evolution equations and levy processes on quantum groups in this chapter we consider stochastic processes on quantum groups that are related to evolution equations of the form partial t u lu with some difference differential operator l for the equations considered in section 7 1 u is an element of a quantum or braided group a we recall that solutions of these equations can be given as appell systems or shifted moments of the associated process and show how these can be calculated explicitly on the q affine group and on a braided analogue of the heisenberg weyl group

random walk and the heat equation on superspace and anyspace random walks are used to study diffusion on anyspace anyspace is characterized by coordinate with n 0 and statistics e2 i n between independent copies anyonic integration and anyonic dirac functions are introduced and reduced to familiar results for supersymmetry when n 2 these ingredients are then used to formulate and solve the resulting anyonic diffusion equation

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

overlapping and non overlapping domain decomposition methods for large scale meshless efg simulations abstract meshless methods have a number of virtues in problems concerning crack growth and propagation strain localization dynamic shear band propagation projectile penetration among others the main drawback of such methods is that the resulting matrices are more densely populated and the computational cost for the formulation and solution of the problem is much higher than the conventional fem this is the reason for their limited application to academic and small scale problems until now in this paper we introduce a novel approach for reducing the computational cost of meshless element free galerkin efg methods by employing domain decomposition techniques on the physical as well as on the algebraic domains specifically we propose the implementation of the dual domain decomposition feti family of methods that have been successfully used with fem in many problems in computational mechanics we address the issues of dividing an overlapping meshfree efg domain to several overlapping subdomains following the algebraic decomposition of the feti method and alternatively on non overlapping subdomains by modifying the required displacement compatibility conditions and by improving the accuracy of the solution through a stress update procedure

dispersion and pollution of meshless solutions for the helmholtz equation it is well known today that the standard finite element method fem is unreliable to compute approximate solutions of the helmholtz equation for high wavenumbers due to the pollution effect consisting mainly of the dispersion i e the numerical wavelength is longer than the exact one unless highly refined meshes are used fem solutions lead to unacceptable solutions in terms of precision while the use of very refined meshed increases the cost in terms of computational times the paper presents an application of the element free galerkin method efgm and focuses on the dispersion analysis in 2d it shows that it is possible to choose the parameters of the method in order to minimize the dispersion and to get extremely good results in comparison with the stabilized fem moreover the present meshless formulation is not restricted to regular distribution of nodes and a simple but real life problem is investigated in order to show the improvement in the accuracy of the numerical results w r fem results

calculation of sound radiation in infinite domain using a meshless method a meshless method coupling with a variable order infinite acoustic wave envelope element for sound radiation calculation in infinite domain is presented with the aim of accurately calculating the acoustic radiation and improving computational efficiency it is based on using the element free galerkin method in the inner region enclosing the radiator and a variable order infinite acoustic wave envelope element in the outer region for the proper modeling of the pressure amplitude decay the details are provided for the derivation and implementation of this method the factors of influencing the performance of the method which include the shape function constructing the number of integration points the weight functions and the support domain are discussed a hybrid adaptive gauss legendre quadrature is devised to obtain good integration accuracy the suitable radius of the support domain for the acoustic field calculation in free space is also determined by use of numerical experiments a complex structure is designed for simulation to validate the method the results illustrate the accuracy applicability and effectiveness of this method

algebra of screening operators for the deformed wn algebra we construct a family of intertwining operators screening operators between various fock space modules over the deformed wn algebra they are given as integrals involving a product of screening currents and elliptic theta functions we derive a set of quadratic relations among the screening operators and use them to construct a felder type complex in the case of the deformed w3 algebra

a completeness relation for the q analogue coherent states by q integration q integration is defined for the q oscillator realization of quantum groups this is used to prove a completeness relation for the q analogue of the usual coherent states these states are overcomplete

dispersion analysis of the meshfree radial point interpolation method for the helmholtz equation when numerical methods such as the finite element method fem are used to solve the helmholtz equation the solutions suffer from the so called pollution effect which leads to inaccurate results especially for high wave numbers the main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution which is known as dispersion in order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences in this paper a meshfree method namely the radial point interpolation method rpim is investigated with respect to the pollution effect in the 2d case it is shown that this methodology is able to reduce the dispersion significantly two modifications of the rpim namely one with polynomial reproduction and another one with a problem dependent sine cosine basis are also described and tested numerical experiments are carried out to demonstrate the advantages of the method compared with the fem for identical discretizations the rpim yields considerably better results than the fem copyright 2008 john wiley sons ltd

quantum diffusions and appell systems within the algebraic framework of hopf algebras random walks and associated diffusion equations master equations are constructed and studied for two basic operator algebras of quantum mechanics i e the heisenberg weyl algebra hw and its q deformed version hw q this is done by means of functionals determined by the associated coherent state density operators the ensuing master equations admit solutions given by hw and hw q valued appell systems

a q deformed completely integrable bose gas model the authors construct the hamiltonian of a new quantum integrable q boson lattice model in 1 1 dimensions which has q bosons as dynamical variables and solve it for its energy eigenstates and energy eigenvalues under periodic boundary conditions of finite period this model can be regarded as the q deformation of the integrable lattice bose gas the quantum lattice nonlinear schrodinger quantum lattice nls model in appropriate continuum limits both of these lattice models become the quantum repulsive nls model the bose gas in the lattice case the distinction between the q bosons and ordinary bosons leads to significant new features particularly in the energy spectrum for which the q boson lattice model has an optical branch and bound state spectra as well as the one acoustic branch of the quantum lattice nls model it is argued that this sort of distinction is generic for discrete systems with a finite number of degrees of freedom as opposed to continuum field theories like the bose gas where quantization by quantum group quantization or by canonical quantization leads to the same results various properties of the q bosons are given and used in particular a q boson primakov holstein transformation is given which connects the q boson lattice model with the xxz models of arbitrary spin

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

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

optimized meshfree methods for acoustics abstract when the helmholtz equation is solved by numerical methods as e g the finite element method fem the solution suffers from the so called pollution effect the pollution is mainly caused by the dispersion meaning that the numerical wave number disagrees with the wave number of the exact solution this leads to inaccurate results especially for high wave numbers in order to obtain acceptable results also for higher wave numbers either a high element resolution or elements of a higher order can be used for either option the consequence is an increased computation time and memory capacity meshfree methods as the element free galerkin method efgm and the radial point interpolation method rpim are not dispersion free either but it has been shown that meshfree methods are able to reduce the dispersion significantly both methods offer several parameters which can be modified in order to obtain optimal results with respect to the dispersion effect this work presents an exhaustive parameter study on both the efgm and the rpim it is shown that the methods can be significantly improved if certain parameters as e g weighting functions shape parameters size of the influence domain are chosen appropriately

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

ordering relations for q boson operators continued fraction techniques and the q cbh enigma ordering properties of boson operators have been very extensively studied and q analogues of many of the relevant techniques have been derived these relations have far reaching physical applications and at the same time provide a rich and interesting source of combinatorial identities and of their g analogues an interesting exception involves the transformation from symmetric to normal ordering which for conventional boson operators can most simply be effected using a special case of the campbell baker hausdorff cbh formula to circumvent the lack of a suitable q analogue of the cbh formula two alternative procedures are proposed based on a recurrence relation and on a double continued fraction respectively these procedures enrich the repertoire of techniques available in this field for conventional bosons they result in an expression that coincides with that derived using the cbh formula

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

quantum 401 1401 1401 1algebras and macdonald polynomials we derive a quantum deformation of thew n algebra and its quantum miura transformation whose singular vectors realize the macdonald polynomials

quantum groups and generalized statistics in integrable models the paper deals with the integrable massive models of quantum field theory it is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups this invariance provides the possibility to construct quasi local operators parafermions possessing generalized statistics which interpolates the physical particles for the particular examples of sg rsg models and scaling 3 state potts model the parafermions are described completely all their matrix elements in the space of states are presented

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

theta function solutions of the quantum knizhnik zamolodchikov bernard equation for a face model we consider the quantum knizhnik zamolodchikov bernard equation for a face model with elliptic weights the sos model we provide explicit solutions as theta functions on the so called combinatorial line in which the model is equivalent to the three colour model these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions

bosonization of vertex operators for the face model we present a free boson realization of the vertex operators and their duals for the solvable sos lattice model of type we discuss a possible connection with the calculation of correlation functions

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

a general formula for soliton form factors in the quantum sine gordon model a general formula for soliton form factors of several local fields in the quantum sine gordon model is presented

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

connection problems for quantum affine kz equations and integrable lattice models cherednik attached to an affine hecke algebra module a compatible system of difference equations called quantum affine knizhnik zamolodchikov kz equations in the case of a principal series module we construct a basis of power series solutions of the quantum affine kz equations relating the bases for different asymptotic sectors gives rise to a weyl group cocycle which we compute explicitly in terms of theta functions

one dimensional dispersion analysis for the element free galerkin method for the helmholtz equation the standard finite element method fem is unreliable to compute approximate solutions of the helmholtz equation for high wave numbers due to the dispersion unless highly refined meshes are used leading to unacceptable resolution times the paper presents an application of the element free galerkin method efg and focuses on the dispersion analysis in one dimension it shows that if the basis contains the solution of the homogenized helmholtz equation it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods for the general case it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion finally theoretical developments are validated by numerical experiments showing that for the same distribution of nodes the element free galerkin method solution is much more accurate than the finite element one copyright 2000 john wiley sons ltd

an edge based smoothed finite element method es fem for static free and forced vibration analyses of solids abstract this paper presents an edge based smoothed finite element method es fem to significantly improve the accuracy of the finite element method fem without much changing to the standard fem settings the es fem can use different shape of elements but prefers triangular elements that can be much easily generated automatically for complicated domains in the es fem the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles intensive numerical results demonstrated that the es fem possesses the following excellent properties 1 the es fem model possesses a close to exact stiffness it is much softer than the overly stiff fem and much stiffer than the overly soft ns fem model 2 the results are often found superconvergence and ultra accurate much more accurate than the linear triangular elements of fem and even more accurate than those of the fem using quadrilateral elements with the same sets of nodes 3 there are no spurious non zeros energy modes found and hence the method is also temporally stable and works well for vibration analysis and 4 the implementation of the method is straightforward and no penalty parameter is used and the computational efficiency is better than the fem using the same sets of nodes in addition a novel domain based selective scheme is proposed leading to a combined es ns fem model that is immune from volumetric locking and hence works very well for nearly incompressible materials these properties of the es fem are confirmed using examples of static free and forced vibration analyses of solids

an improved partition of unity finite element model for diffraction problems the partition of unity finite element method pufem is explored and improved to deal with practical diffraction problems efficiently the use of plane waves as an external function space allows an efficient implementation of an approximate exterior non reflective boundary condition improving the original proposed by higdon for general diffraction problems a virtually analytical integration procedure is introduced for multi dimensional high frequency problems which exhibits a dramatic decrease in the number of operations for a given error compared with standard integration methods suitable conjugate gradient type solvers for the whole range of wavenumbers are used including such cases in which pufem can produce nearly singular matrices caused by round off limits copyright 2001 john wiley sons ltd

assessment and applications of point interpolation methods for computational mechanics two meshfree point interpolation methods pims which are based on the polynomial and the radial basis functions have been proposed recently in addition to the earlier work with the moving least squares mls approximation for the field function approximation however it is found that pims cannot automatically ensure the compatibility of the solution when they are used together with the energy principles in this paper issues related to the compatibility of pims are studied a technique of background cell based nodal selections and a penalty method are proposed to enforce the compatibility of the solution of pims the patch test is studied in great detail the convergences and performances are investigated for both conforming and non conforming pims it is found that those methods of the pim family are very easy to implement and are very efficient in obtaining numerical solutions for problems of computational mechanics copyright 2004 john wiley sons ltd

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

a residual based finite element method for the helmholtz equation a new residual based finite element method for the scalar helmholtz equation is developed this method is obtained from the galerkin approximation by appending terms that are proportional to residuals on element interiors and inter element boundaries the inclusion of residuals on inter element boundaries distinguishes this method from the well known galerkin least squares method and is crucial to the resulting accuracy of this method in two dimensions and for regular bilinear quadrilateral finite elements it is shown via a dispersion analysis that this method has minimal phase error numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods it is found that even for unstructured meshes this method retains a high level of accuracy copyright 2000 john wiley sons ltd

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

quantum scr w algebras and elliptic algebras we define a quantum open image in new window algebra associated to mathfrak s mathfrak l n as an associative algebra depending on two parameters for special values of the parameters this algebra becomes the ordinary open image in new window algebra of mathfrak s mathfrak l n or theq deformed classical open image in new window algebra algebra of mathfrak s mathfrak l n we construct free field realizations of the quantum open image in new window algebra and the screening currents we also point out some interesting elliptic structures arising in these algebras in particular we show that the screening currents satisfy elliptic analogues of the drinfeld relations in open image in new window

time space domain mesh free finite difference based on least squares for 2d acoustic wave modeling abstractfinite difference fd methods approximate derivatives through a weighted summation of function values from neighboring nodes traditionally these neighboring nodes are assumed to be distributed regularly such as in square or rectangular lattices to improve geometric flexibility one option is to develop fd in a mesh free discretization in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra etc these mesh free fds have had successful applications especially in computational geoscience however they are all space domain fd schemes in which fd coefficients are derived by approximating spatial derivatives individually in the space domain for acoustic wave modeling it has been proven that space domain fd methods normally have higher dispersion error than time space domain fd methods which determine fd coefficients by approximating the time space domain disp

a new hierarchical partition of unity formulation of efg meshless methods abstract meshless methods have shown increased accuracy and better convergence rates compared to other well known simulation methods in a variety of computational mechanics problems their computational cost is relatively higher especially in adaptivity analysis when new nodes are inserted at each step where the construction of updated shape functions requires a significant amount of computational effort in this paper two hierarchical formulations are proposed in the context of an h type refinement scheme the addition of new nodes and subsequently the re calculation of the influenced moment matrices that are necessary for obtaining the shape functions and their derivatives and subsequently for the construction of the stiffness matrix are properly addressed both hierarchical schemes do not need the recalculation of the initial stiffness matrix but only the additional node contributions to each shape function field while the second scheme produces purely hierarchically refined stiffness matrices leaving the initial stiffness matrix unmodified

path integrals for quantum algebras and the classical limit coherent states path integral formalism for the simplest quantum algebras q oscillator suq 2 and suq 1 1 is introduced in the classical limit canonical structure is derived with a modified symplectic and riemannian metric non constant deformation induced curvature for the phase spaces is obtained

higher spin polynomial solutions of quantum knizhnik zamolodchikov equation we provide explicit formulae for highest weight to highest weight correlation functions of perfect vertex operators of u q widehat mathfrak sl 2 in arbitrary integer level l they are given in terms of certain macdonald polynomials we apply this construction to the computation of the ground state of higher spin vertex models spin chains spin l 2 xxz or loop models in the root of unity case q e i pi ell 2

assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics several shape function families currently used in the spectral element and the finite element method fem are reviewed widely known shape functions such as the conventional lagrange functions various p fem shapes and recent developments in the area of spectral methods are compared especially shape functions derived from bernstein polynomials are discussed due to their ease of formulation for almost any geometric entity their applicability and efficiency for acoustic simulations are assessed in particular the efficiency is measured in terms of performance of current krylov solvers

an edge based smoothed finite element method es fem for analyzing three dimensional acoustic problems abstract it is well known that one key issue of solving the helmholtz equation using finite element method fem is the accuracy deterioration in the solution with increasing wave number due to the numerical dispersion error such a numerical dispersion error is essentially caused by the overly stiff nature of the fem model to overcome this problem this paper presents an edge based smoothed finite element method es fem for analyzing acoustic problems using linear triangular and tetrahedron elements that can be generated automatically respectively for complicated two dimensional and three dimensional domains the discretized linear system equations for es fem are established using the smoothed galerkin weak form with smoothing domains associated with the edges of the triangles or surfaces of the tetrahedrons the edge based gradient smoothing operation provides proper softening effect makes the es fem model much softer than the overly stiff fem model and hence significantly reduces the numerical dispersion error numerical examples including a 2d problem of acoustic pressure distribution in a vehicle passenger compartment and a 3d problem about the acoustic pressure distribution in an engine chamber have been studied using the present es fem the results demonstrate that the es fem possesses the following advantages compared with the standard fem using the same meshes first es fem achieves similar convergence rate but better accuracy especially at high frequency second es fem is less sensitive to the mesh distortion meaning that the quality of mesh has less effect on the solution of es fem third it works well for triangular types of meshes and thus for the problems with complicated geometry

connection coefficients for basic harish chandra series basic harish chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems the associated connection coefficients are explicitly computed in terms of jacobi theta functions we interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of cherednik s root system analogs of the quantum knizhnik zamolodchikov equations they thus give rise to explicit elliptic solutions of root system analogs of dynamical yang baxter and reflection equations applications to quantum c functions basic hypergeometric functions reflectionless difference operators and multivariable baker akhiezer functions are discussed

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

maximum entropy methods for time harmonic acoustics abstract this paper explores the application of maximum entropy methods max ent to time harmonic acoustic problems max ent basis functions are mesh free approximants that are constructed observing an equivalence between basis functions and discrete probability distributions and applying jaynes s maximum entropy principle they are c continuous and therefore they are particularly suited for the resolution of helmholtz problems where classical finite element methods show a poor accuracy in the high frequency region in addition it was recently shown that max ent approximants can be blended with isogeometric basis functions on the boundary of the domain this preserves the correct representation of the boundary like in isogeometric analysis with the advantage that the discretization of the interior of the domain is straightforward in this paper the max ent mathematical formulation is reviewed and then some numerical applications are studied including a 2d car cavity geometry defined by b spline curves in all cases if the same nodal discretization is used finite elements results are significantly improved

diffusions on braided spaces the notion of q brownian motion introduced by majid is extended to braided spaces corresponding to a generic r matrix and combined with the theory of quantum probability this leads to a definition of diffusions on these spaces the corresponding heat equations difference differential equations are solved in terms of appell polynomials i e shifted moment systems some examples of interest for applications are given

new formulae for solutions of quantum knizhnik zamolodchikov equations on level 4 we present a new form of solution to the quantum knizhnik zamolodchikov equation qkz on level 4 in a special case corresponding to the heisenberg xxx spin chain our form is equivalent to the integral representation obtained by jimbo and miwa in 1996 7 an advantage of our form is that it is reduced to the product of single integrals this fact is deeply related to a cohomological nature of our formulae our approach is also based on the deformation of hyper elliptic integrals and their main property deformed riemann bilinear relation jimbo and miwa also suggested a nice conjecture which relates solution of the qkz on level 4 to any correlation function of the xxx model this conjecture together with our form of solution to the qkz makes it possible to prove a conjecture that any correlation function of the xxx model can be expressed in terms of the riemann function at odd arguments and rational coefficients suggested in 8 9 this issue will be discussed in our next publication

a q analogue of bargmann space and its scalar product a q analogue of bargmann space is defined using the properties of coherent states associated with a pair of q deformed bosons the space consists of a class of entire functions of a complex variable z and has a reproducing kernel on this space the q boson creation and annihilation operators are represented as multiplication by z and q differentiation with respect to z respectively a q integral analogue of bargmann s scalar product is defined involving the q exponential as a weight function associated with this is a completeness relation for the q coherent states

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

free field construction for the abf models in regime ii the wakimoto construction for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 admits a reduction to the q deformed parafermion algebras we interpret the latter theory as a free field realization of the andrews baxter forrester models in regime ii we give multi particle form factors of some local operators on the lattice and compute their scaling limit where the models are described by a massive field theory with mathbb z k symmetric minimal scattering matrices

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

quantum group symmetries and non local currents in 2d qft we construct and study the implications of some new non local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions including the sine gordon theory and its generalization to affine toda theory these non local currents provide a non perturbative formulation of the theories the symmetry algebras correspond to the quantum affine kac moody algebras thes matrices are completely characterized by these symmetries formals matrices for the imaginary coupling affine toda theories are thereby derived the application of theses matrices to perturbed coset conformal field theory is studied non local charges generating the finite dimensional quantum group in the liouville theory are briefly presented the formalism based on non local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d

particle field duality and form factors from vertex operators using a duality between the space of particles and the space of fields we show how one can compute form factors directly in the space of fields this introduces the notion of vertex operators and form factors are vacuum expectation values of such vertex operators in the space of fields the vertex operators can be constructed explicitly in radial quantization furthermore these vertex operators can be exactly bosonized in momentum space we develop these ideas by studying the free fermion point of the sine gordon theory and use this scheme to compute some form factors of some non free fields in the sine gordon theory this work further clarifies earlier work of one of the authors and extends it to include the periodic sector

element free galerkin solutions for helmholtz problems fomulation and numerical assessment of the pollution effect abstract the element free galerkin method efgm a particular case of the meshless methods is examined in its application to acoustic wave propagation addressed by the helmholtz equation dispersion and pollution effects two problems encountered by the classical numerical methods are reviewed numerical tests on two dimensional problems focus on the parameters governing the formulation of the efgm they also demonstrate that the efgm is affected by dispersion and pollution effects as well as fem but these effects are rather low showing that the efgm is a promising method

dispersion and pollution of the fem solution for the helmholtz equation in one two and three dimensions for high wave numbers the helmholtz equation suffers the so called pollution effect this effect is directly related to the dispersion a method to measure the dispersion on any numerical method related to the classical galerkin fem is presented this method does not require to compute the numerical solution of the problem and is extremely fast numerical results on the classical galerkin fem p method is compared to modified methods presented in the literature a study of the influence of the topology triangles is also carried out the efficiency of the different methods is compared the numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well the most effective modified method is the qsfem 1 2 but it is also very complicated in the general setting the residual free bubble 3 4 is effective in one dimension but not in higher dimensions the least square method 1 5 approach lowers the dispersion but relatively little the results for triangular meshes show that the best topology is the criss cross pattern copyright 1999 john wiley sons ltd

correlation functions of disorder fields and parafermionic currents in zn ising models we study the correlation functions of parafermionic currents and disorder fields in the zn symmetric conformal field theory perturbed by the first thermal operator following the ideas of al zamolodchikov zamolodchikov 1991 nucl phys b 348 619 zamolodchikov 1990 nucl phys b 342 695 and zamolodchikov 1995 int j mod phys a 10 1125 we develop for the correlation functions the conformal perturbation theory at small scales and the form factors spectral decomposition at large ones for all n there is an agreement between the data at the intermediate distances we consider the problems arising in the description of the space of scaling fields in perturbed models such as null vector relations equations of motion and a consistent treatment of fields related by a resonance condition

determinant representation for a quantum correlation function of the lattice sine gordon model we consider a completely integrable lattice regularization of the sine gordon model with discrete space and continuous time we derive a determinant representation for a correlation function which in the continuum limit turns into the correlation function of local fields the determinant is then embedded into a system of integrable integro differential equations the leading asymptotic behaviour of the correlation function is described in terms of the solution of a riemann hilbert problem rhp related to the system of integro differential equations the leading term in the asymptotical decomposition of the solution of the rhp is obtained

on the optimal shape parameters of radial basis functions used for 2 d meshless methods abstract a radial point interpolation meshless or radial pim method was proposed by authors to overcome the possible singularity associated with only polynomial basis the radial pim used multiquadric mq or gaussian as basis functions these two radial basis functions all included shape parameters although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed how these shape parameters affect the accuracy of the radial pim has not been studied yet this paper studied the effect of shape parameters on the numerical accuracy of radial pim a range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix error of energy and irregularity of node distribution it is observed that the widely used shape parameters for mq and reciprocal mq basis are not even close to their optimums the optimal shape parameters are found in this paper to be simply q 1 03 and r 1 42 for mq basis and c 0 003 0 03 for gaussian basis

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

vertex operators in solvable lattice models the basic properties of q vertex operators are formulated in the context of the andrews baxter forrester abf series as an example of face interaction models the q difference equations satisfied by their correlation functions are derived and their connection with representation theory established the q difference equations of the kashiwara miwa km series are discussed as an example of edge interaction models next the ising model the simplest special case of both abf and km series is studied in more detail using the jordan wigner fermions in particular all matrix elements of vertex operators are calculated

robustness and dispersion analysis of the partition of unity finite element method applied to the helmholtz equation abstract the partition of unity finite element method pufem has been widely used for the numerical simulation of the helmholtz equation in different physical settings in fact it is a numerical pollution free alternative method to the classical piecewise polynomial based finite element methods taking into account a plane wave enrichment of the piecewise linear finite element method the main goal of this work is focused on the derivation of the numerical dispersion relation and the robustness analysis of the pufem discretization when a spurious perturbation is presented in the wave number value used in the enrichment definition from the one dimensional helmholtz equation the discrete wave number is estimated based on a bloch s wave analysis and a priori error estimates are computed explicitly in terms of the mesh size the wave number and the perturbation value