decidable sentences of church rosser congruences abstract a decision procedure for a class of true sentences of congruences generated by finite monadic church rosser systems is developed using this decision procedure it is shown that if m t is the monoid presented by such a system t then i it is decidable given t whether m t is a group ii it is decidable given t and a finite set a whether the submonoid generated by a is a group or a left right two sided ideal and iii green s relations for m t are decidable

classical solutions of the chiral model unitons and holomorphic vector bundles this paper deals with classical solutions of thesu 2 chiral model on 2 and of a generalized chiral model on 2 1 such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space with an appropriate boundary condition the solutions called 1 unitons in 9 correspond to bundles over a compact 2 dimensional complex manifold and the problem becomes one of algebraic geometry

a note on the index bundle over the moduli space of monopoles donaldson has shown that the moduli space of monopolesmk is diffeomorphic to the spaceratk of based rational maps from the two sphere to itself we use this diffeomorphism to give an explicit description of the bundle onratk obtained by pushing out the index bundle frommk this gives an alternative and more explicit proof of some earlier results of cohen and jones

on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

kac moody monopoles and periodic instantons it is shown that calorons or periodic instantons are the same as monopoles with the loop group as their structure group their twistor correspondences and spectral data are defined the spectral data is shown to determine the general caloron

bezout hankel and loewner matrices abstract relations between the classes of bezout hankel and loewner matrices and of their inverses are investigated all these classes appear as matrix realizations of one operator in suitable bases different bases in the space of all polynomials of degree not exceeding n 1 are considered these bases correspond to different expansions of the polynomial w z p x p z w x z x we introduce a matrix k corresponding to this polynomial this matrix together with its limit forms may be used to represent all the classes of matrices mentioned in particular we obtain a new interpretation of the variant of generalized loewner matrices introduced recently by the second named author

calorons nahm s equations on s1 and bundles over mathbb p 1 times mathbb p 1 the moduli space of solutions to nahm s equations of rank k k j on the circle and hence of su 2 calorons of charge k j is shown to be equivalent to the moduli of holomorphic rank 2 bundles on mathbb p 1 times mathbb p 1 trivialized at infinity infty times mathbb p 1 cup mathbb p 1 times infty with c 2 k and equipped with a flag of degree j along mathbb p 1 times 0 an explicit matrix description of these spaces is given by a monad construction

monopoles and rational maps a note on a theorem of donaldson in a recent paper donaldson gave a description of the moduli space of su 2 monopoles in terms of rational maps this was done indirectly via the associated solution of nahm s equations we give here an interpretation of these rational maps in terms of the monopole s spectral curve and then as scattering data for the monopole itself

algebraic cycles and infinite loop spaces in this paper we use recent results about the topology of chow varieties to answer an open question in infinite loop space theory that is we construct an infinite loop space structure on a certain product of eilenberg maclane spaces so that the total chern map is an infinite loop map an analogous result for the total stiefel whitney map is also proved further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined

when is a monoid a group the church rosser case is tractable abstract the following question is shown to be tractable given a finite monadic church rosser thue system t and a finite set a is the submonoid generated by a of the monoid presented by t a group

nahm s transformation for instantons we describe in mathematical detail the nahm transformation which maps anti self dual connections on the four torus s1 4 onto anti self dual connections on the dual torus this transformation induces a map between the relevant instanton moduli spaces and we show that this map is a hyperkahler isometry

cyclic monopoles affine toda and spectral curves we show that any cyclically symmetric monopole is gauge equivalent to nahm data given by sutcliffe s ansatz and so obtained from the affine toda equations further the direction the ercolani sinha vector and base point of the linearising flow in the jacobian of the spectral curve associated to the nahm equations arise as pull backs of toda data a theorem of accola and fay then means that the theta functions arising in the solution of the monopole problem reduce to the theta functions of toda

monopoles on asymptotically euclidean 3 manifolds on considere une generalisation de la theorie de yang mills higgs sur r 3 euclidien a des sommes convexes de r 3 avec des 3 varietes closes compactes

stratifying monopoles and rational maps a stratification of the moduli space of monopoles and of the space of rational maps into a flag variety is presented it is shown that the map associating a rational map to a monopole preserves these strata these strata explain some problems in the intepretation of the parameters of the moduli space in terms of superpositions of fundamental monopoles this interpretation is not valid on the individual strata the space of fundamental monopoles is described and shown to be the same as the corresponding space of rational maps

