conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

inversion formulas and fast algorithms for lowner vandermonde matrices abstract matrices consisting of two parts one of vandermonde and the other of lowner type are considered inversion formulas also for one sided inverses are presented moreover in the nonsingular case the parameters of the inversion formulas are described in terms of the evaluation of numerator and denominator polynomials which constitute the solution of associated rational interpolation problems these results can be used to establish an algorithm for the solution of lowner vandermonde systems of equations with o n log 2 n complexity

the hamiltonian spectrum of directed percolation the authors calculate the spectrum of a quantum hamiltonian which is in the universality class of the reggeon field theory and directed percolation they find that its low lying states follow a self similarity close to the one encountered in the exactly solvable transverse xy model the spectrum can in the finite size scaling limit approximately be described by excitations with a dispersion relation ktheta on the lattice instead of the k2 behaviour of the transverse xy model

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

singularities of hermitian yang mills connections and harder narasimhan seshadri filtrations this is the first of a series of papers where we relate tangent cones of hermitian yang mills connections at a singularity to the complex algebraic geometry of the underlying reflexive sheaf in this paper we work on the case when the sheaf is locally modeled on the pullback of a holomorphic vector bundle from the projective space and we shall impose an extra assumption that the graded sheaf determined by the harder narasimhan seshadri filtrations of the vector bundle is reflexive in general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a harder narasimhan seshadri filtration of an algebraic tangent cone which is a certain torsion free sheaf on the projective space in this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable

a singularity removal theorem for yang mills fields in higher dimensions the purpose of this paper is to investigate the small energy behavior of weakly yang mills fields in rn for n 4 and in particular to extend the singularity removal theorem of uhlenbeck 7 to higher dimensions fix n 4 and let ft be some bounded domain in rn typically we shall restrict our attention to the cubes ft 1 2 n or ft 0 l n let g be a fixed finite dimensional compact lie group it will be convenient to consider g as embedded in some large unitary group u n let g be the lie algebra of g we define a connection on ft to be a section a of t ft q i e a g valued 1 form which is locally l2 for any connection a let 1 f a da aaa

superintegrable chiral potts model thermodynamic properties an inverse model and a simple associated hamiltonian the partition function of then state superintegrable chiral potts model is obtained exactly and explicitly if not completely rigorously for a finite lattice with particular boundary conditions this yields the bulk and surface free energies and horizontal and vertical correlation lengths and interfacial tensions the critical exponents are 1 2 n hor hor 2 n and vert vert 1 and the finite size corrections are obtained at criticality the eigenvalue spectrum of the column to column transfer matrix is that of a direct product ofn byn matrices inverting this matrix gives a related solvable model which is a generalization of the free fermion model the associated hamiltobian has a very simple form suggesting there may be a more direct algebraic method perhaps a generalized clifford algebra for obtaining its eigenvalues

a fast hankel solver based on an inversion formula for loewner matrices abstract we propose a new o n2 algorithm for solving complex n n linear systems that have hankel structure via ffts the hankel system is transformed into a loewner system an inversion formula enables us to calculate the inverse of the loewner matrix explicitely the parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle we present an o n2 algorithm to solve these interpolation problems one of the advantages of this algorithm is that it incorporates pivoting we have implemented our hankel solver in fortran 90 numerical examples are included they show the effectiveness of our pivoting strategy

look ahead levinson and schur algorithms for non hermitian toeplitz systems we present generalizations of the nonsymmetric levinson and schur algorithms for non hermitian toeplitz matrices with some singular or ill conditioned leading principal submatrices the underlying recurrences allow us to go from any pair of successive well conditioned leading principal submatrices to any such pair of larger order if the look ahead step size between these pairs is bounded our generalized levinson and schur recurrences require o n 2 operations and the schur recurrences can be combined with recursive doubling so that an o n log 2 n algorithm results the overhead in operations and storage of look ahead steps is very small there are various options for applying these algorithms to solving linear systems with toeplitz matrix

minimal vector pade approximation abstract we describe the minimal vector pade approximation problem which consists in finding pade approximants with a common denominator for a number of series these approximants are minimal in the sense that for a given order of approximation and a given discrepancy in numerator and denominator degrees the degree of the rational approximant is minimal properties and solution methods are derived from an associated partial realization problem

inverses of lowner matrices lowner matrices are special matrices which were first used by lowner in 4 to characterize the so called monotone matrix functions lowner showed also their connection with the problem of rational interpolation for these questions see also donoghue 2 belevitch 1 and fiedler 3 lowner matrices also occur in applications for example in the theory of electric networks in this paper we show that for a nonsingular lowner matrix l ci djyi zjn 1i j 0 the matrix l 1 t can be written in the form l 1 t d1l d2 where d1 and d2 are diagonal matrices and l is again a lowner matrix of the form yi jyi zjn 1i j 0 for a more detailed description of all forms l 1 d1l d2 possible we introduce the notion of d classes of the lowner matrices

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

a look ahead algorithm for the solution of general hankel systems the solution of systems of linear equations with hankel coefficient matrices can be computed with onlyo n 2 arithmetic operations as compared too n 3 operations for the general cases however the classical hankel solvers require the nonsingularity of all leading principal submatrices of the hankel matrix the known extensions of these algorithms to general hankel systems can handle only exactly singular submatrices but not ill conditioned ones and hence they are numerically unstable in this paper a stable procedure for solving general nonsingular hankel systems is presented using a look ahead technique to skip over singular or ill conditioned submatrices the proposed approach is based on a look ahead variant of the nonsymmetric lanczos process that was recently developed by freund gutknecht and nachtigal we first derive a somewhat more general formulation of this look ahead lanczos algorithm in terms of formally orthogonal polynomials which then yields the look ahead hankel solver as a special case we prove some general properties of the resulting look ahead algorithm for formally orthogonal polynomials these results are then utilized in the implementation of the hankel solver we report some numerical experiments for hankel systems with ill conditioned submatrices

rational interpolation and state variable realizations abstract the problem is considered of passing from interpolation data for a real rational transfer function matrix to a minimal state variable realization of the transfer function matrix the tool is a loewner matrix which is a generalization of the standard hankel matrix of linear system realization theory and which possesses a decomposition into a product of generalized observability and controllability matrices

finite size scaling for quantum chains with an oscillatory energy gap the authors show that the existence of zeros of the energy gap for finite quantum chains is related to a non vanishing wavevector finite scaling ansatze are formulated for incommensurable and oscillatory structures the ansatze are verified in the one dimensional xy model in a transverse field

compactification of the moduli spaces of vortices and coupled vortices vortices and coupled vortices arise from yang mills higgs theories and can be viewed as generalizations or analogues to yang mills connections and in particular hermitianyang mills connections we proved an analytic compactification of the moduli spaces

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

inversion of a block lo wner matrix in this paper we give a fast algorithm to compute the parameters of an inversion formula for any nonsingular block lowner matrix the connection with matrix rational interpolation is given

quasidirect decompositions of hankel and toeplitz matrices decompositions over an algebraically closed field of a hankel matrix into a sum of hankel matrices the sum of the ranks of which is equal to the rank of the original matrix are completely described similar results hold for toeplitz matrices

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

infinite range mean field percolation transfer matrix study of longitudinal correlation length two infinite range directed percolation models equivalent also to epidemic models are considered for a finite population finite number of sites n at each time directed axis step the general features of the transfer matrix spectrum evolution operator spectrum for the epidemic are studied numerically and compared with analytical predictions in the limitn one of the models is devised to allow numerical results to be obtained forn as high as nearly 800 for the largest longitudinal percolation correlation length relaxation time for epidemic the finite n behavior of this correlation length is studied in detail including scaling near the percolation transition exponential divergence withn above the percolation transition as well as other noncritical and critical point properties

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

the yang mills flow and the atiyah bott formula on compact kahler manifolds we study the yang mills flow on a holomorphic vector bundle e over a compact k ahler manifold x along a solution of the flow we show that the curvature endomorphism i lambda f a t approaches in l 2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the harder narasimhan filtration of e this proves a sharp lower bound for the hermitian yang mills functional and thus the yang mills functional generalizing to arbitrary dimension a formula of atiyah and bott first proven on riemann surfaces furthermore we show any reflexive extension to all of x of the limiting bundle e infty is isomorphic to rm gr rm hns e verifying a conjecture of bando and siu

transfer matrix study of the chiral clock model in the hamiltonian limit the logarithm of the transfer matrix of the two dimensional three state chiral clock model is shown to be equivalent to a one dimensional non hermitian quantum hamiltonian in the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite size scaling a single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase the critical exponents nu x nu tau for the correlation lengths obtained from finite size scaling point towards the existence of a lifshitz point at finite chirality the results of the series expansion for the exponent nu x however are inconsistent with finite size scaling for larger chirality the authors finite size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase they analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations

series and monte carlo studies of 2 and 3 dimensions for axial hyperscaling in directed percolation abstract the relationship between the exponent 0 characterizing the divergence at criticality of the cluster mass measured on the anisotropic axis of directed percolation and the susceptibility exponent while well established in two dimensions both by series and simulation work has been doubtful for a decade for three and higher dimensions in the present paper we furnish extensive simulations for the body centered cubic site problem that focus on that quantity and study further the ascending moments of the number of sites on the axis these numerical results are highly compatible with the scaling relation 0 d where d 2 is the transverse dimensionality they also indicate the existence of a fixed gap exponent for the size moments of this quantity extensive series expansions in two dimensions are in acceptable agreement with the same picture with d 1 above while direct determination of 0 is still not satisfactory for some of the highly connected three dimensional lattices we have adopted the existence of a fixed gap exponent is confirmed in fairly reasonable agreement with the concurrent evidence of the bcc monte carlo studies

a vector bundle version of the monge ampere equation abstract we introduce a vector bundle version of the complex monge ampere equation motivated by a desire to study stability conditions involving higher chern forms we then restrict ourselves to complex surfaces provide a moment map interpretation of it and define a positivity condition ma positivity which is necessary for the infinite dimensional symplectic form to be kahler on rank 2 bundles on compact complex surfaces we prove two consequences of the existence of a positively curved solution to this equation stability involving the second chern character and a kobayashi lubke bogomolov miyaoka yau type inequality finally we prove a kobayashi hitchin correspondence for a dimensional reduction of the aforementioned equation

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

bott chern currents and complex immersions in the whole paper we will say that i m m is an immersion of smooth manifolds if m is a submanifold of m and if i is the corresponding injection map in particular the topology of m is the topology induced by the topology of m in differential geometry such maps i m m are also called embeddings

a canonical matrix continued fraction solution of the minimal partial realization problem abstract a canonical algorithm for minimal partial realization of mimo systems is given the paper starts from the work of dickinson et al and anderson et al but reformulates it in the style of antoulas so that the elements to construct a matrix continued fraction are obtained it gives a detailed explicit algorithm to compute a uniquely defined canonical form of this continued fraction a number of remarkable properties of this canonical form are proved an example is included

analytic cycles bott chern forms and singular sets for the yang mills flow on kahler manifolds abstract it is shown that the singular set for the yang mills flow on unstable holomorphic vector bundles over compact kahler manifolds is completely determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle we assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves we derive a singular bott chern formula relating the second chern form of a smooth metric on the bundle to the chern current of an admissible metric on the associated graded sheaf this is used to show that the multiplicities of the top dimensional bubbling locus defined via the yang mills density agree with the corresponding multiplicities for the harder narasimhan seshadri filtration the set theoretic equality of singular sets is a consequence

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

self affine fractal clusters conceptual questions and numerical results for directed percolation in this paper we address the question of the existence of a well defined non trivial fractal dimensiond of self affine clusters in spite of the obvious relevance of such clusters to a wide range of phenomena this problem is still open since thedifferent published predictions ford have not been tested yet an interesting aspect of the problem is that a nontrivial global dimension for clusters is in contrast with the trivial global dimension of self affine functions as a much studied example of self affine structures we investigate the infinite directed percolation cluster at the threshold we measuredd ind 2 dimensions by the box counting method using a correction to scaling analysis we obtainedd 1 765 10 this result does not agree with any of the proposed relations but it favorsd 1 1 where and are the correlation length exponents and is a fisher exponent in the cluster scaling

on the blow up set of the yang mills flow on kahler surfaces the yang mills flow on a kahler surface with holomorphic initial data converges smoothly away from a singular set determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle

rational interpolation and recursive solution of lowner vandermonde systems of equations abstract rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure the first columns are of vandermonde type whereas the last columns form a lowner matrix here three term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix on this base an o n2 algorithm for the solution of n n nonhomogeneous lowner vandermonde systems is obtained

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

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

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