dimensionally dependent tensor identities by double antisymmetrization some years ago lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure and could all be considered consequences in n dimensional space of a pair of fundamental identities involving trace free p p forms where 2p n we generalize lovelock s results and by using the fact that associated with any tensor in n dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n 1 indices we establish a very general master identity for all trace free k l forms we then show how various other special identities are direct and simple consequences of this master identity in particular we give direct application to maxwell lanczos ricci bel and bel robinson tensors and also demonstrate how relationships between scalar invariants of the riemann tensor can be investigated in a systematic manner

pointwise estimates and stability for dispersive diffusive shock waves we study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with constant diffusion and dispersion along with the usual lax shocks such equations are known to admit undercompressive shocks we unify the treatment of these two cases by introducing a new wave tracking method based on instantaneous projection giving improved estimates even in the lax case another important feature connected with the introduction of dispersion is the treatment of a non sectorial operator an immediate consequence of our pointwise estimates is a simple spectral criterion for stability in all l p norms p 1 for the lax case and p 1 for the undercompressive case our approach extends immediately to the case of certain scalar equations of higher order and would also appear suitable for extension to systems

asymptotic stability of traveling wave solutions for perturbations with algebraic decay abstract for a class of scalar partial differential equations that incorporate convection diffusion and possibly dispersion in one space and one time dimension the stability of traveling wave solutions is investigated if the initial perturbation of the traveling wave profile decays at an algebraic rate then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate and optimal intermediate results that combine temporal and spatial decay are obtained the proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations

the 3 d elastodynamic boundary element method semi analytical integration for linear isoparametric triangular elements a semi analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically this new scheme can be applied to linear triangular elements for use in steady state elastodynamic bem problems and is particularly useful for predicting displacement to high accuracy close to surfaces for a spectrum of frequencies the scheme involves the application of taylor expansions to formulate the integrals into two parts one part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral the line integral is solved numerically using gauss legendre quadrature this approach caters for all the integral types that appear in steady state elastodynamic boundary elements but in particular no special treatment is required for the evaluation of the cauchy principal value singular integrals numerical tests are performed on a simple test problem for which a known analytical solution exists the results obtained using the semi analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods copyright 1999 john wiley sons ltd

a lanczos potential in kerr geometry a lanczos potential is a tensor of rank 3 for which a certain combination of derivatives always equals the weyl tensor general existence theorems have been proved before and some explicit expressions for lanczos potentials in certain spacetimes have been found in this paper we explicitly study a lanczos potential in kerr spacetime obtained from a previously studied flat connection as an application it is shown that the mass can be expressed as an integral of the lanczos potential

the weyl lanczos equations and the lanczos wave equation in four dimensions as systems in involution the weyl lanczos equations in four dimensions form a system in involution we compute its cartan characters explicitly and use janet riquier theory to confirm the results in the case of all space times with a diagonal metric tensor and for the plane wave limit of space times we write the lanczos wave equation as an exterior differential system and with assistance from janet riquier theory we compute its cartan characters and find that it forms a system in involution we compare these cartan characters with those of the weyl lanczos equations all results hold for the real analytic case

necessary and sufficient conditions for n dimensional conformal einstein spaces via dimensionally dependent identities listing has recently extended results of kozameh newman and tod for four dimensional space times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an einstein metric in all semi riemannian spaces of dimension n 4 subject to a nondegeneracy restriction on the weyl tensor by exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications the four dimensional case is discussed in detail and examples are also given in five and six dimensions

bach flat asymptotically locally euclidean metrics we obtain a volume growth and curvature decay result for various classes of complete noncompact riemannian metrics in dimension 4 in particular our method applies to anti self dual or kahler metrics with zero scalar curvature and metrics with harmonic curvature similar results were obtained for einstein metrics in and89 bkn89 tia90 but our analysis differs from the einstein case in that 1 we consider more generally a fourth order system in the metric and 2 we do not assume any pointwise ricci curvature bound

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

remarks on the stability of shock profiles for conservation laws with dissipation two remarks are made the first is to establish the stability of monotone shock profiles of the kdv burgers equation based on an energy method of goodman the second remark illustrates specifically in burgers equation that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms introduction we will prove two results the first is to establish the orbital stability of monotone shock profile solutions 4 x st of the kdv burgers equation 1 1 ut uux uxx uxxx 0 0 the second is to show that for shock profile solutions of viscous conservation laws and specifically for burgers equation 1 2 ut uux uxx one cannot expect perturbations to decay at any uniform rate algebraic or otherwise in ordinary norms such as lp the first result is related to goodman s conditional stability result in 5 for small amplitude shock profiles x st in systems of genuinely nonlinear conservation laws with viscosity u f u x buxx u e rm the second is intended to be relevant to the question of full stability for shock profiles of such systems the nonlinear stability to small perturbations of constant state solutions u0 x c of the system u f u x b u ux x where b c is a strictly stable viscosity matrix see 9 can be demonstrated based on a uniform rate of decay of solutions to the linearized equation the method used is similar to that developed in 8 and 10 for systems in n 1 space dimensions faster decay rates in higher space dimensions make the method easier to apply when n is large but it does just work for conservation laws with stable viscosities in one space dimension see 13 thus for example assuming c 0 if iiuo iih6 ll is sufficiently small then iiu t llh4 ct 1 4iiuoii h4 n la in this note we want to point out that such uniform decay rates cannot be valid for perturbations of shock profiles in particular for the shock profile solution of burgers equations we construct a family of perturbations which decay to zero but do so at no uniform rate in most ordinary norms including all lp norms thus for received by the editors october 15 1984 and in revised form december 19 1984 1980 mathematics subject classification primary 35q20 35l65 1985 american mathematical society 0002 9947 85 1 00 25 per page

constructing bach flat manifolds of signature 2 2 using the modified riemannian extension we use the modified riemannian extension of an affine surface to construct bach flat manifolds as all these examples have vanishing scalar invariants we shall construct scalar invariants which are not of weyl type to distinguish them we illustrate these phenomena in the context of homogeneous affine surfaces we use the modified riemannian extension of an affine surface to construct bach flat manifolds as all these examples have vanishing scalar invariants we shall construct scalar invariants which are not of weyl type to distinguish them we illustrate these phenomena in the context of homogeneous affine surfaces

einstein four manifolds of pinched sectional curvature abstract in this paper we obtain classification of four dimensional einstein manifolds with positive ricci curvature and pinched sectional curvature in particular the first result deals with an upper bound on the sectional curvature improving a theorem of e costa the second is a generalization of d yang s result assuming an upper bound on the difference between sectional curvatures

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

dynamic stress concentration studies by boundary integrals and laplace transform the dynamic stress field and its concentrations around holes of arbitrary shape in infinitely extended bodies under plane stress or plane strain conditions are numerically determined the material may be linear elastic or viscoelastic while the dynamic load consists of plane compressional waves of harmonic or general transient nature the method consists of applying the laplace transform with respect to time to the governing equations of motion and formulating and solving the problem numerically in the transfomed domain by the boundary integral equation method the stress field can then be obtaind by a numerical inversion of the trasformed solution the correspondence principle is invoked for the case of viscoelastic material behavious the method is simplified for the case of harmonic waves where no numerical inversion is involved

some solutions of the lanczos vacuum wave equation in general relativity the non local part of the gravitational field is described by the 10 degrees of freedom of the weyl conformal curvature tensor c abcd in every space time the weyl field c abcd is derivable from a potential l abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions l smallmatrix ab s s endsmallmatrix 0 are imposed the potential l abc discovered by lanczos was shown by illge to have an extremely simple vacuum wave equation namely square l abc equiv g sm l abc s m 0 using tensor spinor and spin coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields

the lanczos potential for the weyl curvature tensor existence wave equation and algorithms in the last few years renewed interest in the 3 tensor potential l abc proposed by lanczos for the weyl curvature tensor has not only clarified and corrected lanczos9s original work but generalized the concept in a number of ways in this paper we first of all carefully summarize and extend some aspects of these results make some minor corrections and clarify some misunderstandings in the literature the following new results are also presented the computer checked complicated second order partial differential equation for the 3 potential in arbitrary gauge for weyl candidates satisfying bianchi type equations is given in those n dimensional spaces with arbitrary signature for which the potential exists this is easily specialized to lanczos potentials for the weyl curvature tensor it is found that it is only in four dimensional spaces with arbitrary signature that the nonlinear terms disappear and that certain awkward second order derivative terms cancel for four dimensional spacetimes with lorentz signature this remarkably simple form was originally found by illge using spinor methods it is also shown that for most four dimensional vacuum spacetimes any 3 potential in the lanczos gauges which satisfies a simple homogeneous wave equation must be a lanczos potential for the non zero weyl curvature tensor of the background vacuum spacetime this result is used to prove that the form of a possible lanczos potential recently proposed by dolan and kim for a class of vacuum spacetimes is in fact a genuine lanczos potential for these spacetimes

corner treatment by assigning dual tractions to every node for elastodynamics in td bem in this study a corner treatment on the boundary for time domain boundary element method td bem by assigning dual tractions to every node is proposed the proposed corner treatment on boundary mimics the real mechanical situation on the physical boundary for elastodynamic problem and the computation accuracy is good taking the rectangular boundary as an example the numerical implementation of the proposed corner treatment on boundary by assigning dual tractions to every node is demonstrated two verification examples one dimensional rod in finite medium and two dimensional circular cavity in infinite medium under dynamic loads are conducted in which the results from the td bem based on the proposed corner treatment on boundary are compared with the results from the analytical solutions good agreement between the analytical results and the td bem results based on the proposed corner treatment on boundary for td bem is achieved for the two verification examples

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

manifolds with positive curvature operators are space forms the ricci flow was introduced by hamilton in 1982 h1 in order to prove that a compact three manifold admitting a riemannian metric of positive ricci curvature is a spherical space form in dimension four hamilton showed that compact four manifolds with positive curvature operators are spherical space forms as well h2 more generally the same conclusion holds for compact four manifolds with 2 positive curvature operators che recall that a curvature operator is called 2 positive if the sum of its two smallest eigenvalues is positive in arbitrary dimensions huisken hu described an explicit open cone in the space of curvature operators such that the normalized ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature hamilton conjectured that in all dimensions compact riemannian manifolds with positive curvature operators must be space forms in this paper we confirm this conjecture more generally we show the following

integral pinched shrinking ricci solitons abstract we prove that a n dimensional 4 n 6 compact gradient shrinking ricci soliton satisfying a l n 2 pinching condition is isometric to a quotient of the round s n the proof relies mainly on sharp algebraic curvature estimates the yamabe sobolev inequality and an improved rigidity result for integral pinched einstein metrics

pointwise green s function approach to stability for scalar conservation laws we study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law obtaining an estimate independent of shock strength we find that for a perturbation with initial data decaying algebraically or slower the perturbation decays in time at the rate of decay of the integrated initial data in any l p norm p 1 stability in any l p norm is a direct consequence the approach taken is that of obtaining pointwise estimates on the perturbation through a duhamel s principle argument that employs recently developed pointwise estimates on the green s function for the linearized equation c 1999 john wiley sons inc

the identities of the algebraic invariants of the four dimensional riemann tensor this paper extends the investigation of the invariants of the riemann tensor to include the invariants that are of odd degree in the trace free ricci tensor it is shown that these invariants can be expressed in terms of 15 such invariants that are irreducible as a consequence it is possible to write down a complete set of invariants of the riemann tensor several syzygies for these invariants have been found and examples of these are given these syzygies suggest there may be several new syzygies of invariants with even degree in the trace free ricci tensor a large number of these have also been found and are discussed in the paper

on closed manifolds with harmonic weyl curvature abstract we derive point wise and integral rigidity gap results for a closed manifold with harmonic weyl curvature in any dimension in particular there is a generalization of tachibana s theorem for non negative curvature operator the key ingredients are new bochner weitzenbock lichnerowicz type formulas for the weyl tensor which are generalizations of identities in dimension four

linear independence of renormalization counterterms in curved space times of arbitrary dimensionality the counterterms in the lagrangian of a renormalizable quantum field theory that involve the riemann curvature tensor are considered it is proved that for six or fewer dimensions the counterterms not containing derivatives of the riemann tensor are linearly independent it is shown that there appears to be a maximum space time dimension for which identities can exist among invariants involving the product of n riemann tensors without derivatives acting on them space times that would require these products as counterterms for a renormalizable theory have a dimensionality which is one higher than this maximum dimension this makes it plausible that the required set of counterterms not involving derivatives of the riemann tensor is linearly independent in arbitrary dimensions in the appendix a relation cubic in the weyl tensor which relates invariants cubic in the riemann tensor in five dimensions is proved

some characterizations of space forms integral conditions on the traceless ricci tensor are used to characterize euclidean and hyperbolic spaces among complete locally conformally flat manifolds of constant scalar curvature the main tools are vanishing type results for l p solutions of a large class of differential inequalities further applications of the technique are also given

a novel time marching scheme using numerical green s functions a comparative study for the scalar wave equation abstract the present paper presents the formulation of a novel time marching method based on the explicit green s approach exga to solve scalar wave propagation problems by means of the weighted residual method in both time and space the time integral expression concerning the exga is readily established the arising exga time integral expression is spatially discretized in a finite element sense and a recursive scheme that employs time domain numerical green s function matrices is adopted to evaluate the displacement and the velocity vectors these green s matrices are computed by the time discontinuous galerkin finite element method only at the first time step the system of coupled equations originated from the time discontinuous galerkin method is then solved by an iterative predictor multicorrector algorithm once the green s matrices are computed no iterative process is required to obtain the displacement and the velocity vectors at any time level at the end of the paper numerical examples are presented in order to compare the proposed approach with other approaches

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

the wave equation for the lanczos potential i the non local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor c abcd of weyl for this field lanczos found a tensor potential l abc with 16 independent components we can make l abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions l smallmatrix ab s s endsmallmatrix 0 both fields c abcd l abc satisfy wave equations the wave equation satisfied by l abc is nonlinear even in vacuo however a linear spinor wave equation for the lanczos potential has been found by illge but no correct tensor wave equation for l abc has yet been published here we derive a correct tensor wave equation for l abc and when it is simplified with the aid of some four dimensional identities it is equivalent to illge s wave equation we also show that the nonlinear spinor wave equation of penrose for the weyl field can be derived from illge s spinor wave equation a set of analogues of well known results of classical electromagentic radiation theory can now be given we indicate how a green s function approach to gravitational radiation could be based on our tensor wave equation when a global study of space time is attempted

a comparative study on three boundary element method approaches to problems in elastodynamics in this study three boundary element method approaches are compared by solving a typical problem in linear elastodynamics the first approach formulates and solves the problem in the real time domain in conjunction with a time stepping algorithm the other two approaches formulate and solve the problem in the laplace and the fourier transformed domains which necessitates a numerical inversion of the results to the time domain the results obtained compare favourably with available analytic solutions a detailed tabulation of the computer time and memory requirements of each approach is presented

two boundary integral equation methods for linear elastodynamics problems on unbounded domains abstract we consider transient 3d elastic wave propagation problems in unbounded isotropic homogeneous media which can be reduced to corresponding 2d ones for their solution we propose and compare two boundary integral equation approaches both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization in the first approach the pde problem is preliminarily replaced by the equivalent well known vector space time boundary integral equation formulation while in the second the same pde is replaced by a system of two coupled wave equations each one of which is then represented by the associated boundary integral equation the construction of these two approaches is described and discussed some numerical testing are also presented

a boundary integral equation method for radiation and scattering of elastic waves in three dimensions etude du rayonnement et de la diffusion d ondes elastiques par des obstacles de forme arbitraire

on effective constraints for the riemann lanczos system of equations there have been conflicting points of view concerning the riemann lanczos problem in three and four dimensions using direct differentiation on the defining partial differential equations massa and pagani in four dimensions and edgar in dimensions n 3 have argued that there are effective constraints so that not all riemann tensors can have lanczos potentials using cartan s criteria of integrability of ideals of differential forms bampi and caviglia have argued that there are no such constraints in dimensions n 4 and that in these dimensions all riemann tensors can have lanczos potentials in this article we give a simple direct derivation of a constraint equation confirm explicitly that known exact solutions of the riemann lanczos problem satisfy it and argue that the bampi and caviglia conclusion must therefore be flawed in support of this we refer to the recent work of dolan and gerber on the three dimensional problem by a method closely related to that of bampi and caviglia they have fou

rigidity of shrinking ricci solitons we show that a compact ricci soliton is rigid if and only if the weyl conformal tensor is harmonic in the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the ricci tensor is bounded from below

identities of the scalars of the four dimensional riemannian manifold a general algorithm is presented for the calculation of identities relating scalars constructed from appropriate contractions of the riemann tensor ricci tensor and curvature scalar the algorithm is applied to determining the identities among the quartic scalars

old and new results for superenergy tensors from dimensionally dependent tensor identities it is known that some results for spinors and in particular for superenergy spinors are much less transparent and require a lot more effort to establish when considered from the tensor viewpoint in this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4 dimensional identities by straightforward tensor methods in a signature independent manner in particular we consider the quadratic identity for the bel robinson tensor tabcxtabcy xy tabcdtabcd 4 and also the new conservation law for the chevreton tensor both of which have been obtained by spinor means both of these results are rederived by tensor means for 4 dimensional spaces of any signature using dimensionally dependent identities and moreover we are able to conclude that there are no direct higher dimensional analogs in addition we demonstrate a simple way to show the nonexistense of such identities via counter examples in particular we show that there is no nontrivial bel ten

generalized kronecker and permanent deltas their spinor and tensor equivalents and applications the aim of this paper is fourfold i to introduce a generalized permanent delta on an equal footing with the generalized kronecker delta to use for the symmetries of any tensor or spinor ii to cite an ancillary reference source of comprehensive tensorial and spinorial combinatorial formulas for both iii to table spinor equivalents of these individual tensors and give examples of their usage and iv to tabulate the tensor equivalents of various useful combinations of their spinor forms

rigidity of einstein metrics as critical points of quadratic curvature functionals on closed manifolds abstract in this paper we prove some rigidity results for the einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds characterized by some point wise inequalities moreover we also provide a few rigidity results that involve the weyl curvature the trace less ricci curvature and the yamabe invariant accordingly

integral pinched gradient shrinking ρ einstein solitons abstract the gradient shrinking einstein soliton is a triple m n g f such that r i j f i j r g i j where m n g is a riemannian manifold 0 r 0 and f is the potential function on m n in this paper using algebraic curvature estimates and the yamabe sobolev inequality we prove some integral pinching rigidity results for compact gradient shrinking einstein solitons