topological noetherianity of polynomial functors we prove that any finite degree polynomial functor over an infinite field is topologically noetherian this theorem is motivated by the recent resolution by ananyan hochster of stillman s conjecture and a recent noetherianity proof by derksen eggermont snowden for the space of cubics via work by erman sam snowden our theorem implies stillman s conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees

gl equivariant modules over polynomial rings in infinitely many variables consider the polynomial ring in countably infinitely many variables over a field of characteristic zero together with its natural action of the infinite general linear group g we study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible g action we define and prove finiteness properties for analogues of hilbert series systems of parameters depth local cohomology koszul duality and regularity we also show that this category is built out of a simpler more combinatorial quiver category which we describe explicitly our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions for example the paper by church ellen berg and farb on the category of fi modules which is equivalent to our category along the way we see several connections with the character polynomials from the representation theory of the symmetric groups several examples are given to illustrate that the invariants we introduce are explicit and computable

kp hierarchy for hodge integrals abstract starting from the elsv formula we derive a number of new equations on the generating functions for hodge integrals over the moduli space of complex curves this gives a new simple and uniform treatment of certain known results on hodge integrals like witten s conjecture virasoro constrains faber s g conjecture etc among other results we show that a properly arranged generating function for hodge integrals satisfies the equations of the kp hierarchy

commutators and images of noncommutative polynomials abstract let a be an algebra and let f be a nonconstant noncommutative polynomial in the first part of the paper we consider the relationship between a a the linear span of commutators in a and span f a the linear span of the image of f in a in particular we show that a a a implies span f a a in the second part we establish some waring type results for images of polynomials for example we show that if c is a commutative unital algebra over a field f of characteristic 0 a is the matrix algebra m n c and the polynomial f is neither an identity nor a central polynomial of m n f then every commutator in a can be written as a difference of two elements each of which is a sum of 7788 elements from f a if c f is an algebraically closed field then 4 elements suffice similar results are obtained for some other algebras in particular for the algebra b h of all bounded linear operators on a hilbert space h

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

grobner methods for representations of combinatorial categories given a category c of a combinatorial nature we study the following fundamental question how does the combinatorial behavior of c affect the algebraic behavior of representations of c we prove two general results the first gives a combinatorial criterion for representations of c to admit a theory of grobner bases from this we obtain a criterion for noetherianity of representations the second gives a combinatorial criterion for a general rationality result for hilbert series of representations of c this criterion connects to the theory of formal languages and makes essential use of results on the generating functions of languages such as the transfer matrix method and the chomsky schutzenberger theorem our work is motivated by recent work in the literature on representations of various specific categories our general criteria recover many of the results on these categories that had been proved by ad hoc means and often yield cleaner proofs and stronger statements for example we give a new more robust proof that fi modules originally introduced by church ellenberg farb and a family of natural generalizations are noetherian we give an easy proof of a generalization of the lannes schwartz artinian conjecture from the study of generic representation theory of finite fields we significantly improve the theory of modules introduced by snowden in connection to syzygies of segre embeddings and we establish fundamental properties of twisted commutative algebras in positive characteristic

finiteness properties of congruence classes of infinite by infinite matrices abstract we look at affine spaces of n n matrices over a field k and consider varieties that are stable under simultaneous row and column operations we prove that up to symmetry any of these varieties is defined by finitely many equations

the character of the infinite wedge representation abstract we study the character of the infinite wedge projective representation of the algebra of differential operators on the circle we prove quasi modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions these correlation functions are sums of determinants built from genus 1 theta functions and their derivatives

finite generation of symmetric ideals let be a commutative noetherian ring and let be the polynomial ring in an infinite collection of indeterminates over let be the group of permutations of the group acts on in a natural way and this in turn gives the structure of a left module over the group ring we prove that all ideals of invariant under the action of are finitely generated as modules the proof involves introducing a certain well quasi ordering on monomials and developing a theory of grobner bases and reduction in this setting we also consider the concept of an invariant chain of ideals for finite dimensional polynomial rings and relate it to the finite generation result mentioned above finally a motivating question from chemistry is presented with the above framework providing a suitable context in which to study it

the possible dimensions of the higher secant varieties let x c p be an irreducible closed subvariety of projective space the kth higher secant variety of x denoted x is the closure of the union of all linear spaces spanned by k points of x let a ix dim x dim x l it is known that the sequence ty x is nonincreasing and that there may be at most one 1 in such a sequence we prove that these two conditions are also sufficient conditions for an arbitrary sequence a of nonnegative integers to be the fl x s for an irreducible variety x to this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals this allows us to compute their dimensions showing that there is in fact always a rational normal scroll x such that ak x a for any sequence a satisfying the two conditions above adlandsvik al a2 and flenner and vogel f v much of this work has centered on the behavior of the differences ak x dimx dimx 1 of the dimensions of consecutive higher secant varieties it is well known that this sequence is nonincreasing and that there is at most one 1 in the sequence palatini claimed moreover that once ak x 1 were counterexamples a2 we answer here the question of what sequences actually appear as the ai x s of some variety x some of the guiding examples of varieties with degenerate higher secant varieties come from rational normal scrolls explicit equations for the scrolls themselves have been known for some time cf acgh we give a determi nantal presentation of the ideals of their higher secant varieties generalizing a result for rational normal scrolls in w this presentation enables us to compute the dimensions of all the higher secant varieties of a rational normal scroll and gives examples of varieties x with a prescribed nonincreasing ak x with at most one 1

integral similarity and commutators of integral matrices abstract let f be a field m n f the algebra of n n matrices over f and a m n f with trace a 0 the following facts are well known i if a is not a scalar then a is similar over f to a matrix with zero diagonal ii a p q pq qp for some p q m n f we consider the situation when f is replaced by the ring of integers z we show that ii holds in this case for every n 1 this result has been proved for n 2 by lissner and vaserstein independently we show also that i holds if f is replaced by z for n 2 provided a ai mod p for all integers a and primes p

an algebraic theory of graph reduction we show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules the method is constructive in the sense that we describe an algorithm which will produce from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth k a set of rewrite rules that reduces any member of the class to one of finitely many graphs in a number of steps bounded by the size of the graph this reduction system corresponds to an algorithm that runs in time linear in the size of the graph

phylogenetic complexity of the kimura 3 parameter model abstract in algebraic statistics the kimura 3 parameter model is one of the most interesting and classical phylogenetic models we prove that the ideals associated to this model are generated in degree four confirming a conjecture by sturmfels and sullivant

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

parallel algorithms with optimal speedup for bounded treewidth we describe the first parallel algorithm with optimal speedup for constructing minimum width tree decompositions of graphs of bounded treewidth on n vertex input graphs the algorithm works in o log n 2 time using o n operations on the erew pram we also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a variety of problems on graphs of bounded treewidth including all decision problems expressible in monadic second order logic on n vertex input graphs the algorithms use o n operations together with o log n log n time on the erew pram or o log n time on the crcw pram

a finite volume method with linearisation in time for the solution of advection reaction diffusion systems the numerical solution in one space dimension of advection reaction diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models this work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest the basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method the resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations

representations under ring extensions latimer macduffee and taussky correspondences c g latimer and c c macduffee defined in 8 a bijective correspondence between similarity classes of nonsingular rational integral n x n matrices having a common separable characteristic polynomialf x and the classes of faithful ideals in the ring z x f x z x z the ring of rational integers several authors have used this simple duality with the ideas indigenous to one theory to obtain valuable consequences in the other see 14 and 15 f or a survey thereby motivating further analysis of the correspondence as in 1 2 4 17 and 181 nevertheless the theory has generally been catagorized as a subset of module theory of orders in semisimple algebras following zassenhaus s proof of jordan s theorem on the finiteness of such similarity classes see 7 14 and 21 i the purpose of this note is to replace the semisimple assumption present in these previous studies with the weaker condition that the modules considered become trivial under a suitable extension of the base ring the ease in which matrix and ideal theory are transferable under the latimer macduffee duality prompted our adherence to their approach when possible in this development over arbitrary commutative rings we review prior matrix ideal correspondences in section 2 preliminary to the general framework in section 3 there an n x ii matrix class is replaced by the isomorphism class of a rank n projective r module m which is also a faithful module over a rank n projective r algebra a the ideal theory is obtained by extending the base ring r of scalars to a commutative ring s for which m becomes trivial s ll mn s or a as s or a modules and m is then identified with a fractional a ideal in s or a in the classical case s is the field of quotients of a domain r theorem 1 reduces to the latimer macduffee duality in the latter case theorem 2 shows that the theory is equally adaptable to any commutative rank iz projective r algebra a which is isomorphic to its dual over a suitable extension of r as an application we show that a nonderogatory matrix is similar to the companion matrix of its

algebraic algorithms for sampling from conditional distributions we construct markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic examples include contingency tables logistic regression and spectral analysis of permutation data the algorithms involve computations in polynomial rings using grobner bases

reduction algorithms for graphs of small treewidth this paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth s arnborg b courcelle a proskurowski and d seese j assoc comput mach 40 1134 1164 1993 have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth using a finite set of reduction rules these algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first we show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth including all problems definable in monadic second order logic we also show that a variant of these reduction algorithms can be used to solve constructive optimization problems in o n time for example optimization and construction variants of independent set and hamiltonian completion number can be solved in this way on graphs of small treewidth additionally we show that the results of h l bodlaender and t hagerup siam j comput 27 1725 1746 1998 can be applied to our reduction algorithms which results in parallel reduction algorithms that use o n operations and o log n log n time on an erew pram or o log n time on a crcw pram

finiteness for the k factor model and chirality varieties this paper deals with two families of algebraic varieties arising from applications first the k factor model in statistics consisting of n times n covariance matrices of n observed gaussian variables that are pairwise independent given k hidden gaussian variables second chirality varieties inspired by applications in chemistry a point in such a chirality variety records chirality measurements of all k subsets among an n set of ligands both classes of varieties are given by a parameterisation while for applications having polynomial equations would be desirable for instance such equations could be used to test whether a given point lies in the variety we prove that in a precise sense which is different for the two classes of varieties these equations are finitely characterisable when k is fixed and n grows

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

meta kernelization in a parameterized problem every instance i comes with a positive integer k the problem is said to admit a polynomial kernel if in polynomial time one can reduce the size of the instance i to a polynomial in k while preserving the answer in this work we give two meta theorems on kernelization the first theorem says that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus these theorems unify and extend all previously known kernelization results for planar graph problems

ideals of bounded rank symmetric tensors are generated in bounded degree we study the syzygies of secant ideals of veronese subrings of a fixed commutative graded algebra over a field of characteristic 0 one corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any veronese embedding of a projective scheme x can be bounded by a constant that only depends on i r and x and not on the choice of the veronese embedding

a finite volume formulation of compact central schemes on arbitrary structured grids in the present paper a formulation allowing the use of compact schemes in the finite volume context on arbitrary meshes is presented the proposed formulation is based on the use of an implicit formula to evaluate the fluxes on the cell faces a series of numerical experiments for a 2d model convection equation a flat plate a subsonic vortical problem as well as the les simulation of channel flow has been carried out the results indicate an important improvement in obtained accuracy compared to a standard central finite volume formulation

comment on a comparison of the dispersion relations for anisotropic elastodynamic finite difference grids h bernth and c chapman 2011 geophysics 76 no 3 wa43 wa50 i have read with interest bernth and chapman 2011 and congratulate the authors for much careful hard work however i would like to bring to the authors attention to three maybe misleading points the first point concerns the definition of the grid cell figure 1b of bernth and chapman 2011 illustrates the correct elementary cell of the rotated staggered grid saenger et al 2000 saenger and bohlen 2004 one set of model parameters density and 21 stiffnesses defines the properties of the illustrated cube however figure 1a of berneth and

