cohomogeneity one manifolds with positive ricci curvature one of the central problems in riemannian geometry is to determine how large the classes of manifolds with positive nonnegative sectional ricci or scalar curvature are see gr for scalar curvature the situation is fairly well understood by comparison special surgery constructions as in sy wr and bundle constructions as in na have resulted in a large number of interesting manifolds with positive ricci curvature so far the only known obstructions to have positive ricci curvature come from obstructions to have positive scalar curvature see li and rs and from the classical bonnet myers theorem which implies that a closed manifold with positive ricci curvature must have finite fundamental group it is well known that among homogeneous manifolds g h this is also a sufficient condition see e g the proof of corollary 3 5 or br in this paper we prove that this is true as well when the manifold admits an action by a compact lie group g with orbits of codimension one

generalized besicovitch spaces and applications to deterministic homogenization the purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems the starting point is the theory of algebras with mean value very often in physics from very simple experimental data one gets complicated structure phenomena these phenomena are represented by functions which are permanent in mean but complicated in detail in addition the functions are subject to the verification of a functional equation which in general is nonlinear the problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties they are functions that represent a phenomenon on a large scale and which vary irregularly undergoing nonperiodic oscillations on a fine scale in this work we study the qualitative properties of spaces of such functions which we call generalized besicovitch spaces and we prove general compactness results related to these spaces we then apply these results in order to study some new homogenization problems one important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo monotone parabolic type operator we also give the answer to the question raised by frid and silva in their paper 35 h frid j silva homogenization of nonlinear pde s in the fourier stieltjes algebras siam j math anal 41 4 2009 1589 1620 as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the fourier stieltjes algebra

fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix geostatistical simulations often require the generation of numerous realizations of a stationary gaussian process over a regularly meshed sample grid omega this paper shows that for many important correlation functions in geostatistics realizations of the associated process over m 1 equispaced points on a line can be produced at the cost of an initial fft of length 2m with each new realization requiring an additional fft of the same length in particular the paper first notes that if an m 1 times m 1 toeplitz correlation matrix r can be embedded in a nonnegative definite 2m times2m circulant matrix s exact realizations of the normal multivariate y sim cal n 0 r can be generated via ffts of length 2m theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which m m is nonnegative definite extensions to simulations of stationary fields in higher dimensions are also provided and illustrated

probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation we present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data for which uncertainties are taken into account by using the nonparametric probabilistic approach such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters consequently the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible two formulations are proposed the first one is the mean square method for which a usual differentiable objective function and an unusual non differentiable objective function are proposed the second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics

isoparametric functions on exotic spheres abstract this paper extends widely the work in 11 existence and non existence results of isoparametric functions on exotic spheres and eells kuiper projective planes are established in particular every homotopy n sphere n 4 carries an isoparametric function with certain metric with 2 points as the focal set in strong contrast to the classification of cohomogeneity one actions on homotopy spheres 26 only exotic kervaire spheres admit cohomogeneity one actions besides the standard spheres as an application we improve a beautiful result of berard bergery 2 see also pp 234 235 of 3

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

non gaussian positive definite matrix valued random fields for elliptic stochastic partial differential operators this paper deals with the construction of a class of non gaussian positive definite matrix valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled the properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given such a matrix valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters the class of random fields which can be approximated is presented and their experimental identification is analyzed an example is given in three dimensional linear elasticity for which the fourth order elasticity tensor valued random field is constructed for a random non homogeneous anisotropic elastic material

ito sde based generator for a class of non gaussian vector valued random fields in uncertainty quantification this paper is concerned with the derivation of a generic sampling technique for a class of non gaussian vector valued random fields such an issue typically arises in uncertainty quantification for complex systems where the input coefficients associated with the elliptic operators must be identified by solving statistical inverse problems specifically we consider the case of non gaussian random fields with values in some arbitrary bounded or semi bounded subsets of r n the approach involves two main features the first one is the construction of a family of random fields converging at a user controlled rate towards the target random field each of these auxialiary random fields can be subsequently simulated by solving a family of ito stochastic differential equations the second ingredient is the definition of an adaptive discretization algorithm the latter allows refining the integration step on the fly and prevents the scheme from diverging the proposed strategy is finally exemplified on three examples each of which serving as a benchmark either for the adaptivity procedure or for the convergence of the diffusions

