on a two grid finite element scheme combined with crank nicolson method for the equations of motion arising in the kelvin voigt model in this paper we present a second order accurate crank nicolson scheme applied to a system arising from galerkin finite element two grid approximations to the equations of motion described by the kelvin voigt viscoelastic fluid flow model optimal error estimates in l l 2 norm and l h 1 norm for velocity and l l 2 norm for pressure are established these estimates are shown to be uniform in time numerical results are provided to verify the theoretical estimates

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

generating a residual intersection let i be an ideal of height g 0 in a commutative noetherian ring r and let 2i be a proper sub ideal of i generated by 3 g elements if the ideal j 21 i has height at least s then j is called an s residual intersection of i see artin and nagata 1 huneke 19 or huneke and ulrich 22 the case s g corresponds to the theory of linkage and is well understood at least from the homolo cal point of view the cornerstone of this theory is that if i is perfect then a free resolution of r j can be produced from a free resolution of r z and the koszul complex on a set of generators of cu very little is known about resolutions of s residual intersections for s g a few special cases have been resolved s 10 301 and the end of the resolution or in other words a set of generators for the canonical module of r j has been described 22 however in general even the generators of j remain unknown here we show that j can be approximated by a sum of links of i if s g 1 and we give conditions under which this approximation in fact gives a full set of generators of j since these generators arise from comparison maps from koszul complexes to known resolutions at least assuming that the resolution of r i is in some sense known it is feasible to count them compute their degrees in the graded case or even to give explicit formulas for them after stating our main results theorem 2 15 and 2 16 and corollary 2 18 precisely in section 2 we lay out the argument in sections 3 4 and 5 in section 6 we are concerned with the question of when a g t 1 residual intersection j is

a spectral element method for the navier stokes equations with improved accuracy we present a procedure to enhance the accuracy of spectral element methods for the evolutionary navier stokes equations the approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete stokes problem only once when the time evolution is completed the analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure the choice of a discrete pressure space endowed with an inf sup condition independent of the discretization parameter leads to optimal error bounds for the pressure

tight double linkage of gorenstein algebras abstract a gorenstein ideal k in a local ring r is in the class if there is a sequence of linked ideals i 0 i 1 i 2 n k with i 0 a complete intersection and i 2 i gorenstein for all i if k is in then k is linked to a complete intersection by a sequence of tight double links this means there is a sequence i 0 j 0 j 1 j 2m k such that for each p j 2 p 1 is an almost complete intersection of the form b y w and its immediate neighbors are linked to it by the similar regular sequences b y and b w respectively in this sense j 2 p and j 2 p 2 are tightly linked gorenstein ideals let k be an algebraically closed field and a k x 1 x q k with k in the k algebra a is rigid if and only if it is linked to a regular complete intersection by a sequence of semi generic tight double links it follows that if a is rigid then it is regular in codimension six

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

linkage and sums of ideals it is shown under mild conditions that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection for a sum of geometrically linked cohen macaulay ideals we compute the depths of the conormal module arld the first koszul homology as applications we construct general examples of ideals which are strongly cohenmacaulay strongly nonobstructed but not in the linkage class of a complete intersection and gorenstein ideals which are strongly nonobstructed but not syzygetic

an alternating matrix and a vector with application to aluffi algebras abstract let x be a generic alternating matrix t be a generic row vector and j be the ideal pf 4 x i 1 tx we prove that j is a perfect gorenstein ideal of grade equal to the grade of pf 4 x plus two this result is used by ramos and simis in their calculation of the aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface we also prove that j defines a domain or a normal ring or a unique factorization domain if and only if the base ring has the same property the main object of study in the present paper is the module n which is equal to the column space of x calculated mod pf 4 x the module n is a self dual maximal cohen macaulay module of rank two furthermore j is a bourbaki ideal for n the ideals which define the homogeneous coordinate rings of the plucker embeddings of the schubert subvarieties of the grassmannian of planes are used in the study of the module n

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

analysis of two grid method for semi linear elliptic equations by new mixed finite element scheme this article considers two grid stable mixed finite element method based on the less regularity of flux velocity in practice for the semi linear elliptic equations approximated by the p 0 p 1 velocity pressure pair which satisfies the inf sup condition this method involves solving a small semi linear system on a coarse mesh with mesh size h and a linear system problem on a fine mesh with mesh size h o h which can still maintain the asymptotically optimal accuracy it provides an approximate solution with the convergence rate of the same order as the usual mixed finite element solution solving the semi linear elliptic equations on a fine mesh hence the two grid scheme can reduce the computational cost finally numerical tests confirm the theoretical results of the present method

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

classification of the tor algebras of codimension four almost complete intersections let r m k be a local ring in which 2 is a unit assume that every element of k has a square root in k we classify the algebras tor r r j k as j varies over all grade four almost complete intersection ideals in r the analogous classification has already been found when j varies over all grade four gorenstein ideals 21 and when j varies over all ideals of grade at most three 5 30 the present paper makes use of the classification in 21 of the tor algebras of codimension four gorenstein rings as well as the usually nonminimal dg algebra resolution of a codimension four almost complete intersection which is produced in 25 and 26

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

on a conjecture of auslander and reiten abstract in studying nakayama s 1958 conjecture on rings of infinite dominant dimension auslander and reiten proposed the following generalization let be an artin algebra and m a generator such that ext i m m 0 for all i 1 then m is projective this conjecture makes sense for any ring we establish auslander and reiten s conjecture for excellent cohen macaulay normal domains containing the rational numbers and slightly more generally

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

a second order accuracy for a full discretized time dependent navier stokes equations by a two grid scheme we study a second order two grid scheme fully discrete in time and space for solving the navier stokes equations the two grid strategy consists in discretizing in the first step the fully non linear problem in space on a coarse grid with mesh size h and time step t and in the second step in discretizing the linearized problem around the velocity u h computed in the first step in space on a fine grid with mesh size h and the same time step the two grid method has been applied for an analysis of a first order fully discrete in time and space algorithm and we extend the method to the second order algorithm this strategy is motivated by the fact that under suitable assumptions the contribution of u h to the error in the non linear term is measured in the l 2 norm in space and time and thus has a higher order than if it were measured in the h 1 norm in space we present the following results if h 2 h 3 t 2 then the global error of the two grid algorithm is of the order of h 2 the same as would have been obtained if the non linear problem had been solved directly on the fine grid

postprocessing fourier galerkin method for the navier stokes equations a full discrete two level postprocessing fourier galerkin scheme for the unsteady navier stokes equations with a periodic boundary condition is proposed in this paper by defining a new projection the interaction between the large and small eddies is reflected by the associated space splitting to some extent therefore a weakly coupled system of the large and small eddies is obtained stability and error estimates for the weakly coupled two level scheme are established in the paper the proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse level subspace by solving the coarse level standard galerkin equation and the matrix of the linear algebraic equations arising in each fine level time stepping is a sparse matrix especially for large scale computations as a result the new scheme saves a lot of cpu time and memory in comparison with the standard fourier galerkin method for deriving an approximation of prescribed accuracy

investigations on two kinds of two level stabilized finite element methods for the stationary navier stokes equations abstract this article considers two kinds of two level stabilized finite element methods based on local gauss integral technique for the two dimensional stationary navier stokes equations approximated by the lowest equal order elements which do not satisfy the inf sup condition the two level methods consist of solving a small non linear system on the coarse mesh and then solving a linear system on the fine mesh the error analysis shows that the two level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the navier stokes equations on a fine mesh for a related choice of mesh widths therefore the two level methods are of practical importance in scientific computation finally the performance of two kinds of two level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments the conclusion is that simple two level stabilized method is a viable choice for the lowest equal order approximations of the stationary navier stokes problem

on improving the efficiency of tensor voting this paper proposes two alternative formulations to reduce the high computational complexity of tensor voting a robust perceptual grouping technique used to extract salient information from noisy data the first scheme consists of numerical approximations of the votes which have been derived from an in depth analysis of the plate and ball voting processes the second scheme simplifies the formulation while keeping the same perceptual meaning of the original tensor voting the stick tensor voting and the stick component of the plate tensor voting must reinforce surfaceness the plate components of both the plate and ball tensor voting must boost curveness whereas junctionness must be strengthened by the ball component of the ball tensor voting two new parameters have been proposed for the second formulation in order to control the potentially conflictive influence of the stick component of the plate vote and the ball component of the ball vote results show that the proposed formulations can be used in applications where efficiency is an issue since they have a complexity of order o 1 moreover the second proposed formulation has been shown to be more appropriate than the original tensor voting for estimating saliencies by appropriately setting the two new parameters

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

semi invariants for quivers with relations for a given quiver q and a dimension vector let si q be the ring of semi invariants for dimensional representations schofield defined in s for any dimensional representation v with 0 a semi invariant c si q we recall that we proved in dw that the ring of semi invariants si q is generated by such schofield semi invariants in this note we generalize the results of dw to quivers with relations in the case when the base field has characteristic zero we make this assumption throughout the paper let q be a quiver and let i be an ideal generated by admissible relations the variety rep q i of representations of q satisfying the relations from i does not have to be irreducible we are interested in the generators of rings of semiinvariants on the irreducible components of rep q i it is clear that the ring of semi invariants on such components will be generated by the restrictions of the semi invariants c where v is a representation of q however such description is not very useful we might have a case when the quiver q with relations i is of a finite or tame type but q is wild in such case one would still like to describe the semi invariants of q i explicitly but the description of indecomposable representations of q is impossible therefore it

on the variety of complexes publisher summary this chapter focuses on the characteristic free study of the buchsbaum eisenbud varieties of complexes in detail with the use of young diagrams and young tableaux to give an explicit basis for buchsbaum and eisenbud have studied the varieties of complexes from the point of view of resolutions and have proven the results for the variety w k 1 a young tableau is a filling of the boxes of a young diagram with integers out of 1 2 and is called the shape of the young tableau the chapter presents a proposition that states that ach element in can be written as a linear combination of standard multitableaux

dimensionality estimation manifold learning and function approximation using tensor voting we address instance based learning from a perceptual organization standpoint and present methods for dimensionality estimation manifold learning and function approximation under our approach manifolds in high dimensional spaces are inferred by estimating geometric relationships among the input instances unlike conventional manifold learning we do not perform dimensionality reduction but instead perform all operations in the original input space for this purpose we employ a novel formulation of tensor voting which allows an n d implementation tensor voting is a perceptual organization framework that has mostly been applied to computer vision problems analyzing the estimated local structure at the inputs we are able to obtain reliable dimensionality estimates at each instance instead of a global estimate for the entire data set moreover these local dimensionality and structure estimates enable us to measure geodesic distances and perform nonlinear interpolation for data sets with varying density outliers perturbation and intersections that cannot be handled by state of the art methods quantitative results on the estimation of local manifold structure using ground truth data are presented in addition we compare our approach with several leading methods for manifold learning at the task of measuring geodesic distances finally we show competitive function approximation results on real data

hilbert function of powers of ideals of low codimension we study the hilbert function of the powers of homogeneous ideals which are either cohen macaulay of codimension 2 or gorenstein of codimension 3 we show that that if i is an ideal in one of these classes and it is of linear type then for all k the hilbert function of i k depends only on the hilbert function of i in other words if i andj are ideals in one of the above mentioned classes which are both of linear type and they have the same hilbert function then also i k and j k have the same hilbert function for all k

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

a finite element variational multiscale method for incompressible flow in this paper we present a numerical scheme prove stability existence of solutions uniqueness and convergence of the incompressible navier stokes equations it has the advantage of being defined from strictly algebraic considerations a significant feature of the present method is that the structure of the stabilization term based on the multiscale enrichment and derived from the navier stokes problem itself ample numerical experiments are carried out to confirm the theory and illustrate the effectiveness of the scheme on incompressible fluid

on a hierarchical basis multilevel method with nonconforming p1 elements we develop a hierarchical basis multilevel method for symmetric second order elliptic boundary value problems in two dimensional polygonal domains based on nonconforming p1 triangular elements the main result is that the condition number of the hierarchical discretization is bounded byo k wherek is the number of refinement levels for comparison yserentant s result yields the sharp boundo k 2 for the corresponding hierarchical discretization with conforming linear elements

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

error estimates of two level finite element method for smagorinsky model two level finite element method for simulating smagorinsky model in large eddy simulation is investigated in this two level algorithm a linearized discrete problem is solved by using the variational multiscale method on the coarse mesh corresponding to newton linearization method the linearized smagorinsky model is solved on the fine mesh the error estimates derived in this paper implies that by choosing appropriate mesh sizes and the radius of the spatial filter used in smagorinsky model the two level method provides the optimal convergence rates for the velocity in h1 norm and the pressure in l2 norm meanwhile the two different numerical experiments are given to support the optimal convergence rates and the high efficiency of two level algorithm

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

on the divisor class group of rees algebras let r be a normal noetherian domain and let i be an ideal whose associated graded ring gr r is reduced barshay 2 showed that under these conditions the rees algebra s r it r 0 it 0 izt2 a e r t is normal a much simpler proof was given later by huneke 111 let p r it t be the extended rees algebra then pt e r t t 1 is normal and p tp e gr r is reduced it follows that p itself is normal this implies easily the normality of s as well we are interested in the class group cl s of s to obtain a neat result we strengthen our hypothesis and assume that gr r is a domain hochster 6 showed in this case that p is factorial if r is so the only tool needed to prove this is the following elementary but powerful theorem of nagata

inference of surfaces 3d curves and junctions from sparse noisy 3d data we address the problem of obtaining dense surface information from a sparse set of 3d data in the presence of spurious noise samples the input can be in the form of points or points with an associated tangent or normal allowing both position and direction to be corrupted by noise most approaches treat the problem as an interpolation problem which is solved by fitting a surface such as a membrane or thin plate to minimize some function we argue that these physical constraints are not sufficient and propose to impose additional perceptual constraints such as good continuity and cosurfacity these constraints allow us to not only infer surfaces but also to detect surface orientation discontinuities as well as junctions all at the same time the approach imposes no restriction on genus number of discontinuities number of objects and is noniterative the result is in the form of three dense saliency maps for surfaces intersections between surfaces i e 3d curves and 3d junctions respectively these saliency maps are then used to guide a marching process to generate a description e g a triangulated mesh making information about surfaces space curves and 3d junctions explicit the traditional marching process needs to be refined as the polarity of the surface orientation is not necessarily locally consistent these three maps are currently not integrated and this is the topic of our ongoing research we present results on a variety of computer generated and real data having varying curvature of different genus and multiple objects

linkage theory for algebras with pure resolutions abstract let s be a graded cohen macaulay quotient r i of a polynomial ring r k x 1 x n over an infinite field k if s is in the linkage class of a complete intersection then its koszul homology and twisted conormal module are cohen macaulay by passing to zero and one dimensional specializations of s one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection we prove that if s is generically a complete intersection and the minimal free solution of s is pure and almost linear then s is not in the linkage class of a complete intersection apart from a few obvious exceptions for algebras of small codimension the results are stronger if s has codimension three and has a pure resolution then s is in the linkage class of a complete intersection if and only if s is gorenstein if s is a codimension four gorenstein algebra with a pure resolution then s is in the linkage class of a complete intersection if and only if s is a complete intersection

