spectral flow is the integral of one forms on the banach manifold of self adjoint fredholm operators abstract one may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 vancouver icm address of i m singer our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded breuer fredholm operators in a semifinite von neumann algebra these formulae have a geometric interpretation which derives from the proof namely we define a family of banach submanifolds of all bounded self adjoint breuer fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path we emphasise that our methods do not give a single globally defined one form on the self adjoint breuer fredholms whose integral along all paths is spectral flow rather as the choice of the plural forms in the title suggests we need a family of such one forms in order to confirm singer s idea the original context for this result concerned paths of unbounded self adjoint fredholm operators we therefore prove analogous formulae for spectral flow in the unbounded case as well the proof is a synthesis of key contributions by previous authors whom we acknowledge in detail in the introduction combined with an additional important recent advance in the differential calculus of functions of non commuting operators

chern simons invariants in kk theory abstract for a unitary representation of the fundamental group of a compact smooth manifold atiyah patodi singer defined the so called invariant of using the chern simons invariants in this article using traces on c algebras we give an intrinsic definition of an element in kk with real coefficients theory whose pullback by the representation is the invariant

a globally convergent version of the polak ribiere conjugate gradient method in this paper we propose a new line search algorithm that ensures global convergence of the polak ribiere conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions in particular we show that with this line search every limit point produced by the polak ribiere iteration is a stationary point of the objective function moreover we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite hessian matrix under strong convexity assumptions the known global convergence results can be reobtained as a special case from a computational point of view we may expect that an algorithm incorporating the step size acceptance rules proposed here will retain the same good features of the polak ribiere method while avoiding pathological situations

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

the index of operators on foliated bundles abstract we compute the equivariant cohomology connes karoubi character of the index of elliptic operators along the leaves of the foliation of a flat bundle the proof uses techniques similar to those developed in algebraic topology in the study of non commutative algebras

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

convergence of the polak ribiere polyak conjugate gradient method abstract in this paper we consider the global convergence of the polak ribiere polyak abbreviated prp conjugate gradient method for unconstrained optimization problems a new armijo type line search is proposed for the original prp method and some convergence properties are given under some mild conditions the new armijo type line search can make the prp method choose a suitable initial step size so as to decrease the function evaluations at each iteration and improve the performance of the prp method numerical results show that the prp method with the new armijo type line search is more efficient than other similar methods in practical computation

toeplitz operators on flows abstract let r act continuously on a compact hausdorff space x giving rise to a flow on x let c x and let t x denote the toeplitz operator on h 2 r determined by the function x on r defined by x t x t in this paper we investigate the relation between the spectral properties of t x the dynamical properties of the flow and the value distribution theory of the analysis proceeds by imbedding t x in a type ii factor and computing the real valued index of the operator a la connes our sharpest invertibility result asserts that if the flow is strictly ergodic and if the asymptotic cycle determined by the flow is injective on h 1 x z then t x is invertible if and only if does not vanish on x and determines the zero element in h 1 x z this generalizes the classical result of gohberg and krein and its extension to toeplitz operators with almost periodic symbols due to coburn douglas schaeffer and singer when is analytic in the sense that x belongs to h r for all x we relate the ii index of the toeplitz operator determined by with the density of the zeros of x in the upper half plane much of our efforts to achieve this result are devoted to generalizing to arbitrary flows the value distribution theory of analytic almost periodic functions developed by bohr jessen and tornehave and others

a higher lefschetz formula for flat bundles in this paper we prove a fixed point formula for flat bundles to this end we use cyclic cocycles which are constructed out of closed invariant currents we show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact lie groups this enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation

induced cyclic cocycles and higher eta invariants abstract we construct some cyclic cocycles on the foliation algebra and show that the result of pairing them with a leafwise dirac operator is a spectral invariant this leads to a notation of higher eta invariants

eta invariants of dirac operators on foliated manifolds we define the eta function of dirac operators on foliated manifolds we show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds under the hypothesis of invertibility of the operator in question we prove the atiyah singer relative index theorem for dirac operators on foliated manifolds then we discuss the homotopy invariance of the index with respect to secondary data

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

stationary and almost stationary iterative k l step methods for linear and nonlinear systems of equations we first discuss the solution of a fixed point equationx x of a frechet differentiable self mapping by iterative methods of the general form x m left phi left sum limits i 0 m 1 x i gamma i m 1 right sum limits j 0 m 1 x j beta j m right frac 1 beta m m m 1 2 ldots defined by two infinite nonsingular upper triangular matricesb j m andc j m with column sums 1 we show that two such methods defined byb c and tilde b tilde c respectively are in a certain sense equivalent if and only if c 1 b tilde c 1 tilde b in particular hat b c 1 and the unit matrix replacingc define an equivalent so called semiiterative method we introduce k l step methods as those whereb andc have upper bandwidthk andl 1 respectively they require storing max k l previous iterates only for stationary methodsb andc have toeplitz structure except for their first row which is chosen such that the column sum condition holds an euler method which may require to store all iterates is equivalent to a stationary k l step method if and only if the underlying conformal mapg is a rational function of the formg w w 0 k k v 0 v l 1 w 1 by choosingg withg 1 1 such that for some onto the exterior of some continuums known to contain the eigen values of the frechet derivative of one obtains a feasible procedure for designing a locally converging stationary k l step method custom made for a set of problems in the case x tx d with a linear operatort where one wants to solve a linear system of equations we show that the residual polynomials of a stationary semiiterative method are generalized faber polynomials with respect to a particular weight function using another weight function leads to what we call almost stationary methods the classical chebyshev iteration is an example of such a method we define equivalent almost stationary k l step methods and give a corresponding convergence result

the localized longitudinal index theorem for lie groupoids and the van est map abstract we define the localized index of longitudinal elliptic operators on lie groupoids associated with lie algebroid cohomology classes we derive a topological expression for these numbers using the algebraic index theorem for poisson manifolds on the dual of the lie algebroid underlying the definition and computation of the localized index is an action of the hopf algebroid of jets around the unit space and the characteristic map it induces on lie algebroid cohomology this map can be globalized to differentiable groupoid cohomology giving a definition of the global index that can be computed by localization this correspondence between the global and localized index is given by the van est map for lie groupoids

new properties of a nonlinear conjugate gradient method this paper provides several new properties of the nonlinear conjugate gradient method in 5 firstly the method is proved to have a certain self adjusting property that is independent of the line search and the function convexity secondly under mild assumptions on the objective function the method is shown to be globally convergent with a variety of line searches thirdly we find that instead of the negative gradient direction the search direction defined by the nonlinear conjugate gradient method in 5 can be used to restart any optimization method while guaranteeing the global convergence of the method some numerical results are also presented

deformations of super kms functionals we investigate the stability of the super kms property under deformations we show that a family of continuous deformations of the super derivation in the quantum algebra yields a continuous family of deformed super kms functionals these functionals define a family of cohomologous entire cocycles in this paper we investigate the super kms skms property of functionals co on a quantum algebra our interest in skms functionals was inspired by work of kastler and by conversations with alain connes ci k jlo2 the skms construction relies on the existence of a super derivation d acting on a dense subalgebra of a c algebra d the square of d is the infinitesimal generator of a continuous one parameter automorphism group c of the quantum algebra the usual kms property relates the cyclicity of a state co to the analytic continuation of a group et of automorphisms the skms property also involves invariance under the super derivation d whose square generates the automorphism group er it is known that an skms functional on a quantum algebra defines an entire cyclic cocycle z this is just the chern character which jaffe lesniewski and osterwalder defined in the context ofsupertrace functionals on a quantum algebra jlo 1 the skms property ensures that the functional co and the cocycle z which is derived from it are invariant under this group action in this paper we study the stability of this structure under perturbations of d we study only bounded perturbations which arise from the graded super commutator with an odd element q of the algebra d we show that such perturbations dq of d can be used to define a deformation coq of co which satisfies the skms property furthermore the corresponding family of cocycles z are cohomologous of course more singular unbounded perturbations can lead to

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

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

global invariants for measured foliations new exotic invariants for measured foliations are constructed which we call the u classes of a pair f u the dependence of the s classes on the geometry of the foliation gis examined and the dynamics of a foliation is shown to determine the u classes in many cases we use the u classes to study the classifying space bfsl of foliations with a transverse invariant volume form and we show the homotopy groups of bfsl are uncountably generated starting in degrees q 3 new invariants for groups of volume preserving diffeomorphisms also arise from the t classes these invariants are nontrivial and related to the geometric aspects of the group action relations between the tl classes and the secondary classes of a foliation are exhibited 0 introduction in this paper we construct new exotic characteristic classes for foliations which have a transverse invariant measure these invariants which we call the t classes are similar to the secondary classes and take values in the real cohomology h m of the foliated manifold m theorem 1 6 the values of the m classes are strongly related to the geometry of the foliation proposition 3 4 the u classes have applications to the study of foliations with a transverse invariant volume form theorems 5 1 and 5 2 to the computations of the homology groups of the discrete group of volume preserving diffeomorphisms of rw theorem 7 4 and to the problem of relating the secondary classes of a foliation with its global geometry 15 201 the a classes arose from an attempt to obtain general invariants of a foliation which are intermediate between the leaf and secondary classes they are a special case of a family of universal invariants associated to a c2 foliation the weil operators in codimension one the introduction of the weil operator is a decisive ingredient in duminy s elegant proof of the sullivan conjecture about the godbillon vey class 9 the properties of the weil operators in higher codimensions are given in 15 where they are used to study the residuable secondary classes we briefly describe the construction of the weil operators and their relations with the l classes and other foliation invariants in 2 for a foliation with all leaves compact we note that the weil operators are identically zero proposition 2 6 we denote by f a c2 foliation of codimension q on a paracompact manifold m and i denotes a transverse invariant measure for sf recall that ti assigns to each received by the editors june 28 1982 and in revised form november 9 1982 1980 mathematics subject classification primary 57r30 57r20 58f1 1 secondary 58h10 28d15

quantum k theory i the chern character we construct a cocycle on an infinite dimensional generalization of ap summable fredholm module our framework is related to connes cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds thep summability condition is characteristic of dimensiono p we replace this assumption by the requirement that there exists an underlying heat kernel which is trace class then we use the heat kernel to regularize states in dimension independent fashion our cocycle may be interpreted as an infinite dimensional chern character