signatures and higher signatures of s 1 quotients we define and study the signature widehat a genus and higher signatures of the quotient space of an s 1 action on a closed oriented manifold we give applications to questions of positive scalar curvature and to an equivariant novikov conjecture

eventual practical stability of impulsive differential equations with time delay in terms of two measurements in this paper we introduce a new stability eventual practical stability for impulsive differential equations with time delay by using lyapunov functions and comparison principle we will get some criteria of eventual practical stability eventual practical quasistability and strong eventual practical stability for impulsive differential equations with time delay in terms of two measurements

etale groupoids eta invariants and index theory let g be a discrete finitely generated group let m t be a g equivariant fibration with fibers dieomorphic to a fixed even dimensional manifold with boundary z we assume that g m m g is a galois covering of a compact manifold with boundary letd y ya t be a g equivariant family of dirac type operators under the assumption that the boundary family is l 2 invertible we define an index class in k0 c 0 t zr g if in addition g is of polynomial growth we define higher indices by pairing the index class with suitable cyclic cocycles our main result is then a formula for these higher indices the structure of the formula is as in the seminal work of atiyah patodi and singer with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family under similar assump tions we extend our theorem to any g proper manifold with g an etale groupoid we employ this generalization in order to establish a higher atiyah patodi singer index formula on certain foliations with boundary fundamental to our work is a suitable generalization of melrose b pseudodierential calculus as well as the superconnection proof of the index theorem on g proper manifolds recently given by gorokhovsky and lott in 10

on the gluing problem for the eta invariant we solve the gluing problem for the invariant consider a generalized dirac operator d over a compact riemannian manifold m that is partitioned by a compact hypersurface n such that m m un m2 we assume that the riemannian metric of m and d have a product structure near n i e d i d d dn with some dirac operator dn on n using boundary conditions of atiyah patodi singer type parametrized by lagrangian subspaces l of kerd we define selfadjoint extensions d i 1 2 over m we express the invariant of d in terms of the invariants of di an invariant m l l2 of the pair of the lagrangian subspaces l l2 which is related to the maslov index and an integer valued term j in the adiabatic limit i e if a tubular neighborhood of n is long enough the vanishing of j is shown under certain regularity conditions we apply this result in order to prove cutting and pasting formulas for the invariant a wall nonadditivity result for the index of atiyah patodi singer boundary value problems and a splitting formula for the spectral flow 1 the gluing problem and applications 1 1 introduction we solve the gluing problem for the invariant of generalized dirac operators consider a closed compact riemannian manifold carrying a dirac bundle with a generalized dirac operator assume that this manifold is separated into two pieces by a compact hypersurface the gluing problem for the invariant consists in expressing the invariant of the original dirac operator in terms of the invariants of the dirac operators living on the pieces the selfadjoint operators on the two components with boundary depend on a boundary condition given by lagrangian subspaces lx l2 of a certain symplectic vector space the gluing formula also contains an additional real valued term m l l2 which is nicely related to the symplectic geometry and the maslov index

hermite interpolation of nonsmooth functions preserving boundary conditions this article is devoted to the construction of a hermite type regularization operator transforming functions that are not necessarily c 1 into globally c 1 finite element functions that are piecewise polynomials this regularization operator is a projection it preserves appropriate first and second order polynomial traces and it has approximation properties of optimal order as an illustration it is used to discretize a nonhomogeneous navier stokes problem with tangential boundary condition

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

stability of impulsive functional differential equations abstract in this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied by using lyapunov functions and razumikhin techniques some criteria of stability asymptotic stability and practical stability for impulsive functional differential equations in which the state variables on the impulses are related to the time delay are provided some examples are also presented to illustrate the efficiency of the results obtained

stability of impulsive stochastic differential equations in terms of two measures via perturbing lyapunov functions abstract in this paper we study the stability of nonlinear impulsive stochastic differential equations in terms of two measures the concept of perturbing lyapunov functions is introduced to discuss stability properties of solutions of nonlinear impulsive stochastic differential equations in terms of two measures by using perturbing lyapunov functions and comparison method some sufficient conditions for the above stability are given

stability for time varying linear dynamic systems on time scales we study conditions under which the solutions of a time varying linear dynamic system of the form x t a t x t are stable on certain time scales we give sufficient conditions for various types of stability including lyapunov type stability criteria and eigenvalue conditions on slowly varying systems that ensure exponential stability finally perturbations of the unforced system are investigated and an instability criterion is also developed

a nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity an optimal order multigrid method is developed for the pure displacement problem in two dimensional linear elasticity it is based on a nonconforming mixed formulation where the displacement is approximated by weakly continuous piecewise linear vector functions and the pressure is approximated by piecewise constants the full multigrid convergence is proved the performance of the multigrid method does not deteriorate as the material becomes nearly incompressible

on the practical stability of the solutions of impulsive systems of differential difference equations with variable impulsive perturbations in the present paper the question of the practical stability of the solutions of impulsive systems of differential difference equations with variable impulsive perturbations is discussed in the investigations piecewise continuous functions are used which are analogues of lyapunov s functions and a technique such that certain minimal subsets of a suitable space of piecewise continuous functions are chosen by the elements of which the derivatives of lyapunov s functions are estimated

adaptive sampling control of high gain stabilizable systems it is well known that proportional output feedback control can stabilize any relative degree one minimum phase system if the sign of the feedback is correct and the proportional gain is high enough moreover there exist simple adaptation laws for tuning the proportional gain so called high gain adaptive controllers which do not need to know the system and do not attempt to identify system parameters the authors consider sampled versions of the high gain adaptive controller the motivation for sampling arises from the possibility that the output of a system may not be available continuously but only at sampled times the main point of interest is the need to develop techniques for adapting the sampling rate since the stiffness of the system increases as the proportional gain is increased our main result shows that adaptive sampling stabilization is possible if the product hk of the decreasing sampling interval h and the increasing proportional gain k decreases at a rate proportional to 1 log k

robust multigrid method for the planar linear elasticity problems we consider the solution of the system of equations that arise from the higher order conforming finite element scott vogelius element discretizations of the boundary value problems associated with the differential operator 2 2 div where and are nonzero parameters robust multigrid method is constructed i e the convergence rate of multigrid method is optimal with respect to the mesh size the number of levels and weights on the two terms in the aforementioned differential operator

stability criteria of impulsive differential systems the notion of eventual stability has been recently discussed we extend this notion to impulsive systems of differential equations our technique depends on liapunov s direct method

a second order virtual node algorithm for nearly incompressible linear elasticity in irregular domains we present a cut cell method in r 2 for enforcing dirichlet and neumann boundary conditions with nearly incompressible linear elastic materials in irregular domains virtual nodes on cut uniform grid cells are used to provide geometric flexibility in the domain boundary shape without sacrificing accuracy we use a mixed formulation utilizing a mac type staggered grid with piecewise bilinear displacements centered at cell faces and piecewise constant pressures at cell centers these discretization choices provide the necessary stability in the incompressible limit and the necessary accuracy in cut cells numerical experiments suggest second order accuracy in l we target high resolution problems and present a class of geometric multigrid methods for solving the discrete equations for displacements and pressures that achieves nearly optimal convergence rates independent of grid resolution

impulsive control of a nuclear spin generator in this paper several new theorems on the stability of impulsive control systems are presented these theorems are then used to find the conditions under which the nuclear spin generator can be asymptotically controlled to the equilibrium point by using impulsive control given the parameters of the nuclear spin generator and the impulsive control law we also present a theory of impulsive synchronization of two nuclear spin generators moreover a larger upper bound of impulsive intervals for the stabilization and synchronization can be obtained

stability and instability for dynamic equations on time scales in this paper we examine the stability and instability of the equilibrium solution x 0 to the first order system of dynamic equations x d f t x t t 0 x d r n where t is from a so called time scale t with t 0 tand d is a compact set our methods involve the existence of a positive definite liapunov function v such that its delta derivative v d satisfies certain integral definite or semidefinite sign properties finally we use liapunov functions to develop an invariance principle regarding solutions to the above dynamic equation

asymptotic behavior of solutions of a third order nonlinear dynamic equation on time scales in this paper we will establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation c t a t x t q t f x t 0 t t0 oscillates or converges to zero

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

a new comparison principle for impulsive differential systems with variable impulsive perturbations and stability theory in this paper the stability properties in terms of two measures for impulsive differential systems with variable impulses are investigated a new comparison principlem which allows trajectories strike the same hypersurface finite times is established and then be applied to obtain a stability criterion finally one example is worked out to show the advantage of our results

stability and input to state stability for stochastic systems and applications this paper is concerned with establishing some stability and input to state stability iss properties in terms of two different measures h0 and h for nonlinear systems of stochastic differential equations of ito type to analyze these properties classical lyapunov s method and a comparison principle are used to justify the proposed theoretical results applications to state estimating systems of luenberger type and feedforward or cascade systems enhanced with numerical examples and simulations are presented

stability criteria for set dynamic equations on time scales very recently a new theory known as set dynamic equations on time scales has been built in this paper notions of stability for the solutions of set dynamic equations on time scales using lyapunov like functions are considered criteria for the equistability equiasymptotic stability uniform and uniform asymptotic stability are developed

a sharp convergence estimate for the method of subspace corrections for singular systems of equations this paper is devoted to the convergence rate estimate for the method of successive subspace corrections applied to symmetric and positive semidefinite singular problems in a general hilbert space setting a convergence rate identity is obtained for the method of subspace corrections in terms of the subspace solvers as an illustration the new abstract theory is used to show uniform convergence of a multigrid method applied to the solution of the laplace equation with pure neumann boundary conditions

stability criteria for impulsive systems on time scales in this paper we study stability of impulsive system on time scales by using comparison method lyapunov function and analysis method the asymptotic stability criteria for system with impulses at fixed times and impulses at variable times on time scales are obtained respectively an example is presented to illustrate the efficiency of proposed result

uniform eventual lipschitz stability of impulsive systems on time scales in this paper we discuss the uniform eventual lipschitz stability of impulsive system on time scales by using comparison method lyapunov function and analysis technology some criteria of such stability for system with impulses on time scales are obtained an example is presented to illustrate the efficiency of proposed results

l2 topological invariants of 3 manifolds we give results on thel2 betti numbers and novikov shubin invariants of compact manifolds especially 3 manifolds we first study the betti numbers and novikov shubin invariants of a chain complex of hilbert modules over a finite von neumann algebra we establish inequalities among the novikov shubin invariants of the terms in a short exact sequence of chain complexes our algebraic results along with some analytic results on geometric 3 manifolds are used to compute thel2 betti numbers of compact 3 manifolds which satisfy a weak form of the geometrization conjecture and to compute or estimate their novikov shubin invariants

less conservative conditions for asymptotic stability of impulsive control systems this note studies the stability of impulsive control systems a new comparison theorem for asymptotic stability of impulsive differential system is presented based on the new result we derive some less conservative conditions for asymptotic stability of impulsive control systems with impulses at fixed times and the results are used to design impulsive control for a class of nonlinear systems the class of nonlinear systems considered is also enlarged

hybrid systems on time scales in this paper we study hybrid dynamic systems on time scales using lyapunov like functions we obtain sufficient conditions for the practical stability of hybrid dynamic systems on time scales

on gain adaptation in adaptive control the adaptive high gain output feedback strategy u t k t y t d dt k t spl par y t spl par sup 2 is well established in the context of linear minimum phase m input m output systems a b c with the property that spec cb spl sub spl copf sub the strategy applied to any such linear system achieves the performance objectives of 1 global attractivity of the zero state and 2 convergence of the adapting gain to a finite limit here these results are generalized in three aspects first the class of systems is enlarged to a class n sub h spl mu encompassing nonlinear systems modeled by functional differential equations where the parameter h spl ges 0 quantifies system memory and the continuous function spl mu 0 spl infin spl rarr 0 spl infin with spl mu 0 0 relates to the allowable system nonlinearities next the linear control law is replaced by u t k t y t spl mu spl par y t spl par spl par y t spl par y t wherein the additional nonlinear term counteracts the system nonlinearities then the quadratic adaptation law is replaced by the law d dt k t spl psi spl par y t spl par where the continuous function spl psi satisfies certain growth conditions determined by spl mu in particular cases e g linear systems a bounded function spl psi is admissible the above performance objectives 1 and 2 are shown to persist in the generalized framework

parallel subspace correction methods for nearly singular systems abstract in this paper we consider the parallel subspace correction psc methods for the nearly singular systems we apply the psc methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods our focus is to estimate the condition number of the preconditioned systems we deduce the parameter independent estimates on the psc preconditioners for the nearly singular systems under appropriate assumptions on subspace decomposition the main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces our estimates can be applied into actual problems and two examples are analyzed in this paper one is the elliptic problem with large jumps in the coefficients the other is the planar nearly incompressible elasticity problem with the scott vogelius finite element discretization we prove that the related parallel multilevel methods for both examples are convergent uniformly with respect to the coefficients and the mesh size

smooth dense subalgebras of reduced group c algebras schwartz cohomology of groups and cyclic cohomology abstract we introduce the notion of schwartz co homology of a discrete group with a length function which is the natural generalization of the bounded cohomology of a discrete group we prove that if the length function is chosen to be a word length on a group of polynomial growth then its schwartz cohomology is canonically isomorphic to the usual group cohomology with complex coefficients using this result we will show that the group cohomology of a discrete group of polynomial growth is of polynomial growth in the sense of connes and moscovici and that there are no nontrivial idempotents in the group c algebra of a torsion free group of polynomial growth by the method of cyclic cohomology of connes

boundedness of the solutions of impulsive differential systems with time varying delay in this paper we investigate boundedness of the solutions of impulsive functional differential equations with time varying delay by using lyapunov functions and razumikhin techniques we will see that impulses do contribute to the solutions boundedness property

new razumikhin type theorems for impulsive functional differential equations in this paper we investigate stability and boundedness for a class of impulsive functional differential equations by using liapunov functions and razumikhin technique some new razumikhin type theorems on stability and boundedness are obtained the corresponding results obtained recently in j shen j yan nonlinear anal 33 1998 519 531 x liu j shen to appear are improved

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

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

dirac index classes and the noncommutative spectral flow abstract we present a detailed proof of the existence theorem for noncommutative spectral sections see the noncommutative spectral flow unpublished preprint 1997 we apply this result to various index theoretic situations extending to the noncommutative context results of booss wojciechowski melrose piazza and dai zhang in particular we prove a variational formula in k c r for the index classes associated to 1 parameter family of dirac operators on a covering with boundary this formula involves a noncommutative spectral flow for the boundary family next we establish an additivity result in k c r for the index class defined by a dirac type operator associated to a closed manifold m and a map r m b when we assume that m is the union along a hypersurface f of two manifolds with boundary m m f m finally we prove a defect formula for the signature index classes of two cut and paste equivalent pairs m1 r1 m1 b and m2 r2 m2 b where m 1 m f 1 m m 2 m f 2 m and j diff f the formula involves the noncommutative spectral flow of a suitable 1 parameter family of twisted signature operators on f we give applications to the problem of cut and paste invariance of novikov s higher signatures on closed oriented manifolds

global stability and stabilization of switched boolean network with impulsive effects the present paper investigates the global stability and stabilization of switched boolean network with impulsive effects by using the semi tensor product the algebraic form of the switched boolean network with impulsive effects is obtained then sufficient and necessary condition is presented for the global stability of the switched boolean network with impulsive effects furthermore sufficient and necessary conditions are provided for the stabilization of the switched boolean network with impulsive effects under two types of controls finally examples are presented to illustrate the main results

existence uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales abstract in this paper we study the existence and uniqueness of solution and stability results of mixed integral dynamic system with instantaneous and noninstantaneous impulses on time scales by using the fixed point method the main tools to establish our results are the gronwall s inequality on time scales picard operator and abstract gronwall lemma some assumptions are made to overcome the hurdles in achieving our results at the end an example is given that shows the validity of our main results

instability results for slowly time varying linear dynamic systems on time scales abstract we develop eigenvalue criteria under which the solutions of a slowly time varying linear dynamic system of the form x t a t x t are unstable

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

impulsive control of discrete systems with time delay this note deals with impulsive control of discrete systems with time delay it s the first time that an impulsive controller was introduced for discrete systems with time delay firstly some stability criteria for delay discrete systems with impulse effects are obtained by using lyapunov functionals and razumikhin technique these stability criteria show that if the impulses satisfy some conditions then the original stable systems can keep their stability property even with impulse effects then by using lyapunov functionals or lyapunov functions together with razumikhin technique two stability criteria for the discrete systems with time delay under impulsive control are given these two results show that unstable systems can be stable under impulsive control some examples are also presented to illustrate the efficiency of the obtained results

stability criteria of delay impulsive systems on time scales abstract in this paper we study stability of delay impulsive systems on time scales by using lyapunov functional and analysis method some stability and instability criteria for the delay system with impulses at fixed times on time scales are obtained two examples are presented to illustrate the efficiency of results proposed in this paper

a new approach to practical stability of impulsive functional differential equations in terms of two measures in this work we consider a new approach to the practical stability theory of impulsive functional differential equations with lyapunov functionals and razumikhin technique we use a new technique in the division of lyapunov functions given by shunian zhang and obtain conditions sufficient for the uniform practical asymptotical stability of impulsive delay differential equations an example is also discussed to illustrate the advantage of the proposed results

high gain robust adaptive controllers for multivariable systems abstract this paper extends the switching free high gain stabilizing adaptive control rules of byrnes and willems to a wide class of adaptive schemes capable of tolerating nonlinear state feedback perturbations

multigrid methods for a parameter dependent problem in primal variables in this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials we construct and analyze multilevel projection algorithms which can be applied to the mixed as well as to the equivalent non conforming finite element scheme in primal variables for proper norms we prove that the smoothing property and the approximation property hold with constants that are independent of the small parameter thus we obtain robust and optimal convergence rates for the w cycle and the variable v cycle multigrid methods the numerical results pretty well conform the robustness and optimality of the multigrid methods proposed

on stability accuracy and fast solvers for finite element approximations of the axisymmetric stokes problem by hood taylor elements we provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric stokes problem by continuous piecewise polynomials of degree kappa 1 for the velocity and continuous piecewise polynomials of degree kappa for the pressure with any kappa geq1 new techniques are designed so that in this perspective by a simple transformation the existing theory developed in three dimensional cartesian coordinates can be effectively exploited in fact this perspective provides a new way of developing theories for the axisymmetric stokes problems and it can be applied potentially to other problems as well a simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems sample numerical experiments have been presented as well to confirm the theoretical results

extensions of razumikhin type stability theorems for nonlinear time delay systems on time scales this paper focuses on the stability analysis of nonlinear time delay systems on time scales which are of generality as they can include not only the traditional continuous and discrete ones but a

on the homotopy invariance of higher signatures for manifolds with boundary if m is a compact oriented manifold with boundary whose fundamental group is virtually nilpotent or gromov hyperbolic we show that the higher signatures of m are oriented homotopy invariants we give applications to the question of when higher signatures of closed manifolds are cut and paste invariant

stability criteria of impulsive differential systems this paper discusses the uniform eventual lipschitz stability and uniform eventual asymptotic stability of impulsive differential system some criteria of such stability for impulsive differential system are given compared with the existing result the condition is less conservative our technique depends on lyapunov s direct method and comparison principle

new hybridized mixed methods for linear elasticity and optimal multilevel solvers in this paper we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions n 2 3 which yields a conforming and strongly symmetric approximation for stress applying mathcal p k 1 mathcal p k as the local approximation for the stress and displacement the mixed methods achieve the optimal order of convergence for both the stress and displacement when k ge n for the lower order case n 2 le k n the stability and convergence still hold on some special grids the proposed mixed methods are efficiently implemented by hybridization which imposes the inter element normal continuity of the stress by a lagrange multiplier then we develop and analyze multilevel solvers for the schur complement of the hybridized system in the two dimensional case provided that no nearly singular vertex on the grids the proposed solvers are proved to be uniformly convergent with respect to both the grid size and poisson s ratio numerical experiments are provided to validate our theoretical results

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

novel α stability criterion of linear systems with multiple time delays abstract in this paper the stability is discussed for a class of linear systems with multiple time delays based on the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi technique new sufficient condition is derived for the exponential stability with a given convergence rate the obtained result is less conservative than those in the literature two numerical examples are given to illustrate the superiority of our one to the previous

on reducing the splitting error in yosida methods for the navier stokes equations with grad div stabilization abstract this paper analyzes the accuracy of the discretize then split yosida solver for incompressible flow problems when divergence free elements are used together with grad div stabilization with parameter the yosida method uses an inexact block lu factorization to create linear algebraic systems that are easier to solve but at the expense of accuracy we prove the difference between solutions of the exact and approximated linear algebraic systems is o 2 in the natural norms of the associated finite element problems and thus that full accuracy can be obtained by the yosida method if large is used 10 is sufficient in our numerical examples the proof is based on transforming the yosida inexact linear algebraic system into finite element problems and analyzing these problems with finite element techniques based on pointwise divergence free subspaces and their orthogonal complements