large deviations of empirical neighborhood distribution in sparse random graphs consider the erd s renyi random graph on n vertices where each edge is present independently with probability lambda n with lambda 0 fixed for large n a typical random graph locally behaves like a galton watson tree with poisson offspring distribution with mean lambda here we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences the associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees we also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices or the uniform random graph with given degree sequence to prove our results we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution provided the latter is supported on trees we also introduce a new class of unimodular random trees which generalizes the usual galton watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth these generalized galton watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right

the eikonal equation in flat space null surfaces and their singularities i the level surfaces of solutions to the eikonal equation define null or characteristic surfaces in this paper we study in minkowski space properties of these surfaces in particular we are interested both in the singularities of these surfaces which can in general self intersect and be only piecewise smooth and in the decomposition of the null surfaces into a one parameter family of two dimensional wavefronts which can also have self intersections and singularities we first review a beautiful method for constructing the general solution to the flat space eikonal equation it allows for solutions either from arbitrary cauchy data or for time independent stationary solutions of the form s t s0 x y z we then apply this method to obtain global asymptotically spherical null surfaces that are associated with shearing bad two dimensional cuts of null infinity the surfaces are defined from the normal rays to the cut this is followed by a study of the caustics and singularities of these surfa

on the symplectic formalism for general relativity a covariant formalism for the hamiltonian formulation of general relativity in arbitrary dimensions is presented specifically a presymplectic form on the solution space for the vacuum equations in n dimensions is given the basic variables are taken to be a soldering form and a torsion free connection on an so p q bundle over the space time manifold m it is shown how the present formalism is related to the standard adm formalism

null cones from i and legendre submanifolds it is shown that the main variable z of the null surface formulation of gr is the generating family of a constrained lagrange submanifold that lives on the energy surface h 0 and that its level surfaces z const yield legendre submanifolds on that energy surface thus the singularity structure of past null cones with apex at i is obtained by studying the projection map of the legendre submanifolds to the configuration space the behavior of the coordinate system defined by the variable z at the caustic points is analyzed it is shown that a single function z xa cannot generate the conformal structure of an asymptotically flat space time that satisfies the generic and weak energy condition

applying multiquadric quasi interpolation to solve burgers equation abstract in this paper we develop a kind of univariate multiquadric mq quasi interpolation and use it to solve burgers equation with viscosity at first we construct the mq quasi interpolation which possesses the properties of linear reproducing and preserving monotonicity next we obtain the numerical scheme by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable then we verify our method for two examples with distinguishing initial value condition one example is tested for three reynolds number that is r 10 r 100 and r 10 000 from the numerical experiments we see that the presented method in this paper is valid although the accuracy of the technique is not higher than hon and mao s one another example is used to examine the travelling of the shock we can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate mq quasi interpolation the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

cauchy surfaces in a globally hyperbolic space time with the help of volume functions it is shown that a globally hyperbolic space time possesses a cauchy surface which is a three dimensional connected spacelike c hypersurface

a conformal boundary for space times based on light like geodesics the 3 dimensional case a new causal boundary which we will term the l boundary inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space times of any dimension m 3 proposed by one of the authors r j low the space of null geodesics and a new causal boundary lecture notes in physics 692 springer 2006 pp 35 50 is analyzed in detail for space times of dimension 3 under some natural assumptions it is shown that the completed space time becomes a smooth manifold with boundary and its relation with geroch kronheimer penrose causal boundary is discussed a number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided

operators on radial functions a general theory is provided that allows to write multi variate fourier transforms or convolutions of radial functions as very simple univariate operations as a byproduct an interesting group of operators i x r with i i i i i is defined it contains the classical derivatives as i1 d dr and is intimately connected to the fourier transform applications to the construction of new positive definite radial and to new identities for special functions are included

