loop star decomposition of basis functions in the discretization of the efie a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder the proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions the completeness of the loop star basis is discussed employing the presented scheme the issue of the irrotational property of the nonsolenoidal functions is addressed

conformal field theory and elliptic cohomology abstract in this paper we use conformal field theory to construct a generalized cohomology theory which has some properties of elliptic cohomology theory which was some properties of elliptic cohomology a part of our presentation is a rigorous definition of conformal field theory following segal s axioms and some examples such as lattice theories associated with a unimodular even lattice we also include certain examples and formulate conjectures on modular forms and monstrous moonshine related to the present work

closed and open conformal field theories and their anomalies we describe a formalism allowing a completely mathematical rigorous approach to closed and open conformal field theories with general anomaly we also propose a way of formalizing modular functors with positive and negative parts and outline some connections with other topics in particular elliptic cohomology

on the relationship between monstrous moonshine and the uniqueness of the moonshine module we consider the relationship between the conjectured uniqueness of the moonshine module open image in new window and monstrous moonshine the genus zero property of the modular invariance group for each monster group thompson series we first discuss a family of possiblezn meromorphic orbifold constructions of open image in new window based on automorphisms of the leech lattice compactified bosonic string we reproduce the thompson series for all 51 non fricke classes of the monster groupm together with a new relationship between the centralisers of these classes and 51 corresponding conway group centralisers generalising a well known relationship for 5 such classes assuming that open image in new window is unique we consider meromorphic orbifoldings of open image in new window and show that monstrous moonshine holds if and onlyzr if the only meromorphic orbifoldings of open image in new window are open image in new window itself or the leech theory this constraint on the meromorphic orbifoldings of open image in new window therefore relates monstrous moonshine to the uniqueness of open image in new window in a new way

an accurate scheme for the solution of the time domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation despite the numerous advances made in increasing the computational efficiency of time domain integral equation tdie based solvers the stability and accuracy of tdie solvers remain problematic this paper introduces a new numerical method for the accurate solution of tdies for scattering from arbitrary perfectly conducting surfaces the work described in this paper uses the higher order divergence conforming basis functions of graglia et al for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations since the basis functions used for the temporal representation are noncausal an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization

hybrid integration scheme for the evaluation of strongly singular and near singular integrals in surface integral equations the accurate and efficient evaluation of surface source integrals is a key step in obtaining reliable solutions to electromagnetic problems using integral equation formulations in this paper we propose to combine two of the most extensively used schemes to efficiently evaluate strongly singular and near singular source integrals such as those arising in the mathcal k operator in the proposed approach singularity subtraction is first applied to remove the most dominant singular part singularity cancellation using the radial angular variable transformation is then employed to treat the remaining integrand the method is compared to other common numerical schemes to demonstrate its effectiveness

a mesh adapted closed form regular kernel for 3d singular integral equations the green s functions employed in the method of moments mom diverge when observation and source points coincide this is at the origin of the difficulties in computing the mom matrix entries and in handling the near field interactions in fast fourier transform fft based fast methods and other sampling based methods in this paper we show that this singularity can be avoided and a modified regular green s function can be used instead this latter is obtained from the spectral representation of the usual green s function via windowing of its spectrum the width of the spectral window depends on the size of the mesh employed for discretizing the problem so that the proposed regular green s function is a mesh adapted regular kernel we address a general 3d problem we relate the mom reaction integrals to the 2d fourier spectrum of the green s function that allows to discuss the necessary spectral bandwidth for the windowed green s function we employ a tapered window and present a closed form expression for the spatial green s function numerical results are presented for 3d antenna and scattering problems discretized with rao wilton glisson rwg functions and for uniform and nonuniform meshing they show that the proposed method yields accurate solutions also for the antenna input impedance the meaning of the regularized green s function is also discussed and put in perspective

spectral elements for the integral equations of time harmonic maxwell problems we present a novel high order method of moments mom with interpolatory vector functions on quadrilateral patches the main advantage of this method is that the hdiv conforming property is enforced and at the same time it can be interpreted as a point based scheme we apply this method to field integral equations fies to solve time harmonic electromagnetic scattering problems our approach is applied to the first and second nedelec families of hdiv conforming elements it consists in a specific choice of the degrees of freedom dof made in order to allow a fast integral evaluation in this paper we describe these two sets of dof and their corresponding quadrature rules sample numerical results on fie confirm the good properties of our schemes faster convergence rate and cheap matrix calculation we also present observations on the choice of the discretization method depending on the fie selected

