quadratic algebra structure and spectrum of a new superintegrable system in n dimension we introduce a new superintegrable kepler coulomb system with non central terms in n dimensional euclidean space we show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates we present the wave function in terms of special functions we give an algebraic derivation of spectrum of the superintegrable system we show how the symmetry algebra of the n dimensional kepler coulomb system is deformed to a quadratic algebra with only three generators and structure constants involving a casimir operator of lie algebra we construct the quadratic algebra and the casimir operator we show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum

harmonic polynomials hyperspherical harmonics and atomic spectra the properties of monomials homogeneous polynomials and harmonic polynomials in d dimensional spaces are discussed the properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic projection many important properties of spherical harmonics gegenbauer polynomials and hyperspherical harmonics follow from these formulas harmonic projection also provides alternative ways of treating angular momentum and generalised angular momentum several powerful theorems for angular integration and hyperangular integration can be derived in this way these purely mathematical considerations have important physical applications because hyperspherical harmonics are related to coulomb sturmians through the fock projection and because both sturmians and generalised sturmians have shown themselves to be extremely useful in the quantum theory of atoms and molecules

stabilized finite element methods with shock capturing for advection diffusion problems abstract stabilized fem of streamline diffusion type for advection diffusion problems may exhibit local oscillations in crosswind direction s as a remedy a shock capturing variant of such stabilized schemes is considered as an additional consistent but nonlinear stabilization we prove existence of discrete solutions then we present some a priori and a posteriori estimates finally we address the efficient solution of the arising nonlinear discrete problems

an implementation of the streamline upwind petrovgalerkin method for linear triangular elements abstract the weighting function of the streamline upwind petrov galerkin method contains a coefficient which has the dimension of time and is a function of element parameters e g element dimension element peclet number etc in this paper a formula for the coefficient applicable to linear triangular elements is presented the formula does not consist of the individual element parameters but rather of the element matrices only and so it is not only simple but also cost effective the validity of the formula is shown by several test problems and numerical examples

a natural derivation of discontinuity capturing operator for convection diffusion problems abstract the concept of effective transport velocity is introduced to derive a discontinuity capturing operator for convection diffusion problems the effective transport velocity which depends both on the flow velocity and on the local solution gradient is used to modify the classical representation of the convective term at the continuum level as a result a discontinuity capturing operator arises naturally in the derivation of lax wendroff taylor galerkin and least squares type approximations of the convection diffusion equation the numerical examples presented demonstrate the effectiveness of the proposed approach these include the classical problem of the advection of a steep profile skew to the mesh and the computation of the temperature field in a free convection problem

the multivariate krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states an algebraic interpretation of the bivariate krawtchouk polynomials is provided in the framework of the three dimensional isotropic harmonic oscillator model these polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in three dimensions many of their properties are derived by exploiting the group theoretic setting the bivariate tratnik polynomials of krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters it is explained how the approach generalizes naturally to d 1 dimensions and allows us to interpret multivariate krawtchouk polynomials as matrix elements of so d 1 unitary representations indications are given on the connection with other algebraic models for these polynomials

the multivariate meixner polynomials as matrix elements of so d 1 representations on oscillator states the multivariate meixner polynomials are shown to arise as matrix elements of unitary representations of the so d 1 group on oscillator states these polynomials depend on d discrete variables and are orthogonal with respect to the negative multinomial distribution the emphasis is put on the bivariate case for which the so 2 1 connection is used to derive the main properties of the polynomials orthogonality relation raising lowering relations generating function recurrence relations and difference equations as well as explicit expressions in terms of standard univariate krawtchouk and meixner polynomials it is explained how these results generalize directly to d variables

a numerical study of a posteriori error estimators for convection diffusion equations this paper presents a numerical study of a posteriori error estimators for convection diausion equations the study involves the gradient indicator an a posteriori error estimator which is based on gradient recovery three residual based error estimators for diaerent norms and two error estimators which are defined by solutions of local neumann problems they are compared with respect to the reliable estimation of the global error and with respect to the accuracy of the computed solutions on adaptively refined grids the numerical study shows for both criteria of comparison that none of the considered error estimators works satisfactorily in all tests 2000 elsevier science s a all rights reserved msc 65n50 65n30

a multigrid preconditioner for stabilised discretisations of advection diffusion problems abstract this paper is concerned with the iterative solution of linear systems arising from stabilised discretisations of advection diffusion problems on stretched finite element grids using nonuniform grids of this type leads in general to very badly conditioned matrix problems we therefore consider using gmres in conjunction with a multigrid mg preconditioning strategy in particular we show that in order to achieve the grid size independent convergence which is characteristic of mg methods it is essential to use an effective stabilisation strategy at each level of the mg structure

crosswind smear and pointwise errors in streamline diffusion finite element methods for a model convection dominated singularly perturbed convection diffusion prob lem it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion the ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5 4 under local smoothness assumptions 1 introduction the streamline diffusion method is a finite element method for convection dominated convection diffusion problems which combines formal high accuracy with decent stability properties the method was introduced in the case of stationary problems by hughes and brooks 7 cf raithby and torrance 14 and wahlbin 15 for earlier thoughts in this direction the mathematical analysis of the method was started in johnson and navert 8 and continued with extensions to e g time dependent problems in navert 12 johnson navert and pitkaranta 9 and johnson and saranen 10 in these papers local error estimates in l2 of order o h k 1 2 in regions of smoothness with piecewise polynomial finite elements of degree k were derived together with estimates stating as a typical example that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0 h1 2 essentially the purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0 h3 4 the improvement from 0 h1 2 to 0 h3 4 is obtained by adding a small amount 0 h3 2 of artificial crosswind diffusion to the method in the piecewise linear case k 1 this does not destroy the known o h3 2 accuracy in l2 in smooth regions using our first result we then obtain our second main result localized pointwise error estimates of order 0 h5 4 in regions of smoothness the previously known best pointwise error estimate in the piecewise linear situation is 0 h1 2 another consequence is a global l estimate of order 0 h 2 in the presence of typical crosswind and downwind singularities we shall consider the model problem of finding u u x y such that

pointwise error estimates for a streamline diffusion finite element scheme pointwise error estimates for a streamline diffusion scheme for solving a model convection dominated singularly perturbed convection diffusion problem are given these estimates improve pointwise error estimates obtained by johnson et al 5

necessary l2 uniform convergence conditions for difference schemes for two dimensional convection diffusion problems we derive necessary conditions for the uniform convergence with respect to the perturbation parameter in l2 of schemes for singularly perturbed convection diffusion problems applying these conditions we show that the convergence order of the streamline diffusion finite element method with piecewise linear trial functions cannot be better than 12 uniformly in the singular perturbation parameter

interbasis expansions for the isotropic 3d harmonic oscillator and bivariate krawtchouk polynomials an explicit expression for the general bivariate krawtchouk polynomials is obtained in terms of the standard krawtchouk and dual hahn polynomials the bivariate krawtchouk polynomials occur as matrix elements of the unitary reducible representations of so 3 on the energy eigenspaces of the three dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components the decomposition entails expanding the cartesian basis states in the spherical bases that span irreducible so 3 representations the overlap coefficients are obtained from the clebsch gordan problem for the lie algebra

an improved algorithm for computing topological entropy a new algorithm is presented for computing the topological entropy of a unimodal map of the interval the accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

streamline design of stability parameters for advection diffusion problems the dependence of the computation of advective diffusive transport phenomena on the orientation of the mesh with respect to the flow direction is analyzed poor performance of the classical galerkin finite element method in the convection dominated regime is alleviated by stabilization we propose definitions of the stability parameter that rationally incorporate the flow direction numerical tests compare the performance of the proposed methods with established techniques

9 j symbols of the oscillator algebra and krawtchouk polynomials in two variables a simple generating function for the 9j symbols of the oscillator algebra is found on the basis of this function it is shown that corresponding 9j symbols are expressed in terms of polynomials in two discrete variables which are orthogonal with respect to the trinomial distribution these polynomials depend on four independent parameters and can be considered as a two dimensional analogue of the krawtchouk polynomials difference difference relations the factorization chain the duality property and the rodriguez formula for these polynomials are obtained it is shown that the polynomials are eigenfunctions of two commuting difference operators these polynomials are also covariant with respect to two commuting difference derivation operators in a special symmetric case these polynomials admit a simple factorized expression in terms of two distinct ordinary krawtchouk polynomials in another special case we obtain an explicit expression of 9j symbols in terms of the appell hypergeometric function in two variables

some multivariable orthogonal polynomials of the askey tableau discrete families a multivariable generalization is presented for all the discrete families of the askey tableau this significantly extends the multivariable hahn polynomials introduced by karlin and mcgregor the latter are recovered as a limit case from a family of multivariable racah polynomials

computing the topological entropy of maps we give an algorithm for determining the topological entropy of a unimodal map of the interval given its kneading sequence we also show that this algorithm converges exponentially in the number of letters of the kneading sequence

further considerations on residual free bubbles for advective diffusive equations abstract we further consider the galerkin method for advective diffusive equations in two dimensions the finite dimensional space employed is of piecewise polynomials enriched with residual free bubbles rfb we show that in general this method does not coincide with the supg method unless the piecewise polynomials are spanned by linear functions furthermore a simple stability analysis argument displays the effect of the rfb on the reduced space of piecewise polynomials which in some situations is not equivalent to streamline diffusion for bilinears

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

an analysis of smoothing effects of upwinding strategies for the convection diffusion equation using a technique for constructing analytic expressions for discrete solutions to the convection diffusion equation we examine and characterize the effects of upwinding strategies on solution quality in particular for grid aligned flow and discretization based on bilinear finite elements with streamline upwinding we show precisely how the amount of upwinding included in the discrete operator affects solution oscillations and accuracy when different types of boundary layers are present this analysis provides a basis for choosing a streamline upwinding parameter which also gives accurate solutions for problems with non grid aligned and variable speed flows in addition we show that the same analytic techniques provide insight into other discretizations such as a finite difference method that incorporates streamline diffusion and the isotropic artificial diffusion method

piecewise invertible dynamical systems the aim of the paper is the investigation of piecewise monotonic maps t of an interval x the main tool is an isomorphism of x t with a topological markov chain with countable state space which is described by a 0 1 transition matrix m the behavior of the orbits of points in x under t is very similar to the behavior of the paths of the markov chain every irreducible submatrix of m gives rise to a t invariant subset l of x such that l is the set x of all limit points of the orbit of an x x the topological entropy of l is the logarithm of the spectral radius of the irreducible submatrix which is a l 1 operator besides these sets l there are two t invariant sets y and p such that for all x x the set x is either contained in one of the sets l or in y or in p the set p is a union of periodic orbits and y is contained in a finite union of sets y with y x and has topological entropy zero this isomorphism of x t with a topological markov chain is also an important tool for the investigation of t invariant measures on x results in this direction which are published elsewhere are described at the end of the paper furthermore a part of the proofs in the paper is purely topological without using the order relation of the interval x so that some results hold for more general dynamical systems x t

the multivariate hahn polynomials and the singular oscillator karlin and mcgregor s d variable hahn polynomials are shown to arise in the dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in cartesian and hyperspherical coordinates these polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution the focus is put on the d 2 case for which the connection with the three dimensional singular oscillator is used to derive the main properties of the polynomials forward backward shift operators orthogonality relation generating function recurrence relations bispectrality difference equations and explicit expression in terms of the univariate hahn polynomials the extension of these results to an arbitrary number of variables is presented at the end of the paper

on the estimation of topological entropy we study a method for estimating the topological entropy of a smooth dynamical system our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system we present two algorithms for this purpose and we analyze each according to its strengths and pitfalls we also contrast these with a method based on the definition of topological entropy using n spanning sets

on the complete weight distribution of subfield subcodes of algebraic geometric codes in this paper we first study deviations of the complete weight distribution of a linear code from that of a random code then we consider a large family of subfield subcodes of algebraic geometric codes over prime fields which include bch codes and goppa codes and prove that the complete weight distribution is close to that of a random code if the code length is large compared with the genus of the curve and the degree of the divisor defining the code

a stable petrov galerkin method for convection dominated problems in this paper a new petrov galerkin formulation for solving convection dominated problems is presented the method developed achieves the quasi optimal convergence rates when the solution is regular and provides the necessary stability to avoid spurious oscillations when strong gradients are present such important properties allow the use of p refinement to improve the solution in regions with discontinuities because of the stability engendered by the new petrov galerkin method in this matter a proper evaluation of the intrinsic time scale function appearing in the design of this method is crucial to guarantee the required accuracy

computing the topological entropy of maps of the interval with three monotone pieces an algorithm is presented for computing the topological entropy of a piecewise monotone map of the interval having three monotone pieces the accuracy of the algorithm is discussed and some graphs of the topological entropy obtained using the algorithm are displayed some of the ideas behind the algorithm have application to piecewise monotone functions with more than three monotone pieces

hidden symmetry separation of variables and interbasis expansions in the two dimensional hydrogen atom expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined rcpresentations of the elliptic basis as a superposition of polar and parabolic states are obtained certain interesting limiting cases are investigated

finite difference approximations of multidimensional convection diffusion reaction problems with small diffusion on a special grid a numerical scheme for the convection diffusion reaction cdr problems is studied herein we propose a finite difference method on a special grid for solving cdr problems particularly designed to treat the most interesting case of small diffusion we use the subgrid nodes in the link cutting bubble lcb strategy 5 to construct a numerical algorithm that can easily be extended to the higher dimensions the method adapts very well to all regimes with continuous transitions from one regime to another we also compare the performance of the present method with the streamline upwind petrov galerkin supg and the residual free bubbles rfb methods on several benchmark problems the numerical experiments confirm the good performance of the proposed method

calculating topological entropy the attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem the algorithm introduced here is motivated by the fact that if f 0 1 0 1 is a piecewise monotone map of the unit interval into itself thenh f limn 1 n log var f n where h f is the topological entropy off and var f n is the total variation off n we show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function because of the slow convergence however a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper we prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices with shift of origin in those cases for which the function is markov and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one parameter families of test functions

bivariate krawtchouk polynomials in this paper we present a recurrent procedure to solve an inversion problem for monic bivariate krawtchouk polynomials written in vector column form giving its solution explicitly as a by product a general connection problem between two vector column of monic bivariate krawtchouk families is also explicitly solved moreover in the non monic case and also for krawtchouk families several expansion formulas are given but for polynomials written in scalar form

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

computing the topological entropy for piecewise monotonic maps on the interval a new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented it uses a transition matrix associated with the transformation for this matrix we give a spectral theorem this can be used for an estimation of the accuracy of the algorithm

a new stabilized finite element formulation for scalar convection diffusion problems the streamline and approximate upwind petrov galerkin method abstract a new stabilized and accurate finite element formulation for convection dominated problems is herein developed the basis of the new formulation is the choice of a new upwind function the upwind function chosen for the new method provokes its degeneration into the supg or cau methods depending on the approximate solution s regularity the accuracy and stability of the new formulation for the linear and scalar advection diffusion equation is demonstrated in several numerical examples

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

the multivariate charlier polynomials as matrix elements of the euclidean group representation on oscillator states a family of multivariate orthogonal polynomials generalizing the standard univariate charlier polynomials is shown to arise in the matrix elements of the unitary representation of the euclidean group e d on oscillator states these polynomials in d discrete variables are orthogonal on the product of d poisson distributions the accent is put on the d 2 case and the group theoretical setting is used to obtain the main properties of the polynomials orthogonality and recurrence relations difference equation raising lowering relations generating function hypergeometric and integral representations and explicit expression in terms of standard charlier and krawtchouk polynomials the approach is seen to extend straightforwardly to an arbitrary number of variables the contraction of so 3 to e 2 is used to show that the bivariate charlier polynomials correspond to a limit of the bivariate krawtchouk polynomials

some properties of hyperspherical harmonics a general formula is given for the canonical decomposition of a homogeneous polynomial of order in m variables into a sum of harmonic polynomials this formula which involves successive applications of the generalized laplace operator is proved in the appendix it is shown that the group theoretical method for constructing irreducible cartesian tensors follows from the general formula for canonical decomposition the relationship between harmonic polynomials and hyperspherical harmonics is discussed and an addition theorem for hyperspherical harmonics is derived an expansion of a many dimensional plane wave in terms of gegenbauer polynomials and bessel functions is derived and used to construct bicenter expansions of arbitrary functions in many dimensional spaces finally a formula is derived for the 3 coefficients of hyperspherical harmonics these coefficients give the values of integrals involving the products of three harmonics

oscillation absorption finite element methods for convection diffusion problems this paper proposes a new approach to eliminating overshoots and undershoots when using finite element methods fem for convection dominated convection diffusion problems the first scheme involves an additional nonlinear absorption term which frequently improves the solution quality of both the usual galerkin fem and the streamline diffusion fem mathematical support for the method is given in the form of global estimates for the overshoot undershoot and the basic error in the approximate solution although nonlinear the additional term is in fact monotone another approach is to strongly impose zero overshoot and undershoot via a variational inequality formulation finally we consider the possibility of variational inequality postprocessing to remove overshoots and undershoots

topological entropy of one dimensional maps approximations and bounds we present a method for computing the topological entropy of one dimensional maps as an approximation scheme the algorithm converges rapidly and provides both upper and lower bounds

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

computing the topological entropy of general one dimensional maps a matrix theoretic method for computing the topological entropy of continuous piecewise monotonic maps of the interval is presented the method results in a constructive procedure which is easily implemented on the computer examples for families of unimodal nonunimodal and discontinuous maps are presented

bivariate raising and lowering differential operators for eigenfunctions of a 2d fourier transform we define a two dimensional 2d fourier transform that self reproduces a one parameter family of bivariate hermite functions these are eigenfunctions of a hamiltonian differential operator of second order whose exponential is that transform we find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2d fourier transform