preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

zeros of l functions attached to maass forms this paper considers psl 2 automorphic cusp forms the maass forms and their associated dirichlet series there are two major results the first uses real variable techniques to give an explicit reconstruction of the cusp form in terms of the dirichlet series this representation is valid in the entire upper half plane the second uses this representation to show that the dirichlet series has many zeros on its critical line this is the analogue of the hardy littlewood result for the riemann zeta function

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

finite dimensional irreducible representations of the quantum superalgebra u q gl n 1 it is shown that every finite dimensional irreducible module over the general linear lie superalgebragl n 1 can be deformed to an irreducible module ofu q gl n 1 aq analogue of the universal enveloping algebra ofgl n 1 the results are extended also to all kac modules which in the atypical cases remain indecomposible within each module expressions for the transformations of the gel fand zetlin basis under the action of the algebra generators are written down an analogoue of the poincare birkhoff witt theorem is formulated

q oscillator realizations of the metaplectic representation of quantum osp 3 2 q oscillator representations of the quantum superalgebra ospq 3 2 are discussed it is pointed out that the metaplectic representation of ospq 3 2 can be equivalently realized either in terms of one q boson and one q fermion or with one q boson and two q fermions

finite dimensional representations of the quantum superalgebra uq gl 2 2 ii nontypical representations at generic q the construction approach proposed in the previous paper n a ky j math phys 35 2583 1994 allows us there and in the present paper to construct at generic deformation parameter q all finite dimensional representations of the quantum lie superalgebra uq gl 2 2 the finite dimensional uq gl 2 2 modules wq constructed in the previous paper are either irreducible or indecomposable if a module wq is indecomposable i e when the condition 4 41 in the previous paper does not hold there exists an invariant maximal submodule of wq say iqk such that the factor representation in the factor module wq iqk is irreducible and called nontypical here in this paper indecomposable representations and nontypical finite dimensional representations of the quantum lie superalgebra uq gl 2 2 are considered and classified as their module structures are analyzed and the matrix elements of all nontypical representations are written down explicitly

finite dimensional representations of the quantum superalgebra uq gl 2 2 i typical representations at generic q in the present paper all typical finite dimensional representations of the quantum lie superalgebra uq gl 2 2 are constructed at generic deformation parameter q as in the nondeformed case the finite dimensional uq gl 2 2 module wq obtained is irreducible and can be decomposed into finite dimensional irreducible uq gl 2 gl 2 submodules vqk

representations of quantum superalgebra uq gl 2 1 in a coherent state basis and generalization the coherent state method has proved to be useful in quantum physics and mathematics this method more precisely the vector coherent state method has been used by some authors to construct representations of superalgebras but almost to our knowledge it has not yet been extended to quantum superalgebras except uq osp 1 2 one of the smallest quantum superalgebras in this article the method is applied to a bigger quantum superalgebra namely uq gl 2 1 in constructing q boson fermion realizations and finite dimensional representations which when irreducible are classified into typical and nontypical representations this construction leads to a more general class of q boson fermion realizations and finite dimensional representations of uq gl 2 1 and thus at q 1 of gl 2 1 both gl 2 1 and uq gl 2 1 have found different physics applications therefore it is meaningful to construct their representations

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

an additive problem in the fourier coefficients of cusp forms we establish an estimate on sums of shifted products of fourier coefficients coming from holomorphic or maass cusp forms of arbitrary level and nebentypus these sums are analogous to the binary additive divisor sum which has been studied extensively as an application we derive extending work of duke friedlander and iwaniec a subconvex estimate on the critical line for l functions associated to character twists of these cusp forms

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

on clebsch gordan coefficients and matrix elements of representations of the quantum algebra uq su2 clebsch gordan coefficients and matrix elements of irreducible representations of the quantum algebra uq su2 were considered in several papers in particular a few expressions for them were derived an approach to clebsch gordan coefficients and to matrix elements of representations of uq su2 on the base of the theory of basic hypergeometric functions is given this approach allows one to obtain q analogs of all well known classical expressions for clebsch gordan coefficients most of them were absent new symmetry relations generating functions and recurrence formulas for clebsch gordan coefficients of uq su2 are obtained unlike other papers clebsch gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions in fact without using the notion of a c algebra and of a hopf algebra

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

classification of the quantum deformations of the superalgebra gl 1 1 we present a classification of the possible quantum deformations of the supergroup and its lie superalgebra in each case the super commutation relations and the hopf structures are explicitly computed for each r matrix one finds two inequivalent co products whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras in the braided case one recovers the classical algebra for suitable limits of the deformation parameters but this is no longer true in the unbraided case

finite dimensional representations of the lie superalgebra gl 2 2 in a gl 2 gl 2 basis ii nontypical representations all finite dimensional irreducible representations of the general linear lie superalgebra gl 2 2 are written down in matrix form the basis within each representation space is chosen in such a way that it makes evident the decomposition of gl 2 2 into irreducible representations of its even subalgebra gl 2 gl 2 special attention is devoted to the analysis of all nontypical representations and some indecomposible representations

induced representations of the two parametric quantum deformation upq gl 2 2 the two parametric quantum superalgebra up q gl 2 2 and its induced representations are considered a method for constructing all finite dimensional irreducible representations of this quantum superalgebra is also described in detail it turns out that finite dimensional representations of the two parametric up q gl 2 2 even at generic deformation parameters are not simply trivial deformations from those of the classical superalgebra gl 2 2 unlike the one parametric cases

representations of quantum affine superalgebras we study the quantum affine superalgebra u q mathcal l mathfrak sl m n and its finite dimensional representations we prove a triangular decomposition and establish a system of poincare birkhoff witt generators for this superalgebra both in terms of drinfel d currents we define the weyl modules in the spirit of chari pressley and prove that these weyl modules are always finite dimensional and non zero in consequence we obtain a highest weight classification of finite dimensional simple representations when m ne n some concrete simple representations are constructed via evaluation morphisms

finite dimensional irreducible representations of the quantum supergroup uq gl m n finite dimensional irreducible representations of the quantum supergroup uq gl m n at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction the representation theory is rather similar to that of gl m n at generic q but drastically different when q is a root of unity in the latter case atypicality conditions of highest weight irreducible representations irreps are substantially altered and such finite dimensional irreps arise that do not have highest weight and or lowest weight vectors as concrete examples the irreps of uq gl 2 1 are classified

the mean square of the error term for the fourth power moment of the zeta function we prove that j e2 t 2 dt t 2 log c t where e2 t denotes the error term in the asymptotic formula for fo k of dt the proof depends on the second author s explicit formula for a modified version of the latter integral several related results are also proved including the analogue of the above result for the binary additive divisor problem

irreducible representations of up q gl 2 2 the two parametric quantum superalgebra up q gl 2 2 and its representations are considered all finite dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and non typical ones according to a proposition proved in the present paper this proposition is a non trivial deformation from the one for the classical superalgebra gl 2 2 unlike the case of one parametric deformations