categorified central extensions etale lie 2 groups and lieʼs third theorem for locally exponential lie algebras abstract lie s third theorem asserting that each finite dimensional lie algebra is the lie algebra of a lie group fails in infinite dimensions the modern account on this phenomenon is the integration problem for central extensions of infinite dimensional lie algebras which in turn is phrased in terms of an integration procedure for lie algebra cocycles this paper remedies the obstructions for integrating cocycles and central extensions from lie algebras to lie groups by generalising the integrating objects those objects obey the maximal coherence that one can expect moreover we show that they are the universal ones for the integration problem the main application of this result is that a mackey complete locally exponential lie algebra e g a banach lie algebra integrates to a lie 2 group in the sense that there is a natural lie functor from certain lie 2 groups to lie algebras sending the integrating lie 2 group to an isomorphic lie algebra

a numerical scheme to couple subdomains with different time steps for predominantly linear transient analysis abstract this paper generalizes the explicit implicit time integration algorithms pioneered by belytschko hughes and their respective co workers and the feti domain decomposition methods introduced by farhat and his co workers to the case where the same newmark scheme but different and coefficients and different time steps are specified in each subdomain building upon the work of farhat crivelli and geradin it considers various interface boundary constraints and performs a stability analysis of the proposed time integration algorithms

a conservative space time mesh refinement method for the 1 d wave equation part i construction we propose a new method for space time refinement for the 1 d wave equation this method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme our approach results in a non interpolatory scheme whose stability condition is not affected by the transition between the two grids

s 1 actions and elliptic genera a proof is given of witten s conjectures for the rigidity of the index of the dirac ramond operator on the loop space of a spin manifold which admits ans1 symmetry

conservative space time mesh refinement methods for the fdtd solution of maxwell s equations a new variational space time mesh refinement method is proposed for the fdtd solution of maxwell s equations the main advantage of this method is to guarantee the conservation of a discrete energy that implies that the scheme remains l 2 stable under the usual cfl condition the only additional cost induced by the mesh refinement is the inversion at each time step of a sparse symmetric positive definite linear system restricted to the unknowns located on the interface between coarse and fine grid the method is presented in a rather general way and its stability is analyzed an implementation is proposed for the yee scheme in this case various numerical results in 3 d are presented in order to validate the approach and illustrate the practical interest of space time mesh refinement methods

high order explicit local time stepping methods for damped wave equations locally refined meshes impose severe stability constraints on explicit time stepping methods for the numerical simulation of time dependent wave phenomena local time stepping methods overcome that bottleneck by using smaller time steps precisely where the smallest elements in the mesh are located starting from classical adams bashforth multi step methods local time stepping methods of arbitrarily high order of accuracy are derived for damped wave equations when combined with a finite element discretization in space with an essentially diagonal mass matrix the resulting time marching schemes are fully explicit and thus inherently parallel numerical experiments with continuous and discontinuous galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time stepping methods

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

coupling discontinuous galerkin methods and retarded potentials for transient wave propagation on unbounded domains this work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities to do so we propose to combine a discretization based on a discontinuous galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain the coupling formula enforces a discrete energy identity ensuring the stability under the usual cfl condition in the interior moreover the scheme allows to use a smaller time step in the interior domain yielding to quasi optimal discretization parameters for both methods the aliasing phenomena introduced by the local time stepping are reduced by a post processing by averaging in time obtaining a stable and second order consistent in time coupling algorithm we compute the numerical rate of convergence of the method for an academic problem the numerical results show the feasibility of the whole discretization procedure

generalized witten genus and vanishing theorems we construct a generalized witten genus for spinc manifolds which takes values in level 1 modular forms with integral fourier expansion on a class of spinc manifolds called stringc manifolds we also construct a mod 2 analogue of the witten genus for 8k 2 dimensional spin manifolds the landweber stong type vanishing theorems are proven for the generalized witten genus and the mod 2 witten genus on stringc and string generalized complete intersections in product of complex projective spaces respectively

the dirac ramond operator on loops in flat space in this paper a rigorous construction of the s1 equivariant dirac operator i e dirac ramond operator on the space of mean zero loops in rd is given and its equivariant l2 index computed essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra

a mortar based frictional contact formulation for large deformations using lagrange multipliers abstract in this work a lagrange multiplier method is proposed to solve 2d coulomb frictional contact problems in the context of large deformations as the proposed formulation is based on the mortar method the constraints are imposed in a weak integral sense along the contact surface in order to compute the contact integrals we use a numerical integration based on the definition of the kinematical variables gap slip and their variations at the quadrature points the linearization of non linear equations virtual work and contact constraints is developed in order to apply a newton s method the examples show that the numerical integration still preserves the optimal rate of convergence of the finite element solution

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

explicit and provably stable spatiotemporal fdtd refinement abstract in this paper we introduce an explicit and provably conditionally stable finite difference time domain fdtd algorithm for maxwell s equations with local refinement in both the spatial discretization length and in the time step spatiotemporal refinement this enables local spatial refinement with a locally reduced time step

runge kutta based explicit local time stepping methods for wave propagation locally refined meshes severely impede the efficiency of explicit runge kutta rk methods for the simulation of time dependent wave phenomena by taking smaller time steps precisely where the smallest elements are located local time stepping lts methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh starting from classical or low storage explicit rk methods explicit lts methods of arbitrarily high accuracy are derived when combined with an essentially diagonal finite element mass matrix the resulting time marching schemes retain the high accuracy stability and efficiency of the original rk methods while circumventing the geometry induced stiffness numerical experiments with continuous and discontinuous galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these lts rk methods

spin rm c manifolds with pin 2 action we study spin rm c manifolds with pin 2 action the main tool is a vanishing theorem for certain indices of twisted spin rm c dirac operators this theorem is used to show that the witten genus vanishes on such manifolds provided the first chern class and the first pontrjagin class are torsion we apply the vanishing theorem to cohomology complex projective spaces and give partial evidence for a conjecture of petrie for example we prove that the total pontrjagin class of a cohomology mathbb c p 2n with s 3 action has standard form if the first pontrjagin class has standard form we also determine the intersection form of certain 4 manifolds with pin 2 action

obstructions to positive curvature and symmetry we discuss new obstructions to positive sectional curvature and symmetry the main result asserts that the index of the dirac operator twisted with the tangent bundle vanishes on a 2 connected manifold of dimension 8 if the manifold admits a metric of positive sectional curvature and isometric effective s1 action the proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds

local time stepping and discontinuous galerkin methods for symmetric first order hyperbolic systems we present a new non conforming space time mesh refinement method for the symmetric first order hyperbolic system this method is based on the one hand on the use of a conservative higher order discontinuous galerkin approximation for space discretization and a finite difference scheme in time on the other hand on appropriate discrete transmission conditions between the grids we use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method

an error analysis of conservative space time mesh refinement methods for the one dimensional wave equation we study two space time mesh refinement methods as the one introduced in f collino t fouquet and p joly numer math 95 2003 pp 197 221 the stability of such methods is guaranteed by construction through the conservation of a discrete energy in this paper we show the l2 convergence of these schemes and provide optimal error estimates the proof is based on energy techniques and bootstrap arguments our results are validated with numerical simulations and compared with results from plane wave analysis f collino t fouquet and p joly numer math 95 2003 pp 223 251

efficient algorithms for problems with friction in this paper a nonconforming discretization method for the frictional contact between two bodies subjected to antiplane shear deformation is considered the method is based on a mixed variational formulation where dual basis functions are used for the discretization of the lagrange multiplier under some regularity assumptions on the solution an optimal a priori error estimate is obtained to solve the discrete nonlinear problem an inexact primal dual active set strategy is introduced finally numerical examples confirming the theoretical result for the a priori error estimate and illustrating the performance of the algorithm are presented the results can easily be generalized to the case of coulomb friction for this case a numerical example is given

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

space time mesh refinement for elastodynamics numerical results we propose here a generalization to the elastodynamic equations of the space time mesh refinement technique intro duced in raffinement de maillage spatio temporel pour les equations de maxwell ph d thesis universitede dau phine paris 2000 for the maxwells equations this method uses a discrete energy conservation to ensure the stability the method is presented in a variational way applicable to other type of hyperbolic systems several numerical experiments are provided to show the efficiency of this approach 2004 elsevier b v all rights reserved

a primal dual active set strategy for non linear multibody contact problems abstract non conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a non linear multibody contact problem we use the mortar approach with a dual lagrange multiplier space to handle the non linearity of the contact conditions we apply a primal dual active set strategy to find the actual contact zone the algorithm can be easily generalized to multibody contact problems a suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case using an inexact primal dual active set strategy in combination with a multigrid method yields an efficient iterative solver different numerical examples for one and multibody contact problems illustrate the performance of the method

a conservative space time mesh refinement method for the 1 d wave equation part ii analysis we introduced in 2 a new method for space time refinement for the 1 d wave equation this method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme in this second part we analyse the accuracy of this scheme in a detailed way by means of a plane wave analysis and numerical experiments that permit us to point out spurious numerical phenomena and explain how to control them