on the geometry of sasakian einstein 5 manifolds on simply connected five manifolds sasakian einstein metrics coincide with riemannian metrics admitting real killing spinors which are of great interest as models of near horizon geometry for three brane solutions in superstring theory 24 we expand on the recent work of demailly and kollar 14 and johnson and kollar 20 who give methods for constructing kahler einstein metrics on log del pezzo surfaces by a previous result of the first two authors 9 circle v bundles over log del pezzo surfaces with kahler einstein metrics have sasakian einstein metrics on the total space of the bundle here these simply connected 5 manifolds arise as links of isolated hypersurface singularities which by the well known work of smale 36 together with 11 must be diffeomorphic to s 5 l s 2 s 3 more precisely using methods from mori theory in algebraic geometry we prove the existence of 14 inequivalent sasakian einstein structures on s 2 s 3 and infinite families of such structures on l s 2 s 3 with 2 l 7 we also discuss the moduli problem for these sasakian einstein structures

a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

constructions in sasakian geometry we first generalize the join construction described previously by the first two authors 4 for quasi regular sasakian einstein orbifolds to the general quasi regular sasakian case this allows for the further construction of specific types of sasakian structures that are preserved under the join operation such as positive negative or null sasakian structures as well as sasakian einstein structures in particular we show that there are families of sasakian einstein structures on certain 7 manifolds homeomorphic to s 2 s 5 we next show how the join construction emerges as a special case of lerman s contact fibre bundle construction 32 in particular when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric sasakian manifold finally we study toric sasakian manifolds in dimension 5 and show that any simply connected compact oriented 5 manifold with vanishing torsion admits regular toric sasakian structures this is accomplished by explicitly constructing circle bundles over the equivariant blow ups of hirzebruch surfaces

an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

existence and uniqueness of periodic solutions for a kind of lienard equation with two deviating arguments in this paper a kind of lienard equation with two deviating arguments of the formx t f x t x t g 1 t x email protected 1 t g 2 t x email protected 2 t p t is considered we derive some new sufficient conditions for checking uniqueness of t periodic solutions of this equation by using coincidence degree theory we also establish some new results on the existence and uniqueness of t periodic solutions for this equation which are new and complement previously known results

adaptive time stepping for incompressible flow part i scalar advection diffusion even the simplest advection diffusion problems can exhibit multiple time scales this means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally the performance of the second order trapezoid rule using an explicit adams bashforth method for error control is assessed in this work this combination is particularly well suited to long time integration of advection dominated problems herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations specifically diffusion problems with rough initial data and general advection diffusion problems with different physical time scales governing the system evolution

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

toric geometry sasaki einstein manifolds and a new infinite class of ads cft duals recently an infinite family of explicit sasaki einstein metrics yp q on s2 s3 has been discovered where p and q are two coprime positive integers with q p these give rise to a corresponding family of calabi yau cones which moreover are toric aided by several recent results in toric geometry we show that these are kahler quotients open image in new window namely the vacua of gauged linear sigma models with charges p p p q p q thereby generalising the conifold which is p 1 q 0 we present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold open image in new window for all q p with fixed p we hence find that the yp q manifolds are ads cft dual to an infinite class of open image in new window superconformal field theories arising as ir fixed points of toric quiver gauge theories with gauge group su n 2p as a non trivial example we show that y2 1 is an explicit irregular sasaki einstein metric on the horizon of the complex cone over the first del pezzo surface the dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its ir fixed point using the ads cft correspondence the value we obtain is a quadratic irrational number and remarkably agrees with a recent purely field theoretic calculation using a maximisation

kahler einstein metrics for some quasi smooth log del pezzo surfaces recently johnson and kollar determined the complete list of anti canonically embedded quasi smooth log del pezzo surfaces in weighted projective 3 spaces they also proved that many of those surfaces admit a kahler einstein metric and that some of them do not have tigers the aim of this paper is to settle the question of the existence of kahler einstein metrics and tigers for those surfaces for which the question was left open in order to do so we will use techniques developed earlier by nadel demailly and kollar

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

a new stabilization of adaptive step trapezoid rule based on finite difference interrupts the adaptive step trapezoid rule tr is a generally effective numerical integrator but it is prone to ringing instability and solution stall we introduce a new stabilization algorithm based on finite difference interrupts fdi that reduces ringing and prevents stall unlike previously reported stabilization schemes our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion tr with fdi is at least as stable and accurate as existing methods when solving the prototypical scalar problem we demonstrate that it has better stability accuracy and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids though we demonstrate tr with fdi in the context of a finite element method of lines it is applicable to any tr based algorithm

time integration for ale simulations of fluid structure interaction problems stepsize and order selection based on the bdf abstract we present an adaptive algorithm for time integration of fluid structure integration problems the method relies on a fully coupled procedure to solve fsi problems in which a naturally gcl compliant ale formulation for the finite element spatial discretization is used the main originality of the proposed solution procedure is that time integration is performed using automatic order and stepsize selections hp adaptivity based on the backward differentiation formulas bdf the stepsize selection is based on a local error estimate an error controller and a step rejection mechanism it guarantees that the solution precision is within the user targeted tolerance the order selection is based on a stability test and a quarantine mechanism the selection is performed to ensure that no other methods within the family of 0 stable bdf methods would produce a solution of the targeted precision for a larger stepsize and thus a lower computational time to improve efficiency the time integration procedure also relies on a modified newton method and a predictor the time adaptive algorithm behaviors and performances are assessed on the vortex induced translational and rotational vibrations of a square cylinder and on the wake induced vibrations of 3 cylinders in an in line arrangement the algorithm yields substantial cpu time savings compared to constant stepsize and order integration while delivering solutions of prescribed accuracies

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

segregated runge kutta methods for the incompressible navier stokes equations in this work we propose runge kutta time integration schemes for the incompressible navier stokes equations with two salient properties first velocity and pressure computations are segregated at the time integration level without the need to perform additional fractional step techniques that spoil high orders of accuracy second the proposed methods keep the same order of accuracy for both velocities and pressures the segregated runge kutta methods are motivated as an implicit explicit runge kutta time integration of the projected navier stokes system onto the discrete divergence free space and its re statement in a velocity pressure setting using a discrete pressure poisson equation we have analysed the preservation of the discrete divergence constraint for segregated runge kutta methods and their relation in their fully explicit version with existing half explicit methods we have performed a detailed numerical experimentation for a wide set of schemes from first to third order including implicit and imex integration of viscous and convective terms for incompressible laminar and turbulent flows further segregated runge kutta schemes with adaptive time stepping are proposed

levi parallel contact riemannian manifolds we study the riemannian geometry of contact manifolds with respect to a fixed admissible metric making the reeb vector field unitary and orthogonal to the contact distribution under the assumption that the levi tanaka form is parallel with respect to a canonical connection with torsion

fast iterative solvers for buoyancy driven flow problems we outline a new class of robust and efficient methods for solving the navier stokes equations with a boussinesq model for buoyancy driven flow we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the efficiency of the chosen preconditioning schemes with respect to the discretization parameters

on conserved quantities of certain cohomogeneity one ricci flat equations we give a classification of continuously differentiable on shell integrals which are linear or quadratic in momenta for a hamiltonian system with constraint associated to the ricci flat cohomogeneity one einstein equations for two special classes of principal orbits hypersurfaces relations to known on shell integrals and to painleve analysis are discussed

adaptive time stepping for incompressible flow part ii navier stokes equations we outline a new class of robust and efficient methods for solving the navier stokes equations we describe a general solution strategy that has two basic building blocks an implicit time integrator using a stabilized trapezoid rule with an explicit adams bashforth method for error control and a robust krylov subspace solver for the spatially discretized system we present numerical experiments illustrating the potential of our approach

periodic solutions for a kind of lienard equation with a deviating argument abstract by means of continuation theorem of coincidence degree theory we study a kind of lienard equation with a deviating argument as follows x t f x t x t g x t t x t e t some new results on the existence of periodic solutions are obtained

aladins an algebraic splitting time adaptive solver for the incompressible navier stokes equations we address a time adaptive solver specifically devised for the incompressible navier stokes ins equations one of the challenging issues in this context is the identification of a reliable a posteriori error estimator typical strategies are based on the combination of the solutions computed either with two different time steps or two schemes with different accuracy in this paper we move from the pressure correction algebraic factorizations formerly proposed by saleri veneziani 2005 these schemes feature an intrinsic hierarchical nature such that an accurate solution for the pressure is obtained by computing intermediate low order guesses the difference between the two estimates provide a natural a posteriori estimator after introducing the incremental formulation of the pressure correction schemes we address the properties of this approach including extensive implementation details numerical results presented refer to 2d and 3d unstructured problems with a particular emphasis on cardiovascular problems which are expected to largely benefit from time adaptive solvers in memory of f saleri 1965 2007

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument abstract by means of mawhin s continuation theorem we study some second order differential equations with a deviating argument x t f t x t x t t x t e t some new results on the existence of periodic solutions are obtained the interest is that we allow the degree with respect to the variables x 0 x 1 x 2 of f t x 0 x 1 x 2 to be greater than 1 and also the result theorem 3 2 is related to the deviating argument t meanwhile we give an example to demonstrate our result

periodic solutions for a kind of second order differential equation with multiple deviating arguments by means of continuation theorem of coincidence degree theory some new results on the existence and nonexistence of periodic solutions for a kind of second order functional differential equation with multiple deviating arguments are obtained

the existence of kahler einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group on utilise la methode de continuite pour demontrer l existence de metriques de kahler einstein pour des varietes de kahler de fibre linge positif anticanonique sous l hypothese additionnelle de l existence d un bon groupe de symetries fini ou compact

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

a parallel block multi level preconditioner for the 3d incompressible navier stokes equations the development of robust and efficient algorithms for both steady state simulations and fully implicit time integration of the navier stokes equations is an active research topic to be effective the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence in particular they should be insensitive to parameters in the problem such as mesh size time step and reynolds number in this context we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an oseen nonlinear iteration for the primitive variable formulation of the system the key to this preconditioner is the approximation of a certain schur complement operator by a technique first proposed by kay loghin and wathen siam j sci comput 2002 and silvester elman kay and wathen j comput appl math 128 2001 261 the resulting operator entails subsidiary computations solutions of pressure poisson and convection diffusion subproblems that are similar to those required for decoupled solution methods however in this case these solutions are applied as preconditioners to the coupled oseen system one important aspect of this approach is that the convection diffusion and poisson subproblems are significantly easier to solve than the entire coupled system and a solver can be built using tools developed for the subproblems in this paper we apply smoothed aggregation algebraic multigrid to both subproblems previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial two dimensional steady state computations employing geometric multi grid methods we focus on extending these methods to large scale parallel three dimensional transient and steady state simulations employing algebraic multigrid amg methods our results display nearly optimal convergence rates for steady state solutions as well as for transient solutions over a wide range of cfl numbers on the two dimensional and three dimensional lid driven cavity problem

uniqueness and examples of compact toric sasaki einstein metrics in 11 it was proved that given a compact toric sasaki manifold with positive basic first chern class and trivial first chern class of the contact bundle one can find a deformed sasaki structure on which a sasaki einstein metric exists in the present paper we first prove the uniqueness of such einstein metrics on compact toric sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure and secondly remark that the result of 11 implies the existence of compatible einstein metrics on all compact sasaki manifolds obtained from the toric diagrams with any height or equivalently on all compact toric sasaki manifolds whose cones have flat canonical bundle we further show that there exists an infinite family of inequivalent toric sasaki einstein metrics on s 5 sharp k s 2 times s 3 for each positive integer k

harmonicity of sections of sphere bundles we consider the energy functional on the space of sections of a sphere bundle over a riemannian manifold m langle cdot cdot rangle equipped with the sasaki metric and discuss the characterising condition for critical points furthermore we provide a useful method for computing the tension field in some particular situations such a method is shown to be adequate for many tensor fields defined on manifolds m equipped with a g structure compatible with langle cdot cdot rangle this leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from m langle cdot cdot rangle into its sphere bundle

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

a low rank solver for the stochastic unsteady navier stokes problem abstract we study a low rank iterative solver for the unsteady navier stokes equations for incompressible flows with a stochastic viscosity the equations are discretized using the stochastic galerkin method and we consider an all at once formulation where the algebraic systems at all the time steps are collected and solved simultaneously the problem is linearized with picard s method to efficiently solve the linear systems at each step we use low rank tensor representations within the krylov subspace method which leads to significant reductions in storage requirements and computational costs combined with effective mean based preconditioners and the idea of inexact solve we show that only a small number of linear iterations are needed at each picard step the proposed algorithm is tested with a model of flow in a two dimensional symmetric step domain with different settings to demonstrate the computational efficiency

einstein manifolds of dimension five with small first eigenvalue of the dirac operator soit m n une variete de spin compacte d einstein a courbure scalaire positive r 0 et on note par d t s t s l operateur de dirac agissant sur des sections du fibre spinoriel on etudie le probleme de la classification des varietes pour le cas a 5 dimensions

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

on eta einstein sasakian geometry a compact quasi regular sasakian manifold m is foliated by one dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact kahler orbifold in the case when the transverse space is also einstein the corresponding sasakian manifold m is said to be sasakian einstein in this article we study einstein geometry as a class of distinguished riemannian metrics on contact metric manifolds in particular we use a previous solution of the calabi problem in the context of sasakian geometry to prove the existence of einstein structures on many different compact manifolds including exotic spheres we also relate these results to the existence of einstein weyl and lorenzian sasakian einstein structures

cohomogeneity one einstein sasaki 5 manifolds we consider hypersurfaces in einstein sasaki 5 manifolds which are tangent to the characteristic vector field we introduce evolution equations that can be used to reconstruct the 5 dimensional metric from such a hypersurface analogous to the nearly hypo and half flat evolution equations in higher dimensions we use these equations to classify einstein sasaki 5 manifolds of cohomogeneity one

examples of asymptotically conical ricci flat kahler manifolds previously the author has proved that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r these manifolds can be considered to be generalizations of the ricci flat ale kahler spaces known by the work of p kronheimer d joyce and others this article considers further the problem of constructing examples we show that every 3 dimensional gorenstein toric kahler cone admits a crepant resolution for which the above theorem applies this gives infinitely many examples of asymptotically conical ricci flat manifolds then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit ricci flat kahler cone metrics by the work of c boyer k galicki j kollar and others we concentrate on 3 dimensional examples two families of hypersurface examples are given which are distinguished by the condition b3 y 0 or b3 y 0