reconstruction of graphs based on random walks the analysis of complex networks is of major interest in various fields of science in many applications we face the challenge that the exact topology of a network is unknown but we are instead given information about distances within this network the theoretical approaches to this problem have so far been focusing on the reconstruction of graphs from shortest path distance matrices often however movements in networks do not follow shortest paths but occur in a random fashion in these cases an appropriate distance measure can be defined as the mean length of a random walk between two nodes a quantity known as the mean first hitting time in this contribution we investigate whether a graph can be reconstructed from its mean first hitting time matrix and put forward an algorithm for solving this problem a heuristic method to reduce the computational effort is described and analyzed in the case of trees we can even give an algorithm for reconstructing graphs from incomplete random walk distance matrices

submatrices of non tree realizable distance matrices abstract we deal with distance matrices of real this means not necessarily integer numbers it is known that a distance matrix d of order n is tree realizable if and only if all its principal submatrices of order 4 are tree realizable we discuss bounds for the number denoted qi d of non tree realizable principal submatrices of order i 4 of a non tree realizable distance matrix d of order n i and we construct some distance matrices which meet extremal conditions on qi d our starting point is a proof that a non tree realizable distance matrix of order 5 has at least two non tree realizable principal submatrices of order 4 optimal realizations by graphs with circuits of distance matrices which are not tree realizable are not yet as well known as optimal realizations which are trees using as starting point the optimal realization of the arbitrary distance matrix of order 4 we investigate optimal realizations of non tree realizable distance matrices with the minimum number of non tree realizable principal submatrices of order 4

the pro nilpotent group topology on a free group abstract in this paper we study the pro nilpotent group topology on a free group first we describe the closure of the product of finitely many finitely generated subgroups of a free group in the pro nilpotent group topology and then present an algorithm to compute it we deduce that the nil closure of a rational subset of a free group is an effectively constructible rational subset and hence has decidable membership we also prove that the g nil kernel of a finite monoid is computable and hence pseudovarieties of the form v g nil have decidable membership problem for every decidable pseudovariety of monoids v finally we prove that the semidirect product j g nil has a decidable membership problem

trees tight extensions of metric spaces and the cohomological dimension of certain groups a note on combinatorial properties of metric spaces abstract the concept of tight extensions of a metric space is introduced the existence of an essentially unique maximal tight extension t x the tight span being an abstract analogon of the convex hull is established for any given metric space x and its properties are studied applications with respect to 1 the existence of embeddings of a metric space into trees 2 optimal graphs realizing a metric space and 3 the cohomological dimension of groups with specific length functions are discussed

distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

hyperconvexity and tight span theory for diversities abstract the tight span or injective envelope is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded the concept has stimulated a large body of theory and has applications to metric classification and data visualisation here we introduce a generalisation of metrics called diversities and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity

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

subtree prune and regraft a reversible real tree valued markov process we use dirichlet form methods to construct and analyze a reversible markov process the stationary distribution of which is the brownian continuum random tree this process is inspired by the subtree prune and regraft spr markov chains that appear in phylogenetic analysis a key technical ingredient in this work is the use of a novel gromov hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure also the investigation of the dirichlet form hinges on a new path decomposition of the brownian excursion

diversities hyperconvexity and fixed points abstract diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated in this work we take up a number of questions about hyperconvexity diversities and fixed points of nonexpansive mappings most of these questions are motivated by the study of the connection between a hyperconvex diversity and its induced metric space for which we provide some answers examples are given for instance showing that such a metric space need not be hyperconvex but still we prove as our main result that they enjoy the fixed point property for nonexpansive mappings provided the diversity is bounded and that this boundedness condition cannot be transferred from the diversity to the induced metric space

