local and pointwise error estimates of the local discontinuous galerkin method applied to the stokes problem we prove local and pointwise error estimates for the local discontinuous galerkin method applied to the stokes problem in two and three dimensions by using techniques originally developed by a schatz math comp 67 1998 877 899 to prove pointwise estimates for the laplace equation we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries

pointwise error estimates for differences in piecewise linear finite element approximations we consider piecewise linear finite element approximations uh to u the solution of an elliptic boundary value problem new estimates for the differences e x1 e x2 where e x u x u h x is the error and x1 and x2 are any two points in the domain are obtained in terms of weighted l infty norms as a consequence so called asymptotic expansion inequalities are derived that have been applied to obtain asymptotically exact a posteriori estimators for the gradient on each element

generalized binary udaya siddiqi sequences in this correspondence we present a family of binary 2n sequences of period 2 2n 1 where n is an integer which can be seen as a generalization of nonlinear binary sequences obtained from z4 sequences and recently constructed gkw gold kasami and welch like sequences the sequences have low correlations and are useful in code division multiple access cdma communication systems and cryptography

new families of binary sequences with low correlation for a positive integer n new families s and u of binary sequences of period 2 sup n 1 with low correlations are proposed where for some positive integer e s is defined for odd n e and u for even n e the family s has four valued correlations and is a generalization of the family of gold like sequences introduced by s boztas and p v kumar see ibid vol 40 p 532 7 1994 the family u which is also a generalization of the sequence family defined by p udaya polyphase and frequency hopping sequences obtained from finite rings ph d dissertation dept elec eng indian inst technol kanpur 1992 has six valued correlations the relationship between gold like sequences and gold sequences is the same as the relationship between the family s and the family constructed from the binary sequences partially contributed by r gold see ibid vol it 14 p 154 6 1968 t kasami see coordinated sci lab univ of illinois urbana champaign tech rep r 285 ad 632574 1966 and welch using a lifting idea no j s and kumar p v ibid vol 35 p 371 9 1989 for the families s and u families of binary sequences with the same correlation distributions and large linear span are also constructed

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

extrapolation of numerical solutions for elliptic problems on corner domains we study the extrapolation method for the numerical solution of elliptic boundary value problems on corner domains most papers on the extrapolation method for solving differential equations require smoothness assumptions on exact solutions which do not hold for boundary value problems on corner domains we conjecture that asymptotic error expansions exist as long as there is enough smoothness on the input data except the domain this is confirmed by our theoretical result on asymptotic error expansions for model problems on a rectangle numerical examples suggest that similar results hold for problems on more general corner domains although it seems very difficult to prove them

buildings and hecke algebras abstract in this paper we establish a strong connection between buildings and hecke algebras by studying two algebras of averaging operators on buildings to each locally finite regular building we associate a natural algebra b of chamber set averaging operators and when the building is affine we also define an algebra a of vertex set averaging operators we show that for appropriately parametrised hecke algebras h and h the algebra b is isomorphic to h and the algebra a is isomorphic to the centre of h on the one hand these results give a thorough understanding of the algebras a and b on the other hand they give a nice geometric and combinatorial understanding of hecke algebras and in particular of the macdonald spherical functions and the centre of affine hecke algebras our results also produce interesting examples of association schemes and polynomial hypergroups in later work we use the results here to study random walks on affine buildings

asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions to define a discretization we introduce a set f h of grid points see 2 2 and replace the derivatives in 1 1 by symmetric divided differences with respect to those grid points this device is unproblematic as long as the direct grid neighbours of interior grid points are either interior grid points or boundary points by a proper choice of h that may be obtained e g for rectangular shape regions f2 for all inner grid points for general domains f2 this assumption is no longer satisfied and direct grid neighbours will not always be in and therefore symmetric divided differences are not directly applicable in some points so called irregular mesh points there is a long history to solve this problem 1930 gerschgorin 153 has proposed to use for the unknown function value in a grid neighbour outside of f2 next to an irregular grid point the boundary value of the next boundary point on that grid line obtaining a method of order 1 compared with order 2 available in cases without irregular grid points in 1933 collatz 11 improved this method by using linear interpolation between the irregular grid point and its neighbouring boundary point obtaining a method of order 2 wasow 22 showed that for the methods in 11 15 no asymptotic expansions of the discretization error can exist the 1938 approach of shortley weller 20 may be interpreted as using polynomials of degree 2 interpolating in the

the hall polynomial revisited abstract the hall polynomial g p counts subgroups of type and cotype in a finite abelian p group of type we derive a formula for g p along the lines of t klein s original formula but much simpler which shows that its expansion in powers of p 1 has nonnegative coefficients

a family of five weight cyclic codes and their weight enumerators cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper a family of p ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed the weight distribution of this family of cyclic codes is determined it turns out that the proposed cyclic codes have five nonzero weights