the generalized method of moments for electromagnetic boundary integral equations the generalized method of moments gmm is a partition of unity based technique for solving electromagnetic and acoustic boundary integral equations past work on gmm for electromagnetics was confined to geometries modeled by piecewise flat tessellations and suffered from spurious internal line charges in the present article we redesign the gmm scheme and demonstrate its ability to model scattering from pec scatterers composed of mixtures of smooth and non smooth geometrical features furthermore we demonstrate that because the partition of unity provides both functional and effective geometrical continuity between patches gmm permits mixtures of local geometry descriptions and approximation function spaces with significantly more freedom than traditional moment methods

improved conditioning of finite element matrices using new high order interpolatory bases the condition number of finite element matrices constructed from interpolatory bases will grow as the polynomial degree of the basis functions is increased the worst case scenario for this growth rate is exponential and in this paper we demonstrate through computational example that the traditional set of uniformly distributed interpolation points yields this behavior we propose a set of nonuniform interpolation points which yield a much improved polynomial growth rate of condition number these points can be used to construct several types of popular hexahedral basis functions including the 0 form standard lagrangian 1 form curl conforming and 2 form divergence conforming varieties we demonstrate through computational example the benefits of using these new interpolatory bases in finite element solutions to maxwell s equations in both the frequency and time domain

singular basis functions and curvilinear triangles in the solution of the electric field integral equation basis functions are formulated that account for singularities in the charge density near an edge on a conducting body the formulation is general and the basis functions are valid for planar as well curvilinear geometries in principle singularities of any order can be treated but best results are obtained for so called knife edge singularities results are compared with exact solutions or measurements where available for some simple problems

singular higher order divergence conforming bases of additive kind and moments method applications to 3d sharp wedge structures we present new subsectional singular divergence conforming vector bases that incorporate the edge conditions for conducting wedges the bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood singular bases of this kind complete to arbitrarily high order are described in a unified and consistent manner for curved quadrilateral and triangular elements the higher order basis functions are obtained as the product of lowest order functions and silvester lagrange interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties are discussed and these bases are proved to be fully compatible with the standard high order regular vector bases used in adjacent elements our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components adequate modelling of the divergence and removal of spurious solutions these singular high order bases provide more accurate and efficient numerical solutions of surface integral problems several test case problems are considered in the paper thereby obtaining highly accurate numerical results for the current and charge density induced on 3d sharp wedge structures the results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problems

monstrous moonshine from orbifolds we consider the monster module of frenkel lepowsky and meurman as az2 orbifold of a bosonic string compactified by the leech lattice we show that the main conway and norton monstrous moonshine properties stating that the thompson series for each monster group conjugacy class has a modular invariance group of genus zero follow from an orbifold construction based on an orbifold group composed of monster group elements it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each thompson series it is also shown that the power map formula of conway and norton follows from the same vacuum structure finally we demonstrate the validity of the vacuum conjectures for sectors twisted by leech lattice automorphisms in many cases

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

hierarchical curl conforming nedelec elements for triangular prism cells new curl conforming hierarchical vector basis functions are developed for triangular prism cells these bases are constructed by taking the product of base vectors with scalar orthogonal polynomials and are shown to maintain linear independence as their polynomial order is increased the procedure for constructing these basis functions is described and specific functions are tabulated for orders up to 6 5 these bases are compatible with recently proposed hierarchical bases for tetrahedral and hexahedral cells and will maintain tangential continuity with those bases if used in a mesh containing multiple cell shapes

nearly hypersingular integrals by double arctan transformation in higher order geometry modeling in order to directly utilize the dyadic green s functions in surface integral equations sies a novel double arctan transformation for nearly hypersingular integrals is proposed in this communication this new transformation is flexible and applicable to nearly hypersingular integrals in the forms of hat r hat r r 3 hat r r 3 and 1 r 3 over the curved surfaces by a fully numerical method with the help of the sigmoidal transformation to improve the stability of this new singularly handling method there results an efficient solution for the third order near singularity problems in sies moreover the proposed method is also effective for the lower orders of the nearly singular integral kernels with typical testing cases the performance of this method is fairly evaluated and its validity and stability is well demonstrated

hierarchal vector boundary elements and p adaption for 3 d electromagnetic scattering this paper describes the development of new vector boundary elements for solving electromagnetic em scattering problems the new elements are suitable for the magnetic field integral equation mfie electrical field integral equation efie or the combination field integral equation cfie the basis functions are assigned to the edges of an element rather than to its nodes the new element guarantees the continuity of the normal component of the surface current across element edges furthermore the basis functions are hierarchical from linear to higher order which enables one to use the new elements in a p adaption scheme

on discrete field theory properties of the dimer and ising models and their conformal field theory limits we study various mathematical aspects of discrete models on graphs specifically the dimer and the ising models we focus on proving gluing formulas for individual summands of the partition function we also obtain partial results regarding conjectured limits realized by fermions in rational conformal field theories

perturbation technique for non self adjoint systems with repeated eigenvalues the reanalysis of non self adjoint systems is computationally very expensive in the design of structural dynamics an effective even if approximate evaluation of the eigensolutions of a modified system is valuable to designers in a paper by meirovitch and ryland meirovitch l and ryland g a perturbation technique for gyroscopic systems with small internal and external damping journal of sound and vibration vol 100 no 3 1985 pp 393408 a perturbation technique was used to derive the perturhed solutions of a damped gyroscopic system with distinct eigenvalues that approach is extended to the case of systems with repeated eigenvalues a numerical example is presented to verify the present method and satisfactory results are observed

numerical analysis of single microstrip antenna and antennas array in spherically layered media this communication presents a comprehensive numerical analysis of single microstrip antenna and antennas array in spherically layered media using the method of moments mom the curvilinear rao wilton glisson crwg triangular basis functions are employed to simulate the equivalent current on the antenna surface first the dyadic green s functions using debye potential are deduced then their efficient computation procedures are carried out to improve calculation efficiency finally the characteristic basis function method is employed to save memory and calculation time by reducing dimension of the impedance matrix the results of this study are in good agreement with those of the feko software and they show that the method proposed in this communication is more efficient than feko in the analysis process

regularized integral equations and curvilinear boundary elements for electromagnetic wave scattering in three dimensions the boundary integral equations bies in their original forms which govern the electromagnetic em wave scattering in three dimensional space contain at least a hypersingularity 1 r sup 3 or a cauchy singularity 1 r sup 2 usually both thus obtaining reliable numerical solutions using such equations requires considerable care especially when developing systematic numerical integration procedures for realistic problems regularized bies for the numerical computation of time harmonic em scattering fields due to arbitrarily shaped scatterers are introduced two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration both approaches differ from all existing approaches to em scattering problems both work for integral equations initially containing either hypersingularities or cauchy singularities without the need to introduce surface divergences or other derivatives of the em fields on the boundary also neither approach is limited to flat surfaces nor flat element models of curved surfaces the muller linear combination of the electric and magnetic field integral equations efie and mfie is used to avoid the resonance difficulty that is usually associated with integral equation based formulations some preliminary numerical results for em scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions

singular hierarchical scalar basis functions for triangular cells a new family of hierarchical basis functions is proposed for triangular cells these functions incorporate singular behavior to model fields with corner singularities the functions are of the additive kind and combine a traditional polynomial complete representation on triangles with additional singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest the condition numbers arising from the use of these functions is investigated

volumetric testing parallel to the boundary surface for a nonconforming discretization of the electric field integral equation the volumetric monopolar rwg discretization of the electric field integral equation efie imposes no continuity constraint across edges in the surface discretization around a closed conductor the current is expanded with the monopolar rwg set and the electric field is tested over a set of tetrahedral elements attached to the boundary surface this scheme is facet oriented and therefore well suited for the scattering analysis of nonconformal meshes or composite objects the observed accuracy though is only competitive with respect to the rwg discretization for a restricted range of heights of the tetrahedral elements in this communication we introduce a novel implementation of the volumetric monopolar rwg discretization of the efie with testing over a set of wedges we show with rcs and near field results that this scheme offers improved accuracy for a wider range of heights than the approach with tetrahedral testing the application of the wedge testing to the even surface odd volumetric monopolar rwg discretization of the efie edge oriented and therefore less versatile shows similar accuracy as with tetrahedral testing which is a sign of robustness

first order singular basis functions for corner diffraction analysis using the method of moments this paper presents a set of divergence conforming basis functions to model the behavior of surface currents near the edges and corners of an infinitely thin perfectly conducting polygonal flat plate the basis functions are derived by imposing the edge and corner conditions on the first order basis functions current distributions obtained using the new set of basis functions are compared with the exact solutions for several scattering configurations even though the proposed basis functions are not highest order complete they provide accurate results without incurring complexity in the analysis or additional computational requirements

laplaza sets or how to select coherence diagrams for pseudo algebras we define a general concept of pseudo algebras over theories and 2 theories a more restrictive such a notion was introduced in po hu igor kriz conformal field theory and elliptic cohomology adv math 189 2 2004 325 412 http www math lsa umich edu ikriz but as noticed by m gould did not capture the desired examples the approach taken in this paper corrects the mistake by introducing a more general concept allowing more flexibility in selecting coherence diagrams for pseudo algebras

open string vertex algebras tensor categories and operads we introduce notions of open string vertex algebra conformal open string vertex algebra and variants of these notions these are open string theoretic noncommutative generalizations of the notions of vertex algebra and of conformal vertex algebra given an open string vertex algebra we show that there exists a vertex algebra which we call the meromorphic center inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center this result gives us a general method for constructing open string vertex algebras besides obvious examples obtained from associative algebras and vertex super algebras we give a nontrivial example constructed from the minimal model of central charge open image in new window we establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable vertex operator algebra and the grading restricted conformal open string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra we also give a geometric and operadic formulation of the notion of grading restricted conformal open string vertex algebra we prove two isomorphism theorems and in particular we show that such an algebra gives a projective algebra over what we call the swiss cheese partial operad

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

an accurate approach for the calculation of mom matrix elements the method of moments solution on integral equations for electromagnetic scattering and radiation problems with rao wilton glisson rwg and n spl times rwg basis functions requires the calculation of singular double surface integrals over triangular subdomains in this paper an approach is presented to deal with the singularity and almost singularity in which the triangle subdomain is divided into several subtriangles and then the singular or almost singular integral is calculated over each of the subtriangles this approach is mainly based on the projection process and the local polar frame it can be used to deal with the singular double surface integrals of order 1 r and 1 r sup 2 in which vector functions may be of zeroth order and higher order numerical examples are provided to verify the validity of the method presented in this paper for dealing with the singularity and almost singularity

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

hierarchical divergence conforming bases for edge singularities in quadrilateral cells singular divergence conforming bases have been proposed for the solution of integral equations although they have seen only occasional use in practical applications the existing singular bases are not hierarchical which prevents their use in adaptive p refinement applications in this paper a new family of singular hierarchical basis functions is proposed for quadrilateral cells these functions model the singularities associated with current and charge density at edges and are more convenient for modeling such singularities than triangular bases of the same kind the basis functions are of the additive kind and combine a hierarchical polynomial representation on quadrilaterals with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest moreover the added singular basis functions are computed on the fly on the basis of various reported numerical results this paper also illustrates the difficulties the advantages the accuracy and the cost of using such bases in the method of moment solutions of integral equations

optimal modeling parameters for higher order mom sie and fem mom electromagnetic simulations general guidelines and quantitative recipes for adoptions of optimal higher order parameters for computational electromagnetics cem modeling using the method of moments and the finite element method are established and validated based on an exhaustive series of numerical experiments and comprehensive case studies on higher order hierarchical cem models of metallic and dielectric scatterers the modeling parameters considered are electrical dimensions of elements h refinement polynomial orders of basis functions p refinement orders of gauss legendre integration formulas integration accuracy and geometrical curvature orders of elements in the model the goal of the study which is the first such study of higher order parameters in cem is to reduce the dilemmas and uncertainties associated with the great modeling flexibility of higher order elements basis and testing functions and integration procedures this flexibility is the principal advantage but also the greatest shortcoming of the higher order cem and to ease and facilitate the decisions to be made on how to actually use them by both cem developers and practitioners

lattices and θ function identities ii theta series the gluing construction of lattices is used to generate and study a number of theta function densities it is shown for example that riemann s formula a fundamental degree 4 identity can be derived from a degree 2 identity it is proved that all in a large class of identities can be derived from these lattice techniques a list of 24 independent quadratic identities in the jacobi functions 1 and 3 conjectured to be complete with all but two of them seeming to be new is given the theta series of glue classes is also investigated and it is shown that all of them as well as the functions a b satisfy polynomials whose coefficients are linear combinations of theta series of lattices these polynomials have some interesting properties

singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra a numerical solution of integral equations typically requires calculation of integrals with singular kernels the integration of singular terms can be considered either by purely numerical techniques e g duffy s method polar co ordinate transformation or by singularity extraction in the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically this method has been well established for linear and constant shape functions in this paper we extend the method for polynomial shape functions of arbitrary order we present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the helmholtz equation or its gradient and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra the presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high order basis functions numerical experiments show that the developed method leads to a more accurate and robust integration scheme and in many cases also a faster method than for example duffy s transformation copyright 2003 john wiley sons ltd

scattering from frequency selective surfaces an efficient set of v dipole basis functions in this paper a novel set of v dipole basis functions is introduced these basis functions are used to approximate the induced surface current density on an infinite plane frequency selective surface fss the elements of the fss are supposed to consist of straight sections and bends two groups of elements which the present v dipole basis functions can be applied to are identified namely the center connected elements and the loop type elements using the established spectral galerkin method where the method of moment mom procedure is carried out in the spectral domain we determine the reflection and transmission coefficients of the fss the convergence of the solution is demonstrated both for existing bases and the present v dipole basis functions it is found that the double infinite floquet sum diverges when existing discontinuous basis functions are used but that convergence is obtained for the present basis functions therefore care needs to be exercised and it would seem discontinuous bases should be avoided

low frequency scattering through an aperture in a rigid screen some numerical results abstract the penetration of a plane wave through an aperture in a rigid screen is investigated the first three terms of the low frequency expansion are evaluated numerically for both simply connected and annular apertures in the numerical method use is made of a division of the aperture into arbitrary triangles the singular behaviour at the edges is incorporated in the shape functions which yields good accuracy and fast convergence special attention has been given to the thin ring which can serve as a model for e g imperfectly shielded windows

application of curvilinear basis functions and mlfma for radiation and scattering problems involving curved pec structures the curvilinear surface and wire basis functions have been applied to solve radiation and scattering problems of various shapes of curved metallic structures with the method of moments however a suitable form of junction basis function that can connect the curvilinear wire and the curvilinear rao wilton glisson rwg surface basis functions has not been found we present a novel curvilinear junction basis function that can fill the gap the convergence tests reveal that the curvilinear junction basis function in conjunction with curvilinear wire and surface basis functions can reduce the radar cross section rcs error by one order of magnitude more than their reduced linear forms for a curved structure the curvilinear basis functions are applied to solve the radiation from wire antennas attached to curved surfaces we will demonstrate that the curvilinear basis functions with the multilevel fast multipole algorithm can significantly reduce the computational resources for simulating electrically large objects with curved features

iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations frequency selective surfaces fss and other periodic gratings are often analyzed under the assumption that they are infinite in extent most existing methods for analyzing periodic structures are based on the use of a floquet type representation of the fields in a unit cell whose dimensions are typically comparable to the wavelength in this work a finite truncated version of an infinite periodic structure is dealt with directly without the benefit of the assumption that the structure is periodic this in turn requires the handling of a large number of unknowns and makes it difficult to solve the problem using conventional matrix methods two different iteration approaches to solving the finite fss problem are discussed in the paper both of which employ the spectral domain formulation the first of these employs the spectral iteration technique and the second one uses the conjugate gradient cg iteration algorithm convergence characteristics of both of these methods are investigated and the results are reported

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

second order nedelec curl conforming prismatic element for computational electromagnetics a systematic approach to obtaining mixed order curl conforming basis functions for a triangular prism is presented focus is made on the second order case space of functions for the prism is given basis functions are obtained as the dual basis with respect to suitably discretized nedelec degrees of freedom functionals acting on elements of the space thus the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed different strategies for the finite element assembly of the basis are discussed numerical results showing the verification procedure of the correctness of the implemented basis functions are given numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown comparison with other representative sets of basis functions for prisms is included

an explicit a posteriori error indicator for electromagnetic finite element boundary integral analysis within the finite element method fem there is always an error associated with the solution quantifying this error is a very important aspect of fe analysis since it can increase confidence in a solution and can be used to drive adaptive refinements of the discretization optimizing the quality of the solution relative to the number of degrees of freedom and in turn reducing computational cost this paper presents an a posteriori error indicator for time harmonic electromagnetic analysis in three dimensions when using a galerkin fe approach with curl conforming elements and including a nonlocal boundary integral boundary condition as well as a dominant mode port boundary condition the error indicator is residual based of an explicit nature and constitutes a bound on an error energy measure the error indicator is evaluated by presenting results relating to common microwave engineering applications which shows that it can be used successfully to estimate the relative error distribution and that it can successfully serve within an adaptive algorithm

multilevel fast multipole method for higher order discretizations the multi level fast multipole method mlfmm for a higher order ho discretization is demonstrate do n high frequency hf problems illustrating for the first time how an efficient mlfmm for ho can be achieved even for very large groups applying several novel ideas beneficial to both lower order and higher order discretizations results from a low memory high speed mlfmm implementation of a ho hierarchical discretization are shown these results challenge the general view that the benefits of ho and hf mlfmm cannot be combined

volume integral equation with higher order hierarchical basis functions for analysis of dielectric electromagnetic scattering an efficient method of moments mom solution of volume electric field integral equation v efie with a kind of higher order basis functions is presented to model scattering from a dielectric object with arbitrary shapes and inhomogenuity this method is based on both higher order geometrical modeling and higher order current modeling which is realized using a family of higher order hierarchical vector hhov basis functions to expand the equivalent volume electric currents within curvilinear volume elements this higher order scheme of mom results in a significant reduction in meshing density and number of unknowns leading to a remarkable reduction of memory and cpu time in comparison with conventional low order mom while the accuracy of the solution remains at a comparable level numerical results agree with analytical solutions or with simulation results obtained by commercial software demonstrating the accuracy convergence efficiency flexibility and adaptability of the new approach

robust solution of time domain integral equations using loop tree decomposition and bandlimited extrapolation a stabilization method that eradicates low frequency instabilities in the solution of the time domain integral equations of electromagnetic scattering is presented the method uses the loop tree decomposition originally suggested for the solution of low frequency scattering problems via the method of moments specifically a temporally differentiated form of the pertinent integral equation is tested using tree basis functions and the undifferentiated form is tested using solenoidal basis functions the underlying solution method uses bandlimited interpolation functions blifs and the higher order divergence conforming vector bases of graglia et al for the temporal and spatial discretization of the integral equations respectively an extrapolation technique has been implemented to overcome the noncausality introduced into the system by the blifs numerical results will demonstrate the stability and accuracy of the proposed technique

time domain bie analysis of large three dimensional electromagnetic scattering problems a time domain boundary integral equation bie solution of the magnetic field integral equation mfie for large electromagnetic scattering problems is presented it employs isoparametric curvilinear quadratic elements to model fields geometry and time dependence eliminating staircasing problems the approach is implicit which seems to provide both stability and permits arbitrary local mesh refinement to model geometrically difficult regions without the significant cost penalty explicit methods suffer error dependence on discretization is investigated accurate results are obtained with as few as five nodes per wavelength the performance both on large scatterers and on low radar cross section rcs scatterers is demonstrated including the six wavelength nasa almond two spheres a thirteen wavelength missile and a high q cavity

a perturbation technique for gyroscopic systems with small internal and external damping abstract the response of linear damped gyroscopic systems can be obtained by means of techniques of linear systems theory which involves the computation of the transition matrix the response is in terms of complex quantities which is likely to cause computational difficulties as the order of the system increases in the absence of damping it is possible to derive the response of a linear gyroscopic system with relative ease by working with real quantities alone when damping is small one can use a perturbation approach to produce the response by regarding the undamped gyroscopic system as the unperturbed system in a previous paper a perturbation analysis was used to derive the response of a gyroscopic system with small internal damping this paper extends the approach to the case of external damping which is characterized not only by symmetric coefficients multiplying velocities but also by skew symmetric coefficients multiplying displacements where the latter terms are known as circulatory a numerical example is presented

lattices and θ function identities i theta constants the gluing construction of lattices is used to generate and study a number of theta constant identities it is shown for example that the jacobi identity a well known degree 4 identity can be derived from a degree 2 identity a list of all quadratic identities in 3 derivable from this lattice method containing over 30 algebraically independent identities and conjectured to yield all quadratic identities in 3 derivable by any means is included in contrast only two independent quadratic identities in 3 have been found elsewhere in the mathematical literature the theta constants of lattices and their glue classes are investigated and the fact that the theta constant of each glue class is a root of a polynomial whose coefficients are linear combinations of theta constants of lattices is shown

electromagnetic scattering from homogeneous dielectric bodies using time domain integral equations this paper presents a stable and accurate method to compute the electromagnetic scattering from homogeneous isotropic and nondispersive bodies using time domain integral equations tdies unlike previous tdie based scattering work the formulation presented here is based on the equations of poggio miller chang harrington wu and tsai formulation the method employs the higher order divergence conforming basis functions described by graglia et al and bandlimited interpolation functions to effect the spatial and temporal discretization of the integral equations respectively as the temporal basis functions are noncausal an extrapolation mechanism is used to modify the noncausal system of equations to a form solvable by standard marching on in time procedure this work also explains the reason for late time low frequency instabilities encountered in current tdie implementations and details a stabilization technique employed to overcome them numerical results demonstrate the accuracy and stability of the proposed technique

derivatives of complex eigenvectors with distinct and repeated eigenvalues in this paper we investigate the computation of the first order derivatives of complex eigenvectors for general non defective eigensystems a new normalization condition is proposed with which we can compute unique first order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues we also present an efficient algorithm to compute the particular solutions to the governing equations of the first order derivatives of eigenvectors finally numerical examples are included to demonstrate the validity of the proposed method copyright 2007 john wiley sons ltd

representation and character theory in 2 categories we develop a 2 categorical generalization of the theory of group representations and characters we categorify the concept of the trace of a linear transformation associating to any endofunctor of any small category a set called its categorical trace in a linear situation the categorical trace is a vector space and we associate to any two commuting self equivalences a number called their joint trace for a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors we call this function the 2 character of such functions of commuting pairs and more generally n tuples appear in the work of hopkins kuhn and ravenel michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic on equivariant morava e theories we define the concept of induced categorical representation and show that the corresponding 2 character is given by the same formula as was obtained in michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic for the transfer map in the second equivariant morava e theory

recursive algorithms for calculating the scattering from n strips or patches using the definition of recursive relations for the reflection operator for n strips or patches two easily programmable recursive algorithms are developed to calculate the electromagnetic scattering by n strips or patches one algorithm is for arbitrary excitation and the other is for a fixed excitation the recursive algorithms require the inversion of small matrices at each stage and hence are suitable for programming on smaller computers if the n strips or patches are identical and equally spaced symmetry can be exploited to speed up the algorithms a program was developed to calculate scattering by n strips and the result is shown to converge to scattering by a large strip when the n strips are contiguous

computation of resonant modes for axisymmetric maxwell cavities using hp version edge finite elements the computation of the resonant frequencies for closed cavities is not a trivial task multi materials and sharp corners all give rise to highly singular eigenfunctions however an approach using hp finite elements is well suited to such problems and with the correct combination of h and p refinements it yields the theoretically predicated exponential rates of convergence in this paper we present a novel approach to the solution of axisymmetric cavity problems which uses a hierarchic h1 and h curl conforming finite element basis a selection of numerical examples is included and these demonstrate that the exponential rates of convergence are achieved in practice copyright 2005 john wiley sons ltd

derivatives of eigenvalues and eigenvectors of a general complex matrix a survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non hermitian matrices is presented in addition a modification of one method based on a better normalizing condition is proposed methods are classified as direct or adjoint and are evaluated for efficiency operation counts are presented in terms of matrix size number of design variables and number of eigenvalues and eigenvectors of interest the effect of the sparsity of the matrix and its derivatives is also considered and typical solution times are given general guidelines are established for the selection of the most efficient method

singular higher order complete vector bases for finite methods this paper presents new singular curl and divergence conforming vector bases that incorporate the edge conditions singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements the higher order basis functions are obtained as the product of lowest order functions and silvester lagrange interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties are discussed and these bases are proved to be fully compatible with the standard high order regular vector bases used in adjacent elements the curl divergence conforming singular bases guarantee tangential normal continuity along the edges of the elements allowing for the discontinuity of normal tangential components adequate modeling of the curl divergence and removal of spurious modes solutions these singular high order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems sample numerical results confirm the faster convergence of these bases on wedge problems

an application of the boundary element method to the magnetic field integral equation a boundary element method bem for the solution of electromagnetic scattering problems using the magnetic field integral equation mfie is discussed the discretized form of the mfie is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation by considering several different types of boundary elements it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution it is shown that a higher order approximation for the current could significantly improve the accuracy of the numerical solution the superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower order approximations the bem results are compared to the results obtained using the dielectric bodies of revolution dbr code

zeroth order complete discretizations of integral equation formulations involving conducting or dielectric objects at very low frequencies we present the self loop basis functions a divergence conforming set with zero charge density these basis functions allow a rearrangement of the linear linear basis functions set to overcome the low frequency breakdown we show the accuracy and stability at very low frequencies of the linear linear discretizations of i electric field integral equation for perfectly conducting objects and ii the poggio miller chang harrington wu tsai formulation for dielectric bodies

multiresolution analysis of printed antennas and circuits a dual isoscalar approach a new approach is presented for the integral equation analysis of multilayer printed circuits antennas and arrays it is based on the introduction of new multiresolution vector functions with properties similar to those of the scalar wavelets and flexible enough to accommodate for general shapes the resulting method of moments mom impedance matrix shows a good conditioning that allows its safe sparsification and the efficient use of iterative methods for the solution of the linear system the results of the analysis of various real life structures show the advantage of the method

an accurate method for the calculation of singular integrals arising in time domain integral equation analysis of electromagnetic scattering an accurate method for the evaluation of the cauchy principal value integrals arising in time domain electromagnetic wave scattering is presented this is applied to a boundary integral equation bie method employing quadratic curvilinear surface elements where such singularities do not vanish as they generally do when simpler but less accurate discretizations are employed the technique involves weakening the singularity in the original kernel to a degree where conventional integration methods may be employed and transforming the strong singularity to a line integral in a form which allows cancellation of its singular components the effectiveness of the method is demonstrated by scattering calculations on a variety of targets and the computational cost of the approach is trivial

power operations in elliptic cohomology and representations of loop groups the first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve the second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups the third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part

design of quadrature rules for muntz and muntz logarithmic polynomials using monomial transformation a method for constructing the exact quadratures for muntz and muntz logarithmic polynomials is presented the algorithm does permit to anticipate the precision machine precision of the numerical integration of muntz logarithmic polynomials in terms of the number of gauss legendre gl quadrature samples and monomial transformation order to investigate in depth the properties of classical gl quadrature we present new optimal asymptotic estimates for the remainder in boundary element integrals this quadrature rule can be applied to evaluate singular functions with end point singularity singular kernel as well as smooth functions the method is numerically stable efficient easy to be implemented the rule has been fully tested and several numerical examples are included the proposed quadrature method is more efficient in run time evaluation than the existing methods for muntz polynomials

machine precision evaluation of singular and nearly singular potential integrals by use of gauss quadrature formulas for rational functions a new technique for machine precision evaluation of singular and nearly singular potential integrals with 1 r singularities is presented the numerical quadrature scheme is based on a new rational expression for the integrands obtained by a cancellation procedure in particular by using library routines for gauss quadrature of rational functions readily available in the literature this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions the rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed and several numerical examples are provided

universal perturbation technique for reanalysis of non self adjoint systems a novel matrix perturbation technique for modal dynamic reanalysis of non self adjoint systems is presented this technique is established first by performing complex eigensubspace condensation and orthogonal decomposition procedure the lower order perturbations of eigensolutions i e complex eigenvalues complex left and right eigenvectors are obtained by solving two reduced eigenvalue problems the higher order perturbations of eigensolutions are then derived by implementing the successive approximation process the proposed technique is a kind of universal dynamic reanalysis method because it is universally applicable to systems with all of the three cases distinct repeated and closely spaced eigenvalues illustrative examples are given to verify the present method and satisfactory results are observed

moment method calculations on apertures using basis singular functions the transmission properties of perforated perfectly conducting screens are of practical interest the treatment of nonperiodical structures by numerical techniques such as the method of moments is very computer intensive in this paper it is shown that using basis functions that incorporate the edge as well as the corner singularities the number of unknowns can be drastically reduced the advantages and limitations of the method are discussed numerical results are presented illustrating the transmission properties of arrays of square and rectangular apertures