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

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

locally conformal dirac structures and infinitesimal automorphisms we study the main properties of locally conformal dirac bundles which include dirac structures on a manifold and locally conformal symplectic manifolds it is proven that certain locally conformal dirac bundles induce jacobi structures on quotient manifolds furthermore we show that given a locally conformal dirac bundle over a smooth manifold m there is a lie homomorphism between a subalgebra of the lie algebra of infinitesimal automorphisms and the lie algebra of admissible functions we also show that dirac manifolds can be obtained from locally conformal dirac bundles by using an appropriate covering map finally we extend locally conformal dirac bundles to the context of lie algebroids

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

critical size of neural population for reliable information transmission it is known that the probability of decoding error has a phase transition at information rate equal to the channel capacity the corresponding thermodynamic limit requires infinite coding dimension hence making the actual decoding practically impossible in this rapid communication we analyze finite size effects that occur in limited neural populations we report that the achievable rate approaches the asymptote in a remarkably nonlinear manner with the population size qualitatively our findings do not seem to depend on the details of the model

asymptotical statistics of misspecified hidden markov models this paper deals with the problem of modeling data generated by an ergodic stochastic process as the output of a hidden markov model hmm more specifically we consider the problem of fitting a parametric family of hmm with continuous output to an ergodic stochastic process with continuous values which does not necessarily belong to the family in this context we derive the main asymptotic results almost sure consistency of the maximum likelihood estimator asymptotic normality of the estimation error and the exact rates of almost sure convergence

upper bounds on the capacities of noncontrollable finite state channels with without feedback noncontrollable finite state channels fscs are fscs in which the channel inputs have no influence on the channel states i e the channel states evolve freely since single letter formulas for the channel capacities are rarely available for general noncontrollable fscs computable bounds are usually utilized to numerically bound the capacities in this paper we take the delayed channel state as part of the channel input and then define the directed information rate from the new channel input including the source and the delayed channel state sequence to the channel output sequence with this technique we derive a series of upper bounds on the capacities of noncontrollable fscs with without feedback these upper bounds can be achieved by conditional markov sources and computed by solving an average reward per stage stochastic control problem arscp with a compact state space and a compact action space by showing that the arscp has a uniformly continuous reward function we transform the original arscp into a finite state and finite action arscp that can be solved by a value iteration method under a mild assumption the value iteration algorithm is convergent and delivers a near optimal stationary policy and a numerical upper bound

easily computed lower bounds on the information rate of intersymbol interference channels provable lower bounds are presented for the information rate i x x s n where x is the symbol drawn independently and uniformly from a finite size alphabet s is a discrete valued random variable rv and n is a gaussian rv it is well known that with s representing the precursor intersymbol interference isi at the decision feedback equalizer dfe output i x x s n serves as a tight lower bound for the symmetric information rate sir as well as capacity of the isi channel corrupted by gaussian noise when evaluated on a number of well known finite isi channels these new bounds provide a very similar level of tightness against the sir to the conjectured lower bound by shamai and laroia at all signal to noise ratio snr ranges while being actually tighter when viewed closed up at high snrs the new lower bounds are obtained in two steps first a mismatched mutual information function is introduced which can be proved as a lower bound to i x x s n secondly this function is further bounded from below by an expression that can be computed easily via a few single dimensional integrations with a small computational load

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

bounds on the information rate of intersymbol interference channels based on mismatched receivers we consider the problem of bounding the information rate of intersymbol interference channels via simulation based algorithms the adopted approach which is based on a general class of reduced complexity receivers that includes several previously studied receivers as special cases leads to provable upper and lower bounds on the information rate of interest as a by product of the information theoretic investigations novel insights on the design of efficient reduced complexity receivers are also provided since the proposed lower bounds are known to be achievable by practical receivers in many scenarios our novel approach significantly outperforms the existing ones for all practical values of the signal to noise ratio

a new approach to synthesis problems in linear system theory we consider a rational nonlinear equation which is involved in the general synthesis design problem of linear multivariable systems the complete solution of this equation subject to causality stability and minimality constraints is provided formal power series and in particular the theory of partial realizations prove of central importance for the design problem

disturbance rejection a polynomial approach a detailed analysis of the disturbance rejection problem in single input single output linear systems is presented the analysis is based on an external polynomial model and employs the algebra of polynomials all solutions are expressed in a parametric form and solvability conditions are given for a number of constraints including those of internal stability and arbitrary pole placement measurable and unmeasurable disturbances are distinguished and the effect of output feedback is investigated a simple design procedure is then proposed which provides a stable dynamical solution and which can be extended so as to minimize its order the procedure consists of solving two linear polynomial equations and yields the compensator in a compact form suitable for direct realization

algebraic aspects of linear control system stability after formulating the problem of linear control system stability in very general terms the problem is resolved by 1 establishing a necessary and sufficient condition for stability 2 exploring in some detail the nature of this condition and 3 developing a parametric characterization of all controllers which stabilize a given plant the problem formulation is based on a ring theoretic setting for linear systems and is distinguished by the use of a novel control system configuration any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems furthermore no other configuration has this property several benefits and extensions are accrued as a result of using this configuration first a completely general yet physically meaningful problem formulation results consequently state space and operator theoretic approaches to linear control system stability are thoroughly reconciled this reconciliation embraces continuous time as well as discrete time systems lumped as well as distributed systems and 1 d as well as n d systems second the controller parametrization affords four degrees of freedom two more than any existing parametrization the additional degrees of freedom make explicit the design opportunity associated with controller implementation and thus determine several important control system characteristics this result is expected to provide a theoretical basis for the development of comprehensive control system design methodologies wherein design considerations related to controller implementation are addressed

simulation based computation of information rates for channels with memory the information rate of finite state source channel models can be accurately estimated by sampling both a long channel input sequence and the corresponding channel output sequence followed by a forward sum product recursion on the joint source channel trellis this method is extended to compute upper and lower bounds on the information rate of very general channels with memory by means of finite state approximations further upper and lower bounds can be computed by reduced state methods

nambu dirac manifolds we present a generalization of the dirac structure in the direction of the nambu poisson structure it is shown that with every integrable nambu dirac structure there is associated a leibniz algebroid which yields a singular foliation endowed with a closed form the examples of nambu dirac manifolds are dirac manifolds nambu poisson manifolds and manifolds with closed forms a different leibniz algebroid structure associated with a nambu poisson structure is adopted for this generalization

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

disturbance localization and pole shifting by dynamic compensation this paper concerns the problem of disturbance localization and pole shifting by dynamic compensation in multivariable linear systems with inaccessible state vector a necessary and sufficient condition for the solvability of the problem is given two synthesis procedures of the compensator are presented one is based on an observer and the other is based on a dual observer the observer designed by the proposed procedure acts as as unknown input observer for a linear function an improved procedure of designing a reduced order unknown input observer is also given the result may be applicable to reducing the compensator order

optimization of information rate upper and lower bounds for channels with memory we consider the problem of minimizing upper bounds and maximizing lower bounds on information rates of stationary and ergodic discrete time channels with memory the channels we consider can have a finite number of states such as partial response channels or they can have an infinite state space such as time varying fading channels we optimize recently proposed information rate bounds for such channels which make use of auxiliary finite state machine channels fsmcs our main contribution in this paper is to provide iterative expectation maximization em type algorithms to optimize the parameters of the auxiliary fsmc to tighten these bounds we provide an explicit iterative algorithm that improves the upper bound at each iteration we also provide an effective method for iteratively optimizing the lower bound to demonstrate the effectiveness of our algorithms we provide several examples of partial response and fading channels where the proposed optimization techniques significantly tighten the initial upper and lower bounds finally we compare our results with results obtained by the conjugate gradient optimization algorithm and an improved variation of the simplex algorithm called soblex while the computational complexities of our algorithms are similar to the conjugate gradient method and less than the soblex algorithm our algorithms robustly find the tightest bounds interestingly from a channel coding decoding perspective optimizing the lower bound is related to increasing the achievable mismatched information rate i e the information rate of a communication system where the decoder at the receiver is matched to the auxiliary channel and not to the original channel

