weakly lefschetz symplectic manifolds for a symplectic manifold the harmonic cohomology of symplectic divisors introduced by donaldson 1996 and of the more general symplectic zero loci introduced by auroux 1997 are compared with that of its ambient space we also study symplectic manifolds satisfying a weakly lefschetz property that is the s lefschetz property in particular we consider the symplectic blow ups cp m of the complex projective space cp m along weakly lefschetz symplectic submanifolds m cp m as an application we construct for each even integer s 2 compact symplectic manifolds which are s lefschetz but not s 1 lefschetz

batalin vilkovisky quantization and the algebraic index abstract into a geometric setting we import the physical interpretation of index theorems via semi classical analysis in topological quantum field theory we develop a direct relationship between fedosov s deformation quantization of a symplectic manifold x and the batalin vilkovisky bv quantization of a one dimensional sigma model with target x this model is a quantum field theory of aksz type and is quantized rigorously using costello s homotopic theory of effective renormalization we show that fedosov s abelian connections on the weyl bundle produce solutions to the effective quantum master equation moreover bv integration produces a natural trace map on the deformation quantized algebra this formulation allows us to exploit a rigorous localization argument in quantum field theory to deduce the algebraic index theorem via semi classical analysis i e one loop feynman diagram computations

the steiner tree problem i formulations compositions and extension of facets in this paper we give some integer programming formulations for the steiner tree problem on undirected and directed graphs and study the associated polyhedra we give some families of facets for the undirected case along with some compositions and extensions we also give a projection that relates the steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph this is used to show that the lp relaxation of the directed formulation is superior to the lp relaxation of the undirected one

lichnerowicz jacobi cohomology in this paper we extend the notion of lichnerowicz poisson cohomology to jacobi manifolds we study the relation of the so called lichnerowicz jacobi cohomology with the basic de rham cohomology and the cohomology of the lie algebra of functions relative to the representation defined by the hamiltonian vector fields a natural pairing with the canonical homology is constructed the relation between the lichnerowicz poisson cohomology of a quantizable poisson manifold and the lichnerowicz jacobi cohomology of the total space of a prequantization bundle is obtained particular cases of cosymplectic contact and locally conformal symplectic manifolds are discussed finally the lichnerowicz jacobi cohomology of a non transitive example is studied

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

induced c complexes in metaplectic geometry for a symplectic manifold admitting a metaplectic structure and for a kuiper map we construct a complex of differential operators acting on exterior differential forms with values in the dual of kostant s symplectic spinor bundle defining a hilbert c structure on this bundle for a suitable c algebra we obtain an elliptic c complex in the sense of mishchenko fomenko its cohomology groups appear to be finitely generated projective hilbert c modules the paper can serve as a guide for handling differential complexes and pdes on hilbert bundles

solving group steiner problems as steiner problems abstract the generalized spanning tree or group steiner problem gsp is a generalization of the steiner problem in graphs spg one requires a tree spanning at least one vertex of each subset given in a family of vertex subsets while minimizing the sum of the corresponding edge costs specialized solution procedures have been developed for this problem in this paper we investigate the performance of a known but so far neglected transformation to the undirected steiner problem in graphs when combined with a recent metaheuristic for the spg this straightforward approach compares favorably with specialized gsp heuristics thus we set a standard for future algorithms

zero energy fields on complex projective space we consider complex projective space with its fubini study metric and the x ray transform defined by integration over its geodesics we identify the kernel of this transform acting on symmetric tensor fields

generalized cohomologies and supersymmetry we show that the complex cohomologies of bott chern and aeppli and the symplectic cohomologies of tseng and yau arise in the context of type ii string theory specifically they can be used to count a subset of scalar moduli fields in minkowski compactification with rr fluxes in the presence of either o5 d5 or o6 d6 brane sources respectively further we introduce a new set of cohomologies within the generalized complex geometry framework which interpolate between these known complex and symplectic cohomologies the generalized complex cohomologies play the analogous role for counting massless fields for a general supersymmetric minkowski type ii compactification with ramond ramond flux

