from concentration to logarithmic sobolev and poincare inequalities abstract we give a new proof of the fact that gaussian concentration implies the logarithmic sobolev inequality when the curvature is bounded from below and also that exponential concentration implies poincare inequality under null curvature condition our proof holds on non smooth structures such as length spaces and provides a universal control of the constants we also give a new proof of the equivalence between dimension free gaussian concentration and talagrand s transport inequality

estimates of the derivatives for parabolic operators with unbounded coefficients we consider a class of second order uniformly elliptic operators a with unbounded coefficients in r n using a bernstein approach we provide several uniform estimates for the semigroup t t generated by the realization of the operator a in the space of all bounded and continuous or holder continuous functions in r n as a consequence we obtain optimal schauder estimates for the solution to both the elliptic equation u au f 0 and the nonhomogeneous dirichlet cauchy problem d t u au g then we prove two different kinds of pointwise estimates of t t that can be used to prove a liouville type theorem finally we provide sharp estimates of the semigroup t t in weighted l p spaces related to the invariant measure associated with the semigroup

stability of the bakry emery theorem on rn abstract we establish quantitative stability estimates for the bakry emery bound on logarithmic sobolev and poincare constants of uniformly log concave measures more specifically we show that if a 1 uniformly log concave measure has almost the same logarithmic sobolev or poincare constant as the standard gaussian measure then it almost splits off a gaussian factor our results are dimension free leading to dimension free stability estimates for gaussian concentration of lipschitz functions the proofs are based on stein s method optimal transport and an approximate integration by parts identity relating measures and near extremals in the associated functional inequality

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

a theoretical investigation into the performance of the hopfield model an analysis is made of the behavior of the hopfield model as a content addressable memory cam and as a method of solving the traveling salesman problem tsp the analysis is based on the geometry of the subspace set up by the degenerate eigenvalues of the connection matrix the dynamic equation is shown to be equivalent to a projection of the input vector onto this subspace in the case of content addressable memory it is shown that spurious fixed points can occur at any corner of the hypercube that is on or near the subspace spanned by the memory vectors analysed is why the network can frequently converge to an invalid solution when applied to the traveling salesman problem energy function with these expressions the network can be made robust and can reliably solve the traveling salesman problem with tour sizes of 50 cities or more

the sharp quantitative sobolev inequality for functions of bounded variation abstract the classical sobolev embedding theorem of the space of functions of bounded variation bv r n into l n r n is proved in a sharp quantitative form

finding a maximum clique in an arbitrary graph we describe a new type of branch and bound procedure for finding a maximum clique in an arbitrary graph g v e the two main ingredients both of o v e time complexity are i an algorithm for finding a maximal triangulated induced subgraph of g and ii an algorithm for finding a maximal k chromatic induced subgraph of g we discuss computational experience on randomly generated graphs with up to 400 vertices and 30 000 edges

a two sided estimate for the gaussian noise stability deficit the gaussian noise stability of a set a subset mathbb r n is defined by begin aligned mathcal s rho a mathbb p left x in a y in a right end aligned where x y are standard jointly gaussian vectors satisfying mathbb e x i y j delta ij rho borell s inequality states that for all 0 rho 1 among all sets a subset mathbb r n with a given gaussian measure the quantity mathcal s rho a is maximized when a is a half space we give a novel short proof of this fact based on stochastic calculus moreover we prove an almost tight two sided dimension free robustness estimate for this inequality by introducing a new metric to measure the distance between the set a and its corresponding half space h namely the distance between the two centroids we show that the deficit mathcal s rho h mathcal s rho a can be controlled from both below and above by essentially the same function of the distance up to logarithmic factors as a consequence we also establish the conjectured exponent in the robustness estimate proven by mossel neeman which uses the total variation distance as a metric in the limit rho rightarrow 1 we obtain an improved dimension free robustness bound for the gaussian isoperimetric inequality our estimates are also valid for a generalized version of stability where more than two correlated vectors are considered

sharp dimension free quantitative estimates for the gaussian isoperimetric inequality we provide a full quantitative version of the gaussian isoperimetric inequality the difference between the gaussian perimeter of a given set and a half space with the same mass controls the gap between the norms of the corresponding barycenters in particular it controls the gaussian measure of the symmetric difference between the set and the half space oriented so to have the barycenter in the same direction of the set our estimate is independent of the dimension sharp on the decay rate with respect to the gap and with optimal dependence on the mass

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

faber krahn inequalities in sharp quantitative form the classical faber krahn inequality asserts that balls uniquely minimize the first eigenvalue of the dirichlet laplacian among sets with given volume in this paper we prove a sharp quantitative enhancement of this result thus confirming a conjecture by nadirashvili and bhattacharya weitsman more generally the result applies to every optimal poincare sobolev constant for the embeddings w 1 2 0 omega hookrightarrow l q omega

isoperimetric constants for product probability measures a dimension free lower bound is found for isoperimetric constants of product probability measures from this some analytic inequalities are derived

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

symmetry of minimizers of a gaussian isoperimetric problem we study an isoperimetric problem described by a functional that consists of the standard gaussian perimeter and the norm of the barycenter the second term is in competition with the perimeter balancing the mass with respect to the origin and because of that the solution is not always the half space we characterize all the minimizers of this functional when the volume is close to one by proving that the minimizer is either the half space or the symmetric strip depending on the strength of the barycenter term as a corollary we obtain that the symmetric strip is the solution of the gaussian isoperimetric problem among symmetric sets when the volume is close to one as another corollary we obtain the optimal constant in the quantitative gaussian isoperimetric inequality

on the relation of kinetic gelation and percolation we analyse the results of kinetic gelation and propose a diagram compatible with the existing data we introduce the concept of limited and full universality the former refers to systems with an identical subset of critical exponents the latter refers to systems with an identical set of critical exponents the evidence suggests that kinetic gelation and random percolation are in the same limited universality class it appears that the concentration of initiators plays the role of an active parameter similar to that of the interplane coupling in quasi two dimensional king models

computer simulation of a model for irreversible gelation a random mixture of bifunctional and tetrafunctional units is placed on a simple cubic lattice permanent bonds between these units are then formed by the random motion of active centres radicals resulting in a model for the gelation of polyacrylamide and similar processes the largest of the clusters formed by this kinetic percolation process is identified with the gel fraction the mean cluster size with the weight average degree of polymerisation the critical exponent gamma of the mean cluster size is roughly the same as for random percolation that is different from that of the classical theory of flory 1941 and stockmayer 1943 but the corresponding amplitude ratio that is the ratio of average molecular weight on both sides of the gel point differs strongly from its value in random percolation thus this kinetic gelation model seems to belong to a universality class of its own different from both that of random percolation and that of classical gelation theories

extremal properties of central half spaces for product measures we deal with the isoperimetric and the shift problem for subsets of measure 1 2 in product probability spaces we prove that the canonical central half spaces are extremal in particular cases products of log concave measures on the real line satisfying precise conditions and products of uniform measures on spheres or balls as a corollary we improve the known log sobolev constants for euclidean balls we also give some new results about the related question of estimating the volume of sections of unit balls of lp sums of minkowski spaces

a boolean hebb rule for binary associative memory design a binary associative memory design procedure that gives a hopfield network with a symmetric binary weight matrix is introduced in this paper the proposed method is based on introducing the memory vectors as maximal independent sets to an undirected graph which is constructed by boolean operations analogous to the conventional hebb rule the parameters of the resulting network is then determined via the adjacency matrix of this graph in order to rind a maximal independent set whose characteristic vector is close to the given distorted vector we show that the method provides attractiveness for each memory vector and avoids spurious memories whenever the set of given memory vectors satisfy certain compatibility conditions which implicitly imply sparsity the applicability of the design method is finally investigated by a quantitative analysis of the compatibility conditions

approximating maximum clique with a hopfield network in a graph a clique is a set of vertices such that every pair is connected by an edge max clique is the optimization problem of finding the largest clique in a given graph and is np hard even to approximate well several real world and theory problems can be modeled as max clique in this paper we efficiently approximate max clique in a special case of the hopfield network whose stable states are maximal cliques we present several energy descent optimizing dynamics both discrete deterministic and stochastic and continuous one of these emulates as special cases two well known greedy algorithms for approximating max clique we report on detailed empirical comparisons on random graphs and on harder ones mean field annealing an efficient approximation to simulated annealing and a stochastic dynamics are the narrow but clear winners all dynamics approximate much better than one which emulates a naive greedy heuristic

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

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

stereo correspondence through feature grouping and maximal cliques the authors propose a method for solving the stereo correspondence problem the method consists of extracting local image structures and matching similar such structures between two images linear edge segments are extracted from both the left and right images each segment is characterized by its position and orientation in the image as well as its relationships with the nearby segments a relational graph is thus built from each image for each segment in one image as set of potential assignments is represented as a set of nodes in a correspondence graph arcs in the graph represent compatible assignments established on the basis of segment relationships stereo matching becomes equivalent to searching for sets of mutually compatible nodes in this graph sets are found by looking for maximal cliques the maximal clique best suited to represent a stereo correspondence is selected using a benefit function numerous results obtained with this method are shown

bv functions in abstract wiener spaces abstract functions of bounded variation in an abstract wiener space i e an infinite dimensional banach space endowed with a gaussian measure and a related differential structure have been introduced by m fukushima and m hino using dirichlet forms and their properties have been studied with tools from analysis and stochastics in this paper we reformulate in an integral geometric vein and with purely analytical tools the definition and the main properties of bv functions and investigate further properties

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

harnack and functional inequalities for generalized mehler semigroups abstract harnack inequalities are established for a class of generalized mehler semigroups which in particular imply upper bound estimates for the transition density moreover poincare and log sobolev inequalities are proved in terms of estimates for the square field operators furthermore under a condition well known in the gaussian case we prove that generalized mehler semigroups are strong feller the results are illustrated by concrete examples in particular we show that a generalized mehler semigroup with an stable part is not hyperbounded but exponentially ergodic and that the log sobolev constant obtained by our method in the special gaussian case can be sharper than the one following from the usual curvature condition moreover a harnack inequality is established for the generalized mehler semigroup associated with the dirichlet heat semigroup on 0 1 we also prove that this semigroup is not hyperbounded

approximating the maximum weight clique using replicator dynamics given an undirected graph with weights on the vertices the maximum weight clique problem mwcp is to find a subset of mutually adjacent vertices a clique having the largest total weight this is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph which is the special case of the mwcp when all vertex weights are equal the problem is np hard for arbitrary graphs and so is the problem of approximating it within a constant factor we present a parallel distributed heuristic for approximating the mwcp based on dynamics principles it centers around a continuous characterization of the mwcp a purely combinatorial problem and lets it be formulated in terms of continuous quadratic programming one drawback is the presence of spurious solutions and we present their characterizations to avoid them we introduce a regularized continuous formulation of the mwcp and show how it completely solves the problem the formulation naturally maps onto a parallel distributed computational network whose dynamical behavior is governed by the replicator equations these are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale we present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought experimental results confirm the effectiveness of the proposed approach

a functional form of the isoperimetric inequality for the gaussian measure abstract letgbe a smooth function on r nwith values in 0 1 using the isoperimetric property of the gaussian measure itis proved that 1 eg e 1 g e g conversely this inequality implies the isoperimetric property of the gaussian measure

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

upper bounds on the first eigenvalue for a diffusion operator via bakry emery ricci curvature let l delta nabla varphi cdot nabla be a symmetric diffusion operator with an invariant measure d mu e varphi dx on a complete riemannian manifold in this paper we prove li yau gradient estimates for weighted elliptic equations on the complete manifold with nabla varphi leq theta and dimensional bakry emery ricci curvature bounded below by some negative constant based on this we give an upper bound on the first eigenvalue of the diffusion operator l on this kind manifold and thereby generalize a cheng s result on the laplacian case math z 143 289 297 1975

payoff monotonic game dynamics and the maximum clique problem evolutionary game theoretic models and in particular the so called replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems the approach is centered around a classic result from graph theory that formulates the maximum clique problem as a standard continuous quadratic program and exploits the dynamical properties of these models which under a certain symmetry assumption possess a lyapunov function in this letter we generalize previous work along these lines in several respects we introduce a wide family of game dynamic equations known as payoff monotonic dynamics of which replicator dynamics are a special instance and show that they enjoy precisely the same dynamical properties as standard replicator equations these properties make any member of this family a potential heuristic for solving standard quadratic programs and in particular the maximum clique problem extensive simulations performed on random as well as dimacs benchmark graphs show that this class contains dynamics that are considerably faster than and at least as accurate as replicator equations one problem associated with these models however relates to their inability to escape from poor local solutions to overcome this drawback we focus on a particular subclass of payoff monotonic dynamics used to model the evolution of behavior via imitation processes and study the stability of their equilibria when a regularization parameter is allowed to take on negative values a detailed analysis of these properties suggests a whole class of annealed imitation heuristics for the maximum clique problem which are based on the idea of varying the parameter during the imitation optimization process in a principled way so as to avoid unwanted inefficient solutions experiments show that the proposed annealing procedure does help to avoid poor local optima by initially driving the dynamics toward promising regions in state space furthermore the models outperform state of the art neural network algorithms for maximum clique such as mean field annealing and compare well with powerful continuous based heuristics

