analysis of a novel lattice hydrodynamic model considering density integral and backward looking effect abstract by introducing the backward looking influence and the integration form of the density difference between the front and rear vehicles an improved lattice hydrodynamic model is proposed the linear stability condition is derived by linear stability analysis the results of the linear stability analysis demonstrate that the influence of backward looking and density difference between the front and rear vehicles play a positive part in improving the steadiness of the traffic flow system the nonlinear analysis testifies that the mkdv equation can describe the propagation feature of density wave the results of numerical simulation show that the steadiness of traffic flow can be amenable to improvement effectively by considering these two factors comprehensive analysis demonstrates that the effect of the backward looking and density difference is propitious to improve traffic stability and suppress traffic congestion

a new car following model with consideration of the velocity difference between the current speed and the historical speed of the leading car in this study a new car following model is proposed based on taking the effect of the leading vehicle s velocity difference between the current speed and the historical speed into account the model s linear stability condition is obtained via the linear stability theory the time dependent ginzburg landau tdgl equation and the modified korteweg de vries mkdv equation are deduced through the nonlinear analysis the kink antikink soliton can interpret the traffic jams near the critical point in addition the connection between the tdgl and the mkdv equations is also given numerical simulation shows that the new model can improve the stability of traffic flow which is consistent with the theoretical analysis correspondingly

tdgl and mkdv equations for jamming transition in the lattice models of traffic abstract the lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena they are the lattice versions of the hydrodynamic model of traffic two lattice models are presented one is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other is the difference equation in which both time and space variables are discrete we apply the linear stability theory and the nonlinear analysis to the lattice models it is shown that the time dependent ginzburg landau tdgl equation is derived to describe the traffic flow near the critical point a thermodynamic theory is formulated for describing the phase transitions and critical phenomena it is also shown that the perturbed modified korteweg de vries mkdv equation is derived to describe the traffic jam

analyses of lattice hydrodynamic model using delayed feedback control with passing abstract delayed feedback control method is used for one dimensional lattice model by incorporating the effect of passing in order to check the stability of system under control theory hurwitz criteria is used and h norm of transfer functions are obtained the efficiency of control signal in minimizing the jamming transitions due to passing is shown through bode plot the results show that the control parameter helps in reducing congestion and makes the flow stable even at higher rate of passing simulations are carried out and are found in good agreement with theoretical results

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

linear universal decoding for compound channels over discrete memoryless channels dmc linear decoders maximizing additive metrics afford several nice properties in particular if suitable encoders are employed the use of decoding algorithms with manageable complexities is permitted for a compound dmc decoders that perform well without the channel s knowledge are required in order to achieve capacity several such decoders have been studied in the literature however there is no such known decoder which is linear hence the problem of finding linear decoders achieving capacity for compound dmc is addressed and it is shown that under minor concessions such decoders exist and can be constructed a geometric method based on the very noisy transformation is developed and used to solve this problem

stabilization effect of multiple drivers desired velocities in car following theory abstract in order to reveal the influence of driver s individual behavior on traffic flow more accurately a new car following model is proposed with consideration of multiple drives desired velocities the stability criterion of the new model is derived through linear stability theory and the results show that the current driver s desired velocity can stabilize traffic flow but the preceding driver s desired velocity can damage traffic stability through nonlinear analysis the traffic jamming transition characteristics near the critical point can be described by the kink antikink soliton of the mkdv equation numerical simulation confirms the analytical results which shows that the multiple drivers desired velocities play an important role in traffic evolution

an extended two lane lattice hydrodynamic model for traffic flow on curved road with passing abstract an extended two lane lattice hydrodynamic model is proposed for the purpose of studying the effects of lane changing behavior and passing behavior on curved road the main work of this paper is as follows firstly the stability condition is acquired through the use of linear stability analysis and it reveals that passing effect as well as the angle of the curve road can affect the stability of traffic flow to some extent meanwhile the stability of traffic flow varies greatly under different lane changing coefficients the modified korteweg de vries mkdv equation is deduced to describe the characteristic of traffic jams near the critical point furthermore the numerical simulations are performed and the results coincide well with the theoretical analysis in conclusion the angle of curved road and lane changing behavior have a positive impact on traffic flow stability while passing effect on the contrary will aggravate traffic congestion