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

a posteriori error analysis of a cell centered finite volume method for semilinear elliptic problems in this paper we conduct a goal oriented a posteriori analysis for the error in a quantity of interest computed from a cell centered finite volume scheme for a semilinear elliptic problem the a posteriori error analysis is based on variational analysis residual errors and the adjoint problem to carry out the analysis we use an equivalence between the cell centered finite volume scheme and a mixed finite element method with special choice of quadrature

algebraic factor analysis tetrads pentads and beyond factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables known as factors and all the random variables follow a multivariate normal distribution the parameter space of a factor analysis model is a subset of the cone of positive definite matrices this parameter space is studied from the perspective of computational algebraic geometry grobner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space these polynomials known as model invariants arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix besides revealing the geometry of the factor analysis model the model invariants also furnish useful statistics for testing goodness of fit

similarity and commutators of matrices over principal ideal rings we prove that if is a principal ideal ring and is a matrix with trace zero then is a commutator that is for some this generalises the corresponding result over fields due to albert and muckenhoupt as well as that over due to laffey and reams and as a by product we obtain new simplified proofs of these results we also establish a normal form for similarity classes of matrices over pids generalising a result of laffey and reams this normal form is a main ingredient in the proof of the result on commutators

reply to the comment on a comparison of the dispersion relations for anisotropic elastodynamic finite difference grids h bernth and c chapman 2011 geophysics 76 no 3 wa43 wa50 by e h saenger published in november december 2011 we acknowledge saenger s 2011 comments and welcome the opportunity to clarify the potentially misleading points throughout bernth and chapman 2011 we used the cartesian grid cell as illustrated in their figure 1a saenger 2011 rightly points out that the elementary cell is an octahedron his figure 1 we used the cartesian grid cell for simplicity and so it decomposes directly into virieux cells provided that every reference

noetherianity up to conjugation of locally diagonal inverse limits abstract we prove that the inverse limit of the sequence dual to a sequence of lie algebras is noetherian up to the action of the direct limit of the corresponding sequence of classical algebraic groups when the sequence of groups consists of diagonal embeddings we also classify all conjugation stable closed subsets of the space of n n matrices

equivariant grobner bases and the gaussian two factor model exploiting symmetry in grobner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables this is largely due to the fact that the group elements being invertible cannot preserve a term order by contrast inspired by work of aschenbrenner and hillar we introduce the concept of equivariant grobner basis in a setting where a monoid acts by homomorphisms on monomials in potentially infinitely many variables we require that the action be compatible with a term order and under some further assumptions derive a buchberger type algorithm for computing equivariant grobner bases using this algorithm and the monoid of strictly increasing functions we prove that the kernel of the ring homomorphism y ij i j i j s i t i i y ij s i s j t i t j is generated by two types of polynomials off diagonal 3 x 3 minors and pentads this confirms a conjecture by drton sturmfels and sullivant on the gaussian two factor model from algebraic statistics

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

plucker varieties and higher secants of sato s grassmannian every grassmannian in its plucker embedding is defined by quadratic polynomials we prove a vast qualitative generalisation of this fact to what we call plucker varieties a plucker variety is in fact a family of varieties in exterior powers of vector spaces that like the grassmannian is functorial in the vector space and behaves well under duals a special case of our result says that for each fixed natural number k the k th secant variety of any plucker embedded grassmannian is defined in bounded degree independent of the grassmannian our approach is to take the limit of a plucker variety in the dual of a highly symmetric space known as the infinite wedge and to prove that up to symmetry the limit is defined by finitely many polynomial equations for this we prove the auxilliary result that for every natural number p the space of p tuples of infinite by infinite matrices is noetherian modulo row and column operations our results have algorithmic counterparts every bounded plucker variety has a polynomial time membership test and the same holds for zariski closed basis independent properties of p tuples of matrices

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

finite grobner bases in infinite dimensional polynomial rings and applications abstract we introduce the theory of monoidal grobner bases a concept which generalizes the familiar notion in a polynomial ring and allows for a description of grobner bases of ideals that are stable under the action of a monoid the main motivation for developing this theory is to prove finiteness results in commutative algebra and applications a basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry using this machinery we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of hosten and the second author

hilbert series of symmetric ideals in infinite polynomial rings via formal languages abstract let r be the polynomial ring k x i j where 1 i r and j n and let i be an ideal of r stable under the natural action of the infinite symmetric group s nagel romer recently defined a hilbert series h i s t of i and proved that it is rational we give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series

bounded rank tensors are defined in bounded degree matrices of rank at most k are defined by the vanishing of polynomials of degree k 1 in their entries namely their k 1 k 1 subdeterminants regardless of the size of the matrix we prove a qualitative analogue of this statement for tensors of arbitrary dimension where matrices correspond to two dimensional tensors more specifically we prove that for each k there exists an upper bound d d k such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d regardless of the dimension of the tensor and regardless of its size in each dimension our proof involves passing to an infinite dimensional limit of tensor powers of a vector space whose elements we dub infinite dimensional tensors and exploiting the symmetries of this limit in crucial ways

a compact finite volume method and its extrapolation for elliptic equations with third boundary conditions a fourth order compact finite volume method is constructed for one and two dimensional elliptic equations with third boundary conditions in this paper taking two point boundary value problem of third kind as an example we derive some useful high accuracy post processing formulas to recover the numerical derivatives at the nodes or midpoints of the elements we also improve the accuracy of the compact finite volume scheme from order 4 to 6 based on richardson extrapolation by rigorously proving the scheme has error asymptotic expansion numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme as well as its post processing formulas and extrapolation

a finite volume method for one dimensional reaction diffusion problems abstract a truly conservative finite volume method for one dimensional reaction diffusion equations is presented the method is based on piecewise second degree polynomials approximations that result in linear fluxes a symmetric mass matrix and a semi discrete system of nonlinear ordinary differential equations in time it is shown that the semi discrete method is linearly stable provided that the problem is mathematically well posed it is also shown that implicit discretizations of the time variable are stable whereas the explicit one is conditionally stable and has a smaller domain of linear stability than the standard explicit finite difference method it is also illustrated that upon lumping the mass matrix the method coincides with a finite difference discretization of the original problem a detailed formulation of the dirichlet neumann and robin boundary conditions is presented such a formulation makes use of control volumes which are here referred to as incomplete ones although the finite volume technique presented here uses a node centered formulation it can be easily extended to cell centered ones the technique is also developed for multidimensional problems

a comparison of the dispersion relations for anisotropic elastodynamic finite difference grids several staggered grid schemes have been suggested for performing finite difference calculations for the elastic wave equations in this paper the dispersion relationships and related computational requirements for the lebedev and rotated staggered grids for anisotropic elastic finite difference calculations in smooth models are analyzed and compared these grids are related to a popular staggered grid for the isotropic problem the virieux grid the lebedev grid decomposes into virieux grids two in two dimensions and four in three dimensions which decouple in isotropic media therefore the lebedev scheme will have twice or four times the computational requirements memory and cpu as the virieux grid but can be used with general anisotropy in two dimensions the rotated staggered grid is exactly equivalent to the lebedev grid but in three dimensions it is fundamentally different the numerical dispersion in finite difference grids depends on the direction of propagation and the grid type and parameters a joint numerical dispersion relation for the two grids types in the isotropic case is derived in order to compare the computational requirements for the two grid types the dispersion averaged over propagation direction and medium velocity are calculated setting the parameters so the average dispersion is equal for the two grids the computational requirements of the two grid types are compared in three dimensions the rotated staggered grid requires at least 20 more memory for the field data and at least twice as many number of floating point operations and memory accesses so the lebedev grid is more efficient and is to be preferred