on asymptotic properties of maximum likelihood estimators for parabolic stochastic pde s we investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic spdes of the form du t x a 0 theta a 1 u t x dt dw t x wherea 0 anda 1 are partial differential operators andw is a cylindrical brownian motion we introduce a spectral method for computing mles based on finite dimensional approximations to solutions of such systems and establish criteria for consistency asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity we derive the asymptotic properties of the mle from a condition on the order of the operators in particular the mle is consistent if and only if ord a 1 1 2 ord a 0 a 1 d

homogenization of hamilton jacobi bellman equations with respect to time space shifts in a stationary ergodic medium we consider a family fu t x g 0 of solutions to the equation u t c u 2ch t x ru d0 with the terminal datau t x du x assuming that the dependence of the hamiltonian h t x p on time and space is realized through shifts in a stationary ergodic random medium and that h is convex inp and satisfies certain growth and regularity conditions we show the almost sure locally uniform convergence in time and space of u t x as 0 to the solution u t x of a deterministic averaged equation u t c h ru d 0 u t x d u x the effective hamiltonian h is given by a variational formula

probabilistic equivalence and stochastic model reduction in multiscale analysis this paper presents a probabilistic upscaling of mechanics models a reduced order probabilistic model is constructed as a coarse scale representation of a specified fine scale model whose probabilistic structure can be accurately determined equivalence of the fine and coarse scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained a significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine and coarse scale representations the upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations an illustration in stochastic structural dynamics is provided to demonstrate the proposed framework

kernel density estimation via diffusion we present a new adaptive kernel density estimator based on linear diffusion processes the proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate in addition we propose a new plug in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods we present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability

isoparametric functions on riemannian manifolds i on etablit les proprietes de base des fonctions transnormales et on demontre le lemme du tube on demontre que soit m une variete de riemann lisse complete connexe et f une fonction transnormale sur m alors a les varietes focales de f sont des sous varietes lisses de m b chaque ensemble de niveau regulier de f est un tube sur d autres des varietes focales

ito sde mcmc method for bayesian characterization of errors associated with data limitations in stochastic expansion methods for uncertainty quantification abstract this paper is concerned with the characterization and the propagation of errors associated with data limitations in polynomial chaos based stochastic methods for uncertainty quantification such an issue can arise in uncertainty quantification when only a limited amount of data is available when the available information does not suffice to accurately determine the probability distributions that must be assigned to the uncertain variables the bayesian method for assigning these probability distributions becomes attractive because it allows the stochastic model to account explicitly for insufficiency of the available information in previous work such applications of the bayesian method had already been implemented by using the metropolis hastings and gibbs markov chain monte carlo mcmc methods in this paper we present an alternative implementation which uses an alternative mcmc method built around an ito stochastic differential equation sde that is ergodic for the bayesian posterior we draw together from the mathematics literature a number of formal properties of this ito sde that lend support to its use in the implementation of the bayesian method and we describe its discretization including the choice of the free parameters by using the implicit euler method we demonstrate the proposed methodology on a problem of uncertainty quantification in a complex nonlinear engineering application relevant to metal forming

