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

resolubilite d equations differentielles et representation induites g un groupe de lie connexe h un sous groupe de lie ferme de g u une representation unitaire de h dans un espace de hilbert e vu un ouvert de g h p la projection canonique de g sur g h on pose v p 1 vu et l on considere l espace c v h u forme des fonctions u definies sur l ouvert v de g a valeurs dans e indefiniment differentiables et verifiant x h v h u xh h g 1 2 h u 1 h u x on considere des operateurs differentiels p agissant sur c v h u et pour v c v h u on cherche une solution u c v h u pour l equation pu v

spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces abstract if k is a connected subgroup of a nilpotent lie group g the irreducible decompositionof the action on l2 kg has either pure infinite or boundedly finite multiplicities in the finite case the authors recently proved that the algebra d kg of g invariant differential operators on kg is commutative even if the action is not multiplicity free and produced evidence for the conjecture that d kg is isomorphic to the algebra of all ad k invariant polynomials on the annihilator where is the lie algebra of k here the conjecture is proved for a large class of data k g for such pairs an explicit construction of the isomorphism can be found it is a type of fourier transform with some unusual nonlinear aspects furthermore the operators in d kg have tempered fundamental solutions

analyse harmonique sur les espaces symetriques nilpotents resume soit g un groupe de lie nilpotent connexe et simplement connexe h l ensemble des points fixes d une involution de g nous donnons la formule de plancherel de la representation ind h g 1 et en deduisons que tout operateur differentiel g invariant sur g h non nul admet une solution elementaire temperee h invariante

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

meshless method and convergence analysis for 2 dimensional fredholm integral equation with complex factors in this paper the meshless method is presented for numerically solving the 2 dimensional fredholm integral equation with complex factors first the existence and uniqueness of solution are proved by the fixed point theorem second the approximation solution method of the radial basis function interpolation is given the results of convergence analysis and error estimation are also obtained some numerical examples are presented to verify the validity and applicability of the method

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

iterative method and convergence analysis for a kind of mixed nonlinear volterra fredholm integral equation in this paper the iterative method is presented for numerically solving the nonlinear volterra fredholm integral equation first considering some conditions on k i and l i i 1 2 of the integral equation then we define the equidistance collocation points and the integral part of equation is discretized by newton cotes integration formula finally the approximate solutions of integral equation are obtain by iterative method the convergence analysis of the integral equation is given some numerical examples are given to illustrate the accuracy and dependability of the method

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 numerical scheme for a class of nonlinear fredholm integral equations of the second kind in this paper an iterative approach for obtaining approximate solutions for a class of nonlinear fredholm integral equations of the second kind is proposed the approach contains two steps at the first one we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation solution of the discretized form converges to the exact solution of the problem following that in the next step solution of the discretized form is approximated by an iterative approach we finally on some examples show the efficiency of the proposed approach

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

applying the modified block pulse functions to solve the three dimensional volterra fredholm integral equations the main aim of this work is to give further studies for the multi dimensional integral equations in this work we solve special types of the three dimensional volterra fredholm linear integral equations of the second kind via the modified block pulse functions some theorems are included to show convergence and advantage of the method we solve some examples to investigate the applicability and simplicity of the method the numerical results confirm that the method is efficient and simple

asymptotic error expansion for the nystro m method for a nonlinear volterra fredholm integral equation in this paper we study the numerical solution of general nonlinear mixed volterra fredholm integral equations by the trapezoidal nystrom method it focuses on the derivation and analysis of asymptotic error expansion of the nystrom solution providing the basis for richardson s extrapolation this will greatly increase the accuracy of the numerical solution some numerical results are given to illustrate this theory

fredholm volterra integral equation of the first kind and contact problem regular and singular asymptotic methods are applied to one and two dimensional fredholm volterra integral equation f vie of the first kind that arise in the treatment of various two dimensional axisymmetric and three dimensional problems with mixed boundary conditions in the mechanics of continuous media the solution of the integral equation is obtained in the space l 2 w xc 0 t 0

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

direct and iterative methods for the numerical solution of mixed integral equations abstract in this paper we analyze and compare two classical methods to solve volterra fredholm integral equations the first is a collocation method the second one is a fixed point method both of them are proposed on a particular class of approximating functions precisely the first method is based on a linear spline class approximation and the second one on schauder linear basis we analyze some problems of convergence and we propose some remarks about the peculiarities and adaptability of both methods numerical results complete the work

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

on the numerical solution of nonlinear volterra fredholm integral equations by collocation methods particular cases of nonlinear mixed volterra fredholm integral equations of the second kind arise in the mathematical modeling of the spatio temporal development of an epidemic this paper is concerned with the numerical solution of general integral equations of this type by continuous time and discrete time spline collocation methods its focus is on the derivation and analysis of methods of high order of convergence

calcul symbolique sur les groupes de lie nilpotents et applications abstract for any real nilpotent connected and simply connected lie group g and for any unitary irreducible representation of g we define the generally unbounded operator with left symbol p acting on the hilbert space of the representation the symbol p being some c function defined on the dual g of the lie algebra g of g and prove that the operator thus defined can be viewed as a classical pseudo differential operator whose left symbol r is explicitly related to p we also compute the link between left and weyl symbolic calculus on g as an application we prove that if the symbol p defined on g vanishes on a neighbourhood of the coadjoint orbit associated to by means of kirillov s method then the operator with weyl or left symbol p is a smoothing operator hence we extend the weyl formula proved in previous work to elliptic symbols

lagrange collocation method for solving volterra fredholm integral equations in this paper the lagrange collocation method is used to solve the volterra fredholm integral equations this method transforms the system of the linear integral equations into matrix equations via lagrange collocation points the lagrange collocation solution of the three kinds of volterra fredholm integral equations and the error estimate of these equations are given some numerical examples are presented to illustrate the accuracy of the method

fonctions generalisees spheriques induites sur gc gr et applications resume soit g un groupe de lie semi simple complexe d algebre de lie g soit h une forme reelle de g et soit h le sous groupe analytique de g d algebre de lie h par un procede d induction on construit une fonction generalisee spherique p indgl a partir d une fonction generalisee spherique definie sur un espace symetrique plus petit l l h lorsque est le coefficient d une representation on montre que p indgl est un coefficient de representation paraboliquement induite en appliquant ceci a l espace gl n c gl n r on obtient la formule de plancherel sur cet espace a partir de la formule d inversion demontree par s sano

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

numerical solutions of the nonlinear volterra fredholm integral equations by using homotopy perturbation method abstract a numerical method for solving the nonlinear volterra fredholm integral equations is presented the method is based upon homotopy perturbation theory the results reveal that the homotopy perturbation method is very effective and simple and gives the exact solution

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

least squares approximation method for the solution of volterra fredholm integral equations in this paper an efficient numerical method is developed for solving the volterra fredholm integral equations by least squares approximation method which is based on a polynomial of degree n to compute an approximation to the solution of volterra fredholm integral equations the convergence analysis of the approximation solution relative to the exact solution of the integral equation is proved the reliability and efficiency of the proposed method are demonstrated by some numerical experiments