a construction of f 1 as automorphisms of a 196 883 dimensional algebra abstract in this note i announce the construction of the finite simple group f1 whose existence was predicted independently in 1973 by bernd fischer and by me the group has order 246320597611213317 19 23 29 31 41 47 59 71 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 and is realized as a group of automorphisms of a 196 883 dimensional commutative nonassociative algebra over the rational numbers which has an associative form equivalently it is a group of automorphisms of a cubic form in 196 883 variables it turns out that all the relevant arguments and calculations may be done by hand furthermore existence of the group f1 implies the existence of a number of other sporadic simple groups for which existence proofs formerly depended on work with computers we are beginning to look upon this group as a friendly giant

calculation of acoustic radiation using equivalent sphere methods among the methods generally used to solve a problem in the domain of acoustic radiation the equivalent sources method offers an interesting alternative it consists in replacing the vibrating surface with a distribution of acoustic sources placed inside the structure the contribution of each source is determined in such a way that the acoustic field radiated by these sources verifies the same boundary conditions on the structure the number of unknowns in the problem is no longer directly linked to the number of mesh points on the structure as with boundary elements methods but to the number of equivalent sources employed in the model the equivalent source method is therefore of major interest if the acoustic radiation of the structure can be approximated with a sufficiently low number of sources this paper proposes its application when the equivalent source is a sphere in this case the number of unknowns is equal to the number of modes in contrast to the one point multipole the sphere has a surface surrounding a closed volume to express the boundary conditions although sphere multipole equivalence has been demonstrated the surface of the sphere allows normalization of the functions used leading to stabilization of the system to be resolved first the main acoustic radiation characteristics of a sphere and of the linear system verified by the modal coefficients of an equivalent sphere are presented the different parameters of the model are then studied position and radius of the equivalent sphere truncation of the series and influence of the spatial sampling mesh in the same vein a second approach is presented it consists of making each point of the structure correspond to a point of the sphere and the vibrating field at the surface of the sphere is deduced from that of the structure by simple geometric projection results can be obtained very quickly as no matrix inversion is required the accuracy of the results depends on the distance between the sphere and the structure finally an experimental validation that uses both methods is presented and shows interesting results when the structure is closed and when its shape is not too far removed from a sphere

an effective method for solving the hyper singular integral equations in 3 d acoustics the application of the boundary integral methods to the problem of acoustics exterior to a three dimensional surface suffers in general from the nonexistence of nonuniqueness of its solutions at frequencies that are characteristic of the associated interior problem the formulation that is most suitable for numerical implementation still appears to be that proposed by burton and miller proc r soc london ser a 323 201 210 1971 however the hypersingular kernels present in such a formulation render it computationally unattractive previous attempts to regularize such hypersingular kernels involved the use of double surface integrals or implicit use of tangential operators or closed form evaluations of hypersingular integrals the method of double surface integrals is computationally highly inefficient even though it allows higher order interpolation schemes on the surface the other two approaches are more conducive to the assumption of a constant value for each of the variables pressure a

an even unimodular 72 dimensional lattice of minimum 8 an even unimodular 72 dimensional lattice having minimum 8 is constructed as a tensor product of the barnes lattice and the leech lattice over the ring of integers in the imaginary quadratic number field with discriminant 7 the automorphism group of contains the absolutely irreducible rational matrix group psl2 7 sl2 25 2

conformal generalized near field acoustic holography for axisymmetric geometries the normal velocity field on the boundary of a radiating source is reconstructed with high resolution by using pressure data measured in the very near field to include evanescent components fast decaying with the distance the measurement points belong to a surface conformal to the reconstruction is performed by identifying the boundary normal velocity distribution generating the pressure field best mean square fitting the measurement data the ensuing normal equations are solved by resorting to the singular value decomposition svd of the transformation from the boundary normal velocity to the pressure at the measurement points the reconstructed boundary normal velocity is represented as a linear combination of basis functions each associated with a singular value sv the errors in the estimation of the coefficients of the linear combination are proportional to the inverses of the corresponding svs to achieve robustness the svd is truncated to retain the terms corresponding to a limited dynamic range in turn related to the dynamic range of the measurement system for axisymmetric but otherwise general surfaces the numerical complexity is reduced because separate svds of greatly reduced dimensionality must be performed on a number of matrices each associated with one of the harmonics of the fourier expansion of the field around the symmetry axis in a numerical experiment making use of an available computer model by processing very near field pressure samples the normal velocity was reconstructed with excellent accuracy on the surface of a finite compliant cylinder with flat endcaps excited by point forces the viability of the method was then confirmed in a measurement tank experiment in which the distribution of the boundary normal velocity was reconstructed on a finite compliant cylinder driven by point shakers comparison with accelerometer data shows good agreement on a wide frequency band

the use of chief to obtain unique solutions for acoustic radiation using boundary integral equations this report is concerned with the problem of obtaining a unique solution for radiation problems at characteristic wavenumbers when a boundary integral equation formulation is used it is shown that the combined helmholtz integral equation formulation chief method works even when numerous interior points lie exactly on nodal surfaces of a mode corresponding to an eigenfrequency of the related interior problem when at least one interior point does not lie on a nodal surface the use of chief is illustrated with numerical radiation experiments for the sphere and the finite cylinder chief is compared to the helmholtz gradient formulation hgf for circumventing nonuniqueness and is found to yield a more accurate solution in at least one published example a procedure is used to indicate the solution error when using integral equation methods with or without a technique to circumvent nonuniqueness this procedure uses the velocity potential of an interior point as an indicator of solution error in all cases considered the interior potential correctly indicated a good or bad solution whereas the matrix condition number falsely indicated a bad solution in several instances work supported in part by onr

on the holographic reconstruction of vibroacoustic fields using equivalent sources and inverse boundary element method in using the near field acoustical holography based on the inverse boundary element method bem for the reconstruction of vibroacoustic source parameters an enormous number of measurements required in practice have limited the extended application of this method to obtain the sufficient field data with a small number of actual measured data the regeneration method of the radiated field using the multipoint equivalent sources is attempted in this paper in using the equivalent source method a vibrating source can be represented by distributed spherical sources inside the actual source surface in this paper the radiated field is expressed with a series of spherical hankel functions and spherical harmonics for suppressing the adverse effect of high order spherical functions spatial filtering of coefficients and wave vector components by a regularization scheme is adopted restored field data appended with actual measured data can be used as input for the inverse bem to reconstruct the source field

on the uniqueness of solution for all wavenumbers in acoustic radiation the integral relations equivalent to helmholtz s equation with closed boundary conditions possess nonunique exterior solutions at eigenwavenumbers of the corresponding interior problems if the green s function for helmholtz s equation can be expanded into a series of orthogonal functions then an infinite set of integral equations at the boundary results these null field equations are shown to possess unique solutions for all wavenumbers furthermore these solutions can be used to represent the solutions in the exterior domain as algebraic series

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

development of an energy boundary element formulation for computing high frequency sound radiation from incoherent intensity boundary conditions an energy boundary element analysis ebea formulation is presented for calculating sound radiation at high frequency from a radiator with arbitrary shape the ebea development presented in this paper compliments previous energy finite element analysis efea developments for computing high frequency vibration of structures immersed in an acoustic medium from the efea solution the structural vibrational energy throughout the structure and the acoustic power radiated from each surface element in contact with the surrounding fluid are computed the ebea utilizes the information for the acoustic power radiated from each element of the outer surface of the structure as a boundary condition in order to compute the acoustic field in an unbound medium with no damping that surrounds the radiating structure since the structural vibration between different elements is considered as incoherent in the efea the acoustic intensity associated with each surface element that radiates in the surrounding medium is also considered as incoherent between different elements in the ebea formulation the governing integral equation of ebea is established and a numerical solution is presented the present method is validated by comparing ebea results to analytical solutions and to numerical results from an axi symmetric hp finite element code with infinite acoustic elements suitable for structural acoustic computations good correlation is demonstrated for all the analyses

the wave superposition method as a robust technique for computing acoustic fields the wave superposition method has recently been advocated koopmann et al a method for computing acoustic fields based on the principle of wave superposition j acoust soc am 86 2433 2438 1989 as a reliable and accurate technique for computing the acoustic fields generated by arbitrary shaped radiators this study examines in depth the robustness and numerical stability of the method the implementation of the method requires the placement of a finite number of point sources on a surface interior to the body of the radiator the magnitude of these point sources can be evaluated in terms of the prescribed velocity distribution on the surface of the radiator once calculated the magnitude of these point sources allow the pressure distribution exterior to the radiator to be evaluated the first improvement made to the robustness of the method is the use of a hybrid combination of monopole and dipole sources this modification allows the inherent problem of nonuniqueness of the method at certa

a weighted residual formulation for the chief method in acoustics in this article the combined helmholtz integral equation formulation chief proposed by schenck is greatly improved so that it will alleviate the annoying nodal surface problem the constraint provided by each chief equation and its first order derivatives is enforced in a weighted residual sense over a small interior region rather than only at a single point consequently chief blocks rather than chief points are used in the numerical implementation although the corresponding interior problem may have nodal surfaces at characteristic frequencies it has no such things as nodal blocks therefore the chief block chief b method is expected to work even if all of the chief blocks fall on the intersection of several nodal surfaces this new formulation is implemented in an isoparametric element environment numerical examples in spherical radiation and scattering at intermediate frequencies are given to test the new formulation

on efficient preconditioners for iterative solution of a galerkin boundary element equation for the three dimensional exterior helmholtz problem the paper presents a galerkin numerical method for solving the hyper singular boundary integral equations for the exterior helmholtz problem in three dimensions with a neumann s boundary condition previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and galerkin methods are not available due to the presence of a hypersingular integral operator this paper proposes a high order galerkin method by using singularity subtraction technique to reduce the hyper singular operator to a weakly singular one moreover we show here how to extend the previous work j appl numer math 36 4 2001 475 489 on sparse preconditioners to the galerkin case leading to fast convergence of two iterative solvers the conjugate gradient normal method and the generalised minimal residual method a comparison with the collocation method is also presented for the helmholtz problem with several wavenumbers

a lattice theoretical construction of a gab of the suzuki sporadic simple group geometries that are almost buildings or gabs are buekenhout tits geometries z in which all rank two residual geometries are generalized polygons except that they need not satisfy the intersection property 2 sect 6 or 51 there exist two known examples of finite gabs with flagtransitive automorphism groups and the diagram of afline type arising from sporadic simple groups in s kantor constructed one of them from the lyons sims group the other was associated with the suzuki sporadic simple group suz and was constructed by ronan and smith s as a subgeometry of a geometry of the conway group co 1 which is fixed by some element of order 3 of co 1 it seems that these geometries deserve further studies because they are quotients of certain infinite buildings as is shown by a theorem of tits 12 the main purpose of this paper is to provide a non group theoretical description of the above gab associated with suz using natural notions with respect to the complex leech lattice for this purpose we give a new presentation of the complex leech lattice a for its usual definition using the ternary golay code see 11 or 10 5 8 61 since suz is a factor group of the automorphism group of the lattice 1 this gives another existence proof for suz the subsidiary aim of this paper is to provide explicit and elementary proofs for some fundamental properties of suz which are based only on this presentation of a the details are as follows let r be the ring z w where w 1 fl 2 we first consider some r submodule r in r4 the complex e lattice see 2 1 and define the complex leech lattice n as an r submodule of r3 198 0021 8693 88 3 00

acoustic scattering by two spheres multiple scattering and symmetry considerations abstract acoustic scattering by two identical spheres is theoretically numerically and experimentally studied by highlighting the role of the symmetries of the scatterer incident and scattered fields are expanded over the different irreducible representations of d h the continuous symmetry group of the scatterer then from the boundary conditions one obtains for each irreducible representation an infinite system of linear complex algebraic equations where the unknown scattering coefficients are uncoupled this feature greatly simplifies the treatment of the problem and speeds up calculations farfield form functions are computed in the cases of neumann boundary conditions rigid spheres and elastic boundary conditions elastic spheres immersed in water a series of experiments based on ultrasonic spectroscopy is performed in the case of two stainless steel spheres immersed in water the comparison between the theoretical and the experimental results provides quite a good agreement

spurious and true eigensolutions of helmholtz bies and bems for a multiply connected problem the spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary element methods bems and circumvented by using the burton miller approach do they also occur in other formulations continuous formulations such as the singular and hypersingular boundary integral equations bies the null field bies and the fictitious bies or such discrete formulations as the null field bems and the fictitious bems for the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper by using the degenerate kernels and the fourier series it is demonstrated analytically for the six continuous formulations of bies that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary this conclusion can be extended to the six discrete formulations of bems to filter out the spurious eigenvalues the chief combined helmholtz integral equation formulation method is used here instead of the burton miller approach the optimum number and appropriate positions of the chief points are also addressed it is then shown that in the null field and fictitious bems the spurious and true eigenvalues can be detected and distinguished by using the singular value decomposition updating techniques in conjunction with the fredholm alternative theorem illustrative examples show the validity of the proposed methodologies

minimal vectors in the second exterior power of a lattice w x in c we studied the rankin k invariants which were defined by rankin w x in r as a natural generalization of the classical hermite invariant of a lattice in the following way let l be a lattice of maximal rank in an n dimensional euclidean space e we denote by d l the minimal k determinant of k dimensional sublattices of l and put g l s k k r n d l r det l the upper bound of g l for all lattices l in e is k k denoted by g n k it is readily verified that the k dimensional sublattice spanned by a k tuple of linearly independent vectors x x in l depends only on its 1 k image x n n x in the k th exterior power h l moreover h e 1 k inherits a canonical euclidean structure from the original scalar product on e which is defined for split multivectors that is vectors which can be written as x n n x by 1 k

evaluation of 2 d green s boundary formula and its normal derivative using legendre polynomials with an application to acoustic scattering problems this paper presents the non singular forms in a global sense of two dimensional green s boundary formula and its normal derivative the main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain that is the proposed element free method requires only nodal data the approach includes expressing the unknown function as a truncated fourier legendre series together with transforming the integration interval a b to 1 1 the series coefficients are thus to be determined the hypersingular integral interpreted in the hadamard finite part sense and some weakly singular integrals can be evaluated analytically the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point the effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions the regularization is further applied to acoustic scattering problems the well known burton miller method using a linear combination of the surface helmholtz integral equation and its normal derivative is adopted to overcome the non uniqueness problem a general non singular form of the composite equation is derived comparisons with analytical solutions for acoustically soft and hard circular cylinders are made copyright 2001 john wiley sons ltd

a boundary element method for acoustic radiation valid for all wavenumbers a boundary element method for solving the exterior acoustic radiation problem that is valid for all wavenumbers has been developed the new formulation derives from the earlier work of burton and miller proc r soc london ser a 323 201 210 1971 which uses a linear combination of the helmholtz integral equation and its gradient to overcome the uniqueness problem a main feature of this formulation is the absence of integral singularities by restricting the field points of the kernels of the integrals to interior regions well away from radiating boundaries the usual complication of dealing with singular kernels is circumvented the new formulation has been named chi for coupled helmholtz integrals a corresponding computer program chi 4 35 can be used to compute the surface acoustic pressure on an arbitrary body for a given specified surface normal velocity results generated with chi 4 35 are presented for the spherical monopole and dipole along with results for parametric studies on the acc

a fast and stable solver for acoustic scattering problems based on the nonuniform grid approach a fast and stable boundary element method bem algorithm for solving external problems of acoustic scattering by impenetrable bodies is developed the method employs the burton miller integral equation which provides stable convergence of iterative solvers and a generalized multilevel nonuniform grid mlng algorithm for fast evaluation of field integrals the mlng approach is used here for the removal of computational bottlenecks involved with repeated matrix vector multiplications as well as for the low order basis function regularization of the hyper singular integral kernel the method is used for calculating the fields scattered by large acoustic scatterers including nonconvex bodies with piece wise smooth surfaces as a result the algorithm is capable of accurately incorporating high frequency effects such as creeping waves and multiple edges diffractions in all cases stable convergence of the method is observed high accuracy of the method is demonstrated by comparison with the traditional b

application of the finite element method to two dimensional radiation problems acoustic fields radiated by vibrating elastic bodies immersed in an infinite fluid domain are in general quite difficult to compute this paper demonstrates in the two dimensional 2 d case that the radiated near field can be easily obtained using the finite element method if dipolar damping elements are attached to the mesh external circular boundary these elements are specifically designed to absorb completely the first two components of the asymptotic expansion of the radiated field then the paper provides a new extrapolation method to compute far field pressures from near field pressures using the 2 d helmholtz equation and its solution obeying the sommerfeld radiation condition these developments are valid for any radiation problem in 2d finally two test examples are described the oscillating cylinder of order m and a finite width planar source mounted in a rigid or a soft baffle this approach is the generalization to 2 d problems of a previously described approach devoted to axisymmetrical and three dimensional 3 d problems r bossut et al j acoust soc am 86 1234 1244 1989 it has been implemented in the atila code it is well suited to the modeling of high frequency transducers for imaging and nondestructive testing

a new boundary finite element method for fluid structure interaction problems in order to study problems on fluid structure interaction we have used a mixed formulation which couples the classical functional of the structure with a new variational formulation by integral equations for the fluid this formulation has the advantage over the finite element methods of avoiding the discretization of the fluid domain furthermore unlike collocation methods the explicit calculation of the hadamard finite part of the singular integrals is avoided this leads after discretization by boundary finite elements to a small and symmetrical algebraic system typical examples are presented that demonstrate the efficiency of this variational formulation by studying the sound transmission through a baffled plane structure and through a flexible panel backed by a rigid cavity these include the calculation of the transmission loss factor and the determination of which modes dominate the noise transmission good agreement is obtained between numerical results and analytical results found in the literature

orbifolds of lattice vertex operator algebras at d 48 and d 72 abstract motivated by the notion of extremal vertex operator algebras we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self dual lattices in d 48 and d 72 in this way we construct about one hundred new examples of holomorphic voas with a small number of low weight states

a simple method avoiding non uniqueness in the boundary element method for acoustic scattering problem the boundary element method bem is widely used for sound field analysis problems however it has a non uniqueness problem in the exterior domain various methods to avoid this problem have been developed however these are not easily applied to the bem in this paper a simple method called the ica ring inner cavity ringing method is proposed for avoiding the non uniqueness problem and this method is applied to the bem in both single and plural domains the concept of the ica ring method is that a scatterer in free space is hollowed as a shell and the volume is smaller the eigenfrequencies are shifted to a higher range next the mechanism of the non uniqueness problem in plural domains and a reason of the application of the ica ring method to the case of plural domains are explained finally some results calculated by the bem using the ica ring method are shown the calculational condition is that a cylinder with radius 0 125m floats in two dimensional free space in this case no calculationa

a clarification of nonexistence problems with the superposition method the purpose of this paper is to clarify the question of nonexistence of solutions for the superposition method in acoustic radiation and scattering problems recently a number of authors have implemented this method and have claimed that nonexistence or nonuniqueness difficulties do not arise numerical evidence appears to support this view however this paper shows that the superposition method is a discrete approximation to an equation that does suffer from nonexistence of its solution at certain frequencies nonexistence of solutions is demonstrated with some numerical examples and an improved method is discussed

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

acoustic scattering by a hard or soft body across a wide frequency range by the helmholtz integral equation method in this paper the acoustic scattering by an obstacle across a wide frequency range of sound waves is investigated on the basis of the helmholtz integral formulation to overcome the nonuniqueness difficulties the methods proposed by burton npl report nac 30 jan 1973 and by burton and miller proc r soc london ser a 323 201 210 1971 are adopted for the dirichlet and neumann problems respectively the aim of this paper is twofold the first is to bring together completely regular formulations of the helmholtz integral equation and its normal derivative the second is to extend these formulations to treat the higher frequency problems the weakly singular integrals are regularized by subtracting out one term and adding it back depending on the problem concerned the additional integral can finally be expressed in an explicit form or results in solving a surface source distribution of the equipotential body the hypersingular kernels are regularized by the method of using some properties of the associated laplace equation originally proposed by chien et al j acoust soc am 88 918 937 1990 the completely regularized integral equations are amenable to computation by direct use of the standard quadrature methods to study the acoustic scattering due to higher frequency waves filon s quadrature method proc r soc edinburgh 49 38 47 1928 is extended to treat the rapidly oscillatory integrands numerical examples consist of acoustic scattering from a hard or soft sphere of radius a across a wide spectrum of wave numbers ka 20 comparisons of the numerical results with the exact solutions demonstrate the validity and efficiency of the implementation

a boundary integral equation method for two dimensional acoustic scattering problems this work formulates the singularity free integral equations to study 2 d acoustic scattering problems to avert the nonuniqueness difficulties burton s and burton and miller s methods are employed to solve the dirichlet and neumann problems respectively the surface helmholtz integral equations and their normal derivative equations in bounded form are derived the weakly singular integrals are desingularized by subtracting a term from the integrand and adding it back with an exact value depending on the relevant problem the additional integral can finally be either expressed in an explicit form or transformed to form a surface source distribution of the related equipotential body the hypersingular kernel is desingularized further using some properties of an interior laplace problem the new formulations are advantageous in that they can be computed by directly using standard quadrature formulas also discussed is the contour a unique feature of 2 d problems instead of dividing the boundary surface into several small elements a parametric representation of a 2 d boundary curve is further proposed to facilitate a global numerical implementation calculations consist of acoustic scattering by a hard and a soft circular cylinder respectively comparing the numerical results with the exact solutions demonstrates the proposed method s effectiveness

conformal mapping for the helmholtz equation acoustic wave scattering by a two dimensional inclusion with irregular shape in an ideal fluid the complex variables method with mapping function was extended to solve the linear acoustic wave scattering by an inclusion with sharp smooth corners in an infinite ideal fluid domain the improved solutions of helmholtz equation shown as bessel function with mapping function as the argument and fractional order bessel function were analytically obtained based on the mapping function the initial geometry as well as the original physical vector can be transformed into the corresponding expressions inside the mapping plane as all the physical vectors are calculated in the mapping plane this method can lead to potential vast savings of computational resources and memory in this work the results are validated against several published works in the literature the different geometries of the inclusion with sharp corners based on the proposed mapping functions for irregular polygons are studied and discussed the findings show that the variation of angles and frequencies of the incident waves ha

solving the hypersingular boundary integral equation in three dimensional acoustics using a regularization relationship regularization of the hypersingular integral in the normal derivative of the conventional helmholtz integral equation through a double surface integral method or regularization relationship has been studied by introducing the new concept of discretized operator matrix evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices such a treatment greatly improves the computational efficiency as the number of frequencies to be computed increases the computational cost of solving the composite helmholtz integral equation is comparable to that of solving the conventional helmholtz integral equation in this paper the detailed formulation of the proposed regularization method is presented the computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems the radiation of a pulsating sphere an oscillating sphere and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements it is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied

analysis of the near field acoustic radiation characteristics of two radially vibrating spheres by the helmholtz integral equation formulation and a critical study of the efficacy of the chief overdetermination method in two body problems the formulation of acoustic radiation from multiple vibrating bodies of arbitrary shape by the helmholtz integral equation is presented a computer code was developed to calculate the sound pressure field around an arbitrary number of three dimensional vibrating bodies of arbitrary shape the near field of two dilating spheres determined by using this code is presented in the form of equal pressure contours the paper also presents a critical study of the efficacy of the chief method in acoustic radiation problems involving more than one vibrating body

on extremal even unimodular 72 dimensional lattices by computer search we show that the lattice from 9 is the unique extremal even unimodular 72 dimensional lattices that can be constructed as proposed in 6

split level iteration many techniques are used for solving large sparse linear systems some of the most popular methods for solving the more difficult problems combine direct and iterative algorithms to yield efficient multistage procedures although each stage is standard the manner in which stages are combined is often novel both multigrid and preconditioned conjugate gradients fall into this class of numerical methods in lo two promising algorithms were described with supportive numerical studies the first was cyclic application of model problem adi iteration followed by a conjugate gradient iteration on the deviation of the model problem from the actual problem some new developments relating to this procedure are described in a companion paper in this journal 121 the second was a two level iteration in which a variational correction with a coarse grid basis was applied cyclically to the result of conventional iteration over a fine grid this was the first published implementation of a multigrid method based on the variational techniques that provide the foundations for finite element computation in these initial two level variational studies certain advantages of additive over multiplicative coarse mesh correction were not recognized the coefficient matrix for additive correction does not depend on the fine mesh iterate and need thus be computed only once moreover it need only be factored once for direct solution the right hand side of the correction equations varies each cycle brandt 4 and others have exploited this in multigrid and related methods some numerical comparisons of additive and multiplicative correction for two level iteration are given in 111 also it was not recognized that under suitable conditions one may use the two level iteration as a preconditioner for conjugate gradients thus combining the two initially distinct methods as suggested subsequently in 1 7 9 some insight into the nature of such an iteration may be gained by a qualitative analysis of error behavior the fine mesh iteration acts primarily on high mcde error components and the coarse mesh correction reduces low mode components in some implementations the two iteration matrices commute but in most they do not in the commuting case careful choice of iteration parameters precludes benefits from the conjugate gradient third stage in the noncommuting case the conjugate gradient stage should act on error persisting because of component mixing between the other two stages the automatic extrapolation in the conjugate gradient method greatly reduces convergence sensitivity to choice of iteration parameters for the fine mesh iteration

a comparison among four direct boundary integral methods four distinct direct boundary integral equation bie schemes two of which are new are derived and are used to solve multibody scattering problems these schemes are similar in that each is a composite of integral operators that are completely continuous and are dissimilar in that they are either a mixture of equations of the first and second kind or are strictly of the second kind their relative usefulness is determined by analysis and numerical experiments suggesting that the optimal scheme is governed by the tradeoff between stability and computational efficiency to partially verify these bie methods each was used to compute the scattering from a cylindrical obstacle and results of each method were compared to the analytic solution in addition one of these schemes was used to calculate the line source responses to half space and two layer models the numerical and analytic solutions are in favorable agreement

some cyclo quaternionic lattices abstract the faithful lattices of rank 2 p 1 of the groups sl 2 p are described for small primes p these and related lattices are investigated by computer in particular a new extremal even unimodular lattice of rank 48 is constructed

an integral equation approach to three dimensional acoustic radiation and scattering problems this paper provides numerical implementation and tests for a boundary integral equation method proposed recently by krutitskii ima j appl math 64 259 269 2000 for evaluating three dimensional acoustic radiation and scattering fields in the exterior neumann problem for several obstacles auxiliary boundaries with certain prescribed boundary conditions are introduced inside the obstacles for the purpose of eliminating the fictitious characteristic frequencies applying the simple source formulation on all boundaries leads to a set of uniquely solvable integral equations with weakly singular kernels to facilitate the numerical implementation the modified integral equations are further transformed to ordinary ones that contain nonsingular kernels only test cases consist of the acoustic radiation and scattering by one sphere and a pair of equal spheres comparisons are made with analytical solutions and schenck s chief method numerical results demonstrate the efficiency robustness and accuracy of

a method for computing acoustic fields based on the principle of wave superposition a method for computing the acoustic fields of arbitrarily shaped radiators is described that uses the principle of wave superposition the superposition integral which is shown to be equivalent to the helmholtz integral is based on the idea that the combined fields of an array of sources interior to a radiator can be made to reproduce a velocity prescribed on the surface of the radiator the strengths of the sources that produce this condition can in turn be used to compute the corresponding surface pressures the results of several numerical experiments are presented that demonstrate the simplicity of the method also the advantages that the superposition method has over the more commonly used boundary element methods are discussed these include simplicity of generating the matrix elements used in the numerical formulation and improved accuracy and speed the latter two being due to the avoidance of uniqueness and singularity problems inherent in the boundary element formulation

a retracted boundary integral equation for exterior acoustic problem with unique solution for all wave numbers a regular integral equation based on the surface source distribution method is developed to solve the exterior acoustic radiation and scattering problems this integral equation formed by seeking the equivalent mixed layer potential distributed over an auxiliary surface retracted from the actual boundary is uniquely solvable and has nonsingular kernel function hence it is more amenable to numerical implementation and can circumvent the difficulties of singularity nonuniqueness and slope discontinuity of the conventional integral equation formulations in contrast to the behavior of fredholm integral equations of the second kind the discretization of a regular integral equation will not result in a diagonally dominant coefficient matrix to overcome the undesirable numerical instability the optimal selection of the interior source surface is investigated through error estimate of the numerical integration the versatility and accuracy of the proposed formulation are demonstrated by numerical examples involving spheres prolate spheroids and finite cylinders

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

combined integral equation formulation and null field method for the exterior acoustic problem the numerical implementation of a combined integral equation and null field method used to solve the exterior neumann problem d s jones q j mech appl math xxvii 129 1974 is presented here the exterior helmholtz integral equation is solved on the radiating or scattering surface and the irregular frequencies are eliminated up to a given irregular frequency fm through the use of m additional null field equations an impedance matrix defined on the object surface is then obtained that can be used as an exact radiation condition in a finite element code the program and the numerical examples presented here are specialized to axisymmetrical problems a purely null field method is implemented and simple rules are defined that display its failure when applied to high aspect ratio objects and or in the high frequency range similar but less restricting rules are used to specify the numerical limitations of jones technique besides a few theoretical considerations clarify the role played by t

acoustic radiation from vibrating surfaces at characteristic frequencies the application of boundary integral methods to the problem of acoustic radiation suffers from nonexistence or nonuniqueness problems at the characteristic frequencies of the associated interior dirichlet problem to overcome this difficulty an iterative overdetermination scheme is developed first the surface integral helmholtz equation is discretized and at noncharacteristic frequencies the resulting system of linear algebraic equations can be solved directly yielding the values of the acoustic pressure at the boundary nodal points at characteristic frequencies the system matrix becomes ill conditioned and the direct solution is not possible an approximate surface impedance function is then assumed to serve as a basis for the computation of approximate farfield pressures these are then utilized to form an overdetermined system which is solved with the least squares procedure this solution can be used to compute the next order approximation and the process is continued the method is applied to the cylindrical transducer problem and proves to yield good results at characteristic frequencies it is also used to solve a structural radiation problem for which no characteristic frequencies are found nonexistence of such frequencies for problems of this type is conjectured although no general proof is given

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

a generalized network formulation for acoustic scattering the discretized helmholtz and equivalence integral equations yield the impedance representation of a generalized network formulation for acoustic scattering trans and self impedances relate the ensonification the scattered field and the pressure and velocity on the scatterer s surface s the scatterer s dry impedance loads s this formulation suggests applying analysis tools from electrical circuit theory network element abstraction and change of field variable basis the scatterer s wet admittance impedance corresponds to the augmented driving point admittance impedance from electrical network theory the lower branch modes of the scattering from a submerged spherical shell occur when the reactance looking into the fluid from s negates that looking into the shell change of basis on s transforms the impedance representation into the admittance arbitrary reference and network scattering representations the impedance admittance and arbitrary reference representations naturally separate hard

evaluation of the helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions abstract this paper presents a means of evaluating singular integrals in the helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions the subtraction addition technique is applied to the singular integral equations to convert the singular integrals to either ordinary integrals with bounded integrands or modified singular integrals including hypersingular integrals with exact integration values this regularization is performed before any discretization the modified integral equations can be calculated by directly applying standard quadrature rules over the entire integration domain numerical computations involve evaluating the acoustic field associated with a radiating inverse elliptic cylinder the velocity potential is obtained by applying the burton miller method which linearly combines the helmholtz boundary integral equation with its normal derivative to treat the fictitious characteristic frequencies further substituting the velocity potential into the regularized tangential derivative yields the surface tangential velocity comparing the numerical results with the analytical solutions verifies the effectiveness of the presented approach

a boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions this paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions the difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface this process gives rise to a set of uniquely solvable boundary integral equations distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation the modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only this implementation allows direct application of standard quadrature formulas over the entire integration domain that is the collocation points are exactly the positions at which the integration points are located selecting the interior surface is an easy task moreover only a few corresponding interior nodal points are sufficient for the computation numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders comparisons with analytical solutions are made numerical results demonstrate the efficiency and accuracy of the current solution method