a 2 adic approach to the analysis of cyclic codes this paper describes how 2 adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over z sub 2 a a spl ges 2 the ring of integers modulo 2 sup a it provides a 2 adic proof of a theorem of mceliece that characterizes the possible hamming weights that can appear in a binary cyclic code a generalization of this theorem is derived that applies to cyclic codes over z sub 2 a that are obtained from binary cyclic codes by a sequence of hensel lifts this generalization characterizes the number of times a residue modulo 2 sup a appears as a component of an arbitrary codeword in the cyclic code the limit of the sequence of hensel lifts is a universal code defined over the 2 adic integers this code was first introduced by calderbank and sloane 1995 and is the main subject of this paper binary cyclic codes and cyclic codes over z sub 2 a are obtained from these universal codes by reduction modulo some power of 2 a special case of particular interest is cyclic codes over z sub 4 that are obtained from binary cyclic codes by means of a single hensel lift the binary images of such codes under the gray isometry include the kerdock preparata and delsart goethals codes these are nonlinear binary codes that contain more codewords than any linear code presently known fundamental understanding of the composition of codewords in cyclic codes over z sub 4 is central to the search for more families of optimal codes this paper also constructs even unimodular lattices from the hensel lift of extended binary cyclic codes that are self dual with all hamming weights divisible by 4 the leech lattice arises in this way as do extremal lattices in dimensions 32 through 48

new estimates for ritz vectors the following estimate for the rayleigh ritz method is proved u u au u sin u u u 1 here a is a bounded self adjoint operator in a real hilbert euclidian space u one of its eigenpairs u a trial subspace for the rayleigh ritz method and u a ritz pair this inequality makes it possible to analyze the fine structure of the error of the rayleigh ritz method in particular it shows that u u c 2 if an eigenvector u is close to the trial subspace with accuracy and a ritz vector u is an approximation to another eigenvector with a different eigenvalue generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested and estimates of approximation of eigenspaces and invariant subspaces are proved

an interpretive model of line continuation in human visual perception abstract continuation is a fundamental intuitive property of perceptual grouping in line segregation to explain this property a hypothetical model from the standpoint of computer engineering is presented in this paper this model includes four mechanisms i e signal beaming mechanism feedback transmitting mechanism image partitioning mechanism and image revising mechanism and a temporary 3 d space called rho space or quotient space the map of excitatory and inhibitory signals from an input image is first constructed by applying some functions of the so called simple cortical cells which are sensitive to the position and orientation of the stimulus the signal beaming mechanism and feedback transmitting mechanism interact in human behaviours such as walking to resolve problems such as continuity transition fuzziness and ambiguity by using these two mechanisms the constructed map is then converted into a three dimensional quotient space in which all the members voxels satisfy the equivalence relationships so that the image partitioning mechanism can accomplish the work of line segregation the image revising mechanism is further used to eliminate the jaggedness caused by the signal transmission so that the final result of line segregation is consistent with the one from human observation this interpretive model was built on a 32 bits workstation sun 3 160 the results of several image processing simulations suggest that the hypothetical model is feasible and reasonable

new a priori fem error estimates for eigenvalues we analyze the ritz galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates for a simple eigenvalue we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants for a multiple eigenvalue we prove in addition what is apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace these estimates reflect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities thus resulting in different levels of errors for the approximate eigenvalues for clustered eigenvalues we derive eigenvalue error bounds that do not depend on the width of the cluster our results are readily applicable to the classical ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators

a posteriori error estimators for convection diffusion eigenvalue problems a posteriori error estimators for convection diffusion eigenvalue model problems are discussed in heuveline and rannacher 2001 17 in the context of the dual weighted residual method dwr this paper directly addresses the variational formulation rather than the non linear ansatz of becker and rannacher for some convection diffusion model problem and presents a posteriori error estimators for the eigenvalue error based on averaging techniques two different postprocessing techniques attached to the dwr paradigm plus two new dual weighted a posteriori error estimators are also presented the first new estimator utilises an auxiliary raviart thomas mixed finite element method and the second exploits an averaging technique in combination with ideas of dwr the six a posteriori error estimators are compared in three numerical examples and illustrate reliability and efficiency and the dependence of generic constants on the size of the eigenvalue or the convection coefficient

decoding of linear codes over galois rings we present a method for decoding an arbitrary linear code over a galois ring spl rscr by a process of lifting decoding algorithms for a family of linear codes over a finite field spl kscr forming an spl alpha element chain where spl kscr is the quotient field of spl rscr and spl rscr has characteristic p sup spl alpha as a new result this method also works for linear codes over spl rscr which are nonfree spl rscr modules

a note on self dual group codes we classify group algebras over galois rings containing self dual ideals i e ideals c which satisfy c c sup spl perp with respect to the natural nondegenerate bilinear form given on group algebras

p adic estimates of hamming weights in abelian codes over galois rings a generalization of mceliece s theorem on the p adic valuation of hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by wilson along with a technique known as trace averaging introduced here the original theorem of mceliece concerned cyclic codes over prime fields delsarte and mceliece later extended this to abelian codes over finite fields calderbank li and poonen extended mceliece s original theorem to cover cyclic codes over the rings zopf2 d wilson strengthened their results and extended them to cyclic codes over zopf p d and katz strengthened wilson s results and extended them to abelian codes over zopfp d it is natural to ask whether there is a single analogue of mceliece s theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers in this paper this question is answered affirmatively a single theorem for abelian codes over galois rings is presented this theorem contains all previously mentioned results and more

cyclic codes over gr 4 sup m which are also cyclic over spl zopf sub 4 let gr 4 sup m be the galois ring of characteristic 4 and cardinality 4 sup m and spl alpha spl alpha sub 0 spl alpha sub 1 spl alpha sub m 1 be a basis of gr 4 sup m over spl zopf sub 4 when we regard gr 4 sup m as a free spl zopf sub 4 module of rank m define the map d sub spl alpha from gr 4 sup m z z sup n 1 into spl zopf sub 4 z z sup mn 1 by d spl alpha a z spl sigma sub i 0 sup m 1 spl sigma sub j 0 sup n 1 a sub ij z sup mj i where a z spl sigma sub j 0 sup n 1 a sub j z sup j and a sub j spl sigma sub i 0 sup m 1 a sub ij spl alpha sub i a sub ij spl isin spl zopf sub 4 then for any linear code c of length n over gr 4 sup m its image d sub spl alpha c is a spl zopf sub 4 linear code of length mn in this article for n and m being odd integers it is determined all pairs spl alpha c such that d sub spl alpha c is spl zopf sub 4 cyclic where spl alpha is a basis of gr 4 sup m over spl zopf sub 4 and c is a cyclic code of length n over gr 4 sup m

a convergent adaptive method for elliptic eigenvalue problems we prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second order symmetric elliptic partial differential operators the weak form is assumed to yield a bilinear form which is bounded and coercive in h 1 each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation the error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem because of this nonlinearity the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine

a transform approach to permutation groups of cyclic codes over galois rings berger and charpin see ibid vol 42 p 2194 2209 1996 and des codes cuyptogr vol 18 no 1 3 p 29 53 1999 devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes we extend this method to cyclic and extended cyclic codes over the galois ring gr p sup a m developing a generalization of the mattson solomon polynomial in particular we classify all affine invariant codes of length 2 sup m over z sub 4 thus generalizing the corresponding result of kasami lin and peterson 1967 and giving an alternative proof to abdukhalikov we give a large class of codes over z sub 4 with large permutation groups which include generalizations of bose chaudhuri hocquenghem bch and reed muller rm codes

algebraic description of curve structure the authors propose a compact and concise method of describing curves in terms of the quasi topological features and the structure of each singular point the quasi topological features are the convexity loop and connectivity the quasi topological structure is analyzed in a hierarchical way and algebraic structure is presented explicitly for each representation level the lower level representations are integrated into the higher level one in a systematic way when a curve has singular points branch points the curve is decomposed into components where each is a simple arc or a simple closed curve by decomposing each singular point the description scheme is applied to character recognition

consta abelian codes over galois rings we study n length consta abelian codes a generalization of the well known abelian codes and constacyclic codes over galois rings of characteristic p sup a where n and p are coprime a twisted discrete fourier transform dft is used to generalize transform domain results of abelian and constacyclic codes to consta abelian codes further we characterize consta abelian codes invariant under two kinds of monomials whose underlying permutations are effected by i multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions all the codes studied here belong to the class of quasi twisted codes which are known to contain some good codes we show that the dual of a consta abelian code invariant under the two monomials is also a consta abelian code closed under both monomials

p adic valuation of weights in abelian codes over spl zopf p sup d counting polynomial techniques introduced by wilson are used to provide analogs of a theorem of mceliece mceliece s original theorem relates the greatest power of p dividing the hamming weights of words in cyclic codes over gf p to the length of the smallest unity product sequence of nonzeroes of the code calderbank li and poonen presented analogs for cyclic codes over zopf 2d using various weight functions hamming lee and euclidean weight as well as count of occurrences of a particular symbol some of these results were strengthened by wilson who also considered the alphabet zopf pd for p an arbitrary prime these previous results new strengthened versions and generalizations are proved here in a unified and comprehensive fashion for the larger class of abelian codes over zopf pd with p any prime for abelian codes over zopf 4 combinatorial methods for use with counting polynomials are developed these show that the analogs of mceliece s theorem obtained by wilson for hamming weight lee weight and symbol counts and the analog obtained here for euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose fourier transforms have a specified support

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

a generalized dft for abelian codes over z sub m a generalized discrete fourier transform defined over an appropriate extension ring is given that is suitable to characterize abelian codes over residue class integer rings z sub m the characterization is in terms of generalized discrete fourier transform components taking values from certain ideals of the extension ring it is shown that the results known for cyclic codes over z sub m like the simple characterization of dual and self dual codes and the nonexistence of self dual codes for certain values of code parameters extend to abelian codes over z sub m as well

abelian codes over galois rings closed under certain permutations we study n length abelian codes over galois rings with characteristic p sup a where n and p are relatively prime having the additional structure of being closed under the following two permutations i permutation effected by multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions a code is t quasi cyclic t qc if t is an integer such that cyclic shift of a codeword by t positions gives another codeword we call the abelian codes closed under the first permutation as unit invariant abelian codes and those closed under the second as quasi cyclic abelian qca codes using a generalized discrete fourier transform gdft defined over an appropriate extension of the galois ring we show that unit invariant abelian and qca codes can be easily characterized in the transform domain for t 1 qca codes coincide with those that are cyclic as well as abelian the number of such codes for a specified size and length is obtained and we also show that the dual of an unit invariant t qca code is also an unit invariant t qca code unit invariant abelian hence unit invariant cyclic and t qca codes over galois field f sub p sup l and over the integer residue rings are obtainable as special cases

hp finite elements for elliptic eigenvalue problems error estimates which are explicit with respect to lambda h and p convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form if the mesh width h is fine enough then the eigenvalues resp eigenfunctions converge at some well defined rate in this paper we will determine the maximal mesh width h 0 more precisely the minimal dimension of a finite element space so that the asymptotic convergence estimates hold for h leq h 0 this mesh width will depend on the size and spacing of the exact eigenvalues the spatial dimension and the local polynomial degree of the finite element space for example in the one dimensional case the condition lambda 3 4 h 0 lesssim1 is sufficient for piecewise linear finite elements to compute an eigenvalue lambda with optimal convergence rates as h 0 geq h rightarrow0 it will turn out that the condition for eigenfunctions is slightly more restrictive furthermore we will analyze the dependence of the ratio of the errors of the galerkin approximation and of the best approximation of an eigenfunction on lambda and h in this paper the error estimates for the eigenvalue function are limited to the selfadjoint case however the regularity theory and approximation property cover also the nonselfadjoint case and hence pave the way towards the error analysis of nonselfadjoint eigenvalue function problems

stroke segmentation by bernstein be zier curve fitting abstract stroke segmentation is essential to the recognition of handwritten chinese characters here some new and reliable techniques are proposed first the thinning process is applied to each character to obtain the skeleton then a maximum circle technique is used to remove the spurious branches and merge the split 4 fork points all possible pairs of stroke segments connected at the same fork point are considered and the bernstein bezier curve is used to fit each pair to smooth the data and find its trend then from the result of this curve fitting we can decide which pair belongs to the same stroke finally the inflection points with very small radius of curvature are found and the stroke segmentation is carried out based on these inflection points experiments on some chinese characters show that the proposed new techniques are reliable and time saving compared with the direction method

an adaptive homotopy approach for non selfadjoint eigenvalue problems this paper presents adaptive algorithms for eigenvalue problems associated with non selfadjoint partial differential operators the basis for the developed algorithms is a homotopy method which departs from a well understood selfadjoint problem apart from the adaptive grid refinement the progress of the homotopy as well as the solution of the iterative method are adapted to balance the contributions of the different error sources the first algorithm balances the homotopy discretization and approximation errors with respect to a fixed stepsize in the homotopy the second algorithm combines the adaptive stepsize control for the homotopy with an adaptation in space that ensures an error below a fixed tolerance e the outcome of the analysis leads to the third algorithm which allows the complete adaptivity in space homotopy stepsize as well as the iterative algebraic eigenvalue solver all three algorithms are compared in numerical examples

finite difference approximation of space fractional diffusion problems a mathematical analysis is presented to establish the convergence of the matrix transformation or matrix transfer method for the finite difference approximation of space fractional diffusion problems combined this with an implicit euler time discretization the optimal order convergence is proved with respect to the discrete l2 and the maximum norm the analysis is performed on general two and three dimensional domains with homogeneous boundary conditions the corresponding error estimates are illustrated with some numerical experiments

sharp p divisibility of weights in abelian codes over bbz p d bbz a theorem of mceliece on the p divisibility of hamming weights in cyclic codes over fp is generalized to abelian codes over zopf p dzopf this work improves upon results of helleseth kumar moreno shanbhag calderbank li poonen wilson and katz these previous attempts are not sharp in general i e do not report the full extent of the p divisibility except in special cases nor do they give accounts of the precise circumstances under which they do provide best possible results this paper provides sharp results on p divisibilities of hamming weights and counts of any particular symbol for an arbitrary abelian code over zopf p dzopf it also presents sharp results on 2 divisibilities of lee and euclidean weights for abelian codes over zopf 4zopf

a chinese character stroke extraction algorithm based on contour information abstract in this paper we present a new stroke extraction algorithm that integrates all levels of contour information including boundary points dominant points corner points segments cross section sequence graph and character structure to extract strokes of chinese characters in the algorithm first the boundary points are extracted then the dominant and corner points are detected third the character structure including singular and regular regions are extracted by the contour information and a modified cross section sequence graph cssg finally a bezier curve taking dominant points and corner points as inputs is used to check the continuity of strokes experimental results show that the proposed algorithm can correctly extract the strokes up to 95 from printed and handwritten test samples based on the human perception compared with a typical thinning approach the proposed algorithm gives better results in terms of both stroke smoothness and the precise number of stroke extractions

the p and hp versions of the virtual element method for elliptic eigenvalue problems abstract we discuss the p and h p versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes an application of this model is provided by the schrodinger equation with a pseudo potential term as an interesting byproduct we present for the first time in literature an explicit construction of the stabilization of the mass matrix we present in detail the analysis of the p version of the method proving exponential convergence in the case of analytic eigenfunctions the theoretical results are supplied with a wide set of experiments we also show numerically that in the case of eigenfunctions with finite sobolev regularity an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p refinements importantly the geometric flexibility of polygonal meshes is exploited in the construction of the h p spaces

efficient resonance computations for helmholtz problems based on a dirichlet to neumann map we present an efficient procedure for computing resonances and resonant modes of helmholtz problems posed in exterior domains the problem is formulated as a nonlinear eigenvalue problem nep whe

an oscillation free adaptive fem for symmetric eigenvalue problems a refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first order adaptive finite element method afem is presented the h 1 stability of the l 2 projection provides reliability and efficiency of the edge contribution of standard residual based error estimators for p 1 finite element methods in fact the volume contributions and even oscillations can be omitted for courant finite element methods this allows for a refined averaging scheme and so improves mao et al in adv comput math 25 1 3 135 160 2006 the proposed afem monitors the edge contributions in a bulk criterion and so enables a contraction property up to higher order terms and global convergence numerical experiments exploit the remaining l 2 error contributions and confirm our theoretical findings the averaging schemes show a high accuracy and the afem leads to optimal empirical convergence rates