scalar curvature of a metric with unit volume the problem of finding riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of kazdan and warner in 1970 s 10 and references cited in it in this paper we shall consider the same problem under a constraint on the volume for this purpose it is useful to introduce an invariant p m of a smooth closed manifold m which will be called the yamabe number of m defined as the supremum of m c of all conformal classes c of riemannian metrics on m

boundary penalty finite element methods for blending surfaces ii biharmonic equations abstract in this paper the biharmonic equations are discussed and the boundary penalty finite methods bp fems using piecewise cubic hermite elements are chosen to seek their approximate solutions satisfying the normal derivative and periodical boundary conditions theoretical analysis is made to discover that when the penalty power 2 3 or 4 and 0

large deviations for stochastic pde with levy noise abstract we prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small levy noise we use general large deviations theorems of varadhan and bryc coupled with the techniques of feng and kurtz 2006 15 viscosity solutions of integro partial differential equations in hilbert spaces and deterministic optimal control methods the laplace limit is identified as a viscosity solution of a hamilton jacobi bellman equation of an associated control problem we also establish exponential moment estimates for solutions of stochastic evolution equations driven by levy noise general results are applied to stochastic hyperbolic equations perturbed by subordinated wiener process

aubin s lemma for the yamabe constants of infinite coverings and a positive mass theorem aubin s lemma says that if the yamabe constant of a closed conformal manifold m c is positive then it is strictly less than the yamabe constant of any of its non trivial finite conformal coverings we generalize this lemma to the one for the yamabe constant of any m c of its infinite conformal coverings provided that 1 m has a descending chain of finite index subgroups tending to 1 m moreover if the covering m is normal the limit of the yamabe constants of the finite conformal coverings associated to the descending chain is equal to that of m c for the proof of this we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

smooth yamabe invariant and surgery we prove a surgery formula for the smooth yamabe invariant sigma m of a compact manifold m assume that n is obtained from m by surgery of codimension at least 3 we prove the existence of a positive number lambda n depending only on the dimension n of m such that sigma n geq min sigma m lambda n

linear bounds for stochastic dispersion it has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid we define a stochastic flow by the equations begin eqnarray phi 0 x x d phi t x f dt phi t x end eqnarray where f t x is a field of semimartingales on x in mathbb r d for d geq 2 whose local characteristics are bounded and lipschitz the particles are points in a bounded set mathscr x and we ask how far the substance has spread in a time t that is we define phi t sup x in mathscr x sup 0 leq t leq t phi t x and seek to bound p phi t z without drift when f cdot x are required to be martingales although single points move on the order of sqrt t it is easy to construct examples in which the supremum phi t still grows linearly in time that is lim inf t to infty phi t t 0 almost surely we show that this is an upper bound for the growth that is we compute a finite constant k 0 depending on the bounds for the local characteristics such that limsup t to infty frac phi t t leq k 0 text almost surely a linear bound on growth holds even when the field itself includes a drift term

a limit shape theorem for periodic stochastic dispersion we consider the evolution of a connected set on the plane carried by a space periodic incompressible stochastic flow while for almost every realization of the stochastic flow at time t most of the particles are at a distance of order t away from the origin there is a measure zero set of points that escape to infinity at the linear rate we study the set of points visited by the original set by time t and show that such a set when scaled down by the factor of t has a limiting nonrandom shape c 2004 wiley periodicals inc

boundary penalty techniques we examine techniques for enforcing dirichlet boundary data in finite element solution of boundary value problems the common programming strategy of adding a large number to certain diagonal entries and scaling the right side is compared with penalty methods by selectively underintegrating the continuous penalty formulation and selecting the penalty parameter e 1 optimally as a function of mesh size h we deduce the large number strategy and hence show that the optimal rate is preserved numerical experiments to study the rates for continuous penalty underintegrated penalty and large number strategies corroborate the argument

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

moduli spaces of critical riemannian metrics in dimension four abstract we obtain a compactness result for various classes of riemannian metrics in dimension four in particular our method applies to anti self dual metrics kahler metrics with constant scalar curvature and metrics with harmonic curvature with certain geometric non collapsing assumptions the moduli space can be compactified by adding metrics with orbifold like singularities similar results were obtained for einstein metrics in j amer math soc 2 3 1989 455 invent math 97 2 1989 313 invent math 101 1 1990 101 but our analysis differs substantially from the einstein case in that we do not assume any pointwise ricci curvature bound