stability analysis of a new lattice hydrodynamic model by considering lattice s self anticipative density effect abstract in this paper a new lattice hydrodynamic model with consideration of the density difference of a lattice s current density and its anticipative density is proposed the influence of lattice s self anticipative density on traffic stability is revealed through linear stability theory and it shows that lattice s self anticipative density can improve the stability of traffic flow to describe the phase transition of traffic flow the mkdv equation near the critical point is derived by using nonlinear analysis method the propagating behavior of density wave in the unstable region can be described by the kink antikink soliton of the mkdv equation numerical simulation validates the analytical results which shows that traffic jam can be suppressed efficiently by considering lattice s self anticipative density in the modified lattice hydrodynamic model

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

an extended macro traffic flow model accounting for the driver s bounded rationality and numerical tests abstract in this paper we propose a macro traffic flow model to explore the effects of the driver s bounded rationality on the evolutions of traffic waves which include shock and rarefaction waves and small perturbation and on the fuel consumption and emissions that include co hc and no x during the evolution process the numerical results illustrate that considering the driver s bounded rationality can prominently smooth the wavefront of the traffic waves and improve the stability of traffic flow which shows that the driver s bounded rationality has positive impacts on traffic flow but considering the driver s bounded rationality reduces the fuel consumption and emissions only at the upstream of the rarefaction wave while enhances the fuel consumption and emissions under other situations which shows that the driver s bounded rationality has positive impacts on the fuel consumption and emissions only at the upstream of the rarefaction wave while negative effects on the fuel consumption and emissions under other situations in addition the numerical results show that the driver s bounded rationality has little prominent impact on the total fuel consumption and emissions during the whole evolution of small perturbation

universal decoding for frequency selective fading channels we address the problem of universal decoding in unknown frequency selective fading channels using an orthogonal frequency division multiplexing ofdm signaling scheme a block fading model is adopted where the bands fading coefficients are unknown yet assumed constant throughout the block given a codebook we seek a decoder independent of the channel parameters whose worst case performance relative to a maximum likelihood ml decoder that knows the channel is optimal specifically the decoder is selected from a family of quadratic decoders and the optimal decoder is referred to as a quadratic minimax qmm decoder for that family as the qmm decoder is generally difficult to find a suboptimal qmm decoder is derived instead despite its suboptimality the proposed decoder is shown to outperform the generalized likelihood ratio test glrt which is commonly used when the channel is unknown while maintaining a comparable complexity the qmm decoder is also derived for the practical case where the fading coefficients are not entirely independent but rather satisfy some general constraints simulations verify the superiority of the proposed qmm decoder over the glrt and over the practically used training sequence approach

analysis of social optimum for staggered shifts in a single entry traffic corridor with no late arrivals abstract this paper investigates the traffic flow dynamics under the social optimum so principle in a single entry traffic corridor with staggered shifts from the analytical and numerical perspectives the lwr lighthill whitham and richards model and the greenshield s velocity density function are utilized to describe the dynamic properties of traffic flow the closed form so solution is analytically derived and some numerical examples are used to further testify the analytical solution the optimum proportion of the numbers of commuters with different desired arrival times is further discussed where the analytical and numerical results both indicate that the cumulative outflow curve under the so principle is piecewise smooth

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

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

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

combined effects of drivers disturbance risk preference heterogeneity and the nearest following vehicle headway on traffic flow instability analytical studies abstract driver s disturbance risk preference is defined as the driver s reaction degree to the disturbance which is a kind of the driver s attributes and can be expressed by the disturbance risk preference coefficient this paper proposes a new car following model and firstly attempts to analytically investigate the combined effects of drivers disturbance risk preference heterogeneity and the nearest following vehicle headway on traffic flow instability under some specific traffic scenarios solution of the model yields many insights including the following 1 when drivers disturbance risk preference increases from the following driver to the preceding driver traffic flow instability increases and the bigger the disturbance risk preference coefficient ratio of the preceding driver s coefficient to the following driver s coefficient the bigger the increasing effect and vice versa at this scenario the nearest following vehicle headway increases traffic flow instability 2 when drivers disturbance risk preference is homogeneous drivers disturbance risk preference and the nearest following vehicle headway have no effects on traffic flow instability 3 when drivers disturbance risk preference reduces from the following driver to the preceding driver for the general traffic situation traffic flow instability reduces and the smaller the disturbance risk preference coefficient ratio of the preceding driver s coefficient to the following driver s coefficient the bigger the reducing effect and vice versa at this scenario if the coefficient ratio reduces from the following to the preceding or is homogeneous the nearest following vehicle headway increases traffic flow instability otherwise the effects of the nearest following vehicle headway on traffic flow instability cannot be confirmed

fading channels how perfect need perfect side information be the analysis of flat fading channels is often performed under the assumption that the additive noise is white and gaussian and that the receiver has precise knowledge of the realization of the fading process these assumptions imply the optimality of gaussian codebooks and of scaled nearest neighbor decoding here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process we quantify the degradation in performance that results from such estimation errors and demonstrate the lack of robustness of this scheme for some situations we suggest the rule of thumb that in order to avoid degradation the estimation error should be negligible compared to the reciprocal of the signal to noise ratio snr

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

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

a novel lattice hydrodynamic model considering the optimal estimation of flux difference effect on two lane highway abstract the optimal estimation of flux difference effect is introduced to construct a new lattice hydrodynamic model for two lane highway through linear stability analysis it is found that the new consideration plays an important influence upon the stability of two lane traffic flow moreover the mkdv equation near the critical point is derived from nonlinear analysis which describes the propagation behavior of traffic jam in two lane traffic system simulation results show that the stability of two lane traffic system can be increased and the emergence of traffic jams are effectively relieved by the optimal estimation of flux difference effect which is in good agreement with theoretical analysis

nonlinear analysis of a new lattice hydrodynamic model with the consideration of honk effect on flux for two lane highway abstract influence of the honk on traffic flux is taken to construct a new lattice model for two lane freeway the linear stability analysis is applied to obtain the stability condition involving the honk effect on traffic flux in two lane system nonlinear analysis is implemented to get the mkdv equation for the phase transition of two lane lattice traffic flow numerical simulation is worked out to authenticate that the impact of honk on traffic flux strengthens the stability of traffic flow and suppresses the traffic congestion effectively under lane changing in two lane lattice traffic flow which is accordance with the analytic results

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

modified kdv equation for jamming transition in the continuum models of traffic abstract continuum models of traffic are proposed to describe the jamming transition in traffic flow on a highway they are the simplified versions of the hydrodynamic model of traffic two continuum models are presented one is described by the partial differential equations and the other is the discrete lattice version the linear stability theory and the nonlinear analysis are applied to the continuum models the modified korteweg de vries equation kdv near the critical point is derived using the reduction perturbation method it is shown that the jamming transition and the density wave in the congested traffic flow are described by the modified kdv equation the solutions of the kdv equations obtained from the two models are compared with that of the optimal velocity model car following model

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

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

chaotic jam and phase transition in traffic flow with passing the lattice hydrodynamic model is presented to take into account the passing effect in one dimensional traffic flow when the passing constant gamma is small the conventional jamming transition occurs between the uniform traffic and kink density wave flows when passing constant gamma is larger than the critical value the jamming transitions occur from the uniform traffic flow through the chaotic density wave flow to the kink density wave flow with an increasing delay time the chaotic region increases with passing constant gamma the neutral stability line is derived from the linear stability analysis the neutral stability line coincides with the transition line between the uniform traffic and density wave flows the modified korteweg de vries equation describing the kink jam is derived for small values of gamma by use of a nonlinear analysis

the backward looking effect in the lattice hydrodynamic model abstract the novel lattice hydrodynamic model is presented by incorporating the backward looking effect the stability condition for the the model is obtained using the linear stability theory the result shows that considering one following site in vehicle motion leads to the stabilization of the system compared with the original lattice hydrodynamic model and the cooperative driving lattice hydrodynamic model the korteweg de vries kdv for short equation near the neutral stability line is derived by using the reductive perturbation method to show the traffic jam which is proved to be described by kdv soliton solution obtained from the kdv equation the simulation result is consistent with the nonlinear analysis

study on the interval integration effect of vehicle s self delayed velocity on traffic stability in micro traffic modeling abstract a new car following model is proposed in this paper to study the interval integration effect of vehicle s self delayed velocity information on traffic flow the stability condition of the new model is derived through linear analysis and the result shows that the traffic stability can be improved when a longer interval of vehicle s self delayed velocity information is taken into account also it is revealed that the traffic information in the time interval which is far from the present time is more positive in stabilizing traffic flow comparing to the traffic information in the time interval which is close to the present time through nonlinear analysis the mkdv equation is derived to describe the propagation properties of density wave in the unstable region finally the theoretical results are confirmed by numerical simulation

a new lattice model of traffic flow with the consideration of the traffic interruption probability abstract in this paper we present a new lattice model which involves the effects of traffic interruption probability to describe the traffic flow on single lane freeways the stability condition of the new model is obtained by the linear stability analysis and the modified korteweg de vries kdv equation is derived through nonlinear analysis thus the space will be divided into three regions stable metastable and unstable the simulation results also show that the traffic interruption probability could stabilize traffic 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

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

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

study on varying time delay on traffic stability in a novel lattice hydrodynamic model abstract a novel traffic lattice hydrodynamic model with consideration of varying time delay is proposed the asymptotic stability condition of the new model is derived in the form of linear matrix inequality through lyapunov stability theory and the traffic disturbance appearing in the new model can be suppressed under this stable condition numerical simulation shows that varying time delay affects the stability of traffic flow importantly and the traffic flow with consideration of varying time delay is more likely to develop into traffic congestion

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

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

minimum rates of approximate sufficient statistics given a sufficient statistic for a parametric family of distributions one can estimate the parameter without access to the data however the memory or code size for storing the sufficient statistic may nonetheless still be prohibitive indeed for n independent samples drawn from a k nomial distribution with d k 1 degrees of freedom the length of the code scales as d log n o 1 in many applications we may not have a useful notion of sufficient statistics e g when the parametric family is not an exponential family and we may also not need to reconstruct the generating distribution exactly by adopting a shannon theoretic approach in which we allow a small error in estimating the generating distribution we construct various approximate sufficient statistics and show that the code length can be reduced to d 2 log n o 1 we consider errors measured according to the relative entropy and variational distance criteria for the code constructions we leverage rissanen s minimum description length principle which yields a non vanishing error measured according to the relative entropy for the converse parts we use clarke and barron s formula for the relative entropy of a parameterized distribution and the corresponding mixture distribution however this method only yields a weak converse for the variational distance we develop new techniques to achieve vanishing errors and we also prove strong converses the latter means that even if the code is allowed to have a non vanishing error its length must still be at least d 2 log n

a converse coding theorem for mismatched decoding at the output of binary input memoryless channels an upper bound on the maximal transmission rate over binary input memoryless channels provided that the decoding decision rule is given is derived if the decision rule is equivalent to the maximum likelihood decoding matched decoding then the bound coincides with the channel capacity otherwise mismatched decoding it coincides with a known lower bound

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

the impact of the density difference memory integral on traffic stability in two lane lattice hydrodynamic model abstract the density difference memory integral ddmi term can be applied to describe traffic density information of the driver s historical memory during a continuous period of time in this paper a new lattice model is provided with the consideration of the ddmi effect for two lane freeway the linear stability condition of two lanes has been derived by considering the ddmi effect through linear stability theory which shows that the ddmi effect can improve traffic stability under lane changing according to the observation of simulation the ddmi effect conduces to the stability of traffic flow for two lane freeway

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

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

cellular automaton for migration in ecosystem application of traffic model to a predator prey system in most cases physicists have studied the migration of biospecies by the use of random walk in the present article we apply cellular automaton of traffic model for simplicity we deal with an ecosystem contains a prey and predator and use one dimensional lattice with two layers preys stay on the first layer but predators uni directionally move on the second layer the spatial and temporal evolution is numerically explored it is shown that the migration has the important effect on populations of both prey and predator without migration the phase transition between a prey phase and coexisting phase occurs in contrast the phase transition disappears by migration this is because predator can survive due to migration we find another phase transition for spatial distribution in one phase prey and predator form a stripe pattern of condensation and rarefaction while in the other phase they uniformly distribute the self organized stripe may be similar to the migration patterns in real ecosystems

social optimum for evening commute in a single entry traffic corridor with no early departures abstract in this paper we investigate the evening commute behaviors on the social optimum so state in a single entry traffic corridor with no early departures differing from the previous studies on evening commute the dynamic properties of traffic flow are analyzed with the lwr lighthill whitham richards model the properties of optimum cumulative inflow curve with general desired departure time distribution curve are deduced and then the analytic solutions for common desired departure time in so are obtained three numerical examples are carried out to capture the characteristics of evening commuting behaviors under different values of time the analytic and numerical results both indicate that the rarefaction wave originating from the first entry point influences the whole or part of the outflow curve no shock wave exists through the commuting process in addition the cost curves show that the trip cost increases and the departure delay cost decreases with departure time whereas the travel time cost first increases then decreases with departure time under the so principle

a lattice hydrodynamic model based on delayed feedback control considering the effect of flow rate difference abstract in this paper a lattice hydrodynamic model is derived considering not only the effect of flow rate difference but also the delayed feedback control signal which including more comprehensive information the control method is used to analyze the stability of the model furthermore the critical condition for the linear steady traffic flow is deduced and the numerical simulation is carried out to investigate the advantage of the proposed model with and without the effect of flow rate difference and the control signal the results are consistent with the theoretical analysis correspondingly

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

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

effect of optimal estimation of flux difference information on the lattice traffic flow model abstract in this paper a new lattice model is proposed by considering the optimal estimation of flux difference information the effect of this new consideration upon the stability of traffic flow is examined through linear stability analysis furthermore a modified korteweg de vries mkdv equation near the critical point is constructed and solved by means of nonlinear analysis method and thus the propagation behavior of traffic jam can be described by the kink antikink soliton solution of the mkdv equation numerical simulation is carried out under periodical condition with results in good agreement with theoretical analysis therefore it is verified that the new consideration can enhance the stability of traffic systems and suppress the emergence of traffic jams effectively

the influence of continuous historical velocity difference information on micro cooperative driving stability abstract in this paper a new micro cooperative driving car following model is proposed to investigate the effect of continuous historical velocity difference information on traffic stability the linear stability criterion of the new model is derived with linear stability theory and the results show that the unstable region in the headway sensitivity space will be shrunk by taking the continuous historical velocity difference information into account through nonlinear analysis the mkdv equation is derived to describe the traffic evolution behavior of the new model near the critical point via numerical simulations the theoretical analysis results are verified and the results indicate that the continuous historical velocity difference information can enhance the stability of traffic flow in the micro cooperative driving process

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

the capacity of finite abelian group codes over symmetric memoryless channels the capacity of finite abelian group codes over symmetric memoryless channels is determined for certain important examples such as m psk constellations over additive white gaussian noise awgn channels with m a prime power it is shown that this capacity coincides with the shannon capacity i e there is no loss in capacity using group codes this had previously been known for binary linear codes used over binary input output symmetric memoryless channels on the other hand a counterexample involving a three dimensional geometrically uniform constellation is presented in which the use of abelian group codes leads to a loss in capacity the error exponent of the average group code is determined and it is shown to be bounded away from the random coding error exponent at low rates for finite abelian groups not admitting galois field structure

on the universality of the lz based decoding algorithm a universal decoder for a family of channels is a decoder that can be designed without prior knowledge of the particular channel in the family over which transmission takes place and it yet attains the same random coding error exponent as the optimal decoder tuned to the channel in use we study ziv s 1985 decoding algorithm which is based on lempel ziv 1978 incremental string parsing and demonstrate that while it was originally proposed as a universal decoder for the family of finite state channels with deterministic but unknown transitions it is in fact universal for the broader class of all finite state channels we also demonstrate that the generalized likelihood decoder may not be universal even for finite families for which a universal decoder always exists

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

jamming transitions and the effect of interruption probability in a lattice traffic flow model with passing abstract a new lattice hydrodynamic model is proposed by considering the interruption probability effect on traffic flow with passing and analyzed both theoretically and numerically from linear and non linear stability analysis the effect of interruption probability on the phase diagram is investigated and the condition of existence for kink antikink soliton solution of mkdv equation is derived the stable region is enhanced with interruption probability and the jamming transition occurs from uniform flow to kink flow through chaotic flow for higher and intermediate values of non interruption effect of passing it is also observed that there exists conventional jamming transition between uniform flow and kink flow for lower values of non interruption effect of passing numerical simulations are carried out and found in accordance with the theoretical findings which confirm that the effect of interruption probability plays an important role in stabilizing traffic flow when passing is allowed