lie theory of multiplicative tensors we study tensors on lie groupoids suitably compatible with the groupoid structure called multiplicative our main result gives a complete description of these objects only in terms of infinitesimal data special cases include the infinitesimal counterparts of multiplicative forms multivector fields and holomorphic structures obtained through a unifying and conceptual method we also give a full treatment of multiplicative vector valued forms particularly nijenhuis operators and related structures

integration of holomorphic lie algebroids we prove that a holomorphic lie algebroid is integrable if and only if its underlying real lie algebroid is integrable thus the integrability criteria of crainic fernandes theorem 4 1 in crainic fernandes in ann math 2 157 2003 do also apply in the holomorphic context without any modification as a consequence we prove that a holomorphic poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real poisson manifold

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

an initial boundary value problem of a nonlinear klein gordon equation we present a fully implicit and discrete energy conserving finite difference scheme for the solution of an initial boundary value problem of the nonlinear klein gordon equation a theoretical analysis is performed and it has been demonstrated that the numerical scheme is particularly attractive when long time solutions are sought

difference schemes for solving the generalized nonlinear schrodinger equation this paper studies finite difference schemes for solving the generalized nonlinear schrodinger gnls equationiut uxx q u 2 u f x t u a new linearlized crank nicolson type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the gnls equation several schemes including crank nicolson type schemes hopscotch type schemes split step fourier scheme and pseudospectral scheme are adopted for solving three model problems of gnls equation which arise from many physical problems withq s s2 q s ln 1 s andq s 4s 1 s respectively the numerical results demonstrate that the linearized crank nicolson scheme is efficient and robust

stability analysis of difference schemes for variable coefficient schro dinger type equations we consider the stability of difference schemes for the solution of the initial boundary value problem for the equation u 1 left a x t u x right x b x t u c x t u f x t where u a b c and f are complex valued functions using energy methods we establish the stability of a general two level scheme which includes euler s method crank nicolson s method and the backward euler method if the coefficient a x t is purely imaginary the explicit euler method is unconditionally unstable for this case we propose a new scheme with appropriately chosen artificial dissipation which we prove to be conditionally stable

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

multiplicative forms at the infinitesimal level we describe arbitrary multiplicative differential forms on lie groupoids infinitesimally i e in terms of lie algebroid data this description is based on the study of linear differential forms on lie algebroids and encompasses many known integration results related to poisson geometry we also revisit multiplicative multivector fields and their infinitesimal counterparts drawing a parallel between the two theories

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

analysis of some new conservative schemes for nonlinear schrodinger equation with wave operator abstract some new conservative finite difference schemes are presented for an initial boundary value problem of schrodinger equation with wave operator they have the advantages that there are some discrete energies which are conserved respectively the existence of the solution of the finite difference schemes are proved by leray schauder fixed point theorem and the uniqueness stability and convergence of difference solutions with order o h2 2 are proved in the energy norm results of numerical experiment demonstrate the efficiency of the new scheme

a compact finite difference scheme for the nonlinear schrodinger equation with wave operator abstract in this paper a compact finite difference scheme is presented for an periodic initial value problem of the nonlinear schrodinger nls equation with wave operator this is a scheme of three levels with a discrete conservation law the unconditional stability and convergence in maximum norm with order o h 4 2 are proved by the energy method a numerical experiment is presented to support our theoretical results

a class of infinite dimensional representations of the lie superalgebra mathfrak o mathfrak s mathfrak p 2m 1 2n and the parastatistics fock space an orthogonal basis of weight vectors for a class of infinite dimensional representations of the orthosymplectic lie superalgebra osp 2m 1 vertical bar 2n is introduced these representations are particular lowest weight representations v p with a lowest weight of the form p 2 p 2 vertical bar p 2 p 2 p being a positive integer explicit expressions for the transformation of the basis under the action of algebra generators are found since the relations of algebra generators correspond to the defining relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations the parastatistics fock space of order p is also explicitly constructed furthermore the representations v p are shown to have interesting characters in terms of supersymmetric schur functions and a simple character formula is also obtained

local classification of generalize complex structures we give a local classification of generalized complex structures about a point a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic poisson manifold we use a nash moser type argument in the style of conn s linearization theorem

holomorphic jacobi manifolds and holomorphic contact groupoids this paper belongs to a series of works aiming at exploring generalized complex geometry in odd dimensions holomorphic jacobi manifolds were introduced and studied by the authors in a separate paper as special cases of generalized contact bundles in fact generalized contact bundles are nothing but odd dimensional analogues of generalized complex manifolds in the present paper we solve the integration problem for holomorphic jacobi manifolds by proving that they integrate to holomorphic contact groupoids a crucial tool in our proof is what we call the homogenization scheme which allows us to identify holomorphic jacobi manifolds with homogeneous holomorphic poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids

contact reduction and groupoid actions we introduce a new method to perform reduction of contact manifolds that extends willett s and albert s results to carry out our reduction procedure all we need is a complete jacobi map j m 0 from a contact manifold to a jacobi manifold this naturally generates the action of the contact groupoid of 0 on m and we show that the quotients of fibers j 1 x by suitable lie subgroups x are either contact or locally conformal symplectic manifolds with structures induced by the one on m we show that willett s reduced spaces are prequantizations of our reduced spaces hence the former are completely determined by the latter since a symplectic manifold is prequantizable iff the symplectic form is integral this explains why willett s reduction can be performed only at distinguished points as an application we obtain kostant s prequantizations of coadjoint orbits finally we present several examples where we obtain classical contact manifolds as reduced spaces

a conservative difference scheme for the zakharov equations a new conservative difference scheme is presented for the periodic initial value problem of zakharov equations the scheme can be implicit or semi explicit depending on the choice of a parameter the discretization of the initial condition is of second order accuracy which is consistent with the accuracy of the scheme on the basis of a priori estimates and an inequality about norms convergence of the difference solutions is proved in the energy norm numerical experiments with the schemes are done for several test cases computational results demonstrate that the new semi explicit scheme with a new initial approximation is more accurate and computationally efficient

vector bundles over lie groupoids and algebroids abstract we study vb groupoids and vb algebroids which are vector bundles in the realm of lie groupoids and lie algebroids through a suitable reformulation of their definitions we elucidate the lie theory relating these objects i e their relation via differentiation and integration we also show how to extend our techniques to describe the more general lie theory underlying double lie algebroids and la groupoids

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

multiplicative forms and spencer operators motivated by our attempt to recast cartan s work on lie pseudogroups in a more global and modern language we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem from this point of view the novelty of this paper is that we study forms with coefficients however the main contribution of this paper is conceptual the discovery of the relationship between multiplicative forms and cartan s work is explored here to provide a completely new approach to integrability theorems for multiplicative forms the multiplicative point of view shows that modulo lie s functor the cartan pfaffian system itself a multiplicative form with coefficients is the same thing as the classical spencer operator

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes