hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

equivariant basic cohomology of riemannian foliations the basic cohomology of a riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space in this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non closed leaves in the simply connected case or more generally for killing foliations we show that the total basic betti number of the union c of the closed leaves is smaller than or equal to the total basic betti number of the foliated manifold and we give sufficient conditions for equality if there is a basic morse bott function with critical set equal to c we can compute the basic cohomology explicitly another case in which the basic cohomology can be determined is if the space of leaf closures is a simple convex polytope our results are based on molino s observation that the existence of non closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set c we introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of lie group actions enabling us to transfer many results from the theory of lie group actions to riemannian foliations the prominent role of the fixed point set in the theory of torus actions explains the relevance of the set c in the basic setting

lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

cartan decomposition subgroups of 2 for g sl 3 r and g so 2 n we give explicit practical conditions that determine whether or not a closed connected subgroup h of g has the property that there exists a compact subset c of g with chc g to do this we fix a cartan decomposition g ka k of g and then carry out an approximate calculation of khk n a for each closed connected subgroup h of g

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

evaluation of regular and singular domain integrals with boundary only discretization theory and fortran code in this paper a set of boundary integrals are derived based on a radial integration technique to accurately evaluate two dimensional 2d and three dimensional 3d regular and singular domain integrals a self contained fortran code is listed and described for numerical implementation of these boundary integrals the main feature of the theory is that only the boundary of the integration domain needs to be discretized into elements this feature cannot only save considerable efforts in discretizing the integration domain into internal cells as in the conventional method but also make computational results for singular domain integrals more accurate since the integrals have been regularized some examples are provided to verify the correctness of the presented formulations and the included code

numerical evaluation of two dimensional singular boundary integrals theory and fortran code in this paper an approach is presented for the numerical evaluation of weakly strongly hyper and supersingular boundary integrals which exist in the cauchy principal value sense in two dimensional problems in this approach the singularities involved in integration kernels are analytically removed by expressing the nonsingular parts of the integration kernels as polynomials of the distance r a self contained fortran code is listed and described for implementation of the proposed approach the attached code is also able to evaluate general regular integrals using gaussian quadrature which enables the code to evaluate any two dimensional boundary integral some examples are provided to verify the correctness of the presented formulations and the included code

boundary values as hamiltonian variables i new poisson brackets the ordinary poisson brackets in field theory do not fulfill the jacobi identity if boundary values are not reasonably fixed by special boundary conditions it is shown that these brackets can be modified by adding some surface terms to lift this restriction the new brackets generalize a canonical bracket considered by lewis marsden montgomery and ratiu for the free boundary problem in hydrodynamics the definition of poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals this construction is applied to any local form of poisson brackets

finiteness of the basic intersection cohomology of a killing foliation we prove that the basic intersection cohomology mathbb h overline p m mathcal f where mathcal f is the singular foliation determined by an isometric action of a lie group g on the compact manifold m is finite dimensional

hochschild cohomology and deformation quantization of affine toric varieties abstract for an affine toric variety spec a we give a convex geometric description of the hodge decomposition of its hochschild cohomology under certain assumptions we compute the dimensions of the hodge summands t i 1 a generalizing the existing results about the andre quillen cohomology group t 1 1 a we prove that every poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization

some properties of a cohomology group associated to a totally geodesic foliation we consider an oriented totally geodesic foliation o of dimension p on a compact connected riemannian manifold m there is a homological characte rization of the p l case due to d sullivan 16 the codimension one case has been completely classified by e ghys 9 in collaboration with e ghys we have given a detailed description of the p 2 case for 4 manifolds 5 the geometry of the arbitrary dimension p case is examined in 4 continuing this work we consider in this paper the complex at of tangential forms these are the forms along the leaves of that are invariant in directions perpendicu lar to the leaves the exterior derivative on m induces a differential operator d v in a w

weakly singular stress bem for 2d elastostatics a weakly singular stress bem is presented in which the linear state regularizing field is extended over the entire surface the algorithm employs standard conforming c0 elements with lagrangian interpolations and exclusively uses gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space the linear state stress bie on which the algorithm is based has no free term so that the bem treatment of external corners requires no special consideration other than to admit traction discontinuities the self regularizing nature of the somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary a key observation is the relation between bie density functions and successful interpolation orders numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations additionally the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions copyright 1999 john wiley sons ltd

