the capelli identity tube domains and the generalized laplace transform the key result in this paper is a proof of a large class of identities generalizing the capelli identity cl the capelli identity is a centerpiece of 19th century invariant theory it asserts the equality of two differential operators on the n2 dimensional space m n r of n x n real matrices let e denote the ijth elementary matrix let xii denote the linear functions on m n r dual to e and write 3 for the partial differential operator a ax then this identity says

comparison results between linear and quasilinear elliptic boundary value problems on considere des resultats de comparaison du type suivant on suppose que u c 2 c satisfait un probleme aux valeurs limites elliptique quasi lineaire div a x u u u b x u u dans u sur et on suppose que v est la solution classique du probleme lineaire v const dans v sur on etudie l existence de situations ou u x v x x

the capelli eigenvalue problem for lie superalgebras for a finite dimensional unital complex simple jordan superalgebra j the tits kantor koecher construction yields a 3 graded lie superalgebra mathfrak g flat cong mathfrak g flat 1 oplus mathfrak g flat 0 oplus mathfrak g flat 1 such that mathfrak g flat 1 cong j set v mathfrak g flat 1 and mathfrak g mathfrak g flat 0 in most cases the space mathscr p v of superpolynomials on v is a completely reducible and multiplicity free representation of mathfrak g and there exists a direct sum decomposition mathscr p v bigoplus lambda in omega v lambda where left v lambda right lambda in omega is a family of irreducible mathfrak g modules parametrized by a set of partitions omega in these cases one can define a natural basis left d lambda right lambda in omega of capelli operators for the algebra mathscr pd v mathfrak g of mathfrak g invariant superpolynomial differential operators on v in this paper we complete the solution to the capelli eigenvalue problem which asks for the determination of the scalar c mu lambda by which d mu acts on v lambda we associate a restricted root system varsigma to the symmetric pair mathfrak g mathfrak k that corresponds to j which is either a deformed root system of type mathsf a m n or a root system of type mathsf q n we prove a necessary and sufficient condition on the structure of varsigma for mathscr p v to be completely reducible and multiplicity free when varsigma satisfies the latter condition we obtain an explicit formula for the eigenvalue c mu lambda in terms of sergeev veselov s shifted super jack polynomials when varsigma is of type mathsf a m n and okounkov ivanov s factorial schur q polynomials when varsigma is of type mathsf q n along the way we prove that the natural map from the centre of the enveloping algebra of mathfrak g into mathscr pd v mathfrak g is surjective in all cases except when j cong f where f is the 10 dimensional exceptional jordan superalgebra

deformed quantum calogero moser problems and lie superalgebras the deformed quantum calogero moser sutherland problems related to the root systems of the contragredient lie superalgebras are introduced the construction is based on the notion of the generalized root systems suggested by v serganova for the classical series a recurrent formula for the quantum integrals is found which implies the integrability of these problems the corresponding algebras of the quantum integrals are investigated the explicit formulas for their poincare series for generic values of the deformation parameter are presented

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

on fourth order elliptic boundary value problems with nonmonotone nonlinear function this paper is concerned with the fourth order elliptic boundary value problems with nonmonotone nonlinear function the existence and uniqueness of a solution is proven by the method of upper and lower solutions a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution

on symmetric invariants of centralisers in reductive lie algebras let g be a finite dimensional simple lie algebra of rank l over an algebraically closed field of characteristic 0 let e be a nilpotent element of g and let ge be the centraliser of e in g in this paper we study the algebra s ge ge of symmetric invariants of ge we prove that if g is of type a or c then s ge ge is always a graded polynomial algebra in l variables and we show that this continues to hold for some nilpotent elements in the lie algebras of other types in type a we prove that the invariant algebra s ge ge is freely generated by a regular sequence in s ge and describe the tangent cone at e to the nilpotent variety of g

nonlinear fourth order elliptic equations with nonlocal boundary conditions this paper is concerned with a class of fourth order nonlinear elliptic equations with nonlocal boundary conditions including a multi point boundary condition in a bounded domain of rn also considered is a second order elliptic equation with nonlocal boundary condition and the usual multi point boundary problem in ordinary differential equations the aim of the paper is to show the existence of maximal and minimal solutions the uniqueness of a positive solution and the method of construction for these solutions our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations the monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

