efficient root finding algorithm with application to list decoding of algebraic geometric codes a list decoding for an error correcting code is a decoding algorithm that generates a list of codewords within a hamming distance t from the received vector where t can be greater than the error correction bound in previous work by m shokrollahi and h wasserman see ibid vol 45 p 432 7 march 1999 a list decoding procedure for reed solomon codes was generalized to algebraic geometric codes recent work by v guruswami and m sudan see ibid vol 45 p 1757 67 sept 1999 gives improved list decodings for reed solomon codes and algebraic geometric codes that work for all rates and have many applications however these list decoding algorithms are rather complicated r roth and g ruckenstein see ibid vol 46 p 246 57 jan 2000 proposed an efficient implementation of the list decoding of reed solomon codes in this correspondence extending roth and ruckenstein s fast algorithm for finding roots of univariate polynomials over polynomial rings i e the reconstruct algorithm we present an efficient algorithm for finding the roots of univariate polynomials over function fields based on the extended algorithm we give an efficient list decoding algorithm for algebraic geometric codes

on weierstrass semigroups and the redundancy of improved geometric goppa codes improved geometric goppa codes have a smaller redundancy and the same bound on the minimum distance as ordinary algebraic geometry codes for an asymptotically good sequence of function fields we give a formula for the redundancy

algebraic function fields over finite fields with many rational places algebraic function fields or equivalently algebraic curves provide a useful tool for coding theory for instance algebraic geometric codes and trace codes but also for other branches of information theory in these applications the number of rational places of a function field plays a crucial role one is particularly interested in function fields having a large number of rational places after a short introduction into the mathematical theory of algebraic functions the paper gives a survey of old and new results on the number of rational places of function fields

approximate reduction of dynamic systems abstract the reduction of dynamic systems has a rich history with many important applications related to stability control and verification reduction of nonlinear systems is typically performed in an exact manner as is the case with mechanical systems with symmetry which unfortunately limits the type of systems to which it can be applied the goal of this paper is to consider a more general form of reduction termed approximate reduction in order to extend the class of systems that can be reduced using notions related to incremental stability we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors i e when it is behaviourally similar to a dynamic system on a lower dimensional space these concepts are illustrated on a series of examples

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

codes on garcia stichtenoth curves with true distance greater than feng rao distance we give codes on all members after level 4 of the family of garcia stichtenoth see invent math vol 121 p 211 22 1995 curves with the property d sub true d sub fr these codes also give an improvement on tsfasman vladut zink bound

correlated algebraic geometric codes improved list decoding over bounded alphabets we define a new family of error correcting codes based on algebraic curves over finite fields and develop efficient list decoding algorithms for them our codes extend the class of algebraic geometric ag codes via a nonobvious generalization of the approach in the recent breakthrough work of parvaresh and vardy 2005 our work shows that the pv framework applies to fairly general settings by elucidating the key algebraic concepts underlying it also more importantly ag codes of arbitrary block length exist over fixed alphabets thus enabling us to establish new trade offs between the list decoding radius and rate over a bounded alphabet size the work of parvaresh and vardy 2005 was extended in guruswami and rudra 2006 to give explicit codes that achieve the list decoding capacity optimal trade off between rate and fraction of errors corrected over large alphabets a similar extension of this work along the lines of guruswami and rudra could have substantial impact indeed it could give better trade offs than currently known over a fixed alphabet say gf 2 12 which in turn upon concatenation with a fixed well understood binary code could take us closer to the list decoding capacity for binary codes this may also be a promising way to address the significant complexity drawback of the result of guruswami and rudra and to enable approaching capacity with bounded list size independent of the block length the list size and decoding complexity in their work are both n 1 e where e is the distance to capacity similar to algorithms for ag codes from guruswami and sudan 1999 and 2001 our encoding decoding algorithms run in polynomial time assuming a natural polynomial size representation of the code for codes based on a specific optimal algebraic curve we also present an expected polynomial time algorithm to construct the requisite representation this in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for ag codes

monad transformers as monoid transformers the incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a pre existing monad a complication of this approach is that the operations associated to the pre existing monad need to be lifted to the new monad in a companion paper by jaskelioff the lifting problem has been addressed in the setting of system email protected here we recast and extend those results in a category theoretic setting we abstract and generalize from monads to monoids in a monoidal category and from monad transformers to monoid transformers the generalization brings more simplicity and clarity and opens the way for lifting of operations with applicability beyond monads

unconditionally stable meshless integration of time domain maxwell s curl equations grid based methods coupled with an explicit approach for the evolution in time are traditionally adopted in solving pdes in computational electromagnetics the discretization in space with a grid covering the problem domain and a stability step size restriction must be accepted evidence is given that efforts need for overcoming these heavy constraints the connectivity laws among the points scattered in the problem domain can be avoided by using meshless methods among these the smoothed particle electromagnetics gives an interesting answer to the problem overcoming the limit of the grid generation in the original formulation an explicit integration scheme is used providing spatial and time discretization strictly interleaved and mutually conditioned in this paper a formulation of the alternating direction implicit scheme is proposed into the meshless framework the developed formulation preserves the leapfrog marching on in time of the explicit integration scheme studies on the systems matrices arising at each temporal step are reported referring to the meshless discretization the new method not constrained by a grid in space and unconditionally stable in time is validated by numerical simulations

list decoding of algebraic geometric codes we generalize sudan s see j compl vol 13 p 180 93 1997 results for reed solomon codes to the class of algebraic geometric codes designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error correction bound d 1 2 d being the minimum distance of the code our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields for the latter problem we design a polynomial time algorithm and show that the resulting overall list decoding algorithm runs in polynomial time under some mild conditions several examples are included

imposing various boundary conditions on positive definite kernels abstract this work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly it can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the kernel based pseudospectral method for handling the differential equations with more complicated boundary conditions in the proposed technique some new kernels are constructed using the positive definite kernels in a manner that they satisfy the required conditions in addition we prove the positive definiteness of the newly constructed kernel and also the non singularity of the collocation matrix is proved under some conditions the proposed method is verified through the numerical solution of some benchmark problems such as a singularly perturbed steady state convection diffusion problem two and three dimensional poissons equations with various boundary conditions

repairing algebraic geometry codes minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable for short codes thus the number of nodes is upper bounded by 2 mathfrak b where mathfrak b is the bits of data stored in each node from both theoretical and practical points of view see the details in section 1 it is natural to consider regenerating codes that nearly have minimum storage of data and meanwhile the number of nodes is unbounded one of the candidates for such regenerating codes is an algebraic geometry code in this paper we generalize the repairing algorithm of reed solomon codes given by guruswami and wotters to algebraic geometry codes and present a repairing algorithm for arbitrary one point algebraic geometry codes by applying our repairing algorithm to the one point algebraic geometry codes based on the garcia stichtenoth tower one can repair a code of rate 1 varepsilon and length n over mathbb f q with bandwidth n 1 1 tau log q for any varepsilon 2 tau 1 2 log q with a real tau in 0 1 2 in addition storage in each node for an algebraic geometry code is close to the minimum storage due to nice structures of hermitian curves repairing of hermitian codes is also investigated as a result we are able to show that algebraic geometry codes are regenerating codes with good parameters

fast decoding of multipoint codes from algebraic curves multipoint codes are a broad class of algebraic geometry codes derived from algebraic functions which have multiple poles and or zeros on an algebraic curve thus they are more general than one point codes which are an important class of algebraic geometry codes in the sense that they can be decoded efficiently using the berlekamp massey sakata algorithm we present a fast method for decoding multipoint codes from a plane curve particularly a hermitian curve our method with some adaptation can be applied to decode multipoint codes from a general algebraic curve embedded in the n dimensional affine space fqn over a finite field fq so that those algebraic geometry codes can be decoded efficiently if the dimension n of the affine space including the defining curve is small

on the splitting of places in a tower of function fields meeting the drinfeld vladut bound a description of how places split in an asymptotically optimal tower of function fields studied by garcia and stichtenoth 1995 is provided and an exact count of the number of places of degree one is given this information is useful in the setting up of generator matrices for algebraic geometry codes constructed over this function field tower these long codes have performance that asymptotically improves upon the gilbert varshamov bound

new uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields we obtain new uniform upper bounds for the tensor rank of the multiplication in the extensions of the finite fields mathbb f q for any prime power q moreover these uniform bounds lead to new asymptotic bounds as well in addition we also give purely asymptotic bounds which are substantially better by using a family of shimura curves defined over mathbb f q with an optimal ratio of mathbb f q t rational places to their genus where q t is a square

an implicit explicit runge kutta chebyshev finite element method for the nonlinear lithium ion battery equations abstract we introduce a numerical method to integrate the nonlinear system of equations that model the physical chemical laws of the lithium ion batteries the mathematical model formulated in doyle et al j electrochem soc 140 1993 1526 1533 is a system of strongly coupled nonlinear parabolic and elliptic equations our numerical method combines a second order implicit explicit runge kutta chebyshev scheme for time discretization of the parabolic equations governing the dynamics of the variables ce and cs with linear finite element space discretization of the system the implicit explicit numerical formulation of the parabolic equations allows us to decouple the strong nonlinear reaction terms which are treated implicitly from the linear diffusion terms treated explicitly this approach is computationally efficient because a it has an extended stability region and b the coupled system of algebraic equations becomes a single variable nonlinear equation per mesh point which is easily solved by newton method

torsion limits and riemann roch systems for function fields and applications the ihara limit or constant a q has been a central problem of study in the asymptotic theory of global function fields or equivalently algebraic curves over finite fields it addresses global function fields with many rational points and so far most applications of this theory do not require additional properties motivated by recent applications we require global function fields with the additional property that their zero class divisor groups contain at most a small number of d torsion points we capture this with the notion of torsion limit a new asymptotic quantity for global function fields it seems that it is even harder to determine values of this new quantity than the ihara constant nevertheless some nontrivial upper bounds are derived apart from this new asymptotic quantity and bounds on it we also introduce riemann roch systems of equations it turns out that this type of equation system plays an important role in the study of several other problems in each of these areas arithmetic secret sharing symmetric bilinear complexity of multiplication in finite fields frameproof codes and the theory of error correcting codes finally we show how our new asymptotic quantity our bounds on it and riemann roch systems can be used to improve results in these areas

combining effects sum and tensor we seek a unified account of modularity for computational effects we begin by reformulating moggi s monadic paradigm for modelling computational effects using the notion of enriched lawvere theory together with its relationship with strong monads this emphasises the importance of the operations that produce the effects effects qua theories are then combined by appropriate bifunctors on the category of theories we give a theory for the sum of computational effects which in particular yields moggi s exceptions monad transformer and an interactive input output monad transformer we further give a theory of the commutative combination of effects their tensor which yields moggi s side effects monad transformer finally we give a theory of operation transformers for redefining operations when adding new effects we derive explicit forms for the operation transformers associated to the above monad transformers

an explicit construction of a sequence of codes attaining the tsfasman vladut zink bound the first steps we present a sequence of codes attaining the tsfasman vladut zink bound the construction is based on the tower of artin schreier extensions described by garcia and stichtenoth 1995 we also determine the dual codes the first steps of the constructions are explicitly given as generator matrices

asymptotic bound for multiplication complexity in the extensions of small finite fields in 1986 d v chudnovsky and g v chudnovsky first employed algebraic curves over finite fields to construct bilinear multiplication algorithms implicitly through supercodes introduced by shparlinski tsfasman vladut or equivalently multiplication friendly codes that we will introduce in this paper this idea was further developed by shparlinski tsfasman vladut in order to study the asymptotic behavior of multiplication complexity in extension fields later on ballet et al further investigated the method and obtained some improvements recently ballet and pieltant made use of curves over an extension field of to obtain an improvement on the complexity of multiplications in extensions of the binary field in this paper we develop the multiplication friendly splitting technique and then apply this technique to study asymptotic behavior of multiplications in extension fields by combining this with the idea of using algebraic function fields we are able to improve further the asymptotic results of multiplication complexity in particular the improvement for small fields such as the binary and ternary fields is substantial

a posteriori error estimation for deim reduced nonlinear dynamical systems in this work an efficient approach for a posteriori error estimation for pod deim reduced nonlinear dynamical systems is introduced the considered nonlinear systems may also include time and parameter affine linear terms as well as parametrically dependent inputs and outputs the reduction process involves a galerkin projection of the full system and approximation of the system s nonlinearity by the deim method s chaturantabut and d c sorensen siam j sci comput 32 2010 pp 2737 2764 the proposed a posteriori error estimator can be efficiently decomposed in an offline online fashion and is obtained by a one dimensional auxiliary ode during reduced simulations key elements for efficient online computation are partial similarity transformations and matrix deim approximations of the nonlinearity jacobians the theoretical results are illustrated by application to an unsteady burgers equation and a cell apoptosis model

curves of every genus with many points ii asymptotically good families we resolve a 1983 question of serre by constructing curves with many points of every genus over every finite field more precisely we show that for every prime power q there is a positive constant cq with the following property for every integer g 0 there is a genus g curve over fq with at least cqg rational points over fq moreover we show that there exists a positive constant d such that for every q we can choose cq d log q we show also that there is a constant c 0 such that for every q and every n 0 and for every sufficiently large g there is a genus g curve over fq that has at least cg n rational points and whose jacobian contains a subgroup of rational points isomorphic to z nz for some r cg n

preserving lagrangian structure in nonlinear model reduction with application to structural dynamics this work proposes a model reduction methodology that preserves lagrangian structure and achieves computational efficiency in the presence of high order nonlinearities and arbitrary parameter dependence as such the resulting reduced order model retains key properties such as energy conservation and symplectic time evolution maps we focus on parameterized simple mechanical systems subjected to rayleigh damping and external forces and consider an application to nonlinear structural dynamics to preserve structure the method first approximates the system s lagrangian ingredients the riemannian metric the potential energy function the dissipation function and the external force and subsequently derives reduced order equations of motion by applying the forced euler lagrange equation with these quantities from the algebraic perspective key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness matri

efficient multi point local decoding of reed muller codes via interleaved codex reed muller codes are among the most important classes of locally correctable codes currently local decoding of reed muller codes is based on decoding on lines or quadratic curves to recover one single coordinate to recover multiple coordinates simultaneously the naive way is to repeat the local decoding for recovery of a single coordinate this decoding algorithm might be more expensive i e require higher query complexity in this paper we focus on reed muller codes with usual parameter regime namely the total degree of evaluation polynomials is d theta q where q is the code alphabet size in fact d can be as big as q 4 in our setting by introducing a novel variation of codex i e interleaved codex the concept of codex has been used for arithmetic secret sharing we are able to locally recover arbitrarily large number k of coordinates of a reed muller code simultaneously with error probability exp omega k at the cost of querying merely o q 2 k coordinates it turns out that our local decoding of reed muller codes shows perhaps surprisingly that accessing k locations is in fact cheaper than repeating the procedure for accessing a single location for k times precisely speaking to get the same success probability by repeating the local decoding algorithm of a single coordinate one has to query omega qk 2 coordinates thus the query complexity of our local decoding is smaller for k omega q if we impose the same query complexity constraint on both algorithm our local decoding algorithm yields smaller error probability when k omega q q in addition our local decoding is efficient i e the decoding complexity is mathrm poly k q construction of an interleaved codex is based on concatenation of a codex with a multiplication friendly pair while the main tool to realize codex is based on algebraic function fields or more precisely algebraic geometry codes

explicit construction of ag codes from a curve in the tower of bassa beelen garcia stichtenoth we present multi point algebraic geometric codes overstepping the gilbert varshamov bound the construction is based on a curve in the tower introduced by bassa et al these codes are described in detail by constructing generator matrices it turns out that these codes have nice properties similar to those of hermitian codes the duals are also such codes and can be expressed as an explicit formula

an improvement of the construction of the d v and g v chudnovsky algorithm for multiplication in finite fields from an interpolation method on algebraic curves due to d v chudnovsky and g v chudnovsky we give a new method for the construction of bilinear algorithms for multiplication in the extensions of finite fields we obtain algorithms better than known ones

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

high rate locally correctable codes via lifting we present a general framework for constructing high rate error correcting codes that are locally correctable and hence locally decodable if linear with a sublinear number of queries based on lifting codes with respect to functions on the coordinates our approach generalizes the lifting of affine invariant codes of guo kopparty and sudan and its generalization automorphic lifting alluded to in the work of ben sasson et al but distinct from their degree lifting which lifts algebraic geometry codes with respect to a group of automorphisms of the code our notion of lifting is a natural alternative to the degree lifting of ben sasson et al and it carries two advantages first it overcomes the rate barrier inherent in degree lifting second it requires no special properties e g linearity and invariance of the base code and requires a very little structure on the set of functions on the coordinates of the code as an application we construct new explicit families of locally correctable codes by lifting algebraic geometry codes like the multiplicity codes of kopparty saraf yekhanin and the affine lifted codes of guo kopparty and sudan our codes of block length n can achieve n epsilon query complexity and 1 alpha rate for any given epsilon alpha 0 while correcting a constant fraction of errors in contrast to the reed muller codes and the degree lifted ag codes of ben sasson et al which face a rate barrier of epsilon o 1 epsilon however like the degree lifted ag codes our codes are over an alphabet significantly smaller than that obtained by reed muller codes affine lifted codes and multiplicity codes

a matrix deim technique for model reduction of nonlinear parametrized problems in cardiac mechanics abstract when relying on newton iterations to solve nonlinear problems in the context of reduced basis rb methods the assembling of the rb arrays during the online stage depends on the dimension of the underlying high fidelity approximation this is more of an issue when dealing with fully nonlinear problems for which the global jacobian matrix has to be entirely reassembled at each newton step in this paper the discrete empirical interpolation method deim and its matrix version mdeim are combined to evaluate both the residual vector and the jacobian matrix very efficiently in the case of complex parametrized nonlinear mechanical problems we compare this strategy with the classical deim approach and we derive a posteriori error estimates on the solution accounting for the contribution of deim mdeim errors the effectiveness of the proposed framework is assessed on quasi static nonlinear problems in particular we consider a nonlinear elasticity problem defined on a cube and a mechanical model describing heart contraction for an idealized left ventricle geometry the latter is a coupled problem in which the activation of the heart contraction is given by the solution of an electrophysiology model our numerical results show that mdeim is preferable to the classical deim both in terms of efficiency and accuracy

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

erasure list decodable codes from random and algebraic geometry codes erasure list decoding was introduced to correct a larger number of erasures by outputting a list of possible candidates in this paper we consider both random linear codes and algebraic geometry codes for list decoding from erasures the contributions of this paper are twofold first for arbitrary 0 and epsilon 0 r and epsilon are independent we show that with high probability a q ary random linear code of rate r is an erasure list decodable code with constant list size q o 1 epsilon that can correct a fraction 1 r epsilon of erasures i e a random linear code achieves the information theoretic optimal tradeoff between information rate and fraction of erasures second we show that algebraic geometry codes are good erasure list decodable codes precisely speaking a q ary algebraic geometry code of rate r from the garcia stichtenoth tower can correct 1 r 1 sqrt q 1 1 q epsilon fraction of erasures with list size o 1 epsilon this improves the johnson bound for erasures applied to algebraic geometry codes furthermore list decoding of these algebraic geometry codes can be implemented in polynomial time note that the code alphabet size q in this paper is constant and independent of epsilon

multiscale empirical interpolation for solving nonlinear pdes in this paper we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations the proposed method combines empirical interpolation techniques and local multiscale methods such as the generalized multiscale finite element method gmsfem to solve nonlinear equations the gmsfem is used to represent the solution on a coarse grid with multiscale basis functions computed offline computing the gmsfem solution involves calculating the system residuals and jacobians on the fine grid we use empirical interpolation concepts to evaluate these residuals and jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse scale problem rather than the fully resolved fine scale one the empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times the coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system the proposed multiscale empirical interpolation techniques 1 divide computing the nonlinear function into coarse regions 2 evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced order representation of the solution and 3 introduce multiscale proper orthogonal decomposition techniques to find appropriate interpolation vectors we demonstrate the effectiveness of the proposed methods on several nonlinear multiscale pdes that are solved with newton s methods and fully implicit time marching schemes our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale pdes on a coarse grid with bounded error and significant computational cost reduction

kernel based approximation in sobolev spaces with radial basis functions in this paper we study several radial basis function approximation schemes in sobolev spaces we obtain an optional error estimate by using a class of smoothing operators we also discussed sufficient conditions for the smoothing operators to attain the desired approximation order we then construct the smoothing operators by some compactly supported radial kernels and use them to approximate sobolev space functions with optimal convergence order these kernels can be simply constructed and readily applied to practical problems the results show that the approximation power depends on the precision of the sampling instrument and the density of the available data

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

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

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

the generalized empirical interpolation method stability theory on hilbert spaces with an application to the stokes equation the generalized empirical interpolation method geim is an extension first presented by maday and mula in 1 in 2013 of the classical empirical interpolation method presented in 2004 by barrault maday nguyen and patera in 2 where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of banach spaces as outlined in 1 this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach in this paper we present a thorough analysis of the concept of stability condition of the generalized interpolant the lebesgue constant by relating it to an inf sup problem in the case of hilbert spaces in the second part of the paper it will be explained how geim can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number optimally placed measurements from the processes with their mathematical models parameter dependent pdes this idea is illustrated through a parameter dependent stokes problem in which it is shown that the pressure and velocity fields can e ciently be reconstructed with a relatively low dimensional interpolation space

constructions of codes from number fields we define number theoretic error correcting codes based on algebraic number fields thereby providing a generalization of chinese remainder codes akin to the generalization of reed solomon codes to algebraic geometric codes our construction is very similar to and in fact less general than the one given by lenstra 1986 but the parallel with the function field case is more apparent since we only use the non archimedean places for the encoding we prove that over an alphabet size as small as 19 there even exist asymptotically good number field codes of the type we consider this result is based on the existence of certain number fields that have an infinite class field tower in which some primes of small norm split completely

proper orthogonal decomposition based model order reduction via radial basis functions for molecular dynamics systems summary model order reduction for molecular dynamics md systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi atomic interactions in high dimensions in the present work we introduce a proper orthogonal decomposition based method in conjunction with the radial basis function rbf approximation of the nonlinear and nonlocal potential energies and inter atomic forces for md systems this approach avoids coordinate transformation between the physical and reduced order coordinates and allows the potentials and inter atomic forces to be calculated directly in the reduced order space the rbf approximated potential energies and inter atomic forces in the reduced order space are discretized on the basis of the smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method the good approximation properties of rbfs in interpolating scattered data make them ideal candidates for the reduced order approximation of md inter atomic force calculations the proposed approach is validated by performing the reduced order simulations of dna molecules under various external loadings copyright 2013 john wiley sons ltd

disjoint linear codes from algebraic function fields in this correspondence we study disjoint linear codes and give constructions of families of disjoint linear codes based on algebraic function fields it turns out that for some parameters our constructions improve on a result of johansson and pasalic

the localized reduced basis multiscale method for two phase flows in porous media summary in this work we propose a novel model order reduction approach for two phase flow in porous media by introducing a global pressure formulation in which the total mobility which realizes the coupling between saturation and pressure is regarded as a parameter to the pressure equation in our approach the mobility itself is an optimal fit of mobility profiles that are precomputed using the time of flight for the initial saturation using this formulation and the localized reduced basis multiscale method we obtain a low dimensional surrogate of the high dimensional pressure equation by applying ideas from model order reduction for parametrized partial differential equations we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two phase flow simulation which is thereby largely accelerated usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online steps to reach an ideal runtime at acceptable error increase for the two phase flow simulation this is achieved by constructing reduced dimensional local spaces that lead to a non conforming global approximation space as one example for a coupling on the global space we introduce a discontinuous galerkin scheme copyright 2014 john wiley sons ltd

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

pod galerkin reduced order modeling with adaptive finite element snapshots we consider model order reduction by proper orthogonal decomposition pod for parametrized partial differential equations where the underlying snapshots are computed with adaptive finite elements we address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces we propose a method to create a pod galerkin model without interpolating the snapshots onto their common finite element mesh the error of the reduced order solution is not necessarily galerkin orthogonal to the reduced space created from space adapted snapshot we analyze how this influences the error assessment for pod galerkin models of linear elliptic boundary value problems as a numerical example we consider a two dimensional convection diffusion equation with a parametrized convective direction to illustrate the applicability of our techniques to non linear time dependent problems we present a test case of a two dimensional viscous burgers equation with parametrized initial data

surrogate modeling of multiscale models using kernel methods this work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods the key element is to learn the interface between the different scales using a fast surrogate for the microscale model which is given by multivariate kernel expansions the expansions are computed using statistically representative samples of in and output of the microscale model as learning methods we apply both support vector machines and a vectorial kernel greedy algorithm we demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines first we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and second a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

transitive and self dual codes attaining the tsfasman vla spl breve dut 80 zink bound a major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good in this correspondence as a generalization of cyclic codes the notion of transitive codes is introduced see definition 1 4 in section i and it is shown that the class of transitive codes is asymptotically good even more transitive codes attain the tsfasman vlabrevedut 80 zink bound over fq for all squares q l2 it is also shown that self orthogonal and self dual codes attain the tsfasman vlabrevedut 80 zink bound thus improving previous results about self dual codes attaining the gilbert varshamov bound the main tool is a new asymptotically optimal tower e0subee1 subee2sube middotmiddotmiddot of function fields over fq with q lscr2 where all extensions en e0 are galois

finding the defining functions for one point algebraic geometry codes an iterative algorithm is given for producing the parity check functions for one point algebraic geometry ag codes in particular but not limited to those related to the two towers of function fields introduced by garcia and stichtenoth 1995 this simple method based on the linearity of the qth power map over f sub q is an alternative to the method of using laurent series expansions examples are given for small values of m with characteristic p 2 for both towers

a dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction in this article we introduce a new dimensional reduction approach which is based on the application of reduced basis rb techniques in the hierarchical model reduction hmr frame work considering problems that exhibit a dominant spatial direction the idea of hmr is to perform a galerkin projection onto a reduced space which combines the full solution space in the dominant direction with a reduction space in the transverse direction the latter is spanned by modal or thonormal basis functions while so far the basis functions in the hmr approach have been chosen a priori s perotto a ern and a veneziani multiscale model simul 8 2010 pp 1102 1127 for instance as legendre or trigonometric polynomials in this work a highly nonlinear approximation is employed for the construction of the reduction space to this end we first derive a lower dimen sional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction exploiting the good ap proximation properties of rb methods we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation for an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article we introduce our method for general elliptic problems such as advection diffusion equations in two space dimensions numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach

good recursive towers over prime fields exist we give a construction and equations for good recursive towers over any finite field mathbf f q with q ne 2 and 3

a unified theory of radial basis functions native hilbert spaces for radial basis functions ii abstract this contribution provides a new formulation of the theory of radial basis functions in the context of integral operators instead of fourier transforms the most important tools now are expansions into eigenfunctions this unifies the theory of radial basis functions in r d with the theory of zonal functions on the sphere sd 1 and the theory of kernel functions on riemannian manifolds new characterizations of native spaces and positive definite functions are included the paper is a self contained continuation of an earlier survey r schaback international series of numerical mathematics vol 132 birkhauser basel 1999 pp 255 282 over the native spaces associated to not necessarily radial basis functions

efficient a posteriori error estimation for nonlinear kernel based reduced systems abstract in this paper we consider the topic of model reduction for nonlinear dynamical systems based on kernel expansions our approach allows for a full offline online decomposition and efficient online computation of the reduced model in particular we derive an a posteriori state space error estimator for the reduction error a key ingredient is a local lipschitz constant estimation that enables rigorous a posteriori error estimation the computation of the error estimator is realized by solving an auxiliary differential equation during online simulations estimation iterations can be performed that allow a balancing between estimation sharpness and computation time numerical experiments demonstrate the estimation improvement over different estimator versions and the rigor and effectiveness of the error bounds

multiplication algorithm in a finite field and tensor rank of the multiplication abstract we generalize the multiplication algorithm of d v and g v chudnovsky using the new algorithm we improve the upper bounds of the bilinear complexity of multiplication in all extensions of a finite field of characteristic 2 on the other hand we also improve in characteristic 2 the asymptotic upper bounds of this complexity

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

on the number of correctable errors of the feng rao decoding algorithm for ag codes in this correspondence we give geometric codes on one of the curves of garcia stichtenoth 1995 with the property that the true distances are bigger than feng rao distances moreover an example for which the feng rao algorithm cannot correct more than d sub fr 1 2 1 d sub true 1 2 errors is given

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

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

are the snapshot difference quotients needed in the proper orthogonal decomposition this paper presents a theoretical and numerical investigation of the following practical question should the time difference quotients dqs of the snapshots be used to generate the proper orthogonal decomposition pod basis functions the answer to this question is important since some published numerical studies use the time dqs whereas other numerical studies do not the criterion used in this paper to answer this question is the optimality of the convergence rate of the error of the reduced order model with respect to the number of pod basis functions since to the best of our knowledge a definition of the optimality of the convergence rate is not available we propose one in definition 3 1 in this paper two cases are considered the no dq case in which the snapshot dqs are not used and the dq case in which the snapshot dqs are used for each case two types of pod bases are used the l 2 pod basis in which the basis is generated in the l 2 norm and the h 1 pod basis in which the basi

categories of rough sets and textures abstract it is known that the theories of rough sets and fuzzy sets have successful applications in computing textures as a theoretical model provide a new perspective for both rough sets and fuzzy sets indeed recent papers have shown that there is a natural link between rough sets and textures while a texture is an alternative point set based setting for fuzzy sets relations are representatives of information systems and induce approximation operators therefore the first step for the categorical discussions on rough sets involves the category rel of sets and relations in this context we observe that power sets and pairs of rough set approximation operators form a category denoted by r apr in particular we prove that r apr is isomorphic to a full subcategory of the category cdrtex whose objects are complemented textures and morphisms are complemented direlations therefore cdrtex may be regarded as a suitable abstract model of rough set theory here we show that r apr and cdrtex are new examples of dagger symmetric monoidal categories

a posteriori error analysis of the reduced basis method for nonaffine parametrized nonlinear pdes in this paper we present the a posteriori error analysis for the reduced basis method rbm applied to nonlinear variational problems that depend on a parameter in a nonaffine manner to this end we generalize the analysis by veroy and patera int j numer methods fluids 47 2005 pp 773 788 to nonaffine parametrized partial differential equations we use the empirical interpolation method eim in order to approximate the nonaffine parameter dependencies by a linear combination of affine functions we also investigate a standard dual problem formulation in particular for the computation of a general output functional also in combination with the eim first we study the well posedness in terms of the brezzi rappaz raviart theory then we develop a posteriori error estimates and investigate offline online decompositions the a posteriori error analysis allows us to introduce an adaptive sampling procedure for the choice of the modes numerical experiments for a convection diffusion problem around a rotating propeller show the effectivity of the scheme

algebraic geometry codes with complementary duals exceed the asymptotic gilbert varshamov bound it was shown by massey that linear complementary dual lcd codes are asymptotically good in 2004 sendrier proved that lcd codes meet the asymptotic gilbert varshamov gv bound until now the gv bound still remains to be the best asymptotical lower bound for lcd codes in this paper we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an lcd code and consequently there exists a family of lcd codes that are equivalent to algebraic geometry codes and exceed the asymptotical gv bound

notions of computation and monads abstract the calculus is considered a useful mathematical tool in the study of programming languages since programs can be identified with terms however if one goes further and uses conversion to prove equivalence of programs then a gross simplification is introduced programs are identified with total functions from values to values that may jeopardise the applicability of theoretical results in this paper we introduce calculi based on a categorical semantics for computations that provide a correct basis for proving equivalence of programs for a wide range of notions of computation

reduced order modelling of parameter dependent linear and nonlinear dynamic partial differential equation models in this paper we develop reduced order models for dynamic parameter dependent linear and nonlinear partial differential equations using proper orthogonal decomposition pod the main challenges are to accurately and efficiently approximate the pod bases for new parameter values and in the case of nonlinear problems to efficiently handle the nonlinear terms we use a bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values computational efficiency is ensured by using manifold learning to perform the emulation in a low dimensional space the accuracy of the method is demonstrated on a linear and a nonlinear example with comparisons with a global basis approach

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

the semantics of second order lambda calculus abstract in the second order polymorphic typed lambda calculus lambda abstraction over type variables leads to terms denoting polymorphic functions straightforward cardinality considerations show that a naive set theoretic interpretation of the calculus is impossible we give two definitions of semantic models for this language and prove them equivalent our syntactical environment model definition and a more algebraic combinatory model definition for the polymorphic calculus correspond to analogous model definitions for untyped lambda calculus soundness and completeness theorems are proved using the environment model definition we verify that some specific interpretations of the calculus proposed in the literature indeed yield models in our sense

curves of every genus with many points i abelian and toric families abstract let nq g denote the maximal number of f q rational points on any curve of genus g over f q ihara for square q and serre for general q proved that lim supg nq g g 0 for any fixed q here we prove limg nq g more precisely we use abelian covers of p 1 to prove lim infg nq g g log g 0 and we use curves on toric surfaces to prove lim infg nq g g1 3 0 we also show that these results are the best possible that can be proved using these families of curves

bases for riemann roch spaces of one point divisors on an optimal tower of function fields for applications in algebraic geometric codes an explicit description of bases of riemann roch spaces of divisors on function fields over finite fields is needed we give an algorithm to compute such bases for one point divisors and weierstrass semigroups over an optimal tower of function fields we also explicitly compute weierstrass semigroups till level eight

towers of function fields and iterated means towers of function fields meeting the drinfeld vladut bound with equality were constructed by gorcal stichtenoth 1995 1996 modular uniformizations thereof were pointed out by elkies 1996 we connect this fact to hypergeometric analogs of the arithmetic geometric mean and related theta series identities discovered by borweins and garven

an improvement of the gilbert varshamov bound over nonprime fields the gilbert varshamov bound guarantees the existence of families of codes over the finite field bbf ell with good asymptotic parameters we show that this bound can be improved for all nonprime fields bbf ell with ell geq 49 except possibly ell 125 we observe that the same improvement even holds within the class of transitive codes and within the class of self orthogonal codes

efficient model reduction of parametrized systems by matrix discrete empirical interpolation in this work we apply a matrix version of the so called discrete empirical interpolation mdeim for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced order models roms however in many cases such an affine decomposition is not readily available and must be recovered through often intrusive procedures such as the empirical interpolation method eim and its discrete variant deim in this paper we show that mdeim represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non intrusive efficient and purely algebraic way we propose different strategies to combine mdeim with a state approximation resulting either from a reduced basis greedy approach or proper orthogonal decomposition a posteriori error estimates accounting for the mdeim error are also developed in the case of parametrized elliptic and parabolic equations finally the capability of mdeim to generate accurate and efficient roms is demonstrated on the solution of two computationally intensive classes of problems occurring in engineering contexts namely pde constrained shape optimization and parametrized coupled problems

locally recoverable codes on algebraic curves a code over a finite alphabet is called locally recoverable lrc code if every symbol in the encoding is a function of a small number at most r of other symbols of the codeword in this paper we introduce a construction of lrc codes on algebraic curves extending a recent construction of the reed solomon like codes with locality we treat the following situations local recovery of a single erasure local recovery of multiple erasures and codes with several disjoint recovery sets for every coordinate the availability problem for each of these three problems we describe a general construction of codes on curves and construct several families of lrc codes we also describe a construction of codes with availability that relies on automorphism groups of curves we also consider the asymptotic problem for the parameters of the lrc codes on curves we show that the codes obtained from asymptotically maximal curves for instance garcia stichtenoth towers improve upon the asymptotic versions of the gilbert varshamov bound for lrc codes