min max theory for the yang mills higgs equations in each monopole sector there exist an infinite number of finite energy solutions to the prasad sommerfield limit of the su 2 yang mills higgs equations on 3 whose energy is greater than any finite number

monadic thue systems abstract certain infinite thue systems over a finite alphabet are studied in particular systems s e such that for each a e the set u u a s is a context freelanguage the syntactic structure of sets of ancestors and sets of descendants is considered as well as that of unions of congruence classes taken over infinite context free languages or regular sets the common descendant problem is shown to be tractable while the common ancestor problem is shown to be undecidable even for finite systems the word problem for confluent systems of this type is shown to be tractable the question of whether an infinite system of this type is confluent is shown to be undecidable as is the question of whether the congruence generated by such a system has a confluent presentation

tensored polynomial models the theme of the present paper is the study of two different versions of tensor products of functional models and present some applications to various problems related to system theory among those are stability of higher order systems tangent spaces spaces of intertwining maps invariant factors of tensor products of linear transformations and the solvability of sylvester equations

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

the classification of monopoles for the classical groups by studying a construction of nahm we compute the moduli spaces of monopoles with maximal symmetry breaking at infinity forsu n so n andsp n these are found to be equivalent to spaces of holomorphic maps from 1 into flag manifolds

instantons and geometric invariant theory we show that the yang mills instantons can be described in terms of certain holomorphic bundles on the projective plane the proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries

instantons in two and four dimensions it is shown that yang mills instantons in four dimensions can naturally be identified with the instantons of a two dimensional theory with values in the loop group

polynomial bezoutian matrix with respect to a general basis we introduce a class of so called polynomial bezoutian matrices the operator representation relative to a pair of dual bases and the generalized barnett factorization formula for this kind of matrix are derived an intertwining relation and generalized bezoutian reduction via polynomial vandermonde matrix are presented

on bezoutians van der monde matrices and the lienard chipart stability criterion we show how the bezoutian can be reduced by congruence to diagonal form with the congruence given by a van der monde matrix this result is applied to obtain new proofs of some classical stability criteria and in particular of the lienardchipart theorem

some polynomial time algorithms for finite monadic church rosser thue systems abstract based on a careful analysis of reduction sequences in monadic thue systems we show that some uniform decision problems among them the uniform conjugacy problem are decidable in polynomial time for finite monadic church rosser thue systems on the other hand an example of a decision problem is exhibited that is undecidable even for this class of thue systems

on the construction of monopoles for the classical groups forg a classical group an equivalence is exhibited between a g monopoles over 3 with maximal symmetry breaking at infinity b families of rank g algebraic curves int 1 along with divisors on those curves satisfying certain constraints c solutions of nahm s equations over rank g intervals satisfying the appropriate boundary conditions a and b are linked by twistor techniques b and c via the krichever method for solving non linear differential equations and a and c via the adhmn construction providing a unified picture of techniques for solution amongst other things an asymptotic formula for the higgs field of the monopole is computed

on the complete integrability of the discrete nahm equations the discrete nahm equations a system of matrix valued difference equations arose in the work of braam and austin on half integral mass hyperbolic monopoles

monopoles and spectral curves for arbitrary lie groups the definition of the spectral curve of a monopole is extended to any connected compact simple lie groupk it is found there are rankk curves whose degrees are related to the topological weights of the monopole

on the construction of monopoles we show that any self dual su 2 monopole may be constructed either by ward s twistor method or nahm s use of the adhm construction the common factor in both approaches is an algebraic curve whose jacobian is used to linearize the non linear ordinary differential equations which arise in nahm s method we derive the non singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions

bezoutian representation via vandermonde matrices abstract the result on benoutian representation via vandermonde matrices is provided with a proof which is based on polynomial models

periodic instantons and the loop group we construct a large class of periodic instantons conjecturally we produce all periodic instantons this confirms a conjecture of garland and murray that relates periodic instantons to orbits of the loop group acting on an extension of its lie algebra

symmetric calorons and the rotation map we study su 2 calorons also known as periodic instantons and consider invariance under isometries of s1 r3 coupled with a non spatial isometry called the rotation map in particular we investigate the fixed points under various cyclic symmetry groups our approach utilises a construction akin to the adhm construction of instantons what we call the monad matrix data for calorons derived from the work of charbonneau and hurtubise to conclude we present an example of how investigating these symmetry groups can help to construct new calorons by deriving nahm data in the case of charge 2

on matrix hurwitz type polynomials and their interrelations to stieltjes positive definite sequences and orthogonal matrix polynomials abstract this paper is a direct continuation of the author s recent investigations 4 on the non degenerate truncated matricial stieltjes problem inspired by a characterization of scalar hurwitz polynomials via continued fractions see e g gantmacher 15 a class of matrix hurwitz type polynomials is introduced it is shown that this class is intimately related with stieltjes positive definite sequences of matrices and can be expressed in terms of the associated stieltjes quadruple of sequences of left orthogonal matrix polynomials it is shown that a monic matrix polynomial is a matrix hurwitz type polynomial if and only if the sequence of its markov parameters is a stieltjes positive definite sequence

a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type in this note a generalized bezoutian matrix with respect to a polynomial sequence of interpolatory type is introduced the operator representation relative to a pair of dual bases and the generalized barnett type factorization formula are derived an intertwining relation and a bezoutian reduction to a block diagonal form by congruence via a generalized vandermonde matrix are presented fujiwara hermite and routh hurwitz criteria in terms of this generalized bezout matrix are obtained

spectral curves and the mass of hyperbolic monopoles the moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass and this leads to a nontrivial mass dependence of the holomorphic data spectral curves rational maps holomorphic spheres associated to hyperbolic multi monopoles in this paper we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two in addition we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries spectral curves of euclidean monopoles are recovered from our results via an infinite mass limit

monopoles braid groups and the dirac operator using the relation between the space of rational functions on the space ofsu 2 monopoles on 3 and the classifying space of the braid group see 10 we show how the index bundle of the family of real dirac operators coupled tosu 2 monopoles can be described using permutation representations of artin s braid groups we also show how this implies the existence of a pair consisting of a gauge fielda and a higgs field on 3 whose corresponding dirac equation has an arbitrarily large dimensional space of solutions

generalized polynomial bezoutian with respect to a jacobson chain basis over an arbitrary field abstract we introduce a so called generalized polynomial bezoutian with respect to a jacobson chain basis over an arbitrary field some characterization of this kind of matrix such as the barnett type factorization and the intertwining relation with generalized hypercompanion matrix are obtained the diagonal reduction formula via the generalized confluent vandermonde matrix similar to that of classical bezoutian is presented the method used is based on polynomial module and operator representation

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

platonic hyperbolic monopoles we construct a number of explicit examples of hyperbolic monopoles with various charges and often with some platonic symmetry the fields are obtained from instanton data in mathbb r 4 that are invariant under a circle action and in most cases the monopole charge is equal to the instanton charge a key ingredient is the identification of a new set of constraints on adhm instanton data that are sufficient to ensure the circle invariance unlike for euclidean monopoles the formulae for the squared higgs field magnitude in the examples we construct are rational functions of the coordinates using these formulae we compute and illustrate the energy density of the monopoles we also prove for particular monopoles that the number of zeros of the higgs field is greater than the monopole charge confirming numerical results established earlier for euclidean monopoles we also present some one parameter families of monopoles analogous to known scattering events for euclidean monopoles within the geodesic approximation

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

elements of finite order for finite monadic church rosser thue systems a thue system t over z is said to allow nontrivial elements of finite order if there exist a word u e 2 and integers n 0 and k 1 such that u 9 tx

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

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

toda flows inverse spectral transform and realization theory abstract in this paper we establish some new relationships between the solution techniques for completely integrable nonlinear differential equations of the toda type and topics in realization theory

the geodesic approximation for the yang mills higgs equations in this paper we consider the dynamics of the monopole solutions of yang mills higgs theory on minkowski space the monopoles are solutions of the yang mills higgs equations on three dimensional euclidean space it is of interest to understand how they evolve in time when considered as solutions of the yang mills higgs equations on minkowski space i e the time dependent equations it was suggested by manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain metric on the space of guage equivalence classes of monopoles the moduli space the metric is defined by taking thel 2 inner product of tangent vectors to this space in this paper we will prove that manton s approximation is indeed valid in the right circumstances which correspond to the slow motion of monopoles the metric on the moduli space of monopoles was analysed in a book by atiyah and hitchin so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in yang mills higgs theory is obtained the strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions and then to use energy estimates to prove their validity for the case of monopoles to be considered here there is a technical obstacle to be overcome when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin this is overcome by using a norm introduced by taubes in a discussion of index theory for the yang mills higgs functional

nahm s equations and the classification of monopoles solutions of nahm s system of ordinary differential equations are produced by variational methods this leads to an explicit parametrisation of the solutions to the bogomolny equation over 3

closed geodesics on the space of stable two monopoles we show that the atiyah hitchin metric on the space of stable two monopoles admits closed geodesics