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

an efficient approximation algorithm for finding a maximum clique using hopfield network learning in this article we present a solution to the maximum clique problem using a gradient ascent learning algorithm of the hopfield neural network this method provides a near optimum parallel algorithm for finding a maximum clique to do this we use the hopfield neural network to generate a near maximum clique and then modify weights in a gradient ascent direction to allow the network to escape from the state of near maximum clique to maximum clique or better the proposed parallel algorithm is tested on two types of random graphs and some benchmark graphs from the center for discrete mathematics and theoretical computer science dimacs the simulation results show that the proposed learning algorithm can find good solutions in reasonable computation time

exact relations between kinetic gelation and percolation an exact relation between one of the recently introduced models for kinetic gelation and random percolation is displayed this explains the frequently observed identity between the exponents of the two models moreover it strongly suggests that any further universal properties of percolation are also shared by kinetic gelation in particular a rigorous inequality due to van den berg and kesten 1985 can be used to display an inconsistency between the observed values of the exponents beta and nu and the observed value of the fractal dimension of the backbone in kinetic gelation the more complex issue of the observed variation of the amplitude ratio cannot be easily resolved however should the observed discrepancies be real they would according to the author s results indicate that the amplitude ratio can indeed vary depending upon the underlying lattice structure

labeled point pattern matching by delaunay triangulation and maximal cliques abstract in many problems of pattern recognition objects in an image may be efficiently represented by a set of labeled points this paper describes a method for matching labeled point patterns with one of the prototypes the delaunay triangulation is used to partition a point pattern into a set of triangles and the largest maximal clique of the consistency graph for each triangle is used to obtain the largest set of mutually consistent point pairs the proposed method is invariant under affine transformation of point patterns and allows for additions and deletions of points and some random perturbations in their relative locations the method is applied to the matching of constellations

from u bounds to isoperimetry with applications to h type groups abstract in this paper we study u bounds in relation to l 1 type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space r n d we prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework the usefulness of our approach is illustrated by an application to the setting of h type groups and an extension to infinite dimensions

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

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

tail estimates for markovian rough paths we work in the context of markovian rough paths associated to a class of uniformly subelliptic dirichlet forms 26 and prove a better than exponential tail estimate for the accumulated local p variation functional which has been introduced and studied in 17 we comment on the significance of these estimates to a range of currently studied problems including the recent results of ni hao 32 and chevyrev and lyons 18

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

an isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in gauss space we prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to talagrand as a consequence the gaussian isoperimetric inequality is derived

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

weak poincare inequalities and l2 convergence rates of markov semigroups abstract in order to describe l 2 convergence rates slower than exponential the weak poincare inequality is introduced it is shown that the convergence rate of a markov semigroup and the corresponding weak poincare inequality can be determined by each other conditions for the weak poincare inequality to hold are presented which are easy to check and which hold in many applications the weak poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes some typical examples are given to illustrate the general results in particular our results are applied to the stochastic quantization of field theory in finite volume moreover a sharp criterion of weak poincare inequalities is presented for poisson measures on configuration spaces

isoperimetric inequality for radial probability measures on euclidean spaces abstract we generalize the poincare limit which asserts that the n dimensional gaussian measure is approximated by the projections of the uniform probability measure on the euclidean sphere of appropriate radius to the first n coordinates as the dimension diverges to infinity the generalization is done by replacing the projections with certain maps using this generalization we derive a gaussian isoperimetric inequality for an absolutely continuous probability measure on euclidean spaces with respect to the lebesgue measure whose density is a radial function

second fundamental form and gradient of neumann semigroups abstract the second fundamental form for the boundary of a compact riemannian manifold is described by a short time behavior for the gradient of neumann semigroups as an application we prove that the manifold is convex if and only if the neumann heat semigroup p t satisfies the gradient estimate p t f e k t p t f for some constant k r and all t 0 f c 1

a geometrical approach to monotone functions in r n this paper is concerned with the fine properties of monotone func tions on r n we study the continuity and differentiability properties of these functions the approximability properties the structure of the distributional derivatives and of the weak jacobians moreover we exhibit an example of a monotone function u which is the gradient of a c 1 convex function and

properties of isoperimetric functional and transport entropy inequalities via concentration various properties of isoperimetric functional transport entropy and concentration inequalities are studied on a riemannian manifold equipped with a measure whose generalized ricci curvature is bounded from below first stability of these inequalities with respect to perturbation of the measure is obtained the extent of the perturbation is measured using several different distances between perturbed and original measure such as a one sided l bound on the ratio between their densities wasserstein distances and kullback leibler divergence in particular an extension of the holley stroock perturbation lemma for the log sobolev inequality is obtained and the dependence on the perturbation parameter is improved from linear to logarithmic second the equivalence of transport entropy inequalities with different cost functions is verified by obtaining a reverse jensen type inequality the main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting of independent interest is a new dimension independent characterization of transport entropy inequalities with respect to the 1 wasserstein distance which does not assume any curvature lower bound

the growing self avoiding walk the authors introduce a new self avoiding walk with one step probabilities which depend on the local environment as a consequence this walk is irreversible and models the growth process of a linear polymer in a good solvent to calculate its properties they have performed exact enumerations up to 22 steps on the square lattice and on the diamond lattice this gives for the critical indices the values v 0 68 gamma 1 16 in two dimensions and v 0 525 and gamma 1 in three dimensions

entropy bounds for gibbs measures with non gaussian tails abstract we introduce and study relative entropy bounds which under some mixing conditions allow us to classify the distribution tails in the infinite dimensional setting

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

heat kernel analysis on infinite dimensional heisenberg groups abstract we introduce a class of non commutative heisenberg like infinite dimensional lie groups based on an abstract wiener space the ricci curvature tensor for these groups is computed and shown to be bounded brownian motion and the corresponding heat kernel measures t t 0 are also studied we show that these heat kernel measures admit 1 gaussian like upper bounds 2 cameron martin type quasi invariance results 3 good l p bounds on the corresponding radon nikodym derivatives 4 integration by parts formulas and 5 logarithmic sobolev inequalities the last three results heavily rely on the boundedness of the ricci tensor

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

energy function based approaches to graph coloring we describe an approach to optimization based on a multiple restart quasi hopfield network where the only problem specific knowledge is embedded in the energy function that the algorithm tries to minimize we apply this method to three different variants of the graph coloring problem the minimum coloring problem the spanning subgraph k coloring problem and the induced subgraph k coloring problem though hopfield networks have been applied in the past to the minimum coloring problem our encoding is more natural and compact than almost all previous ones in particular we use k state neurons while almost all previous approaches use binary neurons this reduces the number of connections in the network from nk sup 2 to n sup 2 asymptotically and also circumvents a problem in earlier approaches that of multiple colors being assigned to a single vertex experimental results show that our approach compares favorably with other algorithms even nonneural ones specifically developed for the graph coloring problem

curvature dimension inequalities on sub riemannian manifolds obtained from riemannian foliations part ii using the curvature dimension inequality proved in part i we look at consequences of this inequality in terms of the interaction between the sub riemannian geometry and the heat semigroup p t corresponding to the sub laplacian we give bounds for the gradient entropy a poincare inequality and a li yau type inequality these results require that the gradient of p t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold

isoperimetric rigidity and distributions of 1 lipschitz functions abstract we prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal then the space is foliated by minimal geodesics where the observable variance is defined to be the supremum of the variance of 1 lipschitz functions on the space our result can be considered as a variant of cheeger gromoll s splitting theorem and also of cheng s maximal diameter theorem as an application we obtain a new isometric splitting theorem for a complete weighted riemannian manifold with a positive bakry emery ricci curvature

a class of degenerate elliptic equations and a dido s problem with respect to a measure in this paper we consider the following class of linear elliptic problems div a x u xnkexp x 22 f x in u 0on xn 0 where k 0 is a domain possibly unbounded of r n x x1 xn rn xn 0 f belongs to a suitable weighted lebesgue space and a x aij x ij is a symmetric matrix with measurable coefficients satisfying xnkexp x 22 2 aij x i j cxnkexp x 22 2 we compare the solution to such a problem with the solution to a symmetric one dimensional problem belonging to the same class our approach use classical symmetrization methods adapted to a relative isoperimetric inequality with respect to a measure related to the structure of the equation

gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities by using probabilistic approaches some uniform gradient estimates are obtained for dirichlet heat semigroups on a riemannian manifold with boundary as an application lower bound estimates of isoperimetric constants are presented in terms of functional inequalities