vortex sheets and diffeomorphism groupoids abstract in 1966 v arnold suggested a group theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right invariant l 2 metric on the group of volume preserving diffeomorphisms of the flow domain here we propose geodesic group theoretic and hamiltonian frameworks to include fluid flows with vortex sheets it turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume preserving diffeomorphisms with common interface we also develop a general framework for euler arnold equations for geodesics on groupoids equipped with one sided invariant metrics

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

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

internal deformation of lie algebroids and symplectic realizations given a lie algebroid and a bundle over its base which is endowed with a localizable poisson structure and a flat connection we construct an extended bundle whose dual is endowed with an almost poisson structure that is a quadratic poisson structure when a certain compatibility property is satisfied this new formalism on lie algebroids describes systems with internal degrees of freedom

nonholonomic versus vakonomic dynamics on a riemann cartan manifold for the chaplygin s nonholonomic constrained systems the constraint manifold can be endowed with riemann cartan geometric structure by nonholonomic mapping into a riemann manifold the two kinds of existing dynamics nonholonomic dynamics and vakonomic dynamics are compared in the framework of riemann cartan geometry it is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the riemann cartan constraint manifold respectively if the metricity condition of riemann cartan connection is satisfied the torsion contorsion of the riemann cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations

on the differential geometry of the euler lagrange equations and the inverse problem of lagrangian dynamics the conditions for a system of second order differential equations to be derivable from a lagrangian the conditions of self adjointness in the terminology of santilli 1978 and others are related in the time independent case to the differential geometry of the tangent bundle of configuration space these conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second order differential equations necessary and sufficient conditions for such a vector field to be derivable from a lagrangian may be stated as the existence of a two form with certain properties it is interesting that it is a deduction not an assumption that this two form is closed and thus defines a symplectic structure some other differential geometric properties of euler lagrange second order differential equations are described

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

geometric quantization of dirac manifolds we extend a prequantization procedure to dirac manifolds by using singular distributions obtained from 2 cocycles associated with dirac structures given a dirac manifold m d we describe a prequantization formula in terms of a lie algebroid connection and show that it is a representation of a poisson algebra consisting of admissible functions on m d on the space of global sections of a hermitian line bundle over m if and only if the curvature derived from the lie algebroid connection is represented by a skew symmetric operation which is naturally defined for m d moreover we describe a necessary and sufficient condition for the prequantization formula to be the representation in terms of a lie algebroid cohomology we introduce the notion of a polarization for m d and construct a representation of a subalgebra of admissible functions lastly we discuss procedures for quantization in two cases where m is compact and where m is not compact

dirac lie systems and schwarzian equations abstract a lie system is a system of differential equations admitting a superposition rule i e a function describing its general solution in terms of any generic set of particular solutions and some constants following ideas going back to dirac s description of constrained systems we introduce and analyze a particular class of lie systems on dirac manifolds called dirac lie systems which are associated with dirac lie hamiltonians our results enable us to investigate constants of the motion superposition rules and other general properties of such systems in a more effective way several concepts of the theory of lie systems are adapted to this dirac setting and new applications of dirac geometry in differential equations are presented as an application we analyze solutions of several types of schwarzian equations but our methods can be applied also to other classes of differential equations important for physics

improvements on khrapchenko s theorem abstract we present an improvement of khrapchenko s theorem which gives lower bounds for the size of boolean formulae our main theorem gives a better lower bound than the original khrapchenko s theorem or at least the same although we know of no function where it gives an improvement factor larger than two this lower bound is the largest eigenvalue of a certain matrix associated with the formula moreover we give an approximation of this bound which is easier to compute and is never smaller than the bound given by khrapchenko s theorem

coded modulation using superimposed binary codes in this correspondence we investigate in a comprehensive fashion a one layer coding shaping scheme resembling a perfectly cooperated multiple access system at the transmitter binary data are encoded by either single level or multilevel codes the coded bits are first randomly interleaved and then entered into a signal mapper at each time the signal mapper accepts as input multiple binary digits and delivers as output an amplitude signal where the input are first independently mapped into 2 pam signals possibly having different amplitudes and then superimposed to form the output the receiver consists of an iterative decoding demapping algorithm with an entropy based stopping criterion in the special cases when all the 2 pam signals have equal amplitudes based on an irregular trellis we propose an optimal soft input soft output siso demapping algorithm with quadratic rather than exponential complexity in the general cases when multilevel codes are employed we propose power allocation strategies to facilitate the iterative decoding dempaping algorithm using the unequal power allocations and the gaussian approximation based suboptimal demapping algorithm with linear complexity coded modulation with high bandwidth efficiency can be implemented

lagrangian systems with constraints a geometric approach to the method of lagrange multipliers a geometric approach to the method of lagrange multipliers is presented using the framework of the tangent bundle geometry the nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then and using this study of the nonholonomic mechanical systems as a previous result the holonomic systems are considered the lagrangian inverse problem is also analysed and finally the theory is illustrated with several examples

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

dirac structures and poisson homogeneous spaces poisson homogeneous spaces for poisson groupoids are classified in terms of dirac structures for the corresponding lie bialgebroids applications include drinfel d s classification in the case of poisson groups and a description of leaf spaces of foliations as homogeneous spaces of pair groupoids

clifford finsler algebroids and nonholonomic einstein dirac structures we propose a new framework for constructing geometric and physical models on nonholonomic manifold provided both with clifford lie algebroid symmetry and nonlinear connection structure explicit parametrizations of generic off diagonal metrics and linear and nonlinear connections define different types of finsler lagrange and or riemann cartan spaces a generalization to spinor fields and dirac operators on nonholonomic manifolds motivates the theory of clifford algebroids defined as clifford bundles in general enabled with nonintegrable distributions defining the nonlinear connection in this work we elaborate the algebroid spinor differential geometry and formulate the scalar proca graviton spinor and gauge field equations on lie algebroids the paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exact solutions with generic off diagonal metrics and generalized symmetries in modern gravity defi

integrability of lie brackets in this paper we present the solution to a longstanding problem of differential geometry lie s third theorem for lie algebroids we show that the integrability problem is controlled by two computable obstructions as applications we derive explain and improve the known integrability results we establish integrability by local lie groupoids we clarify the smoothness of the poisson sigma model for poisson manifolds and we describe other geometrical applications

a geometrical framework for the study of non holonomic lagrangian systems a geometrical framework is presented for the treatment of a class of dynamical systems which are modelled by a system of second order differential equations coupled with first order equations linear in the derivatives such problems in particular make their appearance in the study of lagrangian systems with non holonomic constraints among other things we discuss the concepts of symmetry and adjoint symmetry for such systems and identify for that purpose an appropriate notion of dynamical covariant derivative and jacobi endomorphism the intrinsic tools which are being developed further allow a direct geometrical construction of the dynamics of non holonomic systems the vertically rolling disc is chosen as an illustrative example for the newly proposed formalism and results

lagrangian submanifolds and higher order mechanical systems higher order lagrangian and hamiltonian systems time dependent or independent are interpreted in terms of lagrangian submanifolds of sympletic higher order tangent bundles the relation between both formalisms is given

reduction of poisson nijenhuis lie algebroids to symplectic nijenhuis lie algebroids with a nondegenerate nijenhuis tensor we show how to reduce under certain regularity conditions a poisson nijenhuis lie algebroid to a symplectic nijenhuis lie algebroid with a nondegenerate nijenhuis tensor we generalize the work done by magri and morosi for the reduction of poisson nijenhuis manifolds the choice of the more general framework of lie algebroids is motivated by the geometrical study of some reduced bi hamiltonian systems an explicit example of the reduction of a poisson nijenhuis lie algebroid is also provided

properties of the maximum a posteriori path estimator in hidden markov models a hidden markov model hmm consists of a nonobservable markov chain x x0 x1 and a measured process y y 0 y1 whose distribution is determined by x to estimate the hidden path of x up to time n x0 x1 xn by the observations y 0 y1 yn usually the maximum a posteriori probability path estimator map path estimator is applied an effective means for calculating this estimator is the viterbi algorithm which is widely employed in the fields of coding theory correction of intersymbol interference and text recognition here properties of the map estimator are derived under a certain condition c it is shown that the limiting process u u0 u1 is a regenerative process particularly this means that u has an asymptotic distribution satisfies the central limit theorem and possesses a mean error furthermore condition c is satisfied for a broad class of hmms including the most important case for applications the hmm with additive white gaussian noise

