flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

a bundle view of boundary value problems generalizing the gardner jones bundle abstract holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter dependent boundary conditions are considered from a geometrical viewpoint the gardner jones bundle which was introduced for linearized reaction diffusion equations is generalized and applied to this abstract class of dependent boundary value problems where is a complex eigenvalue parameter the fundamental analytical object of such boundary value problems bvps is the characteristic determinant and it is proved that any characteristic determinant on a jordan curve can be characterized geometrically as the determinant of a transition function associated with the gardner jones bundle the topology of the bundle represented by the chern number then yields precise information about the number of eigenvalues in a prescribed subset of the complex plane this result shows that the gardner jones bundle is an intrinsic geometric property of such dependent bvps the bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex ginzburg landau equation

construction of starting algorithms for the rk gauss methods in this paper starting algorithms for the numerical solution of stage equations in runge kutta gauss formulae with 2 3 and 4 stages are constructed for each of these formulae three types of starting algorithms are given according to their requirement of none one or two additional function evaluations per step numerical experiments with hamiltonian systems are presented to show the superior performance of the new starting algorithms of high order

an automatic orthonormalization method for solving stiff boundary value problems abstract a new initial value method is described based on a remark by drury for solving stiff linear differential two point cigenvalue and boundary value problems the method is extremely reliable it is especially suitable for high order differential systems and it is capable of accommodating realms of stiffness which other methods cannot reach the key idea behind the method is to decompose the stiff differential operator into two non stiff operators one of which is nonlinear the nonlinear one is specially chosen so that it advances an orthonormal frame indeed the method is essentially a kind of automatic orthonormalization the second is auxiliary but it is needed to determine the required function the usefulness of the method is demonstrated by calculating some eigenfunctions for an orr sommerfeld problem when the reynolds number is as large as 10

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

grassmannian spectral shooting we present a new numerical method for computing the pure point spectrum associated with the linear stability of coherent structures in the context of the evans function shooting and matching approach all the relevant information is carried by the flow projected onto the underlying grassmann manifold we show how to numerically construct this projected flow in a stable and robust manner in particular the method avoids representation singularities by in practice choosing the best coordinate patch representation for the flow as it evolves the method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed for large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization we demonstrate this by comparing the two methods in three applications boussinesq solitary waves autocatalytic travelling waves and the ekman boundary layer

stabilization of daes and invariant manifolds many methods have been proposed for the stabilization of higher index differential algebraic equations daes such methods often involve constraint differentiation and problem stabilization thus obtaining a stabilized index reduction a popular method is baumgarte stabilization but the choice of parameters to make it robust is unclear in practice here we explain why the baumgarte method may run into trouble we then show how to improve it we further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature our approach is to i consider stabilization of odes with invariants ii discretize the stabilizing term in a simple way generally different from the ode discretization and iii use orthogonal projections whenever possible the best methods thus obtained are related to methods of coordinate projection we discuss them and make concrete algorithmic suggestions

twisted holomorphic chains and vortex equations over non compact kahler manifolds abstract in this paper we study twisted holomorphic chains and related gauge equations over non compact kahler manifolds we use the heat flow method to solve the dirichlet boundary problem for the related gauge equations and prove a hitchin kobayashi type correspondence for twisted holomorphic chain over some non compact kahler manifolds

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

on cyclic higgs bundles in this paper we derive a maximum principle for a type of elliptic systems and apply it to analyze the hitchin equation for cyclic higgs bundles we show several domination results on the pullback metric of the possibly branched minimal immersion f associated to cyclic higgs bundles also we obtain a lower and upper bound of the extrinsic curvature of the image of f as an application we give a complete picture for maximal sp 4 mathbb r representations in the 2g 3 gothen components and the hitchin components

optimization algorithms exploiting unitary constraints this paper presents novel algorithms that iteratively converge to a local minimum of a real valued function f x subject to the constraint that the columns of the complex valued matrix x are mutually orthogonal and have unit norm the algorithms are derived by reformulating the constrained optimization problem as an unconstrained one on a suitable manifold this significantly reduces the dimensionality of the optimization problem pertinent features of the proposed framework are illustrated by using the framework to derive an algorithm for computing the eigenvector associated with either the largest or the smallest eigenvalue of a hermitian matrix

an elementary proof that the biharmonic green function of an eccentric ellipse changes sign p r garabedian showed in 1951 that the green function for the biharmonic boundary value problem with vanishing dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse this refuted a conjecture made by j hadamard in 1908 the proof of garabedian was based on kernel functions the present note gives an elementary proof

computing the maslov index of solitary waves part 2 phase space with dimension greater than four this paper extends the theory of the maslov index of solitary waves in part 1 to the case where the phase space is of dimension greater than four the starting point is hamiltonian pdes in one space dimension and time whose steady part is a hamiltonian ode with a phase space of dimension six or greater this steady hamiltonian ode is the main focus of the paper homoclinic orbits of the steady ode represent solitary waves of the pde and one of the properties of the homoclinic orbits is the maslov index we develop formulae for the maslov index the maslov angle and its subangles in an exterior algebra framework and develop numerical algorithms to compute them in addition a new numerical approach based on a discrete qr algorithm is proposed the maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time dependent problem the theory is applied to a class of reaction diffusion equations the longwave shortwave resonance equations and the seventh order kdv equation

sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the cauchy problem for the anisotropic plate equation abstract we prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation among these we prove a stability estimate for the cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate the paper is self contained and the carleman estimate from which the sharp three sphere inequality is derived is proved in an elementary and direct way based on standard integration by parts

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

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

conformal limits and the bialynicki birula stratification of the space of λ connections abstract the bialynicki birula decomposition of the space of connections restricts to the morse stratification on the moduli space of higgs bundles and to the partial oper stratification on the de rham moduli space of holomorphic connections for both the morse and partial oper stratifications every stratum is a holomorphic lagrangian fibration over a component of the space of complex variations of hodge structure in this paper we generalize known results for the hitchin section and the space of opers to arbitrary strata these include the following a biholomorphic identification of the fibers of the two strata over a stable variation of hodge structure via the conformal limit of gaiotto a proof that the fibers of the morse and partial oper stratifications are transverse at the base point and an explicit parametrization of the fibers as half dimensional affine spaces

computing lyapunov exponents on a stiefel manifold the problem of numerical computation of a few lyapunov exponents les of finite dimensional dynamical systems is considered from the viewpoint of the differential geometry of stiefel manifolds whether one computes one many or all les of a continuous dynamical system by time integration discrete or continuous orthonormalization is essential for stable numerical integration a differential geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a stiefel manifold however the stiefel manifold is not in general an attracting submanifold of the ambient euclidean space it is a constraint manifold with a weak numerical invariant new numerical algorithms for this problem are then designed which use the fiber bundle characterization of these manifolds this framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew symmetry property numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few les which preserve orthonormality to machine accuracy this idea is also taken a step further by making the stiefel manifold an attracting invariant manifold in which case standard explicit runge kutta algorithms can be used this leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization these class of methods should be particularly effective for computing a few les for large dimension dynamical systems the new schemes are straightforward to implement a test case is presented for illustration and an example from dynamical systems is presented where a few les are computed for an array of coupled oscillators

a reliable algorithm for computing the topological degree of a mapping in r2 abstract in this paper we present a method to reliably compute the topological degree of a mapping in the plane over a simple closed polygon the method is based on henrici s argument principle and computes the degree using the winding number concept in range arithmetic the algorithm is then applied to the root computation of a univariate polynomial the proposed algorithms are demonstrated with examples

long time existence for yang mills flow we establish that finite time singularities do not occur in four dimensional yang mills flow confirming the conjecture of schlatter et al am j math 120 117 128 1998 the proof relies on a weighted energy identity and sharp decay estimates in the neck region

global stability for an inverse problem in soil structure interaction we consider the inverse problem of determining the winkler subgrade reaction coefficient of a slab foundation modelled as a thin elastic plate clamped at the boundary the plate is loaded by a concentrated force and its transversal deflection is measured at the interior points we prove a global holder stability estimate under mild regularity assumptions on the unknown coefficient

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

on the compuation of lyapunov exponents for continuous dynamical systems in this paper we consider discrete and continuous qr algorithms for computing all of the lyapunov exponents of a regular dynamical system we begin by reviewing theoretical results for regular systems and present general perturbation results for lyapunov exponents we then present the algorithms give an error analysis of them and describe their implementation finally we give several numerical examples and some conclusions

yang mills fields on cylindrical manifolds and holomorphic bundles i we give complex holomorphic descriptions of yang mills instantons on tubular four manifolds with nontrivial circle bundles over riemann surfaces as section

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

energy of sections of the deligne hitchin twistor space we study a natural functional on the space of holomorphic sections of the deligne hitchin moduli space of a compact riemann surface generalizing the energy of equivariant harmonic maps corresponding to twistor lines we show that the energy is the residue of the pull back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by hitchin as a byproduct we show the existence of a hyper kahler potentials for new components of real holomorphic sections of twistor spaces of hyper kahler manifolds with rotating s 1 action additionally we prove that for a certain class of real holomorphic sections of the deligne hitchin moduli space the energy functional is basically given by the willmore energy of corresponding equivariant conformal map to the 3 sphere as an application we use the functional to distinguish new components of real holomorphic sections of the deligne hitchin moduli space from the space of twistor lines

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

numerical implementation of complex orthogonalization parallel transport on stiefel bundles and analyticity abstract numerical integration of complex linear systems of odes depending analytically on an eigenvalue parameter are considered complex orthogonalization which is required to stabilize the numerical integration results in non analytic systems it is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non analytic characteristic function the orthonormal systems are constructed using the geometry of stiefel bundles different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one form on the stiefel bundle for the numerical integration gauss legendre runge kutta algorithms are the principal choice for preserving orthogonality and performance results are shown for a range of glrk methods the theory and methods are tested by application to example boundary value problems including the orr sommerfeld equation in hydrodynamic stability

a class of intrinsic schemes for orthogonal integration numerical integration of odes on the orthogonal stiefel manifold is considered points on this manifold are represented as n k matrices with orthonormal columns of particular interest is the case when n gg k mainly two requirements are imposed on the integration schemes first they should have arithmetic complexity of order nk2 second they should be intrinsic in the sense that they require only the ode vector field to be defined on the stiefel manifold as opposed to for instance projection methods the design of the methods makes use of retractions maps two algorithms are proposed one where the retraction map is based on the qr decomposition of a matrix and one where it is based on the polar decomposition numerical experiments show that the new methods are superior to standard lie group methods with respect to arithmetic complexity and may be more reliable than projection methods owing to their intrinsic nature

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