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

the atiyah singer index theorem as passage to the classical limit in quantum mechanics a new approach to the atiyah singer index theorem is described using the technique of continuous fields ofc algebras the proof is given in the case of elliptic pseudodifferential operators on n

on scaling laws of diversity schemes in decentralized estimation this paper is concerned with decentralized estimation of a gaussian source using multiple sensors we consider a diversity scheme where only the sensor with the best channel sends their measurements over a fading channel to a fusion center using the analog amplify and forwarding technique the fusion centre reconstructs a minimum mean squared error mmse estimate of the source based on the received measurements a distributed version of the diversity scheme where sensors decide whether to transmit based only on their local channel information is also considered we derive asymptotic expressions for the expected distortion of the mmse estimate at the fusion centre of these schemes as the number of sensors becomes large for comparison asymptotic expressions for the expected distortion for a coherent multiaccess scheme and an orthogonal access scheme are derived it is seen that as opposed to the coherent multiaccess scheme and the orthogonal scheme where the expected distortion decays as 1 m m being the number of sensors the expected distortion decays only as 1 ln m for the diversity schemes this reduction of the decay rate can be seen as a tradeoff between the simplicity of the diversity schemes and the strict synchronization and large bandwidth requirements for the coherent multiaccess and the orthogonal schemes respectively we study for the diversity schemes the optimal power allocation for minimizing the expected distortion subject to average power constraints the effect of optimizing the probability of transmission on the expected distortion in the distributed scenario is also studied it is proved that for rayleigh fading optimal sensor transmit power allocation achieves the same asymptotic scaling law as the constant power allocation scheme whereas it is observed that optimizing the sensor transmission probability with or without optimal power allocation in the distributed case makes very little difference to the asymptotic scaling laws

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

minimal energy decentralized estimation via exploiting the statistical knowledge of sensor noise variance we study the problem of minimal energy decentralized estimation via sensor networks with the best linear unbiased estimator fusion rule while most of the existing solutions require the knowledge of instantaneous noise variances for energy allocation the proposed approach instead relies on an associated statistical model the minimization of total energy is subject to a performance constraint in terms of the reciprocal of mean square errors averaged over the considered distribution a closed form formula for such a mean distortion metric as well as an associated tractable lower bound is derived by imposing a target distortion constraint in terms of this bound and further through feasible set relaxation the problem can be reformulated in the form of convex optimization and is then analytically solved the proposed method shares several attractive features of the existing designs via instantaneous noise variances through simulations it is seen to significantly improve the energy efficiency against the uniform allocation scheme

a homology theory for etale groupoids etale groupoids arise naturally as models for leaf spaces of foliations for orbifolds and for orbit spaces of discrete group actions in this paper we introduce a sheaf homology theory for etale groupoids we prove its invariance under morita equivalence as well as verdier duality between haeiger cohomology and this homology we also discuss the relation to the cyclic and hochschild homologies of connes convolution algebra of the groupoid and derive some spectral sequences which serve as a tool for the computation of these homologies

fedosov differentials and catalan numbers the aim of the paper is to establish a non recursive formula for the general solution of fedosov s quadratic fixed point equation fedosov 1994 j diff geom 40 213 38 fedosov s geometrical fixed point equation for a differential is rewritten in a form similar to the functional equation for the generating function of catalan numbers this allows us to guess the solution an adapted example for kaehler manifolds of constant sectional curvature is considered in detail also for every connection on a manifold a familiar classical differential will be introduced