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

an analysis of the orthogonality structures of convolutional codes for iterative decoding the structures of convolutional self orthogonal codes and convolutional self doubly orthogonal codes for both belief propagation and threshold iterative decoding algorithms are analyzed on the basis of difference sets and computation tree it is shown that the double orthogonality property of convolutional self doubly orthogonal codes improves the code structure by maximizing the number of independent observations over two successive decoding iterations while minimizing the number of cycles of lengths 6 and 8 on the code graphs thus the double orthogonality may improve the iterative decoding in both convergence speed and error performance in addition the double orthogonality makes the computation tree rigorously balanced this allows the determination of the best weighing technique so that the error performance of the iterative threshold decoding algorithm approaches that of the iterative belief propagation decoding algorithm but at a substantial reduction of the implementation complexity

reduction of symplectic principal mathbb r bundles we describe a reduction process for symplectic principal bundles in the presence of a momentum map these types of structures play an important role in the geometric formulation of non autonomous hamiltonian systems we apply this procedure to the standard symplectic principal bundle associated with a fibration moreover we show a reduction process for non autonomous hamiltonian systems on symplectic principal bundles we apply these reduction processes to several examples

remarks on generalized lie algebroids and related concepts we prove that generalized lie algebroid a geometric object which appeared recently in the literature is a misconception

nonholonomic constraints a new viewpoint the purpose of this paper is to show that at least for lagrangians of mechanical type nonholonomic euler lagrange equations for a nonholonomic linear constraint d may be viewed as nonconstrained euler lagrange equations but on a new generally not lie algebroid structure on d the proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of noether theorem producing actual first integrals in the case of symmetries the purpose of this paper is to show that at least for lagrangians of mechanical type nonholonomic euler lagrange equations for a nonholonomic linear constraint d may be viewed as nonconstrained euler lagrange equations but on a new generally not lie algebroid structure on d the proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of noether theorem producing actual first integrals in the case of symmetries

classification of extensions of principal bundles and transitive lie groupoids with prescribed kernel and cokernel the equivalence of principal bundles with transitive lie groupoids due to ehresmann is a well known result a remarkable generalization of this equivalence given by mackenzie is the equivalence of principal bundle extensions with those transitive lie groupoids over the total space of a principal bundle which also admit an action of the structure group by automorphisms in this paper the existence of suitably equivariant transition functions is proved for such groupoids generalizing consequently the classification of principal bundles by means of their transition functions to extensions of principal bundles by an equivariant form of cech cohomology

pontryagin maximum principle on almost lie algebroids the fundamental theorem of the theory of optimal control the pontryagin maximum principle pmp is extended to the setting of almost lie al algebroids geometrical objects generalizing lie algebroids this formulation of the pmp yields in particular a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics on the other hand in the presented approach the reduced and unreduced pmps are parts of the same universal formalism the framework is based on a very general concept of homotopy of measurable paths and the geometry of al algebroids

the shrinkage exponent of de morgan formulas is 2 we prove that if we hit a de morgan formula of size l with a random restriction from rp then the expected remaining size is at most o p 2 log frac 1 p 3 2 l p sqrt l as a corollary we obtain an omega n 3 o 1 formula size lower bound for an explicit function in p this is the strongest known lower bound for any explicit function in np

on co morphisms of lie pseudoalgebras and groupoids we give a unified description of morphisms and comorphisms of lie pseudoalgebras showing that the both types of morphisms can be regarded as subalgebras of a lie pseudoalgebra called the sum we also provide similar descriptions for morphisms and comorphisms of lie algebroids and groupoids

smallest formulas for the parity of 2 k variables are essentially unique for n 2 k we know that the size of a smallest and or not formula computing the boolean function parity x 1 x n odd x 1 x n is exactly n 2 for any n it is at least n 2 by the classical khrapchenko bound and for n 2 k we easily obtain a formula of size n 2 by writing and recursively expanding odd x 1 x n odd x 1 x n 2 even x n 2 1 x n even x 1 x n 2 odd x n 2 1 x n we show that for n 2 k the formula obtained above is an essentially unique one that computes parity x 1 x n with size n 2 in the equivalent framework of the karchmer wigderson communication game our result means that an optimal protocol for the parity of 2 k variables is essentially unique

dirac sigma models we introduce a new topological sigma model whose fields are bundle maps from the tangent bundle of a 2 dimensional world sheet to a dirac subbundle of an exact courant algebroid over a target manifold it generalizes simultaneously the twisted poisson sigma model as well as the g g wzw model the equations of motion are satisfied iff the corresponding classical field is a lie algebroid morphism the dirac sigma model has an inherently topological part as well as a kinetic term which uses a metric on worldsheet and target the latter contribution serves as a kind of regulator for the theory while at least classically the gauge invariant content turns out to be independent of any additional structure in the twisted poisson case one may drop the kinetic term altogether obtaining the wz poisson sigma model in general however it is compulsory for establishing the morphism property

a lie algebroid framework for non holonomic systems in order to obtain a framework in which both non holonomic mechanical systems and non holonomic mechanical systems with symmetry can be described we introduce in this paper the notion of a lagrangian system on a subbundle of a lie algebroid

abnormal minimizers this paper constructs the first example of a singular abnormal minimizer for the lagrange problem with linear velocity constraints and quadratic definite lagrangian or equivalently for an optimal control system of linear controls with k controls n states and a running cost function that is quadratic positive definite in the controls in the example k 2 n 3 and the system is completely controllable the example is stable if both the control law and cost are perturbed the singular minimizer persists its importance is due in part to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature there the problem is called the problem of finding minimizing sub riemannian geodesics and it has been claimed that all minimizers are normal pontryagin extremals the mathematical theory of optimal processes wiley interscience new york 1962 if the number of states equals the number of controls then the problem is that of finding riemannian geodesics the main difficulty is proving minimality to do this the length cost of the abnormal is compared with all competing normal extremals a detailed asymptotic analysis of the differential equations governing the normals shows that they are all longer

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

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

block markov superposition transmission construction of big convolutional codes from short codes a construction of big convolutional codes from short codes called block markov superposition transmission bmst is proposed the bmst is very similar to superposition block markov encoding sbme which has been widely used to prove multiuser coding theorems the bmst codes can also be viewed as a class of spatially coupled codes where the generator matrices of the involved short codes referred to as basic codes are coupled the encoding process of bmst can be as fast as that of the basic code while the decoding process can be implemented as an iterative sliding window decoding algorithm with a tunable delay more importantly the performance of bmst can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available numerical results show that 1 the lower bounds can be matched with a moderate decoding delay in the low bit error rate ber region implying that the iterative sliding window decoding algorithm is near optimal 2 bmst with repetition codes and single parity check codes can approach the shannon limit within 0 5 db at the ber of 10 5 for a wide range of code rates and 3 bmst can also be applied to nonlinear codes

on the geometry of the hamilton jacobi equation and generating functions in this paper we develop a geometric version of the hamilton jacobi equation in the poisson setting specifically we geometrize what is usually called a complete solution of the hamilton jacobi equation we use some well known results about symplectic groupoids in particular cotangent groupoids as a keystone for the construction of our framework our methodology follows the ambitious program proposed by weinstein in mechanics day waterloo on 1992 volume 7 of fields institute communications american mathematical society providence 1996 in order to develop geometric formulations of the dynamical behavior of lagrangian and hamiltonian systems on lie algebroids and lie groupoids this procedure allows us to take symmetries into account and as a by product we recover results from channell and scovel phys d 50 1 80 88 1991 ge indiana univ math j 39 3 859 876 1990 ge and marsden phys lett a 133 3 134 139 1988 but even in these situations our approach is new a theory of generating functions for the poisson structures considered here is also developed following the same pattern solving a longstanding problem of the area how to obtain a generating function for the identity transformation and the nearby poisson automorphisms of poisson manifolds a direct application of our results gives the construction of a family of poisson integrators that is integrators that conserve the underlying poisson geometry these integrators are implemented in the paper in benchmark problems some conclusions current and future directions of research are shown at the end of the paper

representations up to homotopy of lie algebroids we introduce and study the notion of representation up to homotopy of a lie algebroid paying special attention to examples we use representations up to homotopy to define the adjoint representation of a lie algebroid and show that the resulting cohomology controls the deformations of the structure the weil algebra of a lie algebroid is defined and shown to coincide with kalkman s brst model for equivariant cohomology in the case of group actions the relation of this algebra with the integration of poisson and dirac structures is explained in arias abad crainic ann inst fourier

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

classical field theory on lie algebroids variational aspects the variational formalism for classical field theories is extended to the setting of lie algebroids given a lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to being morphisms of lie algebroids in addition to the standard case our formalism includes as particular examples the case of systems with symmetry covariant euler poincare and lagrange poincare cases sigma models or chern simons theories

lie algebroid structures on double vector bundles and representation theory of lie algebroids abstract a vb algebroid is essentially defined as a lie algebroid object in the category of vector bundles there is a one to one correspondence between vb algebroids and certain flat lie algebroid superconnections up to a natural notion of equivalence in this setting we are able to construct characteristic classes which in special cases reproduce characteristic classes constructed by crainic and fernandes we give a complete classification of regular vb algebroids and in the process we obtain another characteristic class of lie algebroids that does not appear in the ordinary representation theory of lie algebroids

reduced classical field theories k cosymplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k cosymplectic geometry we discuss the relation between lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied mainly on reduction of classical field theories

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

variational calculus with constraints on general algebroids variational calculus on a vector bundle e equipped with a structure of a general algebroid is developed together with the corresponding analogs of euler lagrange equations constrained systems are introduced in the variational and geometrical settings the constrained euler lagrange equations are derived for analogs of holonomic vakonomic and nonholonomic constraints this general model covers the majority of first order lagrangian systems which are present in the literature and reduces to the standard variational calculus and the euler lagrange equations in classical mechanics for e tm

on integrability of infinitesimal actions we use foliations and connections on principal lie groupoid bundles to prove various integrability results for lie algebroids in particular we show under quite general assumptions that the semi direct product associated to an infinitesimal action of one integrable lie algebroid on another is integrable this generalizes recent results of dazord and nistor

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

from atiyah classes to homotopy leibniz algebras a celebrated theorem of kapranov states that the atiyah class of the tangent bundle of a complex manifold x makes t x 1 into a lie algebra object in d x the bounded below derived category of coherent sheaves on x furthermore kapranov proved that for a kahler manifold x the dolbeault resolution omega bullet 1 t x 1 0 of t x 1 is an l algebra in this paper we prove that kapranov s theorem holds in much wider generality for vector bundles over lie pairs given a lie pair l a i e a lie algebroid l together with a lie subalgebroid a we define the atiyah class e of an a module e as the obstruction to the existence of an a compatible l connection on e we prove that the atiyah classes l a and e respectively make l a 1 and e 1 into a lie algebra and a lie algebra module in the bounded below derived category d mathcal a where mathcal a is the abelian category of left mathcal u a modules and mathcal u a is the universal enveloping algebra of a moreover we produce a homotopy leibniz algebra and a homotopy leibniz module stemming from the atiyah classes of l a and e and inducing the aforesaid lie structures in d mathcal a

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

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 bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

on the integrability of lie subalgebroids abstract let g be a lie groupoid with lie algebroid g it is known that unlike in the case of lie groups not every subalgebroid of g can be integrated by a subgroupoid of g in this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible we also consider the problem of integrability by closed subgroupoids and we give conditions under which the closure of a subgroupoid is again a subgroupoid

the berwald type linearization of generalized connections we study the existence of a natural linearization process for generalized connections on an affine bundle it is shown that this leads to an affine generalized connection over a prolonged bundle which is the analogue of what is called a connection of berwald type in the standard theory of connections various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalized connections on such bundles

path partitions and forward only trellis algorithms this is a semitutorial paper on trellis based algorithms we argue that most decoding detection algorithms described on trellises can be formulated as path partitioning algorithms with proper definitions of mappings from subsets of paths to metrics of subsets thereby the only two operations needed are path concatenation and path collection which play the roles of multiplication and addition respectively furthermore we show that the trellis structure permits the path partitioning algorithms to be formulated as forward only algorithms with structures resembling the viterbi 1967 algorithm thus eliminating the need for backward computations regardless of what task needs to be performed on the trellis while all of the actual decoding detection algorithms presented here are rederivations of variations of previously known methods we believe that the exposition of the algorithms in a unified manner as forward only path partitioning algorithms is the most intuitive manner in which to generalize the viterbi algorithm we also believe that this approach may in fact influence the practical implementation of the algorithms as well as influence the construction of other forward only algorithms e g byte wise forward only detection algorithms

note on generalized connections and affine bundles we develop an alternative view on the concept of connections over a vector bundle map which consists of a horizontal lift procedure to a prolonged bundle we further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalized connection

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

reduction of a symplectic like lie algebroid with momentum map and its application to fiberwise linear poisson structures this paper addresses the problem of developing an extension of the marsden weinstein reduction process to symplectic like lie algebroids and in particular to the case of the canonical cover of a fiberwise linear poisson structure whose reduction process is the analog to cotangent bundle reduction in the context of lie algebroids dedicated to the memory of jerrold e marsden

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

a geometric approach to the gibbs appell equations in lagrangian mechanics we present a geometrical derivation of the gibbs appell equations in constrained lagrangian mechanics using the affine structures existing in the spaces of velocities and accelerations the theoretical basis is the gauss principle of least constraint and the necessary conditions for the minimum expressed in terms of the gibbs function give rise to the gibbs appell equations we give a second order formulation of the constraint equation and using the concept of moore penrose inverse we provide an explicit intrinsic construction of the projector on the space of the effective or non ideal forces the gibbs appell equations are expressed in an intrinsic way in terms of the fibre derivative of the gibbs function and the effective part of the applied force

quasi coordinates from the point of view of lie algebroid structures a geometrical description of the lagrangian dynamics in quasi coordinates on the tangent bundle using the lie algebroid framework is given linear nonholonomic systems on lie algebroids are solved in local coordinates adapted to the constraints through lagrangian multipliers and gibbs appell generalized methods

reduced dynamics and lagrangian submanifolds of symplectic manifolds in this paper we will see that the symplectic creed by weinstein everything is a lagrangian submanifold also holds for hamilton poincare and lagrange poincare reduction in fact we show that solutions of the hamilton poincare equations and of the lagrange poincare equations are in one to one correspondence with distinguished curves in a lagrangian submanifold of a symplectic manifold for this purpose we will combine the concept of a tulczyjew triple with marsden weinstein symplectic reduction

weyl s theory in the generalized lie algebroids framework the geometry of the lie algebroid generalized tangent bundle of a generalized lie algebroid is developed formulas of ricci type and identities of cartan and bianchi type are presented introducing the notion of geodesic of a mechanical system with respect to a spray the berwald derivative operator and its mixed curvature we obtain main results to conceptualize the weyl s method in this general framework finally we obtain two new results of weyl type for the geometry of mechanical systems in this way it is proved that the projectively related sprays first have the same geodesics rather to an increasing parameter transformation and second their berwald derivatives verify a respective relation

reduction of some classical non holonomic systems with symmetry two types of nonholonomic systems with symmetry are treated i the configuration space is a total space of a g principal bundle and the constraints are given by a connection ii the configuration space is g itself and the constraints are given by left invariant forms the proofs are based on the method of quasicoordinates in passing a derivation of the maurer cartan equations for lie groups is obtained simple examples are given to illustrate the algorithmical character of the main results

gravity from lie algebroid morphisms inspired by the poisson sigma model and its relation to 2d gravity we consider models governing morphisms from t to any lie algebroid e where is regarded as a d dimensional spacetime manifold we address the question of minimal conditions to be placed on a bilinear expression in the 1 form fields sij x aiaj so as to permit an interpretation as a metric on this becomes a simple compatibility condition of the e tensor s with the chosen lie algebroid structure on e for the standard lie algebroid e tm the additional structure is identified with a riemannian foliation of m in the poisson case e t m with a sub riemannian structure which is poisson invariant with respect to its annihilator bundle for integrable image of s this means that the induced riemannian leaves should be invariant with respect to all hamiltonian vector fields of functions which are locally constant on this foliation this provides a huge class of new gravity models in d dimensions embedding known 2d and 3d models as particular examples

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

matched information rate codes for partial response channels in this paper we design capacity approaching codes for partial response channels the codes are constructed as concatenations of inner trellis codes and outer low density parity check ldpc codes unlike previous constructions of trellis codes for partial response channels we disregard any algebraic properties e g the minimum distance or the run length limit in our design of the trellis code our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a markov process that achieves a high capacity approaching information rate hence we name it a matched information rate mir design we provide a set of five design rules for constructions of capacity approaching mir inner trellis codes we optimize the outer ldpc code using density evolution tools specially modified to fit the superchannel consisting of the inner mir trellis code concatenated with the partial response channel using this strategy we design degree sequences of irregular ldpc codes whose noise tolerance thresholds are only fractions of a decibel away from the capacity examples of code constructions are shown for channels both with and without spectral nulls

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

on the developability of lie subalgebroids abstract for a lie groupoid g with algebroid g one says that a subalgebroid h g is developable if it can be integrated to a closed lie subgroupoid of the universal covering groupoid of g under some additional hypotheses we construct an algebroid b depending on g and h and prove that the developability of h is equivalent to the integrability of b this result extends the almeida molino obstruction to developability of foliations

reduction of constrained systems with symmetries a general model is proposed for constrained dynamical systems on a symplectic manifold which covers among others the description of lagrangian and hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular lagrangian the reduction properties of these systems in the presence of symmetry are investigated within this general framework

lie algebroid structures on a class of affine bundles we introduce the notion of a lie algebroid structure on an affine bundle whose base manifold is fibered over r it is argued that this is the framework which one needs for coming to a time dependent generalization of the theory of lagrangian systems on lie algebroids an extensive discussion is given of a way one can think of forms acting on sections of the affine bundle it is further shown that the affine lie algebroid structure gives rise to a coboundary operator on such forms the concept of admissible curves and dynamical systems whose integral curves are admissible brings an associated affine bundle into the picture on which one can define in a natural way a prolongation of the original affine lie algebroid structure

integrability of poisson brackets we discuss the integration of poisson brackets motivated by our recent solution to the integrability problem for general lie brackets we give the precise obstructions to integrating poisson manifolds describing the integration as a symplectic quotient in the spirit of the poisson sigma model of cattaneo and felder for regular poisson manifolds we express the obstructions in terms of variations of symplectic areas improving on results of alcalde cuesta and hector we apply our results and our point of view to decide about the existence of complete symplectic realizations to the integrability of submanifolds of poisson manifolds and to the study of dual pairs morita equivalence and reduction

classical intertwiner space and quantization given two symplectic realizations a symplectic manifold called the classical intertwiner space is introduced as a classical analogue of an intertwiner space of representations of an associative algebra we describe explicitly how a quantum data on realizations induces a quantum data on their classical intertwiner space

non holonomic lagrangian systems in jet manifolds a geometrical setting in terms of jet manifolds is developed for time dependent non holonomic lagrangian systems an almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics a constrained poincare cartan 2 form is defined if the non holonomic system is singular a constraint algorithm is constructed special attention is devoted to caplygin systems and a reduction theorem is proved