dirac nijenhuis structures in this paper we study the concept of dirac nijenhuis structures we consider them to be a pair where d is a dirac structure defined with respect to a lie bialgebroid a a and is a nijenhuis operator which defines a deformation of the lie algebroid structure of d in a compatible way the transformation can be considered to deform also the double of the lie bialgebroid which leads to the concept of dirac nijenhuis structures of type i or to not affect it leading to dirac nijenhuis structures of type ii we prove that the concept contains poisson nijenhuis structures as a particular case and provides another example for the case of kahler manifolds

worst case expected capacity loss of slow fading channels for delay limited communication over block fading channels the difference between the ergodic capacity and the maximum achievable expected rate for coding over a finite number of coherent blocks represents a fundamental measure of the penalty incurred by the delay constraint this paper introduces a notion of worst case expected capacity loss focusing on the slow fading scenario one block delay the worst case additive and multiplicative expected capacity losses are precisely characterized for the point to point fading channel extension to the problem of writing on fading paper is also considered where both the ergodic capacity and the additive expected capacity loss over one block delay are characterized to within one bit per channel use

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

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given

bv quantization of topological open membranes we study bulk boundary correlators in topological open membranes the basic example is the open membrane with a wz coupling to a 3 form we view the bulk interaction as a deformation of the boundary string theory this boundary string has the structure of a homotopy lie algebra which can be viewed as a closed string field theory we calculate the leading order perturbative expansion of this structure for the 3 form field we find that the c field induces a trilinear bracket deforming the lie algebra structure this paper is the first step towards a formal universal quantization of general quasi lie bialgebroids

variable length coding over an unknown channel burnashev in 1976 gave an exact expression for the reliability function of a discrete memoryless channel dmc with noiseless feedback a coding scheme that achieves this exponent needs in general to know the statistics of the channel suppose now that the coding scheme is designed knowing only that the channel belongs to a family q of dmcs is there a coding scheme with noiseless feedback that achieves burnashev s exponent uniformly over q at a nontrivial rate we answer the question in the affirmative for two families of channels binary symmetric and z for these families we show that for any given fraction there is a feedback coding strategy such that for any member of the family i guarantees this fraction of its capacity as rate and ii guarantees the corresponding burnashev s exponent therefore for these families in terms of delay and error probability the knowledge of the channel becomes asymptotically irrelevant in feedback code design there are blind schemes that perform as well as the best coding scheme designed with the foreknowledge of the channel under use however a converse result shows that in general even for families that consist of only two channels such blind schemes do not exist

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

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

generalized complex geometry generalized complex geometry encompasses complex and symplectic ge ometry as its extremal special cases we explore the basic properties of this geometry including its enhanced symmetry group elliptic deforma tion theory relation to poisson geometry and local structure theory we also dene and study generalized complex branes which interpolate be tween at bundles on lagrangian submanifolds and holomorphic bundles on complex submanifolds

infinitesimal moduli for the strominger system and killing spinors in generalized geometry we construct the space of infinitesimal variations for the strominger system and an obstruction space to integrability using elliptic operator theory we initiate the study of the geometry of the moduli space describing the infinitesimal structure of a natural foliation on this space the associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable killing spinor equations on a courant algebroid as an application we propose a unifying framework for metrics with holonomy mathrm su 3 and solutions of the strominger system

reduction of poisson manifolds and jacobi manifolds via dirac theory we give two kinds of reduction relating poisson manifolds and jacobi manifolds by classifying reducible dirac and jacobi dirac structures into two classes respectively the proof needs only some direct calculation without using the existence of momentum mappings or the introducing of admissible functions etc meanwhile we present some examples and applications

star products on graded manifolds and α corrections to courant algebroids from string theory courant algebroids originally used to study integrability conditions for dirac structures have turned out to be of central importance to study the effective supergravity limit of string theory the search for a geometric description of t duality leads to double field theory dft whose gauge algebra is governed by the c bracket a generalization of the courant bracket in the sense that it reduces to the latter by solving a specific constraint recently in dft deformations of the c bracket and o d d invariant bilinear form to first order in the closed string sigma model coupling were derived by analyzing the transformation properties of the neveu schwarz b field by choosing a particular poisson structure on the drinfel d double corresponding to the courant algebroid structure of the generalized tangent bundle we are able to interpret the c bracket and bilinear form in terms of poisson brackets as a result we reproduce the deformations for a specific solution to the strong constraint of dft as expansion of a graded version of the moyal weyl star product

