introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

on invariants for difference equations and systems of difference equations of rational form abstract c j schinas j math anal appl 216 1997 164 179 has presented some invariants for difference equations and systems of difference equations of rational form with constant and periodic coefficients of certain period we report that the presented invariants as well as their difference equations can be generalized

a minimal residual algorithm for the inconsistent matrix equation axb c over symmetric matrices abstract an iterative method with short recurrences is presented by peng z y peng an iteration method for the least squares symmetric solution of the linear matrix equation axb c appl math comput 170 2005 711 723 for solving the nearness problem associated with the inconsistent matrix equation axb c for symmetric matrices the solution of this nearness problem can be computed with little work and low storage requirements per iteration however this algorithm may be slow in case of the irregular convergence behavior in the residual norm of axb c in order to remedy this problem a modification based on the idea of the classical cg method is presented in this paper and an error bound is given finally numerical experiments are reported

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

accurate solutions of m matrix algebraic riccati equations this paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an m matrix algebraic riccati equations mare xdx ax xb c 0 by which we mean the following conformally partitioned matrix left begin array ll b d c a end array right is a nonsingular or an irreducible singular m matrix it is known that such an mare has a unique minimal nonnegative solution phi it is proved that small relative perturbations to the entries of a b c and d introduce small relative changes to the entries of the nonnegative solution phi thus the smaller entries phi do not suffer bigger relative errors than its larger entries unlike the existing perturbation theory for general algebraic riccati equations we then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute phi as accurately as the input data deserve current study is based on a previous paper of the authors on m matrix sylvester equation for which d 0

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems

a new class of nonsymmetric algebraic riccati equations abstract motivated by the study of linear quadratic differential games we introduce a new class of nonsymmetric algebraic riccati equations it is shown that every equation in this class has a unique stabilizing solution which is the solution required to find the open loop nash equilibrium for the differential game we show that the doubling algorithm can be used to find this solution efficiently the solution may also be found by the schur method and under further assumptions by newton s method and a basic fixed point iteration

mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation abstract in this paper the mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation are considered the explicit expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed some simple numerical experiments show that the mixed and componentwise condition numbers can be sharp

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

brief paper perturbation analysis and condition numbers of symmetric algebraic riccati equations this paper is devoted to the perturbation analysis of symmetric algebraic riccati equations based on our perturbation analysis the upper bounds for the normwise mixed and componentwise condition numbers are presented the results are demonstrated by our preliminary numerical experiments

thompson metric method for solving a class of nonlinear matrix equation based on the elegant properties of the thompson metric we prove that the general nonlinear matrix equation x q a f x a q q 1 always has a unique positive definite solution an iterative method is proposed to compute the unique positive definite solution we show that the iterative method is more effective as q increases a perturbation bound for the unique positive definite solution is derived in the end

integrable hierarchies of nonlinear difference difference equations and symmetries we construct the hierarchy of nonlinear difference difference equations associated with the discrete schrodinger spectral problem as examples of equations contained in this hierarchy we obtain the discrete time toda and volterra lattice equations in the case of the time discrete toda lattice we construct its lie point and generalized symmetries finally we present its backlund transformations and relate it to the already constructed symmetries

symplectic structures of moduli space of higgs bundles over a curve and hilbert scheme of points on the canonical bundle the moduli space of triples of the form e s are considered where e is a higgs bundle on a fixed riemann surface x and s is a nonzero holomorphic section of e such a moduli space admits a natural map to the moduli space of higgs bundles simply by forgetting s if y l is the spectral data for the higgs bundle e then s defines a section of the line bundle l over y the divisor of this section gives a point of a hilbert scheme parametrizing 0 dimensional subschemes of the total space of the canonical bundle k x since y is a curve on k x the main result says that the pullback of the symplectic form on the moduli space of higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the hilbert scheme using the map that sends any triple e s to the divisor of the corresponding section of the line bundle on the spectral curve

on mixed and componentwise condition numbers for moore penrose inverse and linear least squares problems classical condition numbers are normwise they measure the size of both input perturbations and output errors using some norms to take into account the relative of each data component and in particular a possible data sparseness componentwise condition numbers have been increasingly considered these are mostly of two kinds mixed and componentwise in this paper we give explicit expressions computable from the data for the mixed and componentwise condition numbers for the computation of the moore penrose inverse as well as for the computation of solutions and residues of linear least squares problems in both cases the data matrices have full column row rank

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

newton s method for the quadratic matrix equation abstract under suitable condition a quadratic matrix equation is equivalent to a nonlinear matrix equation we apply newton s method to the nonlinear matrix equation for computing the numerical solution of a class of quadratic matrix equations convergence theorems are derived and numerical results are also given

two kinds of condition numbers for the quadratic matrix equation in the situation that the quadratic matrix equation has the dominant or minimal solvents the mixed and componentwise condition numbers of the quadratic matrix equation are considered the explicit approximate expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed the theoretical results are illustrated by using some simple numerical examples

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a p q generalized reflexive or anti reflexive matrix with respect to the generalized reflection matrix dual p q if a paq or a paq this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm has been provided

bps states torus links and wild character varieties a string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line for this case this leads to a conjectural colored generalization of existing results of hausel mereb and wong as well as shende treumann and zaslow

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

perturbation analysis of a quadratic matrix equation associated with an m matrix this paper is devoted to the perturbation analysis for the quadratic matrix equation x 2 e x f 0 where e is a diagonal matrix and f is an m matrix the quadratic matrix equation of this type arises in noisy wiener hopf problems for markov chains the solution of practical interest is a particular m matrix solution in this paper the perturbation bound of the m matrix solution is presented meanwhile the residual bound for an approximate solution to the m matrix solution is obtained the theoretical results are illustrated by using simple numerical examples

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

fast iterative schemes for nonsymmetric algebraic riccati equations arising from transport theory for a class of nonsymmetric algebraic riccati equations arising from transport theory lu siam j matrix anal appl 26 2005 pp 679 685 reformulated them into a couple of fixed point equations in a vector form and constructed an iterative method for finding their minimal positive solutions in this paper based on this fixed point equation we propose several fast and effective iterative methods for solving the above mentioned special type of nonsymmetric algebraic riccati equations these iterative methods can be categorized into the class of nonlinear block relaxation iterations and can also be considered as accelerated variants of the known fixed point iteration according to the principle of using the currently available information as promptly as possible the monotone convergence theorems about these new iterative methods are established and the sharper bounds about the solutions of the nonsymmetric algebraic riccati equations are derived numerical implementations show that the new iterative methods are feasible and effective solvers for the nonsymmetric algebraic riccati equations arising from transport theory

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

gauge theories on four dimensional riemannian manifolds this paper develops the riemannian geometry of classical gauge theories yang mills fields coupled with scalar and spinor fields on compact four dimensional manifolds some important properties of these fields are derived from elliptic theory regularity an energy gap theorem the manifold structure of the configuration space and a bound for the supremum of the field in terms of the energy it is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities this extends a theorem of k uhlenbeck

mixed hodge polynomials of character varieties we calculate the e polynomials of certain twisted gl n character varieties mathcal m n of riemann surfaces by counting points over finite fields using the character table of the finite group of lie type text gl n mathbb f q and a theorem proved in the appendix by n katz we deduce from this calculation several geometric results for example the value of the topological euler characteristic of the associated pgl n character variety the calculation also leads to several conjectures about the cohomology of mathcal m n an explicit conjecture for its mixed hodge polynomial a conjectured curious hard lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver we prove these conjectures for n 2

the intersection form on moduli spaces of twisted pgl_n higgs bundles vanishes hausel and rodriguez villegas conjectured that the intersection form on the moduli space of stable textsf pgl n higgs bundles on a curve vanishes if the degree is coprime to n in this note we prove this conjecture along the way we show that moduli spaces of stable chains are irreducible for stability parameters larger than the stability condition induced form stability of higgs bundles

lie symmetries and linearisation of the qrt mapping abstract we study the map u x 2 1 u x 1 u x 2 u x 1 2 u x 1 u x 3 u x 1 introduced by quispel roberts and thompson qrt we show using lie point symmetries under what conditions the qrt mapping can be linearised requiring that the qrt mapping is invariant under the symmetry vector field x x u x x a x b cu du 2 u where b c and d are constants and x is an arbitrary unit periodic function in x we derive conditions on the unknown functions i in the qrt mapping further for these cases of the qrt mapping we explicitly construct two independent integrals of motion ensuring its integrability we also derive its exact solution

accurate solutions of m matrix sylvester equations this paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an m matrix sylvester equation ax xb c by which we mean both a and b have positive diagonal entries and nonpositive off diagonal entries and p i m otimes a b rm t otimes i n is a nonsingular m matrix and c is entrywise nonnegative it is proved that small relative perturbations to the entries of a b and c introduce small relative errors to the entries of the solution x thus the smaller entries of x do not suffer bigger relative errors than its larger entries unlikely the existing perturbation theory for general sylvester equations we then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute x as accurately as the input data deserve

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

linearization through symmetries for discrete equations we show that one can devise through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete cole hopf transformation if the equation is linearizable then the symmetry provides the linearizing transformation at the end we present a few examples of applications for equations defined on four lattice points

mirror symmetry langlands duality and the hitchin system among the major mathematical approaches to mirror symmetry are those of batyrev borisov and strominger yau zaslow syz the first is explicit and amenable to computation but is not clearly related to the physical motivation the second is the opposite furthermore it is far from obvious that mirror partners in one sense will also be mirror partners in the other this paper concerns a class of examples that can be shown to satisfy the requirements of syz but whose hodge numbers are also equal this provides significant evidence in support of syz moreover the examples are of great interest in their own right they are spaces of flat slr connections on a smooth curve the mirror is the corresponding space for the langlands dual group pglr these examples therefore throw a bridge from mirror symmetry to the duality theory of lie groups and more broadly to the geometric langlands program

a class of iteration methods based on the moser formula for nonlinear equations in markov chains abstract many stochastic models in queueing inventory communications and dam theories etc result in the problem of numerically determining the minimal nonnegative solutions for a class of nonlinear matrix equations various iterative methods have been proposed to determine the matrices of interest we propose a new efficient successive substitution moser method and a newton moser method which use the moser formula which originally is just the schulz method these new methods avoid the inverses of the matrices and thus considerable savings on the computational workloads may be achieved moreover they are much more suitable for implementing on parallel multiprocessor systems under certain conditions we establish monotone convergence of these new methods and prove local linear convergence for the substitution moser method and superlinear convergence for the newton moser method

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c