on almost cylindrical languages and the decidability of the d0l and pwd0l primitivity problems abstract primitive words and their properties have always been of fundamental importance in the study of formal language theory head and lando in periodic d0l languages proposed the idea of deciding whether or not a given d0l language has the property that every word in it is a primitive word after reducing the general problem to the case in which h is injective it will be shown that primitivity is decidable when a h is an almost cylindrical set moreover in this case it is shown that the set of words which generate primitive sequences given a particular d0l scheme is an algorithmically constructible context sensitive language an undecidability result for the pwd0l primitivity problem and decidability results for cases of the rwd0l primitivity problem are also given

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

the equivalence problem of multitape finite automata abstract using a result of b h neumann we extend eilenberg s equality theorem to a general result which implies that the multiplicity equivalence problem of two nondeterministic multitape finite automata is decidable as a corollary we solve a long standing open problem in automata theory namely the equivalence problem for multitape deterministic finite automata the main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group

refinements of the littlewood richardson rule in the prequel to this paper we showed how results of mason involving a new combinatorial formula for polynomials that are now known as demazure atoms characters of quotients of demazure modules called standard bases by lascoux and schutzenberger could be used to dene a new basis for the ring of quasisymmetric functions we call quasisymmetric schur functions qs functions for short in this paper we develop the combinatorics of these polynomials futher by showing that the product of a schur function and a demazure atom has a positive expansion in terms of demazure atoms we use these techniques together with the fact that both a qs function and a demazure character have explicit expressions as a positive sum of atoms to obtain the expansion of a product of a schur function with a qs function demazure character as a positive sum of qs functions demazure characters our formula for the coecients in the expansion of a product of a demazure character and a schur function into demazure characters is similar to known results and includes in particular the famous littlewood richardson rule for the expansion of a product of schur functions in terms of the schur basis a composition weak composition with n parts is a sequence of n positive nonnegative integers respectively a partition is a composition whose parts are monotone nonincreasing if is a weak composition composition or partition we let denote the number of parts of throughout this article is a weak composition with n while and denote compositions and partitions respectively with n n the polynomials in this paper schur functions demazure atoms and characters qs functions depend on a

on dol power series abstract we study d0l power series we show how elementary morphisms introduced by ehrenfeucht and rozenberg can be used in connection with power series characterize the sequences of rational numbers and integers which can appear as coefficients in d0l power series and establish various decidability results

finite traces and representations of the group of infinite matrices over a finite field abstract the article is devoted to the representation theory of locally compact infinite dimensional group glb of almost upper triangular infinite matrices over the finite field with q elements this group was defined by s k a v and andrei zelevinsky in 1982 as an adequate n analogue of general linear groups gl n q it serves as an alternative to gl q whose representation theory is poor our most important results are the description of semifinite unipotent traces characters of the group glb via certain probability measures on the borel subgroup b and the construction of the corresponding von neumann factor representations of type ii as a main tool we use the subalgebra a glb of smooth functions in the group algebra l 1 glb this subalgebra is an inductive limit of the finite dimensional group algebras c gl n q under parabolic embeddings as in other examples of the asymptotic representation theory we discover remarkable properties of the infinite case which does not take place for finite groups like multiplicativity of indecomposable characters or connections to probabilistic concepts the infinite dimensional iwahori hecke algebra h q plays a special role in our considerations and allows to understand the deep analogy of the developed theory with the representation theory of infinite symmetric group s which had been intensively studied in numerous previous papers

on a conjugate gradient type method for solving complex symmetric linear systems abstract we consider large sparse linear systems ax b with complex symmetric coefficient matrices a a t which arise e g from the discretization of partial differential equations with complex coefficients for the solution of such systems we present a new conjugate gradient type iterative method csym which is based on unitary equivalence transformations of a to symmetric tridiagonal form an analysis of csym shows that its convergence depends on the singular values of a and that it has both the minimal residual property and constant costs per iteration step we compare the algorithm with other methods for solving large sparse complex symmetric systems

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

gpbi cg generalized product type methods based on bi cg for solving nonsymmetric linear systems recently bi cgstab as a variant of bi cg has been proposed for solving nonsymmetric linear systems and its attractive convergence behavior has been confirmed in many numerical experiments bi cgstab can be characterized by its residual polynomial which consists of the product of the residual polynomial of bi cg with other polynomials generated from two term recurrence relations in this paper we propose a unified way to generalize a class of product type methods whose residual polynomials can be factored by the residual polynomial of bi cg and other polynomials with standard three term recurrence relations such product type methods which are based on bi cg can be regarded as generalizations of bi cgstab from the unified way the well known variants of the product type methods like cgs bi cgstab bi cgstab2 are reacquired again

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

the bicor and cors iterative algorithms for solving nonsymmetric linear systems we present two iterative algorithms for solving real nonsymmetric and complex non hermitian linear systems of equations and that were developed from variants of the nonsymmetric lanczos method in this paper we give the theoretical background of the two iterative methods and discuss their main computational aspects using a large number of numerical experiments we analyze their convergence properties and we also compare them with other popular nonsymmetric iterative solvers in use today

repetiveness of languages generated by morphisms abstract we study the repetition of subwords in languages generated by morphisms fundamental to our approach is the notion of quasi repetitive elements using these elements we present a new characterization for repetitive morphisms from which we derive a simple proof for the fact that a d0l language is repetitive if and only if it is strongly repetitive ehrenfeucht and rozenberg inform and control 59 1983 13 35 from this proof we obtain a structurally simple polynomial time algorithm for deciding whether such a language is repetitive from further results on quasi repetitive elements we obtain as a consequence a complete characterization for all those morphisms on a two letter alphabet that are repetitive a result which is closely related to a result of seebold bull eatcs 36 1988 137 151 on the d0l periodicity problem finally we characterize those morphisms f on a two letter alphabet for which the language l f generated by f or the language sl f of subwords of l f are context free or even regular

a comparative study of iterative solutions to linear systems arising in quantum mechanics this study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by sherry li of nersc numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed lanczos biconjugate a orthonormalization methods to other classic and popular iterative methods by the way experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence

an analysis of the composite step biconjugate gradient method the composite step biconjugate gradient method csbcg is a simple modification of the standard biconjugate gradient algorithm bcg which smooths the sometimes erratic convergence of bcg by computing only a subset of the iterates we show that 2 2 composite steps can cure breakdowns in the biconjugate gradient method caused by near singularity of principal submatrices of the tridiagonal matrix generated by the underlying lanczos process we also prove a best approximation result for the method some numerical illustrations showing the effect of roundoff error are given

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

bounded d0l languages for an alphabet a and a morphism h a a the set of words w such that the dol language l a h w is a bounded language is shown to be b where b is an effectively computable subset of a consequently boundedness is decidable for dol languages the result depends on the authors recent results on periodic dol languages interpretation of the result for polynomially bounded dol languages is also considered

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

a new iterative method for solving complex symmetric linear systems based on implementation of the quasi minimal residual qmr and biconjugate a orthogonal residual bicor method a new krylov subspace method is presented for solving complex symmetric linear systems the new method can be combined with arbitrary symmetric preconditioners the preconditioned modified hermitian and skew hermitian splitting pmhss preconditioner is used to accelerate the convergence rate of this method numerical experiments indicate that the proposed method and its preconditioned version are efficient and robust in comparison with other krylov subspace methods

regularity of sets of initial strings of periodic d0l systems for an alphabet a and a morphism h a a the set of words w such that the d0l system a h w is periodic is shown to be a noncounting regular language use is made of the author s recent investigations of periodic d0l systems

an iterative boundary potential method for the infinite domain poisson problem with interior dirichlet boundaries an iterative method is developed for the solution of poisson s problem on an infinite domain in the presence of interior boundaries held at fixed potential in three dimensions the method combines pre existing fast multigrid based poisson solvers for data represented on cartesian grids with the fast multipole method interior boundaries are represented with the embedded boundary formalism the implementation is in parallel and uses adaptive mesh refinement examples are presented for a smooth interior boundary for which an analytical result is known and for an irregular interior boundary problem second order accuracy in l 1 with respect to the grid resolution is demonstrated for both problems

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

idr a new generation of krylov subspace methods the induced dimension reduction idr technique developed by sonneveld and van gijzen is a powerful concept resulting in a variety of transpose free krylov subspace methods based on short term recurrences we present the main di erences between and similarities of idr methods and classical krylov subspace methods our tool of trade is the so called generalized hessenberg decomposition the concept of transfer of techniques from the setting of classical krylov subspace methods to the idr based methods is introduced for simplicity we only sketch some recent results in the fields of eigenvalue computations and of solution of linear systems

a shifted complex global lanczos method and the quasi minimal residual variant for the stein conjugate matrix equation x ax b c abstract in this paper based on a complex global lanczos process a shifted complex global lanczos method is proposed to solve the stein conjugate matrix equation x a x b c the new method adopts the shifted technique and is implemented by the original coefficient matrices the direct version of the proposed method is derived by exploiting the special structure of the l u factorization moreover the quasi minimal residual variant is developed to stabilize the convergence behavior finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

an implementation of the qmr method based on coupled two term recurrences recently we proposed a new krylov subspace iteration the quasi minimal residual algorithm qmr 1 for solving general nonsingular non hermitian systems of linear equations ax b 1

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

a cocr method for solving complex symmetric linear systems the conjugate orthogonal conjugate gradient cocg method has been recognized as an attractive lanczos type krylov subspace method for solving complex symmetric linear systems however it sometimes shows irregular convergence behavior in practical applications in the present paper we propose a conjugate a orthogonal conjugate residual cocr method which can be regarded as an extension of the conjugate residual cr method numerical examples show that cocr often gives smoother convergence behavior than cocg

dynamical properties of pwdol systems the definitions of a piecewise deterministic zero lindenmayer pwd0l scheme and system are given and dynamical properties of such systems are introduced harrison 1994 showed that given an arbitrary finite alphabet a the emptiness problem is undecidable for the class of languages which are intersections of a d0l language and a context sensitive language this result is used to prove that many dynamical properties of pwd0l systems such as finiteness periodicity etc are in general undecidable over a two member context sensitive partition of a the idea of a morphic equivalence relation on a a is finite is defined and the idea of a finite morphic refinement is introduced harrison 1994 showed that every regular language and its complement is refined by a finite partition which is induced by a morphic congruence using this theorem it is shown that dynamical properties such as finiteness and periodicity are in general decidable for rwd0l systems i e a pwd0l system over a finite partition of a made up of all regular languages

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

periodicity and ultimate periodicity of dol systems a unified presentation of periodicity and ultimate periodicity of d0l systems is given bounds are provided for the period and index of periodic systems and simplified algorithms are obtained by using elementary morphisms it was shown previously harju and linna 1986 pansiot 1986 that if a h w is a d0l system with h w wy then it is decidable whether the word h w is ultimately periodic i e of the form xu in this paper a decision procedure is given to decide if there are an index i and a period p such that hp hi w hi w y and h p hi w is ultimately periodic given a morphism h a a bounds are obtained for the period and index of ultimately periodic systems it is also shown that the set w a h p hi w is ultimately periodic is a constructable regular language

new look ahead lanczos type algorithms for linear systems a breakdown due to a division by zero can arise in the algorithms for implementing lanczos method because of the non existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate such a breakdown can be avoided by jumping over the polynomials involved this strategy was already used in some algorithms such as the mrz and its variants in this paper we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors independent of the length of the jump these new algorithms are based on horner s rule and on a different way for computing the coefficients of the recurrence relationships moreover these new algorithms seem to be more stable than the old ones and they provide better numerical results numerical examples and comparisons with other algorithms will be given

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

a combinatorial formula for macdonald polynomials abstract in this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types these formulas resemble the formulas of haglund haiman and loehr for macdonald polynomials of type gl n at q 0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded alcove walks and at q t 0 these formulas specialize to the formula for the weyl character in terms of the littelmann path model in the positively folded gallery form of gaussent and littelmann

minres qlp a krylov subspace method for indefinite or singular symmetric systems cg symmlq and minres are krylov subspace methods for solving symmetric systems of linear equations when these methods are applied to an incompatible system that is a singular symmetric least squares problem cg could break down and symmlq s solution could explode while minres would give a least squares solution but not necessarily the minimum length pseudoinverse solution this understanding motivates us to design a minres like algorithm to compute minimum length solutions to singular symmetric systems minres uses qr factors of the tridiagonal matrix from the lanczos process where r is upper tridiagonal minres qlp uses a qlp decomposition where rotations on the right reduce r to lower tridiagonal form on ill conditioned systems singular or not minres qlp can give more accurate solutions than minres we derive preconditioned minres qlp new stopping rules and better estimates of the solution and residual norms the matrix norm and the condition number

a quasi minimal residual variant of the bicorstab method for nonsymmetric linear systems abstract the biconjugate a orthogonal residual stabilized method named as bicorstab was proposed by jing et al 2009 where the numerical experiments therein demonstrate that the bicorstab method converges more smoothly than the bi conjugate gradient stabilized bicgstab method in some circumstances in order to further stabilize the convergence behavior and hopefully to accelerate the convergence speed of the bicorstab algorithm when it has erratic convergence curves a quasi minimal residual variant of the bicorstab algorithm named as qmrcorstab will be developed and investigated for solving nonsymmetric systems of linear equations borrowing the same further smooth effect idea for the qmrcgstab method numerical experiments on typical sets of both sparse and dense matrices will show that the proposed qmrcorstab method shares attractive smoother effect over its basic parent and also outperforms its counterpart

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

code properties and homomorphisms of d0l systems abstract the types of codes dealt with in detail are prefix codes suffix codes and two special classes of biprefix codes called infix codes and outfix codes conditions are given under which polynomially bounded d0l languages form codes of each these types a concept of homomorphism is defined for d0l systems it is demonstrated that when e is a homomorphic image of a d0l system d and l e is infinite then if l e is a code of any of the types listed above then l d is also a code of the same type a concept of derivative is defined for d0l systems that is closely related to a special type of homomorphism based on the erasure of finite symbols code properties of linearly bounded d0l languages are studied in detail the results are then extended to apply to polynomially bounded d0l languages through the use of the newly introduced derivative concept it is shown that for every polynomially bounded d0l language l l 1 is a communtative equivalent of a prefix code every d0l language is shown to be the union of 1 a finite set 2 a finite number of d0l languages each of which has a singleton as alphabet and 3 a commutative equivalent of a prefix code

iterative system solvers for the frequency analysis of linear mechanical systems abstract the paper deals with the numerical treatment of the direct frequency domain dfd analysis of linear mechanical systems attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency strategies for transforming the system into the shifted form t i z d are proposed and discussed where is related to the frequency for two formulations of the shifted system the performance of some krylov sub space iterative solvers is tested and compared to that of a multi frontal direct method advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift frequency numerical experiments on some prototype structural dynamics problems are reported the results shown demonstrate how the devised strategies for the iterative solution can outperform in many cases the direct solver

bicgstab vpastab and an adaptation to mildly nonlinear systems the key equations of bicgstab are summarised to show its connections with pade and vector pade approximation these considerations lead naturally to stabilised vector pade approximation of a vector valued function vpastab and an algorithm for the acceleration of convergence of a linearly generated sequence of vectors a generalisation of this algorithm for the acceleration of convergence of a nonlinearly generated system is proposed here and comparative numerical results are given

pieri formulas for macdonald s spherical functions and polynomials we present explicit pieri formulas for macdonald s spherical functions or generalized hall littlewood polynomials associated with root systems and their q deformation the macdonald polynomials for the root systems of type a our pieri formulas recover the well known pieri formulas for the hall littlewood and macdonald symmetric functions due to morris and macdonald as special cases

on lindenmayerian algebraic sequences abstract we define and study lindenmayerian algebraic sequences these sequences are a generalization of algebraic sequences k regular sequences and automatic sequences

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

vpastab j l an iterative method with look ahead for the solution of large sparse linear systems vpastab j l denotes an iterative method designed to give an accurate numerical solution of a large sparse system of linear equations the iteration comprises l steps and possibly jump steps an l step involves stabilisation minimisation over an l dimensional subspace following l levels of vector pade approximation a jump step look ahead stage avoids near breakdowns by jumping past them by j levels simultaneously a numerical implementation of vpastab 2 l is considered in the context of some well known examples

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

a breakdown free lanczos type algorithm for solving linear systems lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates the various lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae if the denominator in the formula for one of these coefficients is zero then a breakdown occurs in the algorithm and it must be stopped such a breakdown is in fact due to the non existence of some orthogonal polynomial in this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method the resulting algorithm called mrz is equivalent to the nongeneric biodir algorithm which is a look ahead lanczos type algorithm but our derivation is much simpler

on combinatorial formulas for macdonald polynomials a recent breakthrough in the theory of type a macdonald polynomials is due to haglund haiman and loehr who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of young diagrams ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of so called alcove walks these originate in the work of gaussent littelmann and of the author with postnikov on discrete counterparts to the littelmann path model in this paper we relate the above developments by explaining how the ram yip formula compresses to a new formula which is similar to the haglund haiman loehr one but contains considerably fewer terms

periodic d0l languages abstract periodic d0l systems and languages are defined and periodicity is shown to be a decidable property of d0l systems the fundamental tools used are recent results on the representation of stationary words head and lando 1986 and the decidability of ultimate periodicity of words harju and linna 1986 pansiot 1986 the relation of d0l periodicity to local catenativity and to n codes is examined

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

a littlewood richardson rule for macdonald polynomials macdonald polynomials are orthogonal polynomials associated to root systems and in the type a case the symmetric macdonald polynomials are a common generalization of schur functions macdonald spherical functions and jack polynomials we use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine hecke algebra from this we obtain a product formula for macdonald polynomials of general lie type

morphic congruences and dol languages in a short note head 1991 showed that given an arbitrary finitely generated submonoid m of a finitely generated free monoid a and an endomorphism e of a such that m e m there exists an algorithmically computable natural number n such that for any x a l sx m if and only if x ei 0 i n m where sx a e x is a d0l system in this paper the idea of a morphic refinement of a finite partition of a is introduced to allow us to extend head s result to all regular languages in a moreover it is shown that such a bound n can be computed which only depends upon the regular language l and is independent of the endomorphism e

accurate numerical solutions of the time dependent schrodinger equation we present a generalization of the often used crank nicolson cn method of obtaining numerical solutions of the time dependent schr odinger equation the generalization yields numerical solutions accurate to order ensuremath delta x 2r ensuremath 1 in space and ensuremath delta t 2m in time for any positive integers r and m while cn employ r m 1 we note dramatic improvement in the attainable precision circa ten or greater orders of magnitude along with several orders of magnitude reduction of computational time the improved method is shown to lead to feasible studies of coherent state oscillations with additional short range interactions wave packet scattering and long time studies of decaying systems

a transpose free quasi minimal residual variant of the cors method for solving non hermitian linear systems recently the conjugate a orthogonal residual squared cors method is often competitive and superior to other krylov subspace methods for solving complex non hermitian linear systems from many realistic problems however like the conjugate gradient squared cgs method the cors method often exhibits irregular convergence behavior with wild oscillations in the residual norm to try and remedy this problem we develop a transpose free quasi minimal residual tfqmr variant of the cors method which leads to smooth convergence curves with a convergence rate similar to the cors method we also give convergence results for this new method finally numerical examples are reported to illustrate the effectiveness of the proposed method

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

a generalized product type bicor method and its application in signal deconvolution for solving nonsymmetric linear systems we attempt to establish symmetric structures in nonsymmetric systems and handle them through the methods devised for symmetric cases a biconjugate a orthogonal residual method based on biconjugate a orthonormalization procedure has been proposed and nominated as bicor in y f jing t z huang y zhang l li g h cheng z g ren y duan t sogabe b carpentieri lanczos type variants of the cocr method for complex nonsymmetric linear systems j comput phys 228 2009 6376 6394 as many similar characteristics exist between bicor and bicg the strategies of improved variants of bicg such as cgs and bicgstab can be utilized to enhance the algorithm for bicor making use of the product of residual polynomials of bicor and other polynomials cors and bicorstab have been proposed along the same ideas of cgs and bicgstab respectively in the above mentioned paper in this paper a unified generalized framework of product type bicor which is epitomized by the product of residual polynomials and other polynomials is proposed numerical examples are selected from the blurring signal cases and the effect of the generalized product type bicor method is prominent in signal deconvolution

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

the set of strings mapped into a submonoid by iterates of a morphism abstract let s be a finitely generated submonoid of a free monoid a p let h be an endomorphism of a p which maps s into itself let e s be the submonoid of all strings x in a p which are eventually mapped into s by h i e the strings x for which there exists a non negative integer i such that h p i x is in s then e s is a constructable regular language applications of this result are given to the study of periodic behaviors of morphisms

on lindenmayerian algebraic power series abstract we define lindenmayerian algebraic power series and study their properties these series generalize both algebraic series and etol languages

ls galleries the path model and mv cycles we give an interpretation of the path model of a representation 18 of a complex semisimple algebraic group g in terms of the geometry of its affine grassmannian in this setting the paths are replaced by ls galleries in the affine coxeter complex associated to the weyl group of g to explain the connection with geometry consider a demazure hansen bott samelson desingularization of the closure of an orbit g c t in the affine grassmannian the homology of has a basis given by bia lynicki birula cell s which are indexed by the t fixed points in now the points of can be identified with galleries of a fixed type in the affine tits building associated to g and the t fixed points correspond in this language to combinatorial galleries of a fixed type in the affine coxeter complex we determine those galleries such that the associated cell has a non empty intersection with g c t identified with an open subset of and we show that the closures of the strata associated to ls galleries are exactly the mv cycles 24 which form a basis of the representation v for the langland s dual group g

automata and languages generalized to ω continuous semirings abstract we generalize the following two language and automata theoretic results to continuous semirings i the family of languages accepted by finite automata is the smallest class containing all finite languages and closed under union product and star kleene s theorem ii the family of languages accepted by pushdown automata is the family ofcontext free languages

composite step product methods for solving nonsymmetric linear systems first proposed in numer math 66 1993 pp 295 319 numer algorithms 7 1994 pp 1 16 by bank and chan the composite step cs method is a technique for curing the pivot breakdown one of two possible breakdowns in the biconjugate gradient bcg algorithm by skipping over steps in which the iterate is not defined we show how to extend this method to cure the same breakdown inherited in product methods such as conjugate gradients squared cgs siam j sci statist comput 10 1989 pp 36 52 bi cgstab siam j sci statist comput 13 1992 pp 631 644 bi cgstab2 variants of bicgstab for matrices with complex spectrum ips research report no 91 14 eth zurich which are derived from a product of the bcg polynomial with another polynomial of the same degree new methods introduced in this paper are cs cgstab composite step applied to bi cgstab and a more stable variant of this cs cgstab2 which handles a possible additional breakdown problem due to the bi cgstab process in the case where a is skew symmetric cs cgstab2 can be viewed as an improvement over the bi cgstab l algorithm sleijpen and fokkema etna 1 1993 pp 11 32 with l 2 in that although bi cgstab 2 is designed to cure the skew symmetric breakdown it does not handle pivot breakdown as cs cgstab2 does the new methods require only a minor modification to the existing product methods moreover the sizes of the steps taken in our methods can vary as opposed to fixed step methods like bi cgstab l and bi cgstab2 our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates numerical experiments show that the new methods do produce improved performance over those without composite step on practical problems furthermore we extend the best approximation result in numer algorithms 7 1994 pp 1 16 to obtain convergence proofs for cgs and bi cgstab and their composite step stabilized versions

a neumann neumann preconditioned iterative substructuring approach for computing solutions to poisson s equation with prescribed jumps on an embedded boundary in multifluid problems with surface tension the fluid pressure and its derivative are discontinuous at fluid interfaces we present a cartesian grid embedded boundary method for numerically resolving these discontinuities in which we use neumann neumann preconditioned iterative substructuring to solve the governing equations we validate this method by computing several well known poisson problems with discontinuous coefficients and we compare its performance to an approach based on simple iteration by analogy with the conjugate gradient method we hypothesize that the scaling of the neumann neumann preconditioned iterative substructuring is o h dlnh 1 where h is the cell size and d 2 3 is the dimensionality of the problem in contrast we show that the simple iterative procedure scales like o h d 1 and is slower by a factor of 4000 for a small i e 64x64 cell model calculation with physical parameters corresponding to a 1 5mm air bubble in water we present an analytical model to explain the scaling of this iterative procedure

on two sided infinite fixed points of morphisms let s be a finite alphabet and let h s s be a morphism finite and infinite fixed points of morphismsi e those words w such that h w wplay an important role in formal language theory head characterized the finite fixed points of h and later head and lando characterized the one sided infinite fixed points of h our paper has two main results first we complete the characterization of fixed points of morphisms by describing all two sided infinite fixed points of h for both the pointed and unpointed cases second we completely characterize the solutions to the equation h xy yx in finite words

the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings the complex bi conjugate gradient iterative method is applied to an isoparametric boundary integral equation formulation for frequency domain electromagnetic scattering problems it is demonstrated to work well on large and geometrically complex examples including a 20 wavelength slender dipole the nasa almond and a resonant cavity on such problems with asymmetric curvilinear irregular meshes and nontrivial geometries the number of iterations required seems to increase rather more than linearly with body size indicating an overall spl sim sixth power cost scaling this scaling is essentially as for direct methods but with costs still a small fraction of the direct approach a method is proposed for the selection of a termination condition designed to avoid seeking the approximate answer too precisely it typically permits a further halving of costs