tests of mode coupling theory in two dimensions we analyze the glassy dynamics of binary mixtures of hard disks in two dimensions predictions of the mode coupling theory mct are tested with extensive brownian dynamics simulations measuring the collective particle density correlation functions in the vicinity of the glass transition we verify four predicted mixing effects for instance for large size disparities adding a small amount of small particles at a fixed packing fraction leads to a speedup in the long time dynamics while for small size disparities it leads to a slowing down qualitative features of the nonergodicity parameters and the relaxation which both depend in a nontrivial way on the mixing ratio are found in the simulated correlators studying one system in detail we are able to determine its ideal mct glass transition point as c 0 7948 and test mct predictions quantitatively

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

symmetries in generalized kahler geometry we define the notion of a moment map and reduction in both generalized complex geometry and generalized kahler geometry as an application we give very simple explicit constructions of bi hermitian structures on mathbb c mathbb p n hirzebruch surfaces the blow up of mathbb c mathbb p n at arbitrarily many points and other toric varieties as well as complex grassmannians

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

unobstructed k deformations of generalized complex structures and bi hermitian structures abstract we introduce k deformations of generalized complex structures on a compact kahler manifold m x j with an effective anti canonical divisor and show that obstructions to k deformations of generalized complex structures on m always vanish applying unobstructed k deformations and the stability theorem for generalized kahler structures we construct deformations of bi hermitian structures in the form j j t h t on a compact kahler surface with a non zero holomorphic poisson structure then we prove that a compact kahler surface s admits a non trivial bi hermitian structure with the torsion condition and the same orientation if and only if s has a non zero holomorphic poisson structure we also obtain bi hermitian structures j j h on del pezzo surfaces degenerate del pezzo surfaces and some ruled surfaces for which the complex structure j is not equivalent to j under diffeomorphisms

glass transition of hard spheres in high dimensions we have investigated analytically and numerically the liquid glass transition of hard spheres for dimensions d ensuremath rightarrow ensuremath infty in the framework of mode coupling theory the numerical results for the critical collective and self nonergodicity parameters f c k d and f c s k d exhibit non gaussian k dependence even up to d 800 f c s k d and f c k d differ for k ensuremath sim d 1 2 but become identical on a scale k ensuremath sim d which is proven analytically the critical packing fraction ensuremath varphi c d ensuremath sim d 2 2 ensuremath d is above the corresponding kauzmann packing fraction ensuremath varphi k d derived by a small cage expansion its quadratic pre exponential factor is different from the linear one found earlier the numerical values for the exponent parameter and therefore the critical exponents a and b depend on d even for the largest values of d

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

geometry of maurer cartan elements on complex manifolds the semi classical data attached to stacks of algebroids in the sense of kashiwara and kontsevich are maurer cartan elements on complex manifolds which we call extended poisson structures as they generalize holomorphic poisson structures a canonical lie algebroid is associated to each maurer cartan element we study the geometry underlying these maurer cartan elements in the light of lie algebroid theory in particular we extend lichnerowicz poisson cohomology and koszul brylinski homology to the realm of extended poisson manifolds we establish a sufficient criterion for these to be finite dimensional we describe how homology and cohomology are related through the evens lu weinstein duality module and we describe a duality on koszul brylinski homology which generalizes the serre duality of dolbeault cohomology

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

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

an architecture for addition and subtraction of long word length numbers in the logarithmic number system an architecture is described for performing addition and subtraction of numbers in the logarithmic number system using small lookup tables previous implementations require approximately 4 2 sup f words for f bits of precision in the fraction the author shows how to reduce the size of the lookup table to fewer than 18 f 2 sup f 2 words the key to this is the use of a linear approximation of the nonlinear functions stored in the lookup tables the functions involved are highly nonlinear in some regions so variable size regions are used for the approximation the replacement of the f exponential dependence on the number of words by f 2 allows roughly 50 more bits of fractional precision to be obtained for a given amount of rom the architecture is mathematically analyzed yielding explicit expressions for all design parameters the approach is illustrated with an example logarithmic addition and subtraction unit using 32 b words with 30 b exponents containing 22 fractional bits a factor of 118 reduction in table size compared to previous techniques is achieved for this example

