invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

the index of the scattering operator on the positive spectral subspace we construct the scattering operator for a spinor field in a time dependent background by the dyson expansion then we show that the restriction of the scattering operator to the positive spectral subspace with respect to a reference hamiltonian is fredholm the computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero we obtain the index in terms of a cohomological formula by means of the atiyah singer index theorem

impulsive stabilization of two kinds of second order linear delay differential equations in this paper we consider the impulsive stabilization problems for two kinds of 2nd order linear delay differential equations by the method of lyapunov functionals criteria on the exponential stabilization by impulses and exponential stabilization by periodical impulses are gained the control effects of impulses are especially stressed in the definitions and results some examples are also given to show the applications of our results

construction of hadamard states by pseudo differential calculus we give a new construction based on pseudo differential calculus of quasi free hadamard states for klein gordon equations on a class of space times whose metric is well behaved at spatial infinity in particular on this class of space times we construct all pure hadamard states whose two point function expressed in terms of cauchy data on a cauchy surface is a matrix of pseudo differential operators we also study their covariance under symplectic transformations

impulsive effects on stability of cohen grossberg neural networks with variable delays in this paper a model of impulsive cohen grossberg neural networks is first formulated by establishing some impulsive differential inequalities we investigate impulsive effects on the stability of cohen grossberg neural networks with variable delays and obtain some sufficient conditions ensuring global exponential stability of the impulsive delay system our criteria not only show that the stability still remains under certain impulsive perturbations for some continuous stable neural networks but also present an approach to stabilize the unstable neural networks by utilizing impulsive effects the results extend and improve some recent works for impulsive neural networks as well as non impulsive neural networks some examples and their simulations are given for illustration of the theoretical results

persistent impulsive effects on stability of functional differential equations with finite or infinite delay abstract in this paper the stability problem of impulsive functional differential equations ifdes is considered several criteria ensuring the uniform stability of ifdes with finite or infinite delay are derived by establishing some new razumikhin conditions different with the current existing ones our development results can be applied to delay systems with persistent impulsive effects finally an example and its computer simulation is given to show the effectiveness and advantages of the proposed method

adaptive stabilization of non necessarily inversely stable first order systems by using estimates modification based on testing the sylvester determinant this paper presents an adaptive control scheme for first order continuous time systems the estimated plant model is controllable and then the adaptive scheme is free from singularities the singularities are avoided through a modification of the a priori estimated plant parameter vector so that its associated sylvester matrix is guaranteed to be non singular that property is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold in addition the use of a hysteresis switching function is not required for a modification of the estimates the global asymptotic stability of the closed loop scheme is ensured it is also proved that the problem is well posed even if chattering motion exists due to the modification process the results are also extended to the case of presence of uncertainties consisting of bounded noise and a class of unmodelled dynamics so that global stability of the scheme is ensured in this last case relative adaptation zones are used on the a priori estimates for robust stabilization purposes so that the estimation process is frozen when the size of the prediction error is small compared to the contribution of the unmodelled dynamics to the plant output

microlocal analysis of asymptotically hyperbolic and kerr de sitter spaces with an appendix by semyon dyatlov in this paper we develop a general systematic microlocal framework for the fredholm analysis of non elliptic problems including high energy or semiclassical estimates which is stable under perturbations this framework described in sect 2 resides on a compact manifold without boundary hence in the standard setting of microlocal analysis many natural applications arise in the setting of non riemannian b metrics in the context of melrose s b structures these include asymptotically de sitter type metrics on a blow up of the natural compactification kerr de sitter type metrics as well as asymptotically minkowski metrics the simplest application is a new approach to analysis on riemannian or lorentzian or indeed possibly of other signature conformally compact spaces such as asymptotically hyperbolic or de sitter spaces including a new construction of the meromorphic extension of the resolvent of the laplacian in the riemannian case as well as high energy estimates for the spectral parameter in strips of the complex plane these results are also available in a follow up paper which is more expository in nature vasy in uhlmann g ed inverse problems and applications inside out ii 2012 the appendix written by dyatlov relates his analysis of resonances on exact kerr de sitter space which then was used to analyze the wave equation in that setting to the more general method described here

the massive feynman propagator on asymptotically minkowski spacetimes we consider the massive klein gordon equation on asymptoti cally minkowski spacetimes in the sense that the manifold is r 1 d and the metric approaches that of minkowski space at infinity in a short range way jointly in time and space variables in this setup we define feynman and anti feynman scattering data and prove the fredholm property of the klein gordon operator with the associated atiyah patodi singer type boundary conditions at infinite times we then construct a parametrix with compact remainder terms for the fredholm problem and prove that it is also a feynman parametrix in the sense of duistermaat and hormander