linking legendrian linking and causality the set n of all null geodesics of a globally hyperbolic d 1 dimensional spacetime m g is naturally a smooth 2d 1 dimensional contact manifold the sky of an event x in m is the subset x of n consisting of all null geodesics through x and is an embedded legendrian submanifold of n diffeomorphic to s d 1 it was conjectured by low that for d 2 two events x and y are causally related if and only if x and y are linked in an appropriate sense we use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases and suggest that for d 3 smooth linking should be replaced with legendrian linking

lorentzian covering space and homotopy classes we analyze structures of covering space over a lorentzian manifold by use of this we show that if a lorentzian manifold is globally hyperbolic then for any two causally related points p and q the number of homotopy classes of causal curves from p to q is finite and each of its homotopy classes has a causal geodesic from p to q

convergence of the hamiltonian particle mesh method for barotropic fluid flow we prove convergence of the hamiltonian particle mesh hpm method initially proposed by j frank g gottwald and s reich on a pe riodic domain when applied to the irrotational shallow water equations as a prototypical model for barotropic compressible uid ow under appropriate assumptions most notably suciently fast decay in fourier space of the global smoothing operator and a strang fix condition of order 3 for the local par tition of unity kernel the hpm method converges as the number of particles tends to innity and the global interaction scale tends to zero in such a way that the average number of particles per computational mesh cell remains con stant and the number of particles within the global interaction scale tends to innity the classical sph method emerges as a particular limiting case of the hpm algorithm and we nd that the respective rates of convergence are comparable under suitable assumptions since the computational complexity of bare sph is algebraically superlinear and the complexity of hpm is logarithmically su perlinear in the number of particles we can interpret the hpm method as a fast sph algorithm

on smooth cauchy hypersurfaces and geroch s splitting theorem given a globally hyperbolic spacetime m we show the existence of a smooth spacelike cauchy hypersurface s and thus a global diffeomorphism between m and r s

a quasi interpolation method for solving stiff ordinary differential equations based on the idea of quasi interpolation and radial basis functions approximation a numerical method is developed to quasi interpolate the forcing term of differential equations by using radial basis functions a highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions this overcomes the ill conditioning problem resulting from using the radial basis functions as a global interpolant error estimation is given for a particular second order stiff differential equation with boundary layer the result of computations indicates that the method can be applied to solve very stiff problems with the use of multiquadric a special class of radial basis functions it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation copyright 2000 john wiley sons ltd

the geometry of the space of null geodesics the topology and geometry of the space of null geodesics n of a space time m are used to study the causal structure of the space time itself in particular the question of whether the topology of n is hausdorff or admits a compatible manifold structure carries information on the global structure of m and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic

some necessary and sufficient conditions for admitting a continuous sphere order representation of two dimensional space times a space time may be regarded as a partially ordered set by causal relations recently as a poset a sphere order representation of a space time is considered and some interesting results are obtained in this paper we define a weakly causally convex open set and characterize connected open subsets of the n dimensional minkowski space time which admit a continuous n sphere order representation then some necessary and sufficient conditions for admitting a continuous two sphere order representation of two dimensional space times are given as a result it is shown that any simply connected and causally simple space time of dimension two can be causally isomorphically imbedded into the two dimensional minkowski space time

on the regular and the asymptotic characteristic initial value problem for einstein s vacuum field equations the regular characteristic initial value problem for einstein s vacuum field equations where data are given on two intersecting null hypersurfaces is reduced to a characteristic initial value problem for a symmetric hyperbolic system of differential equations this is achieved by making use of the spin frame formalism instead of the harmonic gauge condition the method is applied to the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on part of past null infinity and on an incoming null hypersurface a uniqueness theorem for this problem is proved by showing that a solution of the problem must satisfy a regular symmetric hyperbolic system of differential equations in a neighbourhood of past null infinity