towards a classification of transitive group actions on finite metric spaces abstract based on j r isbell s completion of metric spaces cf comment math helv 39 1964 1965 65 74 the combinatorial dimension dimcombx of a metric space x has been defined in a w m dress adv in math 53 3 1984 321 402 and shown to satisfy dim comb x x 2 in this paper finite metric spaces x with a transitive group g of isometries which satisfy in addition dim comb x x 2 are investigated it is shown that for almost trivial reasons in case x 0 2 there exists a fixed point free involution on x which commutes with the action of g and that vice versa any finite transitive g space admitting a fixed point free g automorphism of order 2 carries a g invariant metric satisfying the above equality while for not so trivial reasons in case x 1 2 the group g must be either cyclic of order x or dihedral of order 2 x as a corollary the thompson feit theorem turns out to be actually equivalent to its simple consequence that any finite simple group acts transitively as a group of isometries on some finite metric space x satisfying dim comb x x 2

a reflected stochastic heat equation as symmetric dynamics with respect to the 3 d bessel bridge abstract we prove that a stochastic heat equation with reflection at 0 on the spatial interval 0 1 with dirichlet boundary conditions and additive white noise admits an explicit symmetrizing invariant measure on c 0 1 the 3 d bessel bridge i e the law of the modulus of a 3 dimensional brownian motion conditioned to be 0 at time 1 a classical measure in probability theory also connected with the theory of excursions of brownian motion this is a non trivial example of a gibbs type measure being singular with respect to the reference gaussian measure and concentrated on the convex set of positive continuous functions on 0 1

euclidean spaces as weak tangents of infinitesimally hilbertian metric measure spaces with ricci curvature bounded below we show that in any infinitesimally hilbertian cd k n space at almost every point there exists a euclidean weak tangent i e there exists a sequence of dilations of the space that converges to a euclidean space in the pointed measured gromov hausdorff topology the proof follows by considering iterated tangents and the splitting theorem for infinitesimally hilbertian cd 0 n spaces

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

a constructive algorithm for realizing a distance matrix the natural metric of a weighted graph is the length of the shortest paths between all pairs of vertices the investigated problem consists in a representation of a given metric by a graph such that the total length of the graph is minimized for that purpose we give a constructive algorithm based on a technique of reduction fusion and deletion we then show some results on a set of various distance matrices whose optimal realization is known

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

exit times for an increasing levy tree valued process we give an explicit construction of the increasing tree valued process introduced by abraham and delmas using a random point process of trees and a grafting procedure this random point process will be used in companion papers to study record processes on levy trees we use the poissonian structure of the jumps of the increasing tree valued process to describe its behavior at the first time the tree grows higher than a given height using a spinal decomposition of the tree similar to the classical bismut and williams decompositions we also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree valued process

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

spinal partitions and invariance under re rooting of continuum random trees we develop some theory of spinal decompositions of discrete and continuous fragmentation trees specifically we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions we prove that for a two parameter poisson dirichlet family of continuous fragmentation trees including the stable trees of duquesne and le gall the fine partition is obtained from the coarse one by shattering each of its parts independently according to the same law as a second application of spinal decompositions we prove that among the continuous fragmentation trees stable trees are the only ones whose distribution is invariant under uniform re rooting

absolutely minimal lipschitz extension of tree valued mappings we prove that every lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal lipschitz extension amle we relate these extensions to a stochastic game called politics a generalization of a game called tug of war that has been used in peres et al j am math soc 22 1 167 210 2009 to study real valued amles

on isometric embeddings of wasserstein spaces the discrete case abstract the aim of this short paper is to offer a complete characterization of all not necessarily surjective isometric embeddings of the wasserstein space w p x where x is a countable discrete metric space and 0 p is any parameter value roughly speaking we will prove that any isometric embedding can be described by a special kind of x 0 1 indexed family of nonnegative finite measures our result implies that a typical non surjective isometric embedding of w p x splits mass and does not preserve the shape of measures in order to stress that the lack of surjectivity is what makes things challenging we will prove alternatively that w p x is isometrically rigid for all 0 p

convergence in distribution of random metric measure spaces λ coalescent measure trees we consider the space of complete and separable metric spaces which are equipped with a probability measure a notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge this topology is metrized following gromov s idea of embedding two metric spaces isometrically into a common metric space combined with the prohorov metric between probability measures on a fixed metric space we show that for this topology convergence in distribution follows provided the sequence is tight from convergence of all randomly sampled finite subspaces we give a characterization of tightness based on quantities which are reasonably easy to calculate subspaces of particular interest are the space of real trees and of ultra metric spaces equipped with a probability measure as an example we characterize convergence in distribution for the ultra metric measure spaces given by the random genealogies of the coalescents we show that the coalescent defines an infinite random metric measure space if and only if the so called dust free property holds