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

sample path properties of the stochastic flows we consider a stochastic flow driven by a finite dimensional brownian motion we show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem the proof uses new estimates of the mixing rates of the multi point motion

isotropic stochastic flows on considere les flots stochastiques isotropes dans un espace euclidien de d dimensions d 2 on relie la tendance des distances entre deux points et des vecteurs tangents a se borner ou s etendre a la dimension et a la proportion du flot solenoidal ou potentiel

large deviations and stochastic flows of diffeomorphisms previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold m are extended and then applied to the analysis of the lyapunov exponents of a stochastic flow of diffeomorphisms of m an approximation argument relates these results to the behavior near the diagonal in m2 of the associated two point motion finally it is shown under appropriate non degeneracy conditions that the two point motion is ergodic on m2 if the top lyapunov exponent is positive

classification of prime 3 manifolds with σ invariant greater than rp 3 in this paper we compute the invariants sometimes also called the smooth yamabe invariants of rp 3 and rp 2 s 1 which are equal and show that the only prime 3 manifolds with larger invariants are s 3 s 2 s 1 and s 2 s 1 the nonorientable s 2 bundle over s 1 more generally we show that any 3 manifold with invariant greater than rp 3 is either s 3 a connect sum with an s 2 bundle over s 1 or has more than one nonorientable prime compo

partially penalized immersed finite element methods for elliptic interface problems this article presents new immersed finite element ife methods for solving the popular second order elliptic interface problems on structured cartesian meshes even if the involved interfaces have nontrivial geometries these ife methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in ife functions with the enhanced stability due to the added penalty not only can these ife methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity but also numerical results indicate that their convergence rates in both the h 1 norm and the l 2 norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic ife methods in some situations trace inequalities are established for both linear and bilinear ife functions that are not only critical for the error analysis of these new ife methods but are also of a great potential to be useful in error analysis for other

hausdorff dimension in stochastic dispersion we consider the evolution of a connected set in euclidean space carried by a periodic incompressible stochastic flow while for almost every realization of the random flow at time t most of the particles are at a distance of order sqrt t away from the origin there is an uncountable set of measure zero of points which escape to infinity at the linear rate in this paper we prove that this set of linear escape points has full hausdorff dimension

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

towards optimal finite element error estimates for the penalized dirichlet problem in a domain with curved boundary we consider the finite element approximation of an elliptic problem with homogeneous dirichlet boundary conditions on a curved boundary and imposed using the penalty method we establish optimal h 1 error estimates with suitable assumptions on the penalty parameter e as a function of the elements size h our focus is on establishing these results with least restrictive assumptions possible on this dependency

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

asymptotic support theorem for planar isotropic brownian flows it has been shown by various authors that the diameter of a given nontrivial bounded connected set x grows linearly in time under the action of an isotropic brownian flow ibf which has a nonnegative top lyapunov exponent in case of a planar ibf with a positive top lyapunov exponent the precise deterministic linear growth rate k of the diameter is known to exist in this paper we will extend this result to an asymptotic support theorem for the time scaled trajectories of a planar ibf which has a positive top lyapunov exponent starting in a nontrivial compact connected set x r2 that is we will show convergence in probability of the set of time scaled trajectories in the hausdorff distance to the set of lipschitz continuous functions on 0 1 starting in 0 with lipschitz constant k

almost sure enhanced dissipation and uniform in diffusivity exponential mixing for advection diffusion by stochastic navier stokes we study the mixing and dissipation properties of the advection diffusion equation with diffusivity 0 kappa ll 1 and advection by a class of random velocity fields on mathbb t d d 2 3 including solutions of the 2d navier stokes equations forced by sufficiently regular in space non degenerate white in time noise we prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity kappa namely that there is a deterministic exponential rate independent of kappa such that all mean zero h 1 initial data decays exponentially fast in h 1 at this rate with probability one this implies almost sure enhanced dissipation in l 2 specifically that there is a deterministic uniform in kappa exponential decay in l 2 after time t rsim left log kappa right both the mathcal o left log kappa right time scale and the uniform in kappa exponential mixing are optimal for lipschitz velocity fields this work is also a major step in our program on scalar mixing and lagrangian chaos necessary for a rigorous proof of the batchelor power spectrum of passive scalar turbulence

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

stabilization by noise for a class of stochastic reaction diffusion equations we prove uniqueness ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction diffusion equations with multiplicative noise in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable to this purpose we show that the l1 norm of the difference of two solutions starting from any two different initial data converges a s to zero as time goes to infinity

on isotropic brownian motions this paper includes a detailed study of isotropic brownian motion on matrices and brownian flows associated with isotropic gaussian fields characteristic exponents stability asymptotic behaviour statistical equilibrium are the topics on which the main results are obtained

enhanced functional evaluation for the finite element penalty method abstract we analyze the impact of using the penalty method on the estimation of a prescribed solution functional or quantity of interest qoi specifically we consider the use of penalty methods to enforce dirichlet boundary constraints focusing our attention on boundary fluxes as qois we propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter we show that the new estimator reduces the error arising from the use of the penalty method from o to o a well posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method errors in the enhanced flux estimator are then controlled using adjoint based techniques several numerical experiments are presented to demonstrate that the enhanced estimator in combination with adjoint error estimation allows one to efficiently control both the discretization and penalty errors in target qois

lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms the lyapunov exponents 1 2 d for a stochastic flow of diffeomorphisms of a d dimensional manifold m with a strongly recurrent one point motion describe the almost sure limiting exponential growth rates of tangent vectors under the flow this paper shows how the lyapunov exponents are related to measure preserving properties of the stochastic flow on m and of the induced stochastic flow on the projective bundle pm relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow the results include the following if m is compact and if the one point motion on m is a non degenerate diffusion with stationary probability measure then 1 d 0 with equality if and only if the flow preserves almost surely if in addition the induced one point motion on pm satisfies a weak non degeneracy condition then 1 d if and only if there is a smooth riemannian structure on m with respect to which the flow is conformal almost surely

eulerian shape design sensitivity analysis and optimization with a fixed grid conventional shape optimization based on the finite element method uses lagrangian representation in which the finite element mesh moves according to shape change while modern topology optimization uses eulerian representation in this paper an approach to shape optimization using eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed a continuum geometric model is defined on the fixed grid of finite elements an active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization in which each element has a unique shape density the shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements using this transformation it has been shown that the shape design problem can be treated as a parameter design problem which is a much easier method than the former a detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary the accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement two design optimization problems are presented to show the feasibility of the proposed design approach

chasing balls through martingale fields we consider the way sets are dispersed by the action of stochastic flows derived from martingale fields under fairly general continuity and ellipticity conditions the following dichotomy result is shown any nontrivial connected set x either contracts to a point under the action of the flow or its diameter grows linearly in time with speed at least a positive deterministic constant a the linear growth may further be identified again almost surely with a much stronger behavior which we call ball chasing if is any path with lipschitz constant smaller than a the ball of radius e around t contains points of the image of x for an asymptotically positive fraction of times t if the ball grows as the logarithm of time there are individual points in x whose images eventually remain in the ball

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

derivative calculation from finite element solutions abstract a technique is considered whereby very accurate derivatives of a finite element solution can be calculated efficiently it is demonstrated here that most of the necessary quantities for this subsidiary computation are available as computed by products in the preceding finite element solution procedure the calculation is shown in this note to be a particular form of a procedure for which superconvergent theoretical error estimates have been proven elsewhere numerical experiments confirm the superior accuracy in the computed derivative stress or flux

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

continuity properties of entropy on donne une borne superieure sur le defaut en semicontinuite superieure des entropies topologiques et metriques d une auto application c r d une variete c r r 1

computational model for 3 d contact problems with friction based on the penalty method the friction forces are assumed to follow the coulomb law with a slip criterion treated in the context of a standard return mapping algorithm consistent linearization of the field equations is performed which leads to a fully implicit scheme with non symmetric tangent stiffness which preserves asymptotic quadratic convergence of the newton raphson method

a deformation of flat conformal structures we consider deformations of flat conformal structures from a viewpoint of connected sum decomposition of conformally flat manifolds

finite element methods for constrained problems in elasticity several different variational formulations of boundary value problems with constraints are discussed with particular reference to constrained problems in elasticity special attention is given to exterior penalty methods a discussion of the conditions necessary for penalty methods to provide a basis for stable and convergent finite element methods is given in particular the use of reduced integration is discussed and criteria on the order of reduced integration rules sufficient to produce stable and convergent schemes are described applications of reduced integration penalty methods to incompressible elasticity problems and contact problems are described

universal mixers in all dimensions abstract we construct universal mixers incompressible flows that mix arbitrarily well general solutions to the corresponding transport equation in all dimensions this mixing is exponential in time i e essentially optimal for any initial condition with at least some regularity and we also show that a uniform mixing rate for all initial conditions cannot be achieved the flows are time periodic and uniformly in time bounded in spaces w s p for a range of s p that includes points with s 1 and p 2

finite element approximation of the dirichlet problem using the boundary penalty method summary this paper considers a finite element approximation of the dirichlet problem for a second order self adjoint elliptic equation a u f in a region f2cn n 2 or 3 by the boundary penalty method if the finite element space defined over d h a union of elements has approximation power h in the l 2 norm then i for f2 d h convex polyhedral we show that choosing the penalty parameter e h x with 2 k yields optimal h 1 and l 2 error bounds if u hk i q

adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives an anisotropic error estimator involving only first order derivatives is proposed for the laplace problem and continuous piecewise linear finite elements upper and lower bounds are presented the involved constants being independent of the mesh aspect ratio provided the error gradient is equidistributed in the directions of maximum and minimum stretching an anisotropic adaptive algorithm is then proposed with aim to equidistribute the error gradient in the directions of maximum and minimum stretching numerical results in two and three space dimensions show that the effectivity index is aspect ratio independent on such adapted meshes

penalty finite element methods for the analysis of stokesian flows a study of a class of finite element methods for the analysis of stokes problem based on the use of exterior penalty formulations is described the effects of selective reduced integration i e the use of quadrature rules for integrating the penalty terms which are of lower order than that required to integrate polynomial approximations of these terms exactly are investigated error estimates are derived and the numerical stability of these methods as depicted by a special babuska brezzi condition is explored in some detail the results of several numerical experiments with these methods are also given