an overlapping domain decomposition preconditioner for a class of discontinuous galerkin approximations of advection diffusion problems we consider a scalar advection diffusion problem and a recently proposed discontinuous galerkin approximation which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency for the corresponding sparse non symmetric linear system we propose and study an additive two level overlapping schwarz preconditioner consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains this is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces related to the lack of continuity of our approximation spaces some interesting new features arise in our generalization which have no analog in the conforming case we prove an upper bound for the number of iterations obtained by using this preconditioner with gmres which is independent of the number of degrees of freedom of the original problem and the number of subdomains the performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions

limitedness theorem on finite automata with distance functions an algebraic proof a distance function on a finite automaton m is defined by assigning to each transition a distance of nonnegative integer value m is said to be limited in distance if there is a nonnegative integer k such that for each accepted string w in the language of m there is an accepting path p for w on which the sum of distances is bounded by k the limitedness theorem 6 9 on finite automata with distance functions states that it is decidable if an arbitrary finite automaton with a distance function is limited in distance in this paper we give an algebraic proof of the theorem and derive from it an exponential time decision algorithm in addition we prove that the decision problem is pspace hard

on 2d elliptic discontinuous galerkin methods we discuss the discretization using discontinuous galerkin dg formulation of an elliptic poisson problem two commonly used dg schemes are investigated the original average flux proposed by bassi and rebay j comput phys 1997 131 267 and the local discontinuous galerkin ldg siam j numer anal 1998 35 2440 2463 scheme in this paper we expand on previous expositions discontinuous galerkin methods theory computation and applications springer berlin 2000 135 146 siam j sci comput 2002 24 2 524 547 int j numer meth engng 2003 58 2 1119 1148 by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the dg method through consideration of standard c0 type expansion bases as opposed to elementally orthogonal expansions with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted the use of c0 type expansions also permits the direct enforcement of dirichlet boundary conditions through a lifting approach where the ldg flux does not require further stabilization in our construction we also adopt a formulation of the continuous dg fluxes that permits a more general interpretation of their numerical implementation in particular it allows us to determine the conditions under which the ldg method provides a near local stencil finally a study of the conditioning and the size of the null space of the matrix systems resulting from the dg discretization of the elliptic problem is undertaken copyright 2005 john wiley sons ltd

preconditioning methods for local discontinuous galerkin discretizations a multilevel interior penalty method is used as an efficient preconditioner for the schur complement of the local discontinuous galerkin ldg discretization of a poisson problem the method is then used in a block triangular preconditioner of the ldg saddle point system the block preconditioner is of the same efficiency as the schur complement version finally the block preconditioner is extended to the discretization of the stokes problem by the ldg method again the preconditioned saddle point problem can be solved in about as many steps as the schur complement the influence of several parameters on the performance of these methods is investigated

on measuring nondeterminism in regular languages abstract it is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language this paper studies situations intermediate between forbidding nondeterminism and allowing it the amount of nondeterminism used by a finite automaton is quantified so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied these intermediate situations are shown always to lie between two extremes 1 there are no savings as the amount of nondeterminism increases incrementally so that savings occur only when the amount of nondeterminism becomes unlimited 2 each increment of nondeterminism results in additional savings the number s of states decreasing approximately as s 1 i until exponential savings have been achieved after about i log s log log s increments

an extension of kleene s and ochman ski s theorems to infinite traces abstract as was noted by mazurkiewicz traces constitute a convenient tool for describing finite behaviour of concurrent systems extending in a natural way mazurkiewicz s original definition infinite traces have recently been introduced enabling one to deal with infinite behaviour of nonterminating concurrent systems in this paper we examine the basic families of recognizable sets and of rational sets of infinite traces the seminal kleene characterization of recognizable subsets of the free monoid and its subsequent extensions to infinite words due to buchi and to finite traces due to ochmanski are the cornerstones of the corresponding theories the main result of our paper is an extension of these characterizations to the domain of infinite traces using recognizing and weakly recognizing morphisms as well as a generalization of the schutzenberger product of monoids we prove various closure properties of recognizable trace languages moreover we establish normal form representations for recognizable and rational sets of infinite traces

some properties of recognizable l subsets abstract we study some properties of recognizable z subsets of a and its subfamilies the simple z subsets the limited z subsets the recognizable m subsets the simple m subsets and the m subsets which are nondeterministic complexities of finite automata at first we study some necessary conditions for membership in each one of these families and after we investigate the closure properties of these families under several operations we also study the relations existing among some subfamilies of recognizable m subsets and the families h p p 0 obtained by simon

recognizable closures and submonoids of free partially commutative monoids abstract the main result of this paper is the presentation of two sufficient conditions for the closure x of x and the closure x of x respectively in any free partially commutative monoid for any regular language x to be recognizable several previously obtained sufficient conditions for x to be recognizable can be seen to imply one of those two conditions

optimal off line detection of repetitions in a string an algorithm is presented to detect within optimal time o n log n and space o n off line on a ram all of the distinct repetitions in a given textstring on a finite alphabet the proposed strategy is self contained as it depends more heavily on algorithmic design considerations than on the combinatorial properties of the output it is based on a new data structure the leaf tree which is particularly suited to exploit simple properties of the suffix tree associated with the string to be analyzed

deflation accelerated preconditioning of the poisson neumann schur problem on long domains with a high order discontinuous element based collocation method a combination of block jacobi and deflation preconditioning is used to solve a high order discontinuous element based collocation discretization of the schur complement of the poisson neumann system as arises in the operator splitting of the incompressible navier stokes equations the preconditioners and deflation vectors are chosen to mitigate the effects of ill conditioning due to highly elongated domains typical of simulations of strongly non hydrostatic environmental flows and to achieve generalized minimum residual method gmres convergence independent of the size of the number of elements in the long direction the ill posedness of the poisson neumann system manifests as an inconsistency of the schur complement problem but it is shown that this can be accounted for with appropriate projections out of the null space of the schur complement matrix without affecting the accuracy of the solution the block jacobi preconditioner is shown to yield gmres convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition the combined deflation and block jacobi preconditioning is compared with two level non overlapping block jacobi preconditioning of the schur problem and while both methods achieve convergence independent of the grid size deflation is shown to require half as many gmres iterations and 25 less wall clock time for a variety of grid sizes and domain aspect ratios the deflation methods shown to be effective for the two dimensional poisson neumann problem are extensible to the three dimensional problem assuming a fourier discretization in the third dimension a fourier discretization results in a two dimensional helmholtz problem for each fourier component that is solved using deflated block jacobi preconditioning on its schur complement here again deflation is shown to be superior to two level non overlapping block jacobi preconditioning requiring about half as many gmres iterations and 15 less time

recognizable subsets of some partially abelian monoids we show that in a free partially abelian monoid generated by a finite alphabet a the subset x of a containing all the words equivalent to a product of words of x is rational if x is a finite set of words each word containing at least one occurrence of any letter of a we suppose that the graph the vertices of which are letters of a and the edges of which correspond to noncommuting pairs of letters is connected

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

rational relations and rational series a rational relation is a rational subset of the direct product of two free monoids r a b consider r as a function of a into the family of subsets of b by posing for all u a r u v b u v r assume r u is a finite set for all u a we study how the cardinality of r u behaves as the length of u tends to infinity and we show that there exists an infinite hierachy of growth functions

two level fourier analysis of a multigrid approach for discontinuous galerkin discretization in this paper we study a multigrid mg method for the solution of a linear second order elliptic equation discretized by discontinuous galerkin dg methods and we give a detailed analysis of the convergence for different block relaxation strategies we find that pointwise block partitioning gives much better results than the classical cellwise partitioning both for the baumann oden method and for the symmetric dg method with and without interior penalty ip the block relaxation methods jacobi gauss seidel and symmetric gauss seidel give excellent smoothing procedures in a classical mg setting independent of the mesh size simple mg cycles give convergence factors of 0 075 0 4 per iteration sweep for the different discretization methods studied

solving discontinuous galerkin formulations of poisson s equation using geometric and p multigrid methods for solving discontinuous galerkin formulations of the poisson equation by coupling p multigrid to geometric multigrid are investigated the simple approach of performing iterative relaxation on solution approximations of decreasing polynomial degree p down to p 0 and then applying geometric multigrid is ineffective the transition from p 1 top 0 causes the performance of the entire iteration to degrade because the long wavelength eigenfunctions of the p 1 discontinuous system are not represented well in thep 0 space a new approach is proposed that coarsens from thep 1 discontinuous space to thep 1 continuous space this approach eliminates the problems caused by the p 1 to p 0 transition furthermore the p 1 continuous space is a standard finite element space for which geometric multigrid is well defined in addition when the discontinuous galerkin equations are restricted to a continuous space one recovers the galerkin formulation of continuous finite elements thus applying geometric multigrid to this system is straightforward and effective numerical tests agree well with the analysis and confirm that the new approach gives rapid convergence rates that are grid independent and only weakly sensitive to the polynomial order

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

on the degree of ambiguity of finite automata we investigate the ambiguity behavior of finite automata in connection with their inner structure we show that the degree of ambiguity of a finitely ambiguous nondeterministic finite automaton nfa with n states is at most 5n2 nn there is a simple criterion which characterizes the infinite degree of ambiguity of an nfa and which is decidable in polynomial time the degree of growth of the ambiguity of an nfa is computable in polynomial time starting from the first result we discuss the maximal finite degree of ambiguity of an nfa with n states and we present subclasses of nfas where this quantity is of order 2 n

on recognizable subsets of free partially commutative monoids on montre que dans un mono de partiellement commutatif libre engendre par un alphabet fini a le sous ensemble x de a forme des mots equivalents a un des mots de x est reconnaissable si x est reconnaissable et si tout facteur iterant de x a un graphe de non commutation connexe

coloring trees in reverse mathematics abstract the tree theorem for pairs tt 2 2 first introduced by chubb hirst and mcnicholl asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2 there is a set of nodes isomorphic to 2 which is homogeneous for the coloring this is a generalization of the more familiar ramsey s theorem for pairs rt 2 2 which has been studied extensively in computability theory and reverse mathematics we answer a longstanding open question about the strength of tt 2 2 by showing that this principle does not imply the arithmetic comprehension axiom aca 0 over the base system recursive comprehension axiom rca 0 of second order arithmetic combined with a recent result of patey s that tt 2 2 is strictly stronger than rt 2 2 this establishes tt 2 2 as the first known example of a natural combinatorial principle to occupy the interval strictly between aca 0 and rt 2 2 the proof of this fact uses an extension of the bushy tree forcing method and develops new techniques for dealing with combinatorial statements formulated on trees rather than on

smoothed aggregation multigrid for the discontinuous galerkin method the aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems that arise from the discontinuous galerkin method the smoothed aggregation multigrid introduced by vanek for the conforming finite element method is applied to low order discretizations of convection diffusion equations for the elliptic model problem the algorithm is shown to be quasi optimal adjustments for the case of nonvanishing advection such as directionally implicit smoothing and a suitable splitting of the operator are discussed several numerical experiments are presented for two dimensional problems including a newton type linearization of the compressible navier stokes equations

scalable two level preconditioning and deflation based on a piecewise constant subspace for sip dg systems for diffusion problems this paper is focused on the preconditioned conjugate gradient cg method for linear systems resulting from symmetric interior penalty discontinuous galerkin sipg discretizations for stationary diffusion problems in particular it concerns two level preconditioning strategies where the coarse space is based on piecewise constant dg basis functions in this paper we show that both the two level preconditioner and the corresponding bnn or adef2 deflation variant yield scalable convergence of the cg method independent of the mesh element diameter these theoretical results are illustrated by numerical experiments we consider a solver for the discontinuous galerkin method we use the symmetric interior penalty variant the deflated solver appears to be scalable our solver is independent of the jump in the coefficients

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

theory of traces abstract the theory of traces originated by a mazurkiewicz in 1977 is an attempt to provide a mathematical description of the behavior of concurrent systems its aim is to reconcile the sequential nature of observations of the system behavior on the one hand and the nonsequential nature of causality between the actions of the system on the other hand one can see the theory of traces to be rooted in formal string language theory with the notion of partial commutativity playing the central role alternatively one can see the theory of traces to be rooted in the theory of labeled acyclic directed graphs or even in the theory of labeled partial orders this paper attempts to present a major portion of the theory of traces in a unified way however it is not a survey in the sense that a number of new notions are introduced and a number of new results are proved although traditionally most of the development in the theory of traces follows the string language theoretic line we try to demonstrate to the reader that the graph theoretic point of view may be more appropriate the paper essentially consists of two parts the first one sections 1 through 4 is concerned with the basic theory of traces the second one section 5 presents applications of the theory of traces to the theory of the behavior of concurrent systems where the basic system model we have chosen is the condition event system introduced by c a petri

two level additive schwarz preconditioners for c 0 interior penalty methods we study two level additive schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from c0 interior penalty methods for fourth order elliptic boundary value problems we show that the condition number of the preconditioned system is bounded by c 1 h3 3 where h is the typical diameter of a subdomain measures the overlap among the subdomains and the positive constant c is independent of the mesh sizes and the number of subdomains

probabilistic computability and choice we study the computational power of randomized computations on infinite objects such as real numbers in particular we introduce the concept of a las vegas computable multi valued function which is a function that can be computed on a probabilistic turing machine that receives a random binary sequence as auxiliary input the machine can take advantage of this random sequence but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful with positive probability the random advice has to be successful we characterize the class of las vegas computable functions in the weihrauch lattice with the help of probabilistic choice principles and weak weak k nig s lemma among other things we prove an independent choice theorem that implies that las vegas computable functions are closed under composition in a case study we show that nash equilibria are las vegas computable while zeros of continuous functions with sign changes cannot be computed on las vegas machines however we show that the latter problem admits randomized algorithms with weaker failure recognition mechanisms the last mentioned results can be interpreted such that the intermediate value theorem is reducible to the jump of weak weak k nig s lemma but not to weak weak k nig s lemma itself these examples also demonstrate that las vegas computable functions form a proper superclass of the class of computable functions and a proper subclass of the class of non deterministically computable functions we also study the impact of specific lower bounds on the success probabilities which leads to a strict hierarchy of classes in particular the classical technique of probability amplification fails for computations on infinite objects we also investigate the dependency on the underlying probability space besides cantor space we study the natural numbers the euclidean space and baire space

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

pattern statistics and vandermonde matrices in this paper we determine some limit distributions of pattern statistics in rational stochastic models we present a general approach to analyze these statistics in rational models having an arbitrary number of strongly connected components we explicitly establish the limit distributions in most significant cases they are characterized by a family of unimodal density functions defined by means of confluent vandermonde matrices

new upper bounds to the limitedness of distance automata abstract a distance automaton is a finite nondeterministic automaton with a distance function which assigns zero or one to each atomic transition and assigns a nonnegative integer to each accepted word by the plus min principle in this paper we prove that the distances of all accepted words of a distance automaton is bounded by some constant if and only if they are bounded by 2 4m 3 m log m 2 m where m is the number of states of the automaton

interior penalty preconditioners for mixed finite element approximations of elliptic problems it is established that an interior penalty method applied to second order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method this relation can be utilized in order to construct preconditioners for the discrete mixed system as an example a family of additive schwarz preconditioners for these systems is constructed numerical examples which confirm the theoretical results are also presented

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

a massively parallel nonoverlapping additive schwarz method for discontinuous galerkin discretization of elliptic problems a second order elliptic problem with discontinuous coefficient in 2 d or 3 d is considered the problem is discretized by a symmetric weighted interior penalty discontinuous galerkin finite element method with nonmatching simplicial elements and piecewise linear functions the resulting discrete problem is solved by a two level additive schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element in this way the method has a potential for a very high level of fine grained parallelism condition number estimate depending on the relative sizes of the underlying grids is provided the rate of convergence of the method is independent of the jumps of the coefficient if its variation is moderate inside coarse grid substructures or on local solvers subdomain boundaries numerical experiments are reported which confirm theoretical results

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

improved limitedness theorems on finite automata with distance functions abstract let a be a finite automaton with a distance function and id a be the set of distances associated with words accepted by a this paper presents an improved upper bound of id a when the upper limit of id a is finite it also presents one necessary and sufficient condition concerning word expressions for the upper limit of id a to be infinite

the limitedness problem on distance automata hashiguchi s method revisited hashiguchi has studied the limitedness problem of distance automata da in a series of paper j comput system sci 24 1982 233 theoret comput sci 72 1990 27 theoret comput sci 233 2000 19 the distance of a da can be limited or unbounded given that the distance of a da is limited hashiguchi has proved in hashiguchi 2000 that the distance of the automaton is bounded by 24n3 n lg n 2 n where n is the number of states in this paper we study again hashiguchi s solution to the limitedness problem we have made a number of simplification and improvement on hashiguchi s method we are able to improve the upper bound to 23n3 n lg n n 1

norm preconditioners for discontinuous galerkin hp finite element methods we consider a norm preconditioning approach for the solution of discontinuous galerkin finite element discretizations of second order partial differential equations with a nonnegative characteristic form our solution method is a norm preconditioned three term gmres routine we find that for symmetric positive definite diffusivity tensors the convergence of our solver is independent of discretization while for the semidefinite case both theory and experiment indicate dependence on both h and p numerical results are included to illustrate performance on several test cases

new results on the star problem in trace monoids we prove that if it is decidable whether x is recognizable for a recognizable subset x of a free partially commutative monoid then it is decidable whether a recognizable subset of a free partially commutative monoid possesses the finite power property we prove that if any trace of a set x is connected we can decide whether x possesses the finite power property finally we also show that if x is a finite set containing at most four traces or at most two connected traces it is decidable whether x is recognizable

thin set theorems and cone avoidance the thin set theorem mathsf rt n infty ell asserts the existence for every k coloring of the subsets of natural numbers of size n of an infinite set of natural numbers all of whose subsets of size n use at most ell colors whenever ell 1 the statement corresponds to ramsey s theorem from a computational viewpoint the thin set theorem admits a threshold phenomenon in that whenever the number of colors ell is sufficiently large with respect to the size n of the tuples then the thin set theorem admits strong cone avoidance let d 0 d 1 dots be the sequence of catalan numbers for n geq 1 mathsf rt n infty ell admits strong cone avoidance if and only if ell geq d n and cone avoidance if and only if ell geq d n 1 we say that a set a is mathsf rt n infty ell encodable if there is an instance of mathsf rt n infty ell such that every solution computes a the mathsf rt n infty ell encodable sets are precisely the hyperarithmetic sets if and only if ell 2 n 1 the arithmetic sets if and only if 2 n 1 leq ell d n and the computable sets if and only if d n leq ell

a unified analysis of balancing domain decomposition by constraints for discontinuous galerkin discretizations the bddc algorithm is extended to a large class of discontinuous galerkin dg discretizations of second order elliptic problems an estimate of c 1 log h h 2 is obtained for the condition number of the preconditioned system where c is a constant independent of h or h or large jumps in the coefficient of the problem numerical simulations are presented which confirm the theoretical results a key component for the development and analysis of the bddc algorithm is a novel perspective presenting the dg discretization as the sum of elementwise local bilinear forms the elementwise perspective allows for a simple unified analysis of a variety of dg methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms additionally this new perspective enables a connection to be drawn between the dg discretization and a related continuous finite element discretization to simplify the analysis of the bddc algorithm

pigeons do not jump high abstract the infinite pigeonhole principle for 2 partitions asserts the existence for every set a of an infinite subset of a or of its complement in this paper we develop a new notion of forcing enabling a fine analysis of the computability theoretic features of the pigeonhole principle we deduce various consequences such as the existence for every set a of an infinite subset of it or its complement of non high degree we also prove that every 0 3 set has an infinite low3 solution and give a simpler proof of liu s theorem that every set has an infinite subset in it or its complement of non pa degree

on the decidability of some problems about rational subsets of free partially commutative monoids let i a b be a partially commutative alphabet such that two letters commute if one of them belongs to a and the other one belongs to b let m a b denote the free partially commutative monoid generated by i we consider the following six problems for rational given by regular expressions subsets x y of m q1 x y o q2 x y di x y q4 x m l25 m x finite x is recognizable it was proved by choffrut see 2 that all these problems are undecidable if card a 1 and card b 1 and they are decidable if card a card b 1 card u denotes the cardinality of u it was conjectured see 2 p 79 that these problems are decidable in the remaining cases where card a 1 and card b 1 in this paper we show that if card a 1 and card b 1 then the problem 01 is decidable and problems q2 06 are undecidable our paper is an application of results concerning reversal bounded nondeterministic multicounter machines and nondeterministic general sequential machines

the metamathematics of stable ramsey s theorem for pairs we show that over the base theory rca0 stable ramsey s the orem for pairs implies neither ramsey s theorem for pairs nor 0 induction

on some equations in free partially commutative monoids abstract we characterize the solutions of all the equations in two unknowns in the free partially commutative monoids and we show that the connected solutions are cyclic next we define the transposition relation m and n are transposed iff m xy and n yx and the conjugacy relation m and n are conjugate iff m n we show that the conjugacy is the transitive closure of the transposition and that the set of conjugacy factors is a recognizable subset of the free partially commutative monoid

membership problems for regular and context free trace languages abstract trace languages have been introduced in order to describe the behaviour of concurrent systems in the same way as usual formal languages do for sequential system they can be defined as subsets of a free partially commutative monoid and a theory of trace languages can be developed generalizing the usual formal languages theory in this paper the time complexity of membership problems for regular and context free trace languages is investigated it is proved that the membership problem for context free trace languages can be solved in time o bm n where is the dimension of the greatest clique of the concurrency relation c and bm n is the time required for multiplying two arbitrary n n boolean matrices for regular trace languages our method gives an algorithm which requires o n time finally the uniform membership problem is shown to be np complete

string matching problems over free partially commutative monoids abstract this paper studies two string matching problems over free partially commutative monoids we analyze these two problems in detail and present two efficient polynomial time algorithms for solving them

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

on uniform relationships between combinatorial problems the enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic with one of the most common frameworks for doing so being reverse mathematics in this setting one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics since the proofs of such implications take place in classical logic they may in principle involve appeals to multiple applications of a particular theorem or to non uniform decisions about how to proceed in a given construction in practice however if a theorem q implies a theorem p it is usually because there is a direct uniform translation of the problems represented by p into the problems represented by q in a precise sense formalized by weihrauch reducibility we study this notion of uniform reducibility in the context of several natural combinatorial problems and compare and contrast it with the traditional notion of implication in reverse mathematics we show for instance that for all n j k 1 if j k then ramsey s theorem for n tuples and k many colors is not uniformly or weihrauch reducible to ramsey s theorem for n tuples and j many colors the two theorems are classically equivalent so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions we also study weak konig s lemma the thin set theorem and the rainbow ramsey s theorem along with a number of their variants investigated in the literature weihrauch reducibility turns out to be connected with sequential forms of mathematical principles where one wishes to solve infinitely many instances of a particular problem simultaneously we exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related

analysis of schwarz methods for a hybridizable discontinuous galerkin discretization schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized for hybridizable discontinuous galerkin hdg methods this is a relatively new field of research because hdg methods impose continuity across elements using a robin condition while classical schwarz solvers use dirichlet transmission conditions robin conditions are used in optimized schwarz methods to get faster convergence compared to classical schwarz methods and this even without overlap when the robin parameter is well chosen we present in this paper a rigorous convergence analysis of schwarz methods for the concrete case of the hybridizable interior penalty iph method we show that the penalization parameter needed for convergence of iph leads to slow convergence of the classical additive schwarz method and we propose a modified solver which leads to much faster convergence our analysis is entirely at the discrete level and thus holds for arbitrary in

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

some logically weak ramseyan theorems abstract we study four families of consequences of ramsey s theorem from the viewpoint of reverse mathematics the first which we call the achromatic ramsey theorem is from a partition relation introduced by erd s hajnal and rado c d r which asserts that for every f r c there exists an infinite h with f h r d the second and third are the free set theorem and the thin set theorem which were introduced by harvey friedman and the last is the rainbow ramsey theorem we show that most theorems from these families are quite weak i e they are strictly weaker than aca 0 over rca 0 interestingly these families turn out to be closely related we establish the so called strong cone avoidance property of most instances of the achromatic ramsey theorem by an induction of exponents then apply this and a similar induction to obtain the strong cone avoidance property of the free set theorem from the strong cone avoidance property of the achromatic ramsey theorem and the free set theorem we derive the strong cone avoidance property of the thin set theorem and the rainbow ramsey theorem it follows easily that a theorem with the strong cone avoidance property does not imply aca 0 over rca 0

probabilistic estimation of the number of prefixes of a trace abstract we present a generating function technique to evaluate the number of strings of a given length recognized by a particular kind of finite state automaton using the method we determine some asymptotic estimations of the number of prefixes in free partially commutative monoids more precisely we prove that for every concurrent alphabet c assuming equiprobable strings all of length n the average number of prefixes of a trace of length n is n k o n k 1 and its variance is o n 2 k 1 where k is the number of components of the dependency graph c c and is a rational constant depending only on c these combinatorial results allow to determine the probabilistic behaviour of some algorithms for problems on trace languages including the membership problems for regular and context free trace languages

finite valued distance automata abstract distance automata are a model of finite state machines which charge to an input word the expense of the cheapest successful computation consuming that word called its distance the distance of such a machine is the maximal distance of an input word recognized by it or is infinite depending on whether or not a maximum exists the valuedness of a distance automaton is introduced by considering this machine to be a transducer with unary output alphabet the distance of a finite valued distance automaton with n states is either infinite or at most 6 n 2 n n 2 1 in the former case its growth is linear in the input length the problem of deciding whether a given finite valued 2 valued respectively distance automaton has infinite distance is pspace complete it is decidable in deterministic double exponential time whether two given finite valued distance automata are equivalent i e every input word has the same distance in both machines there is an inherently infinite valued distance automaton and for each k an inherently k valued distance automaton such that the growth of the distance in all these machines is linear in the input length it is decidable in deterministic polynomial time whether a 2 valued distance automaton given as the disjoint union of two single valued distance automata is inherently 2 valued