poisson harmonic forms kostant harmonic forms and the s1 equivariant cohomology ofk t abstract we characterize the harmonic forms on a flag manifold k t defined by kostant in 1963 in terms of a poisson structure namely they are poisson harmonic with respect to the so called bruhat poisson structure on k t this enables us to give poisson geometrical proofs of many of the special properties of these harmonic forms in particular we construct explicit representatives for the schubert basis of the s 1 equivariant cohomology of k t where the s 1 action is defined by using a simple argument in equivariant cohomology we recover the connection between the kostant harmonic forms and the schubert calculus on k t that was found by kostant and kumar in 1986 by using a family of symplectic structures on k t we also show that the kostant harmonic forms are limits of the more familiar hodge harmonic forms with respect to a family of hermitian metrics on k t

poisson structures on complex flag manifolds associated with real forms for a complex semisimple lie group g and a real form go we define a poisson structure on the variety of borel subgroups of g with the property that all go orbits in x as well as all bruhat cells for a suitable choice of a borel subgroup of g are poisson submanifolds in particular we show that every non empty intersection of a go orbit and a bruhat cell is a regular poisson manifold and we compute the dimension of its symplectic leaves

a stable richardson iteration method for complex linear systems the richardson iteration method is conceptually simple as well as easy to program and parallelize this makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues we change the ordering of the relaxation parameters of a richardson iteration method proposed by eiermann niethammer and varga for the solution of such problems the new method obtained is shown to be stable and to have better convergence properties

the nil hecke ring and cohomology of g p for a kac moody group g abstract let g be the group with borel subgroup b associated to a kac moody lie algebra unk with weyl group w and cartan subalgebra unk then h g b has among others four distinguished structures i an algebra structure ii a distinguished basis given by the schubert cells iii a module for w and iv a module for hecke type operators aw for w unk w we construct a ring r which we refer to as the nil hecke ring which is very simply and explicitly defined as a functor of w together with the w module unk alone and such that all these four structures on h g b arise naturally from the ring r

topology of lattice gauge fields non abelian gauge fields on a four dimensional hypercubic lattice with small action density tr u dot p for su 2 gauge fields are shown to carry an integer topological chargeq which is invariant under continuous deformations of the field a concrete expression forq is given and it is verified thatq reduces to the familiar chern number in the classical continuum limit

su n yang mills solutions with constant field strength on t 4 we study fort4 the class of solutions to the su n yang mills equations with constant field strength the fluctuation spectrum is explicitly calculated in terms of generalized riemann theta functions we show that if these solutions are stable they are necessarily anti selfdual in which case we verify the index theorem

a note on the tau method approximations for the bessel functions y0 z and y1 z abstract this paper is to complete and improve the work reported in 1 2 using the lanczos method in coleman s version to approximate the bessel functions y 0 z and y 1 z we introduce symbolic representations of the scaled faber polynomials on any fan shaped section of the complex plane these faber polynomials are used as the perturbation terms in the method numerical comparison among the power series the chebyshev series and the method are conducted to show the accuracy improvement achieved by this new version of the method some concluding remarks and suggestions on future research are given

on the construction of semi iterative methods given a nonsingular linear system a bf x bf b a splitting a m n leads to the one step iteration 1 bf x m t bf x m 1 bf c with t m 1 n and c m 1 bf b we investigate semiiterative methods with respect to 1 under the assumption that the eigenvalues of t are contained in some compact set u subset mathbb c 1 notin u using results about maximal convergence of polynomials and uniformly distributed nodes from approximation theory we describe semiiterative methods which are asymptotically optimal with respect to u under mildly restrictive assumptions on u we construct semiiterative methods which are asymptotically optimal with respect to u and which can be interpreted as one step nonstationary methods

the poisson geometry of su 1 1 we study the natural poisson structure on the group su 1 1 and related questions in particular we give an explicit description of the ginzburg weinstein isomorphism for the sets of admissible elements we also establish an analog of thompson s conjecture for this group

optimal semi iterative methods for complex sor with results from potential theory we consider the application of semi iterative methods sim to the standard sor method with complex relaxation parameter under the following two assumptions 1 the associated jacobi matrix j is consistently ordered and weakly cyclic of index 2 and 2 the spectrum j of j belongs to a compact subset of the complex plane mathbb c which is symmetric with respect to the origin by using results from potential theory we determine the region of optimal choice of omega in mathbb c for the combination sim sor and settle for a large class of compact sets the classical problem of characterising completely all the cases for which the use of the sim sor is advantageous over the sole use of sor under the hypothesis that sigma j subset sigma in particular our results show that unless the outer boundary of is an ellipse sim sor is always better and furthermore one of the best possible choices is an asymptotically optimal sim applied to the gauss seidel method in addition we derive the optimal complex sor parameters for all ellipses which are symmetric with respect to the origin our work was motivated by recent results of m eiermann and r s varga

some results for su n gauge fields on the hypertorus we show how to prove and to understand the formula for the pontryagin indexp for su n gauge fields on the hypertorust4 seen as a four dimensional euclidean box with twisted boundary conditions these twists are defined as gauge invariant integers modulon and labelled byn v n v in terms of these we can write x2124 p frac 1 16 pi 2 int tr g mu v tilde g mu v d 4 x v left frac n 1 n right cdot frac n mu v tilde n mu v 4 furthermore we settle the last link in the proof of the existence of zero action solutions with all possible twists satisfying frac n mu v tilde n mu v 4 kappa n 0 bmod n for arbitraryn

the conformal bratwurst mapsand associated faber polynomials summary conformal maps from the exterior of the closed unit disk onto the exterior of bratwurst shape sets in the complex plane are constructed using these maps coefficients for the computation of the corresponding faber polynomials are derived a bratwurst shape set is the result of deforming an ellipse with foci on the real axis by conformally mapping the real axis onto the unit circle such sets are well suited to serve as inclusion sets for sets associated with a matrix for example the spectrum field of values or a pseudospectrum hence the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems the main advantage of the approach is that the conformal maps are derived from elementary transformations allowing an easy computation of the associated transfinite diameter asymptotic convergence factor and faber polynomials numerical examples are given

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

an adaptive method for solving nonsymmetric linear systems involving applications of scpack consider the following large system of linear equations rm x it t x bf c eqno 0 where it is assumed that t is n by n real matrix with 1 excluded from the spectrum of t and that c is a real vector of dimension n this dissertation uses the results from function theory approximation theory and conformal mapping theory and constructs an adaptive numerical method to solve equation 0 with the aid of scpack this adaptive method periodically estimates the eigenvalue set to obtain near asymptotically optimal rates of convergence of the associated k l step semiiterative methods

iterative and semi iterative methods for computing stationary probability vectors of markov operators iterative and semi iterative methods for computing stationary prob ability vectors of markov type operators are proposed and their convergence properties are analyzed the methods studied apply to certain classes of prob lems in infinite dimensional spaces as well as to classical n x n stochastic ma trices

the analysis ofk step iterative methods for linear systems from summability theory using the theory of euler methods from summability theory we investigate general iterative methods for solving linear systems of equations in particular for a given euler method a regions of the complex plane is determined such that ak step iterative method converges if the eigenvalues of an iteration operatort are contained ins for a givens optimal methods are described and upper and lower bounds are derived for the associated asymptotic rate of convergence special attention is given to two step methods with complex parameters

richardson s iteration for nonsymmetric matrices abstract to solve the linear n n system 1 ax a for any nonsingular matrix a richardson s iteration 2 x j 1 x j j ax j a j 1 2 n which is applied in a cyclic manner with cycle length n is investigated where the j are free parameters the objective is to minimize the error x n 1 x where x is the solution of 1 if the spectrum of a is known to lie in a compact set s one is led to the chebyshev type approximation problem 3 min p 1 v n max z s p z where v n is the linear span of z z 2 z n if p solves 3 then the reciprocals of the zeros of p are optimal iteration parameters j it is shown that for a real problem 1 the iteration 2 can be carried out with real arithmetic alone even when there are complex j the stationary case n 1 is solved completely i e for all compact sets s the problem 3 is solved explicitly as a consequence the converging stationary iteration processes can be characterized for arbitrary closed disks s the problem 3 can be solved for all n n and a simple proof is provided the lemniscates associated with s are introduced they appear as an important tool for studying the stability of the iteration 2

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

surviving extrema for the action on the twisted su one point lattice we give a simplified construction of twist eating configurations based on a theorem due to frobenius these configurations are defined through the equation u u u u exp 2 in n withu su n 1 tod andn an antisymmetric matrix with integer entries in the twisted eguchi kawai model they yield extrema some of which survive forn comparison is made with the monte carlo data of the internal energy in the small coupling region

application of a block modified chebyshev algorithmto the iterative solution of symmetric linear systemswith multiple right hand side vectors an adaptive richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right hand side vectors this scheme learns about the linear system to be solved by computing inner products of residual matrices during the iterations these inner products are interpreted as block modified moments a block version of the modified chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials the eigenvalues of this block tridiagonal matrix define an interval which determines the choice of relaxation parameters for richardson iteration only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right hand side vectors we outline the changes required

some twisted self dual solutions for the yang mills equations on a hypertorus thesu n yang mills equations are considered in a four dimensional euclidean box with periodic boundary conditions hypertorus gauge invariant twists can be introduced in these boundary conditions to be labeled with integersn n defined modulon the pontryagin number in this space is often fractional whenever this number is zero there are solutions to the equationsg 0 hereg is the covariant curl when this number is not zero we find a set of solutions to the equations g mu nu tilde g mu nu provided that the periodsa of the box satisfy certain relations

a parallelizable sor like method systems with plus shaped and linear spectra abstract the classical sor method replaces an invertible cyclic linear system in b x f with a family of systems i n b x f where for some optimal scalar b the spectrum of b b is circular the eigenvalues of b are linearly placed falling on the real line moreover the spectral radius of b b is less than that of b similarly this paper explores a tensor mechanism which converts more general systems in b x f to equivalent systems i 2n b x f where the spectrum of b may be plus shaped the spectrum of b is circular and x col 0 x as in the sor theory the spectral radius of b is less than that of b among other things we recapture the results of the sor theory improve on some known results and extend our theory directly to matrices in b which need not be symmetric yet have real plus and minus valued eigenvalues

convergence properties of art and sor algorithms art algorithms with relaxation parameters are studied for general consistent or inconsistent linear algebraic systemsrx f and a general convergence theorem is formulated the advantage of severe underrelaxation is reexamined and clarified the relationship to solutions obtained by applying sor methods to the equationrr t y f is investigated

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

the sor method on parallel computers the jacobi overrelaxation jor method is usually cited as a perfect parallel algorithm whereas the successive overrelaxation sor method is considered as quite the opposite for linear systems with dense matrices an algorithm for the sor method is presented which is suited for parallelization nearly in the same way as jor for systems with band matrices an algorithm is described which using a pipeline principle yields a speedups p 1 p 1 p m 1 wherep denotes the number of processors andm the number of sor iterations thus form p there is no speedup whereas the speedup tends to its maximal valuep ifm tends to infinity

theorems of stein rosenberg type iii the singular case abstract in the theory of iterative methods the classical stein rosenberg theorem can be viewed as giving the simultaneous convergence or divergence of the extrapolated jacobi jor matrix j and the successive overrelaxation sor matrix in the case when the jacobi matrix j 1 is nonnegative as has been established recently by buoni and varga necessary and sufficient conditions for the simultaneous convergence or divergence of j and have been established which do not depend on the assumption that j 1 is nonnegative our aim here is to extend these results to the singular case using the notion of semiconvergence in particular for a real singular matrix a with nonpositive off diagonal entries we find conditions theorem 3 4 which imply that j and simultaneously semiconverge for all in the real interval 0 1

on the spectrum of the sor operator for symmetric positive definite matrices one attractive feature of the classical successive overrelaxation sor iterative method is its algorithmic simplicity whereas for linear systems with symmetric positive definite matrices a the theorem of ostrowski guarantees convergence as long as the relaxation parameter 0 2 results on the optimal choice of can only be obtained under certain additional assumtions on a here for positive definite systems with coefficient matrices a i l lt in which l is not necessarily strictly lower triangular we study the behavior of the spectrum of the sor operator l as 0 and 2 and describe enclosure sets which may be used to estimate the spectral radius of l for 0 2 numerical experiments show why results on the optimal relaxation parameter are difficult to obtain in the general case of positive definite systems