the whitham hierarchies reductions and hodograph solutions a general scheme for analysing reductions of whitham hierarchies is presented it is based on a method for determining the s function by means of a system of first order partial differential equations compatibility systems of differential equations characterizing both reductions and hodograph solutions of whitham hierarchies are obtained the method is illustrated by exhibiting solutions of integrable models such as the dispersionless toda equation heavenly equation and the generalized benney system

classical r matrix theory of dispersionless systems ii 2 1 dimension theory a systematic way of constructing 2 1 dimensional dispersionless integrable hamiltonian systems is presented the method is based on the so called central extension procedure and classical r matrix applied to the poisson algebras of formal laurent series results are illustrated with the known and new 2 1 dimensional dispersionless systems

li yau and harnack type inequalities in rcd k n metric measure spaces abstract metric measure spaces satisfying the reduced curvature dimension condition cd k n and where the heat flow is linear are called rcd k n spaces this class of non smooth spaces contains gromov hausdorff limits of riemannian manifolds with ricci curvature bounded below by k and dimension bounded above by n we prove that in rcd k n spaces the following properties of the heat flow hold true a li yau type inequality a bakry qian inequality the harnack inequality

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

simple waves in quasilinear hyperbolic systems i theory of simple waves and simple states examples of applications this paper presents a new method of construction of solutions to nonlinear nonelliptic systems of partial differential equations and especially nonhomogeneous ones these equations have been considered from the point of view of integral elements in particular the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions have been studied these classes consist of what is called simple waves and k waves for homogeneous systems and simple states in the case of nonhomogeneous systems they provide us with a possibility for a selection of simple integral elements from the set of all integral elements analyses have been performed using differential forms and cartan theory of system in involution the problem has been reduced to examining pfaff forms the cauchy problem for pfaff systems has been formulated and solved using the riemann function some remarks concerning the notion of backlund transformations for the case of k waves have been formulated it is shown that in contrast to simple wave the simple state has no gradient catastrophy the technique presented of constructing the solutions in form of simple states has been illustrated by the examples of korteweg and de vries and four dimensional klein gordon sine gordon and liouville equations it has been shown that the known soliton equations are closely connected with the elliptical functions and especially with the p weierstrass functions

relative entropy and variational properties of generalized gibbsian measures we study the relative entropy density for generalized gibbs measures we first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of almost gibbsian measures almost sure continuity of conditional probabilities for quasilocal measures we obtain a full variational principle for the joint measures of the random field ising model we show that the weak gibbs property holds with an almost surely rapidly decaying translation invariant potential for these measures we show that the variational principle fails as soon as the measures lose the almost gibbs property these examples suggest that the class of weakly gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

hydrodynamical limit for a hamiltonian system with weak noise starting from a general hamiltonian system with superstable pairwise potential we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles we prolve that in the scaling limit the time conserved quantities energy momenta and density satisfy the euler equation of conservation laws up to a fixed timet provided that the euler equation has a smooth solution with a given initial data up to timet the strength of the noise term is chosen to be very small but nonvanishing so that it disappears in the scaling limit

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

differential geometric approach to the integrability of hydrodynamic chains the haantjes tensor the integrability of an m component system of hydrodynamic type u t v u u x by the generalized hodograph method requires the diagonalizability of the m m matrix v u this condition is known to be equivalent to the vanishing of the corresponding haantjes tensor we generalize this approach to hydrodynamic chains infinite component systems of hydrodynamic type for which the matrix v u is sufficiently sparse for such systems the haantjes tensor is well defined and the calculation of its components involves finite summations only we illustrate our approach by classifying broad classes of conservative and hamiltonian hydrodynamic chains with the zero haantjes tensor we prove that the vanishing of the haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi hamiltonian hydrodynamic reductions thus providing an easy to verify necessary condition for the integrability

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

the illumination invariant recognition of 3d objects using local color invariants traditional approaches to three dimensional object recognition exploit the relationship between three dimensional object geometry and two dimensional image geometry the capability of object recognition systems can be improved by also incorporating information about the color of object surfaces using physical models for image formation the authors derive invariants of local color pixel distributions that are independent of viewpoint and the configuration intensity and spectral content of the scene illumination these invariants capture information about the distribution of spectral reflectance which is intrinsic to a surface and thereby provide substantial discriminatory power for identifying a wide range of surfaces including many textured surfaces these invariants can be computed efficiently from color image regions without requiring any form of segmentation the authors have implemented an object recognition system that indexes into a database of models using the invariants and that uses associated geometric information for hypothesis verification and pose estimation the approach to recognition is based on the computation of local invariants and is therefore relatively insensitive to occlusion the authors present several examples demonstrating the system s ability to recognize model objects in cluttered scenes independent of object configuration and scene illumination the discriminatory power of the invariants has been demonstrated by the system s ability to process a large set of regions over complex scenes without generating false hypotheses

low temperature dynamics of the curie weiss model periodic orbits multiple histories and loss of gibbsianness we consider the curie weiss model at initial temperature 0 1 in vanishing external field evolving under a glauber spin flip dynamics with temperature 0 1 we study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity bad points we provide a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending earlier work for the case of independent spin flip dynamics

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

on ratio based color indexing the color ratio approach to indexing has been found to be robust and effective in indexing image and video databases in different color spaces and when using transformed color features such as those from the karhunen loeve transform klt or the discrete cosine transform dct however the reason for the superior performance of the color ratio model especially on different color spaces or with transformed color features has at best been speculative this paper develops a generalized form for the color ratio model based on which we characterize the general distribution of the color ratios from the distribution we present a theory that explains and supports the performance of the color ratio approach in image and video indexing it is shown that the same theory accounts for its effectiveness in different color spaces and in the transform domain some general problems encountered in using the original retinex lightness algorithm and some other issues specific to ratio based color indexing are discussed in the light of the theory results are presented which show that the proposed theory is supported by empirical evidence

scaling limit for interacting ornstein uhlenbeck processes the problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts see for example 1 for a recent exposition in 4 one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation the interacting particles evolved according to a generator that was symmetric in equilibrium in this article we consider interacting ornstein uhlenbeck processes here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably we use some ideas that were employed in 3 to extend the central limit theorem from the symmetric to nonsymmetric cases

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

local curvature dimension condition implies measure contraction property abstract we prove that for non branching metric measure spaces the local curvature condition cd loc k n implies the global version of mcp k n the curvature condition cd k n introduced by the second author and also studied by lott and villani is the generalization to metric measure space of lower bounds on ricci curvature together with upper bounds on the dimension this paper is the following step of bacher and sturm 2010 1 where it is shown that cd loc k n is equivalent to a global condition cd k n slightly weaker than the usual cd k n it is worth pointing out that our result implies sharp bishop gromov volume growth inequality and sharp poincare inequality

attractor properties for irreversible and reversible interacting particle systems we consider translation invariant interacting particle systems on the lattice with finite local state space admitting at least one gibbs measure as a time stationary measure the dynamics can be irreversible but should satisfy some mild non degeneracy conditions we prove that weak limit points of any trajectory of translation invariant measures satisfying a non nullness condition are gibbs states for the same specification as the time stationary measure this is done under the additional assumption that zero entropy loss of the limiting measure w r t the time stationary measure implies that they are gibbs measures for the same specification we show how to prove the non nullness for a large number of cases and also give an alternate version of the last condition such that the non nullness requirement can be dropped as an application we obtain the attractor property if there is a reversible gibbs measure our method generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non ergodic ones

whitham modulation theory for the kadomtsev petviashvili equation the genus 1 kadomtsev petviashvili kp whitham system is derived for both variants of the kp equation namely the kpi and kpii equations the basic properties of the kp whitham system including symmetries exact reductions and its possible complete integrability together with the appropriate generalization of the one dimensional riemann problem for the korteweg de vries equation are discussed finally the kp whitham system is used to study the linear stability properties of the genus 1 solutions of the kpi and kpii equations it is shown that all genus 1 solutions of kpi are linearly unstable while all genus 1 solutions of kpii are linearly stable within the context of whitham theory

