hamilton s gradient estimates and a monotonicity formula for heat flows on metric measure spaces abstract in this paper we extend the hamilton s gradient estimates hamilton 1993 and a monotonicity formula of entropy ni 2004 for heat flows from smooth riemannian manifolds to non smooth metric measure spaces with appropriate riemannian curvature dimension condition

ricci measure for some singular riemannian metrics we define the ricci curvature as a measure for certain singular torsion free connections on the tangent bundle of a manifold the definition uses an integral formula and vector valued half densities we give relevant examples in which the ricci measure can be computed in the time dependent setting we give a weak notion of a ricci flow solution on a manifold

measure rigidity of synthetic lower ricci curvature bound on riemannian manifolds abstract in this paper we investigate lott sturm villani s synthetic lower ricci curvature bound on riemannian manifolds with boundary we prove several measure rigidity results related to optimal transport on riemannian manifolds which completely characterize cd k condition and non collapsed cd k n condition on riemannian manifolds with boundary in particular we reveal the measure rigidity of riemannian interpolation inequality proved by cordero erausquin mccann and schmuckenschlager we prove that log semi concave measures are the only reference measures so that displacement convexity holds on riemannian manifolds this is the first measure rigidity result concerning the synthetic dimension free cd condition which is new even on r n using l 1 optimal transportation theory we also prove that cd condition yields geodesical convexity with respect to the usual riemannian distance

non commutative calculus optimal transport and functional inequalities in dissipative quantum systems we study dynamical optimal transport metrics between density matrices associated to symmetric dirichlet forms on finite dimensional c algebras our setting covers arbitrary skew derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for markov chains lindblad equations and the fermi ornstein uhlenbeck semigroup we develop a non nommutative differential calculus that allows us to obtain non commutative ricci curvature bounds logarithmic sobolev inequalities transport entropy inequalities and spectral gap estimates

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

displacement convexity of entropy and related inequalities on graphs we introduce the notion of an interpolating path on the set of probability measures on finite graphs using this notion we first prove a displacement convexity property of entropy along such a path and derive prekopa leindler type inequalities a talagrand transport entropy inequality certain hwi type as well as log sobolev type inequalities in discrete settings to illustrate through examples we apply our results to the complete graph and to the hypercube for which our results are optimal by passing to the limit we recover the classical log sobolev inequality for the standard gaussian measure with the optimal constant

quantitative analysis of financial system fragility based on manifold curvature abstract manifold curvature is an intrinsic attribute of a dynamical system by analyzing curvature characteristics of embedding attractor manifold resistance and resilience of financial dynamic system are investigated in the face of disturbances thus system fragility is discussed quantitatively from a systemic perspective first financial time series data are reconstructed into a high dimensional phase space topologically equivalent to its original dynamical system then manifold learning is employed to extract attractor manifold embedding in financial dynamic system last by numerical calculation the manifold curvature is obtained and further visualized with spatial mapping in empirical study shanghai composite index and dow jones industrial index are adopted as experimental data and then a more comprehensive understanding about financial systems fragility is obtained by exploring the dynamical properties related to manifold curvature

test pattern generation based on reed muller coefficients reed muller coefficients are used to generate a test pattern selection procedure for detecting single stuck at faults this procedure is based on the heuristics deduced from the way in which the spectral coefficients are affected by such faults the changes that the spectral coefficients undergo are also compared with the fault models used most frequently to model defects in combinational circuits the proposed pattern generating procedure does not need a simulation of the circuit for each of the possible stuck at faults and its complexity is proportional to the number of gates in the circuit the proportionality constant increases exponentially as the number of inputs in the circuit increases to evaluate the performance of the proposed method its application to some benchmark circuits including the alu 74181 is presented

a free derivative iteration method of order three having convergence of both point and interval for nonlinear equations a free derivative iteration method of order three having convergence of both points and intervals for nonlinear equations is proposed in this paper our numerical experience show that this new method compares well with the classical method

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

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

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

sectional and intermediate ricci curvature lower bounds via optimal transport abstract the goal of the paper is to give an optimal transport characterization of sectional curvature lower and upper bounds for smooth n dimensional riemannian manifolds more generally we characterize via optimal transport lower bounds on the so called p ricci curvature which corresponds to taking the trace of the riemann curvature tensor on p dimensional planes 1 p n such characterization roughly consists on a convexity condition of the p renyi entropy along l 2 wasserstein geodesics where the role of reference measure is played by the p dimensional hausdorff measure as application we establish a new brunn minkowski type inequality involving p dimensional submanifolds and the p dimensional hausdorff measure

ricci tensor for diffusion operators and curvature dimension inequalities under conformal transformations and time changes abstract within the 2 calculus of bakry and ledoux we define the ricci tensor induced by a diffusion operator we deduce precise formulas for its behavior under drift transformation time change and conformal transformation and we derive new transformation results for the curvature dimension conditions of bakry emery as well as for those of lott sturm villani our results are based on new identities and sharp estimates for the n ricci tensor and for the hessian in particular we obtain bochner s formula in the general setting

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

syndrome testability can be achieved by circuit modification in 1 and 2 savir developed many facets of syndrome testing checking the number of minterms realized by a circuit against the number realized by a fault free version of that circuit and presented evidence showing that syndrome testing can be used in many practical circuits to detect all single faults in some cases where syndrome testing did not detect all single stuck at faults savir showed that by the addition of a small number of additional control inputs and gates one would get a function which is syndrome testable for all single stuck at faults and yet which realizes the original function when the control inputs are fed appropriate values however he left open the question of whether one could always modify a circuit to achieve syndrome testability in this correspondence we show that a combinatorial circuit can always be modified to produce a single fault syndrome testable circuit

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

analytic approaches and harmonic functions on alexandrov spaces with nonnegative ricci curvature abstract in this paper the author gets a sharp dimension estimate of the space of harmonic functions with polynomial growth of a fixed order on alexandrov spaces which extends the result of colding and minicozzi from riemannian manifolds to alexandrov spaces

matrix optimal mass transport a quantum mechanical approach in this paper we describe a possible generalization of the wasserstein 2 metric originally defined on the space of scalar probability densities to the space of hermitian matrices with trace one and to the space of matrix valued probability densities our approach follows a control theoretic optimization formulation of the wasserstein 2 metric having its roots in fluid dynamics and utilizes certain results from the quantum mechanics of open systems in particular the lindblad equation it allows determining the gradient flow for the quantum entropy relative to this matricial wasserstein metric

gradient flows and evolution variational inequalities in metric spaces i structural properties abstract this is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space x d that can be characterized by evolution variational inequalities evi we present new results concerning the structural properties of solutions to the evi formulation such as contraction regularity asymptotic expansion precise energy identity stability asymptotic behavior and their link with the geodesic convexity of the driving functional under the crucial assumption of the existence of an evi gradient flow we will also prove two main results the equivalence with the de giorgi variational characterization of curves of maximal slope the convergence of the minimizing movement jko scheme to the evi gradient flow with an explicit and uniform error estimate of order 1 2 with respect to the step size independent of any geometric hypothesis as upper or lower curvature bounds on d in order to avoid any compactness assumption we will also introduce a suitable relaxation of the minimizing movement algorithm obtained by the ekeland variational principle and we will prove its uniform convergence as well

ricci curvature and the manifold learning problem abstract consider a sample of n points taken i i d from a submanifold of euclidean space we show that there is a way to estimate the ricci curvature of with respect to the induced metric from the sample our method is grounded in the notions of carre du champ for diffusion semi groups the theory of empirical processes and local principal component analysis

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

an improved regula falsi method with quadratic convergence of both diameter and point for enclosing simple zeros of nonlinear equations a new method of regula falsi type with global convergence for finding a simple root x of a nonlinear equation f x 0 in the interval a b is presented in this paper the new method is shown to be quadratically convergent both the sequences of diameters b n a n and the iterative sequence x n x are quadratically convergent to zero the new method has been tested on a series of published examples the numerical results show that the new method is very effective

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

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

bakry emery curvature dimension condition and riemannian ricci curvature bounds the aim of the present paper is to bridge the gap between the bakry emery and the lott sturm villani approaches to provide synthetic and abstract notions of lower ricci curvature bounds we start from a strongly local dirichlet form mathcal e admitting a carre du champ gamma in a polish measure space x mathfrak m and a canonical distance mathsf d mathcal e that induces the original topology of x we first characterize the distinguished class of riemannian energy measure spaces where mathcal e coincides with the cheeger energy induced by mathsf d mathcal e and where every function f with gamma f le1 admits a continuous representative in such a class we show that if mathcal e satisfies a suitable weak form of the bakry emery curvature dimension condition mathrm be k infty then the metric measure space x mathsf d mathfrak m satisfies the riemannian ricci curvature bound mathrm rcd k infty according to duke math j 163 2014 1405 1490 thus showing the equivalence of the two notions two applications are then proved the tensorization property for riemannian energy spaces satisfying the bakry emery mathrm be k n condition and thus the corresponding one for mathrm rcd k infty spaces without assuming nonbranching and the stability of mathrm be k n with respect to sturm gromov hausdorff convergence

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

the weighted syndrome sums approach to vlsi testing with the advent of vlsi testing has become one of the most costly complicated and time consuming problems the method of syndrome testing is applicable toward vlsi testing since it does not require test generation and fault simulation it can also be considered as a vehicle for self testing in order to employ syndrome testing in vlsi we electronically partition the chip into macros in test mode the macros are then syndrome tested in sequence

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 use of linear sums in exhaustive testing the increasing role of exhaustive testing techniques in vlsi has prompted a number of investigations into methods for achieving locally exhaustive testing of the components of a large digital circuit using a comparatively small number of test signals this paper examines the possibility of introducing linear sums of the test signals into this testing process it is shown that with the addition of a relatively small amount of additional circuitry significant reductions in the number of required test signals and tests can often be achieved in particular it is shown that for any specific combinational circuit the number of required test signals can be made to be independent of the number of inputs and to depend only logarithmically on the number of outputs the role of these linear sums in the generation of universal test sets is also discussed

existence and uniqueness of optimal maps on alexandrov spaces the purpose of this paper is to show that in a finite dimensional metric space with alexandrov s curvature bounded below monge s transport problem for the quadratic cost admits a unique solution

convergence rates for empirical barycenters in metric spaces curvature convexity and extendable geodesics this paper provides rates of convergence for empirical generalised barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry we study the validity of variance inequalities in spaces of non positive and non negative aleksandrov curvature in this last scenario we show that variance inequalities hold provided geodesics emanating from a barycenter can be extended by a constant factor we also relate variance inequalities to strong geodesic convexity while not restricted to this setting our results are largely discussed in the context of the 2 wasserstein space

ricci flow entropy and optimal transportation let a smooth family of riemannian metrics g tau satisfy the backwards ricer flow equation on a compact oriented n dimensional manifold m suppose two families of normalized n forms omega tau 0 and omega over bar tau 0 satisfy the forwaids in tau heat equation on m generated by the connection laplacian delta g t if these n forms represent two evolving distributions of particles over m the minimum root mean square distance w 2 omega t omega over tilde t t to transport the particles of omega t onto those of omega over tilde t is shown to be non increasing as a function of t without sign conditions on the curvature of mg t moreover this contractivity property is shown to characterize supersolutions to the ricci flow

on kac s chaos and related problems this paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by m kac cite kac1956 in his study of mean field limit for systems of n undistinguishable particles as n to infty first we quantitatively liken three usual measures of it kac s chaos some involving the all n variables other involving a finite fixed number of variables the cornerstone of the proof is a new representation of the monge kantorovich wasserstein mkw distance for symmetric n particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand and a new estimate on some mkw distance on probability measures spaces endowed with a suitable hilbert norm taking advantage of the associated good algebraic structure next we define the notion of it entropy chaos and it fisher information chaos in a similar way as defined by carlen et al cite ccllv we show that it fisher information chaos is stronger than it entropy chaos which in turn is stronger than it kac s chaos more importantly with the help of the hwi inequality of otto villani we establish a quantitative estimate between these quantities which in particular asserts that it kac s chaos plus it fisher information bound implies it entropy chaos we then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the it kac s spheres revisiting cite ccllv and giving a possible answer to cite open problem 11 ccllv additionally to the above mentioned tool we use and prove an optimal rate local clt in l infty norm for distributions with finite 6 th moment and finite l p norm for some p 1 last we investigate how our techniques can be used without assuming chaos in the context of probability measures mixtures introduced by de finetti hewitt and savage in particular we define the level 3 fisher information for mixtures and prove that it is l s c and affine as that was done in cite rr for the level 3 boltzmann s entropy

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

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

fokker planck equations for a free energy functional or markov process on a graph the classical fokker planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a euclidean space corresponding to a stochastic process there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy in recent years it has been shown that the fokker planck equation is the gradient flow of the free energy functional defined on the riemannian manifold of probability distributions whose inner product is generated by a 2 wasserstein distance in this paper we consider analogous matters for a free energy functional or markov process defined on a graph with a finite number of vertices and edges if n 2 is the number of vertices of the graph we show that the corresponding fokker planck equation is a system of nnonlinear ordinary differential equations defined on a riemannian manifold of probability distributions however in contrast to stochastic processes defined on euclidean spaces the situation is more subtle for discrete spaces we have different choices for inner products on the space of probability distributions resulting in different fokker planck equations for the same process it is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical fokker planck equation on euclidean spaces furthermore both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the riemannian manifolds of probability distributions with different metrics some examples are also discussed

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

the heat flow on metric random walk spaces abstract in this paper we study the heat flow on metric random walk spaces which unifies into a broad framework the heat flow on locally finite weighted connected graphs the heat flow determined by finite markov chains and some nonlocal evolution problems we give different characterizations of the ergodicity and prove that a metric random walk space with positive ollivier ricci curvature is ergodic furthermore we prove a cheeger inequality and as a consequence we show that a poincare inequality holds if and only if an isoperimetric inequality holds we also study the bakry emery curvature dimension condition and its relation with functional inequalities like the poincare inequality and the transport information inequalities

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

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

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

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

cheeger harmonic functions in metric measure spaces revisited let x d be a complete metric measure space with a locally doubling measure that supports a local weak l2 poincare inequality by assuming a heat semigroup type curvature condition we prove that cheeger harmonic functions are lipschitz continuous on x d gradient estimates for cheeger harmonic functions and solutions to a class of non linear poisson type equations are presented

smooth interpolation of covariance matrices and brain network estimation part ii we propose an approach to use the state covariance of autonomous linear systems to track time varying covariance matrices of nonstationary time series following concepts from the riemannian geometry we investigate three types of covariance paths obtained by using different quadratic regularizations of system matrices the first quadratic form induces the geodesics based on the hellinger bures metric related to optimal mass transport omt theory and quantum mechanics the second type of quadratic form leads to the geodesics based on the fisher rao metric from information geometry in the process we introduce a weighted omt interpretation of the fisher rao metric for multivariate gaussian distributions a main contribution of this work is the introduction of the third type of covariance paths which are steered by system matrices with rotating eigenspaces the three types of covariance paths are compared using two examples with synthetic data and real data from resting state functional magnetic resonance imaging respectively

optimal transport maps for monge kantorovich problem on loop groups let g be a compact lie group and consider the loop group leg l c 0 1 g l 0 l 1 e let be the heat kernel measure at the time 1 for any density function f on leg such that ent f we shall prove that there exists a unique optimal transportation map t leg leg which pushes forward to f

bakry emery curvature and model spaces in sub riemannian geometry we prove comparison theorems for the sub riemannian distortion coefficients appearing in interpolation inequalities these results which are equivalent to a sub laplacian comparison theorem for the sub riemannian distance are obtained by introducing a suitable notion of sub riemannian bakry emery curvature the model spaces for comparison are variational problems coming from optimal control theory as an application we establish the sharp measure contraction property for 3 sasakian manifolds satisfying a suitable curvature bound

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

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

superlinear bracketing method for solving nonlinear equations a new method based on muller s algorithm for solving nonlinear equations has been developed the classic muller s method is based on an interpolating polynomial built on the last three points of an iterative sequence unfortunately muller s method is not globally convergent in order to ensure the global convergence a bracketing is introduced the proposed method does not require the use of a derivative of the function and is more rapidly convergent than a classical regula falsi method the method is good alternative to other bracketing methods

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

generalized ricci curvature bounds for three dimensional contact subriemannian manifolds measure contraction property is one of the possible generalizations of ricci curvature bound to more general metric measure spaces in this paper we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang

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

frenetic bounds on the entropy production we give a systematic derivation of positive lower bounds for the expected entropy production ep rate in classical statistical mechanical systems obeying a dynamical large deviation principle the logic is the same for the return to thermodynamic equilibrium as it is for steady nonequilibria working under the condition of local detailed balance we recover there recently studied uncertainty relations for the ep appearing in studies about the effectiveness of mesoscopic machines in general our refinement of the positivity of the expected ep rate is obtained in terms of a positive and even function of the expected current s which measures the dynamical activity in the system a time symmetric estimate of the changes in the system s configuration also underdamped diffusions can be included in the analysis

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

existence and consistency of wasserstein barycenters in this paper based on the frechet mean we define a notion of barycenter corresponding to a usual notion of statistical mean we prove the existence of wasserstein barycenters of random distributions defined on a geodesic space e d we also prove the consistency of this barycenter in a general setting that includes taking barycenters of empirical versions of the distributions or of a growing set of distributions

on third order convergent regula falsi method an improved regula falsi iterative formula for solving nonlinear equations is proposed in this paper this method is effective and promising with its third order convergence of both the sequences of diameters and iterative points sequence

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

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

quadratically convergent multiple roots finding method without derivatives abstract in this paper an iteration method without derivatives for multiple roots is proposed this method proved to be quadratically convergent its efficiency and accuracy are illustrated by numerical experiments

splitting theorems for finsler manifolds of nonnegative ricci curvature we investigate the structure of a finsler manifold of nonnegative weighted ricci curvature including a straight line and extend the classical cheeger gromolllichnerowicz splitting theorem such a space admits a diffeomorphic measurepreserving splitting in general as for a special class of berwald spaces we can perform the isometric splitting in the sense that there is a one parameter family of isometries generated from the gradient vector field of the busemann function a betti number estimate is also given for berwald spaces

exhaustive generation of bit patterns with applications to vlsi self testing one has a shift register of length n and a collection of designated subsets of 0 1 n 1 the problem is to devise a method for feeding a string of bits into the shift register in such an order that for each designated subset s k1 kr if one keeps track of the bit patterns appearing at the corresponding positions k1 krof the shift register all 2r possible bit patterns will ultimately appear at those positions a simple and efficient solution to this problem derived from the connections between polynomials over finite fields and linear feedback shift registers is presented applications of this solution to the problem of vlsi self testing are discussed and illustrated

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