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

on bi hamiltonian deformations of exact pencils of hydrodynamic type in this paper we are interested in nontrivial bi hamiltonian deformations of the poisson pencil deformations are generated by a sequence of vector fields x2 x3 x4 where each xk is homogeneous of degree k with respect to a grading induced by rescaling constructing recursively the vector fields xk one obtains two types of relations involving their unknown coefficients one set of linear relations and an other one which involves quadratic relations we prove that the set of linear relations has a geometric meaning using miura quasitriviality the set of linear relations expresses the tangency of the vector fields xk to the symplectic leaves of 1 and this tangency condition is equivalent to the exactness of the pencil moreover extending the results of lorenzoni p 2002 j geom phys 44 331 75 we construct the nontrivial deformations of the poisson pencil up to the eighth order in the deformation parameter showing therefore that deformations are unobstructed and that both poisson structures are polynomial in the derivatives of u up to that order

a darboux getzler theorem for scalar difference hamiltonian operators in this paper we extend the notion of poisson lichnerowicz cohomology an object encapsulating the building blocks for the theory of deformations of hamiltonian operators to the difference case a local scalar difference hamiltonian operator is a polynomial in the shift operator and its inverse with coefficients in the algebra of difference functions endowing the space of local functionals with the structure of a lie algebra its poisson lichnerowicz cohomology carries information about the center the symmetries and the admissible deformations of such an algebra the analogue notion for the differential case has been widely investigated the first and most important result is the triviality of all but the lowest cohomology for first order hamiltonian differential operators due to getzler we study the poisson lichnerowicz cohomology for the operator k 0 mathcal s mathcal s 1 which is the normal form for 1 1 order scalar difference hamiltonian operators we obtain the same result as getzler did namely h p k 0 0 forall p 1 and explicitly compute h 0 k 0 and h 1 k 0 we then apply our main result to the classification of lower order scalar hamiltonian operators recently obtained by de sole kac valeri and wakimoto

corrigendum to deformation quantization in algebraic geometry adv math 198 2005 383 432 this note contains a correction of the proofs of the main results of 3 namely theorems 0 1 and 0 2 the results are correct as originally stated the mistake in my original proofs was discovered michel van den bergh and i thank him for calling my attention to it the way to fix the proofs is essentially contained in his paper 2 let me begin by explaining the mistake as can be seen in example 0 1 below the mistake itself is of a rather elementary nature but it was obscured by the complicated context suppose k is a field of characteristic 0 and x is a smooth separated n dimensional scheme over k recall that the coordinate bundle coorx is an infinite dimensional bundle over x with free action by the group gln k the quotient bundle is by definition

deformations of affine varieties and the deligne crossed groupoid abstract let x be a smooth affine algebraic variety over a field k of characteristic 0 and let r be a complete parameter k algebra e g r k we consider associative resp poisson r deformations of the structure sheaf o x the set of r deformations has a crossed groupoid i e strict 2 groupoid structure our main result is that there is a canonical equivalence of crossed groupoids from the deligne crossed groupoid of normalized polydifferential operators resp polyderivations of x to the crossed groupoid of associative resp poisson r deformations of o x the proof relies on a careful study of adically complete sheaves in the associative case we also have to use ring theory ore localizations and the properties of the hochschild cochain complex the results of this paper extend previous work by various authors they are needed for our work on twisted deformation quantization of algebraic varieties