a novel algorithm for color constancy color constancy is the skill by which it is possible to tell the color of an object even under a colored light i interpret the color of an object as its color under a fixed canonical light rather than as a surface reflectance function this leads to an analysis that shows two distinct sets of circumstances under which color constancy is possible in this framework color constancy requires estimating the illuminant under which the image was taken the estimate is then used to choose one of a set of linear maps which is applied to the image to yield a color descriptor at each point this set of maps is computed in advance the illuminant can be estimated using image measurements alone because given a number of weak assumptions detailed in the text the color of the illuminant is constrained by the colors observed in the image this constraint arises from the fact that surfaces can reflect no more light than is cast on them for example if one observes a patch that excites the red receptor strongly the illuminant cannot have been deep blue two algorithms are possible using this constraint corresponding to different assumptions about the world the first algorithm crule will work for any surface reflectance crule corresponds to a form of coefficient rule but obtains the coefficients by using constraints on illuminant color the set of illuminants for which crule will be successful depends strongly on the choice of photoreceptors for narrowband photoreceptors crule will work in an unrestricted world the second algorithm mwext requires that both surface reflectances and illuminants be chosen from finite dimensional spaces but under these restrictive conditions it can recover a large number of parameters in the illuminant and is not an attractive model of human color constancy crule has been tested on real images of mondriaans and works well i show results for crule and for the retinex algorithm of land land 1971 land 1983 land 1985 operating on a number of real images the experimental work shows that for good constancy a color constancy system will need to adjust the gain of the receptors it employs in a fashion analagous to adaptation in humans

riemannian ricci curvature lower bounds in metric measure spaces with finite measure in prior work 4 of the first two authors with savare a new riemannian notion of lower bound for ricci curvature in the class of metric measure spaces x d m was introduced and the corresponding class of spaces denoted by rcd k this notion relates the cd k n theory of sturm and lott villani in the case n to the bakry emery approach in 4 the rcd k property is defined in three equivalent ways and several properties of rcd k spaces including the regularization properties of the heat flow the connections with the theory of dirichlet forms and the stability under tensor products are provided in 4 only finite reference measures m have been considered the goal of this paper is twofold on one side we extend these results to general finite spaces on the other we remove a technical assumption appeared in 4 concerning a strengthening of the cd k condition this more general class of spaces includes euclidean spaces endowed with lebesgue measure complete noncompact riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower ricci curvature bounds

loss without recovery of gibbsianness during diffusion of continuous spins we consider a specific continuous spin gibbs distribution t 0 for a double well potential that allows for ferromagnetic ordering we study the time evolution of this initial measure under independent diffusions

integrable equations in 2 1 dimensions deformations of dispersionless limits we classify integrable third order equations in 2 1 dimensions which generalize the examples of kadomtsev petviashvili veselov novikov and harry dym equations our approach is based on the observation that dispersionless limits of integrable systems in 2 1 dimensions possess infinitely many multi phase solutions coming from the so called hydrodynamic reductions in this paper we adopt a novel perturbative approach to the classification problem based on the method of hydrodynamic reductions we first classify integrable quasilinear systems which may potentially occur as dispersionless limits of soliton equations in 2 1 dimensions to reconstruct dispersive deformations we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart this procedure leads to a complete list of integrable third order equations some of which are apparently new

duality on gradient estimates and wasserstein controls abstract we establish a duality between l p wasserstein control and l q gradient estimate in a general framework our result extends a known result for a heat flow on a riemannian manifold especially we can derive a wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound by applying our result to a subelliptic heat flow on a lie group we obtain a coupling of heat distributions which carries a good control of their relative distance

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

the haantjes tensor and double waves for multi dimensional systems of hydrodynamic type a necessary condition for integrability an invariant differential geometric approach to the integrability of 2 1 dimensional systems of hydrodynamic type is developed we prove that the existence of special solutions known as double waves is equivalent to the diagonalizability of an arbitrary matrix of the two parameter familysince the diagonalizability can be effectively verified by calculating the haantjes tensor this provides a simple necessary condition for integrability

local grayvalue invariants for image retrieval this paper addresses the problem of retrieving images from large image databases the method is based on local grayvalue invariants which are computed at automatically detected interest points a voting algorithm and semilocal constraints make retrieval possible indexing allows for efficient retrieval from a database of more than 1 000 images experimental results show correct retrieval in the case of partial visibility similarity transformations extraneous features and small perspective deformations

non reversible stationary measures for infinite interacting particle systems two results concerning the local conditional distributions of a stationary measure for a spin flip process with strictly positive and continuous rates are obtained 1 the local conditional distributions and the rates of the reversed process determine each other 2 either all shift invariant stationary measures are gibbs with the same potential or no shift invariant stationary gibbs measure exist

hydrodynamic reductions of multidimensional dispersionless pdes the test for integrability a d 1 dimensional dispersionless pde is said to be integrable if its n component hydrodynamic reductions are locally parametrized by d 1 n arbitrary functions of one variable the most important examples include the four dimensional heavenly equation descriptive of self dual ricci flat metrics and its six dimensional generalization arising in the context of sdiff 2 self dual yang mills equations given a multidimensional pde which does not pass the integrability test the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which when added to the equation make it an integrable system in fewer dimensions as an example of this phenomenon we discuss the second commuting flow of the dispersionless kp hierarchy considered separately this is a four dimensional pde which does not pass the integrability test however the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full 2 1 dimensional dis

bochner weitzenbock formula and li yau estimates on finsler manifolds abstract we prove the bochner weitzenbock formula for the nonlinear laplacian on general finsler manifolds and derive li yau type gradient estimates as well as parabolic harnack inequalities moreover we deduce bakry emery gradient estimates all these estimates depend on lower bounds for the weighted flag ricci tensor

the characterization of two component 2 1 dimensional integrable systems of hydrodynamic type we obtain the necessary and sufficient conditions for a two component 2 1 dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions these conditions are in involution implying that the systems in question are locally parametrized by 15 arbitrary constants it is proved that all such systems possess three conservation laws of hydrodynamic type and therefore are symmetrizable in godunov s sense moreover all such systems are proved to possess a scalar pseudopotential which plays the role of the dispersionless lax pair we demonstrate that the class of two component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions thus establishing the equivalence of the two possible definitions of integrability explicit linearly degenerate examples are constructed

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

simple waves in quasilinear hyperbolic systems ii riemann invariants for the problem of simple wave interactions in this paper a generalization of the riemann invariant method to the case of a nonhomogeneous system of equations has been formulated we have discussed in detail the necessary and sufficient conditions for the existence of riemann invariants we perform the analysis using the apparatus of differential forms and cartan theory of systems in involution the problem has been reduced to examining pfaff forms we have considered the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions depending on k arbitrary functions of one variable these solutions can be interpreted physically as the interactions between k simple waves on a simple state we have proven that in the case of interaction of many simple waves described by riemann invariants a conservation law for the type and quantity of waves holds it has been also shown that such a solution resulting from the interaction of many simple waves propagating on the simple state

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

on classification of integrable davey stewartson type equations this paper is devoted to the classification of integrable davey stewartson type equations a list of potentially deformable dispersionless systems is obtained through the requirement that such systems must be generated by a polynomial dispersionless lax pair a perturbative approach based on the method of hydrodynamic reductions is employed to recover the integrable systems along with their lax pairs some of the found systems seem to be new

spin flip dynamics of the curie weiss model loss of gibbsianness with possibly broken symmetry we study the conditional probabilities of the curie weiss ising model in vanishing external field under a symmetric independent stochastic spin flip dynamics and discuss their set of points of discontinuity bad points we exhibit a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending the results for the corresponding lattice model where only partial answers can be obtained for initial temperature beta 1 geq 1 we prove that the time evolved measure is always gibbsian for frac 2 3 leq beta 1 1 the time evolved measure loses its gibbsian character at a sharp transition time for beta 1 frac 2 3 we observe the new phenomenon of symmetry breaking in the set of points of discontinuity bad points corresponding to non zero spin average appear at a sharp transition time and give rise to biased non gibbsianness of the time evolved measure these bad points become neutral at a later transition time while the measure stays non gibbs in our proof we give a detailed description of the phase diagram of a curie weiss random field ising model with possibly non symmetric random field distribution based on bifurcation analysis

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

localization and tensorization properties of the curvature dimension condition for metric measure spaces abstract this is an addendum to the paper k bacher k t sturm localization and tensorization properties of the curvature dimension condition for metric measure spaces j funct anal 259 2010 28 56 we prove the tensorization property for the curvature dimension condition add some detailed calculations including explicit dependence of constants and comment on assumptions and conjectures concerning the local to global statement in bacher and sturm 2010 1 and villani 2009 6 respectively