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

a matrix of linear forms which is annihilated by a vector of indeterminates abstract let r k t 1 t f be a standard graded polynomial ring over the field k and be an f g matrix of linear forms from r where 1 g f assume t 1 t f is 0 and that grade i g is exactly one short of the maximum possible grade we resolve r r i g prove that r has a g linear resolution record explicit formulas for the h vector and multiplicity of r and prove that if f g is even then the ideal i g is unmixed furthermore if f g is odd then we identify an explicit generating set for the unmixed part i g unm of i g resolve r i g unm and record explicit formulas for the h vector of r i g unm the rings r i g and r i g unm automatically have the same multiplicity these results have applications to the study of the blow up algebras associated to linearly presented grade three gorenstein ideals

singularities of generic linkage of algebraic varieties let y be a generic link of a subvariety x of a nonsingular variety a we give a description of the grauert riemenschneider canonical sheaf of y in terms of the multiplier ideal sheaves associ ated to x and use it to study the singularities of y as the first application we give a criterion when y has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage as another application we give a quick and simple liaison method to generalize the results of de fernex ein and chardin ulrich on the castelnuovo mumford regularity bound for a projective variety

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

hodge dual operators and model algebras for rational representations of the general linear group abstract in this paper we construct a family of algebras each of whose members is a multiplicity free sum of irreducible rational representations of the general linear group gl n c we then use the properties of a generalized version of the hodge dual operator to determine an explicit basis for each of these algebras and by restriction we obtain an explicit basis for each of the irreducible rational representations of gl n c our results cleanly extends classical algebro combinatorial results on polynomial representations of gl n c to rational representations

inferring global perceptual contours from local features we address the problem of contour inference from partial data as obtained from state of the art edge detectors we argue that in order to obtain more pereeptually salient contours it is necessary to impose generic constraints such as continuity and co curvilinearity the implementation is in the form of a convolution with a mask which encodes both the orientation and the strength of the possible continuations we first show how the mask called the extension field is derived then how the contributions from different sites are collected to produce a saliency map we show that the scheme can handle a variety of input data from dot patterns to oriented edgels in a unified manner and demonstrate results on a variety of input stimuli we also present a similar approach to the problem of inferring contours formed by end points in both cases the scheme is non linear non iterative and unified in the sense that all types of input tokens are handled in the same manner

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

complexes which arise from a matrix and a vector resolutions of divisors on certain varieties of complexes abstract consider the polynomial ring r r0 x y where r0 is a normal domain and x1 g and yg f are matrices of indeterminates the r ideal j i1 xy imin f g y defines a variety of complexes over r0 the divisor class group of r j is isomorphic to cl r0 z i where i is an ideal of r j generated by appropriately chosen lower order minors of y we produce the minimal r free resolution of i i for all integers i 1 if f is greater than or equal to g then j is a generic residual intersection of the generic grade g complete intersection i1 x the resolutions that we produce in this case are in many ways analogous to resolutions of divisors on generic residual intersections of grade two perfect ideals or grade three gorenstein ideals

high rank linear syzygies on low rank quadrics we study the linear syzygies of a homogeneous ideal i subseteq s rm sym bf k v focussing on the graded betti numbers b i i 1 rm dim bf k rm tor i s i bf k i 1 for a variety x and divisor d with v h 0 d what conditions on d ensure that b i i 1 ne 0 eisenbud has shown that a decomposition d sim a b such that a and b have at least two sections gives rise to determinantal equations and corresponding syzygies in i x and conjectured that if i 2 is generated by quadrics of rank le 4 then the last nonvanishing b i i 1 is a consequence of such equations we describe obstructions to this conjecture and prove a variant the obstructions arise from toric specializations of the rees algebra of koszul cycles and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank this gives one answer to a question posed by eisenbud and koh

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

on the invariant theory for acyclic gentle algebras in this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions along the way we exhibit for such fields of rational invariants a transcendence basis in terms of schofield s determinantal semi invariants we also show that the moduli spaces of modules over regular irreducible component are just a products of projective spaces contents

on the minimal free resolution of the universal ring for resolutions of length two abstract hochster established the existence of a commutative noetherian ring c and a universal resolution u of the form 0 c e c f c g 0 such that for any commutative noetherian ring s and any resolution v equal to 0 s e s f s g 0 there exists a unique ring homomorphism c s with v u c s in the present paper we assume that f e g and we find the minimal resolution of c z q by free b modules where b is a polynomial ring over the field of rational numbers the modules of the resolution are described in terms of schur functors the graded strands of the differential are described in terms of pieri maps