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

diffusion processes and heat kernels on metric spaces processes x p on any given locally compact metric space x d tx equipped with a radon measure m these processes are associated with local regular dirichlet forms which are obtained as limits of approxi mating nonlocal dirichlet forms this general method works without any restrictions on x d m and yields processes which are well defined for quasi every starting point the second main topic of this paper is to formulate and exploit the so called measure contraction property this is a condition on the original data x d m which can be regarded as a generalization of curvature bounds on the metric space x d it is a bound for distortions of the measure m under contractions of the state space x along suitable geodesics or quasi geodesics w r t the metric d in the case of rieman nian manifolds this condition is always satisfied several other examples will be discussed including uniformly elliptic operators operators with weights certain subelliptic operators manifolds with boundaries or cor ners and glueing together of manifolds the measure contraction property implies upper and lower gaussian estimates for the heat kernel and a harnack inequality for the associated harmonic functions therefore the above mentioned diffusion processes are strong feller processes and are well defined for every starting point

the geometry of profinite graphs with applications to free groups and finite monoids we initiate the study of the class of profinite graphs defined by the following geometric property for any two vertices v and w of there is a unique smallest connected profinite subgraph of containing them such graphs are called tree like profinite trees in the sense of gildenhuys and ribes are tree like but the converse is not true a profinite group is then said to be dendral if it has a tree like cayley graph with respect to some generating set a bass serre type characterization of dendral groups is provided also such groups including free profinite groups are shown to enjoy a certain small cancellation condition we define a pseudovariety of groups h to be arboreous if all finitely generated free pro h groups are dendral with respect to a free generating set our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids we prove for arboreous pseudovarieties h a pro h analog of the ribes and zalesskii product theorem for the profinite topology on a free group also arboreous pseudovarieties are characterized as precisely the solutions h to the much studied pseudovariety equation j h j h

half integrality of node capacitated multiflows and tree shaped facility locations on trees in this paper we establish a novel duality relationship between node capacitated multiflows and tree shaped facility locations we prove that the maximum value of a tree distance weighted maximum node capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree and the maximum is attained by a half integral multiflow utilizing this duality we show that a half integral optimal multiflow and an optimal location can be found in strongly polynomial time these extend previously known results in the maximum free multiflow problems we also show that the set of tree distance weights is the only class having bounded fractionality in maximum node capacitated multiflow problems

recovering symbolically dated rooted trees from symbolic ultrametrics abstract a well known result from cluster theory states that there is a 1 to 1 correspondence between dated compact rooted trees and ultrametrics in this paper we generalize this result yielding a canonical 1 to 1 correspondence between symbolically dated trees and symbolic ultrametrics using an arbitrary set as the set of possible dates or values it turns out that a rather unexpected new condition is needed to properly define symbolic ultrametrics so that the above correspondence holds in the second part of the paper we use our main result to derive as a corollary a theorem by h j bandelt and m a steel regarding a canonical 1 to 1 correspondence between additive trees and metrics satisfying the 4 point condition both taking their values in abelian monoids

fractional poincare and logarithmic sobolev inequalities for measure spaces abstract we prove generalizations of the poincare and logarithmic sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure the class of such spaces includes but is not limited to spaces of homogeneous type with doubling measures several examples and applications are given including poincare inequalities for graph laplacians fractional poincare inequalities of mouhot russ and sire 2011 17 and implications for recent work of the author and r m strain on the boltzmann collision operator gressman and strain 2010 2011 2011 9 10 11