on dual pairs in dirac geometry in this note we discuss weak dual pairs in dirac geometry we show that this notion appears naturally when studying the problem of pushing forward a dirac structure along a surjective submersion and we prove a dirac theoretic version of libermann s theorem from poisson geometry our main result is an explicit construction of self dual pairs for dirac structures this theorem not only recovers the global construction of symplectic realizations from crainic and m rcu j symplectic geom 9 4 435 444 2011 but allows for a more conceptual understanding of it yielding a simpler and more natural proof as an application of the main theorem we present a different approach to the recent normal form theorem around dirac transversals from bursztyn et al j fur die reine und angewandte mathematik crelles j doi 10 1515 crelle 2017 0014 2017

comments on an architecture for addition and subtraction of long word length numbers in the logarithmic number system by d m lewis d m lewis ibid vol 39 no 11 p 1325 36 nov 1990 recently described an architecture for performing addition and subtraction in the logarithmic number system using a taylor series approximation and rom tables here it is shown that a 25 improvement can be achieved in the table used to implement addition this improvement is significant when lewis technique is used to implement variations of the logarithmic number system which do not require the subtraction logarithm

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

leibniz algebras courant algebroids and multiplications on reductive homogeneous spaces we show that the skew symmetrized product on every leibniz algebra s can be realized on a reductive complement to a subalgebra in a lie algebra as a consequence we construct a nonassociative multiplication on s which when s is a lie algebra is derived from the integrated adjoint representation we apply this construction to realize the bracket operations on the sections of courant algebroids and on the omni lie algebras recently introduced by the second author 1 introduction skew symmetric bilinear operations which satisfy weak ened versions of the jacobi identity arise from a number of constructions in algebra and differential geometry the purpose of this paper is to show how cer tain of these operations in particular the courant brackets on the doubles of lie bialgebroids can be realized in a natural way on the tangent spaces of reduc tive homogeneous spaces we use our construction to take steps toward finding group like objects which integrate these not quite lie algebras the main ideas behind our construction come from work of k nomizu k yamaguti and m kikkawa nomizu 18 showed that affine connections with parallel torsion and curvature are locally equivalent to invariant connec tions on reductive homogeneous spaces and that each such space has a canonical connection for which parallel translation along geodesies agrees with the natu ral action of the group yamaguti 22 characterized the torsion and curvature tensors of nomizu s canonical connection as pairs of algebraic operations one bilinear and the other trilinear satisfying axioms defining what he called a gen eral lie triple system and what kikkawa later called a lie triple algebra in this paper we will call these objects lie yamaguti algebras when the trilinear operation is zero the bilinear operation is a lie algebra operation and the homo geneous space is locally a lie group on which the connection is the one which makes left invariant vector fields parallel finally kikkawa 7 showed how to integrate lie yamaguti algebras to nonassociative multiplications on reductive homogeneous spaces and he characterized these multiplications axiomatically unfortunately kikkawa s construction when applied in our setting does not quite reproduce the multiplication on a lie group when the curvature is zero rather it

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

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

non commutative batalin vilkovisky algebras homotopy lie algebras and the courant bracket we consider two different constructions of higher brackets first based on a grassmann odd nilpotent operator we define a non commutative generalization of the higher koszul brackets which are used in a generalized batalin vilkovisky algebra and we show that they form a homotopy lie algebra secondly we investigate higher so called derived brackets built from symmetrized nested lie brackets with a fixed nilpotent lie algebra element q we find the most general jacobi like identity that such a hierarchy satisfies the numerical coefficients in front of each term in these generalized jacobi identities are related to the bernoulli numbers we suggest that the definition of a homotopy lie algebra should be enlarged to accommodate this important case finally we consider the courant bracket as an example of a derived bracket we extend it to the big bracket of exterior forms and multi vectors and give closed formulas for the higher courant brackets

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

redundant logarithmic arithmetic a number system that offers advantages in some situations over conventional floating point and sign logarithmic number systems is described redundant logarithmic arithmetic like conventional logarithmic arithmetic relies on table lookups to make the arithmetic unit simpler than an equivalent floating point unit the cost of 32 bit subtraction in a redundant logarithmic number system is lower than previously published logarithmic subtraction methods the total memory requirement for a 29 bit redundant logarithmic unit is 16 k words compared to 22 k words by the best previously published conventional sign logarithm unit assuming similar addition techniques are employed a redundant logarithmic number system can be implemented with online arithmetic which would be impractical for a conventional sign logarithm number system the disadvantages of redundant arithmetic are typical of redundant number systems first the redundancy doubles the storage requirements for data values second the representation can become ill conditioned especially as a result of iterated multiplications third division and square root operations are more difficult to implement in redundant logarithmic arithmetic

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

beyond floating point un systeme numerique est propose pour l arithmetique d ordinateur basee sur des fonctions exponentielles iterees l avantage principal est d extirper le sous passement et le depassement de capacite mais il y a quelques autres avantages et ceux ci sont decrits et discutes

q manifolds and mackenzie theory double lie algebroids were discovered by kirill mackenzie from the study of double lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for lie algebroids and double vector bundles in this paper we establish a simple alternative characterization of double lie algebroids in a supermanifold language namely we show that a double lie algebroid in mackenzie s sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights our approach helps to simplify and elucidate mackenzie s original definition we show how it fits into a bigger picture of equivalent structures on neighbor double vector bundles it also opens ways for extending the theory to multiple lie algebroids which we introduce here

lns with co transformation competes with floating point the logarithmic number system has been proposed as an alternative to floating point arithmetic multiplication division and square root operations are accomplished with inexpensive fixed point methods but addition and subtraction are considerably more challenging recent work has demonstrated that these operations too can be done with similar speed and accuracy to their fp equivalents but the necessary circuitry is complex in particular it is dominated by the need for large rom tables for the storage and interpolation of non linear functions we describe a new co transformation procedure that eliminates much of the rom space and allows the easy synthesis of the remainder in logic and we then evaluate several interpolation methods that might benefit from it synthesised 32 bit implementations are compared with floating point units and show substantial reductions in delay with equivalent accuracy and area

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

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

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

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

improved addition for the logarithmic number system the logarithmic number system lns which offers a wide dynamic range with an independently choosable signal to noise ratio is discussed to realize addition voluminous lookup tables are needed a method is proposed to reduce the storage requirements of these tables by means of chebyshev approximation with unequally spaced partition points

fpga designs with optimized logarithmic arithmetic using a general polynomial approximation approach we present an arithmetic library generator for the logarithmic number system lns the generator produces optimized lns arithmetic libraries that improve significantly over previous lns designs on area and latency we also provide area cost estimation and bit accurate simulation tools that facilitate comparison between lns and floating point designs

quasiclassical lian zuckerman homotopy algebras courant algebroids and gauge theory we define a quasiclassical limit of the lian zuckerman homotopy bv algebra quasiclassical lz algebra on the subcomplex corresponding to light modes i e the elements of zero conformal weight of the semi infinite brst cohomology complex of the virasoro algebra associated with vertex operator algebra voa with a formal parameter we also construct a certain deformation of the brst differential parametrized by a constant two component tensor such that it leads to the deformation of the a subalgebra of the quasiclassical lz algebra altogether this gives a functor the category of voa with a formal parameter to the category of a algebras the associated generalized maurer cartan equation gives the analogue of the yang mills equation for a wide class of voas applying this construction to an example of voa generated by systems we find a remarkable relation between the courant algebroid and the homotopy algebra of the yang mills theory

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

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

jacobi structures revisited jacobi algebroids i e graded lie brackets on the grassmann algebra associated with a vector bundle which satisfy a property similar to that of the jacobi brackets are introduced they turn out to be equivalent to generalized lie algebroids in the sense of iglesias and marrero jacobi bialgebroids are defined in the same manner a lifting procedure of elements of this grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding lie brackets is developed this gives the possibility of associating canonically a lie algebroid with any local lie algebra in the sense of kirillov

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

generalized moser lemma we show how the classical moser lemma from symplectic geometry extends to generalized complex structures gcs on arbitrary courant algebroids for this we extend the notion of a lie derivative to sections of the tensor bundle i e j e with respect to sections of the courant algebroid e using the dorfman bracket we then give a cohomological interpretation of the existence of one parameter families of gcs on e and of flows of automorphims of e identifying all gcs of such a family in the particular case of symplectic manifolds we recover the results of moser finally we give a criterion to detect the local triviality of arbitrary gcs which generalizes the darboux weinstein theorem

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

input output decoupling of hamiltonian systems the linear case in this note we give necessary and sufficient conditions for a linear hamiltonian system to be input output decouplable by hamiltonian feedback i e feedback that preserves the hamiltonian structure in a second paper we treat the same problem for nonlinear hamiltonian systems

poisson integrators for lie poisson structures on mathbb r 3 this paper is concerned with the study of poisson integrators we are interested in lie poisson systems on first we focus on poisson integrators for constant poisson systems and the transformations used for transforming lie poisson structures to constant poisson structures then we construct local poisson integrators for lie poisson systems on finally we present the results of numerical experiments for two lie poisson systems and compare our poisson integrators with other known methods

performance of digital linear regulators which use logarithmic arithmetic this paper analyzes the performance of digital linear regulators which use finite precision logarithmic arithmetic a linear equation is derived which describes the steady state covariance matrix of such a closed loop regulator system the utility of this covariance matrix in designing regulators which use logarithmic arithmetic is illustrated through several numerical examples and the presentation of a regulator design methodology

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

nambu jacobi structures and jacobi algebroids we describe a nambu jacobi structure as a nambu poisson structure on a certain jacobi algebroid it is shown that the matched pair of leibniz algebroids for a nambu jacobi structure is the leibniz algebroid associated with this nambu poisson structure we also see that a different leibniz algebroid is associated with a nambu jacobi structure which is a natural generalization of the lie algebroid associated with a jacobi structure

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

hamiltonian formalism for general relativistic adiabatic fluids abstract we derive the hamiltonian structures of three theories non relativistic special relativistic and general relativistic adiabatic fluids each in the eulerian representation in riemannian space or lorentzian spacetime all by the same procedure using standard variational principles the evolution in each case is generated by a hamiltonian that is equivalent to that obtained from a canonical analysis for the gravitational variables the poisson bracket has the usual canonical symplectic structure however for the fluid variables the three theories all share the same lie poisson bracket when expressed in the appropriate spaces of physical variables constructed here this shared lie poisson bracket is associated to the dual of the semidirect product lie algebra of vector fields acting on differential forms an immediate consequence of this shared structure is that each of these theories possesses an infinite family of conservation laws the so called casimirs that belong to the kernel of the lie poisson bracket the role of these casimirs in the study of lyapunov stability or dynamic stability for fluid equilibria is discussed the relationship of this approach to other approaches in the literature is also discussed

non antisymmetric versions of nambu poisson and algebroid brackets we show that we can skip the skew symmetry assumption in the definition of nambu poisson brackets in other words an n ary bracket on the algebra of smooth functions which satisfies the leibniz rule and an n ary version of the jacobi identity must be skew symmetric a similar result holds for a non antisymmetric version of lie algebroids

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

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

poisson structures and integrable systems a poisson coalgebra is used to construct integrable hamiltonian systems we consider a poisson structure given by the bivector which does not form a lie algebra with respect to the poisson bracket xa xb p pab x we prove that this coalgebra may be used to generate integrable hamiltonian systems as an example we give the poisson tensor and we show that it is linked with the calogero system

canonical functions differential graded symplectic pairs in supergeometry and alexandrov kontsevich schwartz zaboronsky sigma models with boundaries consistent boundary conditions for alexandrov kontsevich schwartz zaboronsky aksz sigma models and the corresponding boundary theories are analyzed as their mathematical structures we introduce a generalization of differential graded symplectic manifolds called twisted qp manifolds in terms of graded symplectic geometry canonical functions and qp pairs we generalize the aksz construction of topological sigma models to sigma models with wess zumino terms and show that all the twisted poisson like structures known in the literature can actually be naturally realized as boundary conditions for aksz sigma models

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

low power logarithmic number system addition subtraction and their impact on digital filters this paper presents techniques for low power addition subtraction in the logarithmic number system lns and quantifies their impact on digital filter vlsi implementation the impact of partitioning the look up tables required for lns addition subtraction on complexity performance and power dissipation of the corresponding circuits is quantified two design parameters are exploited to minimize complexity namely the lns base and the organization of the lns word a roundoff noise model is used to demonstrate the impact of base and word length on the signal to noise ratio of the output of finite impulse response fir filters in addition techniques for the low power implementation of an lns multiply accumulate mac units are investigated furthermore it is shown that the proposed techniques can be extended to cotransformation based circuits that employ interpolators the results are demonstrated by evaluating the power dissipation complexity and performance of several fir filter configurations comprising one two or four mac units simulations of placed and routed vlsi lns based digital filters using a 90 nm 1 0 v cmos standard cell library reveal that significant power dissipation savings are possible by using optimized lns circuits at no performance penalty when compared to linear fixed point two s complement equivalents

pipelined computation of very large word length lns addition subtraction with polynomial hardware cost a novel pipelined method is proposed to compute the addition subtraction in very large word length logarithmic number system lns arithmetic digit parallel additive normalization and digit on line multiplicative normalization methods are adopted to compute the exponential and logarithmic functions respectively in lns addition subtraction these two methods can both be implemented in a pipelined and regular architecture the size of the required lookup tables is now proportional to a third order polynomial function instead of an exponential function of the word length the total size of the tables in a 32 bit lns unit is estimated to be less than 53 5 kbits and the total size of the table s in a 64 bit lns unit will be less than 471 kbits furthermore the hardware cost of the other circuits in the proposed lns unit is only proportional to the square of the word length this study also develops a simple leading zero bits prediction technique that can significantly enhance the precision in lns addition subtraction computation when compared to floating point arithmetic design our approach still suffers from large hardware cost and pipeline latency however the proposed approach has provided a theoretical advancement in the development of very large word length lns arithmetic

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

applying features of ieee 754 to sign logarithm arithmetic various features found in standard floating point arithmetic ieee 754 are examined in light of their appropriateness for sign logarithm arithmetic the emphasis is on a 32 b word size comparable to ieee 754 single precision although other word sizes are possible a multilayer sign logarithm format is considered the lowest layer similar to previous implementations would provide only normalized representations but would not provide representations for zero denormalized values infinities and nans the highest layer would provide most of the features found in ieee 754 including zeros denormalized values infinities and nans novel algorithms for implementing logarithmic denormalized arithmetic are presented simulation results show that the error characteristics of the proposed logarithmic denormalized arithmetic algorithms are similar to those of the denormalized floating point arithmetic in ieee 754

efficient techniques for binary to multidigit multidimensional logarithmic number system conversion using range addressable look up tables the multidimensional logarithmic number system mdlns which has similar properties to the classical logarithmic number system lns provides more degrees of freedom than the lns by virtue of having two or more orthogonal bases and has the ability to use multiple mdlns components or digits unlike the lns there is no monotonic relationship between standard binary representations and mdlns representations using look up tables luts to perform the mapping function can be unrealistic for hardware implementations when large binary ranges or multiple digits are used this work proposes a novel range addressable technique for using look up tables that allows efficient conversion from binary to single or multidigit mdlns with varying accuracies depending on the selected implementation

dimensional study of the caging order parameter at the glass transition the glass problem is notoriously hard and controversial even at the mean field level little is agreed upon regarding why a fluid becomes sluggish while exhibiting but unremarkable structural changes it is clear however that the process involves self caging which provides an order parameter for the transition it is also broadly assumed that this cage should have a gaussian shape in the mean field limit here we show that this ansatz does not hold by performing simulations as a function of spatial dimension d we find the cage to keep a nontrivial form quantitative mean field descriptions of the glass transition such as mode coupling theory density functional theory and replica theory all miss this crucial element although the mean field random first order transition scenario of the glass transition is qualitatively supported here and non mean field corrections are found to remain small on decreasing d reconsideration of its implementation is needed for it to result in a coherent description of experimental observations

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

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

arithmetic co transformations in the real and complex logarithmic number systems the real logarithmic number system which represents a value with a sign bit and a quantized logarithm can be generalized to create the complex logarithmic number system which replaces the sign bit with a quantized angle in a log polar coordinate system although multiplication and related operations are easy in both real and complex systems addition and subtraction are hard especially when interpolation is used to implement the system both real and complex logarithmic arithmetic benefit from the use of co transformation which converts an addition or subtraction from a region where interpolation is expensive to a region where it is easier two co transformations that accomplish this goal are introduced the first is an approximation based on real analysis of the subtraction logarithm the second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction

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

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

mode coupling theory of the glass transition for confined fluids we present a detailed derivation of a microscopic theory for the glass transition of a liquid enclosed between two parallel walls relying on a mode coupling approximation this geometry lacks translational invariance perpendicular to the walls which implies that the density profile and the density density correlation function depends explicitly on the distances to the walls we discuss the residual symmetry properties in slab geometry and introduce a symmetry adapted complete set of two point correlation functions since the currents naturally split into components parallel and perpendicular to the walls the mathematical structure of the theory differs from the established mode coupling equations in bulk we prove that the equations for the nonergodicity parameters still display a covariance property similar to bulk liquids

on a new relation between jacobi and homogeneous poisson manifolds we establish a new very close relationship which links jacobi structures and homogeneous poisson structures defined on the same manifold and study the characteristic foliations of the related structures several examples of this construction are also given

dirac geometry of the holonomy fibration in this paper we solve the problem of giving a gauge theoretic description of the natural dirac structure on a lie group which plays a prominent role in the theory of d branes for the wess zumino witten model as well as the theory of quasi hamiltonian spaces we describe the structure as an infinite dimensional reduction of the space of connections over the circle our insight is that the formal poisson structure on the space of connections is not an actual poisson structure but is itself a dirac structure due to the fact that it is defined by an unbounded operator we also develop general tools for reducing courant algebroids and morphisms between them allowing us to give a precise correspondence between hamiltonian loop group spaces and quasi hamiltonian spaces

poisson quasi nijenhuis manifolds we introduce the notion of poisson quasi nijenhuis manifolds generalizing poisson nijenhuis manifolds of magri morosi we also investigate the integration problem of poisson quasi nijenhuis manifolds in particular we prove that under some topological assumption poisson quasi nijenhuis manifolds are in one one correspondence with symplectic quasi nijenhuis groupoids as an application we study generalized complex structures in terms of poisson quasi nijenhuis manifolds we prove that a generalized complex manifold corresponds to a special class of poisson quasi nijenhuis structures as a consequence we show that a generalized complex structure integrates to a symplectic quasi nijenhuis groupoid recovering a theorem of crainic

sign logarithm arithmetic for fft implementation sign logarithm arithmetic is applicable to a variety of numerical applications where wide dynamic range and small wordsize are required in this paper the basic sign logarithm arithmetic operations required for signal processing i e addition subtraction and multiplication are reviewed the computational errors are analyzed for fft realization and simulation results are presented which serve to verify the analysis it is shown that the sign logarithm approach provides improved arithmetic quantization error performance for a given word size over fft s implemented with conventional fixed or floating point arithmetic and that the sign logarithm implementation is faster and less complex than conventional approaches

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

interleaved memory function interpolators with application to an accurate lns arithmetic unit this paper describes a new method for polynomial interpolation in hardware with advantages demonstrated by its application to an accurate logarithmic number system lns arithmetic unit the use of an interleaved memory reduces storage requirements by allowing each stored function value to be used in interpolation across several segments this strategy can be shown to always use fewer words of memory than an optimized polynomial with stored polynomial coefficients interleaved memory function interpolators are then applied to the specific goal of an accurate logarithmic number system arithmetic unit many accuracy requirements for the lns arithmetic unit are possible although a round to nearest would be desirable it cannot be easily achieved the goal suggested is to insure that the worst case lns relative error is smaller than the worst case floating point fp relative error using the interleaved memory interpolator the detailed design of an lns arithmetic unit is performed using a second order polynomial interpolator including approximately 91k bits of rom this arithmetic unit has better accuracy and less complexity than previous lns units

integrated circuit logarithmic arithmetic units this correspondence examines integrated circuit logarithmic arithmetic units which include adders subtracters multipliers and dividers the design of these arithmetic units is reviewed and an example arithmetic unit which performs multiplication followed by addition is designed in detail the design results are used to develop a size and speed comparison of integrated circuit logarithmic and fixed point arithmetic units this comparison is exercised through a video signal processing example it is concluded from this comparison that logarithmic arithmetic units are smaller than and as fast as fixed point arithmetic units with comparable capabilities in digital signal processing applications characterized by large dynamic range and moderate computational accuracy requirements further this comparison quantitatively illustrates the interaction of digital signal processing and integrated circuit issues in the design of special purpose digital signal processors

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

semi logarithmic number systems we present a new class of number systems called semi logarithmic number systems that constitute a family of various compromises between floating point and logarithmic number systems this allows trade between the speed of the arithmetic operations and the size of the required tables we give arithmetic algorithms addition subtraction multiplication division for the semi logarithmic number systems and we compare these number systems to the classical floating point or logarithmic number systems

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