submanifold geometry in symmetric spaces the classical local invariants of a submanifold in a space form are the first fundamental form the shape operators and the induced normal connection and they determine the submanifold up to ambient isometry one of the main topics in differential geometry is to study the relation between the local invariants and the global geometry and topology of submanifolds many remarkable results have been developed for submanifolds in space forms whose local invariants satisfy certain natural conditions the study of focal points of a submanifold in an arbitrary riemannian manifold arises from the morse theory of the energy functional on the space of paths in the riemannian manifold joining a fixed point to the submanifold the morse index theorem relates the geometry of a submanifold to the topology of this path space the focal structure is intimately related to the local invariants of the submanifold in the case of space forms one can go backwards and reconstruct the local invariants from the focal structure so it is not too surprising that most of the structure theory of submanifolds can be reformulated in terms of their focal structure what is perhaps surprising is a fact that became increasingly evident to the authors from their individual and joint research over the past decade while extending the theory of submanifolds to ambient spaces more general than space forms proves quite difficult if one tries to use the same approach as for the space forms at least for symmetric spaces it has proved possible to develop an elegant theory based on focal structure that reduces to the classical theory in the case of space forms this paper is an extended report on this theory and the authors believe that the methods developed herein provide important tools for a continuing study of the submanifold geometry in symmetric spaces first we set up some notations let n g be a riemannian manifold m an immersed submanifold of n and m the normal bundle of m the end point map m n of m is the restriction of the exponential map exp to m if v m x is a singular point of and the dimension of the kernel of d v is m then v is called a multiplicity m focal normal and exp v is called a multiplicity m focal point of m with respect to m in n the focal data m is defined to be the set of all pairs v m such that v is a multiplicity m focal normal of m the focal variety v m is the set of all pairs v m with v m m the main purpose of our paper is to study the global geometry and topology

explicit time reversible adaptive step size control adaptive step size control is difficult to combine with geometric numerical integration as classical step size control is based on past information only time symmetry is destroyed and with it the qualitative properties of the method in this paper we develop completely explicit reversible symmetry preserving adaptive step size selection algorithms for geometric numerical integrators such as the stormer verlet method a new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory for integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and in particular the total energy for hamiltonian systems it has the same excellent long term behavior as that obtained when constant steps are used with variable steps however both accuracy and efficiency are greatly improved

η invariant and a problem of berard bergery on the existence of closed geodesics we use the invariant of atiyah patodi singer to compute the eells kuiper invariant for the eells kuiper quaternionic projective plane by combining with a known result of berard bergery it shows that every eells kuiper quaternionic projective plane carries a riemannian metric such that all geodesics passing through a certain point are simply closed and of the same length

reiterated ergodic algebras and applications we redefine the homogenization algebras without requiring the separability assumption we show that this enables one to treat more complicated homogenization problems than those solved by the previous theory in particular we exhibit an example of algebra which contrary to the algebra of almost periodic functions induces no homogenization algebra we prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized reynolds type equations and to some nonlinear hyperbolic equations

identification of bayesian posteriors for coefficients of chaos expansions this article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data the data used for the identification consist of measurements of several realizations of the uncertain quantities that must be characterized the random variables and fields are approximated by a polynomial chaos expansion and the coefficients of this expansion are viewed as unknown parameters to be identified it is shown how the bayesian paradigm can be applied to formulate and solve the inverse problem the estimated polynomial chaos coefficients are hereby themselves characterized as random variables whose probability density function is the bayesian posterior this allows to quantify the impact of missing experimental information on the accuracy of the identified coefficients as well as on subsequent predictions an illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed methodology

on the classification of isoparametric hypersurfaces with four distinct curvatures in spheres in this paper we give a new proof for the classification result in 3 we show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of clifford type provided that the multiplicities m 1 m 2 of the principal curvatures satisfy m 2 2m 1 1 this inequality is satisfied for all but five possible pairs m 1 m 2 with m 1 m 2 our proof implies that for m 1 m 2 1 1 the clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher dimensional of the two focal manifolds for the remaining five possible pairs m 1 m 2 with m 1 m 2 see 13 1 and 15 this stronger form of our result is incorrect for the three pairs 3 4 6 9 and 7 8 there are examples of clifford type such that the associated quadratic forms necessarily vanish on the lower dimensional of the two focal manifolds and for the two pairs 2 2 and 4 5 there exist homogeneous examples that are not of clifford type cf 5 4 3 4 4

a valid matern class of cross covariance functions for multivariate random fields with any number of components we introduce a valid parametric family of cross covariance functions for multivariate spatial random fields where each component has a covariance function from a well celebrated matern class unlike previous attempts our model indeed allows for various smoothnesses and rates of correlation decay for any number of vector components we present the conditions on the parameter space that result in valid models with varying degrees of complexity we discuss practical implementations including reparameterizations to reflect the conditions on the parameter space and an iterative algorithm to increase the computational efficiency we perform various monte carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging the application of the proposed multivariate matern model is illustrated on two meteorological datasets temperature pressure over the pacific northwest bivariate and wind temperature pressure in oklahoma trivariate in the latter case our flexi

functional approximation and projection of stored energy functions in computational homogenization of hyperelastic materials a probabilistic perspective abstract this work is concerned with the construction of a surrogate model for the homogenized stored energy functions defining the effective behavior of nonlinear elastic microstructures here a probabilistic standpoint is adopted and allows for the definition of a nonlinear mapping between the macroscopic deformations and the homogenized potential this functional approximation is specifically obtained by means of a polynomial chaos expansion the coefficients of which are computed through a gauss legendre quadrature rule by invoking well known results related to projections in hilbert spaces closest approximations of arbitrary potentials to standard e g ogden type models are subsequently defined and characterized by appropriate residuals numerical illustrations on various microstructures are finally provided in order to discuss the relevance of the proposed framework in particular it is shown that the surrogate model compares very well with reference results at both the macroscale and the structural scale even for moderate order expansions and that the aforementioned closest approximations constitute very accurate approximation provided that the subset onto which the homogenized response is projected is properly chosen this result readily allows for nonconcurrent coupling using standard constitutive models available in commercial codes and therefore makes the approach very attractive for engineering applications

generalized stochastic approach for constitutive equation in linear elasticity a random matrix model this work is concerned with the construction of stochastic models for random elasticity matrices allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely integrating the statistical dependence between the random stiffness components or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations to this aim we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components having recourse to the maxent principle strategies for random generation and estimation were further reviewed and the approach was exemplified in the case of a material that was transversely isotropic almost surely in a second stage we made use of such derivations to propose a generalized model for random elasticity matrices that took into accountconstraints on both the level of stochastic anisotropy and the level of statistical fluctuations

ricci curvature of double manifolds via isoparametric foliations abstract given a closed manifold m and a vector bundle of rank n over m by gluing two copies of the disc bundle of we can obtain a closed manifold d m the so called double manifold in this paper we firstly prove that each sphere bundle s r of radius r 0 is an isoparametric hypersurface in the total space of equipped with a connection metric and for r 0 small enough the induced metric of s r has positive ricci curvature under the additional assumptions that m has a metric with positive ricci curvature and n 3 as an application if m admits a metric with positive ricci curvature and n 2 then we construct a metric with positive ricci curvature on d m moreover under the same metric d m admits a natural isoparametric foliation for a compact minimal isoparametric hypersurface y n in s n 1 1 which separates s n 1 1 into s n 1 and s n 1 one can get double manifolds d s n 1 and d s n 1 inspired by tang xie and yan s work on scalar curvature of such manifolds with isoparametric foliations cf 25 we study ricci curvature of them with isoparametric foliations in the last part

isoparametric foliation and yau conjecture on the first eigenvalue a well known conjecture of yau states that the first eigenvalue of every closed minimal hypersurface m n in the unit sphere s n 1 1 is just its dimension n the present paper shows that yau conjecture is true for minimal isoparametric hypersurfaces moreover the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non stable range

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

construction of probability distributions in high dimension using the maximum entropy principle applications to stochastic processes random fields and random matrices the construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension this paper deals with the effective construction of the probability distribution in high dimension of a vector valued random variable using the maximum entropy principle the integrals in high dimension are then calculated in constructing the stationary solution of an ito stochastic differential equation associated with its invariant measure a random generator of independent realizations is explicitly constructed in this paper three fundamental applications are presented the first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non stationary random time series random accelerograms for which the velocity response spectrum is given the second one is also a new formulation related to the construction of the probability distribution of positive definite band random matrices finally we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space the third application is then a new formulation related to the construction of the probability distribution of the karhunen loeve expansion of non gaussian positive valued random fields

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

gradient map of isoparametric polynomial and its application to ginzburg landau system abstract in this note we study properties of the gradient map of the isoparametric polynomial for a given isoparametric hypersurface in sphere we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications we find that it should map not only the focal submanifolds to focal submanifolds isoparametric hypersurfaces to isoparametric hypersurfaces but also map isoparametric hypersurfaces to focal submanifolds in particular it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface as an immediate consequence we get the brouwer degree of the gradient map which was firstly obtained by peng and tang with moving frame method following farina s construction another immediate consequence is a counterexample of the brezis question about the symmetry for the ginzburg landau system in dimension 6 which gives a partial answer toward the open problem 2 raised by farina

isoparametric hypersurfaces with g m 6 2 we prove that isoparametric hypersurfaces with g m 6 2 are homogeneous which answers dorfmeister neher s conjecture armatively and solves yau s problem in the case g 6

isoparametric foliations a problem of eells lemaire and conjectures of leung in this paper two sequences of minimal isoparametric hypersurfaces are constructed via representations of clifford algebras based on these we give estimates on eigenvalues of the laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres this improves results of z z tang and w j yan isoparametric foliation and yau conjecture on the first eigenvalue j differential geom 94 2013 521 540 z z tang y q xie and w j yan isoparametric foliation and yau conjecture on the first eigenvalue ii j funct anal 266 2014 6174 6199 eells and lemaire selected topics in harmonic maps c b m s regional conference series in mathematics 50 american mathematical society providence ri 1983 posed a problem to characterize the compact riemannian manifold m for which there is an eigenmap from m to sn as another application of our constructions the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres most importantly by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres we provide infinitely many counterexamples to two conjectures of leung minimal submanifolds in a sphere ii bull london math soc 23 1991 387 390 posed in 1991 on minimal submanifolds in unit spheres note that these conjectures of leung have been proved in the case that the normal connection is flat t hasanis and t vlachos ricci curvatures and minimal submanifolds pacific j math 197 2001 13 24

non gaussian positive definite matrix valued random fields with constrained eigenvalues application to random elasticity tensors with uncertain material symmetries this paper is devoted to the construction of a class of prior stochastic models for non gaussian positive definite matrix valued random fields the proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework having recourse to a particular characterization of material symmetry classes we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties as well as for the development of computational multi scale approaches where the randomness induced by fine scale features may be taken into account we further present a possible strategy for inverse identification relying on the sequential solving of least square optimization problems an application is finally provided copyright c 2011 john wiley sons ltd

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

isoparametric hypersurfaces with four principal curvatures let m be an isoparametric hypersurface in the sphere s n with four distinct principal curvatures munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 m 2 and stolz showed that the pair m1 m 2 must either be 2 2 4 5 or be equal to the multiplicities of an isoparametric hypersurface of fkm type constructed by ferus karcher and munzner from orthogonal representations of clifford algebras in this paper we prove that if the multiplicities satisfy m2 2m1 1 then the isoparametric hypersurface m must be of fkm type together with known results of takagi for the case m1 1 and ozeki and takeuchi for m1 2 this handles all possible pairs of multiplicities except for four cases for which the classification problem remains open

a probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures in this paper we address the construction of a prior stochastic model for non gaussian deterministically bounded positive definite matrix valued random fields in the context of mesoscale modeling of heterogeneous elastic microstructures we first introduce the micromechanical framework and recall in particular huet s partition theorem based on the latter we discuss the nature of hierarchical bounds and define under some given assumptions deterministic bounds for the apparent elasticity tensor having recourse to the maximum entropy principle under the constraints defined by the available information we then introduce two random matrix models it is shown that an alternative formulation of the boundedness constraints further allows constructing a probabilistic model for deterministically bounded positive definite matrix valued random fields such a construction is presented and relies on a class of random fields previously defined we finally exemplify the overall methodology considering an experimental database obtained from ebsd measurements and provide a simple numerical application

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions a microscopic heterogeneous system under random influence is considered the randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium this system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes obstacles or heterogeneities together with random dynamical boundary conditions on the boundaries of these small holes

multiscale young measures in almost periodic homogenization and applications we prove the existence of multiscale young measures associated with almost periodic homogenization we give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure such as scalar conservation laws nonlinear transport equations hamilton jacobi equations and fully nonlinear elliptic equations motivated by the application in nonlinear transport equations we also prove basic results on flows generated by lipschitz almost periodic vector fields which are of interest in their own in our analysis an important role is played by the so called bohr compactification mathbb g n of mathbb r n this is a natural parameter space for the young measures our homogenization results provide also the asymptotic behavior for the whole set of mathbb g n translates of the solutions which is in the spirit of recent studies on the homogenization of stationary ergodic processes

a computational inverse method for identification of non gaussian random fields using the bayesian approach in very high dimension this paper is devoted to the identification of bayesian posteriors for the random coefficients of the high dimension polynomial chaos expansions of non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem must be solved to perform the identification of the random field a complete methodology is proposed to solve this very challenging problem in high dimension which consists in using the first four steps introduced in a previous paper followed by the identification of the posterior model the steps of the methodology are the following 1 introduction of a family of prior algebraic stochastic model pasm 2 identification of an optimal pasm in the constructed family using the partial experimental data 3 construction of a statistical reduced order optimal pasm 4 construction in high dimension of the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal pasm 5 substitution of these deterministic vector valued coefficients by random vector valued coefficients in order to extend the capability of the polynomial chaos expansion to represent the experimental data and for which the joint probability distribution must be identified 6 construction of the prior probability model of these random vector valued coefficients and finally 7 identification of the posterior probability model of these random vector valued coefficients using partial and limited experimental data through the stochastic boundary value problem two methods are proposed to carry out the identification of the posterior model the first one is based on the use of the classical bayesian method the second one is a new approach derived from the bayesian method which is more efficient in high dimension an application is presented for which several millions of random coefficients are identified

weakly almost periodic functions and fourier stieltjes algebras of locally compact groups a noncompact locally compact group g is called an eberlein group if w g b g where w g is the algebra of continuous weakly almost periodic functions on g and b g is the uniform closure of the fourier stieltjes algebra of g we show that if g is a noncompact in group or a noncompact nilpotent group then w g b g contains a linear isometric copy of l in particular g is not an eberlein group on the other hand finite direct products of eucidean motion groups and by a result of w veech noncompact semisimple analytic groups with finite centers are eberlein groups 1 introduction let g be a locally compact group c g the space of bounded continuous complex valued functions on g with the sup norm w g the space of continuous weakly almost periodic w a p functions on g b g the fourier stieltjes algebra of g and b g its uniform closure in c g it is known that b g c w g and if g is compact then c g w g b g therefore in this paper g is assumed to be noncompact in his efforts to characterize w a p functions w f eberlein first raised the question whether for abelian g b g w g w rudin 24 made the initial contribution to this question by showing that b g c w g if g is abelian and contains a discrete subgroup which is not of bounded order and d ramirez 23 by studying discrete abelian groups with bounded orders together with the structure theorem for locally compact abelian groups was able to show that rudin s conclusion holds for all abelian groups see the monographs 4 and 11 for complete accounts of this result in this paper we will extend rudin ramirez result to include many nonabelian groups our main result is the following

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

exotic spheres and curvature since their discovery by milnor in 1956 exotic spheres have provided a fascinating object of study for geometers in this article we survey what is known about the curvature of exotic spheres

differentiable classification of 4 manifolds with singular riemannian foliations in this paper we first prove that any closed simply connected 4 manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4 manifolds mathbb s 4 mathbb cp 2 mathbb s 2 times mathbb s 2 or mathbb cp 2 pm mathbb cp 2 as an application we prove that any closed simply connected 4 manifold admitting a nontrivial singular riemannian foliation is diffeomorphic to a connected sum of copies of standard mathbb s 4 pm mathbb cp 2 and mathbb s 2 times mathbb s 2 a classification of singular riemannian foliations of codimension 1 on all closed simply connected 4 manifolds is obtained as a byproduct in particular there are exactly 3 non homogeneous singular riemannian foliations of codimension 1 complementing the list of cohomogeneity one 4 manifolds

isoparametric foliations diffeomorphism groups and exotic smooth structures abstract in this paper we are concerned with interactions between isoparametric theory and differential topology two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves using differential topology we obtain several results towards the classification problem of isoparametric foliations up to equivalence in particular we show that each homotopy n sphere has the same isoparametric foliations as the standard sphere s n has except for n 4 reducing the classification problem on homotopy spheres to that on the standard sphere moreover we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere s n except for n 5 besides we show that the uniqueness holds on s 5 if and only if 0 diff s 4 z 2 i e pseudo isotopy implies isotopy for diffeomorphisms on s 4 at last some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds

magnetohydrodynamic turbulent flows existence results abstract in this paper we provide a detailed investigation of the problem of existence of solutions to the stochastic magnetohydrodynamic equations which describe the motion of turbulent particles of fluids in a magnetic field the results are proved through a delicate combination of galerkin approximation scheme with some deep compactness results of analytic and probabilistic nature

weak solutions to stochastic 3d navier stokes α model of turbulence α asymptotic behavior abstract we study the asymptotic behavior of weak solutions to the stochastic 3d navier stokes model as approaches zero the main result provides a new construction of the weak solutions of stochastic 3d navier stokes equations as approximations by sequences of solutions of the stochastic 3d navier stokes model

optimal approximation of stochastic differential equations by adaptive step size control we study the pathwise strong approximation of scalar stochastic differential equations with respect to the global error in the l2 norm for equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving brownian motion as a consequence higher order methods do not exist if the global error is analyzed we introduce an adaptive step size control for the euler scheme which performs asymptotically optimally in particular the new method is more efficient than an equidistant discretization this superiority is confirmed in simulation experiments for equations with additive noise as well as for general scalar equations

isoperimetric rigidity and distributions of 1 lipschitz functions abstract we prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal then the space is foliated by minimal geodesics where the observable variance is defined to be the supremum of the variance of 1 lipschitz functions on the space our result can be considered as a variant of cheeger gromoll s splitting theorem and also of cheng s maximal diameter theorem as an application we obtain a new isometric splitting theorem for a complete weighted riemannian manifold with a positive bakry emery ricci curvature

asymptotic sampling distribution for polynomial chaos representation from data a maximum entropy and fisher information approach a procedure is presented for characterizing the asymptotic sampling distribution of estimators of the polynomial chaos pc coefficients of a second order nonstationary and non gaussian random process by using a collection of observations the random process represents a physical quantity of interest and the observations made over a finite denumerable subset of the indexing set of the random process are considered to form a set of realizations of a random vector mathcal y representing a finite dimensional projection of the random process the karhunen loeve decomposition and a scaling transformation are employed to produce a reduced order model mathcal z of mathcal y the pc expansion of mathcal z is next determined by having recourse to the maximum entropy principle the metropolis hastings markov chain monte carlo algorithm and the rosenblatt transformation the resulting pc expansion has random coefficients where the random characteristics of the pc coefficients can be attributed to the limited data available from the experiment the estimators of the pc coefficients of mathcal y obtained from that of mathcal z are found to be maximum likelihood estimators as well as consistent and asymptotically efficient computation of the covariance matrix of the associated asymptotic normal distribution of estimators of the pc coefficients of mathcal y requires knowledge of the fisher information matrix fim the fim is evaluated here by using a numerical integration scheme as well as a sampling technique the resulting confidence interval on the pc coefficient estimators essentially reflects the effect of incomplete information due to data limitation on the characterization of the stochastic process this asymptotic distribution is significant as its characteristics can be propagated through predictive models for which the stochastic process in question describes uncertainty on some input parameters

the two scale convergence method applied to generalized besicovitch spaces the twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems in the present paper we develop this theory to generalized besicovitch spaces

homogenization of a stochastic nonlinear reaction diffusion equation with a large reaction term the almost periodic framework abstract homogenization of a stochastic nonlinear reaction diffusion equation with a large nonlinear term is considered under a general besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation namely the stochastic nonlinear convection diffusion equation which we explicitly derive in terms of appropriate functionals we study some particular cases such as the periodic framework and many others this is achieved under a suitable generalized concept of convergence for stochastic processes