analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

reduced standard modules and cohomology first cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts including wiles s famous paper 1995 internal to group theory 1 cohomology plays a role in the general theory of maximal subgroups of finite groups as developed by aschbacher and scott 1985 one can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible in this case r guralnick 1986 conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups this paper provides the first general positive results on this conjecture proving that the generic 1 cohomology h 1 gen g l lim h 1 g q l see cline parshall scott and van der kallen 1977 of a finite group g q of lie type with absolutely irreducible coefficients l in the defining characteristic of g is bounded by a constant depending only on the root system in all cases we are able to improve this result to a bound on h 1 g q l itself still depending only on the root system the generic h 1 result and related results for ext 1 emerge here as a consequence of a general study of interest in its own right of the homological properties of certain rational modules red red indexed by dominant weights for a reductive group g the modules red and red arise naturally from irreducible representations of the quantum enveloping algebra u of the same type as g at a pth root of unity where p 0 is the characteristic of the defining field for g finally we apply our ext 1 bounds results of bendel nakano and pillen 2006 as well as results of sin 1993 1992 1994 on the ree and suzuki groups to obtain the non generic bounds on h 1 g q l

analysis of a finite pml approximation to the three dimensional elastic wave scattering problem we consider the application of a perfectly matched layer pml technique to approximate solutions to the elastic wave scattering problem in the frequency domain the pml is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain the complement of the scatterer the rapid decay of the pml solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation for the truncated problem we prove existence and uniqueness of the solutions to the infinite domain and truncated domain pml equations provided that the truncated domain is sufficiently large we also show exponential convergence of the solution of the truncated pml problem to the solution of the original scattering problem in the region of interest we then analyze a galerkin numerical approximation to the truncated pml problem and prove that it is well posed provided that the pml damping parameter and mesh size are small enough finally computational results illustrating the efficiency of the finite element pml approximation are presented

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

stability of pole solutions for planar propagating flames i exact eigenvalues and eigenfunctions it is well known that the nonlinear pde describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states such a solution corresponds to a steadily propagating cusp like structure commonly observed in experiments in this work we investigate the linear stability of these equilibrium states the steady coalescent pole solutions in previous similar studies either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized pde was numerically integrated in order to examine the evolution of initially small disturbances in time in contrast our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions in this paper we derive the expressions for the eigenvalues and eigenfunctions their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later

singularities of electromagnetic fields in polyhedral domains in this paper we investigate the singular solutions of time harmonic maxwell equations in a domain which has edges and polyhedral corners it is now well known that in the presence of non convex edges the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the laplace operator 4 5 we show how this type of result can be derived from the classical mellin analysis and how this analysis leads to sharper results concerning the singular parts which belong to h 1 for the singular functions we exhibit simple and explicit formulas based on generalized dirichlet and neumann singularities for the laplace operator these formulas are more explicit than the results announced in our note 10

pml method for electromagnetic scattering problem in a two layer medium the perfectly matched layer pml method is well studied for acoustic scattering problems electromagnetic scattering problems and more recently elastic scattering problems with homogeneous background media the purpose of this paper is to present the stability and exponential convergence of the pml method for a three dimensional electromagnetic scattering problem in a two layer medium the main contributions of this paper are threefold first we establish the well posedness of the original scattering problem for any dirichlet boundary value in boldsymbol h 1 2 rm div gamma d where gamma d stands for the boundary of the scatterer second we propose a new weak formulation for the original problem where the dirichlet to neumann operator is proposed on a truncation boundary inside the pml this argument is favorable in the analysis of the pml dirichlet to neumann operator the inf sup condition is proved for the bilinear form third we establish the well posedness of the pml problem a

an inversion formula for the kazhdan lusztig polynomials for affine weyl groups let w be a coxeter group in 10 kazhdan and lusztig defined for each y w e w certain polynomials py w over 7 they conjectured that in the case where w is the weyl group corresponding to a complex semi simple lie algebra 5 these polynomials determine the characters of the irreducible is modules this conjecture has now been proved see 3 4 later g lusztig 11 formulated a corresponding conjecture for the irreducible modules for a semi simple algebraic group g defined over an algebraically closed field of prime characteristic p involving the same polynomials except that w is now the affine weyl group corresponding to g elementary properties of the polynomials py w show that there exist polynomials q x z w such that the inversion formula

some new examples in 1 cohomology this paper gives some new examples in the 1 cohomology theory of finite groups of lie type obtained from both computer calculations and the use of several theoretical results in particular the paper gives the first known examples of 1 cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group the computer calculations were made originally while checking special cases of lusztig s conjecture on characteristic p representations of algebraic groups and we take this opportunity to announce in print some results in that direction they reinforce lusztig s conjecture even in a strong form suggested by kato 2003 published by elsevier science usa

the cellular nature of hydrodynamic flame instability abstract the nonlinear evolution of a weakly perturbed planar flame under the influence of the landau darrieus instability is studied numerically it is shown that in sufficiently small scale systems spontaneous instability appears in the guise of a cusp shape flame however in sufficiently large scale flames this configuration acquires a fine cellular structure with the cells constantly subdividing and merging in a chaotic manner

absorbing boundary condition for the elastic wave equation velocity stress formulation this paper describes an absorbing boundary condition for finite difference modeling of elastic wave propagation in two and three dimensions the boundary condition is particularly effective for obliquely incident waves typically quite troublesome for absorbing boundaries analytical predictions of the boundary reflection coefficients of a few percent or less for angles of incidence up to 89 are verified in example finite difference applications the algorithm is appropriate for use in a velocity stress finite difference vs fd formulation it is computationally simpler than a similar absorbing boundary given previously for the standard displacement formulation a second algorithm is presented which may be advantageous when the boundary of interest is exposed to strong evanescent waves both algorithms require that the adjacent elastic medium be locally homogeneous

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

new anisotropic perfectly matched layer boundary condition for elastic waves in solids the perfectly matched layer pml is currently a popular absorbing boundary condition for numerical modeling of electromagnetic acoustic and elastic wave problems existing pml formulation for elastic waves uses a field splitting scheme which increases the number of variables in the computation the new anisotropic elastic wave pml modifies the elasticity or compliance tensors to be anisotropic it utilizes fewer variables for both isotropic and anisotropic media so that computing memory and time can be reduced significantly its formulations for cartesian and curvilinear coordinate systems will be discussed moreover this anisotropic pml can be directly used in the finite element method and easily be implemented in the finite difference time domain method as well work supported by nasa nag3 2147 and by nsf cts 9870015

overlapping schwarz methods for maxwell s equations in three dimensions two level overlapping schwarz methods are considered for finite element problems of 3d maxwell s equations nedelec elements built on tetrahedra and hexahedra are considered once the relative overlap is fixed the condition number of the additive schwarz method is bounded independently of the diameter of the triangulation and the number of subregions a similar result is obtained for a multiplicative method these bounds are obtained for quasi uniform triangulations in addition for the dirichlet problem the convexity of the domain has to be assumed our work generalizes well known results for conforming finite elements for second order elliptic scalar equations

stratifying endomorphism algebras associated to hecke algebras abstract let g be a finite group of lie type and let k be a field of characteristic distinct from the defining characteristic of g in studying the non describing representation theory of g the endomorphism algebra s g k end kg j ind g p j k plays an increasingly important role in type a by work of dipper and james s g k identifies with a q schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups this paper presents the first systematic study of the structure and homological algebra of these algebras for g of arbitrary type because s g k has a reinterpretation as a hecke endomorphism algebra it may be analyzed using the theory of hecke algebras its structure turns out to involve new applications of kazhdan lusztig cell theory in the course of this work we prove two stratification conjectures about coxeter group representations made by e cline b parshall and l scott mem amer math soc 591 1996 and we formulate a new conjecture about the structure of s g k we verify this conjecture here in all rank 2 examples

a note on the existence and uniqueness of solutions of frequency domain elastic wave problems a priori estimates in h1 abstract in this note we provide existence and uniqueness results for frequency domain elastic wave problems these problems are posed on the complement of a bounded domain r 3 the scatterer the boundary condition at infinity is given by the kupradze sommerfeld radiation condition and involves different sommerfeld conditions on different components of the field our results are obtained by setting up the problem as a variational problem in the sobolev space h 1 on a bounded domain we use a nonlocal boundary condition which is related to the dirichlet to neumann conditions used for acoustic and electromagnetic scattering problems we obtain stability results for the source problem a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences

a model equation in combustion theory exhibiting an infinite number of secondary bifurcations abstract we investigate periodic solutions of a model equation arising in combustion theory it is known that there is an infinite number of branches of solutions bifurcating from the trivial solution in addition it is shown that each of these primary branches exhibits an infinite number of secondary bifurcations both the primary and secondary branches can be given explicitly in terms of rational trigonometric functions

quantum weyl reciprocity for cohomology we study the relationship between the cohomology of q schur algebras and hecke algebras in type a a key tool which has independent interest involves a specific hecke algebra resolution introduced in a special case by deodhar this resolution is somewhat akin to the koszul resolution for lie algebra cohomology in that it has an explicit description yet applies for all modules an application of the properties of this resolution establishes a quantum weyl reciprocity connecting in a range of degrees the cohomology of the q schur algebra with that of the hecke algebra this work also provides independently of the resolution cohomological inequalities which apply in all degrees and which sometimes can be related to kazhdan lusztig polynomials finally we extend to cohomology a decomposition number method of erdmann and illustrate its usefulness in computations this method enables us to exhibit some properties of symmetric group cohomology which if true would imply the sl n version of lusztig s characteristic p character conjecture

extensions levi subgroups and character formulas abstract this paper consists of three interconnected parts parts i iii study the relationship between the cohomology of a reductive group g and that of a levi subgroup h for example we provide a sufficient condition arising from kazhdan lusztig theory for a natural map ext g l l ext h l h l h to be surjective given irreducible g modules l l and corresponding irreducible h modules l h l h in cohomological degree n 1 the map is always an isomorphism under our hypothesis these results were inspired by recent work of hemmer d hemmer a row removal theorem for the ext 1 quiver of symmetric groups and schur algebras proc amer math soc 133 2 2005 403 414 electronic for g gl n and both extend and improve upon the latter when our condition is met part ii obtains results on lusztig character formulas lcfs for reductive groups obtaining new necessary and sufficient conditions for such formulas to hold in the special case of g gl n these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group this work on gl n improves upon b parshall l scott quantum weyl reciprocity for cohomology proc london math soc 90 3 2005 655 688 which established only sufficient rather than necessary and sufficient conditions for the validity of the lcf in terms of the symmetric group

analysis of a cartesian pml approximation to acoustic scattering problems in r2 and r3 we consider the application of a perfectly matched layer pml technique applied in cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain the pml is viewed as a complex coordinate shift stretching and leads to a variable complex coefficient equation for the acoustic wave posed on an infinite domain the complement of the bounded scatterer the use of cartesian geometry leads to a pml operator with simple coefficients although still complex symmetric non hermitian the pml reformulation results in a problem whose solution coincides with the original solution inside the pml layer while decaying exponentially outside the rapid decay of the pml solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation for the truncated problem this paper provides new stability estimates for the cartesian pml approximations both on the infinite and the truncated domain we first investigate the stability of the infinite pml approximation as a function of the pml strength s 0 this is done for pml methods which involve continuous piecewise smooth stretching as well as piecewise constant stretching functions we next introduce a truncation parameter m which determines the size of the pml layer our analysis shows that the truncated pml problem is stable provided that the product of email protected 0 is sufficiently large in which case the solution of the problem on the truncated domain converges exponentially to that of the original problem in the domain of interest near the scatterer this justifies the simple computational strategy of selecting a fixed pml layer and increasing s 0 to obtain the desired accuracy the results of numerical experiments varying m and s 0 are given which illustrate the theoretically predicted behavior

a new approximation technique for div curl systems in this paper we describe an approximation technique for div curl systems based in l where is a domain in 3 we formulate this problem as a general variational problem with different test and trial spaces the analysis requires the verification of an appropriate inf sup condition this results is a very weak formulation where the solution space is l and the data reside in various negative norm spaces subsequently we consider finite element approximations based on this weak formulation the main approach of this paper involves the development of stable pairs of discrete test and trial spaces with this approach we enlarge the test space so that the discrete inf sup condition holds and use a negative norm least squares formulation to reduce to a uniquely solvable linear system this leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low sobolev regularity e g h with 0 s 1 2 the resulting algebraic equations are symmetric positive definite and well conditioned a second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned some numerical results are also presented

analysis of a cartesian pml approximation to acoustic scattering problems in r2 abstract in this paper we consider a cartesian pml approximation to solutions of acoustic scattering problems on an unbounded domain in r 2 the perfectly matched layer pml technique in a curvilinear coordinate system has been researched for acoustic scattering applications both in theory and computation our goal will be to extend the results of spherical cylindrical pml to pml in cartesian coordinates that is the well posedness of cartesian pml approximation on both the unbounded and truncated domains the exponential convergence of approximate solutions as a function of domain size is also shown we note that once the stability and convergence of the continuous truncated problem has been achieved the analysis of the resulting finite element approximations is then classical finally the results of numerical computations illustrating the theory and efficiency of the cartesian pml approach will be given