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

chern simons theory matrix integrals and perturbative three manifold invariants the universal perturbative invariants of rational homology spheres can be extracted from the chern simons partition function by combining perturbative and nonperturbative results we spell out the general procedure to compute these invariants and we work out in detail the case of seifert spaces by extending some previous results of lawrence and rozansky the chern simons partition function with arbitrary simply laced group for these spaces is written in terms of matrix integrals the analysis of the perturbative expansion amounts to the evaluation of averages in a gaussian ensemble of random matrices as a result explicit expressions for the universal perturbative invariants of seifert homology spheres up to order five are presented

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

the index of scattering operators of dirac equations ii abstract the index formula of scattering operators of the previous paper of the author t matsui the index of scattering operators of dirac equations commun math phys 110 1987 553 571 is shown to hold for a wider class of potentials which contains both gauge potentials with compactly supported energy and instantontype potentials

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

impulsive stabilization of second order nonlinear delay differential systems modeling of many dynamic systems results in second order delay differential systems in this paper we prove that two kinds of delay systems without impulses can be stabilized by proper impulsive control respectively our results have improved and generated some of the known results found in the literature we also give an example to illustrate the obtained results

witten reshetikhin turaev invariants of seifert manifolds for seifert homology spheres we derive a holomorphic function of k whose value at integer k is the sl2 witten reshetikhin turaev invariant zk at q exp 2 i k this function is expressed as a sum of terms which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the witten chern simons path integral the trivial connection contribution is found to have an asymptotic expansion in powers of k 1 which for k an odd prime power converges k adically to the exact total value of the invariant zk at that root of unity evaluations at rational k are also discussed using similar techniques an expression for the coloured jones polynomial of a torus knot is obtained providing a trivial connection contribution which is an analytic function of the colour this demonstrates that the stationary phase expansion of the chern simons witten theory is exact for seifert manifolds and for torus knots in s3 the possibility of generalising such results is also discussed

asymptotics of radiation fields in asymptotically minkowski space we consider a non trapping n dimensional lorentzian manifold endowed with an end structure modeled on the radial compactification of minkowski space we find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes the rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the northern cap of the compactification for small perturbations of minkowski space that fit into our framework our asymptotic expansions yield a rate of decay that improves on the klainerman sobolev estimates

bibo stabilization of time delayed system with nonlinear perturbation time delayed control system with the nonlinear perturbation is considered novel bibo stabilization criteria are established by the lyapunov functional and formulated in terms of existence of a positive definite solution to an auxiliary algebraic riccati matrix equation the robust quadratic stability for such systems is also discussed a numerical example is given to illustrate the effectiveness of our result

micro local approach to the hadamard condition in quantum field theory on curved space time for the two point distribution of a quasi free klein gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space time we prove the equivalence of 1 the global hadamard condition 2 the property that the feynman propagator is a distinguished parametrix in the sense of duistermaat and hormander and 3 a new property referred to as the wave front set spectral condition wfssc because it is reminiscent of the spectral condition in axiomatic quantum field theory on minkowski space results in micro local analysis such as the propagation of singularities theorem and the uniqueness up toc of distinguished parametrices are employed in the proof we include a review of kay and wald s rigorous definition of the global hadamard condition and the theory of distinguished parametrices specializing to the case of the klein gordon operator on a globally hyperbolic space time as an alternative to a recent computation of the wave front set of a globally hadamard two point distribution on a globally hyperbolic curved space time given elsewhere by kohler to correct an incomplete computation in 32 we present a version of this computation that does not use a deformation argument such as that used in fulling narcowich and wald and is independent of the cauchy evolution argument of fulling sweeny and wald both of which are relied upon in kohler s proof this leads to a simple micro local proof of the preservation of hadamard form under cauchy evolution first shown by fulling sweeny and wald relying only on the propagation of singularities theorem in another paper 33 the equivalence theorem is used to prove a conjecture by kay that a locally hadamard quasi free klein gordon state on any globally hyperbolic curved space time must be globally hadamard

hadamard states for the linearized yang mills equation on curved spacetime we construct hadamard states for the yang mills equation linearized around a smooth space compact background solution we assume the spacetime is globally hyperbolic and its cauchy surface is compact or equal mathbb r d

a contribution of the trivial connection to the jones polynomial and witten s invariant of 3d manifolds ii we extend the results of our previous paper 1 from knots to links by using a formula for the jones polynomial of a link derived recently by n reshetikhin we establish a relation between the parameters of this formula and the multivariable alexander polynomial this relation is illustrated by an example of a torus link we check that our expression for the alexander polynomial satisfies some of its basic properties finally we derive a link surgery formula for the loop corrections to the trivial connection contribution to witten s invariant of rational homology spheres

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

analytic hadamard states calderon projectors and wick rotation near analytic cauchy surfaces we consider the klein gordon equation on analytic spacetimes with an analytic cauchy surface in this setting we prove the existence of pure analytic hadamard states the proof is based on considering an elliptic operator obtained by wick rotating the klein gordon operator in a neighborhood of a cauchy hypersurface the cauchy data of hadamard two point functions are constructed as calderon projectors suitably generalized if the hypersurface is non compact for the elliptic operator

global stabilization for linear continuous time varying systems in this paper stabilization problem via static output feedback controls for linear time varying systems is investigated based on the lyapunov function techniques we show that for linear time varying systems the global null controllability guarantees the output feedback stabilization the verifiable stabilizability conditions and output feedback control design are stated the result can be applicable to the output feedback stabilizability of a class of nonlinear time varying systems numerical examples illustrated the conditions are given

hadamard states for the klein gordon equation on lorentzian manifolds of bounded geometry we consider the klein gordon equation on a class of lorentzian manifolds with cauchy surface of bounded geometry which is shown to include examples such as exterior kerr kerr de sitter spacetime and the maximal globally hyperbolic extension of the kerr outer region in this setup we give an approximate diagonalization and a microlocal decomposition of the cauchy evolution using a time dependent version of the pseudodifferential calculus on riemannian manifolds of bounded geometry we apply this result to construct all pure regular hadamard states and associated feynman inverses where regular refers to the state s two point function having cauchy data given by pseudodifferential operators this allows us to conclude that there is a one parameter family of elliptic pseudodifferential operators that encodes both the choice of pure regular hadamard state and the underlying spacetime metric

exponential stability of impulsive cellular neural networks with time delay via lyapunov functionals this paper investigates the problem of exponential stability for a class of impulsive cellular neural networks with time delay by dividing the network state variables into subgroups according to the characters of the neural networks some sufficient conditions for exponential stability are derived by lyapunov functionals and the method of variation of parameters these conditions are given in terms of some blocks of the interconnection matrix which extend and improve some of the known results in the literature our results show that impulses may be used to stabilize the cellular neural networks with time delay if they are not stable or exponentially stable on the other hand if impulses are input disturbances criteria on the magnitude and frequency of the impulses are also established to maintain the stability property of the original system our results generalize and improve some known results two examples with numerical simulations are given to illustrate our results

microlocal analysis and interacting quantum field theories renormalization on physical backgrounds we present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic curved space times we develop a purely local version of the stuckelberg bogoliubov epstein glaser method of renormalization by using techniques from microlocal analysis relying on recent results of radzikowski kohler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space times we construct time ordered operator valued products of wick polynomials of free fields they serve as building blocks for a local perturbative definition of interacting fields renormalization in this framework amounts to extensions of expectation values of time ordered products to all points of space time the extensions are classified according to a microlocal generalization of steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes as a result we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space times finite renormalizations are deferred to a subsequent paper as byproducts we describe a perturbative construction of local algebras of observables present a new definition of wick polynomials as operator valued distributions on a natural domain and we find a general method for the extension of distributions which were defined on the complement of some surface

existence and impulsive stability for second order retarded differential equations we consider certain impulsive second order delay differential equations and give conditions for the existence of solutions moreover we prove that the non impulsive equations can be stabilized by the imposition of proper impulse controls generalizing recent results by li and weng we also comment on some possible applications and give examples

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

existence of local covariant time ordered products of quantum fields in curved spacetime we establish the existence of local covariant time ordered products of local wick polynomials for a free scalar field in curved spacetime our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem so our construction essentially completes the analysis of the existence uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime as a byproduct of our analysis we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times lorentz invariant distributions plus a remainder term of arbitrarily low scaling degree