semiparametric nonlinear mixed effects models and their applications nonlinear mixed effects models nlmms and self modeling nonlinear regression semor models are often used to fit repeated measures data they use a common function shared by all subjects to model variation within each subject and some fixed and or random parameters to model variation between subjects the parametric nlmm may be too restrictive and the semiparametric semor model ignores correlations within each subject in this article we propose a class of semiparametric nonlinear mixed effects models snmms that extend nlmms semor models and many other existing models in a natural way a snmm assumes that the mean function depends on some parameters and nonparametric functions the parameters provide an interpretable data summary the nonparametric functions provide flexibility to allow the data to decide some unknown or uncertain components such as the shape of the mean response over time a second stage model with fixed and random effects is used to model the parameters smoothing splines are us

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

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

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

gradient estimates for heat kernels and harmonic functions abstract let x d be a doubling metric measure space endowed with a dirichlet form e deriving from a carre du champ assume that x d e supports a scale invariant l 2 poincare inequality in this article we study the following properties of harmonic functions heat kernels and riesz transforms for p 2 i g p l p estimate for the gradient of the associated heat semigroup ii r h p l p reverse holder inequality for the gradients of harmonic functions iii r p l p boundedness of the riesz transform p iv g b e a generalised bakry emery condition we show that for p 2 i ii iii are equivalent while for p i ii iv are equivalent moreover some of these equivalences still hold under weaker conditions than the l 2 poincare inequality our result gives a characterisation of li yau s gradient estimate of heat kernels for p while for p 2 it is a substantial improvement as well as a generalisation of earlier results by auscher coulhon duong hofmann 7 and auscher coulhon 6 applications to isoperimetric inequalities and sobolev inequalities are given our results apply to riemannian and sub riemannian manifolds as well as to non smooth spaces and to degenerate elliptic parabolic equations in these settings

on stabilization of equilibria using predictive control with and without pulses consider a chaotic difference equation x n 1 f x n we focus on the problem of control of chaos using a prediction based control pbc method if f has a unique positive equilibrium it is proved that global stabilization of this equilibrium can be achieved under mild assumptions on the map f if f has several positive equilibria we demonstrate that more than one equilibrium can be stabilized simultaneously we also show that it is still possible to stabilize an unstable equilibrium using a strategy of control with pulses that is the control is only applied after a fixed number of iterations we illustrate our main results with several examples mainly from population dynamics

a pde approach to a 2 dimensional matching problem we prove asymptotic results for 2 dimensional random matching problems in particular we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square our technique is based on a rigorous formulation of the challenging pde ansatz by caracciolo et al phys rev e 90 012118 2014 that linearizes the monge ampere equation

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

epidemic outbreaks and its control using a fractional order model with seasonality and stochastic infection abstract a fractional order sir epidemic model is proposed under the influence of both parametric seasonality and the external noise the integer order sir epidemic model originally is stable by introducing seasonality and noise force to the model behaviors of the system is changed it is shown that the system has rich dynamical behaviors with different system parameters fractional derivative order and the degree of seasonality and noise complexity of the stochastic model is investigated by using multi scale fuzzy entropy finally hard limiter controlled system is designed and simulation results show the ratio of infected individuals can converge to a small enough target which means the epidemic outbreak can be under control by the implementation of some effective medical and health measures

self modelling with random shift and scale parameters and a free knot spline shape function the shape invariant model is a semi parametric approach to estimating a function relationship from clustered data multiple observations on each of a number of individuals the common response curve shape over individuals is estimated by adjusting for individual scaling differences while pooling shape information in practice the common response curve is restricted to some flexible family of functions this paper introduces the use of a free knot spline shape function and reduces the number of parameters in the shape invariant model by assuming a random distribution on the parameters that control the individual scaling of the shape function new graphical diagnostics are presented parameter identifiability and estimation are discussed and an example is presented

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

isoperimetric inequality under measure contraction property abstract we prove that if x d m is an essentially non branching metric measure space with m x 1 having ricci curvature bounded from below by k and dimension bounded above by n 1 understood as a synthetic condition called measure contraction property then a sharp isoperimetric inequality a la levy gromov holds true measure theoretic rigidity is also obtained

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

shape invariant modeling of circadian rhythms with random effects and smoothing spline anova decompositions medical studies often collect physiological and or psychological measurements over time from multiple subjects to study dynamics such as circadian rhythms under the assumption that the expected response functions of all subjects are the same after shift and scale transformations shape invariant models have been applied to analyze this kind of data the shift and scale parameters provide efficient and interpretable data summaries while the common shape function is usually modeled nonparametrically to provide flexibility however due to the deterministic nature of the shift and scale parameters potential correlations within a subject are ignored furthermore the shape of the common function may depend on other factors such as disease in this article we propose shape invariant mixed effects models a second stage model with fixed and random effects is used to model individual shift and scale parameters a second stage smoothing spline anova model is used to study potential covariate effects on the common shape function we apply our methods to a real data set to investigate disease effects on circadian rhythms of cortisol a hormone that is affected by stress we find that patients with cushing s syndrome lost circadian rhythms and their 24 hour means were elevated to very high levels patients with major depression had the same circadian shape and phases as normal subjects however their 24 hour mean levels were elevated and amplitudes were dampened for some patients

direct regression models for longitudinal rates of change comparing rates of growth or rates of change across covariate defined subgroups is a primary objective for many longitudinal studies in the special case of a linear trend over time the interaction between a covariate and time will characterize differences in longitudinal rates of change however in the presence of a non linear longitudinal trajectory the standard mean regression approach does not permit parsimonious description or inference regarding differences in rates of change therefore we propose regression methodology for longitudinal data that allows a direct structured comparison of rates across subgroups even in the presence of a non linear trend over time our basic longitudinal rate regression method assumes a proportional difference across covariate groups in the rate of change across time but this assumption can be relaxed rates are compared relative to a generally specified time trend for which we discuss both parametric and non parametric estimating approaches we develop mixed model longitudinal methodology that explicitly characterizes subject to subject variation in rates as well as a marginal estimating equation based method in addition we detail a score test to detect violations of the proportionality assumption and we allow time varying rate effects as a natural generalization simulation results demonstrate potential gains in power for the longitudinal rate regression model relative to a linear mixed effects model in the presence of a non linear trend in time we apply our method to a study of growth among infants born to hiv infected mothers and conclude with a discussion of possible extensions for our methods

latent variable poisson models for assessing the regularity of circadian patterns over time abstractmany researchers in biology and medicine have focused on trying to understand biological rhythms and their potential impact on disease a common biological rhythm is circadian where the cycle repeats itself every 24 hours however a disturbance of the circadian pattern may be indicative of future disease in this article we develop new statistical methodology for assessing the degree of disturbance or irregularity in a circadian pattern for count sequences that are observed over time in a population of individuals we develop a latent variable poisson modeling approach with both circadian and stochastic short term trend autoregressive latent process components that allow for individual variation in the degree of each component a parameterization is proposed for modeling covariate dependence on the proportion of these two model components across individuals in addition we incorporate covariate dependence in the overall mean the magnitude of the trend and the phase shift of the circadian p

controlling one dimensional unimodal population maps by harvesting at a constant rate controlling chaos and periodic oscillations in dynamical systems is a well known problem it has been studied by several types of algorithms some of which were only demonstrated numerically with specific examples we study here a class of one dimensional discrete maps of the form x n 1 f x n where f x is a unimodal function satisfying a few natural conditions as a model for the dynamics of a single species population for managing the population we seek to suppress any possible chaotic or periodic behavior that may emerge the paper proposes a simple rigorously proved algorithm for controlling unimodal maps implemented by harvesting the population at a constant rate this forces the orbits of the map to converge to a limit that we can compute a priori the result holds for any initial conditions within an interval that we specify our control algorithm is easy to implement requires no updated information on the population and no changes in the parameters of the system which in general are fixed by the properties of the population it can therefore be useful for exploiting a population while maintaining it at a fixed density

super ricci flows for metric measure spaces abstract we introduce the notions of super ricci flows and ricci flows for time dependent families of metric measure spaces x d t m t t i the former property is proven to be stable under suitable space time versions of mgh convergence uniformly bounded families of super ricci flows are compact in the spirit of the synthetic lower ricci bounds of lott sturm villani for static metric measure spaces the defining property for super ricci flows is the dynamic convexity of the boltzmann entropy ent m t regarded as a functions on the time dependent geodesic space p x w t t i for ricci flows in addition a nearly dynamic concavity of the boltzmann entropy is requested alternatively super ricci flows will be studied in the framework of the calculus of bakry emery ledoux and equivalence to gradient estimates will be derived for both notions of super ricci flows also enforced versions involving an upper dimension bound n will be presented

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

a flexible model for human circadian rhythms many hormones and other physiological processes vary in a circadian pattern although a sine cosine function can be used to model these patterns this functional form is not appropriate when there is asymmetry between the peak and nadir phases in this paper we describe a semiparametric periodic spline function that can be fit to circadian rhythms the model includes both phase and amplitude so that the time and the magnitude of the peak or nadir can be estimated we also describe tests of fit for components in the model data from an experiment to study immunological responses in humans are used to demonstrate the methods

super ricci flows and improved gradient and transport estimates we introduce brownian motions on time dependent metric measure spaces proving their existence and uniqueness we prove contraction estimates for their trajectories assuming that the time dependent heat flow satisfies transport estimates with respect to every l p kantorovich distance p in 1 infty these transport estimates turn out to characterize super ricci flows introduced by sturm j funct anal 275 12 3504 3569 2015

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

measure rigidity of ricci curvature lower bounds abstract the measure contraction property mcp for short is a weak ricci curvature lower bound conditions for metric measure spaces the goal of this paper is to understand which structural properties such assumption or even weaker modifications implies on the measure on its support and on the geodesics of the space we start our investigation from the euclidean case by proving that if a positive radon measure m over r d is such that r d m verifies a weaker variant of mcp then its support spt m must be convex and m has to be absolutely continuous with respect to the relevant hausdorff measure of spt m this result is then used as a starting point to investigate the rigidity of mcp in the metric framework we introduce the new notion of reference measure for a metric space and prove that if x d m is essentially non branching and verifies mcp and is an essentially non branching mcp reference measure for spt m d then m is absolutely continuous with respect to on the set of points where an inversion plan exists as a consequence an essentially non branching mcp reference measure enjoys a weak type of uniqueness up to densities we also prove a stability property for reference measures under measured gromov hausdorff convergence provided an additional uniform bound holds in the final part we present concrete examples of metric spaces with reference measures both in smooth and non smooth setting the main example will be the hausdorff measure over an alexandrov space then we prove that the following are reference measures over smooth spaces the volume measure of a riemannian manifold the hausdorff measure of an alexandrov space with bounded curvature and the haar measure of the subriemannian heisenberg group

smoothing spline models with correlated random errors abstract spline smoothing techniques are commonly used to estimate the mean function in a nonparametric regression model their performances depend greatly on the choice of smoothing parameters many methods of selecting smoothing parameters such as generalized maximum likelihood gml generalized cross validation gcv and unbiased risk ubr have been developed under the assumption of independent observations they tend to underestimate smoothing parameters when data are correlated in this article i assume that observations are correlated and that the correlation matrix depends on a parsimonious set of parameters i extend the gml gcv and ubr methods to estimate the smoothing parameters and the correlation parameters simultaneously i also relate a smoothing spline model to three mixed effects models these relationships show that the smoothing spline estimates evaluated at design points are best linear unbiased prediction blup estimates and that the gml estimates of the smoothing parameters an

sharp and rigid isoperimetric inequalities in metric measure spaces with lower ricci curvature bounds we prove that if x mathsf d mathfrak m is a metric measure space with mathfrak m x 1 having in a synthetic sense ricci curvature bounded from below by k 0 and dimension bounded above by n in 1 infty then the classic levy gromov isoperimetric inequality together with the recent sharpening counterparts proved in the smooth setting by milman for any k in mathbb r n ge 1 and upper diameter bounds holds i e the isoperimetric profile function of x mathsf d mathfrak m is bounded from below by the isoperimetric profile of the model space moreover if equality is attained for some volume v in 0 1 and k is strictly positive then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions finally we also establish the almost rigidity if the equality is almost attained for some volume v in 0 1 and k is strictly positive then the space must be mgh close to a spherical suspension to our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying ricci curvature lower bounds examples of spaces fitting our assumptions include measured gromov hausdorff limits of riemannian manifolds satisfying ricci curvature lower bounds alexandrov spaces with curvature bounded from below finsler manifolds endowed with a strongly convex norm and satisfying ricci curvature lower bounds the result seems new even in these celebrated classes of spaces

comparison between constant feedback and limiter controllers using symbolic dynamics of the one dimensional unimodal map the chaos stabilization mechanics of the feedback and limiter control schemes are considered for feedback control it is found that the control strength can be efficiently obtained from the superstable parameter of the embedded periodic orbits and the scaling of the control space period doubling bifurcation cascade still obeys the feigenbaum law for sarkovskii orbits the scaling is also consistent with that of the original chaotic system for limiter control a single critical point in the unimodal map is extended to a superstable periodic window and a simple approach for determining the value of the control plateau is found the scaling in the control space of the period doubling bifurcation cascade is indeed superexponential a different scaling for the fine structure of the sarkovskii sequence is also found simple one dimensional unimodal maps can also be used to generate maximum length shift register sequences

sobolev spaces on warped products abstract we study the structure of sobolev spaces on the cartesian warped products of a given metric measure space and an interval our main results are the characterization of the sobolev spaces in such products the proof that under natural assumptions the warped products possess the sobolev to lipschitz property which is key for geometric applications the results of this paper have been needed in the recent proof of the volume cone to metric cone property of rcd spaces obtained by the first author and de philippis

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

semiparametric comparison of regression curves the comparison of nonparametric regression curves is considered it is assumed that there are parametric possibly nonlinear transformations of the axes which map one curve into the other estimation and testing of the parameters in the transformations are studied the rate of convergence is n 1 2 although the nonparametric components of the model typically have a rate slower than that a statistic is provided for testing the validity of a given completely parametric model

fitting nonlinear models with arma errors to biological rhythm data many behavioural and physiological processes are periodic with a period of approximately 24 hours for descriptive purposes linear regression using a simple sinusoid with a fixed 24 hour period is sometimes an adequate tool for analysing data from such processes inference based on regression models under the assumption of independent and identically distributed errors however can often mislead seriously in this paper we present a general class of models for fitting biological rhythms with use of higher order harmonic terms of one or more unknown fundamentals and arma processes for the errors we describe the procedures for model specification and estimation and give the theoretical justification for these procedures analysis of a series of human core body temperature illustrates the methodology

optimal entropy transport problems and a new hellinger kantorovich distance between positive measures we develop a full theory for the new class of optimal entropy transport problems between nonnegative and finite radon measures in general topological spaces these problems arise quite naturally by relaxing the marginal constraints typical of optimal transport problems given a pair of finite measures with possibly different total mass one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals which quantify in some way the deviation of the marginals of the transport plan from the assigned measures as a powerful application of this theory we study the particular case of logarithmic entropy transport problems and introduce the new hellinger kantorovich distance between measures in metric spaces the striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance which lies somehow between the well known hellinger kakutani and kantorovich wasserstein distances

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

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

lusin type approximation of sobolev by lipschitz functions in gaussian and rcd k spaces abstract we establish new approximation results in the sense of lusin of sobolev functions by lipschitz ones in some classes of non doubling metric measure structures our proof technique relies upon estimates for heat semigroups and applies to gaussian and rcd k spaces as a consequence we obtain quantitative stability for regular lagrangian flows in gaussian settings

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

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

weighted littlewood paley inequalities for heat flows in rcd spaces abstract we establish inequalities on vertical littlewood paley square functions for heat flows in the weighted l 2 space over metric measure spaces satisfying the rcd 0 n condition with n 1 and the maximum volume growth assumption in the noncompact setting the later assumption can be removed by showing that the volume of the ball growths at least linearly the estimates are sharp on the growth of the 2 heat weight and the 2 muckenhoupt weight considered the p muckenhoupt weight and the p heat weight are also compared for all p 1

asymmetric unimodal maps some results from q generalized bit cumulants in this study using q generalized bit cumulants q is the nonextensivity parameter of the recently introduced tsallis statistics we investigate the asymmetric unimodal maps x t 1 1 ensuremath a x t z i i 1 2 correspond to x t g0 and x t l0 respectively z i g1 0lal 2 t 0 1 2 dots the study of the q generalized second cumulant c 2 q of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs z 1 z 2 the slope of the c 2 q versus c 2 1 plot where c 2 1 is the standard second cumulant provides the necessary tool to accomplish this task the slope behaves exactly the same as the proper q values say q that were obtained for logisticlike maps z 1 z 2 z by costa et al phys rev e 56 245 1997 it appears that as z 2 ensuremath z 1 ensuremath rightarrow ifmmode pm else textpm fi ensuremath infty this slope approaches unity this behavior is very similar to the behavior of q as a function of the inflection parameter for some z dependent maps namely as stackrel ensuremath rightarrow z ensuremath infty q approaches 1

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

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

on analyzing circadian rhythms data using nonlinear mixed models with harmonic terms summarywang ke and brown 2003 biometrics59 804 812 developed a smoothing based approach for modeling circadian rhythms with random effects their approach is flexible in that fixed and random covariates can affect both the amplitude and phase shift of a nonparametrically smoothed periodic function in motivating their approach wang et al stated that a simple sinusoidal function is too restrictive in addition they stated that although adding harmonics can improve the fit it is difficult to decide how many harmonics to include in the model and the results are difficult to interpret we disagree with the notion that harmonic models cannot be a useful tool in modeling longitudinal circadian rhythm data in this note we show how nonlinear mixed models with harmonic terms allow for a simple and flexible alternative to wang et al s approach we show how to choose the number of harmonics using penalized likelihood to flexibly model circadian rhythms and to estimate the effect of covariates on the rhythms we fit harmonic models to the cortisol circadian rhythm data presented by wang et al to illustrate our approach furthermore we evaluate the properties of our procedure with a small simulation study the proposed parametric approach provides an alternative to wang et al s semiparametric approach and has the added advantage of being easy to implement in most statistical software packages

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

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

weak and strong convergence of derivations and stability of flows with respect to mgh convergence abstract this paper is devoted to the study of weak and strong convergence of derivations and of the flows associated to them when dealing with a sequence of metric measure structures x d m n m n weakly convergent to m in particular under curvature assumptions either only on the limit metric structure x d m or on the whole sequence of metric measure spaces we provide several stability results

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

optimal transport cheeger energies and contractivity of dynamic transport distances in extended spaces abstract we introduce the setting of extended metric topological measure spaces as a general wiener like framework for optimal transport problems and nonsmooth metric analysis in infinite dimension after a brief review of optimal transport tools for general radon measures we discuss the notions of the cheeger energy of the radon measures concentrated on absolutely continuous curves and of the induced dynamic transport distances we study their main properties and their links with the theory of dirichlet forms and the bakry emery curvature condition in particular concerning the contractivity properties and the evi formulation of the induced heat semigroup

a finite diameter theorem on rcd k infty spaces for positive k we prove a finite diameter theorem on rcd k infty spaces with a reasonable condition of the heat distributions the self improving property of the bakry emery condition proven by savare plays a key role in the proof

convergence of pointed non compact metric measure spaces and stability of ricci curvature bounds and heat flows aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition we propose various definitions show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured gromov hausdorff convergence then we show that the curvature conditions cd k and rcd k are stable under this notion of convergence and that the heat flow passes to the limit as well both in the wasserstein and in the l framework we also prove the variational convergence of cheeger energies in the naturally adapted mosco sense and the convergence of the spectra of the laplacian in the case of spaces either uniformly bounded or satisfying the rcd k condition with k 0 when applied to riemannian manifolds our results allow for sequences with diverging dimensions

nonlinear mixed effects models for repeated measures data we propose a general nonlinear mixed effects model for repeated measures data and define estimators for its parameters the proposed estimators are a natural combination of least squares estimators for nonlinear fixed effects models and maximum likelihood or restricted maximum likelihood estimators for linear mixed effects models we implement newton raphson estimation using previously developed computational methods for nonlinear fixed effects models and for linear mixed effects models two examples are presented and the connections between this work and recent work on generalized linear mixed effects models are discussed