multilevel plane wave based near field far field transformation for electrically large antennas in free space or above material halfspace a recently presented fully probe corrected near field far field transformation employing plane wave expansion and diagonal translation operators enables near field far field transformation for arbitrary measurement contours and arbitrary antennas a multilevel extension inspired by the multilevel fast multipole method is presented that is suitable for the efficient transformation of electrically large antennas with a size of tens or even hundreds of wavelengths the measurement points are grouped in a multilevel fashion and translations are carried out to the box centers on the highest level only the plane waves are processed through the different levels to the measurement points using a disaggregation and anterpolation procedure resulting in a reduced overall complexity in the second part of this paper the influence of perfectly conducting ground planes and dielectric halfspaces as an approximation for ground effects in a real measurement setup is investigated as such ground reflected waves are assumed which propagate from the investigated antenna to the field probe and add to the direct wave contributions the far field conditions required for these assumptions are achieved by a source box grouping scheme by this extension ground effects are directly considered within the near field far field transformation transformation results using simulated and measured near field data are shown

numerical properties of a gaussian translation operator for the 2d fmm a diagonal gaussian translation operator for the time harmonic fast multipole method fmm in two dimensions is examined numerically the gaussian translation operator depends on a beam parameter that determines its sharpness when the beam parameter is set to zero the gaussian translation operator reduces to the standard fmm translation operator the sampling rate can be determined straightforwardly to achieve any desired accuracy the directionality of the gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy the required number of plane waves depends strongly on the actual source receiver locations not just on the diameter of the source and receiver regions

unified products for leibniz algebras applications abstract let g be a leibniz algebra and e a vector space containing g as a subspace all leibniz algebra structures on e containing g as a subalgebra are explicitly described and classified by two non abelian cohomological type objects hl g 2 v g provides the classification up to an isomorphism that stabilizes g and hl 2 v g will classify all such structures from the view point of the extension problem here v is a complement of g in e a general product called the unified product is introduced as a tool for our approach the crossed resp bicrossed products between two leibniz algebras are introduced as special cases of the unified product the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem the description and the classification of all complements of a given extension g e of leibniz algebras are given as a converse of the factorization problem they are classified by another cohomological object denoted by ha 2 h g where is the canonical matched pair associated to a given complement h several examples are worked out in details

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

on the self averaging of dispersion for transport in quasi periodic random media in this study we present a numerical analysis for the self averaging of the longitudinal dispersion coefficient for transport in heterogeneous media this is done by investigating the mean square sample to sample fluctuations of the dispersion for finite times and finite numbers of modes for a random field using analytical arguments as well as numerical simulations we consider transport of point like injections in a quasi periodic random field with a gaussian correlation function in particular we focus on the asymptotic and pre asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion and we verify the logarithmic growth of the sample to sample fluctuations as earlier reported in eberhard 2004 j phys a math gen 37 2549 71 we also comment on the choice of the relevant parameters to generate quasi periodic realizations with respect to the self averaging of transport in statistically homogeneous gaussian velocity fields

simulation of near ground long distance radiowave propagation over terrain using nystrom method with phase extraction technique and fmm acceleration a 2d surface integral equation based nystroumlm solver in which a phase extraction technique is utilized to reduce the number of surface unknowns is described as forward scattering is the dominant mechanism for the near ground wave propagation scenario the associated rapidly varying phase components of the integral equation kernel and solution unknowns are deduced and isolated in advance and subsequently built into the solver it is shown that by applying this method when combining with an adaptive surface segmentation routine as few as one to two average unknowns per wavelength is adequate in obtaining accurate solutions this significantly reduces the memory storage and computational expense for the simulation of long distance propagation effects the efficiency of this method is further improved by incorporating it into the framework of the fast multipole scheme the full details of the algorithm are discussed along with performance comparisons of the new solver to a regular nystroumlm solver for terrain surfaces in terms of solution convergence

