finite resolutions of modules for reductive algebraic groups of a suitable g module m by modules which are required to be direct sums of tensor products of exterior powers of the natural representation in particular in z 31 for partitions 1 and p resolutions are constructed for the modules l f and l f the schur functor corresponding to 1 and the skew schur functor corresponding to a p evaluated at f where f is a free module over a field or the integers the purpose of this paper is to characterise the gl modules which admit such a resolution as those modules which have a filtration in which each successive quotient has the form l f for some partition p by contrast with 2 3 we do not produce resolutions explicitly our methods are independent of those of akin and buchsbaum we give a new proof of their result on the existence of a resolution for l f and are algebraic group theoretic in nature we have therefore cast our main result the theorem of section 1 as a statement about resolutions for reductive algebraic groups over an algebraically closed field one may frequently wish to work over a more general coefficient ring see l 3 and so we treat in some detail chevalley group schemes and general linear group schemes over a principal ideal domain 2 6 and 2 7 giving the appropriate versions of our main result and relating it to the work of akin buchsbaum and weyman

error and extrapolation of a compact lod method for parabolic differential equations this paper is concerned with a compact locally one dimensional lod finite difference method for solving two dimensional nonhomogeneous parabolic differential equations an explicit error estimate for the finite difference solution is given in the discrete infinity norm it is shown that the method has the accuracy of the second order in time and the fourth order in space with respect to the discrete infinity norm a richardson extrapolation algorithm is developed to make the final computed solution fourth order accurate in both time and space when the time step equals the spatial mesh size numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm

skew modules for reductive groups skew modules for symmetric groups have been investigated by james and peel 13 farahat and peel 9 and clausen s more recently skew modules for general linear groups have been investigated by akin buchsbaum weyman 3 l 2 and kouwenhoven 161 our aim in this paper is to set up in a rather general context a theory of skew modules for reductive algebraic groups for a partition 1 of it james peel 13 obtain a filtration of the specht module sj viewed as a module for the young subgroup corresponding to a decomposition n a b of n the successive quotients in the filtration have the form s h sj p kronecker product where p runs over the partitions of a such that p is a skew partition and s denotes the skew specht module the analogue of this result is proved for general linear groups by akin buchsbaum and weyman 3 theorem 11 4 1 l and kouwenhoven 16 proposition ii 2 2a the analogue for reductive groups is more or less our definition of skew modules this filtration property is in fact a characterisation of skew modules see 1 3 4 and our definition for general linear groups therefore agrees with that of akin buchsbaum and weyman 3 and kouwenhoven 161 it becomes apparent rather quickly from our point of view that the skew modules have a particular kind of filtration see 2 1 1 which in the case of the general linear group is a filtration by schur modules this is a result of kouwenhoven 16 proposition 11 2 2b a for general linear groups kouwenhoven is able to give an explicit description of such a filtration one may then obtain as in kouwenhoven 16 proposition ii 2 2b b the result of james and peel 131 on the existence of specht series for skew modules for the symmetric groups by specializing to a certain weight space i e applying the schur functor as in ll chap 63 we are also able to prove see 2 3 3 the existence of a certain kind of resolution considered by akin and buchsbaum see 1 sects 83 for skew modules for general linear groups our main point of reference is the representation theory of reductive 465 0021 8693 88 3 00

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

a higher order compact lod method and its extrapolations for nonhomogeneous parabolic differential equations abstract a higher order compact locally one dimensional lod finite difference method for two dimensional nonhomogeneous parabolic differential equations is proposed the resulting scheme consists of two one dimensional tridiagonal systems and all computations are implemented completely in one spatial direction as for one dimensional problems the solvability and the stability of the scheme are proved almost unconditionally the error estimates are obtained in the discrete h 1 l 2 and l norms and show that the proposed compact lod method has the accuracy of the second order in time and the fourth order in space two richardson extrapolation algorithms are presented to increase the accuracy to the fourth order and the sixth order in both time and space when the time step is proportional to the spatial mesh size numerical results demonstrate the accuracy of the compact lod method and the high efficiency of its extrapolation algorithms

on the birman schwinger principle applied to δ m2 m the condition for e 0 to be an eigenvalue of the operator m2 m v is obtained through the use of the birman schwinger principle theorem 3 2 by setting e 2 and using the analyticity of the corresponding birman schwinger kernel theorem 3 1 the series development of 1 is obtained up to second order on theorem 4 1

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

on the numerical solution of multi dimensional parabolic problem by the additive splitting up method several numerical methods arising from the difference methods for certain classes of two dimensional parabolic equation are based on an operator slitting from the theoretical point of view the success and the evaluation of the splitting approach is primarily determined by the accuracy and the stability constrained most of the splitting methods defined in the past decades were of multiplicative non parallel types with respect to the spatial variables in 1994 lui tai and neittaanmaki presented a parallelisable implicit splitting type of methods of different order of approximation and its of additive type with regard to the solution at the advanced time step in this paper the stability analysis will be presented for the implicit splitting methods by lu et al and also we presented a parallelisable explicit splitting algorithm several model problems are solved by the splitting up algorithms to enhance the theoretical results and the concluding remarks

a fourth order compact adi method for solving two dimensional unsteady convection diffusion problems n this article an exponential high order compact ehoc alternating direction implicit adi method in which the crank nicolson scheme is used for the time discretization and an exponential fourth order compact difference formula for the steady state 1d convection diffusion problem is used for the spatial discretization is presented for the solution of the unsteady 2d convection diffusion problems the method is temporally second order accurate and spatially fourth order accurate which requires only a regular five point 2d stencil similar to that in the standard second order methods the resulting ehoc adi scheme in each adi solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal gaussian decomposition and may also be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time the unconditionally stable character of the method was verified by means of the discrete fourier or von neumann analysis numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order adi method of karaa and zhang and the spatial third order compact scheme of note and tan

on coupling constant thresholds and related eigenvalue properties of dirac operators on considere des operateurs de dirac de la forme h h 0 q on etudie les valeurs propres discretes de h en fonction de en particulier la situation ou une ou plusieurs d entre elles approchent un des points e 1 ou 1

high accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients abstract in this article two level compact implicit difference methods of o k2 kh2 h4 using 9 spatial grid points are proposed for the numerical solution of the system of two dimensional nonlinear parabolic equations with variable coefficients subject to the dirichlet boundary conditions where k 0 and h 0 are step lengths in time and space directions respectively the proposed difference method for scalar equation is applied for the solution of the heat conduction equation in polar coordinates to obtain the two level unconditionally stable adi method of o k2 h4 the method having two variables also has been successfully applied on two dimensional unsteady navier stokes model equations in polar coordinates some numerical examples are presented to demonstrate the accuracy of the implementation

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

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

short note a high order pade adi method for unsteady convection diffusion equations a high order alternating direction implicit adi method for computations of unsteady convection diffusion equations is proposed by using fourth order pade schemes for spatial derivatives the present scheme is fourth order accurate in space and second order accurate in time the solution procedure consists of a number of tridiagonal matrix operations which make the computation cost effective the method is unconditionally stable and shows higher accuracy and better phase and amplitude error characteristics than the standard second order adi method d w peaceman h h rachford jr the numerical solution of parabolic and elliptic differential equations journal of the society of industrial and applied mathematics 3 1959 28 41 and the fourth order adi scheme of karaa and zhang high order adi method for solving unsteady convection diffusion problem journal of computational physics 198 2004 1 9

extrapolation algorithm of compact adi approximation for two dimensional parabolic equation abstract in this paper we propose a compact alternating direction implicit adi scheme for solving the two dimensional parabolic equation the maximum norm convergence is proven and the convergence rate is second order in time and fourth order in space we develop a richardson extrapolation algorithm to increase the accuracy to sixth order both in time and space numerical experiments show the effectiveness of the method

puiseux series expansion for an eigenvalue of the generalized friedrichs model with perturbation of rank 1 a family h p 0 of the generalized friedrichs model with perturbation of rank 1 associated with a system of two particles moving on the one dimensional lattice is considered the existence of a unique eigenvalue e p of the operator h p lying below the essential spectrum is proved for any p from a neighborhood of the origin the puiseux series expansion for eigenvalue e p at the point p 0 is found moreover the asymptotics for e p is established as

generalized eigenfunctions for dirac operators near criticality critical dirac operators are those which have eigenfunctions and or resonances for e m we estimate the behavior of the generalized eigenfunctions of critical dirac operators under small perturbations of the potential the estimates are done in the l norm we show that for small k the generalized eigenfunctions are in leading order multiples of the respective eigenfunctions and or resonances we also estimate the k derivatives which are important for estimating decay the method also applies for other differential operators for example schrodinger operators

a two level linearized compact adi scheme for two dimensional nonlinear reaction diffusion equations abstract a novel two level linearized compact alternating direction implicit adi scheme is proposed for solving two dimensional nonlinear reaction diffusion equations the computational cost is reduced by use of the newton linearized method and the adi method the existence and uniqueness of the numerical solutions are proved moreover the error estimates in h 1 and l norms are presented numerical examples are given to illustrate our theoretical results

locally explicit schemes for three dimensional diffusion with a non local boundary specification two finite difference schemes based on the classical three point forward time centred space ftcs method and the five point ftcs formula are used to solve the three dimensional time dependent diffusion equation with a non local boundary specification in these methods locally one dimensional lod procedures are used to extend the one dimensional techniques to solve the three dimensional problem numerical integration procedures are also employed to overcome the non local boundary specification results from some numerical experiments are presented including error estimates derived in the maximum norm the results of numerical testing show that the lod techniques give better results than the fully explicit schemes the cpu times used for the procedures developed in this article are also reported