a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

parameter uniform finite difference schemes for singularly perturbed parabolic diffusion convection reaction problems in this paper parameter uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied parameter explicit theoretical bounds on the derivatives of the solutions are derived the solution is decomposed into a sum of regular and singular components a numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed parameter uniform error bounds for the numerical approximations are established numerical results are given to illustrate the parameter uniform convergence of the numerical approximations

uniformly convergent difference schemes for singularly perturbed parabolic diffusion convection problems without turning points we consider first the initial boundary value problem for the parabolic equation varepsilon y xx a x t y x b x t y d x t y t f x t on 0 1 0 t where 0 0 b 0 andd 0 bounds ony and its derivatives are obtained under certain compatibility assumptions on the data we examine a family of difference schemes for this problem which is exponentially fitted in thex variable and uses classical differencing in thet variable on rectangular grids which are arbitrarily spaced in both thex andt directions under a courant friedrichs lewy type condition the errors at the grid points are shown to be bounded byc h k whereh k is the maximum mesh width in thex t direction andc is a constant independent of h andk the corresponding result for a two point boundary value problem is also derived

analysis of a finite difference scheme for a singularly perturbed problem with two small parameters abstract we study a model linear convection diffusion reaction problem where both the diffusion term and the convection term are multiplied by small parameters e d and e c respectively depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain sharp bounds for the derivatives of the solution are derived using a barrier function technique these bounds are applied in the analysis of a simple upwind difference scheme on shishkin meshes this method is established to be almost first order convergent independently of the parameters e d and e c

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

analytic continuation of the resolvent of the laplacian on symmetric spaces of noncompact type let m g be a globally symmetric space of noncompact type of arbitrary rank and its laplacian we introduce a new method to analyze and the resolvent 1 this has origins in quantum n body scattering but is independent of the classical theory of spherical functions and is analytically much more robust we expect that suitably modified it will generalize to locally symmetric spaces of arbitrary rank as an illustration of this method we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a riemann surface multiply covering the plane we also show how this continuation may be deduced using the theory of spherical functions in summary this paper establishes a long suspected connection between the analysis on symmetric spaces and n body scattering

on the nonvanishing of the central value of the rankin selberg l functions first published in journal of the american mathematical society in volume 17 issue 3 published by the american mathematical society

numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem abstract a singularly perturbed problem involving two singular perturbation parameters is discretized using the classical upwinded finite difference scheme on an appropriate piecewise uniform shishkin mesh scaled discrete derivatives with scaling only used within the layers are shown to be parameter uniformly convergent to the scaled first derivatives of the continuous solution

local shimura correspondence it splits over the maximal compact subgroup let t be the corresponding lift of i the iwahori subgroup of g now fix a faithful character of n f let r be the category of smooth representations of which are generated by its t fixed vectors and such that n f acts via also let h be the hecke algebra of i biinvariant compactly supported functions on satisfying f gq f g for z v obj r let v be the space of i fixed vectors then f f l v is a representation of h t according to bernstein and borel 2 this gives the equivalence of r and the category of representations of h in particular this is the case for g whose hecke algebra will be denoted by h g and category by ri g in this paper we describe h in terms of generators and relations as a consequence in the case of simple group for example we find a group g related to g such that h t h g in particular we get the equivalence of r t and

an efficient numerical scheme for 1d parabolic singularly perturbed problems with an interior and boundary layers in this paper we consider a 1d parabolic singularly perturbed reaction convection diffusion problem which has a small parameter in both the diffusion term multiplied by the parameter e 2 and the convection term multiplied by the parameter in the differential equation e 0 1 0 1 e moreover the convective term degenerates inside the spatial domain and also the source term has a discontinuity of first kind on the degeneration line in general for sufficiently small values of the diffusion and the convection parameters the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain we study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme which combines the implicit euler method defined on a uniform mesh to discretize in time together with the classical upwind finite difference scheme defined on an appropriate nonuniform mesh of shishkin type to discretize in space the numerical scheme converges in the maximum norm uniformly in e and having first order in time and almost first order in space illustrative numerical results corroborating in practice the theoretical results are showed

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

analytic continuation of resolvent kernels on noncompact symmetric spaces let x g k be a symmetric space of noncompact type and let be the laplacian associated with a g invariant metric on x we show that the resolvent kernel of admits a holomorphic extension to a riemann surface depending on the rank of the symmetric space this riemann surface is a branched cover of the complex plane with a certain part of the real axis removed it has a branching point at the bottom of the spectrum of it is further shown that this branching point is quadratic if the rank of x is odd and is logarithmic otherwise in case g has only one conjugacy class of cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of with the only branching point being the bottom of the spectrum

doubling constructions and tensor product l functions the linear case we present an integral representation for the tensor product l function of a pair of automorphic cuspidal representations one of a classical group the other of a general linear group our construction is uniform over all classical groups and is applicable to all cuspidal representations it does not require genericity the main new ideas of the construction are the use of generalized speh representations as inducing data for the eisenstein series and the introduction of a new global and local model which generalizes the whittaker model here we consider linear groups but our construction also extends to arbitrary degree metaplectic coverings this will be the topic of an upcoming work

on open scattering channels for manifolds with ends abstract in the framework of time dependent geometric scattering theory we study the existence and completeness of the wave operators for perturbations of the riemannian metric for the laplacian on a complete manifold of dimension n the smallness condition for the perturbation is expressed intrinsically and coordinate free in purely geometric terms using the harmonic radius therefore the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the ricci curvature as an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the riemannian metric this stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations

theta distinguished representations inflation and the symmetric square l function a theta distinguished representation is a quotient of a tensor of exceptional representations where exceptional is in the sense of kazhdan and patterson we study relations between theta distinguished representations of mathrm gl n and mathrm gspin 2n 1 in the case of mathrm gspin 2n 1 or mathrm so 2n 1 exceptional or small representations were constructed by bump friedberg and ginzburg we prove a rodier type hereditary property a tempered representation tau is distinguished if and only if the representation mathrm i tau induced to mathrm gspin 2n 1 is distinguished and the multiplicity of both quotients is at most one if tau is supercuspidal and distinguished then so is the langlands quotient of mathrm i tau as a corollary we characterize supercuspidal distinguished representations in terms of the pole of the symmetric square l function at s 0

scattering theory for the laplacian on manifolds with bounded curvature abstract in this paper we study the behaviour of the continuous spectrum of the laplacian on a complete riemannian manifold of bounded curvature under perturbations of the metric the perturbations that we consider are such that its covariant derivatives up to some order decay with some rate in the geodesic distance from a fixed point especially we impose no conditions on the injectivity radius one of the main results are conditions on the rate of decay depending on geometric properties of the underlying manifold that guarantee the existence and completeness of the wave operators