two level picard and modified picard methods for the navier stokes equations abstract iterative methods of picard type for the navier stokes equations are known to converge only for quite small reynolds numbers however we study methods involving just one such iteration at general reynolds numbers for the initial approximation a coarse mesh of width h0 is used the corrected approximation is computed by just one picard or modified picard step on a fine mesh of width h1 for example h1 may be of order o h 0 2 when linear velocity elements are used the resulting method requires the solution of a small system of nonlinear equations on the coarse mesh and only one larger linear system on the fine mesh this two level picard method is proven to converge for fixed reynolds number as h 0 further the fine mesh solution satisfies a quasi optimal error bound the error constants grow as re as for the usual finite element method one very heuristic explanation why one step of the divergent picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower order derivatives thus they are approximated with higher accuracy on the coarse mesh this is linked with a smoothing property of the fine mesh step

arithmetic cohen macaulayness and arithmetic normality for schubert varieties let g be a semisimple simply connected chevalley group over a field k let t be a maximal torus b a borel subgroup d t and w the weyl group of g relative to t let p be a maximal parabolic subgroup of classical type cf 7 8 i e a maximal parabolic subgroup such that if w is the associated fundamental weight then i w av i 1 2 w c a is 2 for all roots ae being a w invariant scalar product on x t q where x t character group of t for t e w wp let x t denote the schubert variety in g p associated to t namely x t is the zariski closure btp mod p in g p of the bruhat cell endowed with the canonical reduced structure for the canonical projective embedding g p in p h g p l let rt denote the homogeneous co ordinate ring of x t then the main result of this paper is given by

the minimal free resolutions of the huneke ulrich deviation two gorenstein ideals structure theorems exist for gorenstein ideals of small grade a different parameter one can use for organizing ideals is deviation or complete intersection defect the deviation of the ideal i is the minimal number of generators of i minus the grade of i a deviation zero ideal is a complete intersection kunz s proved that there are no deviation one gorenstein ideals presently people are trying to find all deviation two gorenstein ideals see for example 3 a hypersurface section of a grade g 1 deviation two gorenstein ideal is a grade g deviation two gorenstein ideal we consider this to be a trivial example very few non trivial examples of deviation two gorenstein ideals are known the buchsbaum eisenbud structure theorem 2 describes all grade three gorenstein ideals in particular every deviation two grade three gorenstein ideal is generated by the maximal order pfaflians of a 5 x 5 alternating matrix no non trivial deviation two gorenstein ideals of even grade are known herzog and miller 3 have taken a step toward proving that all grade four deviation two gorenstein ideals are trivial let i be a grade four deviation two gorenstein ideal if i is a generic complete intersection and i i is cohenmacaulay then i is a hypersurface section the only known non trivial deviation 2 gorenstein ideals of grade at least four are the ideals of huneke and ulrich 4 which are the ideals studied in this paper for each integer s let y be a 1 x s matrix of indeterminates x be an s x s alternating matrix of indeterminates g be the product y x and ct be the sequence gp g let a be the ideal of

sums of linked ideals it is shown that the sum of two geometrically linked ideals in the linkage class of a complete intersection is again an ideal in the linkage class of a complete intersection conversely every gorenstein ideal of height at least two in the linkage class of a complete intersection can be obtained as a generalized localization of a sum of two geometrically linked ideals in the linkage class of a complete intersection we also investigate sums of doubly linked gorenstein ideals as an application we construct a perfect prime ideal which is strongly nonobstructed but not strongly cohen macaulay and a perfect prime ideal which is not strongly nonobstructed but whose entire linkage class is strongly cohen macaulay

rigid ideals by deforming quadratic letterplace ideals abstract we compute the deformation space of quadratic letterplace ideals l 2 p of finite posets p when its hasse diagram is a rooted tree these deformations are unobstructed the deformed family has a polynomial ring as the base ring the ideal j 2 p defining the full family of deformations is a rigid ideal and we compute it explicitly in simple example cases j 2 p is the ideal of maximal minors of a generic matrix the pfaffians of a skew symmetric matrix and a ladder determinantal ideal

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded