a new proof of the howe conjecture let j q denote the space of g invariant distributions supported on ad g9 q if g e g set ad g f z f f g zg if q is a locally constant compactly supported function on g resp g define tq q to be the linear functional on j q defined by tq o t t 0 the set q will be fixed throughout the paper so we will abbreviate tq to t the g invariance of the distributions in j q means t ad g f t f if l is a compact open subgroup of g denote as c g l the space of compactly supported functions on g which are right l invariant similary if l is an open compact lattice in g let c g l be the space of compactly supported functions on g which are l invariant in hol howe conjectured that if q is a compact subset of g resp 9 then the restriction of the distributions in j q to c b l resp c g l is finite dimensional a more common formulation of the group version of the howe conjecture replaces the space c g l with the hecke algebra 7 g l of compactly supported l biinvariant functions on g since 7 g l c c g l apriori our formulation with c g l implies the hecke algebra formulation it is a simple matter to show that in fact the two formulations are equivalent if f e c 9 l then the function f g lf kgk 1 dk belongs to the hecke algebra and the map f 1 f is a surjection the linear functional on j q induced by f is a positive multiple of the linear functional induced by f thus the two spaces of linear functionals are equal and so the two formulations are equivalent

restriction to compact subgroups in the cyclic homology of reductive p adic groups restriction of functions from a reductive p adic group g to its compact subgroups defines an operator on the hochschild and cyclic homology of the hecke algebra of g we study the commutation relations between this operator and others coming from representation theory jacquet functors idempotents in the bernstein centre and characters of admissible representations

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

a numerical method for acoustic normal modes for shear flows abstract the normal modes and their propagation numbers for acoustic propagation in wave guides with flow are the eigenvectors and eigenvalues of a boundary value problem for a non standard sturm liouville problem it is non standard because it depends non linearly on the eigenvalue parameter in the classical problem for ducts with no flow the problem depends linearly on the eigenvalue parameter in this paper a method is presented for the fast numerical solution of this problem it is a generalization of a method that was developed for the classical problem a finite difference method is employed that combines well known numerical techniques and a generalization of the sturm sequence method to solve the resulting algebraic eigenvalue problem then a modified richardson extrapolation method is used that dramatically increases the accuracy of the computed eigenvalues the method is then applied to two problems they correspond to acoustic propagation in the ocean in the presence of a current and to acoustic propagation in shear layers over flat plates

convergence of the multilevel full approximation scheme including thev cycle the multilevel full approximation scheme fas ml is a well known solver for nonlinear boundary value problems in this paper we prove local quantitative convergence statements for a class of fas ml algorithms in a general hilbertspace setting this setting clearly exhibits the structure of fas ml we prove local convergence of a nested iteration for a rather concrete class of fas ml algorithms in whichv cycles and only one jacobilike pre and post smoothing on each level are used

convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems the multigrid full approximation scheme fas mg is a well known solver for nonlinear boundary value problems in this paper we restrict ourselves to a class of second order elliptic mildly nonlinear problems and we give local conditions e g a local lipschitz condition on the derivative of the continuous operator under which the fas mg with suitably chosen parameters locally converges we prove quantitative convergence statements and deduce explicit bounds for important quantities such as the radius of a ball of guaranteed convergence the number of smoothings needed the number of coarse grid corrections needed and the number of fas mg iterations needed in a nested iteration these bounds show well known features of the fas mg scheme

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

the base change fundamental lemma for central elements in parahoric hecke algebras let g be an unramified group over a p adic field f and let e f be a finite unramified extension field let denote a generator of gal e f this paper concerns the matching at all semi simple elements of orbital integrals on g f with twisted orbital integrals on g e more precisely suppose belongs to the center of a parahoric hecke algebra for g e this paper introduces a base change homomorphism 7 b taking values in the center of the corresponding parahoric hecke algebra for g f it proves that the functions and b are associated in the sense that the stable orbital integrals for semi simple elements of b can be expressed in terms of the stable twisted orbital integrals of in the special case of spherical hecke algebras which are commutative this result becomes precisely the base change fundamental lemma proved previously by clozel cl90 and labesse lab90 as has been explained in h05 the fundamental lemma proved in this paper is a key ingredient for the study of shimura varieties with parahoric level structure at the prime p

a numerical method for bottom interacting ocean acoustic normal modes in this paper we present a finite difference method to numerically determine the normal modes for the sound propagation in a stratified ocean resting on a stratified elastic bottom the compound matrix method is used for computing an impedance condition at the ocean elastic bottom interface the impedance condition is then incorporated as a boundary condition into the finite difference equations in the ocean yielding an algebraic eigenvalue problem for each fixed mesh size this eigenvalue problem is solved by a combination of efficient numerical methods the richardson mesh extrapolation procedure is then used to substantially increase the accuracy of the computation two applications are given to demonstrate the speed accuracy and efficiency of the method

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

uniform convergence of a nonlinear energy based multilevel quantization scheme a popular vector quantization scheme can be constructed by centroidal voronoi tessellations cvts which also have many other applications in diverse areas of science and engineering the development of efficient algorithms for their construction is a key to the successful applications of cvts in practice this paper studies the details of a new optimization based multilevel algorithm for the numerical computation of cvts the rigorous proof of its uniform convergence in one space dimension and the results of computational simulations are provided they substantiate recent claims on the significant speedup demonstrated by the new scheme in comparison with traditional methods

a numerical method for ocean acoustic normal modes the method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans the propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field such errors can severely degrade the accuracy of the normal mode representation particularly at long ranges in this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes it consists of a combination of well known numerical procedures such as sturm sequences the bisection method newton s and brent s methods richardson extrapolation and inverse iteration we also introduce a modified richardson extrapolation procedure that substantially increases the speed and accuracy of the computation

a note on the relationship between finite difference and shooting methods for ode eigenvalue problems finite difference methods and shooting methods are two standard classes of techniques for solving ordinary differential equation eigenvalue problems we first show that a sturm sequence method often used for solving the finite difference eigenvalue problem may be interpreted in the shooting framework then we find that the sturm sequence procedure is easily extended to more complicated problems this enables the shooting method to guarantee convergence to specified subsets of the modes of the problem without the usual requirement of an initial guess for the eigenvalues

discontinuous galerkin methods for solving elliptic variational inequalities we study discontinuous galerkin methods for solving elliptic variational inequalities of both the first and second kinds analysis of numerous discontinuous galerkin schemes for elliptic boundary value problems is extended to the variational inequalities we establish a priori error estimates for the discontinuous galerkin methods which reach optimal order for linear elements results from some numerical examples are reported

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

orbital integrals on p adic groups a proof of the howe conjecture let g be a connected reductive group over a p adic field f of characteristic 0 let g g f let o be a compact subset of g og the invariant subset gxg1 g e g x e co of g we say that a closed invariant subset of g q is compact modulo conjugation if q c wg for some compact co it is equivalent to assume that the traces of q on all cartan subgroups of g are compact the purpose of this article is to prove the following theorem concerning invariant distributions on g which had been conjectured by howe 10 let of q denote the set of invariant distributions supported in u if k is a compact open subgroup of g let xk be the hecke algebra of compactly supported functions on g with complex values invariant on the left and right by k

unipotent representations and unipotent classes in sl n to achieve the stabilization of the trace formula for a reductive group 18 there are two related problems that must first be solved one is the matching of the sorbital integrals of smooth functions on a reductive group g over a p adic field of characteristic zero with stable orbital integrals on its endoscopic groups h igusa theory clarifies the nature of this first problem 11 12 13 19 22 23 the second problem is the fundamental lemma which asserts the compatibility of the matching of smooth functions with the hecke algebras on g and h buildings have been used to study this problem 25 the statements of the problems are closely related and the main results of this paper give a method of reducing one problem to the other if the first problem matching smooth functions is solved for sl n then the second problem the fundamental lemma for the hecke algebras follows as a consequence at least for the identity of the algebra the solutions of the two problems are linked by making a detailed study of the unipotent representations and the unipotent classes of sl n representations with iwahori fixed vectors on sl n f are called unipotent in this paper to emphasize the close relationship between unipotent conjugacy classes and these representations not long after this paper was written waldspurger refined reformulated and greatly extended the results of this paper this paper made a modest contribution to this subsequent work of his he eliminated the hypothesis that functions on sl n have matching smooth functions by making clever use of a strengthened version of kazhdan s lemma the opinion of the experts is still divided will the solution of the fundamental lemma for other reductive groups come from the matching of smooth functions or will it come from even stronger versions of kazhdan s lemma this much is clear the fundamental lemma no longer needs to be viewed as an isolated combinatorial problem in buildings howe s conjecture uniform germ expansions homogeneity of germs compact traces unipotent orbits and kazhdan s density theorem give us a framework through which the fundamental lemma may be understood in the final portion of the introduction i would like to state a theorem that has considerable interest beyond its usefulness for the fundamental lemma this theorem will be used in section 1 to produce a uniform germ expansion the

on the k0 of a p adic group this article deals with various topics related with grothendieck groups invariant distributions parabolic and compact inductions for a p adic group g the main result is a description of the k0 of the hecke algebra of g in terms of discrete series of levi subgroups which has an interesting behavior with regard to parabolic restriction and induction a similar description but no more compatible with these parabolic functors is obtained for overline mathcal h and the hattori rank map gets an easy description in this dictionary we follow a beautiful idea of j bernstein consisting in comparing two natural filtrations on these objects one of combinatorial nature and one of topological nature the combinatorial filtrations are related to the structure of levi subgroupsin g and have counterparts concerning many classical objects of interest as the grothendieck group of finite length g modules r g the set sr of regular semi simple conjugacy classes and the variety g of infinitesimal characters these filtrations will turn out to be compatible in a sense to be specified with regard to all the classical operations or morphisms between these objects

an algorithm for solving the obstacle problems in this paper we propose an algorithm for solving the obstacle problem we try tofind the approximated region of the contact in the obstacle problem by iteration numerical examples are given for the obstacle problem for a membrane and the elastic plastic torsion problem

residual scaling techniques in multigrid ii practical applications this paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper residual scaling techniques in multigrid i equivalence proof appl math comput 86 283 303 1997 we discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent a heuristic residual analysis technique based on the geometry of the grid points and the relaxation pattern is introduced to estimate the optimal residual scaling factor for a high order multigrid method we compare the performance of a typical pre optimization pre acceleration technique with a typical post optimization post acceleration technique and show that the pre optimization is preferable in both convergence and efficiency our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre optimization technique over the post optimization technique

geometric approach to the local jacquet langlands correspondence in this paper we give a purely geometric approach to the local jacquet langlands correspondence for rm gl n over a p adic field under the assumption that the invariant of the division algebra is 1 n we use the ell adic etale cohomology of the drinfeld tower to construct the correspondence at the level of the grothendieck groups with rational coefficients moreover assuming that n is prime we prove that this correspondence preserves irreducible representations this gives a purely local proof of the local jacquet langlands correspondence in this case we need neither a global automorphic technique nor detailed classification of supercuspidal representations of rm gl n

analysis of a multilevel iterative method for nonlinear finite element equations the multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations we describe how this technique can be combined with a globally convergent approximate newton method to solve nonlinear partial differential equations we show that asymptotically only one newton iteration per level is required thus the complexity for linear and nonlinear problems is essentially equal

cyclic homology and the selberg principle abstract an abstract selberg principle is proved for a smooth compactly supported function f on a connected reductive p adic group which is the trace of some idempotent n n matrix this selberg principle states that the orbital integral of f relative to a non compact element vanishes this generalizes the work of julg and valette who treated the rank one case versions of the abstract selberg principle are proved also for finite groups and for discrete groups computations of the hochschild and cyclic homology of the convolution algebra of compactly supported functions are given

the basic stratum of some simple shimura varieties under simplifying hypotheses we prove a relation between the ell adic cohomology of the basic stratum of a shimura variety of pel type modulo a prime of good reduction of the reflex field and the cohomology of the complex shimura variety in particular we obtain explicit formulas for the number of points in the basic stratum over finite fields we obtain our results using the trace formula and truncation of the formula of kottwitz for the number of points on a shimura variety over a finite fields

an explicit plancherel formula for u 2 1 the admissible duals of quasi split unitary groups over nonarchimedean fields are determined the set of irreducible unitarizable representations and the plancherel measure on the unitary dual is given explicitly