poisson homology in degree 0 for some rings of symplectic invariants abstract let g be a finite dimensional semi simple lie algebra h a cartan subalgebra of g and w its weyl group the group w acts diagonally on v h h as well as on c v the purpose of this article is to study the poisson homology of the algebra of invariants c v w endowed with the standard symplectic bracket to begin with we give general results about the poisson homology space in degree 0 denoted by hp 0 c v w in the case where g is of type b n c n or d n results which support alev s conjecture then we are focusing the interest on the particular cases of ranks 2 and 3 by computing the poisson homology space in degree 0 in the cases where g is of type b 2 so 5 d 2 so 4 then b 3 so 7 and d 3 a 3 so 6 sl 4 in order to do this we make use of a functional equation introduced by y berest p etingof and v ginzburg we recover by a different method the result established by j alev and l foissy according to which the dimension of hp 0 c v w equals 2 for b 2 then we calculate the dimension of this space and we show that it is equal to 1 for d 2 we also calculate it for the rank 3 cases we show that it is equal to 3 for b 3 c 3 and 1 for d 3 a 3

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

the quantum poincare birkhoff witt theorem we define and study a wide class of associative algebras in which the poincare birkhoff witt theorem is valid this class includes numerous quantum algebras which recently appeared

poisson co homology of polynomial poisson algebras in dimension four sklyanin s case abstract in this paper we compute the poisson co homology of a polynomial poisson structure given by two casimir polynomial functions which define a complete intersection with an isolated singularity

a new relaxation method for the generalized minimum spanning tree problem we consider a generalization of the minimum spanning tree problem called the generalized minimum spanning tree problem denoted by gmst it is known that the gmst problem is hard we present several mixed integer programming formulations of the problem based on a new formulation of the problem we give a new solution procedure that finds the optimal solution of the gmst problem for graphs with nodes up to 240 we discuss the advantages of our approach in comparison with earlier methods

twisted poincare duality between poisson homology and poisson cohomology abstract a version of the twisted poincare duality is proved between the poisson homology and cohomology of a polynomial poisson algebra with values in an arbitrary poisson module the duality is achieved by twisting the poisson module structure in a canonical way which is constructed from the modular derivation in the case that the poisson structure is unimodular the twisted poincare duality reduces to the poincare duality in usual sense the main result generalizes the work of launois and richard 8 for the quadratic poisson structures and zhu 25 for the linear poisson structures

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

conformally fedosov manifolds abstract we introduce the notion of a conformally fedosov structure and construct an associated cartan connection when an appropriate curvature vanishes this allows us to construct a family of natural differential complexes akin to the bgg complexes from parabolic geometry

hochschild and cyclic homology of quantum groups for an arbitrary complex linear semisimple lie groupg we consider hopf algebras of the deformations of the formal and algebraic functions ong the hochschild and cyclic homology of these hopf algebras are computed when the value of the deformation parameter is generic

gerstenhaber algebras and bv algebras in poisson geometry the purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well known algebraic structures such as gerstenhaber algebras and bv algebras some applications are discussed in particular we find an explicit connection between the koszul brylinski operator and the modular class of a poisson manifold as a consequence we prove that poisson homology is isomorphic to poisson cohomology for unimodular poisson structures

upper and lower bounding strategies for the generalized minimum spanning tree problem abstract we address the generalized minimum spanning tree problem gmst which requires spanning at least one vertex out of every set of disjoint vertices in a graph we show that the geometric version of this problem is np hard and we propose two stochastic heuristics the first one is a very fast randomized greedy search algorithm and the second one being a genetic algorithm also we investigate some existing integer programming formulations and present an new one a new lagrangian based lower bound is proposed and implemented to assess the performance of the heuristics computational experiments performed on a large set of randomly generated instances with up to 1000 vertices and 10 000 edges provide evidence of the good performance of the proposed heuristics

a fourth order accurate embedded boundary method for the wave equation a fourth order accurate embedded boundary method for the scalar wave equation with dirichlet or neumann boundary conditions is described the method is based on a compact pade type discretization of spatial derivatives together with a taylor series method modified equation in time a novel approach for enforcing boundary conditions is introduced which uses interior boundary points instead of exterior ghost points this technique removes the small cell stiffness problem for both dirichlet and neumann boundary conditions is more accurate and robust than previous methods based on exterior ghost points and guarantees that the solution is single valued when slender bodies are treated numerical experiments are presented to illustrate the stability and accuracy of the method as well as its application to problems with complex geometries

a two level solution approach for solving the generalized minimum spanning tree problem abstract in this paper we are addressing the generalized minimum spanning tree problem denoted by gmstp which is a variant of the classical minimum spanning tree mst problem the main characteristic of this problem is the fact that the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum cost tree spanning a subset of vertices which includes exactly one vertex from each cluster we describe a two level solution approach for solving the gmstp obtained by decomposing the problem into two logical and natural smaller subproblems an upper level global subproblem and a lower level local subproblem and solving them separately the goal of the first subproblem is to determine global trees spanning the clusters using a genetic algorithm with a diploid representation of the individuals while the goal of the second subproblem is to determine the best tree w r t cost minimization for the above mentioned global trees spanning exactly one vertex from each cluster the second subproblem is solved optimally using dynamic programming extensive computational results are reported and discussed for an often used set of benchmark instances the obtained results show an improvement in the quality of the achieved solutions and demonstrate the efficiency of our approach compared to the existing methods from the literature

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

on the cohomology of the lie algebra of hamiltonian vector fields abstract the work of gelfand and fuks concerning the cohomology of the lie algebra of all vector fields on a manifold is extended to the cohomology of the lie algebra of all hamiltonian vector fields on a symplectic manifold the case of an even dimensional algebraic torus is treated in detail

poisson cohomology of the affine plane abstract we compute the poisson cohomology of homogeneous poisson structures on the plane the singular locus of such a poisson structure consists of a family of lines passing through o and we show how the dimensions of the first and second cohomology groups are related to the weight of o as a singular point of

generalized double affine hecke algebras of rank 1 and quantized del pezzo surfaces abstract let d be a simply laced dynkin diagram of rank r whose affinization has the shape of a star i e d 4 e 6 e 7 e 8 to such a diagram one can attach a group g whose generators correspond to the legs of the affinization have orders equal to the leg lengths plus 1 and the product of the generators is 1 the group g is then a 2 dimensional crystallographic group g z l z 2 where l is 2 3 4 and 6 respectively in this paper we define a flat deformation h t q of the group algebra c g of this group by replacing the relations saying that the generators have prescribed orders by their deformations saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots which are deformation parameters the algebra h t q for d 4 is the cherednik algebra of type c c 1 which was studied by noumi sahi and stokman and controls askey wilson polynomials we prove that h t q is the universal deformation of the twisted group algebra of g and that this deformation is compatible with certain filtrations on c g we also show that if q is a root of unity then for generic t the algebra h t q is an azumaya algebra and its center is the function algebra on an affine del pezzo surface for generic q the spherical subalgebra e h t q e provides a quantization of such surfaces we also discuss connections of h t q with preprojective algebras and painleve vi

homologies of cubic artin schelter regular algebras abstract the hochschild homology of cubic artin schelter regular algebras of type a with generic coefficients is computed we follow the method used by van den bergh k theory 8 1994 213 230 in the quadratic case and consider these algebras as deformations of a polynomial algebra with remarkable poisson brackets a new morphism of resolutions is introduced the de rham cohomology cyclic and periodic cyclic homologies and the hochschild cohomology are also computed

hard lefschetz theorem for sasakian manifolds we prove that on a compact sasakian manifold m g of dimension 2n 1 for any 0 p n the wedge product with d defines an isomorphism between the spaces of harmonic forms n p m and n p 1 m therefore it induces an isomorphism between the de rham cohomology spaces h m and h m such isomorphism is proven to be independent of the choice of a compatible sasakian metric on a given contact manifold as a consequence an obstruction for a contact manifold to admit sasakian structures is found

homology of algebras of families of pseudodifferential operators abstract we compute the hochschild cyclic and periodic cyclic homology groups of algebras of families of laurent complete symbols on manifolds with corners we show in particular that the spectral sequence associated with hochschild homology degenerates at e 2 and converges to hochschild homology as a byproduct we identify the space of residue traces on fibrations by manifolds with corners in the process we prove some structural results about algebras of complete symbols on manifolds with corners

uniqueness of multiplicative determinants on elliptic pseudodifferential operators we describe all multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold that are continuous along continuous paths and the restriction to zero order operators of which is of class c1 this boils down to a description of all traces on zero order classical pseudodifferential operators which turn out to be linear combinations of the wodzicki residue w and leading symbol traces introduced in pr1 both of which are continuous consequently multiplicative determinants are parametrized by the residue determinant w87 sc and a new leading symbol determinant both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator

an algebraic index theorem for orbifolds abstract using the concept of a twisted trace density on a cyclic groupoid a trace is constructed on a formal deformation quantization of a symplectic orbifold an algebraic index theorem for orbifolds follows as a consequence of a local riemann roch theorem for such densities in the case of a reduced orbifold this proves a conjecture by fedosov schulze and tarkhanov finally it is shown how the kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem

generalized spanning trees abstract in this paper we propose a definition for the generalized minimal spanning tree gmst of a graph the gmst requires spanning at least one node out of every set of disjoint nodes node partition in a graph the analysis of the gmst problem is motivated by real life agricultural settings related to construction of irrigation networks in desert environments we prove that the gmst problem is np hard and examine a number of heuristic solutions for this problem computational experiments comparing these heuristics are presented

a new linear quotient of c 4 admitting a symplectic resolution we show that the quotient c4 g admits a symplectic resolution for g q 8 times bf z 2 d 8 sf sp 4 bf c here q8 is the quaternionic group of order eight and d8 is the dihedral group of order eight and g is the quotient of their direct product which identifies the nontrivial central elements id of each it is equipped with the tensor product representation bf c 2 boxtimes bf c 2 cong bf c 4 this group is also naturally a subgroup of the wreath product group q 8 2 rtimes s 2 sf sp 4 bf c we compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming c4 g we also discuss preliminary investigations on the more general question of classifying linear quotients v g admitting symplectic resolutions

a tabu search heuristic for the generalized minimum spanning tree problem this paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem gmstp known to be np hard given a graph whose vertex set is partitioned into clusters the gmstp consists of designing a minimum cost tree spanning all clusters an attribute based tabu search heuristic employing new neighborhoods is proposed an extended set of tsplib test instances for the gmstp is generated and the heuristic is compared with recently proposed genetic algorithms the proposed heuristic yields the best results for all instances moreover an adaptation of the tabu search algorithm is proposed for a variation of the gmstp in which each cluster must be spanned at least once

numerical methods for nonlinear second order hyperbolic partial differential equations i time linearized finite difference methods for 1 d problems abstract a three parameter family of finite difference methods for one dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms is presented and shown to result in two time level semi discrete equations second order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three time level linear finite difference equations which are second order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1 2 1 to the past current and future time levels second order accurate in time time linearized three point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth order accurate in space are also presented and their linear stability analyzed an exhaustive comparison between analytical and numerical results indicates that the accuracy of time linearized methods increases as the time step and grid spacing are decreased and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure and the best accuracy is achieved when the three implicitness parameters are equal to 1 2 it is also shown that the accuracy of time linearized methods is a strong function of the wave speed as the wave speed is increased and the wave equation tends to a parabolic one the methods are still three time level techniques which may not preserve the positivity of the solution whereas a crank nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions it is also shown that for one dimensional wave equations with monotonic kinetics and time dependent forcing there are very few differences between second and first order time accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small and the discretization of the reaction terms plays a less important rule than that of the diffusion terms

hochschild co homology of differential operator rings abstract we show that the hochschild homology of a differential operator k algebra e a fu g is the homology of a deformation of the chevalley eilenberg complex of g with coefficients in m formula b moreover when a is smooth and k is a characteristic zero field we obtain a type of hochschild kostant rosenberg theorem for these algebras when a k our complex reduces to the one obtained by c kassel 1988 invent math 91 221 251 for the homology of filtrated algebras whose associated graded algebras are symmetric algebras in the last section we give similar results for the cohomology

examples of non kahler hamiltonian circle manifolds with the strong lefschetz property abstract in this paper we construct six dimensional compact non kahler hamiltonian circle manifolds which satisfy the strong lefschetz property themselves but nevertheless have a non lefschetz symplectic quotient this provides the first known counterexamples to the question whether the strong lefschetz property descends to the symplectic quotient we also give examples of hamiltonian strong lefschetz circle manifolds which have a non lefschetz fixed point submanifold in addition we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong lefschetz manifold we then use it to show the existence of lefschetz four manifolds with non lefschetz finite covering spaces

duality and modular class of a nambu poisson structure in this paper we introduce cohomology and homology theories for nambu poisson manifolds also we study the relation between the existence of a duality for these theories and the vanishing of a particular nambu poisson cohomology class the modular class the case of a regular nambu poisson structure and some singular examples are discussed

cyclic cocycles on deformation quantizations and higher index theorems abstract we construct a nontrivial cyclic cocycle on the weyl algebra of a symplectic vector space using this cyclic cocycle we construct an explicit local quasi isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof we give a new proof of nest tsygan s algebraic higher index theorem by computing the pairing between such cyclic cocycles and the k theory of the formal deformation quantization furthermore we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold as an application we obtain the analytic higher index theorem of connes moscovici and its extension to orbifolds

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

harmonic cohomology of symplectic manifolds abstract mathieu math helv 70 1995 1 introduced a canonic filtration in the de rham cohomology of a symplectic manifold and proved that the middle filtration space is the space of harmonic cohomology classes we give an interpretation of the other filtration spaces prove a kunneth theorem for harmonic cohomology prove poincare duality for harmonic cohomology and show how surjectivity of certain lefschetz type mappings is related to properties of the filtration for a closed symplectic manifold m we also introduce symplectic invariants l m l 0 and show l 0 l m sign m if m is of dimension 2n with n even

stable high order discretization for evolution of the wave equation in 2 1 dimensions in a previous paper it was shown that consistent high order discretization can make a lax wendroff method stable in the presence of small cells near the boundary in one space and one time dimension here i show that a straightforward extension of this approach to two space dimensions is similarly successful

stable high order discretization for evolution of the wave equation in 1 1 dimensions we carry forward the approach of alpert greengard and hagstrom to construct stable high order explicit discretizations for the wave equation in one space and one time dimension they presented their scheme as an integral form of the lax wendroff method our perspective is somewhat different from theirs our focus is on the discretization of the evolution formula rather than on its form integral differential etc a key feature of our approach is the independent computation of three discretizations one for bulk away from boundaries propagation one for propagation near boundaries and a projection operator to enforce boundary conditions this is done in a way that is straightforward to extend to more space dimensions

the lefschetz property formality and blowing up in symplectic geometry in this paper we study the behaviour of the lefschetz property under the blow up construction we show that it is possible to reduce the dimension of the kernel of the lefschetz map if we blow up along a suitable submanifold satisfying the lefschetz property we use this together with results about massey products to construct compact nonformal symplectic manifolds satisfying the lefschetz property

nambu jacobi and generalized jacobi manifolds the geometry of nambu jacobi and generalized jacobi manifolds is studied a large collection of examples is given the characteristic distribution generated by the hamiltonian vector fields on a nambu jacobi manifold is proved to be completely integrable and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated

cohomology and hodge theory on symplectic manifolds iii arxiv 1402 0427v2 math sg 2 may 2014 cohomology and hodge theory on symplectic manifolds iii chung jun tsai li sheng tseng and shing tung yau february 3 2014 abstract we introduce filtered cohomologies of differential forms on symplectic manifolds they generalize and include the cohomologies discussed in paper i and ii as a subset the filtered cohomologies are finite dimensional and can be associated with differential elliptic complexes algebraically we show that the filtered cohomologies give a two sided resolution of lefschetz maps and thereby they are directly related to the kernels and cokernels of the lefschetz maps we also introduce a novel non associative product operation on differential forms for symplectic manifolds this product generates an a algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure as an application we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four manifolds in the context of a circle times a fibered three manifold contents 1 introduction 2 preliminaries operations on differential forms 10 filtered forms and differential operators 14 short exact sequences 16

noncommutative kahler structures on quantum homogeneous spaces abstract building on the theory of noncommutative complex structures the notion of a noncommutative kahler structure is introduced in the quantum homogeneous space case many of the fundamental results of classical kahler geometry are shown to follow from the existence of such a structure allowing for the definition of noncommutative lefschetz hodge kahler dirac and laplace operators quantum projective space endowed with its heckenberger kolb calculus is taken as the motivating example the general theory is then used to show that the calculus has cohomology groups of at least classical dimension

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported

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