factorization and quadratic cost problems in hilbert spaces abstract a factorization perspective to problems of optimal causal estimation and optimal causal control of linear stochastic systems defined on an infinite dimensional hilbert space is presented a separation principle is derived when the system input output map is generated by an abstract evolution operator the factorization formalism allows for an essentially algebraic approach to these problems

combinatorial bounds for list decoding informally an error correcting code has nice list decodability properties if every hamming ball of large radius has a small number of codewords in it we report linear codes with nontrivial list decodability i e codes of large rate that are nicely list decodable and codes of large distance that are not nicely list decodable specifically on the positive side we show that there exist codes of rate r and block length n that have at most c codewords in every hamming ball of radius h sup 1 1 r 1 c spl middot n this answers the main open question from the work of elias 1957 this result also has consequences for the construction of concatenated codes of good rate that are list decodable from a large fraction of errors improving previous results of guruswami and sudan see ieee trans inform theory vol 45 p 1757 67 sept 1999 and proc 32nd acm symp theory of computing stoc portland or p 181 190 may 2000 in this vein specifically for every spl epsi 0 we present a polynomial time constructible asymptotically good family of binary codes of rate spl omega spl epsi sup 4 that can be list decoded in polynomial time from up to a fraction 1 2 spl epsi of errors using lists of size o spl epsi sup 2 on the negative side we show that for every spl delta and c there exists spl tau 0 and an infinite family of linear codes c sub i sub i such that if n sub i denotes the block length of c sub i then c sub i has minimum distance at least spl delta spl middot n sub i and contains more than c sub 1 spl middot n sub i sup c codewords in some hamming ball of radius spl tau spl middot n sub i while this result is still far from known bounds on the list decodability of linear codes it is the first to bound the radius for list decodability by a polynomial sized list away from the minimum distance of the code

upper bounds on generalized distances we derive new asymptotic bounds for generalized distances our approach extends the classical hamming plotkin and elias bounds the latter bound involves extending the definition of generalized distances to nonlinear codes

an extension of the special factorization with applications to wiener hopf equations abstract necessary and sufficient conditions are given for a factorization that generalizes the special factorization applications to wiener hopf equations and filtering are also presented

an explicit a posteriori error estimations for the cell functional minimization scheme of elliptic problems abstract this paper presents an explicit form of a posteriori error estimator for the cell functional minimization scheme of elliptic problems it is shown that this estimator is reliable with respect to an energy type error numerical experiments are conducted to demonstrate the performance of the error estimator in the adaptive mesh refinement algorithm

approximation schemes for the linear quadratic optimal control problem associated with delay equations the linear regulator problem for delay equations is discussed we propose a theoretical solution involving riccati integral equations and then axiomatically discuss a general approximation scheme the details are given for spline and averaging approximations

a cell functional minimization scheme for domain decomposition method on non orthogonal and non matching meshes we are interested in a robust and accurate domain decomposition algorithm with interface conditions of robin type on non matching multiblock grids using a cell functional minimization scheme which has a good performance on non orthogonal meshes in order to treat the non matching grids at the interface we introduce the l 2 l 2 projection operator to ensure weak continuity of the primary unknown and of the normal flux across the non matching interface furthermore we prove the wellposedness of local and global problems and obtain as well an error estimate of first order in a discrete h 1 h 1 norm only using the l 2 l 2 projection operator on the non matching interface as done in the matching case numerical results are presented in confirmation of the theoretical results

a factorization on the semi infinite interval ii applications to stable regulator problems abstract a factorization approach to the synthesis of feedback control laws for stable regulator problems is developed new feedback representations are obtained for hereditary problems with and without control delay also new derivations of representations of the optimal gain for regulator problems in hilbert space are presented

new upper bounds on generalized weights we derive new asymptotic upper bounds on the generalized weights of a binary linear code of a given size we also prove some asymptotic results on the distance distribution of binary codes

geometric approach to higher weights the notion of higher or generalized weights of codes is just as natural as that of the classical hamming weight the authors adopt the geometric point of view and always treat the q ary case some results and proofs being new the main goal is to present a clear picture of what is known on the subject

special factorization and riccati integral equations abstract a class of operator riccati integral equations is associated with a factorization problem in a certain banach algebra recent results concerning factorization in this algebra are used to obtain existence uniqueness and continuous dependence results for the riccati equations

a posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations this paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t nabla cdot u f c varepsilon delta c 0 the estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the l 1 norm in the situation where the diffusion parameter varepsilon is smaller or comparable to the mesh size numerical experiments underline the theoretical results

estimations a posteriori d un schema de volumes finis pour un probleme non lineaire resumedans cet article nous developpons des estimations a posteriori de l erreur commise en approchant un probleme elliptique non lineaire par une methode de volumes finis mvf de type vertex centered les estimateurs obtenus mesurent le residu de l equation forte et l irregularite de la solution discrete qui se traduit par des sauts a travers les inter elelments une condition sur la fonction flux numerique sera imposee pour traiter la non conformite du probleme au cours de l evaluation de l estimation summary we derive a posteriori error estimates for vertex centered finite volume discretizations of a class of non linear elliptic pdes the error estimates are of residual type and incorporate residuals of the pde in its strong form on the control volumes and jumps of the pde s prinicipal part across the element boundaries the non conformity of the discretization is taken into account by a condition on the numerical flux function

a posteriori error estimate and adaptive mesh refinement for the cell centered finite volume method for elliptic boundary value problems we extend a result of nicaise siam j numer anal 43 2005 pp 1481 1503 for the a posteriori error estimation of the cell centered finite volume method for the numerical solution of elliptic problems having computed the piecewise constant finite volume solution u h we compute a morley type interpolant mathcal i u h for the exact solution u the energy error norm nabla mathcal t u mathcal i u h l 2 can be controlled efficiently and reliably by a residual based a posteriori error estimator eta the local contributions of eta are used to steer an adaptive mesh refining algorithm a model example serves the laplace equation in two dimensions with mixed dirichlet neumann boundary conditions

list decoding from erasures bounds and code constructions we consider the problem of list decoding from erasures we establish lower and upper bounds on the rate of a binary linear code that can be list decoded with list size l when up to a fraction p of its symbols are adversarially erased such bounds already exist in the literature albeit under the label of generalized hamming weights and we make their connection to list decoding from erasures explicit our bounds show that in the limit of large l the rate of such a code approaches the capacity 1 p of the erasure channel such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate spl omega spl epsiv sup 2 log 1 spl epsiv that can be efficiently list decoded using lists of size o 1 spl epsiv when an adversarially chosen 1 spl epsiv fraction of symbols are erased for arbitrary spl epsiv 0 this improves previous results in this vein which achieved a rate of spl omega spl epsiv sup 3 log 1 spl epsiv

a posteriori error estimations of some cell centered finite volume methods this paper presents the natural framework to residual based a posteriori error estimation of some cell centered finite volume methods for the laplace equation in r d d 2 or 3 for that purpose we associate with the finite volume solution a reconstructed approximation which is a kind of morley interpolant the error is then the difference between the exact solution and this morley interpolant the residual error estimator is based on the jump of normal and tangential derivatives of the morley interpolant we then prove the equivalence between the discrete h1 seminorm of the error and the residual error estimator numerical tests confirm our theoretical results

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

the decoding error probability of linear codes over the erasure channel in this paper we study the decoding error probability of linear codes over the erasure channel under the list decoding the notion of the q ell incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding by calculating the q ell incorrigible set distributions the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions for the ensemble of all n k linear codes the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined furthermore the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all n nr linear codes

on the list decodability of burst errors burst errors are a type of distortion in many data communications and data storage channels in this paper we consider the list decodability of codes for single burst error case and phased burst error case independently firstly we analyze the list decodability of random codes and we show that the burst list decoding radius and the rate of random codes achieve the singleton bound and the gilbert varshamov bound for single case and phased case respectively second we illustrate that cyclic codes and algebraic geometry codes are good burst list decodable codes for single case and phased case respectively

a factorization on the semi infinite interval i general theory abstract this paper extends the methods of special factorization to treat a class of factorization problems on the half line factorizations involving integral operators with stationary and nonstationary kernels are presented a simple time domain connection between the factorization problem and the stable regulator problem in hilbert space is developed

bounds on the minimum support weights the minimum support weight d sub r c of a linear code c over gf q is the minimal size of the support of an r dimensional subcode of c a number of bounds on d sub r c are derived generalizing the plotkin bound and the griesmer bound as well as giving two new existential bounds as the main result it is shown that there exist codes of any given rate r whose ratio d sub rd sub 1 is lower bounded by a number ranging from q sup r 1 q sup r q sup r 1 to r depending on r

a posteriori error estimator for finite volume methods in this paper first we discuss a technique to compare finite volume method and some well known finite element methods namely the dual mixed methods and nonconforming primal methods for elliptic equations these both equivalences are exploited to give us a posteriori error estimator for finite volume methods this estimator is explicitly given easy to compute and asymptotically exact without any regularity of the solution in unstructured grids

a dual graph norm refinement indicator for finite volume approximations of the euler equations we generalise and apply a refinement indicator of the type originally designed by mackenzie suli and warnecke in 15 and 16 for linear friedrichs systems to the euler equations of inviscid compressible fluid flow the euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in 15 can be applied we discuss the details of the implementation of the refinement indicator into the dlr tau code which is based on a finite volume method of box type on an unstructured grid and present numerical results

on a class of operators on hilbert space with applications to factorization and systems theory abstract a class of operators is defined in a hilbert resolution space setting that offers a new perspective on problems of causal invertibility special factorization and the theory of quadratic cost optimization problems for dynamical systems the major results include an extension of the special factorization to a class of noncompact operators and the definition of an abstract state space these results are then used to obtain an optimal feedback solution to an abstract linear regular quadratic cost problem

a tower of artin schreier extensions of function fields attaining the drinfeld vladut bound for an algebraic function fieldf having a finite constant field letg f resp n f denote the genus off resp the number of places off of degree one we construct a tower of function fields f 1 subseteq f 2 subseteq f 3 subseteq ldots over mathbb f q 2 such that the ration fi g fi tends to the drinfeld vladut boundq 1

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

new lower bounds on the generalized hamming weights of ag codes a sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized then the result is used through the so called feng rao numbers to bound the generalized hamming weights of algebraic geometry codes this is further developed for hermitian codes and the codes on one of the garcia stichtenoth towers as well as for some more general families

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

a lower bound on list size for list decoding a q ary error correcting code c 1 2 q n is said to be list decodable to radius with list size l if every hamming ball of radius contains at most l codewords of c we prove that in order for a q ary code to be list decodable up to radius 1 1 q 1 e n we must have l 1 e2 specifically we prove that there exists a constant cq 0 and a function fq such that for small enough e 0 if c is list decodable to radius 1 1 q 1 e n with list size cq e2 then c has at most fq e codewords independent of n this result is asymptotically tight treating q as a constant since such codes with an exponential in n number of codewords are known for list size l o 1 e2 a result similar to ours is implicit in blinovsky problems of information transmission 1986 for the binary q 2 case our proof is simpler and works for all alphabet sizes and provides more intuition for why the lower bound arises

flux reconstruction and solution post processing in mimetic finite difference methods abstract we present a post processing technique for the mimetic finite difference solution of diffusion problems in mixed form our post processing method yields a piecewise linear approximation of the scalar variable that is second order accurate in the l2 norm on quite general polyhedral meshes including non convex and non matching elements the post processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables this technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied the post processing method is computationally inexpensive optimal performance is confirmed by numerical experiments