the transverse index theorem for proper cocompact actions of lie groupoids given a proper cocompact action of a lie groupoid we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes we prove a cohomological index formula for this pairing by applying the van est map and algebraic index theory finally we discuss in examples the meaning of the index pairing and our index formula

quantum macrostates equivalence of ensembles and an h theorem before the thermodynamic limit macroscopic averages need not commute for a quantum system as a consequence aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis when dealing with several observables we propose an implementation of ideas that go back to john von neumann s writing about the macroscopic measurement we apply our scheme to the relation between macroscopic autonomy and an h theorem and to the problem of equivalence of ensembles in particular we show how the latter is related to the asymptotic equipartition theorem the main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate as the size of the system gets larger on the macroscopic values for the different macroscopic observables deviations from that law are governed by the entropy

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

local index theory over foliation groupoids we give a local proof of an index theorem for a dirac type operator that is invariant with respect to the action of a foliation groupoid g i fm denotes the space of units of g then the input is a g equivariant fiber bundle p m along with a g invariant fiberwise dirac type operator d on p the index theorem is a formula for the pairing of the index of d a s an element of a certain k theory group with a closed graded trace on a certain noncommutative de rham algebra b associated to g the proof is by means of superconnections in the framework of noncommutative geometry

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

formality and kontsevich duflo type theorems for lie pairs newcommand poly operatorname poly bullet newcommand td operatorname td l a nabla frac 1 2 newcommand cx 1 operatorname tot big gamma lambda bullet a vee otimes r mathcal 1 poly big newcommand cy 1 mathbb h bullet operatorname ce a mathcal 1 poly kontsevich s formality theorem states that there exists an l infty quasi isomorphism from the dgla t poly m of polyvector fields on a smooth manifold m to the dgla d poly m of polydifferential operators on m which extends the classical hochschild kostant rosenberg map in this paper we extend kontsevich s formality theorem to lie pairs a framework which includes a range of diverse geometric contexts such as complex manifolds foliations and mathfrak g manifolds the spaces cx t and cx d associated with a lie pair l a each carry an l infty algebra structure canonical up to l infty isomorphism these two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators respectively their corresponding cohomology groups cy t and cy d admit canonical gerstenhaber algebra structures we establish the following formality theorem for lie pairs there exists an l infty quasi isomorphism from cx t to cx d whose first taylor coefficient is equal to operatorname hkr circ td here td acts on cx t by contraction furthermore we prove a kontsevich duflo type theorem for lie pairs the hochschild kostant rosenberg map twisted by the square root of the todd class of the lie pair l a is an isomorphism of gerstenhaber algebras from cy t to cy d as applications we establish formality theorems and kontsevich duflo type theorems for complex manifolds foliations and mathfrak g manifolds in the case of complex manifolds we recover the kontsevich duflo theorem of complex geometry

z stability of crossed products by strongly outer actions ii we consider a crossed product of a unital simple separable nuclear stably finite z stable c algebra a by a strongly outer cocycle action of a discrete countable amenable group under the assumption that a has finitely many extremal tracial states and is elementary amenable we show that the twisted crossed product c algebra is z stable as an application we also prove that all strongly outer cocycle actions of the klein bottle group on z are cocycle conjugate to each other this is the first classification result for actions of non abelian infinite groups on stably finite c algebras

position feedback global tracking control of el systems a state transformation approach the contribution of this paper is a dynamic position feedback global tracking controller for fully actuated euler lagrange el systems the properties we show for the closed loop system are uniform stability and exponential convergence global in the initial tracking errors and semiglobal in the initial estimation errors the novelty of our approach is that our observer and control design are based on a new model for el systems which is linear in the immeasurable velocities this model is shown to fit robot manipulators we also provide some simulation results

local index theory over etale groupoids we give a superconnection proof of connes index theorem for proper cocompact actions of etale groupoids this includes connes general foliation index theo rem for foliations with hausdorholonomy groupoid

approximating macroscopic observables in quantum spin systems with commuting matrices abstract macroscopic observables in a quantum spin system are given by sequences of spatial means of local elements 1 2 n 1 j n n j a i n n i 1 m in a uhf algebra one of their properties is that they commute asymptotically as n goes to infinity it is not true that any given set of asymptotically commuting matrices can be approximated by commuting ones in the norm topology in this paper we show that for macroscopic observables this is true

algebraic constructions in the category of lie algebroids the aim of this paper is to give the first general and abstract treatment of the algebraic properties of lie algebroids the concept of lie algebroid was introduced in 1967 by pradines ls as the basic infinitesimal invariant of a differentiable groupoid the construction of the lie algebroid of a differentiable groupoid is a direct generalization of the construction of the lie algebra of a lie group and 183 described a full lie theory for differentiable groupoids and lie algebroids encompassing many phenomena in the foundations of differential geometry a more detailed account and further references are given in 13 however at this stage the algebraic properties of lie algebroids were not pursued in 13 iii section 2 iv section 11 one of us gave a fairly detailed account of the abstract algebra of lie algebroids over a fixed base lie algebroids are vector bundles with several additional structures and a category of lie algebroids on a given base and morphisms which preserve that base has properties similar to those of the category of lie algebras using this algebra it was demonstrated in 13 that the lie theory of a category of locally trivial differentiable lie groupoids over a fixed base and the corresponding category of transitive lie algebroids over that base is coextensive with the standard theory of connections in principal bundles we now propose to show that the lie functor from the category of all differentiable groupoids over arbitrary bases and arbitrary smooth morphisms to the category of all lie algebroids preserves the basic algebraic constructions known to be possible in the category of differentiable groupoids the main work is to give abstract formulations of the corresponding lie algebroid concepts the groupoid results with which we

adaptive control of robot manipulators via velocity estimated feedback an approach for designing robot controllers using state space feedback is presented the robot model parameters are assumed to be unknown and velocity measurements are assumed not to be available this asymptotically stable control scheme combines an adaptive control law with a sliding observer and does not need additional assumptions on the variation of the inertia matrix eigenvalues state observation and parameter adaptation are performed simultaneously the adaptation law the observer gains and the control law are designed on the reduced order manifold which results from the invariance of the switching surface

an algebraic index theorem for orbifolds abstract using the concept of a twisted trace density on a cyclic groupoid a trace is constructed on a formal deformation quantization of a symplectic orbifold an algebraic index theorem for orbifolds follows as a consequence of a local riemann roch theorem for such densities in the case of a reduced orbifold this proves a conjecture by fedosov schulze and tarkhanov finally it is shown how the kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem

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

a simple nonlinear observer for a class of uncertain mechanical systems a simple nonlinear observer is proposed for a class of uncertain nonlinear multiple input multiple output mimo mechanical systems whose dynamics are first order differentiable the global asymptotic observation of the proposed observer is proved thus the observer can be designed independently of the controller furthermore the proposed observer is formulated without any detailed model knowledge of the system these advantages make it easy to implement numerical simulations are included to illustrate the effectiveness of the proposed observer

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

classification of homomorphisms into simple z stable c algebras abstract we classify unital monomorphisms into certain simple z stable c algebras up to approximate unitary equivalence the domain algebra c is allowed to be any unital separable commutative c algebra or any unital simple separable nuclear z stable c algebra satisfying the uct such that c b is of tracial rank zero for a uhf algebra b the target algebra a is allowed to be any unital simple separable z stable c algebra such that a b has tracial rank zero for a uhf algebra b or any unital simple separable exact z stable c algebra whose projections separate traces and whose extremal traces are finitely many

higher rho invariants and the moduli space of positive scalar curvature metrics abstract given a closed smooth manifold m which carries a positive scalar curvature metric one can associate an abelian group p m to the space of positive scalar curvature metrics on this manifold the group of all diffeomorphisms of the manifold naturally acts on p m the moduli group p m of positive scalar curvature metrics is defined to be the quotient abelian group of this action i e the coinvariant of the action p m measures the size of the moduli space of positive scalar curvature metrics on m in this paper we use the higher rho invariant and the finite part of the k theory of the group c algebra of 1 m to give a lower bound of the rank of the moduli group p m the key idea of our proof is the use of higher rho invariant this is a secondary invariant associated to dirac operators and in particular depends on the choice of riemannian metric we show that the higher rho invariant remains unchanged in a certain k theory group under the action of the diffeomorphism group allowing us to distinguish elements in p m

the shannon mcmillan theorem for ergodic quantum lattice systems we formulate and prove a quantum shannon mcmillan theorem the theorem demonstrates the significance of the von neumann entropy for translation invariant ergodic quantum spin systems on lattices the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes the one dimensional case covers quantum information sources and is basic for coding theorems

cyclic cocycles on deformation quantizations and higher index theorems abstract we construct a nontrivial cyclic cocycle on the weyl algebra of a symplectic vector space using this cyclic cocycle we construct an explicit local quasi isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof we give a new proof of nest tsygan s algebraic higher index theorem by computing the pairing between such cyclic cocycles and the k theory of the formal deformation quantization furthermore we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold as an application we obtain the analytic higher index theorem of connes moscovici and its extension to orbifolds

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

a discontinuous output feedback controller and velocity observer for nonlinear mechanical systems in this paper we develop a new discontinuous output feedback tracking controller for a class of uncertain nonlinear multi input multi output mechanical systems whose dynamics are first order differentiable a novel filter design and lyapunov type stability analysis are used to prove semi global asymptotic tracking as a by product of the proposed framework we also present the design of a new simple discontinuous velocity observer that ensures global asymptotic velocity observation

a globally convergent angular velocity observer for rigid body motion the problem of obtaining the angular velocity of a rigid body from orientation and torque measurements only without noisy numerical differentiation is considered a novel angular velocity angular momentum observer for rigid body motion is presented using euler quaternions and a mechanical energy function approach it is shown that the observer estimates converge globally and that the convergence is eventually exponential it is hoped that a mechanical energy function approach to rigid body control can be combined with the observer presented to lead to a globally stable nonlinear observer based rigid body controller in which the observer and controller errors can be separated in much the same way as one can separate controller and observer poles in the output feedback controllers of linear system theory

a class of globally convergent velocity observers for robotic manipulators we present a method for global estimation of joint velocities in rigid manipulators a class of velocity observers with smooth dynamics are introduced whose structure depend on freely selectable functions and gains giving the flexibility of achieving multiple design objectives at the same time unlike most methods no a priori knowledge of an upper bound for velocity magnitude is used an adaptive version of the observer is also presented to handle a class of structured uncertainties in manipulator model simulation example illustrates low noise sensitivity of the globally convergent observer in comparison with semi globally convergent observers

groups quasidiagonality and interpolation by multiplier projections we relate the following conditions on a a unital c algebra a with the fs property a k1 a 0 b every projection in m a a lifts c the general weyl von neumann theorem holds in m a any selfadjoint element h in m a can be written as h a lpi a for some selfadjoint element a in a some bounded real sequence ai and some mutually orthogonal projections pi in a with ez1 pi 1 d m a has fs and e interpolation by multiplier projections holds for any closed projections p and q in a with pq o there is a projection r in m a such that p b 4 a and that a x b if in addition a is stable combining the above results we obtain counterexamples to the conjecture of g k pedersen a has fs m a has fs for example the stabilized buncededdens algebras hence the generalized weyl von neumann theorem does not generally hold in l ha for a unital c algebras with fs

second order sliding mode observer for mechanical systems the super twisting second order sliding mode algorithm is modified in order to design a velocity observer for uncertain mechanical systems the finite time convergence of the observer is proved thus the observer can be designed independently of the controller a discrete version of the observer is considered and the corresponding accuracy is estimated