derivation of lagrangian and hermitian shape functions for quadrilateral elements this paper introduces a general theory for the derivation of the shape functions for the quadrilateral family of finite elements the first section deals with the lagrangian shape functions for the cases of uniform and boundary described elements two basic procedures are introduced the first by linear combinations of side interpolations and the second by superposition the remainder of the paper introduces a theory for the general uniform hermitian element of any order details for quadrilateral elements with first order derivatives are explained all of the shape functions presented here were derived in the interval 0 1 the shape functions developed by such an engineering approach have been used successfully in the absea finite element system of cranfield institute of technology

global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

pointwise decay for the maxwell field on black hole space times abstract in this article we study the pointwise decay properties of solutions to the maxwell system on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish peeling estimates as well as a t 4 rate of decay on compact regions for all the components of the maxwell tensor

blowup stability at optimal regularity for the critical wave equation abstract we establish strichartz estimates for the radial energy critical wave equation in 5 dimensions in similarity coordinates using these we prove the nonlinear asymptotic stability of the ode blowup in the energy space

resolvent at low energy iii the spectral measure the second and third authors were supported by australian research council discovery grants dp0771826 and dp1095448 and the second author by a future fellowship the first author was partially supported by anr grant anr 09 jcjc 0099 01 and by the pics cnrs progress in geometric analysis and applications and thanks the math department of anu for its hospitality

decay for the wave and schrodinger evolutions on manifolds with conical ends part ii let c n be a compact imbedded riemannian manifold of dimension d 1 and define the d 1 dimensional riemannian manifold m x r x w x with r 0 and smooth and the natural metric ds 2 1 r x 2 dx 2 r 2 x ds 2 we require that m has conical ends r x x o x 1 as x the hamiltonian flow on such manifolds always exhibits trapping dispersive estimates for the schrodinger evolution e it m and the wave evolution e it m are obtained for data of the form f x y n u x where y n are eigenfunctions of with eigenvalues u 2 n in this paper we discuss all cases d n 1 if n 0 there is the following accelerated local decay estimate with 0 1 e it m y n f l m c n m t d 1 2 w 1 f l 1 m where w x x and similarly for the wave evolution our method combines two main ingredients a a detailed scattering analysis of schrodinger operators of the form 2 v 2 1 4 2 u on the line where u is real valued and smooth with u l 0 3 l for all l 0 as and v 0 in particular we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the wronskian between the outgoing jost solutions as the energy tends to zero in particular the division into part i and part ii can be explained by the former being resonant at zero energy where the present paper deals with the nonresonant case b estimation of oscillatory integrals by non stationary phase

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

boolean circuits versus arithmetic circuits abstract we compare the two computational models of boolean circuits and arithmetic circuits in cases where they both apply namely the computation of polynomials over the rational numbers or over finite fields over q and finite fields boolean circuits can simulate arithmetic circuits efficiently with respect to size over finite fields of small characteristic the two models are equally powerful when size is considered but boolean circuits are exponentially more powerful than arithmetic circuits with respect to depth most of the technical results given in this synopis are taken from the literature

a proof of price s law on schwarzschild black hole manifolds for all angular momenta price s law states that linear perturbations of a schwarzschild black hole fall off as t 2 ell 3 for t to infty provided the initial data decay sufficiently fast at spatial infinity moreover if the perturbations are initially static i e their time derivative is zero then the decay is predicted to be t 2 ell 4 we give a proof of t 2 ell 2 decay for general data in the form of weighted l 1 to l infty bounds for solutions of the regge wheeler equation for initially static perturbations we obtain t 2 ell 3 the proof is based on an integral representation of the solution which follows from self adjoint spectral theory we apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates

price s law on nonstationary space times abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate price s law price 1972 54 for linear waves as a corollary we also prove price s law for certain small perturbations of the kerr metric this result was previously established by the second author in tataru 65 on stationary backgrounds the present work was motivated by the problem of nonlinear stability of the kerr schwarzschild solutions for the vacuum einstein equations which seems to require a more robust approach to proving linear decay estimates

microlocal analysis of asymptotically hyperbolic and kerr de sitter spaces with an appendix by semyon dyatlov in this paper we develop a general systematic microlocal framework for the fredholm analysis of non elliptic problems including high energy or semiclassical estimates which is stable under perturbations this framework described in sect 2 resides on a compact manifold without boundary hence in the standard setting of microlocal analysis many natural applications arise in the setting of non riemannian b metrics in the context of melrose s b structures these include asymptotically de sitter type metrics on a blow up of the natural compactification kerr de sitter type metrics as well as asymptotically minkowski metrics the simplest application is a new approach to analysis on riemannian or lorentzian or indeed possibly of other signature conformally compact spaces such as asymptotically hyperbolic or de sitter spaces including a new construction of the meromorphic extension of the resolvent of the laplacian in the riemannian case as well as high energy estimates for the spectral parameter in strips of the complex plane these results are also available in a follow up paper which is more expository in nature vasy in uhlmann g ed inverse problems and applications inside out ii 2012 the appendix written by dyatlov relates his analysis of resonances on exact kerr de sitter space which then was used to analyze the wave equation in that setting to the more general method described here

a local boundary integral equation method for potential problems boundary integral equation boundary element methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed further benefits would be obtained if attention could be confined to information at one small part of the boundary the particular region of interest in a given problem a local boundary integral equation method based on a taylor series expansion of the unknown function is developed to do this for two dimensional potential problems governed by laplace s equation very accurate local values of the function and its derivatives can be obtained the method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques

local decay of waves on asymptotically flat stationary space times author s tataru daniel abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate for linear waves this work was motivated by open problems concerning decay rates for linear waves on schwarzschild and kerr backgrounds where such a decay rate has been conjectured by r price our results apply to both of these cases

phase space analysis on some black hole manifolds abstract the schwarzschild and reissner nordstrom solutions to einstein s equations describe space times which contain spherically symmetric black holes we consider solutions to the linear wave equation in the exterior of a fixed black hole space time of this type we show that for solutions with initial data which decay at infinity and at the bifurcation sphere a weighted l 6 norm in space decays like t 1 3 this weight vanishes at the event horizon but not at infinity to obtain this control we require only an loss of angular derivatives

late time asymptotics for the wave equation on spherically symmetric stationary spacetimes abstract we derive precise late time asymptotics for solutions to the wave equation on spherically symmetric stationary and asymptotically flat spacetimes including as special cases the schwarzschild and reissner nordstrom families of black holes we also obtain late time asymptotics for the time derivatives of all orders and for the radiation field along null infinity we show that the leading order term in the asymptotic expansion is related to the existence of the conserved newman penrose quantities on null infinity as a corollary we obtain a characterization of all solutions which satisfy price s polynomial law 3 as a lower bound our analysis relies on physical space techniques and uses the vector field approach for almost sharp decay estimates introduced in our companion paper in the black hole case our estimates hold in the domain of outer communications up to and including the event horizon our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors

uniform decay of local energy and the semi linear wave equation on schwarzschild space we provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a schwarzschild background our estimate implies that such solutions have asymptotic behavior phi o left r 1 big t r big frac 1 2 right as long as the source term is bounded in the norm 1 frac 2m r 1 cdot 1 t r 1 l 1 big h 3 omega r 2dr d omega big in particular this gives scattering at small amplitudes for non linear scalar fields of the form square g varphi lambda varphi p varphi for all 2 p

hermitian cubic boundary elements for two dimensional potential problems a hermite interpolation based formulation is presented for the boundary element analysis of two dimensional potential problems two three noded hermitian cubic elements hce are introduced for the modelling of corners or points with non unique tangents on the boundary these elements along with the usual two noded hce are used in numerical examples the results obtained show that faster convergence can be achieved using hce compared with using lagrange interpolation type quadratic elements qe for about the same amount of computing resources

semiclassical analysis of low and zero energy scattering for one dimensional schrodinger operators with inverse square potentials this paper studies the scattering matrix s e of the problem 2 x v x x e x for positive potentials v c r with inverse square behavior as x it is shown that each entry takes the form sij e sij 0 e 1 ij e where sij 0 e is the wkb approximation relative to the modified potential v x 24 x 2 and the correction terms ij satisfy ek ij e cke k for all k 0 and uniformly in e 0 e0 0 0 where e0 0 are small constants this asymptotic behavior is not universal if 2 x2 v has a zero energy resonance then s e exhibits different asymptotic behavior as e 0 the resonant case is excluded here due to v 0

factoring polynomials using binary representations of finite fields the factorization of polynomials over finite fields is considered a new deterministic algorithm is proposed that solves the equal degree factorization problem by combining berlekamp s 1970 trace algorithm iterative method with the concept of binary representations of finite fields the interesting aspects of the new algorithm are its simple structure the easy proof of its correctness and its efficiency when an efficient realization of the mapping from the finite field to a binary representation is known some results about binary representations of finite fields are derived to show that the new factoring algorithm is also nonasymptotically efficient for every finite field the only practical drawback may be the precomputation of some constants needed in the binary representation but several suggestions are given to improve this when more about the finite field is known

on the power of straight line computations in finite fields it is shown that a lower hound of n 3 or more on the straight line complexity of a function f over gf 2 n is also a lower bound on the network complexity of f and hence on the product of run time and program size of turing machines it is further shown that most functions over a finite field are hard to compute and that for most hard functions there exists no approximation via an easy algorithm

the cauchy problem for the wave equation in the schwarzschild geometry the cauchy problem is considered for the scalar wave equation in the schwarzschild geometry we derive an integral spectral representation for the solution and prove pointwise decay in time

late time asymptotics for the wave equation on extremal reissner nordstrom backgrounds abstract we derive the precise late time asymptotics for solutions to the wave equation on extremal reissner nordstrom black holes and explicitly express the leading order coefficients in terms of the initial data our method is based on purely physical space techniques we derive novel weighted energy hierarchies and develop a singular time inversion theory which allow us to uncover the subtle contribution of both the near horizon and near infinity regions to the precise asymptotics we introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes our work confirms and in some cases extends the numerical and heuristic analysis of lucietti murata reall tanahashi ori sela and blaksley burko