difference schemes for the singularly perturbed sobolev periodic boundary problem a finite difference method is suggested and analyzed for evolution equations of sobolev type in a single space variable and with boundary layers the convergence and error estimates for an exponentially fitted difference scheme in an equidistant mesh are obtained the numerical methods discussed here are extremely robust in the sense that they have good convergence properties not only for small values of the perturbation parameter but also for moderate and large values

exponentially fitted methods on layer adapted meshes in this paper a new derivation of a uniformly convergent second order method for singularly perturbed linear ordinary differential equations based on the freezing of the coefficients of the differential equation and integration of the resulting equations subject to continuity and smoothness conditions at the nodes is presented the derivation presented here is compared with others based on green s functions when only advection and diffusion processes are considered when solving the homogeneous equations in addition a new method that accounts for advection diffusion and reaction processes when solving the homogeneous equation is also presented the two exponentially fitted techniques presented in the paper are used on layer adapted meshes which are piecewise uniform and concentrate grid points in the boundary layers and their results are compared with those obtained with upwind methods in piecewise uniform meshes it is shown that standard techniques on piecewise uniform meshes are less accurate than exponentially fitted ones and the accuracy of the latter may not improve by employing layer adapted piecewise uniform meshes due to the large change in the step size at the transition points the paper also presents an exponentially fitted method for singularly perturbed periodic two point boundary value problems of ordinary differential equations

the generalised zakharov shabat system and the gauge group action the generalized zakharov shabat systems with complex valued non regular cartan elements and the systems studied by caudrey beals and coifman cbc systems and their gauge equivalent are studied this study includes the properties of fundamental analytical solutions for the gauge equivalent to cbc systems and the minimal set of scattering data the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator related to such systems the hierarchies of hamiltonian structures the results are illustrated in the example of the multi component nonlinear schrodinger equations and the corresponding gauge equivalent multi component heisenberg ferromagnetic type models related to so 5 c algebra

solutions of certain integrable nonlinear pde s describing nonresonant n wave interactions the evolution equations that describe in appropriately coarse grained and slow variables the evolution of the envelopes of n nonresonant dispersive waves can be reduced to the universal form t vn x un x t un x t nm 1 nmum x t in this paper the special case with nm vm vn m which is integrable by quadratures is investigated the subclass of localized solutions i e vanishing as x gives rise to a novel solitonic phenomenology the class of solutions that are asymptotically finite contains a richer solitonic phenomenology including those of novel type which move with the speeds vn and can have any shape and more standard kinks which can move with any speed and have standard shapes the class of rational solutions and the integrable dynamical systems naturally associated with these solutions are also investigated these dynamical systems include and extend known integrable systems in this paper the treatment is confined to 1 1 dimensions at the end togethe

completeness of the eigenfunctions for the caudrey beals coifman system the direct and the inverse scattering problem for the first order linear systems of the type l x i d dx q x j x 0 j h q x gj which generalizes the zakharov shabat system and the systems studied by caudrey beals and coifman cbc is analyzed herein here j is a regular complex constant element of the cartan subalgebra h g of the simple lie algebra g and the potential q x vanishes fast enough for x taking values in the image gj of adj the cbc results are generalized and the fundamental analytic solution m x for any choice of the irreducible finite dimensional representation v of g is constructed four pairwise equivalent minimal sets of scattering data for l invariant with respect to the choice of the representation of g are extracted from the asymptotics of m x for x from m x the resolvent of l is constructed in the adjoint representation vad and the completeness relation is proven for the eigenfunctions of l in vad it is also proven that the discrete spectrum

on a rational differential quadrature method in irregular domains for problems with boundary layers abstract a rational differential quadrature method in irregular domains rdqmid is investigated to deal with a kind of singularly perturbed problems with boundary layers through a transformation the boundary layer which may be not straight is transformed into a segment of a line parallel to one of the cartesian axes the rational differential quadrature method rdqm is applied to discretize the governing equation finally a direct expansion method of the boundary conditions dembc is raised to deal with the boundary conditions numerical experiments show that rdqmid is of high accuracy efficiency and easy to programme

universality and integrability of the nonlinear evolution pde s describing n wave interactions the universality of the equations describing n wave interactions is demonstrated by deriving them from a very large class of nonlinear evolution equations essentially all those whose linear part is dispersive various forms of these equations are displayed the fact that these universal nonlinear evolution equations obtain by an appropriate asymptotic limit from such a large class of nonlinear evolution equations suggests that they should be integrable since for this it is sufficient that the large class from which they are obtainable contain just one integrable equation this expectation is validated in several cases by deriving the equations from known integrable equations in this manner an explanation may be provided of the already known integrable nature of certain equations and new integrable equations may be obtained both s integrable and c integrable equations are discussed namely both equations integrable via an appropriate spectral transform and solvable via an appropriate change of variables in this paper the treatment is limited to equations in 1 1 dimensions

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

on the complete integrability of completely integrable systems the question of complete integrability of evolution equations associated ton n first order isospectral operators is investigated using the inverse scattering method it is shown that forn 2 e g for the three wave interaction additional nonlinear pointwise flows are necessary for the assertion of complete integrability their existence is demonstrated by constructing action angle variables this construction depends on the analysis of a natural 2 form and symplectic foliation for the groupsgl n andsu n

on classification of soliton solutions of multicomponent nonlinear evolution equations we consider several ways of how one could classify the various types of soliton solutions related to multicomponent nonlinear evolution equations which are solvable by the inverse scattering method for the generalized zakharov shabat system related to a simple lie algebra g in doing so we make use of the fundamental analytic solutions the zakharov shabat dressing procedure the reduction technique and other tools characteristic for that method the multicomponent solitons are characterized by several important factors the subalgebras of g and the way these subalgebras are embedded in g the dimension of the corresponding eigensubspaces of the lax operator l as well as by additional constraints imposed by reductions

the generating operator for the n n linear system abstract the class of exactly integrable non linear evolution equations nlee related to the general first order n n linear problem is studied the set of independent scattering data j is determined and trace identities are obtained next we use a special integro differential operator and obtain the expansions of the potential and its variation in the eigenfunctions of these enable us to consider the inverse scattering method as a generalized fourier transform and to write down in a compact form the integrals of motion we prove that there exist a hierarchy of hamiltonian structures generated by at the end we calculate the symplectic form in terms of the scattering data variations

on the classification of automorphic lie algebras the problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants this provides a powerful tool of analysis and it opens the road to new applications of automorphic lie algebras beyond the con text of integrable systems in this paper it is shown that sl2 c based automorphic lie algebras associated to the icosahedral group i the octahedral group o the tetrahedral group t and the dihedral group dn are isomorphic the proof is based on techniques from classical invariant theory and makes use of clebsch gordan decomposition and transvectants molienfunctionsandthetrace form thisresultprovidesacompleteclas sification of sl2 c based automorphic lie algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of automorphic lie algebras

a twisted integrable hierarchy with d2 symmetry a loop algebra approach to the gerdjikov mikhailov valchev gmv equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered using the twisted loop algebra structure we obtain a transparent treatment of the associated scattering and inverse scattering theory and solve the initial value problem for the gmv equation

reduction groups and automorphic lie algebras we study a new class of infinite dimensional lie algebras which has important applications to the theory of integrable equations the construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic lie algebras for automorphic lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly these algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts

approximations with modules having linear resolutions abstract let be a koszul algebra over a field k we study in this paper a class of modules closely related to the koszul modules called weakly koszul modules it turns out that these modules have some special filtrations with modules having linear resolutions and therefore easy to describe minimal projective resolutions we prove that if the koszul dual of a finite dimensional koszul algebra is noetherian then every finitely generated graded module has a weakly koszul syzygy and as a consequence a rational poincare series if is selfinjective koszul we prove that the stable part of each connected component of the graded auslander reiten quiver containing a weakly koszul module is of the form z a and if the koszul dual of is noetherian then every component has its stable part of the form z a

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

reductions of integrable equations dihedral group we discuss the algebraic and analytic structure of rational lax operators with algebraic reductions of lax equations we associate a reduction group a group of automorphisms of the corresponding infinite dimensional lie algebra we present a complete study of dihedral reductions for sl 2 c lax operators with simple poles and corresponding integrable equations in the last section we give three examples of dihedral reductions for sl n c lax operators

rationality of the poincare series for koszul algebras abstract in this paper we study the rationality of the poincare series of a finitely generated graded left module over a koszul connected algebra in particular we show the following result let a be a koszul connected algebra and a be its koszul dual suppose that a is noetherian and having a balanced dualizing complex if a is either 1 an artinian algebra 2 a graded quotient algebra of a noetherian as regular algebra or 3 an fbn algebra e g a noetherian pi algebra then the poincare series of every finitely generated graded left a module is a rational function over the complex numbers