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

efficient computation of antenna coupling and fields within the near field region the theory techniques details of the important equations and description of two computer programs are presented for calculating efficiently the mutual coupling at a single frequency between any two antennas arbitrarily oriented and separated in free space both programs emphasize efficiency and generality and require basically the complex electric far field of each antenna and the eulerian angles designating the relative orientation of each antenna multiple reflections between the antennas are neglected but no other restrictive assumptions are involved if an electric field component is desired instead of coupling the receiving antenna is replaced by a virtual antenna with uniform far field the first computer program computes coupling or fields versus transverse displacement of the antennas in a plane normal to their axis of separation an efficient fast fourier transform fft program was made possible by collapsing the far field input data and showing that inmost cases the spectrum integration need cover only the solid angle mutually subtended by the smallest spheres circumscribing the antennas limiting the integration to this solid angle artifically band limits the coupling function thereby allowing much larger integration increments and reducing run times and storage requirements to a feasible amount for electrically large antennas the second program is based on a spherical wave representation of the coupling function and rapidly computes coupling or fields versus separation distance between the antennas the spherical wave representation emerged naturally from an intriguing characteristic proven for the mutual coupling function it like each rectangular component of electric and magnetic field in free space satisfies the homogeneous wave equation

the global extension problem crossed products and co flag non commutative poisson algebras let p be a poisson algebra e a vector space and e p an epimorphism of vector spaces with v ker the global extension problem asks for the classification of all poisson algebra structures that can be defined on e such that e p becomes a morphism of poisson algebras from a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component all such poisson algebra structures on e are classified by an explicitly constructed classifying set gph2 p v which is the coproduct of all non abelian cohomological objects ph2 p v v v which are the classifying sets for all extensions of p by v v v the second classical poisson cohomology group h2 p v appears as the most elementary piece among all components of gph2 p v several examples are provided in the case of metabelian poisson algebras or co flag poisson algebras over p the latter being poisson algebras q which admit a finite chain of epimorphisms of poisson algebras pn q npn 1 p1 1p0 p such that dim ker i 1 for all i 1 n

numerical investigations on ergodicity of solute transport in heterogeneous aquifers 1 darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields accurate simulations of two dimensional advection dispersion processes were achieved with the global random walk algorithm using 1010 particles in every transport realization reliable statistical estimations were obtained by averaging over 256 realizations the main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem for small initial plumes the ergodic behavior can be expected after thousands of advection timescales when the deviation from the theoretical prediction of the cross section space averaged concentration monotonously decays and falls under 20 the increase of the transverse dimension of the initial plume slows down the approach to the quasi ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities

planar near field to far field transformation using an equivalent magnetic current approach an alternative method is presented for computing far field antenna patterns from near field measurements the method utilizes the near field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna and these currents are used to ascertain the far fields under certain approximations the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near field measurements are made an electric field integral equation efie is developed to relate the near fields to the equivalent magnetic currents the method of moments is used to transform the integral equation into a matrix one the matrix equation is solved with the conjugate gradient method and in the case of a rectangular matrix a least squares solution for the currents is found without explicitly computing the normal form of the equation near field to far field transformation for planar scanning may be efficiently performed under certain conditions numerical results are presented for several antenna configurations

the lazy homology of a hopf algebra to any hopf algebra h we associate two commutative hopf algebras h1l h and h2l h which we call the lazy homology hopf algebras of h these hopf algebras are built in such a way that they are predual to the lazy cohomology groups based on the so called lazy cocycles we establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy homology of the sweedler algebra

iterative solution of high order boundary element method for acoustic impedance boundary value problems abstract a high order boundary element method for time harmonic acoustic impedance boundary value problems is presented the method is based on the galerkin type formulation of the burton miller integral equation bmie with high order polynomial basis and testing functions this formulation has several important advantages it is free of the interior resonance problem the hypersingular integral operator of the traditional bmie formulation can be avoided it leads to faster convergence in terms of the number of unknowns than the low order methods and it shows a good performance with iterative solvers to avoid the numerical difficulties associated to the implementation of the singular surface integral equations the singularity extraction technique is applied to evaluate the integrals in the singular and near singular cases the resulting matrix equation is solved iteratively with the generalized minimal residual method gmres and a simple preconditioner based on the incomplete lu factorization is applied to expedite the convergence numerical results indicate that the bmie formulation with galerkin method and high order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the gmres method is applied to solve the matrix equation iteratively this in turn indicates that the formulation is well suited for an efficient application of fast solution procedures such as the fast multipole method

classifying complements for hopf algebras and lie algebras abstract let a e be a given extension of hopf respectively lie algebras we answer the classifying complements problem ccp which consists of describing and classifying all complements of a in e if h is a given complement then all the other complements are obtained from h by a certain type of deformation we establish a bijective correspondence between the isomorphism classes of all complements of a in e and a cohomological type object ha 2 h a where is the matched pair associated to h the factorization index e a f is introduced as a numerical measure of the ccp for two n th roots of unity we construct a 4 n 2 dimensional hopf algebra whose factorization index over the group algebra is arbitrary large

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

evaluation of the first order approximations for transport in heterogeneous media 1 longitudinal dispersion coefficients in given realizations of the transport computed by two currently used approximations of the first order in velocity variance are compared with accurate global random walk simulations the comparisons are performed for the same ensemble of realizations of the darcy velocity field approximated by a quasiperiodic random field for lognormal hydraulic conductivity with small variance and finite correlation lengths the results show that at finite times of about one dispersion timescale the mean coefficient is underestimated by 20 and the fluctuations are overestimated by 80 at larger times the errors decrease monotonously and the first order approximations yield fairly good predictions for the mean and the fluctuations of the dispersion coefficient

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

zappa szep products of garside monoids a monoid k is the internal zappa szep product of two submonoids if every element of k admits a unique factorisation as the product of one element of each of the submonoids in a given order this definition yields actions of the submonoids on each other which we show to be structure preserving we prove that k is a garside monoid if and only if both of the submonoids are garside monoids in this case these factors are parabolic submonoids of k and the garside structure of k can be described in terms of the garside structures of the factors we give explicit isomorphisms between the lattice structures of k and the product of the lattice structures on the factors that respect the garside normal forms in particular we obtain explicit natural bijections between the normal form language of k and the product of the normal form languages of its factors

memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media 1 for transport in statistically homogeneous random velocity fields with properties that are routinely assumed in stochastic groundwater models the one particle dispersion i e second central moment of the ensemble average concentration for a point source is a memory free quantity independent of initial conditions nonergodic behavior of large initial plumes as manifest in deviations of actual solute dispersion from one particle dispersion is associated with a memory term consisting of correlations between initial positions and displacements of solute molecules reliable numerical experiments show that increasing the source dimensions has two opposite effects it reduces the uncertainty related to the randomness of center of mass but at the same time it yields large memory terms the memory effects increase with the source dimension and depend on its shape and orientation large narrow sources oriented transverse to the mean flow direction yield ergodic behavior with respect to the one particle dispersion of the longitudinal dispersion and nonergodic behavior of the transverse dispersion whereas for large longitudinal sources the longitudinal dispersion behaves nonergodically and the transverse dispersion behaves ergodically such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications for instance calibration of model parameters forecasting and identification of the contaminant source

equivalence principle algorithm with body of revolution equivalence surface for the modeling of large multiscale structures we propose a multiscale solver of equivalence principle algorithm with a body of revolution bor equivalence surface es first the whole object is decomposed into subdomains the ess are defined as proper bors e g spheres in this work to enclose each subdomain the rao wilton glisson rwg and bor basis functions are defined on each sphere respectively second the octree is constructed in each nonzero subdomain the multilevel fast multipole algorithm mlfma is employed to solve the equivalent currents on the es individually and then the equivalent currents are projected from rwg onto the bor basis functions the couplings between neighboring subdomains are evaluated directly using mlfma and the separated subdomains are substituted by the es to es couplings and are evaluated efficiently by the bor method of moments bor mom to solve the equation system a hybrid inner and outer iterative solver is employed where the inner iteration is used to solve each local subdomains and the outer iteration is used to update the global solutions by collecting all the local solutions numerical results and discussions demonstrate the validity of the proposed work

a coupled finite element global random walk approach to advection dominated transport in porous media with random hydraulic conductivity solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high peclet numbers in this paper we develop a new numerical approach to advection dominated transport consisting of coupling an accurate mass conservative mixed finite element method mfem used to solve darcy flows with a particle method stable and free of numerical diffusion for non reactive transport simulations the latter is the efficient global random walk grw algorithm which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single trajectory simulations using traditional particle tracking pt mfem saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields the coupling is achieved by projecting the velocity field from the mfem basis onto the regular grw lattice preliminary results show that mfem grw is tens of times faster than the full mfem flow and transport simulation

theory and practice of the fft matrix inversion technique for probe corrected spherical near field antenna measurements with high order probes a complete antenna pattern characterization procedure for spherical near field antenna measurements employing a high order probe and a full probe correction is described the procedure allows an almost arbitrary antenna to be used as a probe different measurement steps of the procedure and the associated data processing are described in detail and comparison to the existing procedure employing a first order probe is made the procedure is validated through measurements

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

exact complex source point beam expansions for electromagnetic fields complex source point csp beams or cspbs are known to be exact solutions of maxwell s equations hence they can be used as basis functions to represent the electromagnetic fields generated by arbitrary sources in this work it is shown that any fully vector electromagnetic field can be represented exactly as an expansion of cspbs originating from a relatively arbitrary surface enclosing the sources three different variants of the cspb representation are first considered for a spherical expansion surface and their properties are discussed it is shown that the cspb expansion set can be conveniently truncated by selecting only the significantly contributing beams for efficient field calculations the cspb representations are next extended to the case of relatively arbitrary expansion surfaces numerical results are presented to demonstrate and compare the convergence and efficiency of the three approaches

harmonics based inhomogeneous plane wave method hipw a formulation is presented for fast interactions between subdomains in two dimensional 2 d scattering problems the formulation combines inhomogeneous plane waves with cylindrical harmonic decompositions of fields radiated by the subdomains it is shown that the complexity of interactions naturally decays with the distance between subdomains and that very few elementary operations are involved at the lowest level an example of iterative solution for scattering by a collection of cylinders validates the proposed approach

generalized random walk algorithm for the numerical modeling of complex diffusion processes a generalized form of the random walk algorithm to simulate diffusion processes is introduced unlike the usual approach at a given time all the particles from a grid node are simultaneously scattered using the bernoulli repartition this procedure saves memory and computing time and no restrictions are imposed for the maximum number of particles to be used in simulations we prove that for simple diffusion the method generalizes the finite difference scheme and gives the same precision for large enough number of particles as an example simulations of diffusion in random velocity field are performed and the main features of the stochastic mathematical model are numerically tested 2003 elsevier science b v all rights reserved

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

spherical near field scanning with higher order probes a general method for higher order probe correction in spherical scanning is obtained from a renormalized least squares approach the renormalization causes the normal matrix of the least squares problem to closely resemble the identity matrix when most of the energy of the probe pattern resides in the first order modes the normal equation can be solved either with a linear iterative solver leading to an iterative scheme or with a neumann series leading to a direct scheme the computation scheme can handle non symmetric probes requires only the output of two independent ports of a dual polarized probe and works for both and scans the probe can be characterized either by a complex dipole model or by a standard spherical wave representation the theory is validated with experimental data

approximations for transport parameters and self averaging properties for point like injections in heterogeneous media we focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and gaussian random fields the parameters are determined by ensemble averages we study to what extent these averages represent the behaviour in a single realization we calculate the centre of mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation and on the iterative solution of the langevin equation compared with simulations the perturbation theory reproduces the numerical results only poorly whereas the iterative solution yields good results using these approximations we investigate the self averaging properties the ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles the dispersion coefficient is not self averaging in the ensemble of periodic fields for the gaussian ensemble the asymptotic dispersion coefficient is self averaging for finite times however the fluctuations are so large that the average does not represent the behaviour in a single realization

extending structures ii the quantum version abstract let a be a hopf algebra and h a coalgebra we shall describe and classify up to an isomorphism all hopf algebras e that factorize through a and h that is e is a hopf algebra such that a is a hopf subalgebra of e h is a subcoalgebra in e with 1 e h and the multiplication map a h e is bijective the tool we use is a new product we call it the unified product in the construction of which a and h are connected by three coalgebra maps two actions and a generalized cocycle both the crossed product of an hopf algebra acting on an algebra and the bicrossed product of two hopf algebras are special cases of the unified product a hopf algebra e factorizes through a and h if and only if e is isomorphic to a unified product of a and h all such hopf algebras e are classified up to an isomorphism that stabilizes a and h by a schreier type classification theorem a coalgebra version of lazy 1 cocycles as defined by bichon and kassel plays the key role in the classification theorem

on the condition number of a normal matrix in near field to far field transformations in near field scanning the model parameters for the antenna under test can be determined from measured probe outputs by solving a linear system of equations in the least squares sense the model parameters are far field pattern values cylindrical or spherical wave expansion coefficients or equivalent surface source values the normal matrix of this linear system of equations is sometimes extremely ill conditioned this occurs when certain sets of model parameters lie outside the spatial bandwidth of the operator that computes the probe output one remedy is to restrict the sets of model parameters allowed and perform upsampling if needed to achieve the desired accuracy these ideas are illustrated through analysis and examples that involve both 2 d and 3 d scanning geometries

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

application of volume integrals to the solution of partial differential equations abstract a group of algorithms for the numerical solution of elliptic partial differential equations is presented the differential equation is reduced to a fredholm equation of the second kind which in turn is discretized by means of the nystrom algorithm leading to a dense large scale system of linear algebraic equations this linear system is solved by means of the generalized conjugate residual algorithm which requires that the matrix of the system be applied to a sequence of recursively generated vectors since applying a dense matrix to a vector is a process of order n 2 where n is the dimension of the system the resulting algorithms are usually not competitive with finite differences and finite elements in terms of cpu time requirements however it turns out that the matrices of linear systems resulting from many partial differential equations can be applied to vectors in a fast manner i e for the cost proportional to n in some cases and to n log n or n log n 2 in others resulting in extremely efficient algorithms for the solution of certain elliptic partial differential equations the performances of several such algorithms are illustrated by numerical examples

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h

exact gaussian beam theory for outgoing and standing spherical waves application to transmitting and receiving antennas outgoing spherical vector wave functions are expressed in terms of gaussian beams also called complex source point beams radiating in all directions by use of a vector wave expansion the electromagnetic field of an arbitrary source of finite extent is thus expressed in terms of gaussian beams with weights determined directly from the spherical expansion coefficients of the source these outgoing wave formulas allow the field of any transmitting antenna to be expressed in terms of gaussian beams elementary gaussian beam receivers are introduced as the electromagnetic field at complex points in space the outputs of elementary gaussian beam receivers pointing in all directions determine the spherical expansion coefficients in a standing wave expansion these standing wave formulas allow the output of any receiving antenna to be expressed in terms of the outputs of elementary gaussian beam receivers combining the formulas for outgoing and standing waves produces a new antenna antenna transmission formula based solely on gaussian beams the theory is exact and validated through numerical examples