algebraic theory of linear time invariant feedback systems with two input two output plant and compensator this paper present an algebraic theory for linear time invariant multi input multi output systems with two input two output plant and compensator using the factorization approach this system configuration considered by doyle and nett is most general in that any interconnection of two systems can be represented in terms of this scheme other system configurations encountered in compensator design problems are special cases of this system configuration the analysis and synthesis applies to continuous time as well as discrete time systems and to lumped as well as distributed systems since the algebraic setting is completely general the compensator parametrization has four degrees of freedom the paper is self contained and is tutorial in the sense that it develops the main results of nett without introducing the concept of containment

rozansky witten type invariants from symplectic lie pairs we introduce symplectic structures on lie pairs of real or complex lie algebroids as studied by chen et al from atiyah classes to homotopy leibniz algebras arxiv 1204 1075 2012 encompassing homogeneous symplectic spaces symplectic manifolds with a mathfrak g action and holomorphic symplectic manifolds we show that to each such symplectic lie pair are associated rozansky witten type invariants of three manifolds and knots given respectively by weight systems on trivalent and chord diagrams

ddp and model following problem revisited abstract for linear multivariable systems with two input two output plant and compensator the paper solves the disturbance decoupling perfect tracking problems with stability results are obtained for the not necessarily robust as well as for the robust problem

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

manin triples for lie bialgebroids in his study of dirac structures a notion which includes both poisson structures and closed 2 forms t courant introduced a bracket on the direct sum of vector fields and 1 forms this bracket does not satisfy the jacobi identity except on certain subspaces in this paper we systematize the properties of this bracket in the definition of a courant algebroid this structure on a vector bundle e rightarrow m consists of an antisymmetric bracket on the sections of e whose jacobi anomaly has an explicit expression in terms of a bundle map e rightarrow tm and a field of symmetric bilinear forms on e when m is a point the definition reduces to that of a lie algebra carrying an invariant nondegenerate symmetric bilinear form for any lie bialgebroid a a over m a notion defined by mackenzie and xu there is a natural courant algebroid structure on a oplus a which is the drinfel d double of a lie bialgebra when m is a point conversely if a and a are complementary isotropic subbundles of a courant algebroid e closed under the bracket such a bundle with dimension half that of e is called a dirac structure there is a natural lie bialgebroid structure on a a whose double is isomorphic to e the theory of manin triples is thereby extended from lie algebras to lie algebroids our work gives a new approach to bihamiltonian structures and a new way of combining two poisson structures to obtain a third one we also take some tentative steps toward generalizing drinfel d s theory of poisson homogeneous spaces from groups to groupoids

feedback capacity of finite state machine channels we consider a finite state machine channel with a finite memory length e g finite length intersymbol interference channels with finite input alphabets also known as partial response channels for such a finite state machine channel we show that feedback dependent markov sources achieve the feedback capacity and that the required memory length of the markov process matches the memory length of the channel further we show that the whole history of feedback is summarized by the causal posterior channel state distribution which is computed by the sum product forward recursion of the bahl cocke jelinek raviv bcjr baum welch discrete time wonham filtering algorithm these results drastically reduce the space over which the optimal feedback dependent source distribution needs to be sought further the feedback capacity computation may then be formulated as an average reward per stage stochastic control problem which is solved by dynamic programming with the knowledge of the capacity achieving source distribution the value of the capacity is easily estimated using markov chain monte carlo methods when the feedback is delayed we show that the feedback capacity can be computed by similar procedures we also show that the delayed feedback capacity is a tight upper bound on the feedforward capacity by comparing it to tight existing lower bounds we demonstrate the applicability of the method by computing the feedback capacity of partial response channels and the feedback capacity of run length limited rll sequences over binary symmetric channels bscs

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

mismatched decoding revisited general alphabets channels with memory and the wide band limit the mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given possibly suboptimal decoding rule this quantity has been studied extensively for single letter decoding rules over discrete memoryless channels dmcs here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non single letter decoding rules we also study the wide band limit and in particular the mismatch capacity per unit cost and the achievable rates on an additive noise spread spectrum system with single letter decoding and binary signaling

dual capacity upper bounds for binary input single tap isi channels the capacity of noisy channels with memory and finite input has been difficult to characterize explicitly in many cases in this paper we consider single tap binary input gaussian channels with inter symbol interference isi the dual capacity method is used to obtain upper bounds using markov test distributions the bound expressed as a single letter optimization problem is solved numerically for higher memory the notion of cycle basis is used to provide an exponential reduction in the computational complexity of the optimization problem bounds were obtained for the important case of the dicode channel and these are better than the previously known upper bounds and are close to achievable rates over a wide range of snrs

bounds on the information rate for sparse channels with long memory and i u d inputs in this paper we propose new bounds on the achievable information rate for discrete time gaussian channels with intersymbol interference isi and independent and uniformly distributed i u d channel input symbols drawn from finite order modulation alphabets specifically we are interested in developing new bounds on the achievable rates for sparse channels with long memory we obtain a lower bound which can be achieved by practical receivers based on mmse channel shortening and suboptimal symbol detection for a reduced state channel an upper bound is given in the form of a semi analytical solution derived using basic information theoretic inequalities by a grouping of the channel taps into several clusters resulting in a newly defined single input multiple output simo channel we show that the so obtained time dispersive simo channel can be represented by an equivalent single input single output siso channel with a significantly shorter channel memory the reduced computational complexity allows the use of the bcjr algorithm for the newly defined channel the proposed bounds are illustrated through several sparse channel examples and i u d input symbols showing that the upper bound significantly outperforms existing bounds performance of our lower bound strongly depends on the channel structure showing best results for minimum phase and maximum phase systems

disturbance decoupling problems via dynamic output feedback this paper considers the disturbance decoupling problems with or without internal stability and pole placement via dynamic output feedback using polynomial and rational matrix techniques we show that in all three problems considered the central solvability condition can be expressed as a two sided matching problem a bxc where a b and c are the polynomial system matrices of certain natural subsystems of the system model and x is to be determined over various subrings of the rational functions this matching problem can in turn be reduced to certain appropriate zero cancellation conditions on the polynomial system matrices a b and c

the structure of robust disturbance rejection control this paper deals with the problem of disturbance rejection under state feedback in linear multivariable control systems this problem is ill posed under arbitrary state space parameter perturbations however it is shown that with respect to arbitrary perturbations of the plant transfer function coefficients a robust solvability condition and possibly a robust synthesis can be obtained these results are significant because the class of arbitrary state space parameter variations is too rich for practical interest whereas the class of arbitrary variations in transfer functions more closely represents the realistic problem of engineering concern

lower bounds and approximations for the information rate of the isi channel we consider the discrete time intersymbol interference isi channel model with additive gaussian noise and fixed independent identically distributed inputs in this setting we investigate the expression put forth by shamai and laroia as a conjectured lower bound for the input output mutual information after application of a minimum mean square error decision feedback equalizer receiver a low signal to noise ratio snr expansion is used to prove that the conjectured bound does not hold under general conditions and to characterize inputs for which it is particularly ill suited one such input is used to construct a counterexample indicating that the shamai laroia expression does not always bound even the achievable rate of the channel thus excluding a natural relaxation of the original conjectured bound however this relaxed bound is then shown to hold for any finite entropy input and isi channel when the snr is sufficiently high we derive two conditions under which the relaxed bound holds involving compound channel capacity and quasiconvexity arguments finally new simple bounds for the achievable rate are proven and compared with other known bounds information estimation relations and estimation theoretic bounds play a key role in establishing our results