optimal biphase sequences with large linear complexity derived from sequences over z sub 4 new families of biphase sequences of size 2 sup r 1 1 r being a positive integer are derived from families of interleaved maximal length sequences over z sub 4 of period 2 2 sup r 1 these sequences have applications in code division spread spectrum multiuser communication systems the families satisfy the sidelnikov bound with equality on spl theta sub max which denotes the maximum magnitude of the periodic cross correlation and out of phase autocorrelation values one of the families satisfies the welch bound on spl theta sub max with equality the linear complexity and the period of all sequences are equal to r r 3 2 and 2 2 sup r 1 respectively with an exception of the single m sequence which has linear complexity r and period 2 sup r 1 sequence imbalance and correlation distributions are also computed

fourth order eigenvalue approximation by extrapolation on domains with reentrant corners the eigenvalue problem of the laplace operator is considered on a non convex domain composed of rectangles this model problem may be solved by the finite element method with bilinear elements on a rectangular mesh it is known thato h 2 1 convergence can be obtained for the eigenvalues if the mesh haso h 2 points a simple extrapolation scheme is presented which on appropriately graded meshes increases the rate of convergence too h 4

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

the local superconvergence of the trilinear element for the three dimensional poisson problem abstract in this paper we shall combine the finite element theory of green s function presented in this paper the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three dimensional poisson equation

perturbations of forms and error estimates for the finite element method at a point with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point we first derive a variety of local error estimates for u uh at a point x0 where uh belongs to a finite element space shr and is an approximation to u satisfying the local equations a u u h varphi f varphi for all varphi in shr with compact support in a neighborhood of x0 here the a cdot cdot are bilinear forms associated with second order elliptic equations and the f are linear functionals in the case that f equiv 0 our results coincide with those of schatz siam j numer anal 38 2000 pp 1269 1293 but are improvements when f neq 0 we apply these results to improve the superconvergence error estimates obtained by schatz sloan and wahlbin siam j numer anal 33 1996 pp 505 521 at points x0 where the subspaces are symmetric with respect to x0

a new family of p ary sequences with low correlation constructed from decimated sequences in this paper for an odd prime p and an integer n geq 3 a new family of p ary sequences of period p n 1 over 2 with low correlation is constructed the new family is constructed by shifts and additions of two decimated sequences of a p ary m sequence and its family size is 2 p n 1 the complete correlation distribution of this new family is derived it is also shown that the family is optimal with respect to the parameter theta rm rms which denotes the root mean square of all nontrivial correlations compared with the known sequence families our sequence family is new and has a larger family size which is four times of its period

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

a new family of nonbinary sequences with three level correlation property and large linear span in this correspondence we present a new family of nonbinary sequences with three level nontrivial correlations and large linear complexity the sequences may be considered as nonlinear analogues of the well known sequences by trachtenberg and helleseth it is shown that the family is optimal with respect to the welch bound in terms of root mean square of all nontrivial correlations we also determine the correlation distribution of the new family

a partial order on littlewood richardson tableaux abstract for a nilpotent linear transformation f v v of a vector space v of jordan type 1 l i e the sizes of the jordan blocks equal to the conjugate of let x d be the set of f stable subspaces of dimension d x d is partitioned indexed by littlewood richardson tableaux lr tableau t of shapes s with d each of which corresponds to the set s v t consisting of w x d such that the dimension of the vector space f r 1 v f t 1 w f r v f t 1 w f r 1 v f t w is equal to the number of cells squares in the rth row of t filled with the letter t for all r t 1 we define a partial order on the set lr d of lr tableaux of shapes s with d by t t if s v t is contained in the closure of s v t in the grassmaniann g v d we give a certain sufficient condition corollary c and a necessary condition corollary d for t t for this we introduce the notions of generic vectors the generic subspace w t the rational function field f t and the lr tableau t associated to a tableau t and then show the map s v t s v t let 1 w f w where t let 1 is the subtableau of an lr tableau t consisting of the cells filled with the letters greater than 1 is a fiber bundle with fiber isomorphic to a union of schubert cells and s v t is nonsingular theorem a the generic subspace w t of a tableau t is a generic point of s v t whose function field isomorphic to the rational function field f t associated to the lr tableau t theorem b

piecewise linear finite element methods are not localized recent results of schatz show that standard galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error these results do not indicate that such localization occurs when piecewise linear elements are used we show via simple one dimensional examples that schatz s estimates are sharp in that localization indeed does not occur when piecewise linear elements are used

d form sequences families of sequences with low correlation values and large linear spans large families of binary sequences with low correlation values and large linear span are critical for spread spectrum communication systems the author describes a method for constructing such families from families of homogeneous functions over finite fields satisfying certain properties he then uses this general method to construct specific families of sequences with optimal correlations and exponentially better linear span than no sequences no 1988

prime phase sequences with periodic correlation properties better than binary sequences for the case where p is an odd prime n or 2 is an integer and omega is a complex primitive pth root of unity a construction is presented for a family of p sup n p phase sequences symbols of the form omega sup i where each sequence has length p sup n 1 and where the maximum nontrivial correlation value c sub max does not exceed 1 square root p sup n a complete distribution of correlation values is provided as a special case of this construction a previous construction due to sidelnikov 1971 is obtained the family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs the present design has the additional advantage of not requiring an alphabet of size larger than three the new sequences are suitable for achieving code division multiple access and are easily implemented using shift registers they wee discovered through an application of deligne s bound 1974 on exponential sums of the weil type in several variables the sequences are also shown to have strong identification with certain bent functions

distance regularity in buildings and structure constants in hecke algebras in this paper we define generalised spheres in buildings using the simplicial structure and weyl distance in the building and we derive an explicit formula for the cardinality of these spheres we prove a generalised notion of distance regularity in buildings and develop a combinatorial formula for the cardinalities of intersections of generalised spheres motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of hecke operators arising from our generalised distance regularity and prove isomorphisms between these algebras and more well known parabolic hecke algebras we conclude with applications of our main results to non negativity of structure constants in parabolic hecke algebras commutativity of algebras of hecke operators double coset combinatorics in groups with bn pairs and random walks on the simplices of buildings

ultraconvergence of high order fems for elliptic problems with variable coefficients in this paper we investigate local ultraconvergence properties of the high order finite element method fem for second order elliptic problems with variable coefficients under suitable regularity and mesh conditions we show that at an interior vertex which is away from the boundary with a fixed distance the gradient of the post precessed kth k ge 2 k 2 order finite element solution converges to the gradient of the exact solution with order mathcal o h k 2 mathrm ln h 3 o hk 2 lnh 3 the proof of this ultraconvergence property depends on a new interpolating operator some new estimates for the discrete green s function a symmetry theory derived in 26 and the richardson extrapolation technique in 20 numerical experiments are performed to demonstrate our theoretical findings

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients abstract assume that u x satisfies the problem l u x x i a i j u x j f x x u x 0 x in this article using interpolation postprocessing technique we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u x using piecewise polynomials of degrees bi k k 3 over a rectangular partition assume that k 3 is odd and x 0 is an interior vertex satisfying x 0 c using the new interpolation postprocessing formula presented in this study we show that the primal variable and the derivative of the post processed finite element solution using piecewise of degrees bi k k 3 at x 0 converge to the primal variable and the derivative of the exact solution with order o h k 3 ln h under suitable regularity and mesh conditions respectively finally we use numerical experiments to illustrate our theoretical findings

asymptotic expansions and l error estimates for mixed finite element methods for second order elliptic problems asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and richardson extrapolation can be applied to increase the accuracy of the approximations a new procedure which is called the error corrected method is presented as a further application of the asymptotic error expansion for the first order bdm approximation of the scalar field the key point in deriving the asymptotic expansions for the error is an establishment ofl 1 error estimates for mixed finite element approximations for the regularized green s functions as another application of thel 1 error estimates for the regularized green s functions we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems

from elementary calculations to hall polynomials abstract via the computation of cokernels and using the theorem of krull remak and schmidt to decompose a given representation the task of computing hall polynomials for dynkin quivers can be reduced to solving many systems of linear equations the resulting method has been applied to obtain hall polynomials for a quiver of type e 8

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

new nonbinary sequences with ideal two level autocorrelation we find new families of nonbinary sequences of period p sup n 1 with symbols from a finite field f sub p for any prime p spl ges 3 the sequences have two level ideal autocorrelation and are generalizations of previously found ternary sequences with ideal autocorrelation difference sets with parameters p sup n 1 p 1 p sup n 1 1 p 1 p sup n 2 1 p 1 can also be derived from these sequences in a natural way

extrapolation techniques for reducing the pollution effect or reentrant corners in the finite element method standard finite element schemes applied to boundary value problems on domains with reentrant corners suffer from a global loss of accuracy caused by the influence of the corner singularities for a model situation it is shown that this pollution effect can be completely described in terms of an asymptotic error expansion with respect to fractional powers of the mesh size parameter this provides the basis of an extrapolation procedure which recovers the full order of accuracy for the solution as well as for the corresponding stress intensity factors

pointwise error estimates for finite element solutions of the stokes problem in this paper pointwise error estimates for general finite element approximations of the stokes problem are established on quasi uniform grids in r n the results obtained in this paper improve and extend the existing error estimates in the maximum norm for the stokes problem the new pointwise error estimates exhibit a more local dependence of the errors on the true solution and as a by product provide logarithm free bounds for all errors except the error of the velocity approximation of the lowest order

new constructions for optimal sets of frequency hopping sequences in this paper two generic constructions of optimal frequency hopping sequence fhs sets employing d form functions with difference balanced property are presented they generalize the previous constructions of optimal fhs sets using m sequences and produce new optimal fhs sets that cannot be produced by the earlier constructions by choosing appropriate d form functions with difference balanced property both constructions lead to fhss with large linear complexity in addition one of the proposed constructions gives new optimal parameters of fhs sets