calculation of the impedance matrix inner integral to prescribed precision we present a new method for evaluating the inner integral of the impedance matrix element in the traditional rao wilton glisson formulation of the method of moments for perfect conductors in this method we replace the original integrand modified by a constant phase factor by its taylor series and keep enough terms to guarantee a number of significant digits in the integration outcome we develop criteria that relate the number of taylor terms to the number of required significant digits we integrate the leading taylor terms analytically and the rest through iteration formulas we show that the iteration formulas converge for all observation points within a sphere with a radius of half a wavelength and center the triangle s centroid we compare results of our method with existing ones and find them in excellent agreement we also outline a procedure for using cubatures outside the region of convergence

exchangeability and convergence for random sets abstract in many statistical analysis some kind of dependency of random variables may be required and exchangeability as an alternative to the random sample with independent identically distributed random variables gives the study of asymptotic properties of random variables on the other hand statistical estimators are expressible of the linear form of random variables which involves possibly random weights and possibly functions of dependent random variables which are permutation invariant with respect to distributions this paper attempts to provide some properties useful to the limit theorems and laws of large numbers for arrays of banach spaced valued fuzzy random sets are examined with emphasis on permutation invariance among the random sets

a strong law of large numbers for fuzzy random sets abstract an slln for fuzzy random variables in rp including other convergence theorems has been proved by klement puri and ralescu the present paper is motivated by the definitions and properties developed by puri and ralescu and concerned with an slln for banach spaced valued fuzzy random sets as an easy application of the previous results

calculation of cfie impedance matrix elements with rwg and n spl times rwg functions the method of moments mom solution of combined field integral equation cfie for electromagnetic scattering problems requires calculation of singular double surface integrals when galerkin s method with triangular vector basis functions rao wilton glisson functions and the cfie are applied to solve electromagnetic scattering by a dielectric object both rwg and n spl times rwg functions n is normal unit vector should be considered as testing functions robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the cfie with these basis and test functions in computing the impedance matrix elements including the gradient of the green s function we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand in the developed method all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions in addition we present compact iterative formulas for computing the extracted terms in closed form by these formulas we can extract any number of terms from the singular kernels of cfie formulations with rwg and n spl times rwg functions

fast and accurate computation of hypersingular integrals in galerkin surface integral equation formulations via the direct evaluation method hypersingular 4 d integrals arising in the galerkin discretization of surface integral equation formulations are computed by means of the direct evaluation method the proposed scheme extends the basic idea of the singularity cancellation methods usually employed for the regularization of the singular integral kernel by utilizing a series of coordinate transformations combined with a reordering of the integrations the overall algebraic manipulation results in smooth 2 d integrals that can be easily evaluated via standard quadrature rules finally the reduction of the dimensionality of the original integrals together with the smooth behavior of the associated integrands lead up to unmatched accuracy and efficiency

fuzzy specifications and random fuzzy events considered as basic tools for statistical predication abstract fuzzy probability theory based on fuzzy specifications is a new aspect of fuzzy set theory with potential applications in manufacturing and production reliability prediction in this paper the development of the new theory is carried out in parallel with that of classical probability theory in order to highlight their similarities and differences the presentation can be divided into four parts introducing successively non random fuzzy events on 0 1 random fuzzy events on 0 1 non random fuzzy events on an heyting lattice random fuzzy events on an heyting lattice in each section an interpretation of the actual manufacturing data will be refined into appropriate mathematical definitions these definitions will in turn permit the derivation of propositions pertaining to the evaluation of certain manufacturing parameters a simplistic automobile manufacturing example illustrates the theory

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 strong law of large numbers for fuzzy random variables in this paper we obtain a strong law of large numbers for sums of levelwise independent and levelwise identically distributed fuzzy random variables

computation of galerkin double surface integrals in the 3 d boundary element method the galerkin boundary element method bem also known as the method of moments is a powerful method for solving the laplace equation in three dimensions there are advantages to galerkin formulations for integral equations as they treat problems associated with kernel singularity and lead to symmetric and better conditioned matrices however the galerkin method requires the computation of double surface integral over pairs of triangles there are many semianalytical methods to treat these integrals which all have some issues and are discussed in this paper novel methods inspired by the treatment of these kernels in the fast multipole method are presented for computing all the integrals that arise in the galerkin formulation to any accuracy integrals involving completely geometrically separated triangles are nonsingular and are computed using a technique based on spherical harmonics and multipole expansions and translations which require the integration of polynomial functions over the triangles integrals involving cases where the triangles have common vertices edges or are coincident are treated via scaling and symmetry arguments combined with automatic recursive geometric decomposition of the integrals the methods are validated and example results are presented

on the calculation of potential integrals for linear source distributions on triangular domains present analytical formulas for the singular field contributions of linear source distributions on triangular domains the formulas can be used for near field calculations of electric and magnetic source currents especially for the calculation of method of moments mom matrix elements due to the solution of surface integral equations for mom self coupling terms involving the scalar green function of free space the authors present expressions comprising analytical solutions of both the source integrals and the testing integrals

the theory of fuzzy stochastic processes abstract in this paper we deal with the general theory of fuzzy stochastic processes we give the suitable definitions of fuzzy random function fuzzy stochastic process and their fall shadow distribution functions the concepts of separability and measurability of a fuzzy random function are introduced and some properties of these new concepts are derived we investigate the continuity of fuzzy sample function of a fuzzy stochastic process

a central limit theorem for fuzzy random variables abstract fuzzy random variables have been proposed to treat situations in which both random behavior and fuzzy perception must be considered a definition of independence is given for fuzzy random variables as well as a notion of fuzzy gaussian random variables it is shown that a sum or mean of independent fuzzy random variables converges in the limit to a fuzzy gaussian random variable thus providing a fuzzy analogue of the central limit theorem of classical probability theory

on the generalization of directfn for singular integrals over quadrilateral patches a set of fully numerical algorithms for evaluating the 4 d singular integrals arising from galerkin surface integral equation methods over conforming quadrilateral meshes is presented this paper is an extension of directfn which was recently developed for the case of triangular patches utilizing in the same fashion a series of coordinate transformations together with appropriate integration reorderings and is applicable to coincident edge adjacent or vertex adjacent elements the resulting formulas consist of sufficiently smooth kernels and exhibit several favorable characteristics when compared with the vast majority of the methods currently available more specifically they can be applied without modifications to the following challenging cases 1 weakly and strongly singular kernels 2 basis and testing functions of arbitrary order 3 planar and curvilinear patches 4 problem specific green functions e g expressed in spectral integral form and 5 spectral convergence to machine precision finally we show that the overall performance of the fully numerical schemes can be further improved by a judicious choice of the integration order for each dimension

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

numerical integration of the three dimensional green kernel for an electromagnetic problem in this paper we present a method for the numerical integration of the three dimensional green kernel over two triangles this method is compared with the results obtained by other algorithms the comparison proves the efficiency of the technique in removing the numerical singularity furthermore we provide a numerical algorithm for increasing the computation accuracy without refining the finite element mesh

numerical solution of the cfie using vector bases and dual interlocking meshes a numerical solution of the combined field integral equation for wave scattering from homogeneous dielectric bodies is proposed the approach uses a simultaneous representation of the equivalent surface current densities in both curl conforming and divergence conforming bases defined on a dual interlocking mesh representing the scatterer surface the dual mesh based approach is stable at internal resonances and allows the use of the optimum type of basis function for each integral operator within the combined field equation the procedure can also be used to obtain a radiation boundary condition for differential equation formulations

all analytical evaluation of the singular integrals involved in the method of moments surface integral equation methods routinely require the integration of the singular green s function or its gradient over basis functions bf and testing functions tf many techniques have been described in the literature for the fast and accurate computation of these integrals for tf that is located close to the bf in this paper we propose an all analytical formula for the singular part of the integral for both the electric and magnetic field integral equations the method works for any flat polygonal bf and tf of any order and proves to be competitive with existing techniques

on the convergence of fuzzy martingales the purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are lipschitz or continuous fuzzy numbers on rn this approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real valued random variables that appear as their support functions based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales the strong law of large numbers for fuzzy random variables is proved here the convergence is in the uniform metric but not the separable metric on fuzzy number space

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

a novel grid robust higher order vector basis function for the method of moments a set of novel grid robust higher order vector basis functions is proposed for the method of moments mom solution of integral equations for three dimensional 3 d electromagnetic em problems these basis functions are defined over curvilinear triangular patches and represent the unknown electric current density within each patch using the lagrange interpolation polynomials the highlight of these basis functions is that the lagrange interpolation points are chosen to be the same as the nodes of the well developed gaussian quadratures as a result the evaluation of the integrals in the mom is greatly simplified additionally the surface of an object to be analyzed can be easily meshed because the new basis functions do not require the side of a triangular patch to be entirely shared by another triangular patch which is a very stringent requirement for traditional vector basis functions the proposed basis functions are implemented with point matching for the mom solution of the electric field integral equation the magnetic field integral equation and the combined field integral equation numerical examples are presented to demonstrate the higher order convergence and the grid robustness for defective meshes using the new basis functions

bem with linear complexity for the classical boundary integral operators alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as e g galerkin boundary elements including numerical quadrature and panel clustering these representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations low order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered we will present fully discrete galerkin boundary element methods where the storage amount and the cpu time grow only linearly with respect to the number of unknowns

calculation of the impedance matrix inner integral of the magnetic field integral equation to prescribed precision we present a new method to evaluate the inner integral of the impedance matrix elements in the traditional rao wilton glisson formulation of the method of moments for the magnetic field integral equation the evaluation is to a prescribed precision we show that we can write the inner integral in terms of two scalar integrals 1 a surface integral over the integration triangle it and 2 a line integral over the boundary of the it just as with the original integral neither of these integrals can be evaluated analytically in our method we bypass this obstacle by replacing the original integrand modified by a constant phase factor by its taylor series and by keeping enough terms to guarantee a number of significant digits in the integration outcome we have accomplished this for the surface and the line integral we present a systematic derivation of the formulas for the two integrals and we conduct extensive testing of our results

the law of large numbers for fuzzy variables under a general triangular norm extension principle this paper presents a law of large numbers for t noninteractive weakly noninteractive fuzzy variables when using a general t norm extension principle the result is that convergence to a crisp set is obtained for all archimediean t norm intersection operators this generalises a previous result of badard the result is proved by deriving it from a similar result for the law of large numbers for dependency bounds of sums of random variables dependency bounds are used when nothing is known of the joint distribution of two random variables apart from their marginals the bridge used to connect dependency bounds with fuzzy number operations is used to give a probabilistic interpretation of fuzzy number operations some numerical examples are presented and a new method of numerically calculating the sup t convolutions of fuzzy numbers is presented this new method generalises the well known result for the calculation of sup min convolutions in terms of the level sets of membership functions

symmetric triangle quadrature rules for arbitrary functions abstract despite extensive research on symmetric polynomial quadrature rules for triangles as well as approaches to their calculation few studies have focused on non polynomial functions particularly on their integration using symmetric triangle rules in this paper we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions the first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules the second approach better addresses large amounts of points though it is less efficient than the first approach we demonstrate the effectiveness of both approaches on singular integrands which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules

an application of a fuzzy random variable to vulnerability modeling abstract fuzzy random variables have unique value in a modeling context because of their ability to distinguish model components which are incompletely known due to stochastic variation from components which are unknown due to imprecise measurement or inherent vagueness in their quantification in this paper we present a straightforward application of fuzzy methods to an important and ongoing defense problem that of evaluating the susceptibility of an armored vehicle to attack

error covariance bounds for suboptimal filters with lipschitzian drift and poisson sampled measurements abstract we design suboptimal filters for a class of continuous time nonlinear stochastic systems when the measurements are assumed to arrive randomly at discrete times under a poisson distribution the proposed filter is a dynamical system with a differential equation and a reset map which updates the estimate whenever a new measurement is received we analyze the performance of the proposed filter by computing the expected value of the error covariance which is described by a differential equation we study throughly the conditions under which the error covariance remains bounded which depend on the system data and the mean sampling rate associated with the measurement process we also study the particular cases when the error covariance is seen to decrease with the increase in the sampling rate for the particular case of linear filters we can also compare the error covariance bounds with the case when the measurements are continuously available

evaluation of 4 d reaction integrals in the method of moments coplanar element case recently the benefits of simultaneously treating source and testing integrals in the numerical evaluation of 4 d reaction integrals have been reported the reported schemes usually first transform the reaction integral to parametric coordinates and some combination of radial angular and or line segment integrals is then used to treat coincident edge adjacent or vertex adjacent triangular source and test element pairs however advantages of the reported approaches are tempered by their lack of generality and severely degraded performance on poorly shaped elements the latter caused primarily by the parametric transformations severe distortion of the kernel s circularly concentric level contours here for coplanar element pairs and kernels with 1 r singularities we apply the surface divergence theorem twice to obtain a novel formula for 4 d reaction integrals generalizing earlier schemes while retaining their benefits and without distorting the original configuration space numerical results illustrate the method s efficiency which is improved by employing appropriate transformations to further smooth the resulting integrands and hence accelerate their convergence the reaction integral formula can be extended to noncoplanar elements

mom matrix generation based on frequency and material independent reactions fmir mom a novel and efficient method of moments mom matrix generation technique called the frequency and material independent reactions for the method of moments fmir mom technique is presented this new matrix generation algorithm efficiently calculates impedance matrices while sweeping through frequency permittivity conductivity and or permeability values for frequency sweeps it has computational and memory costs comparable to those of interpolation techniques it has the advantage over interpolation techniques in that it does not divide the frequency range into segments and allows one to dynamically update the precision the technique expands the exponential of the green s function into a taylor series this allows the problem to be formulated as a summation where each term consists of a real valued matrix depending only on the geometry discretization multiplied by a scalar dependent on the propagation constant the algorithm s efficiency is obtained by calculating the geometry dependent matrices prior to sweeping through frequency or material parameters

finite width feed and load models we demonstrate a new method of applying the feed model for the method of moments mom formulation for the electric field integral equation efie the model is based around a previously reported magnetic ribbon current model which is accurate and allows for a finite width of the feed port however with proper approximations one can reduce the formulation such that the magnetic field operator can be removed in order to simplify computations arising from the curl of the dyadic green s function and its singularities we show here that the new feed model can also be used to model a lumped element

a strong law of large numbers for fuzzy random variables abstract a limit of a sequence of fuzzy numbers is defined and its some properties are shown based on these concept and properties an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown

equivalent conditions for laws of large numbers for t related l r fuzzy numbers in this paper we give some necessary and sufficient conditions for laws of large numbers for sequence of mutually t related l r fuzzy numbers when t is an continuous archimedean t norm and diameter of support of the fuzzy numbers is not uniformly bounded we also consider some necessary and sufficient conditions for laws of large numbers for l r fuzzy random numbers and generalize earlier results of fuller fuzzy sets and systems 45 1992 299 303 triesch fuzzy sets and systems 58 1993 339 342 and hong and lee fuzzy sets and systems 121 2001 537 543

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

3 d scattering from a pec target buried beneath a dielectric rough surface an efficient pile aca algorithm for solving a hybrid ka efie formulation an efficient hybrid ka efie formulation is deployed to analyze the electromagnetic em scattering from a 3 d perfectly electric conducting pec object buried beneath a 2 d dielectric rough surface in this approach the electric and magnetic current densities on the rough surface are analytically obtained through the current based kirchhoff approximation ka whereas the electric current density on the buried object is rigorously determined by solving the electric field integral equation efie using the galerkin s method of moments mom with rao wilton glisson rwg basis functions the ka efie matrix system is then efficiently solved by the iterative propagation inside layer expansion pile method combined with the algebraic adaptive cross approximation aca the current densities on the dielectric rough surface are thereafter used to handle the bistatic normalized radar cross section nrcs patterns the proposed hybrid approach allows a significant reduction in computation time and memory requirements compared to the rigorous poggio miller chang harrington wu pmchw efie formulation which requires solving a large mom matrix equation moreover the hybridization of the aca algorithm with the pile method improves further the computational cost thanks to the rank deficient propriety of the coupling matrices to validate the hybrid approach we compare its results with those of the rigorous pmchw efie approach

theoretical and numerical treatment of surface integrals involving the free space green s function this paper deals with the problem of the calculation of surface integrals for electromagnetic scattering in the case of the widely popular double triangular basis functions first introduced by rao wilton and glisson 1982 an entire set of formulas is obtained which overrides the difficulties inherent to the singularity of the integrands and results showing the stability accuracy and efficiency of the methods developed are reported in an application of the method of moments to the case of perfectly conducting surfaces and computation of near field in domains including the surface itself furthermore the authors provide insight as regards the capability of triangular basis functions to model near field patterns

a uniform strong law of large numbers for partial sum processes of fuzzy random variables indexed by sets we consider random intervals as measurable mappings from a probability space into the set of intervals of r and prove a uniform strong law of large numbers for sequences of independent and identically distributed random intervals also we consider fuzzy random variables and prove a uniform strong law of large numbers for sequences of fuzzy random variables our results generalize that of bass and pyke ann probab 12 1984 268

convex fuzzy random variables abstract a theory of fuzzy random variables is developed that applies to situations involving both randomness and fuzziness the use of membership functions that are quasi concave play an important role in the theory the expectation of a fuzzy random variable is a fuzzy variable fuzzy set the usual linearity properties of probabilistic expectation carry over to fuzzy random variables a special case of a fuzzy law of large number is proven

integral transformations to handle corner function singularities recent approaches to model corner singularities in electromagnetic analysis require a treatment substantially different from that of edge singularities in this article new algorithms are proposed for handling the combination of corner singularities and green s function singularities on quadrilateral cells in method of moments procedures

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

numerical evaluation via singularity cancellation schemes of near singular integrals involving the gradient of helmholtz type potentials in this paper we present a purely numerical procedure to evaluate strongly near singular integrals involving the gradient of helmholtz type potentials for observation points at finite arbitrarily small distances from the source domain in the proposed approach the source domain is subdivided into a disc plus truncated subtriangles and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity a novel feature of the proposed angular transform is that required discrete values of the inverse transform which is transcendental are determined via a root finding procedure the same idea can also be applied to other transforms that arise in singularity cancellation methods the resulting integral may then evaluated via a low order gauss legendre quadrature scheme

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

embedding calderon multiplicative preconditioners in multilevel fast multipole algorithms calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation efie for perfect electric conductors at lower frequencies previous authors have shown that by using a dense matrix preconditioner based on the calderon identities the low frequency instability is removed while still maintaining the inherent accuracy of the efie it was also demonstrated that the spectral properties of the caldero n preconditioner are conserved during discretization if the efie operator is discretized with rao wilton glisson expansion functions and the preconditioner with buffa christiansen expansion functions in this article we will show how the calderon multiplicative preconditioner cmp can be combined with fast multipole methods to accelerate the numerical solution leading to an overall complexity of o n logn for the entire iterative solution at low frequencies where the cmp is most useful the traditional multilevel fast multipole algorithm mlfma is unstable and we apply the nondirectional stable plane wave mlfma nspwmlfma that resolves the low frequency breakdown of the mlfma the combined algorithm will be called the cmp nspwmlfma applying the cmp nspwmlfma at open surfaces or very low frequencies leads to certain problems which will be discussed in this article

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

considerations on double exponential based cubatures for the computation of weakly singular galerkin inner products highly accurate and efficient cubatures based on the double exponential quadrature rules are presented for the computation of weakly singular integrals arising in galerkin mixed potential integral equation formulations due to their unique ability to handle non smooth kernels the proposed integration schemes can safely replace in a plug n play sense the traditional gauss legendre rules in the existing singularity cancellation and singularity subtraction methods numerical examples using rwg basis functions confirm the excellent performance of the proposed method

an individual ergodic theorem for fuzzy random variables abstract we generalize birkhoff s celebrated individual ergodic theorem to fuzzy random variables in particular it is shown that 1 i the limiting time average exists 2 ii it is invariant and 3 iii can be replaced by the phase average

fast and accurate mom analysis of periodic arrays of multilayered stacked rectangular patches with application to the design of reflectarray antennas the scattering of plane waves by periodic arrays of stacked rectangular patches in multilayered substrates is a problem that has to be solved many times when designing reflectarray antennas made of those patches under the local periodicity assumption the solution to the periodic multilayered problem has been traditionally carried out by means of the galerkin s version of the method of moments mom in the spectral domain this approach involves the computation of double infinite summations and whereas some of these summations converge very fast some other converge very slowly in this paper the slowly convergent summations are computed by making use of an enhanced mixed potential integral equation mpie formulation of the mom in the spatial domain this enhanced formulation is based on the interpolation of the multilayered periodic green s functions and on the efficient computation of the four dimensional 4 d integrals leading to the mom matrix entries both the novel hybrid spectral spatial mom code and the standard spectral domain mom code have been used for the design of a contoured beam reflectarray antenna it has been verified that the spectral spatial mom code requires cpu times that are typically 30 times smaller than those required by the pure spectral domain mom code

advanced numerical schemes for the accurate evaluation of 4 d reaction integrals in the method of moments in this paper we present a simple and efficient fully numerical technique for the evaluation of 4 d reaction integrals arising in the method of moments in the proposed approach the accuracy of the source integration which uses a singularity cancellation scheme is retained in the final 4 d reaction integral through the application of an appropriate variable transformation and numerical quadrature scheme to the test integral several numerical results compared to the usual gauss legendre test quadrature demonstrate the accuracy of the method for both planar and curvilinear discretizations moreover the proposed scheme is independent of test basis functions hence it can be used for higher order basis functions as well as curl conforming or buffa christiansen test basis functions

law of large numbers for uncertain random variables the law of large numbers in probability theory states that the average of random variables converges to its expected value in some sense under some conditions sometimes random factors and human uncertainty exist simultaneously in complex systems and a concept of uncertain random variable has been proposed to study this type of complex systems this paper aims to provide a law of large numbers for uncertain random variables which states that the average of uncertain random variables converges in distribution to an uncertain variable as a byproduct the convergence of a sequence of uncertain variables is also studied

the laws of large numbers for fuzzy random variables the new attempt of weak and strong law of large numbers for fuzzy random variables is discussed in this paper by proposing the convergence in probability and convergence with probability one for fuzzy random variables we first consider the limit properties of fuzzy numbers by invoking the hausdorff metric and then we extend it to the convergence in probability and convergence with probability one for fuzzy random variables we provide the notion of weak and strong convergence in probability and weak and strong convergence with probability one for fuzzy random variables finally we come up with the weak and strong law of large numbers for fuzzy random variables in weak and strong sense

limit theorems for fuzzy random variables this paper deals with limit theorems for fuzzy valued measurable mappings which provide as a whole a foundation of statistical analysis with fuzzy data a strong law of large numbers a central limit theorem and a gliwenko cantelli theorem are proved the results are formulated simultaneously with respect to the lp metrics on the fuzzy sample spaces investigated by diamond and kloeden in particular these versions of the limit theorems are related to identical compatible concepts of convergence and measurability in the fuzzy sample spaces the proofs of the theorems are based heavily on isomorphic isometric embeddings of the fuzzy sample spaces endowed with lp metrics into respective lp spaces which are banach spaces of type 2 these embeddings provide the application of convergence results in banach spaces

