on devising boussinesq type models with bounded eigenspectra the propagation of water waves in the nearshore region can be described by depth integrated boussinesq type equations the dispersive and nonlinear characteristics of the equations are governed by tuneable parameters we examine the associated linear eigenproblem both analytically and numerically using a spectral element method of arbitrary spatial order p it is shown that existing sets of parameters found by optimising the linear dispersion relation give rise to unbounded eigenspectra which govern stability for explicit time stepping schemes the global cfl time step restriction typically requires t p 2 we derive and present conditions on the parameters under which implicitly implicit boussinesq type equations will exhibit bounded eigenspectra two new bounded versions having comparable nonlinear and dispersive properties as the equations of nwogu 1993 and schaffer and madsen 1995 are introduced using spectral element simulations of stream function waves it is illustrated that i the bounded equations capture the physics of the wave motion as well as the standard unbounded equations and ii the bounded equations are computationally more efficient when explicit time stepping schemes are used thus the bounded equations were found to lead to more robust and efficient numerical schemes without compromising the accuracy

an efficient flexible order model for 3d nonlinear water waves the flexible order finite difference based fully nonlinear potential flow model described in h b bingham h zhang on the accuracy of finite difference solutions for nonlinear water waves j eng math 58 2007 211 228 is extended to three dimensions 3d in order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a gmres iterative solution of the discretized laplace problem a robust multigrid method based on gauss seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries and in particular on the sea bottom a new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest linear analysis of the fundamental properties of the scheme with respect to accuracy robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort calculations are made for 3d nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature

development of noncentered wavenumber based optimized interpolation schemes with amplification control for overlapping grids a noncentered interpolation technique has been constructed to perform simulations using overlapping grids for complex geometries high order centered lagrange polynomial interpolations and interpolations optimized in the fourier space are first generalized to the noncentered case these noncentered interpolations either generate significant dispersion errors or strongly amplify high wavenumber components accordingly a noncentered high order wavenumber based optimized interpolation method is developed with the addition of a nonlinear constraint for the control of the amplitude amplification induced by decentering high order piecewise polynomial regressions of the obtained interpolation coefficients are performed the time stability of the method is investigated in the 1 d case when the interpolation method is used in conjunction with explicit high order differencing filtering schemes as well as a 6 step runge kutta time integration algorithm a criterion is formulated to predict its stability as a function of the filtering strength and the courant friedrichs lewy constant finally 1 d convection simulations are presented to illustrate the stability and the accuracy of the developed noncentered interpolations

the use of tricubic interpolation with spectral derivatives to integrate particle trajectories in complicated electromagnetic fields codes that calculate the trajectories of particles in complicated electromagnetic fields often include spectral methods which take advantage of the speed of ffts to rapidly solve for the fields in this case hermite tricubic interpolation can be used to calculate the fields between grid points with spectral derivates used to determine the interpolation coefficients this method is extremely accurate and in the case of electrostatic fields produces an energy conserving force when incorporated into a particle follower code

intermediate wakimoto modules for affine we construct certain boson type realizations of affine that depend on a parameter 0 r n such that when r 0 we get a fock space realization appearing in 6 and when r n they are the wakimoto modules described in the work of feigin and frenkel 7

new irreducible modules for heisenberg and affine lie algebras abstract we study z graded modules of nonzero level with arbitrary weight multiplicities over heisenberg lie algebras and the associated generalized loop modules over affine kac moody lie algebras we construct new families of such irreducible modules over heisenberg lie algebras our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine lie algebras in particular to any function n we associate a highest weight module over the heisenberg lie algebra and a imaginary verma module over the affine lie algebra we show that any imaginary verma module of nonzero level is irreducible

the three point gauge algebra v sl 2 r ωr dr and its action on a fock space abstract the three point current algebra sl 2 r where r c t t 1 u u 2 t 2 4 t and three point virasoro algebra both act on a previously constructed fock space in this paper we prove that the semi direct product i e the gauge algebra acts on the fock space as well

quantization of chern simons gauge theory with complex gauge group the canonical quantization of chern simons gauge theory in 2 1 dimensions is generalized from the case in which the gauge group is a compact lie groupg to the case in which the gauge group is a complex lie groupg though the physical hilbert spaces become infinite dimensional in the latter case the quantization can be described as precisely as for compact gauge groups and using similar methods the special case in which the gauge group issl 2 gives a description of 2 1 dimensional quantum gravity with lorentz signature and positive cosmological constant or with euclidean signature and negative cosmological constant while it is not clear whether there is a 1 1 dimensional conformal field theory related to these 2 1 dimensional models there are natural computable candidates for the central charge and the conformal blocks of such a hypothetical theory

structure of intermediate wakimoto modules abstract we show that our construction of boson type realizations of affine sl n 1 in terms of intermediate wakimoto modules gives representations that are generically isomorphic to certain verma type modules we then use this identification to obtain information about the submodule structure of intermediate wakimoto modules

sum accelerated pseudospectral methods finite differences and sech weighted differences this work continues our previous studies of algorithms for accelerating the convergence of pseudospectral derivative series in order to obtain new differentiation schemes which have the sparsity and low cost of finite differences but the accuracy of spectral methods we develop a general theoretical framework for difference schemes finite differences are close to optimum but can be bettered by a new scheme we have dubbed sech weighted differences through numerical examples we show that sech weighted differences are effective in contrast non linear accelerations like pade approximants and the levin u transform so popular in other applications are inferior for approximating derivatives to linear accelerations like finite differences sech weighted differences and the euler method

permuted difference coefficient realization of fir digital filters a new realization for fir digital filters using permuted difference coefficients is proposed in this paper its coefficients are obtained as the difference between the successive values of the original coefficients reordered in a sequence with falling magnitude the proposed realization can hold desirable properties in fir filters such as an exactly linear phase characteristic and stable implementation and is effectively applied to a wide range filter response quantization error analysis shows that the internal data word lengths must be somewhat increased to maintain the same roundoff noise as in a direct form realization computational complexity becomes about 23 percent and 18 percent for 99th and 299th order filters taking the excess data word lengths into account compared with the direct form

basis functions for triangular and quadrilateral high order elements in this paper we present a unified description of new spectral bases suitable for high order hp finite element discretizations on hybrid two dimensional meshes consisting of triangles and quadrilaterals all bases presented are for c0 continuous discretizations and are described both as modal and as mixed modal nodal expansions general jacobi polynomials of mixed weights are employed that accommodate automatic exact numerical quadratures generalized tensor products and variable expansion order in each element the approximation properties of the bases are analyzed in the context of the projection linear advection and diffusion operators

a reduction theorem for highest weight modules over toroidal lie algebras let open image in new window be a toroidal lie algebra corresponding to a semisimple lie algebra open image in new window we describe all borel subalgebras of open image in new window which contain the cartan subalgebra open image in new window where open image in new window is a fixed cartan subalgebra of open image in new window we show that each such borel subalgebra open image in new window determines a parabolic decomposition open image in new window where open image in new window is a proper toroidal subalgebra of open image in new window and open image in new window our first main result is that for any weight which does not vanish on open image in new window an arbitrary subquotient of the verma module open image in new window is induced from its open image in new window submodule of open image in new window invariant vectors this reduces the study of subquotients of open image in new window to the study of subquotients of verma modules over open image in new window we then introduce categories open image in new window and open image in new window and their respective blocks open image in new window and open image in new window corresponding to a central charge which is nonzero on open image in new window our second main result is that the functors of induction and open image in new window invariants are mutually inverse equivalences of the category open image in new window and the full subcategory of open image in new window whose objects are generated by their open image in new window invariants

new representations of affine kac moody algebras abstract let h be a cartan subalgebra of an affine kac moody algebra without containing the center in this paper we classify the module structures on u h for all affine kac moody algebras which are free u h modules of rank 1 these modules exist only for type a l 1 l 1 or c l 1 l 2 affine kac moody algebras they are generically non weight and simple hence we obtain a class of new simple modules over these affine kac moody algebras

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

quasi ale finite element method for nonlinear water waves this paper presents a newly developed quasi arbitrary lagrangian eulerian finite element method qale fem for simulating water waves based on fully nonlinear potential theory the main difference of this method from the conventional finite element method developed by one of authors of this paper and others see e g q w ma g x wu r eatock taylor finite element simulation of fully non linear interaction between vertical cylinders and steep waves part 1 methodology and numerical procedure and part 2 numerical results and validation int j numer methods fluids 36 2001 265 308 and g x wu z z hu simulation of nonlinear interactions between waves and floating bodies through a finite element based numerical tank proc r soc a 460 2004 2050 3037 3058 is that the complex mesh is generated only once at the beginning and is moved at all other time steps in order to conform to the motion of the free surface and structures this feature allows one to use an unstructured mesh with any degree of complexity without the need of regenerating it every time step which is generally inevitable and very costly due to this feature the qale fem has high potential in enhancing computational efficiency when applied to problems associated with the complex interaction between large steep waves and structures since the use of an unstructured mesh in such a case is likely to be necessary to achieve overall high efficiency the numerical techniques involved in the qale fem are developed including the method to move interior nodes technique to re distribute the nodes on the free surface scheme to calculate velocities and so on the model is validated by water waves generated by a wavemaker in a tank and the interaction between water waves and periodic bars on the bed of tank satisfactory agreement is achieved with analytical solutions experimental data and numerical results from other methods

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

an o 1 integration scheme for three dimensional surface scattering problems we present an accurate method of o 1 complexity with respect to frequency i e a method that to achieve a prescribed error tolerance requires a bounded computational cost for arbitrarily high frequencies for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three dimensional space like the two dimensional counterpart of this algorithm which we introduced recently and which is applicable to scattering by curves in the plane the present method is based on a combination of two main elements 1 a high frequency ansatz for the unknown density in a boundary integral formulation of the problem and 2 an extension of the ideas of the method of stationary phase to allow for o 1 high order accurate integration of oscillatory functions the techniques we introduce to implement an efficient o 1 integrator in the present three dimensional context differ significantly from those used in the earlier two dimensional algorithm in particular in the present text we introduce an efficient canonical hybrid analytic numerical algorithm which in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two dimensional scattering surface we include numerical results illustrating the behavior of the integration algorithm on sound soft spheres with diameters of up to 5000 wavelengths in such cases for a single integral the algorithm yields accuracies of the order of three digits in computational times of less than two seconds in a preliminary full scattering simulation we present a solution with two digits of accuracy in the surface density was obtained in about three hours running time in a single 1 5ghz amd athlon processor for a sphere of 500 wavelengths in diameter

formal group laws for affine kac moody groups and group quantization we describe a method for obtaining formal group laws from the structure constants of affine kac moody groups and then apply a group manifold quantization procedure which permits construction of physical representations by using only canonical structures on the group as an intermediate step we get an explicit expression for two cocycles on loop groups the programme is applied to the affinesu 2 group

a vlsi architecture for simplified arithmetic fourier transform algorithm the arithmetic fourier transform aft is a number theoretic approach to fourier analysis which has been shown to perform competitively with the classical fft in terms of accuracy complexity and speed theorems developed by i s reed et al 1990 for the aft algorithm are used here to derive the original aft algorithm which bruns found in 1903 this is shown to yield an algorithm of less complexity and of improved performance over certain aft algorithms a vlsi architecture is suggested for this simplified aft algorithm this architecture uses a butterfly structure which reduces the number of additions by 25 of that used in the direct method

a fast algorithm for chebyshev fourier and sinc interpolation onto an irregular grid a chebyshev or fourier series may be evaluated on the standard collocation grid by the fast fourier transform fft unfortunately the fft does not apply when one needs to sum a spectral series at n points which are spaced irregularly the cost becomes o n2 operations instead of the ffts o n log n this sort of off grid interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features it is even more crucial to semi lagrangian spectral algorithms for solving convection diffusion and navier stokes problems because off grid interpolation must be performed several times per time step in this work we describe an alternative algorithm the first step is to pad the set of spectral coefficients an with zeros and then take an fft of length 3n to interpolate the chebyshev series to a very fine grid the second step is to apply either the mth order euler sum acceleration or 2m 1 point lagrangian interpolation to approximate the sum of the series on the irregular grid we show that both methods yield full precision with m n allowing an order of magnitude reduction in cost with no loss of accuracy

a harmonic polynomial cell hpc method for 3d laplace equation with application in marine hydrodynamics we propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3d laplace equation the computational domain is discretized by overlapping cells within each cell the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials which are the elementary solutions of laplace equation by its definition the method is named as harmonic polynomial cell hpc method the characteristics of the accuracy and efficiency of the hpc method are demonstrated by studying analytical cases comparisons will be made with some other existing boundary element based methods e g quadratic boundary element method qbem and the fast multipole accelerated qbem fma qbem and a fourth order finite difference method fdm to demonstrate the applications of the method it is applied to some studies relevant for marine hydrodynamics sloshing in 3d rectangular tanks a fully nonlinear numerical wave tank fully nonlinear wave focusing on a semi circular shoal and the nonlinear wave diffraction of a bottom mounted cylinder in regular waves are studied the comparisons with the experimental results and other numerical results are all in satisfactory agreement indicating that the present hpc method is a promising method in solving potential flow problems the underlying procedure of the hpc method could also be useful in other fields than marine hydrodynamics involved with solving laplace equation

verma type modules of level zero for affine lie algebras we study the structure of verma type modules of level zero induced from non standard borel subalgebras of an affine kac moody algebra for such modules in general position we describe the unique irreducible quotients construct a bgg type resolution and prove the bgg duality in certain categories all results are extended to generalized verma type modules of zero level

structure of parabolically induced modules for affine kac moody algebras abstract the main result of the paper establishes the irreducibility of a large family of nonzero central charge induced modules over affine lie algebras for any non standard parabolic subalgebra it generalizes all previously known partial results and provides a construction of many new irreducible modules

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

fourier analysis and signal processing by use of the mobius inversion formula a novel fourier technique for digital signal processing is developed this approach to fourier analysis is based on the number theoretic method of the mobius inversion of series the fourier transform method developed is shown also to yield the convolution of two signals a computer simulation shows that this method for finding fourier coefficients is quite suitable for digital signal processing it competes with the classical fft fast fourier transform approach in terms of accuracy complexity and speed

regular article free surface flow simulation using hp spectral elements computational instabilities inherent in the solution of inviscid incompressible free surface flow have been apparent for some time and different techniques have been used to overcome them in this paper we attempt to explain why these instabilities occur and in doing so a method is developed to study the spatially semi discrete eigenvalues and eigenvectors which govern the stability of the system it is found that the asymmetric spatial discretisation of the fluid domain causes the instabilities after mechanisms for stability are recognised and implemented unsteady inviscid incompressible linear and non linear free surface flow is simulated using a hp spectral element code ensuring fast convergence which incorporates arbitrary lagrangian eulerian ale techniques to decrease deformation of the computational mesh

de aliasing on non uniform grids algorithms and applications we present de aliasing rules to be used when evaluating non linear terms with polynomial spectral methods on non uniform grids analogous to the de aliasing rules used in fourier spectral methods they are based upon the idea of super collocation followed by a galerkin projection of the non linear terms we demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach we begin by deriving from the numerical quadrature rules used by galerkin type projection methods the number of quadrature points and weights needed for quadratic and cubic non linearities we then present a systematic study of the effects of super collocation when using both a continuous galerkin and a discontinuous galerkin method to solve the one dimensional viscous burgers equation we conclude by examining three direct numerical simulation flow examples incompressible turbulent flow in a triangular duct incompressible turbulent flow in a channel at re 395 and compressible flow past a pitching airfoil at re 45 000

a stabilised nodal spectral element method for fully nonlinear water waves we present an arbitrary order spectral element method for general purpose simulation of non overturning water waves described by fully nonlinear potential theory the method can be viewed as a high order extension of the classical finite element method proposed by cai et al 1998 5 although the numerical implementation differs greatly features of the proposed spectral element method include nodal lagrange basis functions a general quadrature free approach and gradient recovery using global l 2 projections the quartic nonlinear terms present in the zakharov form of the free surface conditions can cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep waves we show how the scheme can be stabilised through a combination of over integration of the galerkin projections and a mild spectral filtering on a per element basis this effectively removes any aliasing driven instabilities while retaining the high order accuracy of the numerical scheme the additional computational cost of the over integration is found insignificant compared to the cost of solving the laplace problem the model is applied to several benchmark cases in two dimensions the results confirm the high order accuracy of the model exponential convergence and demonstrate the potential for accuracy and speedup the results of numerical experiments are in excellent agreement with both analytical and experimental results for strongly nonlinear and irregular dispersive wave propagation the benefit of using a high order possibly adapted spatial discretisation for accurate water wave propagation over long times and distances is particularly attractive for marine hydrodynamics applications

hermitian representations of the extended affine lie algebra gl2 cq abstract we use the idea of free fields to obtain highest weight representations for the extended affine lie algebra gl 2 c q coordinatized by the quantum torus c q and go on to construct a contravariant hermitian form we further give a necessary and sufficient condition such that the contravariant hermitian form is positive definite

multipole expansions and pseudospectral cardinal functions a new generalization of the fast fourier transform the polynomial or trigonometric interpolant of an arbitrary function f x may be presented as a critical function series whose coefficients are the values of f x at the interpolation points we show that the cardinal series is identical to the sum of the forces due to a set of n point charges with appropriate force laws it follows that the cardinal series can be summed via the fast multipole method fmm in o nlog sub 2 n operations which is much cheaper than the o n sup 2 cost of direct summation the fmm is slower than the fast fourier transform fft so the latter should always be used where applicable however the multipole expansion succeeds where the fft fails in particular the fmm can be used to evaluate fourier and chebyshev series on an irregular grid as is needed when adaptively regridding in a time integration also the multipole expansion can be applied to basis sets for which the fft is inapplicable even on the canonical grid including legendre polynomials hermite and laguerre functions spherical harmonics and zinc functions 12 refs

a spectral method of characteristics for hyperbolic problems a new numerical technique is introduced for first order hyperbolic equations based on combining the spectral method with explicit timestepping along the characteristics which permits the use of large timesteps at no cost in accuracy the method is unconditionally stable and spectrally accurate the theoretical results are illustrated by numerical experiments

a nonuniform fast fourier transform based on low rank approximation by viewing the nonuniform discrete fourier transform nudft as a perturbed version of a uniform discrete fourier transform we propose a fast and quasi optimal algorithm for computing the nudft based on the fast fourier transform fft our key observation is that an nudft and dft matrix divided entry by entry is often well approximated by a low rank matrix allowing us to express a nudft matrix as a sum of diagonally scaled dft matrices our algorithm is simple to implement automatically adapts to any working precision and is competitive with state of the art algorithms in the fully uniform case our algorithm is essentially the fft we also describe quasi optimal algorithms for the inverse nudft and two dimensional nudfts

field models from two cocycles on infinite dimensional lie groups and symplectic structures 2d gravity and chern simons theory in this paper the connection between field models and infinite dimensional lie groups is widely analysed on the bases of a new group quantization approach we also relate the poincare cartan form of variational calculus to the symplectic current structure of the covariant phase space formulation of higher derivative field theory the virasoro and kac moody groups are considered in the first case the action functional of the 2d induced gravity in the light cone formulation is derived the hidden sl 2 r simply appears as generated by the kernel of the lie algebra two cocycle and plays the role of a gauge type symmetry nevertheless it is shown that a proper space like formulation is out of reach of the virasoro group the corresponding symplectic structure of the nonlocal action functional is determined showing that it is related to the symplectic structure associated with the sl 2 r kac moody group this unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint orbit approach the relation between the kac moody groups and the chern simons gauge theory on a disc in the presence of a source is considered using the new approach

direct operatorial tau method for pantograph type equations abstract the lanczos tau method is applied to find chebyshev polynomial approximations for the solutions of pantograph differential equations the results are accompanied by an error analysis numerical examples calculated by our matlab package chebpack confirm the theory and prove the importance for practice of this approach

a new compact spectral scheme for turbulence simulations we propose a new kind of compact difference scheme for the computation of the first and second derivatives in the simulation of high reynolds number turbulent flows the scheme combines and truncates the pseudospectral representation of derivative for convergence acceleration comparison of the wave resolution property with available optimized compact schemes minimizing the prescribed wave resolution error reveals our scheme s superiority for the same size of stencils without introducing optimization parameters an accompanying boundary scheme is also proposed with the stability analysis the proposed scheme is tested for the evaluation of derivatives of a function that decays very slowly in the wavenumber space and for the simulation of three dimensional isotropic turbulence

fock representations and brst cohomology insl 2 current algebra we investigate the structure of the fock modules overa 1 1 introduced by wakimoto we show that irreducible highest weight modules arise as degree zero cohomology groups in a brst like complex of fock modules chiral primary fields are constructed as brst invariant operators acting on fock modules as a result we obtain a free field representation of correlation functions of thesu 2 wzw model on the plane and on the torus we also consider representations of fractional level arising in polyakov s 2d quantum gravity finally we give a geometrical borel weil like interpretation of the wakimoto construction

fast approximate fourier transforms for irregularly spaced data several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed the algorithms can be viewed as approximate generalizations of the fast fourier transform fft and they are compared with regard to their accuracy and their computational efficiency

a finite element method for fully nonlinear water waves we introduce a numerical method for fully nonlinear three dimensional water surface waves described by standard potential theory the method is based on a transformation of the dynamic water volume onto a fixed domain regridding at each time step is thereby avoided the transformation introduces an elliptic boundary value problem which is solved by a preconditioned conjugate gradient method moreover a simple domain imbedding precedure is introduced to solve problems with an obstacle in the water volume numerical experiments are presented and they show nice convergence properties of the iterative solver as well as convergence of the entire solution towards a reference solution computed by another scheme

whittaker modules for the affine lie algebra a_1 1 abstract we prove the irreducibility of the universal non degenerate whittaker modules for the affine lie algebra sl 2 of type a 1 1 with noncritical level these modules can become simple whittaker modules over sl 2 sl 2 c d with the same whittaker function and central charge we have to modulo a central character for sl 2 to obtain simple degenerate whittaker sl 2 modules with noncritical level in the case of critical level the universal whittaker module is reducible we prove that the quotient of universal whittaker sl 2 module by a submodule generated by a scalar action of central elements of the vertex algebra v 2 sl 2 is simple as sl 2 module we also explicitly describe the simple quotients of universal whittaker modules at the critical level for sl 2 quite surprisingly with the same whittaker function some simple degenerate sl 2 whittaker modules at the critical level have semisimple action of d and others have free action of d at last by using vertex algebraic techniques we present a wakimoto type construction of a family of simple generalized whittaker modules for sl 2 at the critical level this family includes all classical whittaker modules at critical level we also have wakimoto type realization for degenerate whittaker modules for sl 2 at noncritical level

an algorithm for computing the inverse z transform the authors determine an infinite impulse response of a causal system via a sampling algorithm applied to the transform on the unit circle

imaginary verma modules for uq sl 2 ˆ and crystal like bases abstract we consider imaginary verma modules for quantum affine algebra u q sl 2 and define a crystal like base which we call an imaginary crystal basis using the kashiwara algebra k q constructed in earlier work of the authors in particular we prove the existence of imaginary crystal like bases for a suitable category of reduced imaginary verma modules for u q sl 2

a note on the computational complexity of the arithmetic fourier transform it is shown that the number of data points the arithmetic fourier transform aft needs for an n point fourier transform is proportional to n sup 2 thus for example while a standard fast fourier transform algorithm requires 1024 samples to yield 1024 spectral components aft would take more than 300000 samples to do the same job

construction of lighthill s unitary functions the imbricate series of unity the poisson sum theorem implies that every spatially periodic function f x can be written as a so called imbricate series that is as the superposition of an infinite number of evenly space copies of a pattern function a x the pattern is the fourier transform of a function a n that specifies the usual fourier coefficients because there are many a n which coincide with a given set of fourier coefficients aj when n is an integer the pattern function a x is not unique in this work we show how to construct pattern functions of the form a x f x w x where f x is the periodic function itself the imbrication of a x and where w x is what lighthill called a unitary function we derive an explicit formula to construct all possible unitary functions and tabulate eight examples we also prove that all unitary functions are pattern functions of unity that is for the imbricate series that sums to f x 1 x

hermitian symmetric spaces and their unitary highest weight modules abstract the purpose of this article is to determine the set of unitarizable highest weight modules corresponding to hermitian symmetric spaces of the noncompact type the major step is that of proving unitarity at the last possible place with this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem bernstein gelfand gelfand and a diagramatic representation of the set of positive noncompact roots

taming the cfl number for discontinuous galerkin methods on structured meshes the upwind discontinuous galerkin method is an attractive method for solving time dependent hyperbolic conservation laws it is possible to use high order explicit time stepping methods and high order spatial approximations without incurring heavy numerical linear algebra overheads however the courant friedrichs lewy cfl condition for these methods depends on the polynomial order used and there is a somewhat excessive cost for using very high order spatial approximation we discuss the impact of a covolume mesh based filter on the cfl number for these methods and present an algorithm which has a cfl number independent of the spatial order of approximation we present computational results for the advection equation and the wave equation on one dimensional meshes using up to tenth order in space and time

φ imaginary verma modules and their generalizations for the toroidal lie algebras in this paper we first construct a class of weight modules called imaginary verma modules for the toroidal lie algebras then a criterion for the irreducibility of the imaginary verma modules is obtained and the irreducible quotients for the reducible ones are studied furthermore we construct a more general class of n graded modules for the toroidal lie algebras and we discuss their irreducibility this class of modules includes the imaginary verma modules as special examples

rapid convergence of some seismic processing algorithms the rate of convergence of many numerical algorithms can be greatly improved by repeated application of the method of summation by parts cases of interest in seismology arise when we need to resample the spectrum of a function at unevenly spaced frequency values related examples include the time domain evaluation of the hilbert transform and the extrapolation to the real axis of spectra evaluated in the complex domain a formula of this type was first presented by lanczos 1956 the validity of the algorithms depends upon the fact that summation by parts of the cardinal series representation is justified as long as a function is somewhat oversampled

regular subalgebras of affine kac moody algebras we classify regular subalgebras of kac moody algebras in terms of their root systems in the process we establish that a root system of a subalgebra is always an intersection of the root system of the algebra with a sublattice of its root lattice we also discuss applications to investigations of regular subalgebras of hyperbolic kac moody algebras and conformally invariant subalgebras of affine kac moody algebras in particular we provide explicit formulae for determining all virasoro charges in coset constructions that involve regular subalgebras

some remarks on representations of jet groups and gauge groups irreducible unitary representations of the three first jet or leibniz extensions gkm of a compact group g are studied focusing on the neighborhood of the trivial representation some associated irreducible unitary continuous tensor product representations of the gauge or current groups of mappings from a riemannian manifold into gkm are described

geometric realizations of affine kac moody algebras abstract the goal of the present paper is to obtain new free field realizations of affine kac moody algebras motivated by geometric representation theory for generalized flag manifolds of finite dimensional semisimple lie groups we provide an explicit construction of a large class of irreducible modules associated with certain parabolic subalgebras covering all known special cases

comments on a note on the computational complexity of the arithmetic fourier transform by n tepedelenlioglu the commenter states that because of the use of farey sequences the result stated by n tepedelenlioglu ibid vol 37 no 7 p 1146 7 1989 becomes a statement about d sub n the number of elements in a farey sequence of order n tepedelenlioglu s formula is then equivalent to a formula about farey sequences given by m r schroder 1986 c m rader 1977 thus for example while a standard ftt algorithm requires 1024 samples to yield 1024 spectral components aft would take more than 300000 samples to do the same job

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

a new class of unitarizable highest weight representations of infinite dimensional lie algebras ii on donne les details rigoureux pour le probleme de classification on redemontre l unitarite de la representation exceptionnelle on considere les produits tensoriels et les problemes d integrabilite

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

accurate function sinc interpolation and derivative estimations over finite intervals function expansion in terms of finite sinc bases is considered for finite intervals we considered proper transformation and weighted barycentric methods we adjusted the transformation function to comply with the truncation of the expansion this removes the unbounded behavior of the terminal derivatives and yields accurate results efficient weighted barycentric expansions are presented to improve the accuracies of function and derivative estimations a new expression for the estimation of the second derivative is derived the derived expressions are successfully applied to accurately locate the minimum of the solution and solve a singularly perturbed problem