thresholding to the singular values of y we also derive the minimax threshold namely the value lambda rho which is the optimal place to threshold the singular values all these results are obtained for general nonsquare nonsymmetric real matrices comparable results are obtained for square symmetric nonnegative definite matrices

the lipschitz continuity of the distance function to the cut locus let n be a closed submanifold of a complete smooth riemannian manifold m and uv the total space of the unit normal bundle of n for each v c uv let p v denote the distance from n to the cut point of n on the geodesic y with the velocity vector tyv 0 v the continuity of the function p on uv is well known in this paper we prove that p is locally lipschitz on which p is bounded in particular if m and n are compact then p is globally lipschitz on uy therefore the canonical interior metric 6 may be introduced on each connected component of the cut locus of n and this metric space becomes a locally compact and complete length space let n be an immersed submanifold of a complete cc riemannian manifold m and 7 uv n the unit normal bundle of n for each positive integer k and vector v c uv let a number k v denote the parameter value of fy where 7yv denotes the geodesic for which the velocity vector is v at t 0 such that yv ak v is the k th focal point conjugate point for the case in which n is a point of n along y counted with focal or conjugate multiplicities a unit speed geodesic segment 0 a m emanating from n is called an n segment if t d n y t on 0 a by the first variation formula an n segment is orthogonal to n a point 7y to on an n segment 7y v e uv is called a cut point of n if there is no n segment properly containing y o to for each v c uv let p v denote the distance from n to the cut point on y of n whitehead 27 investigated the structure of the conjugate locus and the cut locus of a point on a real analytic finsler manifold he determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood cf also 25 and proved the continuity of the function p in compact symmetric spaces t sakai 19 and m takeuchi 23 determined the detailed structure of the cut locus of a point the detailed structure of the cut locus of a point in a 2 dimensional riemannian manifold has been investigated by poincare myers and others 7 11 13 hartman first tried to show the absolute continuity of the function p when m is 2 dimensional he proved in 8 that if p is of bounded variation then p is absolutely continuous recently hebda 11 and the first named author 13 independently proved ambrose s problem by showing that p is absolutely continuous on a closed arc on which p is bounded when n is a point in a 2 dimensional riemannian manifold therefore the cut locus of a point in a compact 2 dimensional received by the editors october 14 1998 and in revised form april 13 1999 2000 mathematics subject classification primary 53c22 secondary 28a78 supported in part by a grant in aid for scientific research from the ministry of education science sports and culture japan 2000 american mathematical society

bilinear generalized approximate message passing part i derivation in this paper we extend the generalized approximate message passing g amp approach originally proposed for high dimensional generalized linear regression in the context of compressive sensing to the generalized bilinear case which enables its application to matrix completion robust pca dictionary learning and related matrix factorization problems here in part i of a two part paper we derive our bilinear g amp big amp algorithm as an approximation of the sum product belief propagation algorithm in the high dimensional limit where central limit theorem arguments and taylor series approximations apply and under the assumption of statistically independent matrix entries with known priors in addition we propose an adaptive damping mechanism that aids convergence under finite problem sizes an expectation maximization em based method to automatically tune the parameters of the assumed priors and two rank selection strategies in part ii of the paper we will discuss the specializations of em big amp to the problems of matrix completion robust pca and dictionary learning and we will present the results of an extensive empirical study comparing em big amp to state of the art algorithms on each problem

on the cut locus of closed sets abstract let c be a closed subset of a smooth manifold of dimension n m and let m be endowed with a riemannian metric of class c 2 we study the cut locus of c cut c first we show that cut c is a set of measure zero then we assume that c is the boundary of an open bounded set m in particular this assumption implies that cut c 0 we deduce that cut c is invariant w r t the generalized gradient flow associated with the distance function from the set c as a consequence of the invariance we have that cut c has the same homotopy type as the set furthermore if m is a compact manifold then cut c has the same homotopy type as m c finally we show that the closure of the cut locus stays away from c if and only if c is a manifold of class c 1 1

on singularities of value function for bolza optimal control problem abstract we study the set of points of nondifferentiability called the singular set of the value function of a bolza optimal control problem the value function is a viscosity solution to an associated hamilton jacobi equation the method of characteristics associates to this equation a hamiltonian system that in turn can be used to study the propagation of singularities of the value function in particular we obtain an extension of the rankine hugoniot type condition which is well known in the conservation law theory

hardy inequalities for p laplacians with robin boundary conditions in this paper we study the best constant in a hardy inequality for the p laplace operator on convex domains with robin boundary conditions we show in particular that the best constant equals p

propagation of singularities for weak kam solutions and barrier functions this paper studies the structure of the singular set points of nondifferentiability of viscosity solutions to hamilton jacobi equations associated with general mechanical systems on the n torus first using the level set method we characterize the propagation of singularities along generalized characteristics then we obtain a local propagation result for singularities of weak kam solutions in the supercritical case finally we apply such a result to study the propagation of singularities for barrier functions

on a system of partial differential equations of monge kantorovich type abstract we consider a system of pdes of monge kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors the existence of a solution of such system in a regular open domain r n whose construction is based on an asymmetric minkowski distance from the boundary of was already established in g crasta a malusa the distance function from the boundary in a minkowski space trans amer math soc submitted for publication in this paper we prove that this solution is essentially unique a fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form which is of some interest by itself