compactly supported radial functions and the strang fix condition abstract this paper proves some negative results concerning continuous d variate positive definite radial functions 2 for space dimensions d 2 if is compactly supported it cannot satisfy a strang fix condition of positive order if 0 is well defined and nonzero then does not satisfy a refinement equation these two results also hold for finite linear combinations of integer translates of a compactly supported if is conditionally positive definite on r d with a generalized fourier transform that has a laurent expansion around zero and if satisfies a refinement equation then is a polyharmonic spline

causal geometry of einstein vacuum spacetimes with finite curvature flux one of the central difficulties of settling the l2 bounded curvature conjecture for the einstein vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity in this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of enstein vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface

multiscale analysis in sobolev spaces on the sphere we consider a multiscale approximation scheme at scattered sites for functions in sobolev spaces on the unit sphere mathbb s n the approximation is constructed using a sequence of scaled compactly supported radial basis functions restricted to mathbb s n a convergence theorem for the scheme is proved and the condition number of the linear system is shown to stay bounded by a constant from level to level thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points

numerical relativity i the characteristic initial value problem in this the first of a series of papers on numerical relativity the characteristic initial value problem is posed in a form suitable for numerical integration it can be reduced to the solution of two initial value problems for sets of ordinary differential equations on the initial surfaces and the solution of two initial value problems for hyperbolic systems of equations one linear one quasilinear the initial data may be specified freely subsequent papers will develop numerical solutions of einstein s equations with use of this formalism

cauchy problems for the conformal vacuum field equations in general relativity cauchy problems for einstein s conformal vacuum field equations are reduced to cauchy problems for first order quasilinear symmetric hyperbolic systems the hyperboloidal initial value problem where cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two surface is discussed and translated into the conformally related picture it is shown that for conformal hyperboloidal initial data of classhs s 4 there is a unique up to questions of extensibility development which is a solution of the conformal vacuum field equations of classhs it provides a solution of einstein s vacuum field equations which has a smooth structure at past null infinity

on null and causal pseudoconvex space times the condition of pseudoconvexity has important implications for geodesic structures of space time low showed that this condition guarantees the existence of a smooth structure of the space of geodesics recently the place of this property within the causality ladder is in the attention there is a conjecture which states that every strongly causal space time is causally simple if and only if it is null pseudoconvex in this paper we study the difference of causal and null pseudoconvexity in space times and introduce the limit geodesic segment as an equivalence condition to the pseudoconvexity then we show that in the case of open subspaces of n minkowski space time this is causal geodesic connectedness finally we prove a refined form of the conjecture that says every strongly causal space time is causally simple if and only if it is maximal null pseudoconvex

khovanov homology and causality in spacetimes we observe that khovanov homology detects causality in 2 1 dimensional globally hyperbolic spacetimes whose cauchy surface is homeomorphic to r2

cauchy characteristic evolution and waveforms we investigate a new methodology for the computation of waves generated by isolated sources this approach consists of a global spacetime evolution algorithm based on a cauchy initial value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior we match the two schemes at their common interface the characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval so that our numerical solution extends to infinity and accurately models the free space problem the matching interface need not be situated far from the sources the wavefronts may have arbitrary nonspherical geometry and strong nonlinearity may be present in both the interior and the exterior regions stability and second order convergence of the algorithms to the exact solution of the infinite domain problem are established numerically in three space dimensions the matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature for linear problems matching outperformed the local radiation conditions chosen for testing and was about as accurate for the same grid resolution as the exact nonlocal conditions however since the computational cost of the nonlocal conditions is many times that of matching this algorithm may be used with higher grid resolutions yielding a significantly higher final accuracy for strongly nonlinear problems matching was significantly more accurate than all other methods tested this seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field while matching consistently takes nonlinearity into account in both interior and exterior regions

an asymptotic universal focal decomposition for non isochronous potentials galileo in the xvii century observed that the small os cillations of a pendulum seem to have constant period in fact the taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi librium it is well known that for small oscillations of the pendulum and small intervals of time the dynamics of the pendulum can be ap proximated by the dynamics of the harmonic oscillator we study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected we analyze such dynamical behaviour through a renor malization scheme acting on the dynamics of this family of mechanical systems the main theorem states that the asymptotic limit of this renormalization scheme is universal it is the same for all the elements in the considered class of mechanical systems as a consequence we obtain an universal asymptotic focal decomposition for this family of mechanical systems this paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu lum type as a natural application of the focal decomposition associated to the two point boundary value problem

