explicit evaluation of coupling coefficients for the most degenerate representations of so n explicit results for particular coupling coefficients for the most degenerate representations of so n are given the isoscalar factors which allow a recursive calculation of arbitrary coupling coefficients from those of so 3 are derived 6j and 9j symbols for most degenerate representations are also briefly discussed

more on coupling coefficients for the most degenerate representations of so n we present explicit closed form expressions for the general group theoretical factor appearing in the topology of a high temperature expansion of so n symmetric lattice models this object which is closely related to 6j symbols for the most degenerate representation of so n is discussed in detail

a phase based interior penalty discontinuous galerkin method for the helmholtz equation with spatially varying wavenumber this paper is concerned with an interior penalty discontinuous galerkin ipdg method based on a flexible type of non polynomial local approximation space for the helmholtz equation with varying wavenumber the local approximation space consists of multiple polynomial modulated phase functions which can be chosen according to the phase information of the solution we obtain some approximation properties for this space and a priori l2l2 error estimates for the h convergence of the ipdg method using duality argument we also provide ample numerical examples to show that building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces

a least squares non polynomial finite element method for solving the polygonal line grating problem abstract this paper is concerned with the diffraction by a polygonal line grating using the fourier bessel functions that capture the singularities at corners and representing the diffraction field towards infinity by a truncation of the rayleigh expansion we propose a least squares finite element method for solving the problem we analyze the convergence and give an error estimate for the method numerical experiments are also presented to show the effectiveness of our method

high temperature expansions for classical spin models iii series and phase diagrams for the square lattice abstract the concepts of duality and broken symmetry are discussed briefly it is shown how generalized colouring polynomials can be used to calculate the terms in a high temperature expansion effectively these results and those from two previous papers are used to obtain high temperature series for a great variety of models these are given explicitly for all models with two boltzmann factors and for the ashkin teller model many phase diagrams obtained from the series by pade analysis are presented and their salient features discussed a principle suggested by these phase diagrams is applied to the dilute potts model

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

high temperature expansions for classical spin models ii the square and hypercubic lattices and the potts model abstract all graphs and their contributions needed for a high temperature expansion of the partition function of a completely permissible spin model up to order 16 for the square lattice and up to order 11 for the hypercubic lattice are listed for the potts model the corresponding series are derived and analyzed by means of pade approximants in the square lattice case the phase transition of the m state potts model is reproduced to within 1 for m 2

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

a multi level wave based numerical modelling framework for the steady state dynamic analysis of bounded helmholtz problems with multiple inclusions the wave based method wbm is an alternative numerical prediction method for both interior and exterior steady state dynamic problems which is based on an indirect trefftz approach it applies wave functions which are exact solutions of the governing differential equation to describe the dynamic field variables the smaller system of equations and the absence of pollution errors make the wbm very suitable for the treatment of helmholtz problems in the mid frequency range where element based methods are no longer feasible due to the associated computational costs a sufficient condition for convergence of the method is the convexity of the considered problem domain as a result only problems of moderate geometrical complexity can be considered and some geometrical features cannot be handled at all in this paper these limitations are alleviated through the development of a general modelling framework based on existing wbm methodologies which allows for the efficient introduction of inclusion configurations in bounded wbm models for problems governed by one or more helmholtz equations the feasibility and efficiency of the method is illustrated by means of numerical verification studies in which the methodology is applied to two types of dynamic problems on the one hand a single helmholtz equation associated with the steady state dynamic behaviour of acoustic cavities is studied on the other hand the framework is applied to the solution of the navier system of partial differential equations that describe the elastodynamic response of two dimensional perforated solids

numerical treatment of acoustic problems with boundary singularities by the singular boundary method abstract the singular boundary method sbm is a novel boundary type meshless method based on the fundamental solution of the given governing equation the sbm employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions in this paper we investigate the acoustic problems with boundary singularities using the sbm this is achieved by combining the sbm with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution the singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities then the regular solution is obtained by the sbm numerical examples show the excellent performance of the proposed technique

an efficient wave based method for 2d acoustic problems containing corner singularities abstract the wave based method is an efficient alternative prediction technique based on a trefftz approach which as compared to element based methods can be applied in a wider frequency range at a reasonable cost this paper discusses the use of this method for the particular case of 2d acoustic problems where singularities are present in the corners of the acoustic domain the conventional set of expansion functions used to approximate the dynamic pressure field is extended with special purpose functions that incorporate the singular behaviour in the region around the singular corners this enrichment comes with a beneficial convergence rate while only a small increase of computational effort is incurred this is illustrated for various problem settings including different combinations of both homogeneous and non homogeneous boundary conditions and bounded as well as unbounded problems are considered

a least squares method for the helmholtz equation abstract we investigate the use of least squares methods to approximate the helmholtz equation the basis used in the discrete method consists of solutions of the helmholtz equation either consisting of plane waves or bessel functions on each element of a finite element grid unlike previous methods of this type we do not use polynomial based finite elements the use of small elements and relatively few basis functions per element allows us to prove convergence theorems for the method and to some extent control the conditioning of the resulting linear sy stem numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

convergence analysis for finite element discretizations of the helmholtz equation with dirichlet to neumann boundary conditions a rigorous convergence theory for galerkin methods for a model helmholtz problem in bbb r d d in 1 2 3 is presented general conditions on the approximation properties of the approximation space are stated that ensure quasi optimality of the method as an application of the general theory a full error analysis of the classical hp version of the finite element method is presented for the model problem where the dependence on the mesh width h the approximation order p and the wave number k is given explicitly in particular it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

four and eight node hybrid trefftz quadrilateral finite element models for helmholtz problem in this paper four and eight node quadrilateral finite element models which can readily be incorporated into the standard finite element program framework are devised for plane helmholtz problems in these models frame boundary and domain approximations are defined the former is obtained by nodal interpolation and the latter is truncated from trefftz solution sets the equality of the two approximations are enforced along the element boundary both the bessel and plane wave solutions are employed to construct the domain approximation for full rankness a minimal of four and eight domain modes are required for the four and eight node elements respectively by using local coordinates and directions rank sufficient and invariant elements with minimal and close to minimal numbers of domain approximation modes are devised in most tests the proposed hybrid trefftz elements with the same number of nodes yield close solutions in absolute majority of the tests the proposed elements are considerably more accurate than their single field counterparts

least squares trefftz type elements for the helmholtz equation trefftz type elements or t elements are finite elements the internal field of which fulfills the governing differential equations of the problem a priori whereas the prescribed boundary conditions and the interelement continuity must be enforced by some suitable method in this paper the relevant matching is achieved by means of a least squares procedure the so called frameless or least squares t elements for helmholtz s equation related to the scattering of waves by offshore structures in 2 d are developed and studied the required accuracy of the solution can be obtained by increasing the number of either the subdomains or t functions which can be regarded as the h or p type approach respectively convergence studies are performed with much attention to the use of special purpose elements for a doubly connected domain with a circular hole and for an angular corner subdomain the most attractive features of the presented formulation are its simplicity and robustness the matrix of the resulting linear system is always hermitian and positive definite 1998 john wiley sons ltd

a comparison of wave based discontinuous galerkin ultra weak and least square methods for wave problems several numerical methods using non polynomial interpolation have been proposed for wave propagation problems at high frequencies the common feature of these methods is that in each element the solution is approximated by a set of local solutions they can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation there are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated the similarities and discrepancies between several non polynomial numerical methods are discussed in the context of the helmholtz equation the present comparison is concerned with the ultra weak variational formulation uwvf the least squares method lsm and the discontinuous galerkin method with numerical flux dgm an analysis in terms of trefftz methods provides an interesting insight into the properties of these methods second the uwvf and the lsm are reformulated in a similar fashion to that of the dgm this offers a unified framework to understand the properties of several non polynomial methods numerical results are also presented to put in perspective the relative accuracy of the methods the numerical accuracies of the methods are compared with the interpolation errors of the wave bases

a comparison of two trefftz type methods the ultraweak variational formulation and the least squares method for solving shortwave 2 d helmholtz problems trefftz methods for the numerical solution of partial differential equations pdes on a given domain involve trial functions which are defined in subdomains are generally discontinuous and are solutions of the governing pde or its adjoint within each subdomain the boundary conditions and matching conditions between subdomains must be enforced separately an interesting novel result presented in this paper is that the least squares method lsm and the ultraweak variational formulation two methods already established for solving the helmholtz equation can be derived in the framework of the trefftz type methods in the first case the boundary conditions and interelement continuity are enforced by means of a least squares procedure in the second a galerkin type weighted residual method is used another goal of the work is to assess the relative efficiency of each method for solving shortwave problems in acoustics and to study the stability of each method the numerical performance of each scheme is assessed with reference to two 2 d test problems acoustic propagation in an uniform soft walled duct and propagation in an l shaped domain the latter involving singular behaviour at a sharp corner

on matching of intuitionistic fuzzy sets abstract in the paper we describe the new measure of matching two intuitionistic fuzzy sets the operations known in the intuitionistic fuzzy set theory are used and the perturbation of one intuitionistic fuzzy set by another intuitionistic fuzzy set is understood as an impact of one intuitionistic fuzzy set on another intuitionistic fuzzy set the opposite case can also be considered wherein the second intuitionistic fuzzy set is perturbed by the first one the introduced measure of the perturbation of one intuitionistic fuzzy set by another intuitionistic fuzzy set is considered instead of commonly used distance between two intuitionistic fuzzy sets in general the new measure can be asymmetric and therefore can provide more information compare to a distance between intuitionistic fuzzy sets the values of such measures of intuitionistic fuzzy sets perturbation are ranged between 0 and 1 in this paper specific mathematical properties of the measure of intuitionistic fuzzy sets perturbation are studied the presented methodology is explained by several illustrative examples

an approach to dimensionality reduction in time series many methods of dimensionality reduction of data series time series have been introduced over the past decades some of them rely on a symbolic representation of the original data however in this case the obtained dimensionality reduction is not substantial in this paper we introduce a new approach referred to as symbolic essential attributes approximation seaa to reduce the dimensionality of multidimensional time series in such a way we form a new nominal representation of the original data series the approach is based on the concept of data series envelopes and essential attributes generated by a multilayer neural network the real valued attributes are discretized and in this way symbolic data series representation is formed the seaa generates a vector of nominal values of new attributes which form the compressed representation of original data series the nominal attributes are synthetic and while not being directly interpretable they still retain important features of the original data series a validation of usefulness of the proposed dimensionality reduction is carried out for classification and clustering tasks the experiments have shown that even for a significant reduction of dimensionality the new representation retains information about the data series sufficient for classification and clustering of the time series

generalized finite element analysis with t complete boundary solution functions the use of a complete and nonsingular set of trefftz functions in the solution of quasi harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary type approximation procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated

explicit thermostatics of stanley s n vector model on the harmonic chain by fourier analysis thermostatic properties of stanley s classical n vector model on the harmonic chain are studied it is shown that this model belongs to a rather general class of classical spin models in one dimension the members of this class are characterized by spins which are elements of a homogenous space with transformation group g and a g invariant and exchange invariant spin pair interaction for the derivation of basic thermostatic quantities and correlation functions we use the method of abstract fourier analysis this allows to derive rather explicit results for any member of the aforementioned class of spin models we present an exact closed form expression as well as a high and a low temperature expansion for the free energy of stanley s n vector model on the harmonic chain from this basic thermostatic quantities like internal energy entropy and heat capacity are obtained furthermore we present results for expectation values of one spin and two spin functions from the latter it is possible to derive the zero field susceptibility and it also allows for a discussion of magnetostrictive effects

high temperature expansions for classical spin models abstract it is shown that the transitivity of the symmetry group of a classical spin model leads to a high temperature expansion of the free energy of such a model in terms of multiply connected graphs for models with a completely permissible symmetry group e g for ising and potts models these graphs fall into equivalence classes such classes are classified according to their cyclomatic index several different methods for obtaining the high temperature series in a practical way are also discussed the extension of these results to models with an infinite symmetry group is indicated

computational aspects of the ultra weak variational formulation the ultra weak variational formulation uwvf approach has been proposed as an effective method for solving helmholtz problems with high wave numbers however for coarse meshes the method can suffer from instability in this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous helmholtz problem we introduce a method to improve the uwvf scheme and we compare iterative solvers for the resulting linear system computations for the acoustic transmission problem in 2d show that the new approach enables helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

hybrid trefftz displacement element for spectral analysis of bounded and unbounded media the hybrid trefftz displacement element is applied to the elastodynamic analysis of bounded and unbounded media in the frequency domain the displacements are approximated in the domain of the element using local solutions of the wave equation the neumann conditions are enforced directly and the surface forces are approximated on the dirichlet and inter element boundaries of the finite element mesh two alternative elements are developed to model unbounded media namely a finite element with absorbing boundaries and an unbounded element that satisfies explicitly the sommerfeld condition the finite element equations are derived from the fundamental relations of elastodynamics written in the frequency domain the numerical implementation of these equations is discussed and numerical tests are presented to assess the performance of the formulation

an accurate algorithm for computing the eigenvalues of a polygonal membrane abstract for a polygonal domain we consider the eigenvalue problem u u 0 in u 0 on the boundary of is decomposed into subdomains 1 2 on each i u is approximated by a linear combination of functions which satisfy the equation u u 0 and continuity conditions are imposed at the boundaries of the subdomains we propose a non conventional method based on the use of a rayleigh quotient we present numerical examples and a proof of the exponential convergence of the algorithm