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

the generalized assignment problem valid inequalities and facets three classes of valid inequalities based upon multiple knapsack constraints are derived for the generalized assignment problem general properties of the facet defining inequalities are discussed and for a special case the convex hull is completely characterized in addition we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint

geometric structures as deformed infinitesimal symmetries a general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries specifically this model consists of a lie algebroid equipped with an affine connection compatible with the lie algebroid structure the curvature of this connection vanishes precisely when the structure is locally symmetric this model generalizes cartan geometries a substantial class to the intransitive case simple examples are surveyed and corresponding local obstructions to symmetry are identified these examples include foliations riemannian structures infinitesimal g structures symplectic and poisson structures

distributed estimation in energy constrained wireless sensor networks in this paper we consider distributed estimation of a noise corrupted deterministic parameter in energy constrained wireless sensor networks from energy distortion perspective given a total energy budget allowable to be used by all sensors there exists a tradeoff between the subset of active sensors and the energy used by each active sensor in order to minimize the estimation mse to determine the optimal quantization bit rate and transmission energy of each sensor a concept of equivalent unit energy mse function is introduced based on this concept an optimal energy constrained distributed estimation algorithm for homogeneous sensor networks and a quasi optimal energy constrained distributed estimation algorithm for heterogeneous sensor networks are proposed moreover the theoretical energy distortion performance bound for distributed estimation is addressed and it is shown that the proposed algorithm is quasi optimal within a factor 2 of the theoretical lower bound simulation results also show that the proposed method can achieve a significant reduction in the estimation mse when compared with other uniform schemes finally the proposed algorithm is easy to implement in a distributed manner and it adapts well to the dynamic sensor environments

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

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

hyperplane based vector quantization for distributed estimation in wireless sensor networks this paper considers distributed estimation of a vector parameter in the presence of zero mean additive multivariate gaussian noise in wireless sensor networks due to stringent power and bandwidth constraints vector quantization is performed at each sensor to convert its local noisy vector observation into one bit of information which is then forwarded to a fusion center where a final estimate of the vector parameter is obtained within such a context this paper focuses on a class of hyperplane based vector quantizers which linearly convert the observation vector into a scalar by using a compression vector and then carry out a scalar quantization it is shown that the key of the vector quantization design is to find a compression vector for each sensor under the framework of the cramer rao bound crb analysis the compression vector design problem is formulated as an optimization problem that minimizes the trace of the crb matrix such an optimization problem is extensively studied in particular an efficient iterative algorithm is developed for the general case along with optimal and near optimal solutions for some specific but important noise scenarios performance analysis and simulation results are carried out to illustrate the effectiveness of the proposed scheme

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

power scheduling of universal decentralized estimation in sensor networks we consider the optimal power scheduling problem for the decentralized estimation of a noise corrupted deterministic signal in an inhomogeneous sensor network sensor observations are first quantized into discrete messages then transmitted to a fusion center where a final estimate is generated supposing that the sensors use a universal decentralized quantization estimation scheme and an uncoded quadrature amplitude modulated qam transmission strategy we determine the optimal quantization and transmit power levels at local sensors so as to minimize the total transmit power while ensuring a given mean squared error mse performance the proposed power scheduling scheme suggests that the sensors with bad channels or poor observation qualities should decrease their quantization resolutions or simply become inactive in order to save power for the remaining active sensors their optimal quantization and transmit power levels are determined jointly by individual channel path losses local observation noise variance and the targeted mse performance numerical examples show that in inhomogeneous sensing environment significant energy savings is possible when compared to the uniform quantization strategy

lie algebroid foliations and script e1 m dirac structures we prove some general results about the relation between the 1 cocycles of an arbitrary lie algebroid a over m and the leaves of the lie algebroid foliation on m associated with a using these results we show that a 1 m dirac structure l induces on every leaf f of its characteristic foliation a 1 f dirac structure lf which comes from a precontact structure or from a locally conformal presymplectic structure on f in addition we prove that a dirac structure on m can be obtained from l and we discuss the relation between the leaves of the characteristic foliations of l and