on the existence of analytic null asymptotically flat solutions of einstein s vacuum field equations this paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for einstein s field equations for analytic data on past null infinity and on an incoming null hypersurface

stable singularities of wave fronts in general relativity a wave front in a space time m is a family of null geodesics orthogonal to a smooth spacelike two surface in m it is of some interest to know how a wave front can fail to be a smoothly immersed surface in m in this paper we see that the space of null geodesics n of m considered as a contact manifold provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave fronts

the theory of caustics and wave front singularities with physical applications this is intended as an introduction to and review of the theory of lagrangian and legendrian submanifolds and their associated maps developed by arnold and his collaborators the theory is illustrated by applications to hamilton jacobi theory and the eikonal equation with an emphasis on null surfaces and wave fronts and their associated caustics and singularities

celestial spheres light cones and cuts the embedding of celestial spheres in n the space of null geodesics of a strongly causal space time m is studied with particular attention to the contact form on n the algebraic topology of the embedding is also considered light cone cuts with partial cauchy surfaces are examined by regarding n or appropriate submanifolds of it as a bundle over s and projecting down finally a means of reconstructing the original space time up to a conformal factor in the metric from the cuts of light cones with a cauchy surface is given

characteristic initial data and wavefront singularities in general relativity the structure of singularities caustics self intersections of wavefronts null hypersurfaces and wavefront families null coordinates in arbitrary space times is discussed in detail and illustrated by explicit examples of stable wavefront singularities in minkowski space it is shown how characteristic initial data determine the caustics and the self intersections of the characteristics of einstein s field equations

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

improved accuracy of multiquadric interpolation using variable shape parameters given n scattered data points we examined the problem of finding n variable multiquadric mq shape parameters or r2 values because the problem of finding the optimal r2 values is a nonlinear one we optimized these parameters numerically by minimizing the root mean square rms errors the resulting r2 values varied over many orders of magnitude we have tested this approach on a number of univariate and bivariate franke s problems and found that the rms error reduction was substantial

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

linking and causality in globally hyperbolic space times the classical linking number lk is defined when link components are zero homologous in 15 we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes here we apply alk to the study of causality in lorentzian manifolds

the asymptotic characteristic initial value problem for einstein s vacuum field equations as an initial value problem for a first order quasilinear symmetric hyperbolic system the asymptotic characteristic initial value problem for einstein s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown it is delineated how the same method can be applied to the standard cauchy problems for einstein s vacuum and conformal vacuum equations

smoothness of time functions and the metric splitting of globally hyperbolic spacetimes the folk questions in lorentzian geometry which concerns the smoothness of time functions and slicings by cauchy hypersurfaces are solved by giving simple proofs of a any globally hyperbolic spacetime m g admits a smooth time function open image in new window whose levels are spacelike cauchy hyperfurfaces and thus also a smooth global splitting open image in new window if a spacetime m admits a continuous time function t then it admits a smooth time function open image in new window with timelike gradient open image in new window on all m

cardinal interpolation with differences of tempered functions abstract in this paper we investigate the existence and uniqueness of cardinal interpolants associated with functions arising from the k th order iterated discrete laplacian k applied to certain radial basis functions in particular we concentrate on determining for a given radial function which functions k give rise to cardinal interpolation operators which are both bounded and invertible l 2 z 3 in addition to solving the cardinal interpolation problem cip associated with such functions k our approach provides a unified framework and simpler proofs for the cip associated with polyharmonic splines and hardy multiquadrics

conformally embedded spacetimes and the space of null geodesics it is shown that the space of null geodesics of a causally simple spacetime is hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime this provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones