on the maslov index in this paper we give four definitions of maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other moreover relationships of several symplectic and differential geometric analytic and topological invariants including triple maslov indices eta invariants spectral flow and signatures of quadratic forms to the maslov index are developed and formulae relating them are given the broad presentation is designed with a view to applications both in geometry and in analysis 1994 john wiley sons inc

robust pole clustering of structured uncertain systems in union of disjointed circular regions abstract this paper presents a systematic procedure for depicting the robust pole clustering of a system with structured uncertainties different from many previous researches on connected region a union of disjointed circular regions on the complex plane containing either a single nominal pole or multi nominal poles is obtained utilizing matrix operation in addition a criterion for determining the allowable perturbation bounds of the structured uncertain system such that the pole clusters remain within certain pre specified circular regions is also proposed the validity of the proposed approach is confirmed by way of numerical examples

on a spectral flow formula for the homological index abstract consider a selfadjoint unbounded operator d on a hilbert space h and a one parameter norm continuous family of selfadjoint bounded operators a t t r that converges in norm to asymptotes a at then under certain conditions 22 that include the assumption that the operators d t d a t t r all have discrete spectrum the spectral flow along the path d t can be shown to be equal to the index of t d t when the latter is an unbounded fredholm operator on l 2 r h in 16 an investigation of the index spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general in 11 the question of what happens when the fredholm condition is dropped altogether was investigated in these circumstances the fredholm index is replaced by the witten index in this paper we take the investigation begun in 11 much further we show how to generalize a formula known from the setting of the l 2 index theorem to the non fredholm setting restricting back to the case of selfadjoint fredholm operators our formula extends the result of 22 in the sense of relaxing the discrete spectrum condition it also generalizes some other fredholm operator results of 21 16 11 that permit essential spectrum for the operators in the path our result may also apply however when the operators d t have essential spectrum equal to the whole real line our main theorem gives a trace formula relating the homological index of 7 to an integral formula that is known for a path of selfadjoint fredholms with compact resolvent and with unitarily equivalent endpoints to compute spectral flow our formula however applies to paths of selfadjoint non fredholm operators we interpret this as indicating there is a generalization of spectral flow to the non fredholm setting

exact results for a correlated percolation model abstract the equivalence between a correlated percolation model and the syozi model at t 0 has been used to calculate the exact percocation probabilities p as a function of bare p 0 and renormalized p occupation probability for such a percolation process on two dimensional lattices we also find that for both two and three dimensional lattices p p 0 p 0 c near critical point p 0 c where is the magnetic exponent of the corresponding spin model

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

the stable topology of self dual moduli spaces soit m une 4 variete de riemann compacte connexe orientee et soit g un groupe de lie connexe simple on fixe un entier q m q alors l application inclusion i induit un epimorphisme sur les groupes respectifs d homologie et d homotopie a toutes les dimensions inferieures ou egales a q

the topology of instanton moduli spaces i the atiyah jones conjecture in this paper we study the global geometry and topology of the moduli spaces of based su 2 instantons over the 4 sphere s4 these instanton moduli spaces have a rich history and have been analyzed from many points of view originally these spaces which we denote by mk were defined as solution spaces modulo gauge equivalence to certain partial differential equations namely the self duality equations associated to the yang mills functional in su 2 gauge theory they have been successfully studied from this point of view by taubes t1 t2 uhlenbeck u and others an important alternative approach was initiated by ward w who related instantons to certain holomorphic bundles on cp3 and was continued by atiyah and ward in aw this allowed the classification of instantons on 4 in terms of quaternionic linear algebra by atiyah drinfeld hitchin and manin adhm this holomorphic approach was further extended by donaldson d who showed that these bundles were determined by their restriction to a cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in c2r2 hurtubise hul then exploited this fact to study the moduli spaces mk as did atiyah a to show that mk arise naturally in the theory of holomorphic maps into loop groups this latter approach was continued by gravesen g atiyah and jones aj obtained the first results and formulated the foundational questions on the global topology of these moduli spaces recall that an element of mk is a based gauge equivalence class of a connection on the principal su 2 bundle over 4 denoted by pk with second chern class k satisfying the self duality equations there is a natural forgetful map to the based equivalence classes of all connections in pk atiyah and jones aj showed that this target space which we denote by bk is homotopy equivalent

correlated percolation and a sublattice dilute potts model at t 0 abstract it is shown that a sublattice dilute q state potts model at t 0 is equivalent to a correlated percolation which favours subgraphs with larger numbers of disconnected finite clusters the syozi model at t is equivalent to the above model with q 2 in the same limit and thus corresponds to the same correlated percolation a table is listed which reflects the influence of correlation on critical probabilities for such correlated percolation processes on some lattices

dirac operators coupled to instantons on positive definite four manifolds we study the moduli space of instantons on a simply connected positive definite closed four manifold by analyzing the classifying map of the index bundle of a family of dirac operators parameterized by the moduli space as an application we compute the cohomology rings for the charge 1 and 2 moduli spaces in the rank stable limit

moduli spaces of higgs bundles on degenerating riemann surfaces abstract we prove a gluing theorem for solutions a 0 0 of hitchin s self duality equations with logarithmic singularities on a rank 2 vector bundle over a noded riemann surface 0 representing a boundary point of teichmuller moduli space we show that every nearby smooth riemann surface 1 carries a smooth solution a 1 1 of the self duality equations which may be viewed as a desingularization of a 0 0

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

energy gap for yang mills connections i four dimensional closed riemannian manifolds we extend an l2l2 energy gap result due to min oo 28 theorem 2 and parker 30 proposition 2 2 for yang mills connections on principal g bundles p over closed connected four dimensional oriented smooth manifolds x from the case of positive riemannian metrics to the more general case of good riemannian metrics including metrics that are generic and where the topologies of p and x obey certain mild conditions and the compact lie group g is su 2 su 2 or so 3 so 3

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

self adjoint elliptic operators and manifold decompositions part iii determinant line bundles and lagrangian intersection the theory of spectral flows developed in the series 10 11 12 and the present paper has a wide range of applications to important geometric operators on compact manifolds to present our results on spectral flow and manifold decomposition the present paper develops a theory of determinant line bundles and infinitedimensional lagrangians associated to self adjoint elliptic operators on compact manifolds the trace class properties of these infinite lagrangians established here and the precise uniform estimates relating them to finite lagrangians are crucial for such a determinant line bundle approach to analytical questions as an application we elucidate the walker s and other generalizations of casson s su 2 representation theoretic invariant of 3 manifolds in terms of the invariant of certain dirac operators this is carried out by introducing the technique of canonical perturbations of singular lagrangian subvarieties in symplectic geometry at the end of part ii of this series we obtained a formulation of the spectral flow of a family of self adjoint elliptic operators d u l2 e l2 e in terms

a framework for morse theory for the yang mills functional on explique comment la theorie de morse pour la fonctionnelle de yang mills peut se retrouver en examinant la restriction de la fonctionnelle a un ensemble denombrable de varietes non compactes de dimension finie

dilute potts model duality and site bond percolation a number of results are obtained for potts models on an arbitrary planar lattice it is shown that the dual of a dilute potts model is a graph generating function and that the dual of a constrained dilute potts model is an undiluted potts model it is also shown that the dilute potts model generates a correlated site bond percolation for uncorrelated site bond percolations the analysis determines the percolation threshold from a knowledge of the critical point of a potts model this generalises a recent result of kondor 1980 who uses a star triangle transformation to derive this relationship for the honeycomb and triangle lattices

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

robust matrix root clustering analysis through extended kyp lemma this paper is dedicated to robust matrix eigenvalue clustering in a subregion mathcal d of the complex plane a norm bounded uncertainty is considered cd is any intersection and or union of half planes discs and outsides of discs this work aims at computing the associated complex cd stability radius this challenge was already tackled in a recent paper but the technique therein may fail or lead to conservative results when the subregions overlap the present paper proposes to circumvent those limits thanks to an extended version of the kalman yakubovich popov kyp lemma which is deduced from the so called generalized s procedure

twistor spaces and the general adiabatic expansions abstract we investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of dirac operator on a twistor space when the metric is blown up in the base space direction such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants most of whose cores consist of some derivatives of the fundamental solution

burghelea friedlander kappeler s gluing formula for the zeta determinant and its applications to the adiabatic decompositions of the zeta determinant and the analytic torsion the gluing formula of the zeta determinant of a laplacian given by burghelea friedlander and kappeler contains an unknown constant in this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary as applications of this result we prove the adiabatic decomposition theorems of the zeta determinant of a laplacian with respect to the dirichlet and neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions

robust root clustering of a matrix in intersections or unions of regions this paper considers robust stability analysis for a matrix affected by unstructured complex uncertainty a method is proposed to compute a bound on the amount of uncertainty ensuring robust root clustering in a combination intersection and or union of several possibly nonsymmetric half planes discs and outsides of discs in some cases to be detailed this bound is not conservative the conditions are expressed in terms of linear matrix inequalities lmis and derived through lyapunov s second method as a distinctive feature of the approach the lyapunov matrices proving robust root clustering one per subregion are not necessarily positive definite but have prescribed inertias depending on the number of roots in the corresponding subregions as a special case when root clustering in a single half plane disc or outside of a disc is concerned the whole clustering region reduces to only one convex subregion and the corresponding unique lyapunov matrix has to be positive definite as usual

holomorphic curves in loop groups it was observed by atiyah that there is a correspondence between based gauge equivalence classes ofsu n instantons overs4 of charged on the one hand and based holomorphic curves of genus zero in su n of degreed on the other hand in this paper we study the parameter space of such holomorphic curves which have the additional property that they lie entirely in the subgroup algsu n of algebraic loops we describe a cell decomposition of this parameter space and compute its complex dimension to be 2n 1 d

crossing probabilities in critical 2 d percolation and modular forms abstract crossing probabilities for critical 2 d percolation on large but finite lattices have been derived via boundary conformal field theory these predictions agree very well with numerical results however their derivation is heuristic and there is evidence of additional symmetries in the problem this contribution gives a preliminary examination some unusual modular behavior of these quantities in particular the derivatives of the horizontal and horizontal vertical crossing probabilities transform as a vector modular form one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form we include consideration of the interplay between conformal and modular invariance that arises

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

floer homology and splittings of manifolds in f floer defined a mod8 graded homology group i m for an oriented integral homology 3 sphere m it is an invariant of the differentiable structure of m roughly speaking i m is a homology group coming from the morse theory of the chern simons functional f on the infinite dimensional manifold of all the gauge equivalence classes of irreducible connections on the principal su 2 bundle m x su 2 the critical point set of f is the set of the gauge equivalence classes of irreducible flat connections in general these critical sets may be degenerate and f not be a genuine morse functional a suitable perturbation of f is needed to define i m in this paper we consider only non degenerate critical points of f this is partly because we want to avoid some inessential technical complications and partly because it is sufficient for our computations of i nk for homology 3 spheres nk obtained by dehn surgery along the figure eight knot in s for a smooth connection a on m x su 2 a self adjoint fredholm operator da is defined section 2 the gauge equivalence class a of an irreducible flat connection a is a non degenerate critical point of f if and only if ker da 0 in this case a determines a generator of the mod 8 graded chain group of l m the mod 8 degree d a is related to the spectral flow invariant section 2 as follows let ao and a be two irreducible flat connections on m x su 2 such that ker da 0 ker da let at be a smooth path of smooth connections on m x su 2 connecting ao and a1 then d ai d a 0 is the mod 8 reduction of the spectral flow of the path of the self adjoint fredholm operators da the object of this paper is to give a practical method of calculation of the above spectral flow when m is split as m m1 u m2 where m1 n m2 am dm2 x and x is an orientable surface of genus g 2 such a decomposition in 3 dimensional gauge theory was first considered by c h

the atiyah jones conjecture the purpose of this note is to announce our proof of the atiyah jones conjecture concerning the topology of the moduli spaces of based su 2 instantons over s 4 full details and proofs appear in our paper bhmm1

self adjoint elliptic operators and manifold decompositions part i low eigenmodes and stretching this paper is the first of a three part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds in this article we will show how the low eigensolutions of a self adjoint elliptic operator over such a manifold can be studied by a splicing construction this construction yields an approximated solution of the operator whenever we have two l2 solutions on both sides and a common limiting value of two extended l2 solutions in part 11 the present analytic mayer vietoris results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of maslov indices of lagrangians in part i11 after comparing infinite and finite dimensional lagrangians and determinant line bundles and then introducing canonical perturbations of lagrangian subvarieties of symplectic varieties we will study invariants of 3 manifolds including casson s invariant 0 1996 john wiley sons inc

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

on pole placement via eigenstructure assignment approach this note comes back to the hard problem of pole placement by static output feedback let a triplet of matrices a b c be given with n state variables m inputs and p ouputs find a matrix k such that the spectrum of a bkc equals a specified set when mp m p a simple noniterative technique based upon the notion of eigenstructure that in most cases assigns m p roots is proposed it therefore enables the designer to assign the whole of the desired spectrum when m p n mp

eigenstructure perturbation analysis in disjointed domains for linear uncertain systems sufficient bounds are presented on the norm bounded uncertainties for the perturbed eigenvalues to remain in a set of disjointed neighbourhoods of the nominal eigenvalues for such perturbed eigenvalues bounds on the corresponding eigenvector perturbations are also presented the study is used to analyse design methodologies based on eigenstructure assignment with parameter variations in a conrollable pair the left and right eigenvector perturbation analysis carried out in this paper is useful in the time response analysis of linear uncertain systems

finite nyquist and finite inclusions theorems for disjoint stability regions finite nyquist theorem is an important tool in stability analysis and design of linear systems currently finite nyquist theorem can treat only simply connected convex stability regions with some extensions to simply connected non convex regions in this paper we consider the generalization of finite nyquist theorem for the case of union of disjoint convex stability regions based on this result the finite inclusions theorem is also formulated for a union of disjoint convex stability regions