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

on a 145 32 binary cyclic code we discovered that a 145 32 binary cyclic code generated by the product of an irreducible polynomial of degree 28 with exponent 145 and the polynomial x sup 4 x sup 3 x sup 2 x 1 has a minimum distance of 44

torsion free generalized connections and heterotic supergravity this work revisits the notions of connection and curvature in generalized geometry with emphasis on torsion free generalized connections on a transitive courant algebroid as an application we provide a mathematical derivation of the equations of motion of heterotic supergravity in terms of the ricci tensor of a generalized metric inspired by the work of coimbra strickland constable and waldram

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

lagrangian approach to sheaves of vertex algebras we explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of lagrangians the exposition is focused on two examples the wzw model and the 1 1 supersymmetric model we conclude by showing how to construct a family of vertex algebras with base the barannikov kontsevich moduli space thus furnishing the b model moduli for witten s half twisted model

division algebras pseudo orthogonal groups and spinors the groups so nu 1 so nu so nu 1 so nu 1 1 and so nu 2 2 nu 1 2 4 8 and their spin representations are described in terms of the division algebras r c h and o

graded contractions of lie algebras and central extensions we use the concept of grading of lie algebras to investigate the appearance of central charges during the contraction process as for the usual graded contractions one finds simultaneously the central extensions of classes of algebras rather than specific lie algebras to illustrate the method we consider in detail two physical applications the kinematical algebras of spacetime and the u n bosons limits of some lie algebras

group actions on dg manifolds and exact courant algebroids let g be a lie group acting by diffeomorphisms on a manifold m and consider the image of t 1 g and t 1 m of g and m respectively in the category of differential graded manifolds we show that the obstruction to lift the action of t 1 g on t 1 m to an action on a mathbb r n bundle over t 1 m is measured by the g equivariant cohomology of m we explicitly calculate the differential graded lie algebra of the symmetries of the mathbb r n bundle over t 1 m and we use this differential graded lie algebra to understand which actions are hamiltonian we show how split exact courant algebroids could be obtained as the derived leibniz algebra of the symmetries of mathbb r 2 bundles over t 1 m and we use this construction to propose that the infinitesimal symmetries of a split exact courant algebroid should be encoded in the differential graded lie algebra of symmetries of a mathbb r 2 bundle over t 1 m with this setup at hand we propose a definition for an action of a lie group on an exact courant algebroid and we propose conditions for the action to be hamiltonian

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

distortion minimization in gaussian layered broadcast coding with successive refinement a transmitter without channel state information wishes to send a delay limited gaussian source over a slowly fading channel the source is coded in superimposed layers with each layer successively refining the description in the previous one the receiver decodes the layers that are supported by the channel realization and reconstructs the source up to a distortion the expected distortion is minimized by optimally allocating the transmit power among the source layers for two source layers the allocation is optimal when power is first assigned to the higher layer up to a power ceiling that depends only on the channel fading distribution all remaining power if any is allocated to the lower layer for convex distortion cost functions with convex constraints the minimization is formulated as a convex optimization problem in the limit of a continuum of infinite layers the minimum expected distortion is given by the solution to a set of linear differential equations in terms of the density of the fading distribution as the number of channel uses per source symbol tends to zero the power distribution that minimizes expected distortion converges to the one that maximizes expected capacity

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

algebroid structures on para hermitian manifolds we present a global construction of a so called d bracket appearing in the physics literature of double field theory dft and show that if certain integrability criteria are satisfied it can be seen as a sum of two courant algebroid brackets in particular we show that the local picture of the extended space time used in dft fits naturally in the geometrical framework of para hermitian manifolds and that the data of an almost para hermitian manifold is sufficient to construct the d bracket moreover the twists of the bracket appearing in dft can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para hermitian structure we present a global construction of a so called d bracket appearing in the physics literature of double field theory dft and show that if certain integrability criteria are satisfied it can be seen as a sum of two courant algebroid brackets in particular we show that the local picture of the extended space time used in dft fits naturally in the geometrical framework of para hermitian manifolds and that the data of an almost para hermitian manifold is sufficient to construct the d bracket moreover the twists of the bracket appearing in dft can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para hermitian structure

twisted jacobi manifolds twisted dirac jacobi structures and quasi jacobi bialgebroids we study twisted jacobi manifolds a concept that we had introduced in a previous note twisted jacobi manifolds can be characterized using twisted dirac jacobi which are sub bundles of courant jacobi algebroids we show that each twisted jacobi manifold has an associated lie algebroid with a 1 cocycle we introduce the notion of quasi jacobi bialgebroid and prove that each twisted jacobi manifold has a quasi jacobi bialgebroid canonically associated moreover the double of a quasi jacobi bialgebroid is a courant jacobi algebroid several examples of twisted jacobi manifolds and twisted dirac jacobi structures are presented

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

the general solution of the real graded contractions of the general solution of the graded contraction equations for a grading of the real compact simple lie algebra so n 1 is presented in an explicit way it turns out to depend on independent real parameters the structure of the general graded contractions is displayed for the low dimensional cases and kinematical algebras are shown to appear straightforwardly the geometrical or physical meaning of the contraction parameters as curvatures is also analysed in particular for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous spacetimes

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

solutions of the strominger system via stable bundles on calabi yau threefolds we prove that a given calabi yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the strominger system provided that the second chern class of the vector bundle is equal to the second chern class of the tangent bundle if the calabi yau threefold has strict su 3 holonomy then the equations of motion derived from the heterotic string effective action are also satisfied by the solutions we obtain

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

central extensions of the quasi orthogonal lie algebras we determine the central extensions of a whole family of lie algebras obtained by the method of graded contractions from n arbitrary all the inhomogeneous orthogonal and pseudo orthogonal algebras are members of this family as well as a large number of other non semisimple algebras all of which have at least a semidirect structure in some cases two or more the dimensions of their second cohomology groups and the explicit expression of their central extensions are given

frame like geometry of double field theory we relate two formulations of the recently constructed double field theory to a frame like geometrical formalism developed by siegel a self contained presentation of this formalism is given including a discussion of the constraints and its solutions and of the resulting riemann tensor ricci tensor and curvature scalar this curvature scalar can be used to define an action and it is shown that this action is equivalent to that of double field theory

the family of quaternionic quasi unitary lie algebras and their central extensions the family of quaternionic quasi unitary or quaternionic unitary cayley klein algebras is described in a unified setting this family includes the simple algebras sp n 1 and sp p q in the cartan series cn 1 as well as many non semisimple real lie algebras which can be obtained from these simple algebras by particular contractions the algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature this common framework allows one to perform the study of many properties for all these lie algebras simultaneously in this paper the central extensions for all quasi simple lie algebras of the quaternionic unitary cayley klein family are shown to be trivial no matter their dimension

variable rate channel capacity this paper introduces the notions of variable to fixed and fixed to variable channel capacity without feedback for channels that satisfy the strong converse these notions coincide with the conventional shannon capacity for channels that do not behave ergodically the conventional fixed rate shannon capacity only depends on least favorable channel conditions while the variable rate capacity notions are able to capture the whole range of channel states and their likelihood even in the absence of any side information about channel state at the transmitter particular emphasis is placed on memoryless channels that are governed by finitely valued states we show that single user variable to fixed channel capacity is intimately connected to the capacity region of broadcast channels with degraded message sets and we give an expression for the fixed to variable capacity

ricci flow killing spinors and t duality in generalized geometry abstract we introduce a notion of ricci flow in generalized geometry extending a previous definition by gualtieri on exact courant algebroids special stationary points of the flow are given by solutions to first order differential equations the killing spinor equations which encompass special holonomy metrics with solutions of the hull strominger system our main result investigates a method to produce new solutions of the ricci flow and the killing spinor equations for this we consider t duality between possibly topologically distinct torus bundles endowed with courant structures and demonstrate that solutions of the equations are exchanged under this symmetry as applications we give a mathematical explanation of the dilaton shift in string theory and prove that the hull strominger system is preserved by t duality

optimistic shannon coding theorems for arbitrary single user systems the conventional definitions of the source coding rate and of channel capacity require the existence of reliable codes for all sufficiently large block lengths alternatively if it is required that good codes exist for infinitely many block lengths then optimistic definitions of source coding rate and channel capacity are obtained in this work formulas for the optimistic minimum achievable fixed length source coding rate and the minimum spl epsi achievable source coding rate for arbitrary finite alphabet sources are established the expressions for the optimistic capacity and the optimistic spl epsi capacity of arbitrary single user channels are also provided the expressions of the optimistic source coding rate and capacity are examined for the class of information stable sources and channels respectively finally examples for the computation of optimistic capacity are presented

feedback in the non asymptotic regime without feedback the backoff from capacity due to non asymptotic blocklength can be quite substantial for blocklengths and error probabilities of interest in many practical applications in this paper novel achievability bounds are used to demonstrate that in the non asymptotic regime the maximal achievable rate improves dramatically thanks to variable length coding and feedback for example for the binary symmetric channel with capacity 1 2 the blocklength required to achieve 90 of the capacity is smaller than 200 compared to at least 3100 for the best fixed blocklength code even with noiseless feedback virtually all the advantages of noiseless feedback are shown to be achievable even if the feedback link is used only to send a single signal informing the encoder to terminate the transmission stop feedback it is demonstrated that the non asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the end of packet control signal fixed blocklength codes and related questions concerning communicating with a guaranteed delay are discussed in which situation feedback is demonstrated to be almost useless even non asymptotically

dirac structures for generalized lie bialgebroids we study dirac structures for generalized courant algebroids which are doubles of generalized lie bialgebroids the cases investigated include graphs of bivector fields and characteristic pairs of some sub bundles

six dimensional 1 0 superconformal models and higher gauge theory we analyze the gauge structure of a recently proposed superconformal field theory in six dimensions we find that this structure amounts to a weak courant dorfman algebra which in turn can be interpreted as a strong homotopy lie algebra this suggests that the superconformal field theory is closely related to higher gauge theory describing the parallel transport of extended objects indeed we find that under certain restrictions the field content and gauge transformations reduce to those of higher gauge theory we also present a number of interesting examples of admissible gauge structures such as the structure lie 2 algebra of an abelian gerbe differential crossed modules the 3 algebras of m2 brane models and string lie 2 algebras

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

achieving the empirical capacity using feedback memoryless additive models we address the problem of universal communications over an unknown channel with an instantaneous noiseless feedback and show how rates corresponding to the empirical behavior of the channel can be attained although no rate can be guaranteed in advance first we consider a discrete modulo additive channel with alphabet x where the noise sequence zn is arbitrary and unknown and may causally depend on the transmitted and received sequences and on the encoder s message possibly in an adversarial fashion although the classical capacity of this channel is zero we show that rates approaching the empirical capacity log x h emp zn can be universally attained where h emp zn is the empirical entropy of zn for the more general setting where the channel can map its input to an output in an arbitrary unknown fashion subject only to causality we model the empirical channel actions as the modulo addition of a realized noise sequence and show that the same result applies if common randomness is available the results are proved constructively by providing a simple sequential transmission scheme approaching the empirical capacity

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

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

courant algebroids poisson lie t duality and type ii supergravities we reexamine the notions of generalized ricci tensor and scalar curvature on a general courant algebroid reformulate them using objects natural w r t pull backs and reductions and obtain them from the variation of a natural action functional this allows us to prove in a very general setup the compatibility of the poisson lie t duality with the renormalization group flow and with string background equations we thus extend the known results to a much wider class of dualities including the cases with gauging so called dressing cosets or equivariant poisson lie t duality as an illustration we use the formalism to provide new classes of solutions of modified supergravity equations on symmetric spaces

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

cayley klein algebras as graded contractions of so n 1 we study z2 x n graded contractions of the real compact simple lie algebra so n 1 and we identify within them the cayley klein algebras as a naturally distinguished subset

the capacity of channels with feedback in this paper we introduce a general framework for treating channels with memory and feedback first we prove a general feedback channel coding theorem based on massey s concept of directed information second we present coding results for markov channels this requires determining appropriate sufficient statistics at the encoder and decoder we give a recursive characterization of these sufficient statistics third a dynamic programming framework for computing the capacity of markov channels is presented fourth it is shown that the average cost optimality equation acoe can be viewed as an implicit single letter characterization of the capacity fifth scenarios with simple sufficient statistics are described sixth error exponents for channels with feedback are presented

a broadcast approach for a single user slowly fading mimo channel a broadcast transmission strategy for the slowly fading gaussian multiple input multiple output mimo channel is introduced this broadcast strategy is an extension of the single input single output siso broadcast approach perfect channel state information csi is assumed known at the receiver end only this strategy facilitates to adapt the reliably decoded rate to the actual channel state without having any feedback link to the transmitter transmission of layered coded information is motivated by the theory of majorization we derive the basic equations characterizing achievable rates of the strategy several ad hoc approximations to the achievable region are considered and their performance is compared with the siso setting and the ergodic capacity it has been demonstrated that a single layer outage approach is reasonably efficient in the mimo setting in terms of the average reliably decoded rate a multiple access channel mac broadcast approach is also applied for the mimo case and demonstrated to be relatively efficient

joint source channel coding with feedback this paper quantifies the fundamental limits of variable length transmission of a general possibly analog source over a memoryless channel with noiseless feedback under a distortion constraint we consider excess distortion average distortion and guaranteed distortion d semifaithful codes in contrast to the asymptotic fundamental limit a general conclusion is that allowing variable length codes and feedback leads to a sizable improvement in the fundamental delay distortion tradeoff in addition we investigate the minimum energy required to reproduce k source samples with a given fidelity after transmission over a memoryless gaussian channel and we show that the required minimum energy is reduced with feedback and an average rather than maximal power constraint

the first pontryagin class of a quadratic lie 2 algebroid in this paper we first give a detailed study on the structure of a transitive lie 2 algebroid and describe a transitive lie 2 algebroid using a morphism from the tangent lie algebroid t m to a strict lie 3 algebroid constructed from derivations then we introduce the notion of a quadratic lie 2 algebroid and define its first pontryagin class which is a cohomology class in h5 m associated to a clwx 2 algebroid there is a quadratic lie 2 algebroid naturally conversely we show that the first pontryagin class of a quadratic lie 2 algebroid is the obstruction class of the existence of a clwx extension finally we construct a quadratic lie 2 algebroid from a trivial principle 2 bundle with a gamma connection and show that its first pontryagin class is trivial

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

central extensions of the families of quasi unitary lie algebras the most general possible central extensions of two whole families of lie algebras which can be obtained by contracting the special pseudo unitary algebras of the cartan series and the pseudo unitary algebras are completely determined and classified for arbitrary p and q in addition to the and algebras whose second cohomology group is well known to be trivial each family includes many non semisimple algebras their central extensions which are explicitly given can be classified into three types as far as their properties under contraction are involved a closed expression for the dimension of the second cohomology group of any member of these families of algebras is given

generalized kahler geometry generalized kahler geometry is the natural analogue of kahler geometry in the context of generalized complex geometry just as we may require a complex structure to be compatible with a riemannian metric in a way which gives rise to a symplectic form we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure we prove that generalized kahler geometry is equivalent to the bi hermitian geometry on the target of a 2 dimensional sigma model with 2 2 supersymmetry we also prove the existence of natural holomorphic courant algebroids for each of the underlying complex structures and that these split into a sum of transverse holomorphic dirac structures finally we explore the analogy between pre quantum line bundles and gerbes in the context of generalized kahler geometry

even symplectic supermanifolds and double field theory over many decades the word double has appeared in various contexts which at times seem to be unrelated 1 several have some relation to mathematical physics recently this has become particularly striking in double field theory dft two doubles that are particularly relevant are double vector bundles and drinfel d doubles

the graded jacobi algebras and co homology jacobi algebroids i e jacobi versions of lie algebroids are studied in the context of graded jacobi brackets on graded commutative algebras this unifies various concepts of graded lie structures in geometry and physics a method of describing such structures by classical lie algebroids via certain gauging in the spirit of e witten s gauging of the exterior derivative is developed one constructs a corresponding cartan differential calculus graded commutative one in a natural manner this in turn gives canonical generating operators for triangular jacobi algebroids one gets in particular the lichnerowicz jacobi homology operators associated with classical jacobi structures courant jacobi brackets are obtained in a similar way and used to define an abstract notion of a courant jacobi algebroid and dirac jacobi structure