a meshless method with enriched weight functions for fatigue crack growth the meshless method is particularly appropriate to solve crack propagation problems in this paper the fatigue growth of cracks in two dimensional bodies is considered the analysis is based upon paris equation new enriched weight functions are introduced in the meshless method formulation to capture the singularity at the crack tip simple problems show the accuracy and efficiency of this method then it is applied to fatigue analysis of single and multi cracked bodies under mixed mode conditions copyright 2004 john wiley sons ltd

on a q analog of a sahi result we obtain a q analog of a well known sahi result on the joint spectrum of s gln gln invariant differential operators with polynomial coefficients on the vector space of complex n n matrices

a new approach to queer differential equations abstract queer differential equations first arose in the work of harold grad on controlled thermonuclear fusion these relate in particular to models for the slow adiabatic evolution and resistive diffusion of a plasma these are queer in that these share aspects of partial ordinary and functional differential equations in this document the authors give a new way of thinking of these equations by relating these to free boundary problems this is the first of a series of papers intending to demonstrate that solutions of such a queer differential equation can be thought of as limits of solutions of free boundary problems with n free boundaries a long term hope is that this work will complement and further refine existing numerical schemes for finding such solutions

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

some groups of transformations defined by jordan algebras i it is well known that the lie algebra of derivations of the exceptional jordan algebra m 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional lie algebra f 4 of killing cartan and that the lie algebra of linear transformations in m 3 8 leaving a certain cubic norm form defined on m 3 8 invariant is the exceptional lie algebra e 6 chevalley schafer 7 freudenthal 11 the program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple jordan algebras the present paper gives the definition of the groups and lie algebras in the general case and gives their determination for special jordan algebras with some restrictions on the base field in a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple jordan algebras and in a third paper we hope to study the generalized groups of types e 6 which arise from these jordan algebras

a cayley hamilton theorem for the skew capelli elements for the central elements of the universal enveloping algebra of the lie algebra gln named the skew capelli elements a cayley hamilton type formula is given its classical counterpart is an elementary formula for two alternating matrices as a byproduct of the main result the description of the skew capelli elements given by k kinoshita and m wakayama explicit capelli identities for skew symmetric matrices proc edinburgh math soc to appear is deduced naturally

element free galerkin method for wave propagation and dynamic fracture abstract element free galerkin method efg is extended to dynamic problems efg method which is based on moving least square interpolants mls requires only nodal data no element connectivity is needed this makes the method particularly attractive for moving dynamic crack problems since remeshing can be avoided in contrast to the earlier formulation for static problems by authors the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions with this formulation the stiffness matrix is symmetric and positive semi definite and hence the consistency conergence and stability analyses of time integration remain the same as those in finite element method numerical examples are presented to illustrate the performance of this method the relationship between the element free galerkin method and the smooth particle hydrodynamics sph method is also discussed in this paper results are presented for some one dimensional problems and two dimensional problems with static and moving cracks

symmetric subalgebras noncohomologous to zero in a complex semisimple lie algebra and restrictions of symmetric invariants abstract let g be a semisimple involutory lie algebra and g k the corresponding symmetric pair let h be a fundamental cartan subalgebra of g containing a cartan subalgebra t of k the semisimple involutory lie algebras g with the symmetric subalgebra k noncohomologous to zero in g are completely classified by showing that k is noncohomologous to zero in g if and only if the spin representation of g is primary based on this result we then determine the image of the restriction map s h w g h s t w k t where w g h and w k t are the respective weyl groups

