on the hyperbolic kac moody lie algebra ha_1 1 in this paper using a homological theory of graded lie algebras and the representation theory of a we compute the root multiplicities of the hyperbolic kac moody lie algebra ha up to level 4 and deduce some interesting combinatorial identities introduction the hyperbolic kac moody lie algebras have been considered as the next natural objects of study after the affine case while the affine kac moody lie algebras have been extensively studied for their close connections to areas such as combinatorics modular forms and mathematical physics many basic questions regarding the hyperbolic case are still unresolved for example the behavior of the root multiplicities is not well understood feingold and frenkel f f and kac moody and wakimoto k m w made some progress in this area they computed the level 2 root multiplicities for the hyperbolic kac moody lie algebras ha 1 and he 1 respectively in kangl and kang2 we introduced an inductive program to study the higher level root multiplicities and the principally specialized affine characters of a certain class of lorentzian kac moody lie algebras more precisely we realize these lorentzian kac moody lie algebras as the minimal graded lie algebras l efez ln with local part v lo v where lo is an affine kac moody lie algebra v is the basic representation of lo and v is the contragredient of v thus l g i where g nez gn is the maximal graded lie algebra with local part v lo v and i ednez i is the maximal graded ideal of g which intersects the local part trivially by developing a homological theory we determined the structure of homogeneous subspaces ln g ii as modules over the affine kac moody lie algebra lo for certain higher levels the hochschild serre spectral sequences played an important role in determining the structure of i n for the hyperbolic kac moody lie algebra ha 1 we were able to determine the structure of l4 for n 1 2 5 and computed the principally specialized affine characters up to level 5 but the root multiplicities were computed up to level 3 only in this paper using the representation theory of a 1 developed received by the editors june 4 1990 and in revised form september 12 1991 1991 mathematics subject classification primary 1 7b67 1 7b65 secondary 1 7b10 this work was supported by nsf grant dms 8806371 1994 american mathematical society 0002 9947 94 1 00 25 per page

a variational approach to optimal meshes a variational approach for the optimization of triangular or tetrahedral meshes is presented starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity it will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions in addition the injectivity of the deformed mesh is discussed thereby it is possible to construct suitable meshes for various numerical applications

multigrid preconditioning for the biharmonic dirichlet problem a multigrid preconditioning scheme for solving the ciarlet raviart mixed method equations for the biharmonic dirichlet problem is presented in particular a schur complement formulation for these equations which yields noninherited quadratic forms is considered the preconditioning scheme is compared with a standard w cycle multigrid iteration it is proved that a variable v cycle preconditioner leads to problems with uniformly bounded condition numbers however w cycle convergence is proved only if the number of smoothings m is sufficiently large an example is given in which the w cycle diverges unless m geq 8 divergent w cycles are also encountered when solving the morley equations for the biharmonic dirichlet problem although brenner has proved w cycle convergence for sufficiently large m 9 this is illustrated with additional computations while variable v cycles continue to produce excellent preconditioners in this setting certain approximate l 2 inner products are described and a modif

fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows in this paper we investigate the numerical simulation of particulate flows using a new moving mesh method combined with the multigrid fictitious boundary method fbm s turek d c wan l s rivkind the fictitious boundary method for the implicit treatment of dirichlet boundary conditions with applications to incompressible flow simulations challenges in scientific computing lecture notes in computational science and engineering vol 35 springer berlin 2003 pp 37 68 d c wan s turek l s rivkind an efficient multigrid fem solution technique for incompressible flow with moving rigid bodies numerical mathematics and advanced applications enumath 2003 springer berlin 2004 pp 844 853 d c wan s turek direct numerical simulation of particulate flow via multigrid fem techniques and the fictitious boundary method int j numer method fluids 51 2006 531 566 with this approach the mesh is dynamically relocated through a linear partial differential equation to capture the surface of the moving particles with a relatively small number of grid points the complete system is realized by solving the mesh movement and the partial differential equations of the flow problem alternately via an operator splitting approach the flow is computed by a special ale formulation with a multigrid finite element solver and the solid particles are allowed to move freely through the computational mesh which is adaptively aligned by the moving mesh method in every time step one important aspect is that the data structure of the undeformed initial mesh in many cases a tensor product mesh or a semi structured grid consisting of many tensor product meshes is preserved while only the spacing between the grid points is adapted in each time step so that the high efficiency of structured meshes can be exploited numerical results demonstrate that the interaction between the fluid and the particles can be accurately and efficiently handled by the presented method it is also shown that the presented method significantly improves the accuracy of the previous multigrid fbm to simulate particulate flows with many moving rigid particles

dimension formula for graded lie algebras and its applications abstract in this paper we investigate the structure of infinite dimensional lie algebras l l l graded by a countable abelian semigroup satisfying a certain finiteness condition the euler poincare principle yields the denominator identities for the graded lie algebras from which we derive a dimension formula for the homogeneous subspaces l our dimension formula enables us to study the structure of the graded lie algebras in a unified way we will discuss some interesting applications of our dimension formula to the various classes of graded lie algebras such as free lie algebras kac moody algebras and generalized kac moody algebras we will also discuss the relation of graded lie algebras and the product identities for formal power series

indefinite kac moody algebras of classical type the first author is supported in pat by nsf under grant dms 9025111 the 3rd author is supported in part by nsa msp grant mda 90 h 4039

canonical coherent states for the relativistic harmonic oscillator in this paper there are constructed manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given sl 2 r representation once a change of variables z c zd unit disk is performed also introduced are higher order relativistic creation and annihilation operators a a with canonical commutation relation a a 1 rather than the covariant one z z energy and naturally associated with the sl 2 r group the canonical relativistic coherent states are then defined as eigenstates of a finally a canonical minimal representation is constructed in configuration space by means of eigenstates of a canonical position operator

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

on the convergence of recursive trust region methods for multiscale nonlinear optimization and applications to nonlinear mechanics we prove new convergence results for a class of multiscale trust region algorithms originally introduced by gratton sartenaer and toint in siam j optim 19 2008 pp 414 444 to solve unconstrained minimization problems within the euclidean space mathbb r n we will state less restrictive assumptions on the objective function and on the iteratively computed trust region corrections which allow for proving first order convergence moreover we propose a novel projection approach for obtaining the initial coarse level iterates which are needed within the nonlinear multiscale iteration we show the efficiency and robustness of our approach by means of numerical examples from nonlinear continuum mechanics where stored energy functions for materials of ogden type and for materials with visco plastic behavior are minimized

a monolithic geometric multigrid solver for fluid structure interactions in ale formulation summary we present a monolithic geometric multigrid solver for fluid structure interaction problems in arbitrary lagrangian eulerian coordinates the coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill conditioned systems of algebraic equations direct solvers usually are out of question because of memory limitations and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices the use of partitioned preconditioners in krylov subspace iterations is an option but the convergence will be limited by the outer partitioning our proposed solver is based on a newton linearization of the fully monolithic system of equations discretized by a galerkin finite element method approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration preconditioned by a geometric multigrid solver the special character of fluid structure interactions is accounted for by a partitioned scheme within the multigrid smoother only here fluid and solid field are segregated as dirichlet neumann coupling we demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems while 2d problems are well manageable with available direct solvers challenging 3d problems highly benefit from the resulting multigrid solver copyright 2015 john wiley sons ltd

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

a hybrid level set front tracking finite element approach for fluid structure interaction and two phase flow applications we present a hybrid level set front tracking approach suitable for fluid structure interaction and two phase flow applications our approach aims at extending geometrical flexibility of standard mesh moving front tracking methods by introducing an additional implicit level set representation of the geometry under consideration the computational mesh is automatically aligned to the implicitly described geometry by minimizing a nonlinear constrained functional resulting triangulations approximate the geometry accurately while being optimal in a certain sense due to the mesh alignment finite element spaces defined on these triangulations may be easily adjusted to account for special solution properties such as discontinuities across interfaces in order to demonstrate the flexibility of the proposed approach we apply it to a simplified one way coupled fluid structure interaction problem inspired by the flow induced by a moving cardiac valve furthermore we evaluate the approach by solving a two phase flow benchmark problem

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

the monster lie algebra abstract we calculate the multiplicities of all the roots of the monster lie algebra this gives an example of a lie algebra all of whose simple roots and root multiplicities are known and which is not finite dimensional or an affine kac moody algebra there seem to be several similar infinite dimensional lie algebras which to a sympathetic eye appear to correspond to some of the sporadic simple groups

free lie algebras generalized witt formula and the denominator identity abstract let be a countable abelian semigroup satisfying a suitable finiteness condition and let l l be the free lie algebra generated by a graded vector space v over c in this paper from the denominator identity we derive a dimension formula for the homogeneous subspaces of the free lie algebra l l and discuss numerous applications of our dimension formula to various interesting cases our dimension formula will be expressed in terms of the witt partition functions various expressions of the witt partition functions will give rise to a number of interesting combinatorial identities as a special case we obtain a recursive relation for the coefficients of the elliptic modular function j

generalized intelligent states and squeezing the robertson schrodinger uncertainty relation for two observables a and b is shown to be minimized in the eigenstates of the operator a ib being a complex number such states called generalized intelligent states gis can exhibit arbitrarily strong squeezing of a or b the time evolution of gis is stable for hamiltonians which admit linear in a and b invariants systems of gis for the su 1 1 and su 2 groups are constructed and discussed it is shown that su 1 1 gis contain all the perelomov coherent states cs and the barut and girardello cs while the spin cs are a subset of su 2 gis cs for an arbitrary semisimple lie group can be considered as a gis for the quadratures of the weyl generators

kostant s formula for kac moody lie algebras abstract this paper is a generalization of h garland and j lepowsky s paper of 1976 let g a be the kac moody lie algebra defined by a symmetrizable generalized cartan matrix a m the irreducible highest weight module of g a with a dominant integral highest weight kostant s homology and cohomology formulas for the lie algebra g a and the module m are proved without assuming that the reductive part of the parabolic subalgebra of g a is of finite type a resolution of m is constructed for any j the jth term of the resolution has a filtration such that all the factors of the filtration are generalized verma modules of the form vm w p p where w ranges over a certain subset of the weyl group of g a

robertson intelligent states the diagonalization of the uncertainty matrix and the minimization of robertson inequality for n observables are considered it is proved that for even n this relation is minimized in states which are eigenstates of n 2 independent complex linear combinations of the observables in the case of canonical observables this eigenvalue condition is also necessary such minimizing states are called robertson intelligent states ris the group related coherent states cs with maximal symmetry for semisimple lie groups are a particular case of ris for the quadratures of weyl generators explicit constructions of ris are considered for operators of su 1 1 su 2 and sp n r algebras unlike the group related cs ris can exhibit strong squeezing of group generators multimode squared amplitude squeezed states are naturally introduced as sp n r ris it is shown that the uncertainty matrices for quadratures of q deformed boson operators q 0 and of any k power of are positive definite and can be diagonalized by symplectic linear transformations

borcherds superalgebras and a monstrous lie superalgebra this research was supported in part by nondirected research fund korea research foundation and garc kosef at seoul national university

the relativistic hermite polynomial is a gegenbauer polynomial it is shown that the polynomials introduced recently by aldaya bisquert and navarro salas phys lett a 156 381 1991 in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of gegenbauer polynomials this fact is useful in the investigation of the properties of the corresponding wave function some examples are given in particular related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers

the use of solution adaptive grids in solving partial differential equations the grid point distribution used in solving a partial differential equation using a numerical method has a substantial influence on the quality of the solution an adaptive grid which adjusts as the solution changes provides the best results when the number of grid points available for use during the calculation is fixed basic concepts used in generating and applying adaptive grids are reviewed in this paper and examples illustrating applications of these concepts are presented

graded lie algebras of kac moody type the first author is supported in pat by nsf under grant dms 88 00691 the 3rd author is supported in part by nsa msp grant mda 90 h 4039

weyl groups of some kac moody algebras abstract in this paper we prove a lemma about the weyl groups of kac moody algebras and decompose the weyl group of some kac moody algebras into a semi direct product of two subgroups which are coxeter groups

the relativistic bargmann transform an integral transformation generalizing the bargmann transform is established among two sets of wavefunctions associated with the configuration and bargmann fock segal like representations of a relativistic harmonic oscillator the time independent configuration space wavefunctions are also studied and the lack of unitarity occurring when factorizing the time dependence of the wavefunction is solved by modifying the scalar product

deformed kac moody algebras and their representations abstract a class of lie algebras g a associated to generalized cartan matrices a is studied the lie algebras g a have much simpler structure than kac moody algebras but have the same root spaces with g a in particular g a has an abelian subalgebra of half size we show that g a has a non degenerate invariant symmetric bilinear form if and only if a is symmetrizable g x 1 g x 2 if and only if the gcms x 1 and x 2 are the same up to a permutation of rows and columns we study the lowest respectively highest weight verma module v respectively v over g a and obtain the necessary and sufficient conditions for v to be irreducible and also find its maximal proper submodule when v is reducible then using graded dual module of v we deduce the necessary and sufficient conditions for v to be irreducible

moving meshes by the deformation method in this paper we describe the deformation method for generation of moving meshes and its applications in computational fluid dynamics cfd and potential applications for image representation segmentation and landmark matching for image registration we demonstrate that the deformation method can generate meshes of prescribed cell area numerical examples are presented to confirm that

tetrahedral mesh improvement via optimization of the element condition number the authors present a new shape measure for tetrahedral elements that is optimal in that it gives the distance of a tetrahedron from the set of inverted elements this measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume using this shape measure they formulate two optimization objective functions that are differentiated by their goal the first seeks to improve the average quality of the tetrahedral mesh the second aims to improve the worst quality element in the mesh they review the optimization techniques used with each objective function and presents experimental results that demonstrate the effectiveness of the mesh improvement methods they show that a combined optimization approach that uses both objective functions obtains the best quality meshes for several complex geometries

quantization as a consequence of the symmetry group an approach to geometric quantization a method is proposed to obtain the dynamics of a system which only makes use of the group law it incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics the classical or quantum character of the theory is related to the choice of the group avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems i e the classical limit the method is applied to the free particle dynamics and the harmonic oscillator

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

mesh deformation using the biharmonic operator the use of the biharmonic operator for deforming a mesh in an arbitrary lagrangian eulerian simulation is investigated the biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh this allows both the position and the normal mesh spacing along a boundary to be controlled which is important for two fluid interfaces and periodic boundaries at these boundaries we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary in addition results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second order partial differential equations a final advantage is that with the biharmonic operator the integrity of a grid in a moving boundary layer can be preserved as the boundary moves the main disadvantage of the approach is its increased computational expense

dynamics on sl 2 r tilde x u 1 a complete analysis of the quantum dynamical system defined by the quantum group sl 2 r tilde x u 1 is made this quantum group which is primarily related to the relativistic harmonic oscillator is also shown to provide the quantum dynamics of a free particle moving in 1 1 dimensional anti de sitter spacetime the latter interpretation illustrates the capabilities of the present group approach to quantisation in dealing with dynamical systems in more general spacetimes

symmetry and quantization higher order polarization and anomalies the concept of full polarization subalgebra in a group approach to quantization on a lie group g as a generalization of the analogous concept in geometric or standard quantization is discussed the lack of full polarization subalgebras is considered as an anomaly of the corresponding system and related to its more conventional definition a generalization of the subalgebra of full polarization is then provided made out of higher order differential operators in the enveloping algebra of g higher order polarizations can also be used to quantize nonanomalous theories in different representations numerous examples are analyzed including the finite dimensional dynamics associated with the schrodinger group which presents an anomaly and an infinite dimensional anomalous system associated with the virasoro group in the last example the operators in the higher order polarization are in one to one correspondence with the null vectors in the verma module approach

a moving mesh method based on the geometric conservation law a new adaptive mesh movement strategy is presented which unlike many existing moving mesh methods targets the mesh velocities rather than the mesh coordinates the mesh velocities are determined in a least squares framework by using the geometric conservation law specifying a form for the jacobian determinant of the coordinate transformation defining the mesh and employing a curl condition by relating the jacobian to a monitor function one is able to directly control the mesh concentration the geometric conservation law an identity satisfied by any nonsingular coordinate transformation is an important tool which has been used for many years in the engineering community to develop cell volume preserving finite volume schemes it is used here to transform the algebraic expression specifying the jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy it is shown that the resulting method bears a close relation with the lagrangian method advantages of the new approach include the ease of controlling the cell volumes and therefore mesh adaption and a theoretical guarantee for existence and nonsingularity of the coordinate transformation it is shown that the method may suffer from the mesh skewness a consequence resulting from its close relation with the lagrangian method numerical results are presented to demonstrate various features of the new method

classification of finite dimensional solvable lie algebras with nondegenerate invariant bilinear forms abstract in this paper finite dimensional solvable lie algebras with nondegenerate invariant bilinear forms are classified the classification is based on the investigation of the minimal generating sets which are proved to be one of the three types paired type isolated type and mixed type

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

generalized kac moody algebras and the modular function j the generalized kac moody algebras were introdcued by borcherds in his study of the vertex algebras and monstrous moonshine b1 b3 the structure and the representation theory of generalized kac moody algebras are very similar to those of kac moody algebras and a lot of facts about kac moody algebras can be generalized to generalized kac moody algebras the main difference is that the generalized kac moody algebras may have imaginm 3 simple roots with norms 1 for unitarizable irreducible highest weight modules over symmetrizable generalized kac moody algebras borcherds proved a character formula b2 called the weyl kac borcherds formula which generalizes the weylkac formula for integrable highest weight modules over symmetrizable kac moody algebras k as in the case of kac moody algebras the weyl kac borcherds formula when applied to the l dimensional trivial representation yields the denominator identio in this paper using the weyl kac borcherds formula and the denominator identity we obtain a closed form root multiplicity formula for all symmetrizable generalized kac moody algebras theorem 2 1 our formula enables us to study the structure of a symmetrizable generalized kac moody algebra 9 as a representation of a kac moody algebra 0o contained in the choice of 0 gives rise to various expressions of the root multiplicities of 13 which would yield some interesting combinatorial identities in sect 3 we apply this principle to a rank 2 generalized kac moody algebra to obtain a combinatorial identity for the pascal numbers in b3 borcherds proved the moonshine conjecture cni by constructing the monster lie algebra using the monster vertex algebra flm and the vertex algebra associated with the 2 dimensional even lorentzian lattice iil b 1 the monster lie