adelic chern forms and applications let x be a variety over a field of characteristic 0 given a vector bundle e on x we construct chern forms c v t x ax here a x is the sheaf of beilinson adeles and v is an adelic connection when x is smooth wt x a x h r x the algebraic de rham cohomology and c q v are the usual chern classes we include three applications of the construction 1 existence of adelic secondary chern simons characteristic classes on any smooth x and any vector bundle e 2 proof of the bott residue formula for a vector field action and 3 proof of a gauss bonnet formula on the level of differential forms namely in the de rham residue complex 0 introduction let x be a scheme of finite type over a field k according to beilinson be given any quasi coherent x module m and an integer q9 there is a flasque ox module a dcm called the sheaf of adeles this is a gen eralization of the classical adeles of number theory cf example 2 3 moreover there are homomorphisms d aj ed m a m which make ared m into a complex and m ared m is quasi isomorphism now let qx be the algebra of kahler differential forms on x in hy we proved that the sheaf

frobenius manifolds and central invariants for the drinfeld sokolov bihamiltonian structures the drinfeld sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine lie algebra we compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of miura type transformations

hamiltonian pdes deformations integrability solutions we review recent classification results on the theory of systems of nonlinear hamiltonian partial differential equations with one spatial dimension including a perturbative approach to the integrability theory of such systems and discuss universality conjectures describing critical behaviour of solutions to such systems

cpv and hfp integrals and their applications in the boundary element method the subject of this paper is hadamard finite part hfp and cauchy principal value cpv representations of certain singular integrals of primary concern here are hfp interpretations of strongly singular integrals for which cpv interpretations also exist a relationship between the two is stated and proved this matter is then illustrated by two examples and also discussed in the context of singular boundary integral equations bies for linear elasticity

direct computation of cauchy principal value integrals in advanced boundary elements this paper presents a new method for the direct computation of strongly singular integrals existing in the cauchy principal value sense it can be usefully applied in the solution by the direct boundary element method of many different problems initially some considerations are provided in order to summarize the state of the art on this issue then the features of the proposed method are reported the procedure allows the direct calculation of cauchy principal value integrals with first order singularity and it is applicable even in advanced boundary element methods employing high order elements it requires only the use of standard gaussian quadrature formulae plus the computation of a logarithmic term some examples show the effectiveness and efficiency of the procedure

an effective boundary element algorithm for 2d and 3d elastoplastic problems abstract novel methods are described for removing the strong singularities arising in the domain integrals of elastoplasticity and for solving the non linear equation set the former employs a new transformation from domain integrals to cell boundary integrals the number of system equations is minimised by using the plastic multiplier as the primary unknown and an incremental variable stiffness iterative algorithm is developed for solving these equations excellent convergence is achieved and some numerical examples demonstrate the algorithm s effectiveness

an effective method for numerical evaluation of general 2d and 3d high order singular boundary integrals in this paper a robust method is presented for numerical evaluation of weakly strongly hyper and super singular boundary integrals which exist in the cauchy principal value sense in two and three dimensional problems in this method the singularities involved in integration kernels are analytically removed by expressing the non singular parts of the integration kernels as power series in the local distance of the intrinsic coordinate system for three dimensional boundary integrals the radial integration method 1 is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals some examples are provided to verify the correctness and robustness of the presented method

on global deformation quantization in the algebraic case abstract we give a proof of yekutieli s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems instead we use an argument inspired by algebraic de rham cohomology

dispersive deformations of hamiltonian systems of hydrodynamic type in 2 1 dimensions abstract we develop a theory of integrable dispersive deformations of 2 1 dimensional hamiltonian systems of hydrodynamic type following the scheme proposed by dubrovin and his collaborators in 1 1 dimensions our results show that the multi dimensional situation is far more rigid and generic hamiltonians are not deformable as an illustration we discuss a particular class of two component hamiltonian systems establishing the triviality of first order deformations and classifying hamiltonians possessing nontrivial deformations of the second order

deformation quantization of gerbes this is the first in a series of articles devoted to deformation quantization of gerbes we introduce basic definitions interpret deformations of a given stack as maurer cartan elements of a differential graded lie algebra dgla and classify deformations of a given gerbe in terms of maurer cartan elements of the dgla of hochschild cochains twisted by the cohomology class of the gerbe we also classify all deformations of a given gerbe on a symplectic manifold as well as provide a deformation theoretic interpretation of the first rozansky witten class

differential geometry of gerbes abstract we define in a global manner the notion of a connective structure for a gerbe on a space x when the gerbe is endowed with trivializing data with respect to an open cover of x we describe this connective structure in two separate ways which extend from abelian to general gerbes the corresponding descriptions due to j l brylinski and n hitchin we give a global definition of the 3 curvature of this connective structure as a 3 form on x with values in the lie stack of the gauge stack of the gerbe we also study this notion locally in terms of more traditional lie algebra valued 3 forms the bianchi identity which the curvature of a connection on a principal bundle satisfies is replaced here by a more elaborate equation

deformation theory of abelian categories in this paper we develop the basic infinitesimal deformation theory of abelian categories this theory yields a natural generalization of the well known deformation theory of algebras developed by gerstenhaber as part of our deformation theory we define a notion of flatness for abelian categories we show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems

analysis of three dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method abstract in this paper an advanced boundary element method bem is developed for solving three dimensional 3d anisotropic heat conduction problems in thin walled structures the troublesome nearly singular integrals which are crucial in the applications of the bem to thin structures are calculated efficiently by using a nonlinear coordinate transformation method for the test problems studied promising bem results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro scales 10 6 which is sufficient for modeling most thin walled structures as used in for example smart materials and thin layered coating systems the advantages disadvantages as well as potential applications of the proposed method as compared with the finite element method fem are also discussed

an isogeometric boundary element method for three dimensional potential problems abstract isogeometric analysis iga coupled with boundary element method i e igabem received a lot of attention in recent years in this paper we extend the igabem to solve 3d potential problems this method offers a number of key improvements compared with conventional piecewise polynomial formulations firstly the models for analysis in the igabem are exact geometrical representation no matter how coarse the discretization of the studied bodies is thus the igabem ensures that no geometrical errors are produced in the analysis process secondly a meshing process is no longer required which means redundant computations are eliminated to allow analysis to be carried out with greatly reduced pre processing to accurately evaluate the singular integrals appearing in our method the power series expansion method is employed the integration surface is on the real surface of the model rather than the interpolation surface i e no geometrical errors thus the value of integral is more accurate than the traditional boundary element method which can improve the computation accuracy of the igabem some numerical examples for 3d potential problems are used to validate the solutions of the present method with analytical and numerical solutions available

bihamiltonian cohomology of kdv brackets using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of poisson brackets of dispersionless kdv hierarchy in particular this proves a conjecture of liu and zhang about the vanishing of such cohomology groups

an isogeometric boundary element method using adaptive integral method for 3d potential problems due to the merits of exact geometrical representation high accuracy no meshing process and only the boundary description of the problem etc the isogeometric boundary element method i e igabem has achieved rapid development in this paper based on the upper bound of the relative error of the gaussian quadrature formula we presented a 3d igabem using adaptive integration method for potential problems this method offers a number of key improvements compared with conventional igabem firstly the boundary element integrations can be computed easily and effectively at optimal computational cost secondly the adaptive algorithm can cope with the common situation where the sizes of adjacent cells are significantly different moreover the presented method can compute the nearly singular integrals easily owing to the use of subdivision technique to accurately evaluate the singular integrals appearing in our method the power series expansion method is employed the integration surface is on the real surface of the model rather than the interpolation surface i e no geometrical errors thus the value of integral is more accurate than the traditional boundary element method which can improve the computation accuracy of the igabem numerical tests show that the presented method has good performance in both exactness and convergence a general method is proposed to calculate the nearly singular integrals appearing in the 3d igabem the boundary element integrations can be computed easily and effectively at optimal computational cost the adaptive algorithm can be used to cope with the common situation where the sizes of adjacent cells are significantly different the power series expansion method is expanded to compute the singular integrals appearing in 3d igabem

from local to global deformation quantization of poisson manifolds we give an explicit construction of a deformation quantization of the algebra of functions on a poisson manifold based on m kontsevich s local formula the deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection

a decomposition theorem for the spectral sequence of lie foliations for a lie g foliation y on a closed manifold m there is an infinitesimal action of g on m up to homotopy along the leaves in general it is not an action but defines an action of the corresponding connected simply connected lie group on the term e1 of the spectral sequence associated to s9 even though e1 in general is infinite dimensional and non hausdorff with the topology induced by the f topology it is proved that this action can be averaged when is compact obtaining a tensor decomposition theorem of e2 it implies duality in the whole term e2 for riemannian foliations on closed oriented manifolds with compact semisimple structural lie algebra

a general algorithm for evaluating nearly singular integrals in anisotropic three dimensional boundary element analysis abstract this paper presents a new general method for the evaluation of nearly singular boundary element integrals arising in anisotropic three dimensional 3d boundary element analysis it is shown that the original nearly singular integrals can be transformed into using a sinh function an element by element sum of regular integrals each one expressed in terms of intrinsic local coordinates as a consequence the actual computation can be performed by using standard n n gaussian quadrature and the procedure can be easily included in any existing computer code this new method has full generality and therefore can be applied to a wide class of integrals the numerical results demonstrate the accuracy and efficiency of the method along with its insensitivity to the location of the nearly singular points it is shown that several orders of magnitude improvement in relative errors can be obtained using this transformation when compared to a straightforward implementation of gaussian quadrature

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

central extensions of gerbes we introduce the notion of central extension of gerbes on a topological space x we show that there are obstruction classes to lifting objects and isomorphisms in a central extension we also discuss pronilpotent gerbes these results are used in the paper 15 by a yekutieli to study twisted deformation quantization on algebraic varieties

a novel boundary element approach for solving the 2d elasticity problems abstract the presentation is mainly devoted to the research on the regularized bem formulations with indirect unknowns for two dimensional 2d elasticity problems a novel regularization technique in which regularized forms of the gradient equations without involving the direct calculation of cpv and hfp integrals are derived and shown to be independent of displacement equations is pursued in this paper after that a numerically systematic scheme with generality is established by adopting quadratic lagrange s elements moreover considering the special geometric domain such as circular or elliptic arcs a new boundary geometric approximate technique named as exact elements is presented and thus by the utilization of these elements the error of the results will arise mainly from the approximation of boundary quantities numerical examples show that a better precision and high computational efficiency can be achieved

deformation quantization in algebraic geometry abstract we study deformation quantizations of the structure sheaf o x of a smooth algebraic variety x in characteristic 0 our main result is that when x is d affine any formal poisson structure on x determines a deformation quantization of o x canonically up to gauge equivalence this is an algebro geometric analogue of kontsevich s celebrated result

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

boundary element analysis in thermoelasticity with and without internal cells in this paper a set of internal stress integral equations is derived for solving thermoelastic problems a jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations the strongly singular domain integral is then regularized using a semi analytical technique to avoid the requirement of discretizing the domain into internal cells domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method rim two numerical examples for 2d and 3d respectively are presented to verify the derived formulations copyright 2003 john wiley sons ltd

algebroid prestacks and deformations of ringed spaces for a ringed space x o we show that the deformations of the abelian category mod o of sheaves of o modules lowen and van den bergh 2006 are obtained from algebroid prestacks as introduced by kontsevich in case x is a quasi compact separated scheme the same is true for qch o the category of quasi coherent sheaves on x it follows in particular that there is a deformation equivalence between mod o and qch o

deformations of algebroid stacks abstract in this paper we consider deformations of an algebroid stack on an etale groupoid we construct a differential graded lie algebra dgla which controls this deformation theory in the case when the algebroid is a twisted form of functions we show that this dgla is quasiisomorphic to the twist of the dgla of hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe