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

justification of the nonlinear schrodinger approximation for a quasilinear klein gordon equation we consider a nonlinear klein gordon equation with a quasilinear quadratic term the nonlinear schrodinger nls equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet like solutions to the quasilinear klein gordon equation it is the purpose of this paper to present a method that allows one to prove error estimates in sobolev norms between exact solutions of the quasilinear klein gordon equation and the formal approximation obtained via the nls equation the paper contains the first validity proof of the nls approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in sobolev spaces we expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the nls approximation for other quasilinear hyperbolic systems

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

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

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

justification of the nls approximation for a quasilinear water wave model abstract we show how for a quasilinear water wave model the nls approximation can be justified the model presents several new difficulties due to the quadratic terms which have to be eliminated by a normal form transformation due to the quasilinearity of the problem there is some loss of regularity associated with the normal form transformation and there is a nontrivial resonance present in the problem the loss of regularity is dealt with by using a cauchy kowalevskaya like method to treat the initial value problem and the nontrivial resonance is dealt with via a rescaling argument

some remarks on the eigenvalue problem for zn symmetric vertex and face models abstract the eigenvalue solution of the transfer matrix of belavin s z n symmetric vertex model is further extended to the case of the model with the crossing parameter 1 l here l is a nonzero positive integer then the so called cyclic irf model is defined and the eigenvalue solution of its transfer matrices is found the eigenvalue spectrum of the irf model at critical limit is shown to be the same as that of a modified many component six vertex model therefore the spectrums of both models are analysed with the quantum group suq n

leibniz algebroid associated with a nambu poisson structure the notion of a leibniz algebroid is introduced and it is shown that each nambu poisson manifold has associated a canonical leibniz algebroid this fact permits one to define the modular class of a nambu poisson manifold as an appropriate cohomology class extending the well known modular class of poisson manifolds

existence of long time solutions and validity of the nonlinear schrodinger approximation for a quasilinear dispersive equation abstract we consider a nonlinear dispersive equation with a quasilinear quadratic term we establish two results first we show that solutions to this equation with initial data of order o e in sobolev norms exist for a time span of order o e 2 for sufficiently small e secondly we derive the nonlinear schrodinger nls equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave and justify this approximation with the help of error estimates in sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the nls equation the proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal form transforms to justify the nls approximation we have to overcome additional difficulties caused by the occurrence of resonances we expect that the method developed in the present paper will also allow to prove the validity of the nls approximation for a larger class of quasilinear dispersive systems with resonances

eigenvectors of the multi component generalization of the six vertex model abstract the most general multi component solution of the star triangle equations under an ice type restriction is considered the transfer matrix of this model is diagonalized and the eigenvectors are written down explicitly using the quantum inverse scattering formalism the eigenvectors are shown to have simple symmetry properties under an exchange of arguments

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

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

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

dynamics of generalized poisson and nambu poisson brackets a unified setting for generalized poisson and nambu poisson brackets is discussed it is proved that a nambu poisson bracket of even order is a generalized poisson bracket characterizations of poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and lagrangian submanifolds of product and tangent manifolds respectively

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

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

on the fusion of face and vertex models some new solutions of the yang baxter or star triangle relations are found using the fusion procedure for belavin s 1981 zn zn symmetric model the relation between the fusion procedure of the vertex and face models is also constructed

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

diagonalization of the transfer matrix of a nonintersecting string model abstract in this paper we obtain the eigenvectors of the transfer matrix of a nonintersecting string model using the bethe ansatz the result is in agreement with baxter s conjecture of weak equivalence of the present model with the six vertex model

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

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

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

interaction of modulated gravity water waves of finite depth abstract we consider the capillary gravity water wave problem of finite depth with a flat bottom of one or two horizontal dimensions we derive the modulation equations of leading and next to leading order in the hyperbolic scaling for three weakly amplitude modulated plane wave solutions of the linearized problem in the absence of quadratic and cubic resonances we justify the derived system of macroscopic equations in the case of gravity waves using the stability of the finite depth water wave problem on the time scale o 1

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

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

global well posedness of 2d non focusing schrodinger equations via rigorous modulation approximation abstract we consider the long time well posedness of the cauchy problem with large sobolev data for a class of nonlinear schrodinger equations nls on r 2 with power nonlinearities of arbitrary odd degree specifically the method in this paper applies to those nls equations having either elliptic signature with a defocusing nonlinearity or else having an indefinite signature by rigorously justifying that these equations govern the modulation of wave packet like solutions to an artificially constructed equation with an advantageous structure we show that a priori every subcritical inhomogeneous sobolev norm of the solution increases at most polynomially in time global well posedness follows by a standard application of the subcritical local theory

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

deformation quantization and nambu mechanics starting from deformation quantization star products the quantization problem of nambu mechanics is investigated after considering some impossibilities and pushing some analogies with field quantization a solution to the quantization problem is presented in the novel approach of zariski quantization of fields observables functions in this case polynomials this quantization is based on the factorization over of polynomials in several real variables we quantize the infinite dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is abelian associative and distributive this procedure is then adapted to derivatives needed for the nambu brackets which ensures the validity of the fundamental identity of nambu mechanics also at the quantum level our construction is in fact more general than the particular case considered here it can be utilized for quite general defining identities and for much more general star products

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

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

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

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

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

remarks on nambu poisson and nambu jacobi brackets we show that nambu poisson and nambu jacobi brackets can be defined inductively an n bracket n 2 is nambu poisson respectively nambu jacobi if and only if on fixing an argument we obtain an n 1 nambu poisson respectively nambu jacobi bracket as a by product we obtain relatively simple proofs of darboux type theorems for these structures

row transfer matrix spectra of cyclic solid on solid lattice models the eigenvalue spectra of cyclic solid on solid csos row transfer matrices are studied an equivalence is established between the inversion identity and the bethe ansatz functional equations and these equations are solved in the thermodynamic limit by a wiener hopf perturbation technique for the bands of leading excitations thel state csos model with crossing parameter s l possesses a 2 l s fold degenerate largest eigenvalue corresponding to the 2 l s coexisting phases the expressions for the largest eigenvalue and free energy coincide with those of the eight vertex model the string excitations for 2s l admit different classifications and are treated separately the correlation length is calculated in both regimes yielding the critical exponentv l 2s in agreement with the scaling relations

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

hamiltonian of a id quantum chain for belavin s zn zn model the hamiltonian of a 1d quantum chain corresponding to belavin s zn zn symmetric model is derived this hamiltonian is a multicomponent generalisation of the xyz model with n2 coupling constants in general in the limit tau to i infinity it reduces to the generalised xxz model with n 1 n fold degenerate coupling constants and n non degenerate ones the hamiltonian is hermitian only for n 2 or for n 2 with a crossing parameter w restricted to integers there exists a domain of parameters in which the boltzmann weights are positive but the corresponding hamiltonian is non hermitian the relations of the hamiltonian to other models are discussed

on the higher order generalizations of poisson structures the characterization of the nambu poisson n tensors as a subfamily of the generalized poisson ones recently introduced and here extended to the odd order case is discussed the homology and cohomology complexes of both structures are compared and some physical considerations are made

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

a justification of the modulation approximation to the 3d full water wave problem we consider modulational solutions to the 3d inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension for such solutions it is well known that one formally expects the modulation to be a profile traveling at group velocity and governed by a 2d hyperbolic cubic nonlinear schrodinger equation in this paper we justify this fact by providing rigorous error estimates in sobolev spaces we reproduce the multiscale calculation to derive an approximate wave packet like solution to the evolution equations with mild quadratic nonlinearities constructed by sijue wu then we use the energy method along with the method of normal forms to provide suitable a priori bounds on the difference between the true and approximate solutions

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

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

justification of the 2d nls equation for a fourth order nonlinear wave equation quadratic resonances do not matter much in case of analytic initial conditions abstract the 2d nonlinear schrodinger nls equation can be derived by multiple scaling analysis in order to describe slow temporal and spatial modulations of an oscillating wave packet in general dispersive wave systems it is the purpose of this paper to establish estimates for the difference between the so obtained approximation and true solutions of the original system in case that the original system possesses nontrivial quadratic resonances we explain for a fourth order nonlinear wave equation that quadratic resonances do not matter much in case of analytic initial conditions for the nls equation in the sense that independently of whether the resonance is stable or unstable an approximation property can be established on the natural time scale of the nls equation if the set of resonant wave numbers is bounded away from the integer multiples of the basic wave number of the underlying wave packet finally we explain that in case of an approximation by a so called four wave interaction fwi system the same approach works and an even stronger result can be established

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