on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a positive mass theorem in three dimensional cauchy riemann geometry in this note we summarize the results from 6 on the positive mass problem in 3 dimensional cr cauchy riemann geometry

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

positivity of quasi local mass ii a spacetime is a four manifold with a pseudo metric of signature a hypersurface or a 2 surface in a spacetime is spacelike if the induced metric is pos itive definite a quasi local energy momentum vector is a vector in r3 1 associated to a spacelike 2 surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime the time component of the four vector is called quasi local energy mass similar to 10 8 we require the quasi local energy momentum vector to satisfy the following properties 1 it should be zero for the flat spacetime 2 the quasi local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi local mass is evaluated on the spheres 7 we say that two masses mi and m2 are equivalent if there is a universal constant c 0 such that c lm area e 167r 6 the quasi local energy momentum vector should be nonspacelike and the quasi local mass should be nonnegative

a penrose like inequality for the mass of riemannian asymptotically flat manifolds we prove an optimal penrose like inequality for the mass of any asymptotically flat riemannian 3 manifold having an inner minimal 2 sphere and nonnegative scalar curvature our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere as in the original penrose conjecture and some nomalized sobolev ratio as expected the equality case is achieved if and only if the metric is that of a standard spacelike slice in the schwarzschild space

a strictly positive mass theorem we show that the adm 4 momentum of an isolated gravitational system spatially asymptotically flat spacetime satisfying the dominant energy condition cannot be null like unless it is flat together with the positive mass theorem this implies that the adm 4 momentum of an isolated gravitational system must be strictly time like

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

universal positive mass theorems in this paper we develop a general study of contributions at infinity of bochner weitzenbock type formulas on asymptotically flat manifolds inspired by witten s proof of the positive mass theorem as an application we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied

variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems abstract this manuscript presents the formulation and implementation of the variational multiscale enrichment vme method using canopy shaped microscale enrichment functions obtained through the use of a new family of microscale boundary conditions the purpose of the new enrichment functions and the new boundary condition is to relax the overconstraint imposed by the homogeneous microscale boundary condition e g residual free bubbles commonly employed in the variational multiscale literature the formulation and implementation of the method are presented for diffusion and elasticity problems the performance of the proposed method is assessed by comparing with direct numerical simulations on diffusion and deformation problems a boundary parameter identification approach is proposed to obtain near optimal boundary conditions the identification approach is verified in the context of the deformation response of particle reinforced composites

monopoles on asymptotically euclidean 3 manifolds on considere une generalisation de la theorie de yang mills higgs sur r 3 euclidien a des sommes convexes de r 3 avec des 3 varietes closes compactes

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

monopoles and maps from s 2 to s 2 the topology of the configuration space the configuration space for the su 2 yang mills higgs equations on 3 is shown to be homotopic to the space of smooth maps froms 2 tos 2 this configuration space indexes a family of twisted dirac operators the dirac family is used to prove that the configuration space does not retract onto any subspace on which the su 2 yang mills higgs functional is bounded

equivalence of the spinor and tensor methods in the positive energy problem we prove that nontrivial solutions of the sen witten equation with asymptotically flat data set on maximal hypersurface does not equal zero at any point of this hypersurface on this basis we ascertain the equivalence of witten s spinor method and nester s tensor method in the positive energy problem in general relativity

a note on positive energy theorem for spaces with asymptotic susy compactification we extend the higher dimensional positive mass theorem in dai x commun math phys 244 335 345 2004 to the lorentzian setting this includes the original higher dimensional positive energy theorem whose spinor proof is given in witten e commun math phys 80 381 402 1981 and parker t and taubes c commun math phys 84 223 238 1982 for dimension 4 and in zhang x j math phys 40 3540 3552 1999 for dimension 5