ricci ollivier curvature of the rooted phylogenetic subtree prune regraft graph abstract statistical phylogenetic inference methods use tree rearrangement operations such as subtree prune regraft spr to perform markov chain monte carlo mcmc across tree topologies the structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of mcmc motivating the study of the underlying spr graph in greater detail in this paper we investigate the spr graph of rooted trees rspr graph in a new way by calculating the ricci ollivier curvature with respect to uniform and metropolis hastings random walks this value quantifies the degree to which a pair of random walkers from specified points move towards each other negative curvature means that they move away from one another on average while positive curvature means that they move towards each other in order to calculate this curvature we develop fast new algorithms for rspr graph computation we then develop formulas characterizing how the number of rspr neighbors of a tree changes after an rspr operation is applied to that tree these give bounds on the curvature as well as a flatness in the limit theorem indicating that paths of small topology changes are easy to traverse however we find that large topology changes i e moving a large subtree give pairs of trees with negative curvature we show using simulation that mean access time distributions depend on distance degree and curvature demonstrating the relevance of these results to stochastic tree search

injective subsets of l i abstract we prove an explicit characterization of all injective subsets of the model space l i of all bounded real valued functions defined on a non empty set i endowed with the supremum norm since the class of all injective metric spaces coincides with the one of all absolute 1 lipschitz retracts the present work yields a characterization of all the subsets of l i that are absolute 1 lipschitz retracts

the monge problem for distance cost in geodesic spaces we address the monge problem in metric spaces with a geodesic distance x d is a polish space and dl is a geodesic borel distance which makes x dl a non branching geodesic space we show that under the assumption that geodesics are d continuous and locally compact we can reduce the transport problem to 1 dimensional transport problems along geodesics we introduce two assumptions on the transport problem which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w r t the 1 dimensional hausdorff distance induced by dl it is known that this regularity is sufficient for the construction of a transport map we study also the dynamics of transport along the geodesic the stability of our conditions and show that in this setting dl cyclical monotonicity is not sufficient for optimality

monge problem in metric measure spaces with riemannian curvature dimension condition abstract we prove the existence of solutions for the monge minimization problem addressed in a metric measure space x d m enjoying the riemannian curvature dimension condition rcd k n with n for the first marginal measure we assume that 0 m as a corollary we obtain that the monge problem and its relaxed version the monge kantorovich problem attain the same minimal value moreover we prove a structure theorem for d cyclically monotone sets neglecting a set of zero m measure they do not contain any branching structures that is they can be written as the disjoint union of the image of a disjoint family of geodesics

lipschitz quotients from metric trees and from banach spaces containing l1 abstract if x is any separable banach space containing l1 then there is a lipschitz quotient map from x onto any separable banach space y

a canonical decomposition theory for metrics on a finite set abstract we consider specific additive decompositions d d 1 d n of metrics defined on a finite set x where a metric may give distance zero to pairs of distinct points the simplest building stones are the slit metrics associated to splits i e bipartitions of the given set x while an additive decomposition of a hamming metric into split metrics is in no way unique we achieve uniqueness by restricting ourselves to coherent decompositions that is decompositions d d 1 d n such that for every map f x r with f x f y d x y for all x y x there exist maps f 1 f n x r with f f 1 f n and f i x f i y d i x y for all i 1 n and all x y x these coherent decompositions are closely related to a geometric decomposition of the injective hull of the given metric a metric with a coherent decomposition into a weighted sum of split metrics will be called totally split decomposable tree metrics and more generally the sum of two tree metrics are particular instances of totally split decomposable metrics our main result confirms that every metric admits a coherent decomposition into a totally split decomposable metric and a split prime residue where all the split summands and hence the decomposition can be determined in polynomial time and that a family of splits can occur this way if and only if it does not induce on any four point subset all three splits with block size two

geometry of manifolds with densities abstract we study the geometry of complete riemannian manifolds endowed with a weighted measure where the weight function is of quadratic growth assuming the associated bakry emery curvature is bounded from below we derive a new laplacian comparison theorem and establish various sharp volume upper and lower bounds we also obtain some splitting type results by analyzing the busemann functions in particular we show that a complete manifold with nonnegative bakry emery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions while some of our results are new even for the gradient ricci solitons the novelty here is that only a lower bound of the bakry emery curvature is involved in our analysis

the scaling limit of random simple triangulations and random simple quadrangulations let mnmn be a simple triangulation of the sphere s2s2 drawn uniformly at random from all such triangulations with nn vertices endow mnmn with the uniform probability measure on its vertices after rescaling graph distance by 3 4n 1 4 3 4n 1 4 the resulting random measured metric space converges in distribution in the gromov hausdorff prokhorov sense to the brownian map in proving the preceding fact we introduce a labelling function for the vertices of mnmn under this labelling distances to a distinguished point are essentially given by vertex labels with an error given by the winding number of an associated closed loop in the map we establish similar results for simple quadrangulations

tree valued resampling dynamics martingale problems and applications the measure valued fleming viot process is a diffusion which models the evolution of allele frequencies in a multi type population in the neutral setting the kingman coalescent is known to generate the genealogies of the individuals in the population at a fixed time the goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies we encode the genealogy of the population as an isometry class of an ultra metric space which is equipped with a probability measure the space of ultra metric measure spaces together with the gromov weak topology serves as state space for tree valued processes we use well posed martingale problems to construct the tree valued resampling dynamics of the evolving genealogies for both the finite population moran model and the infinite population fleming viot diffusion we show that sufficient information about any ultra metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled individuals we give explicit formulas for the evolution of the laplace transform of the distribution of finite subtrees under the tree valued fleming viot dynamics

subdomain methods in structural mechanics the principle of subdomain methods has been introduced many years ago1 and such methods have been used with success in recent years in this paper we discuss some methods used to solve the condensed system resulting from the schur complement method we will present the hybrid method and a mechanical application the contact problem between two deformable solids in the case of such a problem we propose two methods one for the contact with friction and one for the contact without friction finally the results of validation tests for both proposed approaches are given

the scaling limit of the minimum spanning tree of the complete graph consider the minimum spanning tree mst of the complete graph with nn vertices when edges are assigned independent random weights endow this tree with the graph distance renormalized by n1 3n1 3 and with the uniform measure on its vertices we show that the resulting space converges in distribution as n n to a random compact measured metric space in the gromov hausdorff prokhorov topology we additionally show that the limit is a random binary rr tree and has minkowski dimension 33 almost surely in particular its law is mutually singular with that of the brownian continuum random tree or any rescaled version thereof our approach relies on a coupling between the mst problem and the erd s renyi random graph we exploit the explicit description of the scaling limit of the erd s renyi random graph in the so called critical window established in probab theory related fields 152 2012 367 406 and provide a similar description of the scaling limit for a critical minimum spanning forest contained within the mst in order to accomplish this we introduce the notion of rr graphs which generalise rr trees and are of independent interest

local spectral convergence in rcd k n spaces abstract in this paper we give necessary and sufficient conditions for the validity of the local spectral convergence in balls in the setting of rcd spaces

an explicit computation of the injective hull of certain finite metric spaces in terms of their associated buneman complex an explicit computation of the injective hull of certain finite metric spaces in terms of their associated buneman complex

integration by parts on bolds delta bessel bridges bolds delta 3 and related spdes we study a white noise driven semilinear partial differential equation on the spatial interval 0 1 with dirichlet boundary condition and with a singular drift of the form cu 3 c 0 we prove existence and uniqueness of a non negative continuous adapted solution u on 0 oo x 0 1 for every nonnegative continuous initial datum x satisfying x 0 x 1 0 we prove that the law of the bessel bridge on 0 1 of dimension 3 is the unique invariant probability measure of the process x u with c 1 3 8 and if n that u is the radial part in the sense of dirichlet forms of the r valued solution of a linear stochastic heat equation an explicit integration by parts formula w r t is given for all 3

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

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

on the volume growth of alexandrov spaces with nonnegative curvature abstract recently zhang and zhu introduced a new definition for ricci curvature bounded below on alexandrov spaces in this paper we give a volume growth lower bound estimate of alexandrov spaces with nonnegative ricci curvature in the sense of zhang zhu which extends yau s result from riemannian manifolds to alexandrov spaces

the maximum multiflow problems with bounded fractionality we consider the weighted maximum multiflow problem with respect to a terminal weight we show that if the dimension of the tight span associated with the weight is at most 2 then this problem has a 1 12 integral optimal multiflow for every eulerian supply graph this result solves a weighted generalization of karzanov s conjecture for classifying commodity graphs with finite fractionality in addition our proof technique proves the existence of an integral or half integrality optimal multiflow for a large class of multiflow maximization problems and it gives a polynomial time algorithm

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

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

compact mixed integer linear programming models to the minimum weighted tree reconstruction problem the minimum weighted tree reconstruction mwtr problem consists of finding a minimum length weighted tree connecting a set of terminal nodes in such a way that the length of the path between each pair of terminal nodes is greater than or equal to a given distance between the considered pair of terminal nodes this problem has applications in several areas namely the inference of phylogenetic trees the modeling of traffic networks and the analysis of internet infrastructures in this paper we investigate the mwtr problem and we present two compact mixed integer linear programming models to solve the problem computational results using two different sets of instances one from the phylogenetic area and another from the telecommunications area show that the best of the two models is able to solve instances of the problem having up to 15 terminal nodes

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

a capacitance matrix method for dirichlet problem on polygon region an efficient algorithm for the solution of linear equations arising in a finite element method for the dirichlet problem is given the cost of the algorithm is proportional ton 2log2 n n 1 h where the cost of solving the capacitance matrix equations isnlog2 n on regular grids andn 3 2log2 n on irregular ones

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

topology and curvature of metric spaces abstract we develop a new concept of non positive curvature for metric spaces based on intersection patterns of closed balls in contrast to the synthetic approaches of alexandrov and busemann our concept also applies to metric spaces that might be discrete the natural comparison spaces that emerge from our discussion are no longer euclidean spaces but rather tripod spaces these tripod spaces include the hyperconvex spaces which have trivial cech homology this suggests a link of our geometrical method to the topological method of persistent homology employed in topological data analysis we also investigate the geometry of general tripod spaces

the real tree in this note it is shown that there exists a natural metric on the set formula of bounded subsets of r containing their infimum which endowstwith the structure of an r tree so that for everyt t the number of connected components oft t coincides with the cardinality p r of the set of subsets of r in addition the set of ends oftis explicitly determined and various further features oftare discussed too

growth of levy trees we construct random locally compact real trees called levy trees that are the genealogical trees associated with continuous state branching processes more precisely we define a growing family of discrete galton watson trees with i i d exponential branch lengths that is consistent under bernoulli percolation on leaves we define the levy tree as the limit of this growing family with respect to the gromov hausdorff topology on metric spaces this elementary approach notably includes supercritical trees and does not make use of the height process introduced by le gall and le jan to code the genealogy of sub critical continuous state branching processes we construct the mass measure of levy trees and we give a decomposition along the ancestral subtree of a poisson sampling directed by the mass measure

landscapes on spaces of trees combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology for instance the problem of reconstructing a phylogeny using maximum likelihood or parsimony approaches the collection of possible phylogenetic trees is arranged as a so called robinson graph by means of the nearest neighborhood interchange move the coherent algebra and spectra of robinson graphs are discussed in some detail as their knowledge is important for an understanding of the landscape structure we consider simple model landscapes as well as landscapes arising from the maximum parsimony problem focusing on two complementary measures of ruggedness the amplitude spectrum arising from projecting the cost functions onto the eigenspaces of the underlying graph and the topology of local minima and their connecting saddle points

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models given any regularly varying dislocation measure we identify a natural self similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths as an application we obtain continuum random tree limits of aldous s beta splitting models and ford s alpha models for phylogenetic trees this confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf

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

regenerative tree growth binary self similar continuum random trees and poisson dirichlet compositions we use a natural ordered extension of the chinese restaurant process to grow a twoparameter family of binary self similar continuum fragmentation trees we provide an explicit embedding of ford s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of ford s trees in general the markov branching trees induced by the two parameter growth rule are not sampling consistent so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case we develop here a new approach to establish such limits based on regenerative interval partitions and the urn model description of sampling from dirichlet random distributions ams 2000 subject classifications 60j80

tight span of subsets of the plane with the maximum metric abstract we prove that a nonempty closed and geodesically convex subset of the l plane r 2 is hyperconvex and we characterize the tight spans of arbitrary subsets of r 2 via this property given any nonempty x r 2 a closed geodesically convex and minimal subset y r 2 containing x is isometric to the tight span t x of x

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion