preconditioned krylov solvers for bea the performance of a number of preconditioned krylov methods is analysed for a large variety of boundary element formulations low and high order element two dimensional 2 d and three dimensional 3 d regular singular and hypersingular collocation and symmetric galerkin single and multi zone thermal and elastic continuous and discontinuous boundary formulations with and without condensation are considered preconditioned conjugate gradient cg solvers in standard form and a form effectively operating on the normal equations cgn generalized minimal residual gmres conjugate gradient squared cgs and stabilized bi conjugate gradient bi cgstab krylov solvers are employed in this study both the primitive and preconditioned matrix operators are depicted graphically to illustrate the relative amenability of the alternative formulations to solution via kryiov methods and to contrast and explain their computational performances a notable difference between 2 d and 3 d bea operators is readily visualized in this manner numerical examples are presented and the relative conditioning of the various discrete bea operators is reflected in the performance of the krylov equation solvers a preconditioning scheme which was found to be uncompetitive in the collocation bea context is shown to make iterative solution of symmetric galerkin bea problems more economical than employing direct solution techniques we conclude that the preconditioned krylov techniques are competitive with or superior to direct methods in a wide range of boundary formulated problems and that their performance can be partially correlated with certain problem characteristics

sharp generalizations of the multilinear bohnenblust hille inequality we prove that the multilinear bohnenblust hille inequality is a particular case of a quite general family of optimal inequalities

there exist multilinear bohnenblust hille constants cn n 1 with limn cn 1 cn 0 abstract the n linear bohnenblust hille inequality asserts that there is a constant c n 1 such that the l 2 n n 1 norm of u e i 1 e i n i 1 i n 1 n is bounded above by c n times the supremum norm of u for any n linear form u c n c n c and n n the same holds for real scalars we prove what we call fundamental lemma which brings new information on the optimal constants k n n 1 for both real and complex scalars for instance k n 1 k n 0 87 n 0 473 for infinitely many n s for complex scalars we give a formula of surprisingly low growth in which e and the famous euler mascheroni constant appear k n 1 4 1 e 2 1 2 j 1 n 1 j log 2 e 2 1 2 1 n 2 we study the interplay between the kahane salem zygmund and the bohnenblust hille polynomial and multilinear inequalities and provide estimates for bohnenblust hille type inequality constants for any exponent q 2 n n 1

generic domain decomposition and iterative solvers for 3d bem problems summary in the past two decades considerable improvements concerning integration algorithms and solvers involved in boundary element formulations have been obtained first a great deal of efficient techniques for evaluating singular and quasi singular boundary element integrals have been definitely established and second iterative krylov solvers have proven to be advantageous when compared to direct ones also including non hermitian matrices the former fact has implied in cpu time reduction during the assembling of the system of equations and the latter fact in its faster solution in this paper a triangle polar co ordinate transformation and the telles co ordinate transformation applied in previous works independently for evaluating singular and quasi singular integrals are combined to increase the efficiency of the integration algorithms and so to improve the performance of the matrixassembly routines in addition the jacobi preconditioned biconjugate gradient j bicg solver is used to develop a generic substructuring boundary element algorithm in this way it is not only the system solution accelerated but also the computer memory optimized discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving in the discussion of results expressed in terms of accuracy and cpu time advantages and potential applications of the be code developed are highlighted copyright 2006 john wiley sons ltd

rigid objects in higher cluster categories abstract we study maximal m rigid objects in the m cluster category c h m associated with a finite dimensional hereditary algebra h with n nonisomorphic simple modules we show that all maximal m rigid objects in these categories have exactly n nonisomorphic indecomposable summands and that any almost complete m rigid object in c h m has exactly m 1 nonisomorphic complements we also show that the maximal m rigid objects and the m cluster tilting objects in these categories coincide and that the class of finite dimensional algebras associated with maximal m rigid objects is closed under certain factor algebras

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

coordinatewise multiple summing operators in banach spaces we invent the new notion of coordinatewise multiple summing operators in banach spaces and use it to study various vector valued extensions of the well know bohnenblust hille inequality which originally extended littlewood s 4 3 inequality our results have application on the summability of monomial coefficients of m homogeneous polynomials p l lp as well as for the convergence theory of products of vector valued dirichlet series

iterative solution techniques in boundary element analysis iterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis bea method are discussed continuum structural response analysis problems are considered employing single and multi zone boundary element models with and without zone condensation the impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices resulting in multi zone bea is studied both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods to characterize the potential of these iterative techniques we discuss accuracy storage and timing statistics in comparison with analogous information from direct sparse blocked matrix factorization procedures matrix populations that experience block fill in during the direct decomposition process are included with different degrees of preconditioning iterative equation solving is shown to be competitive with direct methods for the problems considered

a fast 3d dual boundary element method based on hierarchical matrices in this paper a fast solver for three dimensional bem and dbem is developed the technique is based on the use of hierarchical matrices for the representation of the collocation matrix and uses a preconditioned gmres for the solution of the algebraic system of equations the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format special algorithms are developed to deal with crack problems within the context of dbem the structure of dbem matrices has been efficiently exploited and it has been demonstrated that since the cracks form only small parts of the whole structure the use of hierarchical matrices can be particularly advantageous test examples presented show that with the proposed technique substantial increase in number of elements over the crack surfaces leads only to moderate increases in memory storage and solution time

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

on open orbits and their complements 1 1 we fix an algebraically closed field k and a quiver q without oriented cycles whose vertex set q we identify with 1 n for a in the set q of arrows of q we denote by ta and ha the tail and the head of a respectively a representation x of q consists of a family x i in qo of finite dimensional k vector spaces and a family x a x ta x ha a e q1 of k linear maps a morphism f x y between two representations of q is a family f i x i y i in qo of k linear maps satisfying f ha ox a y a of ta for all arrows a we denote the category of representations of q by mod q it is an abelian category of global dimension at most 1 we will denote the only possibly non trivial extension group ext by ext the dimension vector of a representation x of q is the vector dim x dim x l dim x n in n and the dimension of x is the natural number dim x dim x 1 dim x n the support of x is the full subquiver of q whose vertices are i x i zo for a vector d d d e n we define r q d to be the set of representations x of q such that x i k for ie qo thus r q d is the product r q d n hom kdru kdhe

improving the closed formula for subpolynomial constants in the multilinear bohnenblust hille inequalities abstract for k r or c the bohnenblust hille inequality asserts that there exists a sequence of scalars c k m m 1 such that i 1 i m 1 n u e i 1 e i m 2 m m 1 m 1 2 m c k m sup z 1 z m d n tu z 1 z m for all m linear forms u k n k n k and every positive integer n where e i i 1 n denotes the canonical basis of k n and d n represents the open unit polydisc in k n very recently 2012 it was shown that there exist constants c k m m 1 with subpolynomial growth satisfying this inequality however these constants were obtained via a complicated recursive formula we improve the best known closed non recursive approximation for these constants

new upper bounds for the constants in the bohnenblust hille inequality a classical inequality due to bohnenblust and hille states that for every positive integer m there is a constant c m 0 so that sigma n i1 im 1 vertical bar u e i1 e im vertical bar 2m m 1 m 1 2m c where c m m m 1 2m 2 m 1 2 the value of c m was improved to c m 2 m 1 2 by s kaijser and more recently h queffelec and a defant and p sevilla peris remarked that c m 2 root pi m 1 also works the bohnenblust hille inequality also holds for real banach spaces with the constants c m 2 m 1 2 in this note we show that a recent new proof of the bohnenblust hille inequality due to defant popa and schwarting provides in fact quite better estimates for c m for all values of m is an element of n in particular we will also show that for real scalars if m is even with 2 m 24 then c r m 2 1 2 c r m 2 we will mainly work on a paper by defant popa and schwarting giving some remarks about their work and explaining how to numerically improve the previously mentioned constants

from m clusters to m noncrossing partitions via exceptional sequences let w be a finite crystallographic reflection group the generalized catalan number of w coincides both with the number of clusters in the cluster algebra associated to w and with the number of noncrossing partitions for w natural bijections between these two sets are known for any positive integer m both m clusters and m noncrossing partitions have been defined and the cardinality of both these sets is the fuss catalan number cm w we give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category db h for any finite dimensional hereditary algebra h

a note on the polynomial bohnenblust hille inequality abstract recently in a paper published in the annals of mathematics it was shown that the bohnenblust hille inequality for complex homogeneous polynomials is hypercontractive however and to the best of our knowledge there is no result providing nontrivial lower bounds for the optimal constants for n homogeneous polynomials n 2 in this short note we provide lower bounds for these famous constants

a new multilinear insight on littlewood s 4 3 inequality abstract we unify littlewood s classical 4 3 inequality a forerunner of grothendieck s inequality together with its m linear extension due to bohnenblust and hille which originally settled bohr s absolute convergence problem for dirichlet series with a scale of inequalties of bennett and carl in l p spaces which are of fundamental importance in the theory of eigenvalue distribution of power compact operators as an application we give estimates for the monomial coefficients of homogeneous l p valued polynomials on c 0

symmetric galerkin boundary formulations employing curved elements accounts of the symmetric galerkin approach to boundary element analysis bea have recently been published this paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency a new symmetric galerkin theoretical formulation for both the harmonic heat conduction and the biharmonic elasticity problem that employs regularized singular and hypersingular boundary integral equations bies is presented the novel use of regularized bies in the galerkin context is shown to allow straightforward incorporation of curved isoparametric elements a symmetric reusable intrinsic sample point risp numerical integration algorithm is shown to produce a galerkin i e double integration strategy that is competitive with its counterpart i e singular integration procedure in the collocation bea approach when the time saved in the symmetric equation solution phase is also taken into account this new formulation is shown to be capable of employing hypersingular bies while obviating the requirement of c 1 continuity a fact that allows the employment of the popular continuous element technology the behavior of the symmetric galerkin bea method with regard to both direct and iterative equation solution operations is also addressed a series of example problems are presented to quantify the performance of this symmetric approach relative to the more conventional unsymmetric bea in terms of both accuracy and efficiency it is concluded that appropriate implementations of the symmetric galerkin approach to bea indeed have the potential to be competitive with if not superior to collocation based bea for large scale problems

the asymptotic growth of the constants in the bohnenblust hille inequality is optimal the search of sharp estimates for the constants in the bohnenblust hille inequality besides its challenging nature has quite important applications in different fields of mathematics and physics for homogeneous polynomials it was recently shown that the bohnenblust hille inequality for complex scalars is hypercontractive this result interesting by itself has found direct striking applications in the solution of several important problems for multilinear mappings precise information on the asymptotic behavior of the constants of the bohnenblust hille inequality is of particular importance for applications in quantum information theory and multipartite bell inequalities in this paper using elementary tools we prove a quite surprising result the asymptotic growth of the constants in the multilinear bohnenblust hille inequality is optimal besides its intrinsic mathematical interest and potential applications to different areas the mathematical importance of this result also lies in the fact that all previous estimates and related results for the last 80 years such as for instance the multilinear version of the famous grothendieck theorem for absolutely summing operators always present constants c m s growing at an exponential rate of certain power of m

a fast hierarchical dual boundary element method for three dimensional elastodynamic crack problems in this work a fast solver for large scale three dimensional elastodynamic crack problems is presented implemented and tested the dual boundary element method in the laplace transform domain is used for the accurate dynamic analysis of cracked bodies the fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the laplace parameter an aca adaptive cross approximation algorithm is used for the population of the low rank blocks and its performance at varying laplace parameters is investigated a preconditioned gmres is used for the solution of the resulting algebraic system of equations the preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format an original strategy based on the computation of some local preconditioners only is presented and tested to further speed up the overall analysis the reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time

exceptional sequences and clusters abstract we show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse coxeter element in the weyl group we use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type

parallel algorithms for dense linear algebra computations scientific and engineering research is becoming increasingly dependent upon the development and implementation of efficient parallel algorithms on modern high performance computers numerical linear algebra is an indispensable tool in such research and this paper attempts to collect and describe a selection of some of its more important parallel algorithms the purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense banded or block structured problems arising in the major areas of direct solution of linear systems least squares computations eigenvalue and singular value computations and rapid elliptic solvers a major emphasis is given here to certain computational primitives whose efficient execution on parallel and vector computers is essential in order to obtain high performance algorithms

a study of the efficiency of iterative methods for linear problems in structural mechanics the efficiency of iterative methods in linear structural mechanics is studied the efficiency concerns the calculation time the numerical accuracy and the core storage needed we state that iterative methods are effective in connection with hierarchical improvement of a primary approximation three iterative methods are studied the conjugate gradient method preconditioned by a modified incomplete factorization matrix the same method preconditioned by a matrix obtained from natural factors on elemental level and a jacobi integration preconditioned by viscous relaxation split in an element by element way we make comparisons with direct methods gaussian elimination and factorization by use of natural factors

iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation in this paper the diffusion equation is solved in two dimensional geometry by the dual reciprocity boundary element method drbem it is structured by fully implicit discretization over time and by weighting with the fundamental solution of the laplace equation the resulting domain integral of the diffusive term is transformed into two boundary integrals by using green s second identity and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions straight line geometry and constant field shape functions for boundary discretization are employed the described procedure results in systems of equations with fully populated unsymmetric matrices in the case of solving large problems the solution of these systems by direct methods may be very time consuming the present study investigates the possibility of using iterative methods for solving these systems of equations it was demonstrated that krylov type methods like cgs and gmres with simple jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude this paper can be considered as a logical starting point for research of iterative solutions to drbem systems of equations 1998 john wiley sons ltd

exchange graphs and ext quivers abstract we study the oriented exchange graph eg n q of reachable hearts in the finite dimensional derived category d n q of the cy n ginzburg algebra n q associated to an acyclic quiver q we show that any such heart is induced from some heart in the bounded derived category d q via some lagrangian immersion l d q d n q we build on this to show that the quotient of eg n q by the seidel thomas braid group is the exchange graph ceg n 1 q of cluster tilting sets in the higher cluster category c n 1 q as an application we interpret buan thomas coloured quiver for a cluster tilting set in terms of the ext quiver of any corresponding heart in d n q

stability conditions and quantum dilogarithm identities for dynkin quivers abstract we study the fundamental groups of the exchange graphs for the bounded derived category d q of a dynkin quiver q and the finite dimensional derived category d n q of the calabi yau n ginzburg algebra associated to q in the case of d q we prove that its space of stability conditions in the sense of bridgeland is simply connected as an application we show that the donaldson thomas type invariant associated to q can be calculated as a quantum dilogarithm function on its exchange graph in the case of d n q we show that the faithfulness of the seidel thomas braid group action which is known for q of type a or n 2 implies the simply connectedness of the space of stability conditions

defining an m cluster category we show that a certain orbit category considered by keller encodes the combinatorics of the m clusters of fomin and reading in a fashion similar to the way the cluster category of buan marsh reineke reiten and todorov encodes the combinatorics of the clusters of fomin and zelevinsky this allows us to give type uniform proofs of certain results of fomin and reading in the simply laced cases

on the bohnenblust hille inequality and a variant of littlewoodʼs 4 3 inequality the search for sharp constants for inequalities of the type littlewood s 4 3 and bohnenblust hille has lately shown unexpected applications in many fields such as analytic number theory quantum information theory or in results on n dimensional bohr radii recent estimates obtained for the multilinear bohnenblust hille inequality for real scalars have been used as a crucial tool by a montanaro in order to solve problems in quantum xor games here among other results we obtain new upper bounds for the bohnenblust hille constants for complex scalars for bilinear forms we provide optimal constants of variants of littlewood s 4 3 inequality for real scalars when the exponent 4 3 is replaced by any r 4 3 we also prove that the optimal constants in real case are always strictly greater than those from the complex case

an iterative technique for the finite element analysis of near incompressible materials abstract a new preconditioner for the conjugate gradient method applicable to the finite element analysis of near incompressible finite elasticity is presented the boundary value problem is discretized with a displacement pressure mixed formulation the pressure degrees of freedom are eliminated at the element level the problem is linearized using a newton raphson technique each linear system is solved using the preconditioned conjugate gradient the new preconditioner is based on projections onto a subspace associated with the bulk modes of deformation the condition number of the preconditioned stiffness matrix and consequently the rate of convergence of the preconditioned conjugate gradient are totally insensitive to increasing bulk modulus the method has been applied to model problems in linear and nonlinear 2d and 3d using a near incompressible hyperelastic material model the performance of the iterative method compares favorable with direct solvers as well as with conjugate gradient methods for indefinite systems in terms of storage requirements operation count and cpu time

sparse preconditioned iterative methods for dense linear systems two sparse preconditioned iterative methods are presented to solve dense linear systems arising in the solution of two dimensional boundary integral equations in the first method the sparse preconditioner is constructed simply by choosing a small block of elements in the coefficient matrix of a dense linear system the two grid method falls into this category when the dense linear system arises from the nystrom method for a second kind boundary integral equation in the second method the sparse preconditioner is obtained through condensation of the coefficient matrix by discrete fourier transforms which can be implemented efficiently using fast fourier transforms both iterative methods involve only o n 2 arithmetic operations per iteration and converge rapidly when the dense linear systems arise from quadrature methods for boundary integral equations arising in two dimensional problems the author s numerical experiments demonstrate the computational efficiency of each method

a comparison of direct and preconditioned iterative techniques for sparse unsymmetric systems of linear equations in this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric sparse systems of linear algebraic equations these problems occur in finite difference and finite element simulations of semiconductor devices and fluid flow problems we consider five iterative methods that appear to be the most promising for this class of problems the biconjugate gradient method the conjugate gradient squared method the generalized minimal residual method the generalized conjugate residual method and the method of orthogonal minimization each of these methods was tested using similar preconditioning incomplete lu factorization on a set of large sparse matrices arising from finite element simulation of semiconductor devices results are shown where we compare the computation time and memory requirements for each of these methods against one another as well as against a direct method that uses lu factorization to solve these problems the results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large sparse systems of equations and can offer significant savings in storage and cpu time

a generalized sor method for dense linear systems of boundary element equations in this paper an iterative scheme of first degree is developed for the purpose of solving linear systems of boundary element equations of the form hx c where h is a dense square nonsingular matrix the iterative scheme considered is d omega h sl x k 1 d omega h u x k omega c noindent where omega h u and omega h s1 are defined as the upper triangular and strictly lower triangular terms of omega h respectively the parameter matrix omega is selected to minimize the frobenius norm d omega h u f mathematical arguments and numerical experiments are presented to show that minimizing d omega h u f provides for faster convergence numerical tests are performed for systems of boundary element equations generated by three dimensional potential and elastostatic problems computation times are determined and compared against those for gaussian elimination and gauss seidel iteration

a parallel multipolar indirect boundary element method for the neumann interior stokes flow problem a multipolar expansion technique is applied to the indirect formulation of the boundary element method in order to solve the two dimensional internal stokes flow second kind boundary value problems the algorithm is based on a multipolar expansion for the far field and numerical evaluation for the near field due to the nature of the algorithm it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations a parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvements copyright 2000 john wiley sons ltd

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

a unified boundary element method for the analysis of sound and shell like structure interactions ii efficient solution techniques efficient solution methods are investigated in this paper for solving the linear system of equations resulting from the recently developed boundary element method bem for the coupled structural acoustic analysis s h chen and y j liu j acoust soc am 106 pt 1 1247 1254 1999 an iterative solver namely the quasiminimal residual method qmr is selected among others and found to be very favorable over the direct solver for solving the linear systems of equations with complex coefficients generated by the structural acoustic bem four problem dependent preconditioning schemes are developed to facilitate or accelerate the convergence of the iterative solver a new effective preconditioner specially designed for frequency sweep analysis is also presented in this paper with this preconditioner the iterative solver has been found to be stable in a frequency sweep analysis and can converge much faster than the direct solver the double precision arithmetic is also found very useful in improvi

a fast dual boundary element method for 3d anisotropic crack problems in the present paper a fast solver for dual boundary element analysis of 3d anisotropic crack problems is formulated implemented and tested the fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix the admissible low rank blocks are computed by adaptive cross approximation aca the performance of aca against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated focusing on the balance between the kernel representation accuracy and the accuracy required for aca the system solution is computed by a preconditioned gmres and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format the effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated highlighting the accuracy as well as the significant reduction in memory storage and analysis time in particular it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom obtaining up to solution speedups of order 10 for systems of order 104 moreover the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed copyright 2009 john wiley sons ltd

the bohnenblust hille inequality for homogeneous polynomials is hypercontractive the bohnenblust hille inequality says that the 2m m 1 norm of the coefcients of an m homogeneous polynomial p on c n is bounded by kpk1 times a constant independent of n wherekk 1 denotes the supremum norm on the polydisc d n the main result of this paper is that this inequality is hypercontractive i e the constant can be taken to be c m for some c 1 combining this improved version of the bohnenblust hille inequality with other results we obtain the following the bohr radius for the polydisc d n behaves asymptotically as p logn n modulo a factor bounded away from 0 and innity

solving large and sparse linear least squares problems by conjugate gradient algorithms abstract let a e m n with m n and rank a n and b e m 1 be given assume that an approximation to x a b ata 1atb is to be calculated this problem is transformed into an equivalent problem with a symmetric and positive definite matrix c d 1 rt 1atar 1d 1 where d is a diagonal matrix and r is an upper triangular matrix the conjugate gradient cg algorithm is applied in the solution of the system of linear algebraic equations cy d y drx d d 1 rt 1atb if d r i then the cg algorithm is in fact applied to the system of normal equations atax atb and the speed of convergence could be very slow if d and r are obtained by some kind of orthogonalization dr qta with q e m n satisfying qtq i d is often equal to i and q is never used in the cg algorithm and if the calculations are performed without rounding errors then c i and the cg algorithm converges in one iteration only even if the orthogonalization process is carried out with rounding errors the matrix c is normally close to the identity matrix i and the cg algorithm is quickly convergent however for large m and n the orthogonal decomposition is an expensive process both in regard to storage and in regard to computing time therefore it may be profitable to calculate an incomplete orthogonal decomposition this is achieved by introducing a special parameter t a drop tolerance such that all elements which in the course of the computations become smaller in absolute value than t are removed numerical examples are given to illustrate that the cg algorithm applied to system cy d and used with incomplete factors d and r is very efficient for some classes of problems it should be emphasized that matrix c is never calculated explicitly the whole work is carried out by the use of a d and r only

convergence of dirichlet polynomials in banach spaces both authors were supported by the mec project mtm2008 03211 the second author was partially supported by grants pr2007 0384 mec and upv paid 00 07

bohr radii of vector valued holomorphic functions abstract motivated by the scalar case we study bohr radii of the n dimensional polydisc d n for holomorphic functions defined on d n with values in banach spaces

multinode finite element based on boundary integral equations a finite element constructed on the basis of boundary integral equations is proposed this element has a flexible shape and arbitrary number of nodes it also has good approximation properties a procedure of constructing an element stiffness matrix is demonstrated first for one dimensional case and then for two dimensional steady state heat conduction problem numerical examples demonstrate applicability and advantages of the method 1998 john wiley sons ltd

clusters coxeter sortable elements and noncrossing partitions we introduce coxeter sortable elements of a coxeter group for finite we give bijective proofs that coxeter sortable elements are equinumerous with clusters and with noncrossing partitions we characterize coxeter sortable elements in terms of their inversion sets and in the classical cases in terms of permutations

three kinds of mutation for a finite dimensional hereditary algebra h we consider exceptional sequences in mod h silting objects in the bounded derived category d and m cluster tilting objects in the m cluster category cm d 1 m there are mutation operations on both the set of m cluster tilting objects and the set of exceptional sequences it is also possible to define a mutation operation for silting objects we compare these three different notions of mutation