deformations of the hemisphere that increase scalar curvature consider a compact riemannian manifold m of dimension n whose boundary m is totally geodesic and is isometric to the standard sphere s n 1 a natural conjecture of min oo asserts that if the scalar curvature of m is at least n n 1 then m is isometric to the hemisphere s n equipped with its standard metric this conjecture is inspired by the positive mass theorem in general relativity and has been verified in many special cases in this paper we construct counterexamples to min oo s conjecture in dimension n 3

a definition of total energy momenta and the positive mass theorem on asymptotically hyperbolic 3 manifolds i two sets of asymptotically hyperbolic initial data are defined which correspond to the spatial infinity in asymptotically ads spacetimes and to the null infinity in asymptotically minkowski spacetimes respectively the positive mass theorem involving the total energy the total linear momentum and the total angular momentum is established for these initial data sets

moduli spaces of self dual connections over asymptotically locally flat gravitational instantons we investigate yang mills instanton theory over four dimensional asymptotically locally flat alf geometries including gravitational instantons of this type by exploiting the existence of a natural smooth compactification of these spaces introduced by hausel hunsicker mazzeo first referring to the codimension 2 singularity removal theorem of sibner sibner and rade we prove that given a smooth finite energy self dual su 2 connection over a complete alf space its energy is congruent to a chern simons invariant of the boundary three manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly then we introduce framed moduli spaces of self dual connections over ricci flat alf spaces we prove that the moduli space of smooth irreducible rapidly decaying self dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold we calculate its dimension by a variant of the gromov lawson relative index theorem as an application we study yang mills instantons over the flat mathbb r 3 times s 1 the multi taub nut family and the riemannian schwarzschild space

positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature in this paper we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary using a general version of positive mass theorem of schoen yau and witten we prove the following theorem for any compact manifold with boundary and nonnegative scalar curvature if it is spin and its boundary can be isometrically embedded into euclidean space as a strictly convex hypersurface then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in euclidean space moreover equality holds if and only if the manifold is isometric with a domain in the euclidean space conversely under the assumption that the theorem is true then one can prove the adm mass of an asymptotically flat manifold is nonnegative which is part of the positive mass theorem

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

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

a variational multiscale approach to strain localization formulation for multidimensional problems abstract the multiscale approach to strain localization problems developed earlier is generalized to higher dimensions the key idea is the identification of the fine scale field with a component of the displacement that has a large gradient the difference between the total displacement and the fine scale is the coarse scale field the formulation is variationally based with the weak form as the point of departure with suitable assumptions on the space of fine scale fields the weak form is separable into coarse and fine scale parts the weak form of the fine scale problem is used to eliminate the fine scales from the formulation this is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone in this framework fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh it is demonstrated that by correctly accounting for the microstructural fine scale behavior the formulation results in solutions devoid of pathological mesh sensitivity for problems wherein the detailed structure of the localized displacement is of interest recovery of the fine scale components can be performed to reconstruct the entire displacement field several numerical examples are presented that demonstrate the robustness of the formulation

bounds on the redundancy of huffman codes corresp new upper bounds on the redundancy of huffman codes are provided a bound that for 2 9 leq p 1 leq 0 4 is sharper than the bound of gallager when the probability of the most likely source letter p 1 is the only known probability is presented the improved bound is the tightest possible for 1 3 leq p 1 leq 0 4 upper bounds are presented on the redundancy of huffman codes when the extreme probabilities p 1 and p n are known

positive mass conjecture for five dimensional lorentzian manifolds we extend witten s proof on positive mass conjecture to five dimensional lorentzian manifolds with a rigorous treatment in mathematics

on the positive energy theorem involving mass and electromagnetic charges a new theorem relating mass and charges is deduced which can be applied to more general physical systems than those covered by the theorem of gibbons and hull 1

scalar curvature and hammocks scalar curvature is the simplest generalization of gaussian curvature to higher dimensions however there are many questions open with regard to its relation to other geometric quantities and topology here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature namely the one sided affinity for curvature decreasing deformations the first one is concerned with some prescribed decrease of the scalar curvature scal g of some riemannian metric g on a given manifoldmn of dimension 3 we denote the e neighborhood of some set u with respect to g by ue

killing vectors in asymptotically flat space times i asymptotically translational killing vectors and the rigid positive energy theorem we study killing vector fields in asymptotically flat space times we prove the following result implicitly assumed in the uniqueness theory of stationary black holes if the conditions of the rigidity part of the positive energy theorem are met then in such space times there are no asymptotically null killing vector fields except if the initial data set can be embedded in minkowski space time we also give a proof of the nonexistence of nonsingular in an appropriate sense asymptotically flat space times that satisfy an energy condition and that have a null adm four momentum under conditions weaker than previously considered

a new mass for asymptotically flat manifolds abstract in this paper we introduce a mass for asymptotically flat manifolds by using the gauss bonnet curvature we first prove that the mass is well defined and is a geometric invariant if the gauss bonnet curvature is integrable and the decay order satisfies n 4 3 motivated by an elegant idea of lam 35 we then show a positive mass theorem for this new mass for asymptotically flat graphs over r n moreover we obtain various penrose type inequalities in this case in these penrose type inequalities we establish relationship between the new mass and various geometric integrals

multiscale enrichment based on partition of unity a new multiscale enrichment method based on the partition of unity mepu method is presented it is a synthesis of mathematical homogenization theory and the partition of unity method its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible mepu is perfectly suited for enriching the coarse scale continuum descriptions pdes with fine scale features and the quasi continuum formulations with relevant atomistic data numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi continuum formulations copyright 2005 john wiley sons ltd

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

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose

yamabe flow and adm mass on asymptotically flat manifolds in this paper we investigate the behavior of adm mass and einstein hilbert functional under the yamabe flow through studying the yamabe flow by weighted spaces for parabolic operators we show that the asymptotically flat property is preserved under the yamabe flow we also obtain that adm mass is invariant under the yamabe flow and yamabe flow is the gradient flow of einstein hilbert functional on n dimensional n 3 asymptotically flat manifolds with order n 22 for n 3 4 or n 3 for n 4 moreover we show that adm mass and einstein hilbert functional are non increasing under the yamabe flow on n dimensional asymptotically flat manifolds if we only assume the order n 22 for n 4

the minimum codeword length and redundancy in the binary huffman code for uncertain sources bounds to the length of the shortest codeword in a binary huffman code are obtained among them are bounds assuming knowledge of different subsets of the probability distribution each is shown to be the tightest possible bound for the subset of probabilities it presumes two lower bounds to the redundancy of the huffman code are also obtained they are also shown to be the tightest possible bounds

positive mass theorems for higher dimensional lorentzian manifolds we extend witten s proof a new proof of the positive energy theorem commun math phys 80 381 402 1981 of the positive mass theorem to a rigorous proof for lorentzian manifolds of any dimension this includes the original higher dimensional positive mass theorem proved by spinors parker t and taubes c on witten s proof of the positive energy theorem commun math phys 84 223 238 1982 for dimension 4 and zhang x positive mass conjecture for five dimensional lorentzian manifolds j math phys 40 3540 3552 1999 for dimension 5 stimulated by the article of zhang positive mass theorem for hypersurface in 5 dimensional lorentzian manifolds commun anal geom 8 635 652 2000 we weaken the spin condition on the spacelike hypersurface to require only a spinc structure and give a modified positive mass theorem for lorentzian manifolds for dimensions 4 5 and 6

scalar curvature rigidity of asymptotically hyperbolic spin manifolds in this paper we study the hyperbolic version of this result using witten s method and show that a spin manifold which is strongly asymptotically hyperbolic cannot have scalar curvature r n n 1 everywhere unless it is isometric to hyperbolic space

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

a positive mass theorem for spaces with asymptotic susy compactification we prove a positive mass theorem for spaces which asymptotically approach a flat euclidean space times a calabi yau manifold or any special honolomy manifold except the quaternionic kahler this is motivated by the very recent work of hertog horowitz maeda hhm

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

the gbc mass for asymptotically hyperbolic manifolds the paper consists of two parts in the first part by using the gauss bonnet curvature which is a natural generalization of the scalar curvature we introduce a higher order mass the gauss bonnet chern mass m mathbb h k for asymptotically hyperbolic manifolds and show that it is a geometric invariant moreover we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding penrose type inequality for this mass and weighted alexandrov fenchel inequalities in the hyperbolic space mathbb h n in the second part we establish these weighted alexandrov fenchel inequalities in mathbb h n for any horospherical convex hypersurface sigma begin aligned int sigma v sigma 2k 1 d mu ge c n 1 2k 1 omega n 1 left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned where sigma j is the j th mean curvature of sigma subset mathbb h n v cosh r and sigma is the area of sigma as an application we obtain an optimal penrose type inequality for the new mass defined in the first part begin aligned m k mathbb h ge frac 1 2 k left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned for asymptotically hyperbolic graphs with a horizon type boundary sigma provided that a dominant energy type condition tilde l k ge 0 holds both inequalities are optimal

equality in the spacetime positive mass theorem we affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight namely if an asymptotically flat initial data set satisfies the dominant energy condition and has e p then e p 0 where e p is the adm energy momentum vector the dimensional restriction can be removed if we assume the positive mass inequality holds previously the result was only known for spin manifolds beig and chru ciel in j math phys37 4 1939 1961 1996 chru ciel and maerten in j math phys 47 2 2006

the positive mass theorem for manifolds with distributional curvature we formulate and prove a positive mass theorem for n dimensional spin manifolds whose metrics have only the sobolev regularity c 0 cap w rm loc 1 n at this level of regularity the curvature of the metric is defined in the distributional sense only and we propose here a generalized notion of adm mass for such a metric our main theorem establishes that if the manifold is asymptotically flat and has non negative scalar curvature distribution then its generalized adm mass is well defined and non negative and vanishes only if the manifold is isometric to euclidian space prior applications of witten s spinor method by lee and parker and by bartnik required the much stronger regularity w rm loc 2 p with p n our proof is a generalization of witten s arguments in which we must treat the dirac operator and its associated lichnerowicz weitzenbock identity in the distributional sense and cope with certain averages of first order derivatives of the metric over annuli that approach infinity finally we observe that our arguments are not specific to scalar curvature and also allow us to establish a universal positive mass theorem

asymptotically flat conformal structures in the first part of this paper we revisit the theory of weighted spinors on conformal manifolds in the second part we introduce the notions of asymptotically flat weyl structures and of associated mass and we prove a conformal version of the positive mass theorem on conformal spin manifolds

gauge choice in witten s energy expression witten s equation not d psi 0 can be interpreted as a gauge fixing condition for classical supergravity we rigorously prove the existence of asymptotically constant solutions of the more general gauge condition not d psi a psi for almost all endomorphismsa of the spin bundle each gives an expression for the gravitational energy similar to witten s these include the choice a sqrt r which yields the particularly elegant energy expression first noticed by deser

the dirac witten operator on pseudo riemannian manifolds we study the dirac witten operator for spacelike spin submanifolds in pseudo riemannian manifolds when the normal bundles are spin and odd dimensional we derive new eigenvalue lower bounds we also prove the generalized positive mass theorem using this operator

on the average codeword length of optimal binary codes for extended sources corresp although optimal binary source coding using symbol blocks of increasing length must eventually yield a code having an average codeword length arbitrarily close to the source entropy it is known that the sequence of average codeword lengths need not be nonincreasing the sequence is however bounded above by the average codeword length of the source and certain subsequences must be nondecreasing sufficient conditions are obtained describing sources for which a decrease in average codeword length is achieved when coding pairs of symbols a sufficient condition specifying sources for which no such decrease is possible is also obtained

remarks on positive mass theorem we establish a positive mass theorem for time symmetric initial data sets by relaxing the nonnegative scalar curvature to the case where the first neumann eigenvalue of conformal laplacian operators is eventually nonnegative

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry