new kamenev type oscillation criteria for second order neutral nonlinear differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t y t p t y t i 1 n q i t f i y i t 0 t t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

non contraction of heat flow on minkowski spaces we study contractivity properties of gradient flows for functions on normed spaces or more generally on finsler manifolds contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions this is different from the usual convexity along geodesics in non riemannian finsler manifolds as an application we show that the heat flow on minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic wasserstein distance

oscillation of second order functional differential equations with damping abstract using a class of new test functions t s r defined in our recent work y g sun new kamenev type oscillation criteria for second order nonlinear differential equations with damping j math anal appl 291 2004 341 351 we establish some new oscillation criteria for the second order functional differential equation with damping x t p t x t a b q t f x g 1 t x g m t d 0 our results are different from most known ones and can be applied to many cases which are not covered by existing results

stochastic group preference modelling in the multiplicative ahp a model of group consensus this paper proposes a model for the representation of a consensus seeking group decision making process under the multiplicative variant of the analytic hierarchy process ahp in particular it proposes a group preference model which expresses the group s preference intensity judgements as random variables with associated probability distributions and the structure of the multiplicative ahp is exploited to derive interval judgements for the alternatives final impact scores as perceived by the group these interval judgements can be used to determine the probability that a rank reversal may occur amongst alternatives due to the group preference uncertainty i e to assess the stability of the final impact score vector which in turn offers information about the degree of consensus within the group about the decision

probabilistic judgments specified partially in the analytic hierarchy process the analytic hierarchy process ahp is a decision making tool which yields priorities for decision alternatives this paper proposes a new approach to elicit and synthesize expert assessments for the group decision process in the ahp these new elicitations are given as partial probabilistic specifications of the entries of pairwise comparisons matrices for a particular entry of the matrix the partial probabilistic elicitations could arise in the form of either probability assignments regarding the chance of that entry falling in specified intervals or selected quantiles for that entry a new class of models is introduced to provide methods for processing this partial probabilistic information one advantage of this approach is that it allows to generate as many pairwise comparison matrices of the decision alternatives as one desires this in turn allows us to determine the statistical significance of the priorities of decision alternatives

link function selection in stochastic multicriteria decision making models a stochastic formulation of the analytic hierarchy process ahp using an approach based on bayesian categorical data models has been developed however in categorical data models it is known that the selection of the link function may have an impact on the model estimates in particular the selection of the probit link implies an assumption that model error terms are normally distributed and this normality assumption is regularly utilized in other related methods such as the multiplicative ahp we examine model performance with respect to the choice of two model link functions with regard to point estimates it is found that the logit formulation is better able to replicate the estimates obtained by the eigenvector decomposition associated with the original formulation of the ahp by contrast the probit link produces priorities which are consistently more moderate than those of the ahp the results suggest that the logit formulation will be preferred by decision makers who wish to replicate the ahp priorities as closely as possible the results also suggest that the unexamined use of the normality assumption in other stochastic ahp methods may have an impact on priority estimates and thus is worthy of further attention

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

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

the categorical data analysis approach for ratio model of pairwise comparisons abstract in the analytic hierarchy process ahp pairwise comparison values are elicited by a ratio scale representing a set of categories in this article we propose a method of statistical inference in ahp which pertains to cardinal representations of categorical variables relevant estimation and hypothesis testing procedures are addressed in this perspective

oscillation criteria for second order quasi linear neutral delay differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t x t p t x t m 1 x t p t x t q t f x t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel

differential inequalities on finsler manifolds and their geometric applications abstract in this paper we study some differential inequalities on finsler manifolds firstly we prove a liouville type theorem for the solution of a differential inequality secondly we obtain a priori boundedness of solutions of the differential inequality and give some applications to problems in geometry and differential equations

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

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

theoretical estimation of the probability of weight rank reversal in pairwise comparisons abstract pairwise comparison is a key component in multi criteria decision making the probability of rank reversal is a useful measure for evaluating the impact of uncertainty on the final outcome in the context of this paper the type of uncertainty considered is related to the fact that experts have different opinions or that they may perform inconsistent pairwise comparisons we provide a theoretical model for estimating the probability of the consequent rank reversal using the multivariate normal cumulative distribution function the model is applied to two alternative weight extraction methods frequently used in the literature the geometric mean and the eigenvalue method we introduce a reasonable framework for incorporating uncertainty in the decision making process and calculate the mean value and cross correlation of the average weights which are required in the application of the model the theoretical results are compared against numerical simulations and a very good agreement is observed we further show how our model can be extended in applications of a full multi criteria decision making analysis such as the analytic hierarchy process we also discuss how the theoretical model can be used in practice where the statistical properties of the uncertainty induced perturbations are unknown and the only information provided by the pairwise comparison matrices of a small group of experts the methodology presented here can be used to extend the pairwise comparison framework in order to provide some information on the credibility of its outcome

new kamenev type oscillation criteria for second order nonlinear differential equations with damping abstract some new oscillation criteria are established for the nonlinear damped differential equation r t y p t y q t f y 0 that are different from most known ones in the sense that they are based on a class of new functions t s r defined in the sequel our results are sharper than some previous results which can be seen by the examples at the end of this paper

oscillation criteria for second order neutral equations with distributed deviating argument abstract we present new oscillation criteria for the second order neutral delay differential equations with distributed deviating argument r t x t z t a b p t f x g t d 0 t t 0 where z t x t q t x t new oscillation criteria are established which are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

the spherically symmetric finsler metrics with isotropic s curvature abstract in this paper we investigate the spherically symmetric finsler metrics with isotropic s curvature and obtain a characterized equation as an application we prove that these metrics with douglas type must be randers metrics or berwald metrics this result leads to two classification theorems

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

bayesian variable selection in generalized linear models using a combination of stochastic optimization methods abstract in this paper the usage of a stochastic optimization algorithm as a model search tool is proposed for the bayesian variable selection problem in generalized linear models combining aspects of three well known stochastic optimization algorithms namely simulated annealing genetic algorithm and tabu search a powerful model search algorithm is produced after choosing suitable priors the posterior model probability is used as a criterion function for the algorithm in cases when it is not analytically tractable laplace approximation is used the proposed algorithm is illustrated on normal linear and logistic regression models for simulated and real life examples and it is shown that with a very low computational cost it achieves improved performance when compared with popular mcmc algorithms such as the mcmc model composition as well as with vanilla versions of simulated annealing genetic algorithm and tabu search

new kamenev type oscillation criteria for second order differential equations on a measure chain we establish new kamenev type criteria for oscillation of the second order linear differential equation p t x d t d q t x s t 0 on a measure chain our results are extensions of those for differential equations and provide new oscillation criteria for difference equations several examples are given to show the significance of the results

oscillation and nonoscillation for second order linear differential equations abstract new oscillation and nonoscillation theorems are obtained for the second order linear differential equation u p t u 0 where p t c 0 and p t 0 conditions only about the integrals of p t on every interval 2 n t 0 2 n 1 t 0 n 1 2 for some fixed t 0 0 are used in the results

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

revising the owa operator for multi criteria decision making problems under uncertainty abstract multi criteria decision making mcdm problems are usually under uncertainty one of these uncertain parameters is the decision maker dm s degree of optimism which has an important effect on the results fuzzy linguistic quantifiers are used to obtain the assessments of this parameter from dm and then because of its uncertainty it is assumed to have stochastic nature a new approach entitled fsrowa is introduced to combine the fuzzy and stochastic features into a revised owa operator if the dm is not aware of the risk in decision then the decision objective is to maximize the expected combined goodness measure if the dm cares only about the risk then minimizing the variance of the combined goodness measure is his her objective the results of the fsrowa provide the expected value and the variance of the combined goodness measure for each alternative in order to combine these two attributes a composite goodness measure is introduced this measure gives more robust decision outcomes the theoretical results are illustrated in a watershed management problem

some new oscillation theorems for second order nonlinear elliptic equations with damping abstract in this paper some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form a x y b t x y q x f y 0 x where is an exterior domain in r n these criteria are different from most known ones in the sense that they are based on a class of new functions of the form r s l defined in the sequel

interval oscillation criteria for second order nonlinear differential equations with damping new oscillation criteria are established for the nonlinear damped differential equation ry py qf y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are sharper than some previous results and extend and improve the recent interval criteria of kong and huang and can be applied to extreme cases such as t0 q s ds in particular several examples that dwell upon the sharp conditions of our results are also included

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

on general α β metrics with isotropic s curvature abstract the s curvature is an important non riemannian quantity in finsler geometry in this paper we study a class of finsler metrics called general metrics which are defined by a riemannian metric and a 1 form we characterize all of general metrics with isotropic s curvature

note on wong s paper using the method of interval integral average we establish some new oscillation criteria which improve the oscillation criteria due to james s w wong

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

oscillation criteria for second order nonlinear differential equations with damping abstract for a second order nonlinear differential equation with a damping term we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations and both sets of oscillation criteria established in the paper are of independent interest several examples are provided to illustrate the relevance of the new theorems

interval priorities in ahp by interval regression analysis abstract this paper proposes an interval approach for obtaining interval weights of priorities in the analytic hierarchy process ahp reflecting inconsistency of pairwise comparison ratios given by a decision maker even if pairwise comparison values with respect to preference are given as crisp values the priority weights should be estimated as intervals because of a decision maker s uncertainty of judgments we deal with two types of comparison matrices represented as crisp and interval elements this allows some uncertainty of estimations made by a decision maker the proposed approach can obtain the estimated interval priorities since interval regression analysis is based on the concept of possibility the interval weights obtained by the proposed method can be interpreted as the possible ranges estimated from the given data the sum of widths of the obtained intervals can be regarded as the degree of inconsistency included in the given data it is emphasized that interval data of preference can obtain the lower and upper models for the estimated weight intervals

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

reciprocal distributions in the analytic hierarchy process abstract the paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the analytic hierarchy process ahp these distributions called reciprocal satisfy the reciprocity axiom of the ahp methodology first we define and characterise the reciprocal distributions and present some results and properties associated with them we then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector considering the eigenvector method egv in the consistent case and the row geometric mean method rgm as priorization methods and assuming that the judgements follow reciprocal distributions we prove that the elements of the priorities vector are distributed according to reciprocal random variables the knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems the probability that a given alternative would be ranked in the first position and the probability of a given ranking preference structure for the whole set of alternatives in particular we find the analytic expressions for these probabilities when a reciprocal random variable such as the lognormal is used

a bayesian analysis of the multinomial probit model using marginal data augmentation we introduce a set of new markov chain monte carlo algorithms for bayesian analysis of the multinomial probit model our bayesian representation of the model places a new and possibly improper prior distribution directlyon the identi0able parameters and thus is relativelyeasyto interpret and use our algorithms which are based on the method of marginal data augmentation involve onlydraws from standard distributions and dominate other available bay esian methods in that theyare as quick to converge as the fastest methods but with a more attractive prior speci0cation c code along with an r interface for our algorithms is publiclyavailable 1

interval estimation of priorities in the ahp this paper extends and modifies the analytic hierarchy process ahp and the synthetic hierarchy method shm of priority estimation to accommodate random data in the pairwise comparison matrices it employs a cauchy distribution to describe the pairwise comparison of alternatives in saaty matrices and shows how to modify these matrices in order to handle random data the use of random data yields saaty matrices that are not reciprocally symmetrical several variants of the ahp are then modified i to accommodate reciprocally asymmetric matrices and ii to allow each priority estimate to be expressed on an interval of possible values rather than as a single discrete point the merits of interval estimation are illustrated by an example

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

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

some liouville type theorems for harmonic functions on finsler manifolds abstract in this paper we give some liouville type theorems for l p p r harmonic resp subharmonic superharmonic functions on forward complete finsler manifolds moreover we derive a gradient estimate for harmonic functions on a closed finsler manifold as an application one obtains that any harmonic function on a closed finsler manifold with nonnegative weighted ricci curvature ric n n n must be constant

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

interval oscillation of a general class of second order nonlinear differential equations with nonlinear damping abstract the paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions examples are also given to illustrate the results in particular it is shown that under some very mild conditions on k 1 k 2 and f the equation k 1 x x cos tk 2 x x x sin tf x 0 is oscillatory which extends the result obtained by kwong and wong j math anal appl 288 2003 15

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

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

integral averaging techniques for the oscillation of second order nonlinear differential equations abstract in this paper we enlist some known and establish some new oscillation criteria for the second order nonlinear differential equations of the form if a t x t t su p t t q t f x t 0 where the coefficients p t and q t are not assumed to be nonnegative for all large values of t these criteria are obtained by using integral averaging techniques and can be applied to some special cases where other classical oscillation results are not applicable a systematic study is attempted to extend improve and correlate a number of existing results

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

a model for the relative importance of the criteria in the multiplicative ahp and smart abstract we define the relative importance of any pair of criteria as the substitution rate between the relative gains and losses of the alternatives when we move along an indifference curve under the geometric mean aggregation rule in the multiplicative ahp and under the arithmetic mean aggregation rule in smart the relative not the marginal substitution rate depends neither on the performance of the alternatives under the remaining criteria nor on the units of performance measurement hence it provides a sound argument for distributed decision making processes where those who judge the criteria are not the same actors as those who assess the performance of the alternatives the definition has a plausible basis in the psycho physical research on the relationship between physical stimuli and sensory responses which shows that human beings are sensitive not to marginal but to relative changes of the stimulus intensities

stochastic decision making using multiplicative ahp abstract the analytic hierarchy process ahp has found a number of applications in decision making problems its multiplicative version called the multiplicative ahp mahp has been proposed to overcome some of the criticisms of the conventional version both these methodologies operate by obtaining expert judgements on the ratios of perceived importance of objects under consideration the literature on mahp in dealing with these judgements when they are specified without uncertainty is well developed however stochastic aspects of these judgements have not received much consideration in the literature so far stochastic judgements are considered in this paper for use in mahp the fact that weight derivation in mahp can be handled using mathematical programming is exploited and the literature on stochastic programming is adapted to the mahp context

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

a sharp lower bound for the first eigenvalue on finsler manifolds with nonnegative weighted ricci curvature abstract let m f be an n dimensional compact finsler manifold without boundary or with a convex boundary and 1 be the first nonzero closed or neumann eigenvalue of the finsler laplacian on m with nonnegative weighted ricci curvature in this paper we prove that 1 2 d 2 where d is the diameter of m and that the equality holds if and only if m is a 1 dimensional circle or a 1 dimensional segment which generalize the well known zhong yang s sharp estimate in riemannian geometry zhong and yang 1984

caffarelli kohn nirenberg inequality on metric measure spaces with applications we prove that if a metric measure space satisfies the volume doubling condition and the caffarelli kohn nirenberg inequality with the same exponent n ge 3 then it has exactly the n dimensional volume growth as an application if an n dimensional finsler manifold of non negative n ricci curvature satisfies the caffarelli kohn nirenberg inequality with the sharp constant then its flag curvature is identically zero in the particular case of berwald spaces such a space is necessarily isometric to a minkowski space

new oscillation criteria of second order nonlinear differential equations by employing a class of new functions f f t s l and a generalized riccati technique some new oscillation and interval oscillation criteria are established for the second order nonlinear differential equation r t y t q t y t y t 0 the obtained interval oscillation criteria can be applied to equations with forcing term two examples are also included to show the significance of our results

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

on kamenev type oscillation theorems for second order differential equations with damping abstract oscillation criteria for the nonlinear second order ordinary differential equation with damping x p t x f x 0 are given the results are extensions of integral averaging technique of kamenev and include earlier results of yeh yan and chen