poisson transforms on vector bundles let g be a connected real semisimple lie group with finite center and k a maximal compact subgroup of g let r v be an irreducible unitary representation of k and g x k tv the associated vector bundle in the algebra of invariant differential operators on g xk v the center of the universal enveloping algebra of lie g induces a certain commutative subalgebra z we are able to determine the characters of z given such a character we define a poisson transform from certain principal series representations to the corresponding space of joint eigensections we prove that for most of the characters this map is a bijection generalizing a famous conjecture by helgason which corresponds to r the trivial representation introduction let g be a connected real semisimple lie group with finite center and k a maximal compact subgroup of g then g k is a riemannian symmetric space of noncompact type we fix an iwasawa decomposition g kan let m be the centralizer of a in k let g and a be the lie algebras of g and a respectiarely and s 0 a the root system for 0 a let s 0 a be the positiare roots in s a for the ordering giaren by n let d g k be the algebra of inarariant differential operators on g k it is well known that the characters of d g k are parametrized by a e a the complex dual space of a let a g k denote the space of joint eigenfunctions corresponding to a we write g k g exp h g n g for each g e g according to g kan for each x e c k m we define pxx e c g k by p x g t k e p h g lk dk k here p is the half sum of s 0 a including multiplicities it turns out that pxq e v g k one can easily extend the definition of px to the space d k m resp a k m of distributions resp analytic functionals on k m in this paragraph we fix a e a such that 2 a ct ct ct is not in n 0 for each ct c s 0 a it is proared by helgason helg2 that pa defines a bijection from c o k m k finite onto e g k k finite he also proares in the rank one case p is a bijection from a k m onto 6 g k he then conjectured this should be true for the higher rank case the conjecture was earentually proared by six japanese mathematicians in kkmoot it should be mentioned that a representation theoretic proof by schmid starting from the kfinite result is indicated in sch lewis then a student of helgason made the following obseraration let 6 g k be the subspace of e g k whereeach element increases at most exporlentially see 02 for definition received by the editors september 28 1994 and in revised form january 30 1995 1991 mathematics subject classification primary 22e46 secondary 43a85 1998 american mathematical society 857 this content downloaded from 157 55 39 198 on fri 17 jun 2016 05 05 10 utc all use subject to http about jstor org terms

semi direct products of lie algebras and covariants abstract let q be a connected algebraic group with lie algebra q symmetric invariants of q i e the q invariants in the symmetric algebra s q of q is a first approximation to the understanding of the coadjoint action q q and coadjoint q orbits in this article we study a class of non reductive lie algebras where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant theoretic properties if g is a semisimple group with lie algebra g and v is g module then we define q to be the semi direct product of g and v then we are interested in the case where the generic isotropy group for the g action on v is reductive and commutative it turns out that in this case symmetric invariants of q can be constructed via certain g equivariant maps from g to v covariants

sign definite solutions in some linear elliptic systems on considere un ensemble faiblement couple de deux equations aux derivees partielles ou la matrice de couplage a des elements variables et la partie principale de chaque equation est le meme operateur uniformement elliptique on obtient des conditions necessaires pour le decouplage on determine le signe algebrique des composantes de solution

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

comparison theorems for a class of nonlinear elliptic boundary value problems on compare des solutions d une classe de problemes de dirichlet quasi lineaires aux solutions de problemes lineaires d ordre 2 plus simples definis sur le meme domaine

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

explicit hilbert spaces for certain unipotent representations ii letg be the universal cover of the group of automorphisms of a symmetric tube domain and letp ln be its shilov boundary parabolic subgroup this paper attaches an irreducible unitary representation ofg to each of the finitely many l orbits onn the hilbert space of the representation consists of functions on the orbit which are square integrable with respect to a certainl equivariant measure the representation remains irreducible when restricted top and descends to a quotient ofg which is at worst thedouble cover of a linear group if thel orbit isnot open inn the construction gives a unipotent representation ofg

canonical semi invariants and the plancherel formula for parabolic groups a parabolic subgroup of a reductive lie group is called good if the center of the universal enveloping algebra of its nilradical contains an element that is semi invariant of weight proportional to the modular function the good case is characterized here by invariance of the set of simple roots defining the parabolic under the negative of the opposition element of the weyl group in the good case the unbounded dixmier pukanszky operator of the parabolic subgroup is described the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified and an explicit plancherel formula is derived for that parabolic 1 introduction this paper continues our work on the plancherel formula for parabolic subgroups of reductive lie groups earlier we considered maximal parabolic subgroups 7 and 11 minimal parabolic subgroups 4 and 6 and parabolic subgroups whose unipotent radical has square integrable representations modulo its center 7 and 12 in each case we described the representation theory the plancherel measure and the dixmier pukanszky operator that appears in the plancherel formula to compensate for nonunimodularity we now consider arbitrary parabolic subgroups we isolate the common features of all previously treated cases which enabled us to describe the fundamental semi invariant polynomial that gives rise to the dixmier pukanszky operator that is our first main result here the second main result is a simple characterization of which parabolics have the requisite property and the third is the construction in that case of the canonical semi invariant polynomial we explicitly compute the dixmier pukanszky operator and the plancherel formula for parabolics that have the requisite property here is a brief summary of the contents of this paper in 2 we describe the key property see definition 2 1 for parabolic groups that guarantees the existence of a unique appropriate semi invariant theorem 2 2 in the center of the universal enveloping algebra of the nilradical we call such parabolics good let p be a good parabolic and p nam its langlands decomposition we prove proposition 2 4 that the generic stability groups si for

on the coadjoint representation of z2 contractions of reductive lie algebras abstract we study the coadjoint representation of contractions of reductive lie algebras associated with symmetric decompositions let g g 0 g 1 be a symmetric decomposition of a reductive lie algebra g then the semi direct product of g 0 and the g 0 module g 1 is a contraction of g we conjecture that these contractions have many properties in common with reductive lie algebras in particular it is proved that in many cases the algebra of invariants is polynomial we also discuss the so called codim 2 property for coadjoint representations and its relationship with the structure of algebra of invariants

capelli identities for classical lie algebras we extend the capelli identity from the lie algebra frak gl n to the other classical lie algebras frak so n and frak sp n we employ the theory of reductive dual pairs due to howe

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

dirac cohomology unitary representations and a proof of a conjecture of vogan u radu je dokazana slutnja d vogana koja odredjuje infinitesimalni karakter ireducibilnog g k modula koji ima diracovu kohomologiju razlicitu od nula infinitesimalni karakter je dan kao rho translat najvece tezine k tipa u diracovoj kohomologiji diracova kohomologija se definira kao ker d im d cap ker d gdje d oznacava diracov operator na zadanom g k modulu tenzoriranom sa spin modulom

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

the capelli problem for gl m n and the spectrum of invariant differential operators abstract the capelli problem for the symmetric pairs gl gl gl gl o and gl sp is closely related to the theory of jack polynomials and shifted jack polynomials for special values of the parameter see 12 15 14 18 in this paper we extend this connection to the lie superalgebra setting namely to the supersymmetric pairs g k gl m 2 n osp m 2 n and gl m n gl m n gl m n acting on w s 2 c m 2 n and c m n c m n to achieve this goal we first prove that the center of the universal enveloping algebra of the lie superalgebra g maps surjectively onto the algebra pd w g of g invariant differential operators on the superspace w thereby providing an affirmative answer to the abstract capelli problem for w our proof works more generally for gl m n acting on s 2 c m n and is new even for the ordinary cases m 0 or n 0 considered by howe and umeda in 9 we next describe a natural basis d of pd w g that we call the capelli basis using the above result on the abstract capelli problem we generalize the work of kostant and sahi 12 15 20 by showing that the spectrum of d is given by a polynomial c which is characterized uniquely by certain vanishing and symmetry properties we further show that the top homogeneous parts of the eigenvalue polynomials c coincide with the spherical polynomials d which arise as radial parts of k spherical vectors of finite dimensional g modules and which are super analogues of jack polynomials this generalizes results of knop and sahi 14 finally we make a precise connection between the polynomials c and the shifted super jack polynomials of sergeev and veselov 25 for special values of the parameter we show that the two families are related by a change of coordinates that we call the frobenius transform

dynamic fracture using element free galerkin methods the element free galerkin method for dynamic crack propagation is described and applied to several problems this method is a gridless method which facilitates the modelling of growing crack problems because it does not require remeshing the growth of the crack is modelled by extending its surfaces the essential feature of the method is the use of moving least squares interpolants for the trial and test functions in these interpolants the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point the discrete equations are obtained by a galerkin method the procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described

derived functor modules dual pairs and u g k actions abstract derived functors or zuckerman functors play a very important role in the study of unitary representations of real reductive groups these functors are usually applied on highest weight modules in the so called good range and the theory is well understood on the other hand there were several studies on the irreducibility and unitarizability in which derived functors are applied to singular modules see enright et al acta math 1985 for example in this article we apply derived functors to certain modules arising from the formalism of local theta lifting and investigate the irreducible constituent of resulting modules the key technique is to understand u g k actions in the setting of a see saw pair our results suggest that derived functor constructions are compatible with local theta liftings

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

dimensions of spaces of generalized spherical functions generalized spherical functions are defined to be joint eigenfunctions of some invariant differential operators with fixed k type on both sides on a semisimple lie group one of our main results is the same dimension formula for all spaces of generalized spherical functions which is independent of eigenvalues the other main result is proof of the existence of a global basis for all spaces of generalized spherical functions which is holomorphic in the parameter for the eigenvalues as an application we calculate the distribution characters of k bi finite eigenfunctions of the center of enveloping algebra for split groups

classification of linearly compact simple jordan and generalized poisson superalgebras abstract we classify all linearly compact simple jordan superalgebras over an algebraically closed field of characteristic zero as a corollary we deduce the classification of all linearly compact unital simple generalized poisson superalgebras

the polynomiality of the poisson center and semi center of a lie algebra and dixmier s fourth problem abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we provide necessary and also some sufficient conditions in order for its poisson center and semi center to be polynomial algebras over k this occurs for instance if g is quadratic of index 2 with g g g and also if g is nilpotent of index at most 2 the converse holds for filiform lie algebras of type l n q n r n and w n we show how dixmier s fourth problem for an algebraic lie algebra g can be reduced to that of its canonical truncation g moreover dixmier s statement holds for all lie algebras of dimension at most eight the nonsolvable ones among them possess a polynomial poisson center and semi center

accurate fracture modelling using meshless methods the visibility criterion and level sets formulation and 2d modelling fracture modelling using numerical methods is well advanced in 2d using techniques such as the extended finite element method xfem the use of meshless methods for these problems lags somewhat behind but the potential benefits of no meshing particularly in 3d prompt continued research into their development in methods where the crack face is not explicitly modelled as the edge of an element for instance two procedures are instead used to associate the displacement jump with the crack surface the visibility criterion and the diffraction method the visibility criterion is simple to implement and efficient to compute especially with the help of level set coordinates however spurious discontinuities have been reported around crack tips using the visibility criterion whereas implementing the diffraction method in 3d is much more complicated than the visibility criterion in this paper a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved the formulation is based on the use of level set coordinates and the element free galerkin method and is generally applicable for single or multiple crack problems in 2d or 3d the paper explains the formulation and provides verification of the method against a number of 2d crack problems copyright 2011 john wiley sons ltd

gelfand pairs on the heisenberg group and schwartz functions abstract let h n be the 2 n 1 dimensional heisenberg group and k a compact group of automorphisms of h n such that k h n k is a gelfand pair we prove that the gelfand transform is a topological isomorphism between the space of k invariant schwartz functions on h n and the space of schwartz function on a closed subset of r s homeomorphic to the gelfand spectrum of the banach algebra of k invariant integrable functions on h n

multiple crack weight for solution of multiple interacting cracks by meshless numerical methods summary 7 we devise a multiple crack weight mcw method for the accurate and effective solution of strongly interacting cracks by meshless numerical methods the mcw method constructs weight functions 9 around cracks so that they simultaneously characterize all the cracks present in the single nodal domain of influence this approach reduces the number of nodes necessary to achieve sufficient 11 accuracy and consequently it decreases the computational effort numerical examples demonstrate that the method allows an accurate solution of multiple cracks problems convergence of the method is 13 analysed and discussed copyright 2006 john wiley sons ltd

invariant differential operators and eigenspace representations on an affine symmetric space let g h be an affine symmetric space of split rank r let d be a preferred polynomial algebra of g invariant differential operators on g h generated by r elements we show that the space of k finite joint eigenfunctions of d on g h form an admissible g k module which is called an eigenspace representation the main content of this paper is description of the algebras of invariant differential operators and determination of the eigenspace representations on g h we also obtain a poisson transform for t spherical eigenfunctions on g h by eisenstein integrals

functors for unitary representations of classical real groups and affine hecke algebras we define exact functors from categories of harish chandra modules for certain real classical groups to finite dimensional modules over an associated graded affine hecke algebra with parameters we then study some of the basic properties of these functors in particular we show that they map irreducible spherical representations to irreducible spherical representations and moreover that they preserve unitarity in the case of split classical groups we thus obtain a functorial inclusion of the real spherical unitary dual with real infinitesimal character into the corresponding p adic spherical unitary dual

capelli identities for reductive dual pairs abstract analogues of the capelli identity are given for every irreducible reductive dual pair in the complex symplectic group and the complex orthogonal group they describe a natural correspondence between the centers of the two universal enveloping algebras and an algebra of invariant operators

the berezin transform and invariant differential operators the berezin calculus is important to quantum mechanics creation annihilation operators and operator theory toeplitz operators we study the basic berezin transform linking the contravariant and covariant symbol for all bounded symmetric domains and express it in terms of invariant differential operators

calculus of principal series whittaker functions on gl 3 c abstract we study whittaker functions for the principal series representations of sl n r we derive a system of partial differential equations characterizing our whittaker functions we give explicitly power series solutions at the regular singularity of the system and integral representations of the unique moderate growth whittaker functions

an enriched meshless method for non linear fracture mechanics this paper presents an enriched meshless method for fracture analysis of cracks in homogeneous isotropic non linear elastic two dimensional solids subject to mode i loading conditions the method involves an element free galerkin formulation and two new enriched basis functions types i and ii to capture the hutchinson rice rosengren singularity field in non linear fracture mechanics the type i enriched basis function can be viewed as a generalized enriched basis function which degenerates to the linear elastic basis function when the material hardening exponent is unity the type ii enriched basis function entails further improvements of the type i basis function by adding trigonometric functions four numerical examples are presented to illustrate the proposed method the boundary layer analysis indicates that the crack tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered whether for the near tip asymptotic field or for the far tip elastic field numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the j integral for the applied load intensities and material properties considered also the crack mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results furthermore the meshless results show excellent agreement with the experimental measurements indicating that the new basis functions are also capable of capturing elastic plastic deformations at a stress concentration effectively copyright 2003 john wiley sons ltd

invariant differential operators on hermitian symmetric spaces our object of study is the ring of invariant differential operators on a hermitian symmetric space g k of classical and noncompact type here as usual g is a connected noncompact semisimple lie group with finite center and k is a maximal compact subgroup of g it is well known that the ring denoted by g k is isomorphic to a polynomial ring of 1 variables 1 being the rank of g k this is true even in the nonhermitian case if 1 1 it is generated by the laplace beltrami operator z which is essentially self adjoint moreover iz2 is nonnegative in the general case an easily posable problem is to find an explicitly defined set of generators we can go one step further by focusing our attention on the nonnegativity which necessarily limits the range of eigenvalues under a suitable integrability condition on functions it is thus natural to ask whether there exist some canonically defined operators which generate 9 g k and have the property of nonnegativity the main purpose of the present paper is to give an affirmative answer to this question in fact our answer applies to a somewhat more general type of rings of differential operators which includes 9 g k as a special case to be explicit we take an irreducible representation p k gl v with a complex vector space v of finite dimension and consider the set c p of all v valued cx functions f on g such that f xk 1 p k f x for every k e k we then denote by p the ring of left invariant difrrential operators on g which map c p into itself now the complexification g of the lie algebra of g has abelian subalgebras p and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on g k at the origin for any complex vector space w let sr w denote the ring of all complex valued homogeneous polynomial functions on w of degree r and let s w er 0sr w through the adjoint representation of g on g k acts naturally on sr it is a known fact due to hua and schmid that each irreducible constituent of this representation of k has multiplicity one now our principal

the spherical transform of a schwartz function on the heisenberg group suppose thatk u n is a compact lie group acting on the 2n 1 dimensional heisenberg grouphn we say that k hn is a gelfand pair if the convolution algebral1k hn of integrablek invariant functions onhnis commutative in this case the gelfand space k hn is equipped with the godement plancherel measure and the spherical transform l2k hn l2 k hn is an isometry the main result in this paper provides a complete characterization of the set sk hn f f sk hn of spherical transforms ofk invariant schwartz functions onhn we show that a functionfon k hn belongs to sk hn if and only if the functions obtained fromfvia application of certain derivatives and difference operators satisfy decay conditions we also consider spherical series expansions fork invariant schwartz functions onhnmodulo its center

generalised discriminants deformed calogero moser sutherland operators and super jack polynomials abstract it is shown that the deformed calogero moser sutherland cms operators can be described as the restrictions on certain affine subvarieties called generalised discriminants of the usual cms operators for infinite number of particles the ideals of these varieties are shown to be generated by the jack symmetric functions related to the young diagrams with special geometry a general structure of the ideals which are invariant under the action of the quantum cms integrals is discussed in this context the shifted super jack polynomials are introduced and combinatorial formulas for them and for super jack polynomials are given

computing invariants and semi invariants by means of frobenius lie algebras abstract let u be the enveloping algebra of a finite dimensional lie algebra over a field k of characteristic zero z u its center and sz u its semi center a sufficient condition is given in order for sz u to be a polynomial algebra over k surprisingly this condition holds for many lie algebras especially among those for which the radical is nilpotent in which case sz u z u in particular it allows the explicit description of z u for more than half of all complex indecomposable nilpotent lie algebras of dimension at most 7

the poisson center and polynomial maximal poisson commutative subalgebras especially for nilpotent lie algebras of dimension at most seven abstract let g be a finite dimensional lie algebra over an algebraically closed field k of characteristic zero we collect some general results on the poisson center of s g including some simple criteria regarding its polynomiality and also on certain poisson commutative subalgebras of s g these facts are then used to complete our previous work on the subject ooms 2009 o5 5 i e to give an explicit description for the poisson center of all indecomposable nilpotent lie algebras of dimension at most seven among other things we also provide a polynomial maximal poisson commutative subalgebra of s g enjoying additional properties as a by product we show that a conjecture by milovanov is valid in this situation these results easily carry over to the enveloping algebra u g

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

smoothing and accelerated computations in the element free galerkin method two topics in the formulation and implementation of meshless methods are considered the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives these techniques are described in the context of the element free galerkin method but they are applicable to other meshless methods results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant the speedup in calculating the shape functions is about a factor of two

the moment map for a multiplicity free action let k be a compact connected lie group acting unitarily on a finite dimensional complex vector space v one calls this a multiplicity free action whenever the at isotypic components of c v are a irreducible we have shown that this is the case if and only if the moment map t v t for the action is finite to one on a orbits this is equivalent to a result concerning gelfand pairs associated with heisenberg groups that is motivated by the orbit method further details of this work will be published elsewhere

a degree inequality for lie algebras with a regular poisson semi center for lie algebras whose poisson semi center is a polynomial ring we give a bound for the sum of the degrees of the generating semi invariants this bound was previously known in many special cases

bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function in this paper maximum principles are employed to relate solutions of certain classes of nonlinear elliptic problems to solutions of the associated torsion problem by this method a number of new isoperimetric inequalities are derived in special cases solutions of the nonlinear problems are also related to solutions of the clamped membrane problem

new schur function series this paper presents some new product identities for certain summations of schur functions these identities are generalizations of some famous identities known to littlewood and appearing in macdonald s book we refer to these identities as the littlewood type formulas in addition analogues for summations of characters of the other classical groups are also given the littlewood type formulas in this paper are separated into two classes the rational schur function series and the generalized schur function series an application of a rational schur function series to the infinite product representation of the elliptic theta functions is also given we prove these littlewood type formulas using the cauchy binet formula the cauchy binet formula is a basic but powerful tool applicable in the present context which can be derived from our pfaffian formula as we explain q 1998 academic press

complex variable approach to the bem for multiple crack problems a boundary element method for straight multiple center and edge crack problems is developed in this paper the method is constructed upon the systematic use of the elastic singularity solutions in complex variables the crack opening is represented by the continuous distribution of dislocation dipoles and the effect of the non crack boundary by the continuous distributions of point forces and dislocation dipoles the crack tip singularity is embedded into the interpolation using orthogonal polynomials i e chebyshev and jacobi and their associated singular weight functions the proposed analytical integration procedure of the cauchy type integrals defined over the crack eliminates the need for the quadrature formulae for numerical integration streamlines and enhances the accuracy of the traditional singular integral equation method for crack problems the stress intensity factors for the fifteen problems analyzed in this paper have been accurate enough to substitute those given in stress intensity factor handbooks since non crack boundary of arbitrary shape can be introduced at will the method is expected to give accurate stress intensity factors for complex real life problems