inversion of the pieri formula for macdonald polynomials we give the explicit analytic development of macdonald polynomials in terms of modified complete and elementary symmetric functions these expansions are obtained by inverting the pieri formula specialization yields similar developments for monomial jack and hall littlewood symmetric functions

the fast multipole method fmm for electromagnetic scattering problems the fast multipole method fmm developed by v rokhlin 1990 to efficiently solve acoustic scattering problems is modified and adapted to the second kind integral equation formulation of electromagnetic scattering problems in two dimensions the present implementation treats the exterior dirichlet problem for two dimensional closed conducting objects of arbitrary geometry the fmm reduces the operation count for solving the second kind integral equation from o n sup 3 for gaussian elimination to o n sup 4 3 per conjugate gradient iteration where n is the number of sample points on the boundary of the scatterer a sample technique for accelerating convergence of the iterative method termed complexifying k the wavenumber is also presented this has the effect of bounding the condition number of the discrete system consequently the operation count of the entire fmm all iterations becomes o n sup 4 3 computational results for moderate values of ka where a is the characteristic size of the scatterer are given

an efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures an efficient multilevel fast multipole algorithm mlfma formalism to model radiation and scattering by from large planar microwave structures is presented the technique relies on an electric field integral equation efie formulation and a series expansion for the green dyadic based on the use of perfectly matched layers pml in this way a new pml mlfma is developed to efficiently evaluate matrix vector multiplications arising in the iterative solution of the scattering problem the computational complexity of the new algorithm scales down to o n for electrically large structures the theory is validated by means of several illustrative numerical examples

investigation of scattering properties of large scale aperiodic tilings using a combination of the characteristic basis function and adaptive integral methods this paper presents a hybrid approach for efficient analysis of electromagnetic em scattering from large scale aperiodic structures e g aperiodic penrose and danzer tilings which integrates the characteristic basis function method cbfm and the adaptive integral method aim by performing a domain decomposition a series of characteristic basis functions cbfs that are defined on a macro block and comprised of a relatively large number of sub domain basis functions facilitate a substantial reduction in the method of moments mom matrix size enabling the use of a direct solver for large problems the aim is applied to accelerate the calculation of cbfm reduced mom matrices significantly decreasing the cpu time and memory required for solving large scale problems as the size of one block becomes electrically large the original cbfm combined with the aim is employed to generate the initial cbfs by solving a large problem with multiple excitations which results in efficiently constructing the final cbfs afforded by the singular value decomposition svd procedure this methodology produces a two level cbfm aim hybrid algorithm for efficiently characterizing large scale objects the numerical results presented demonstrate the accuracy and efficiency of the proposed hybrid algorithm then the developed solver is applied to investigate em scattering properties of large scale aperiodic tilings the numerical results show that penrose danzer tilings exhibit significantly improved grating lobe suppression as compared to their periodic counterparts

vertex operators symmetric functions and the spin group γn abstract this work provides a vertex operator approach to the symmetric group sn and its double covering group n by generalizing a result of frenkel and sato for sn we formulate a correspondence between the space v of certain twisted vertex operators the ring of symmetric functions over q 2 and the space of nontrivial irreducible characters of n under this identification we show that a distinguished orthogonal basis of v corresponds to the set of nontrivial irreducible characters of n where both are parametrized by partitions with odd integer parts the counterpart of this distinguished basis in the ring over q 2 is the set of schur s q functions which are loosely speaking the square roots of the schur functions the nontrivial part of the character table of n is shown to be given by certain matrix coefficients in v

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

a complete set of linear phase basis functions for scatterers with flat faces and for planar apertures entire domain basis functions are introduced for the analysis of scattering from bodies with flat portions and of radiation from planar apertures they are defined via the application of the sampling theorem of shannon these basis functions are particularly convenient for polygonal contours however arbitrary contours can be treated in the same framework with a very minor pre processing time a non redundant set of basis functions is determined a priori by explicit approximate formulas either for currents and for radiated field numerical results are presented to shown the accuracy of the method

a note on the construction of synthetic basis functions for antenna arrays construction of synthetic basis functions sbfs for analyzing the radiation problems of antenna arrays is discussed the sbfs of a block are generated from two separate solution spaces with one space to account for the effect of the feeding voltages and the other space to account for the responses of the block to incident electromagnetic fields from its environment singular value decomposition svd method is used to extract the characteristic modes of the block examples show that using sbfs constructed in this way may cause better efficiency

solution of electrically large problems with multilevel characteristic basis functions we present a multilevel version to the characteristic basis function method cbfm this technique extends the range of applicability of the conventional cbfm and allows a direct solution of problems even when they exceed 250 000 degrees of freedom in this paper it is shown how we can use the multilevel cbfm which is a natural extension of the conventional approach and which saves both time and memory and yet retains excellent accuracy in the computed far fields also the adaptive cross algorithm is used for fast calculation of the interaction between the blocks in the process of generating the reduced matrix

combining the moment method with geometrical modelling by nurbs surfaces and bezier patches induced current distributions on conducting bodies of arbitrary shape modelled by nurbs non uniform rational b splines surfaces are obtained by using a moment method approach to solve an electric field integral equation efie the nurbs surfaces are expanded in terms of bezier patches by applying the cox de boor transformation algorithm this transformation is justified because bezier patches are numerically more stable than nurbs surfaces new basis functions have been developed which extend over pairs of bezier patches these basis functions can be considered as a generalization of rooftop functions the method is applied to obtain rcs values of several objects modelled with nurbs surfaces good agreement with results from other methods is observed the method is efficient and versatile because it uses geometrical modelling tools that are quite powerful

plethysms replicated schur functions and series with applications to vertex operators specializations of schur functions are exploited to define and evaluate the schur functions s x and plethysms s s x for any integer real or complex plethysms are then used to define pairs of mutually inverse infinite series of schur functions m and l specified by arbitrary partitions these are used in turn to define and provide generating functions for formal characters s of certain groups h thereby extending known results for orthogonal and symplectic group characters each of these formal characters is then given a vertex operator realization first in terms of the seriesm m 0 and various l dual to l and then more explicitly in the exponential form finally the replicated form of such vertex operators are written down the characters of the orthogonal and symplectic groups have been found by schur 34 and weyl 35 respectively the method used is transcendental and depends on integration over the group manifold these characters however may be obtained by purely algebraic methods this algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration littlewood d e 1944 phil trans r soc london ser a 239 809 392 pacs numbers 02 10 v 02 10 de 02 20 a 02 20 hj mathematics subject classification 05e05 17b69 11e57 16w30 20e22 33d52 43a40 1751 8113

efficient algorithm for the evaluation of the physical optics scattering by nurbs surfaces with relatively general boundary condition an adaptive integration algorithm is presented for the computation of the physical optics po electric and magnetic field scattered by electrically large objects modeled by non uniform rational b splines nurbs the algorithm is the customization of a more general purpose result that has been recently published by using a unique formulation both impenetrable e g impedance surfaces coated conductors as well as transparent thin sheet materials e g thin dielectric panels or frequency selective surfaces are treated via their fresnel reflection and transmission coefficients the po radiation integral is evaluated over the nurbs parametric domain since most of the computer aided geometric design cagd tools are based on nurbs the proposed algorithm allows a straightforward electromagnetic analysis of the structures by exploiting the standard available geometrical description with no need of generating new geometrical models furthermore the proposed adaptive sampling requires a number of integration points that is found to be drastically smaller than that resulting from standard nyquist based sampling integration algorithms such reduction of the sampling points is achieved by resorting to high frequency technique concepts and allows a significant reduction of the cpu computational burden therefore the algorithm is efficient and particulary suitable for the electromagnetic characterization of real life electrically large objects

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

vertex operator realization of symplectic and orthogonal s functions it is shown that symplectic and orthogonal schur functions can be realized by the action of the modes of certain vertex operators on the function the properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions the vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating hirota type partial differential equations pdes which have symplectic s functions as functions

linear embedding via green s operators a modeling technique for finite electromagnetic band gap structures we propose a modular electromagnetic modeling procedure for large finite electromagnetic band gap ebg structures called linear embedding via green s operators it is a diakoptic method based on the huygens schelkunoff principle involving equivalent boundary current sources that electromagnetically characterize the enclosed domain of arbitrary shapes as if it were a multiport system in a cascade of embedding steps separate reusable domains are combined to form larger domains device design often involves tuning local medium properties in a compact designated domain with a large environment through an additional embedding step the equivalent sources describing the environment can be transferred to the boundary of the designated domain rendering subsequent design steps very fast this two stage optimization process is applied in the design of an ebg power splitter

fast analysis of a dual band reflectarray using two different numerical approaches based on the moment method this communication presents two efficient approaches for the electromagnetic field analysis of complex 3d bodies the first method can provide a fast and accurate analysis of arbitrary metallic and dielectric magnetic structures and combines the moment method mm the multilevel fast multipole algorithm mlfma physics optics po and the characteristic basis function method cbfm to solve very large scattering problems the second technique is based on a domain decomposition approach that divides the geometry into several parts to minimize the vast computational resources required when applying the mm a parallelization process was carried out by using the message passing interface mpi algorithm to minimize the memory and time requirements for each simulation a dual band reflectarray was successfully analyzed to compare the performance of both codes

bounding the exponent of a finite group with automorphisms a 4 where a s a 1 this follows immediately from the fact that for every prime p dividing g there is an a invariant sylow p subgroup p of g for a which even a stronger decomposition holds p s c a see for ag a p w x example 6 theorems 6 2 2 and 6 2 4 suppose that in this situation n is the least common multiple of the exponents of the centralizers c a g a g a we show that then the exponent of g is bounded by a number that depends only on n and on the group a

fixed points of coprime operator groups abstract let m be a positive integer and a an elementary abelian group of order q r with r 2 acting on a finite q group g we show that if for some integer d such that 2 d r 1 the dth derived group of c g a has exponent dividing m for any a a then g d has m q r bounded exponent and if r 1 c g a has exponent dividing m for any a a then r 1 g has m q r bounded exponent

generation of characteristic basis functions defined over large surfaces by using a multilevel approach an efficient procedure for the generation of characteristic basis functions cbfs over large arbitrary surfaces is presented in this work the first step is to partition the original object into subdomains and then derive the cbfs for these domains by solving for the currents induced by a set of components of the planewave spectrum pws next we aggregate the adjacent domains to obtain enlarged cbfs by expanding the original ones in terms of the weighted planewaves used to compute them this process can be performed iteratively and involves fast non expensive matrix manipulation we obtain a progressive decrease of the number of unknowns as we increase the domain size by following the procedure described above

a family of integral transformations and basic hypergeometric series we present a conjecture giving a family of commuting operators i c in terms of n fold integral transformations for the simplest case n 2 the commutativity is proved by using several summation and transformation formulas for the basic hypergeometric series

entire domain basis functions for scattering from large curved surfaces formulated by transformation optics a complete set of entire domain basis functions are introduced for the analysis of scattering from bodies with curved surfaces they are defined via the application of a generalization of the shannon sampling theorem these basis functions are defined for curved patches with arbitrary contours via a three step procedure first the curved patch is mapped via transformation optics onto a flat parametric domain surrounded by a virtual anisotropic inhomogeneous space next linear phase functions are defined on the parametric flat domain the sufficient and non redundant number of functions is found by using a spectral domain completeness relationship of the delta function finally the back transformation from flat anisotropic to curved isotropic space yields the basis functions for the curved patch the number of basis functions so obtained matches the degrees of freedom of the field known in the literature numerical results show the effectiveness of the representation for both fields and currents

a moment method approach using frequency independent parametric meshes a moment method mm formulation using nonuniform rational b splines nurbs parametric surfaces is introduced to analyze radiation and scattering from arbitrarily shaped conducting bodies a feature of the technique presented is the independence of the geometrical model from the frequency which avoids the regeneration of the geometry model when the frequency changes this feature is obtained by defining new sets of basis and testing functions that extend over subpatches in each nurbs the proposed approach is accurate flexible and efficient

application of the characteristic basis function method utilizing a class of basis and testing functions defined on nurbs patches a novel implementation of the characteristic basis function method cbfm is given in which the high level basis functions called characteristic basis functions cbfs are represented in terms of curved rooftops generated from nonuniform rational b splines nurbs in the parametric u v domain the associated macro testing functions are defined by using curved razor blade functions corresponding to each rooftop the underlying objective of the cbfm is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments mom the result is therefore an approach which can handle many complex cases via direct solvers without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill conditioned matrices as a result of the combination of the cbfm with the special class of low level basis and testing functions directly located over nurbs surfaces complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the cpu time as well as required memory resources

scattering analysis of a large body with deep cavities a numerical scheme is presented for simulating electromagnetic scattering from a large and arbitrarily shaped body coated with inhomogeneous composite materials with large and deep cavities this numerical scheme employs the higher order vector finite element method fem to discretize the fields inside the cavities and coatings and the higher order boundary integral bi method to terminate the fem computational domain a highly efficient special solver is designed to eliminate the unknowns inside the cavities which yields a computed relation cr matrix over the cavity s aperture between the tangential electric and magnetic fields this cr matrix is then combined with the finite element boundary integral fe bi matrix equation to form a complete linear system for the discrete fields everywhere in the computational domain the resulting system is solved iteratively using a novel preconditioner derived by replacing the bi with a corresponding absorbing boundary condition abc

vertex operators and hall littlewood symmetric functions abstract we consider vertex operators on space v with a parameter t their components form an associative algebra which is a generalization of the clifford algebra a distinguished orthogonal basis of v is proved to be the hall littlewood symmetric functions we show that kostka foulkes polynomials or certain kazhdan lusztig polynomials for the affine weyl group of type a are matrix coefficients on the space v we also obtain certain generating functions for the product of hall littlewood functions and the kostka foulkes polynomials

efficient multilevel approach for the generation of characteristic basis functions for large scatters an efficient approach for deriving the characteristic basis functions cbfs for analyzing radiation and scattering problems is proposed the spectrum of the incident plane waves is partitioned into several angular regions that are progressively enlarged in a multilevel procedure to generate orthogonal functions the cbfs are obtained at the final orthogonalization level so that they cover the entire angular range this approach to generating the cbfs is better suited for electrically large objects than the conventional approach which utilizes the entire angular range visible of the plane wave spectrum in a single level

evaluation of physical optics integrals from b spline surfaces by means of a fast locating algorithm of stationary points a novel modeling method based on the parametric domain decomposition is proposed for the b spline surfaces this method combines the advantages of conventional b spline and bezier surfaces moreover a fast algorithm based on the new modeling method is presented to locate the stationary points including inner and boundary stationary points of physical optics po integrals from targets modeled with double quadratic b spline surfaces in this algorithm the locating problem is reduced to a numerical or analytical solution no approximation is involved in the proposed modeling method and locating algorithm the stationary phase method by means of the fast locating algorithm is employed to evaluate po integrals the cpu time is independent of the incident frequency and physical dimension of scattered surfaces the accuracy and performance of the presented methods are validated by comparing them with those of the brute force method

an iterative solution for electrically large problems combining the characteristic basis function method and the multilevel fast multipole algorithm an iterative scheme for the rigorous computation of electrically large problems is presented the approach is based on a combination of the characteristic basis function method cbfm and the multilevel fast multipole algorithm mlfma that can deal with very large problems that require an iterative solution process even considering that the application of the cbfm entails an important reduction of the number of unknowns when compared to method of moments approaches based on subdomain functions this reduction is due to the fact that the number of macro basis functions called characteristic basis functions cbfs is lesser than the number of low level subsectional functions used to sample the geometry in addition the use of the mlfma avoids the need to calculate and store the coupling terms in the reduced matrix that are not on or close to the diagonal thereby optimizing the cpu time and the memory storage requirements non uniform rational b splines nurbs surfaces are employed for the representation of the geometry and the cbfs are described in terms of curved rooftops generated in the parametric space the associated macro testing functions are defined as aggregations of curved razor blade functions

a fast impedance and pattern computation scheme for finite antenna arrays a fast numerical method well suited to the analysis of moderate size arrays made of complex elements is presented it combines macro basis functions and multipole approaches without an iterative procedure this method is exploited to estimate the impedance matrix and active element patterns for the latter an efficient formulation is provided as a series of pattern multiplication problems examples are shown for arrays of broadband dipoles the computational gain obtained for the reduction of the original method of moments system of equations is briefly described

complex multipole beam approach to electromagnetic scattering problems a novel approach to reducing the matrix size associated with the method of moments mom solution of the problem of electromagnetic scattering from arbitrary shaped closed bodies is presented the key step in this approach is to represent the scattered field in terms of a series of beams produced by multipole sources located in a complex space on the scatterer boundary the fields generated by these multipole sources resemble the gabor basis functions by utilizing the properties of the gabor series guidelines for selecting the orders as well as locations of the multipole sources are developed it is shown that the present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure and a low condition number the accuracy of the proposed method is verified by comparing the numerical results with those derived by using the method of moments

generation of excitation independent characteristic basis functions for three dimensional homogeneous dielectric bodies the aim of this paper is the development of characteristic basis functions cbfs for homogeneous dielectric bodies that are modeled with the surface equivalence principle this approach contributes to generalize the characteristic basis function method cbfm for any kind of material the foundations for the generation of these new basis functions are widely revised and the cbfm is adapted for this purpose the validity of the method is illustrated with the systematic analysis of canonical geometries as well as with the computation of the specific absorption rate inside a head like geometry in all the cases the error in the current distribution as well as in the near and far field is considered in addition a detailed comparison of the generation of the cbfs using plane waves and spherical waves is also carried out

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

wave field interaction with complex structures using equivalence principle algorithm a domain decomposition scheme based on the equivalence principle similar to huygens principle for integral equation solvers and the method of moments is introduced the equivalence principle allows the replacement of unknown currents distributed in a volume in space by equivalence currents residing on the surface that bounds the volume it also allows the dissociation of the solution of one region from that of another region in this manner problems of high complexity can be encapsulated by surfaces of simpler shapes using less unknowns it can aid in parallel algorithms reusability of solutions as well as improving the condition number of a matrix system when disparate mesh or adaptive mesh are needed the challenge arises when an equivalence surface intercepts a current carrying conductor because the breakup of the current into separate pieces gives rise to charge singularity a junction basis can be used to mitigate this singularity however a better solution is to introduce a tap basis to model the current that intercepts with the equivalence surfaces using this scheme the current continuity is conserved and the singularity of the charges is avoided the solution is shown to be accurate