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

on the homology of algebras of whitney functions over subanalytic sets in this article we study several homology theories of the algebra e x of whitney functions over a subanalytic set x r n with a view towards noncommutative geometry using a localization method going back to teleman we prove a hochschild kostant rosenberg type theorem for e x when x is a regular subset of r n having regularly situated diagonals this includes the case of subanalytic x we also compute the hochschild cohomology of e x for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories it is shown that the periodic cyclic homology coincides with the de rham cohomology thus generalizing a result of feigin tsygan motivated by the algebraic de rham theory of grothendieck we finally prove that for subanalytic sets the de rham cohomology of e x coincides with the singular cohomology for the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation contents

power constrained linear estimation in wireless sensor networks with correlated data and digital modulation we study the problem of joint quantization and power allocation in wireless sensor networks where spatially distributed sensors observe a gaussian random source quantize the resulting noisy observations and transmit over orthogonal fading channels to a remote fusion center fc the role of the fc is to reconstruct the source with minimal distortion using linear minimum mean square error estimation rule in this paper we undertake the design of joint quantization and power allocation based on the following optimization problem minimize the reconstruction distortion for a given total network power consumption to address this problem at each sensor node uniform scalar quantization is assumed moreover assuming pseudo quantization noise model we show that the problem can be solved using a block coordinate descent type algorithm which iteratively optimizes the quantization bits and the power allocations the algorithm takes into account the spatial correlation the observation noise and the channel quality of the sensors numerical and simulation examples corroborate the analytical results the examples illustrate that the proposed design holds a considerable performance gain compared to a quantization scheme based on uniform power allocation to the sensors

lagrangian reduction by stages for non holonomic systems in a lie algebroid framework the lagrange d alembert equations of a non holonomic system with symmetry can be reduced to the lagrange d alembert poincarequations in a previous contribution we have shown that both sets of equations fall in the category of the so called lagrangian systems on a subbundle of a lie algebroid in this paper we investigate the special case when the reduced system is again invariant under a new symmetry group and so forth via lie algebroid theory we develop a geometric context in which successive reduction can be performed in an intrinsic way we prove that at each stage of the reduction the reduced systems are part of the above mentioned category and that the lie algebroid structure in each new step is the quotient lie algebroid of the previous step we further show that that reduction in two stages is equivalent with direct reduction

hochschild lefschetz class for mathcal d modules we introduce a notion of hochschild lefschetz class for a good coherent mathcal d module on a compact complex manifold and prove that this class is compatible with the direct image functor we prove an orbifold riemann roch formula for a mathcal d module on a compact complex orbifold

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

riemann roch theorems via deformation quantization i abstract we deduce the riemann roch type formula expressing the microlocal euler class of a perfect complex of d modules in terms of the chern character of the associated symbol complex and the todd class of the manifold from the riemann roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization the proof of the latter is contained in the sequel to this paper

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

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

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

integration of twisted dirac brackets given a lie groupoid g over a manifold m we show that multiplicative 2 forms on g relatively closed with respect to a closed 3 form phi on m correspond to maps from the lie algebroid of g into t m satisfying an algebraic condition and a differential condition with respect to the phi twisted courant bracket this correspondence describes as a special case the global objects associated to phi twisted dirac structures as applications we relate our results to equivariant cohomology and foliation theory and we give a new description of quasi hamiltonian spaces and group valued momentum maps

cyclic cohomology of the weyl algebra abstract we give an explicit formula for sp 2 n basic representatives of the cyclic cohomology of the weyl algebra h c a 2 n as an application we prove a generalization of a theorem of nest and tsygan concerning the relation of the todd class and the cyclic cohomology of the differential operators on a complex manifold

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

vector bundles over lie groupoids and algebroids abstract we study vb groupoids and vb algebroids which are vector bundles in the realm of lie groupoids and lie algebroids through a suitable reformulation of their definitions we elucidate the lie theory relating these objects i e their relation via differentiation and integration we also show how to extend our techniques to describe the more general lie theory underlying double lie algebroids and la groupoids

1 k configuration facets for the generalized assignment problem class of facet defining inequalities for the generalized assignment problem is derived these inequalities are based upon multiple knapsack constraints and are derived from 1 k configuration inequalities

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