a robust finite element solver for a multiharmonic parabolic optimal control problem this paper presents the analysis of a distributed parabolic optimal control problem in a multiharmonic setting in particular the desired state is assumed to be multiharmonic after eliminating the control from the optimality system we arrive at the reduced optimality system for the state and the co state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation due to the linearity the state and the co state are multiharmonic as well we discretize the fourier coefficients by the finite element method this leads to a large system of algebraic equations which fortunately decouples into smaller systems each of them defining the cosine and sine fourier coefficients for the state and co state with respect to a single frequency for these smaller systems we construct preconditioners resulting in a fast converging minimal residual solver with a parameter independent convergence rate all these systems can be solved totally in parallel

rich square free words abstract a word w is rich if it has w 1 distinct palindromic factors including the empty word a word is square free if it does not have a factor uu where u is a non empty word pelantova and starosta 2013 16 proved that every infinite rich word contains a square we will give another proof for that result pelantova and starosta marked with r n the length of a longest rich square free word on an alphabet of size n the exact value of r n was left as an open question we will give an upper and a lower bound for r n the lower bound is conjectured to be exact but it is not explicit we will also generalize the notion of repetition threshold for a limited class of infinite words the repetition thresholds for episturmian and rich words are left as an open question

burrows wheeler transform and palindromic richness the investigation of the extremal case of the burrows wheeler transform leads to study the words w over an ordered alphabet a a 1 a 2 a k with a 1

transcendence of formal power series with rational coefficients we give algebraic proofs of transcendence over q x of formal power series with rational coefficients by using inter alia reduction modulo prime numbers and the christol theorem applications to generating series of languages and combinatorial objects are given

the role of the strengthened cauchy buniakowskii schwarz inequality in multilevel methods the basic theory of the strengthened cauchy buniakowskii schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations is surveyed proofs are given both in a finite element context and in purely algebraic form

multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids in this paper we generalize the analysis of classical multigrid and two level overlapping schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition the theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters an assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small we do not need to align them with possible discontinuities in the coefficients the proofs make use of novel stable splittings based on weighted quasi interpolants and weighted poincare type inequalities numerical experiments are

subword complexity and power avoidance abstract we begin a systematic study of the relations between subword complexity of infinite words and their power avoidance among other things we show that the thue morse word has the minimum possible subword complexity over all overlap free binary words and all 7 3 power free binary words but not over all 7 3 power free binary words the twisted thue morse word has the maximum possible subword complexity over all overlap free binary words but no word has the maximum subword complexity over all 7 3 power free binary words if some word attains the minimum possible subword complexity over all square free ternary words then one such word is the ternary thue word the recently constructed 1 2 bonacci word has the minimum possible subword complexity over all symmetric square free ternary words

free subalgebras of graded algebras let k be a field and let a n 1an be a positively graded k algebra we recall that a is graded nilpotent if for every d 1 the subalgebra of a generated by elements of degree d is nilpotent we give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k subalgebra on two generators

algebraic multigrid based on element interpolation amge we introduce amge an algebraic multigrid method for solving the discrete equations that arise in ritz type finite element methods for partial differential equations assuming access to the element stiffness matrices we have that amge is based on the use of two local measures which are derived from global measures that appear in existing multigrid theory these new measures are used to determine local representations of algebraically smooth error components that provide the basis for constructing effective interpolation and hence the coarsening process for amg here we focus on the interpolation process choice of the coarse grids based on these measures is the subject of current research we develop a theoretical foundation for amge and present numerical results that demonstrate the efficacy of the method

a new characteristic property of rich words originally introduced and studied by the third and fourth authors together with j justin and s widmer 2008 rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes several characterizations of rich words have already been established a particularly nice characteristic property is that all complete returns to palindromes are palindromes in this note we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix

extensions of rich words a word w is rich if it has w 1 many distinct palindromic factors including the empty word this article contains several results about rich words particularly related to extending them a word w can be eventually extended richly in n ways if there exists a finite word u and n distinct letters a alph w such that wua is rich we will prove that every non unary rich word can be eventually extended richly in at least two different ways but not always in three or more ways we will also prove that every rich word can be extended to both periodic and aperiodic infinite rich words the defect of a finite word w is defined by d w w 1 pal w this concept has been studied in various papers here we will define a new concept infinite defect for a finite word w the definition is d w min d z zis an infinite word which has factorw we will show that the infinite defect of a finite word is always finite and give some upper bounds for it the difference between defect and infinite defect is also investigated we will also give an upper and a lower bound for the number of rich words a new class of words two dimensional rich words is also introduced

robust multilevel methods for general symmetric positive definite operators a general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed the term robust refers to the convergence rate of the method being independent of discretization parameters i e the problem size and problem parameters important instances of such problem parameters are in particular highly varying coefficients the method belongs to the class of nonlinear algebraic multilevel iterations the crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems the method is analyzed in a rather general setting applying to the scalar elliptic equation the equations of linear elasticity and equations arising in the solution of maxwell s equations numerical results for the scalar elliptic equation are presented showing its robust convergence behavior

balancing and clustering of words in the burrows wheeler transform compression algorithms based on burrows wheeler transform bwt take advantage of the fact that the word output of bwt shows a local similarity and then turns out to be highly compressible the aim of the present paper is to study such clustering effect by using notions and methods from combinatorics on words the notion of balance of a word plays a central role in our investigation empirical observations suggest that balance is actually the combinatorial property of input word that ensure optimal bwt compression moreover it is reasonable to assume that the more balanced the input word is the more local similarity we have after bwt and therefore the better the compression is this hypothesis is here corroborated by experiments on real text by using local entropy as a measure of the degree of balance of a word in the setting of combinatorics on words a sound confirmation of previous hypothesis is given by a result of mantaci et al 2003 27 which states that in the case of a binary alphabet there is an equivalence between circularly balanced words words having a clusterized bwt and the conjugates of standard words in the case of alphabets of size greater than two there is no more equivalence the last section of the present paper is devoted to investigate the relationships between these notions and other related ones as for instance palindromic richness in the case of a general alphabet

complexity for finite factors of infinite sequences abstract we define several notions of language complexity for finite words and use them to define and compute some new complexity functions for infinite sequences in particular they give a new characterization of sturmian sequences and discriminate between sturmian sequences with bounded or unbounded partial quotients

additive schur complement approximation and application to multilevel preconditioning this paper introduces an algorithm for additive schur complement approximation asca which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element fe discretization of partial differential equations pde it is shown how the asca can be used to set up a nonlinear algebraic multilevel iteration amli method this requires the construction of a linear multiplicative two level preconditioner at each level the latter is computed in the course of a simultaneous exact two by two block factorization of local stiffness matrices associated with a covering of the entire domain by overlapping subdomains unlike in schwarz type domain decomposition methods this method does not require a global coarse problem but instead uses local coarse problems to provide global communication a robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid the presented numerical tests demonstrate that the asca whe

amg e based on element agglomeration this paper contains the main ideas for an amge algebraic multigrid for finite elements method based on element agglomeration in the method coarse grid elements are formed by agglomerating fine grid elements compatible interpolation operators are constructed which yield coarse grid basis functions with a minimal energy property heuristics based on interpolation quality measures are used to guide the agglomeration procedure the performance of the resulting method is demonstrated in two level numerical experiments

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

fibonacci morphisms and sturmian words abstract marton kosa 1987 has stated five conjectures devoted to the study of some binary morphisms and the factors of infinite words these morphisms generate positive answers to the first four of these conjectures are given in this paper the main result being the characterization of a class of morphisms whose stationary words are sturmian

spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d

guaranteed lower bounds for cost functionals of time periodic parabolic optimization problems abstract in this paper a new technique is shown for deriving computable guaranteed lower bounds of functional type minorants for two different cost functionals subject to a parabolic time periodic boundary value problem together with previous results on upper bounds majorants for one of the cost functionals both minorants and majorants lead to two sided estimates of functional type for the optimal control problem both upper and lower bounds are derived for the second new cost functional subject to the same parabolic pde constraints but where the target is a desired gradient the time periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure the derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail finally several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained this work provides the basis for an adaptive scheme for time periodic optimization problems

some combinatorial properties of the thue morse sequence and a problem in semigroups abstract some combinatorial properties of the special factors of the thue morse sequence in a two letter alphabet are considered an application of this analysis to a problem in semigroups is given

preconditioners for regularized saddle point problems with an application for heterogeneous darcy flow problems saddle point problems arise in the modeling of many important practical situations preconditioners for the corresponding matrices of block triangular form based on coupled inner outer iteration methods are analyzed and applied to a darcy flow problem possibly with strong heterogeneity and non symmetric saddle point systems using proper regularized forms of the given matrix and its preconditioner it is shown that the eigenvalues cluster about one or two points on the real axis for large values of the regularization parameters and that eigenvalue bounds do not depend on this variation therefore just two outer iterations can suffice on the other hand several iterations may be needed to solve the inner iteration systems

on the subword complexity of thue morse polynomial extractions let the subword complexity of a sequence u u n n 0 over a finite set s be the function m p u m where p u m is the number of distinct blocks of length m in u let t t n n 0 denote the thue morse sequence in this paper we study the complexity of the sequences t h t h n n 0 when h n q n is a polynomial with h n n in particular we solve an open problem of allouche and shallit regarding t n 2 n 0 we also study the vector space over z 2z spanned by the sequences t h

canonical positions for the factors in paperfolding sequences it has been proved by the first author that the number of factors of length k in any paperfolding sequence is equal to 4k once k 7 we prove here that for every integer k there exists a set of integers pk of cardinality 4k for k 7 such that the factors of length k of any paperfolding sequence are exactly the factors beginning at the places indexed by the integers in pk this gives a bijective proof of the result mentioned above then we give an explicit uniform linear upper bound for the recurrence function of paperfolding sequences finally we study the set of all factors of all paperfolding sequences evaluating their number for a given length and studying the language of these words

measuring the clustering effect of bwt via rle abstract the burrows wheeler transform bwt is a reversible transformation on which are based several text compressors and many other tools used in bioinformatics and computational biology the bwt is not actually a compressor but a transformation that performs a context dependent permutation of the letters of the input text that often create runs of equal letters clusters longer than the ones in the original text usually referred to as the clustering effect of bwt in particular from a combinatorial point of view great attention has been given to the case in which the bwt produces the fewest number of clusters cf 5 16 21 23 in this paper we are concerned about the cases when the clustering effect of the bwt is not achieved for this purpose we introduce a complexity measure that counts the number of equal letter runs of a word this measure highlights that there exist many words for which bwt gives an un clustering effect that is bwt produce a great number of short clusters more in general we show that the application of bwt to any word at worst doubles the number of equal letter runs moreover we prove that this bound is tight by exhibiting some families of words where such upper bound is always reached we also prove that for binary words the case in which the bwt produces the maximal number of clusters is related to the very well known artin s conjecture on primitive roots the study of some combinatorial properties underlying this transformation could be useful for improving indexing and compression strategies

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

a finiteness condition for semigroups generalizing a theorem of hotzel in 1979 hotzel proved 1 l the following interesting finiteness condition for semigroups let s be a finitely generated semigroup if s verifies the minimal condition on principal left ideals resp right ideals and all subgroups of s have finite orders then s is finite this theorem solved a question posed by coudrain and schiitzenberger 3 who proved a similar result with the only difference that s satisfies the minimal condition on principal bi ideals instead of left ideals moreover two finiteness conditions for semigroups due respectively to simon 16 and to de luca and restivo 4 are based on hotzel s theorem in this paper we prove the following generalization of hotzel s theorem in which one supposes that only finitely generated subgroups are finite

palindromic richness for languages invariant under more symmetries for a given finite group g consisting of morphisms and antimorphisms of a free monoid a we study infinite words with language closed under the group g we focus on the notion of g richness which describes words rich in generalized palindromic factors i e in factors w satisfying q w w for some antimorphism email protected g we give several equivalent descriptions which are generalizations of known characterizations of rich words in the terms of classical palindromes and show two examples of g rich words

on t extensions of the hankel determinants of certain automatic sequences in 1998 allouche peyriere wen and wen considered the thue morse sequence and proved that all the hankel determinants of the period doubling sequence are odd integers we speak of t extension when the entries along the diagonal in the hankel determinant are all multiplied by t we prove that the t extension of each hankel determinant of the period doubling sequence is a polynomial in t whose leading coefficient is the only one to be an odd integer our proof makes use of the combinatorial set up developed by bugeaud and han which appears to be very suitable for this study as the parameter t counts the number of fixed points of a permutation finally we prove that all the t extensions of the hankel determinants of the regular paperfolding sequence are polynomials in t of degree less than or equal to 3

on element by element schur complement approximations abstract we discuss a methodology to construct sparse approximations of schur complements of two by two block matrices arising in finite element discretizations of partial differential equations earlier results from 2 are extended to more general symmetric positive definite matrices of two by two block form the applicability of the method for general symmetric and nonsymmetric matrices is analysed the paper demonstrates the applicability of the presented method providing extensive numerical experiments

palindrome pattern matching a palindrome is a string that reads the same forward and backward for a string x let pals x be the set of all maximal palindromes of x where each maximal palindrome in pals x is encoded by a pair c r of its center c and its radius r given a text t of length n and a pattern p of length m the palindrome pattern matching problem is to compute all positions i of t such that pals p pals t i i m 1 we present linear time algorithms to solve this problem

using approximate inverses in algebraic multilevel methods this paper deals with the iterative solution of large sparse symmetric positive definite systems we investigate preconditioning techniques of the two level type that are based on a block factorization of the system matrix whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns we analyze the effect of using an approximate inverse instead we derive condition number estimates that are valid for any type of approximation of the schur complement and that do not assume the use of the hierarchical basis they show that the two level methods are stable when using approximate inverses based on modified ilu techniques or explicit inverses that meet some row sum criterion on the other hand we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate these conclusions are numerically illustrated on some examples

on the combinatorics of finite words in this paper we consider a combinatorial method for the analysis of finite words recently introduced in colosimo and de luca special factors in biological strings preprint 97 42 dipt matematica univ di roma for the study of biological macromolecules the method is based on the analysis of right special factors of a given word a factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters we show that in the combinatorics of finite words two parameters play an essential role the first denoted by r represents the minimal integer such that there do not exist special factors of w of length r the second that we denote by k is the minimal length of a factor of w which cannot be extended on the right in a factor of w some new results are proved in particular a new characterization in terms of special factors and of r and k is given for the set per of all words w having two periods p and q which are coprimes and such that w p q 2