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

efficient evaluation of double surface integrals in time domain integral equation formulations a new integration approach is presented for accurately calculating time domain efie mfie and cfie matrix elements over triangular domains it mainly consists of a radial integration scheme for handling weakly singular and near hypersingular inner integrals and some new smoothing techniques for treating outer two dimensional 2 d integrals the proposed approach has sufficient generality and efficiency for solving time domain integral equations tdie with arbitrary types of temporal basis functions and temporal discretization schemes such as marching on in time mot marching on in degree mod and finite difference delay modeling convolution quadrature fddm cq etc the numerical results for calculating some typical integrals are given to demonstrate its capability with high accuracy and rapid convergence rate achieved

on the stability of kalman bucy diffusion processes the kalman bucy filter is the optimal state estimator for an ornstein uhlenbeck diffusion given that the system is partially observed via a linear diffusion type noisy sensor under gaussian assumptions it provides a finite dimensional exact implementation of the optimal bayes filter it is generally the only such finite dimensional exact instance of the bayes filter for continuous state space models consequently this filter has been studied extensively in the literature since the seminal 1961 paper of kalman and bucy the purpose of this work is to review re prove and refine existing results concerning the dynamical properties of the kalman bucy filter so far as they pertain to filter stability and convergence the associated differential matrix riccati equation is a focal point of this study with a number of bounds convergence and eigenvalue inequalities rigorously proven new results are also given in the form of exponential and comparison inequalities for both the filter and the riccati flow

fuzzy sets and typicality theory abstract a set whose elements take grades of membership in the interval 0 1 is known as a fuzzy set in practice assignment of grades of membership to the elements of a fuzzy set is based on statistical considerations this paper is concerned with the problem of identifying what should be meant by a typical element of a fuzzy set because of the subjective nature of membership grades use of the common statistical measures such as the mean or median is theoretically unjustifiable in an effort to solve this problem we draw heavily on the works of kruse kwakernaak ralescu kandel and others in particular we employ the concepts of the fuzzy expected value and fuzzy random variable as the basis for a measure of what is typical of a fuzzy set an approach which we believe shows promise in fields like pattern recognition and expert systems we propose a flexible algorithm which incorporates subjective ideas of typicality and some simple statistical techniques to arrive at the typical element of a fuzzy set no attempt is made to further fuzzify the element of the given fuzzy set with respect to the typical element we believe however that it is an issue worth considering at a later stage the investigation assumes that we are given a well defined fuzzy set i e the grades of membership are known subjective feelings of cohesiveness within a set based on proximity and density for example are incorporated within the algorithm

singularity subtraction technique for high order polynomial vector basis functions on planar triangles in this paper a singularity subtraction technique is developed for computing the impedance matrix elements of various electromagnetic surface integral equation formulations with the galerkin method and high order basis functions analytical closed form formulas for computing surface integrals with r r sup n n spl ges 3 singularities times polynomial nodal shape functions of arbitrary order on a planar triangle are presented

a law of large numbers for fuzzy numbers abstract we study the problem if 1 2 are fuzzy numbers with modal values m1 m2 then what is the strongest t norm for which where mn m1 mn n the arithmetic mean 1 n n i is defined via sup t norm convolution and nes denotes necessity

on the evaluation of the 4 d reaction integral for the scalar potential in galerkin s method of moments despite the great progress made in the evaluation of the 4 d reaction integrals arising in the method of moments applied to the surface integral equation formulations there are still some aspects that deserve to be explored further the focus in this paper lies in the evaluation of the 4 d reaction integral for the scalar potential we demonstrate that unless this integral is computed to machine precision in the whole impedance matrix it is imperative to maintain the trianglewise balance in the evaluation of the scalar potential part a small imbalance even at very small error results in a loss of degrees of freedom dofs for the charge distribution violating the null space property of the divergence operator and yielding potentially spurious solutions the effect although present may go unnoticed at the conventional frequency regimes where the usual mesh sizes can be applied however it becomes more pronounced as we approach the quasi static limit eventually rendering spurious solutions at the frequencies higher than the classical low frequency breakdown would

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

mixed inner outer iteration technique based surface integral equations for fast solving em scattering from penetrable objects in this paper a mixed inner outer iteration technique based on surface integral equation is proposed for solving scattering problems from penetrable and homogeneous dielectric structures the most important novelty of the proposed technique is combining the tangential poggio miller chang harrington wu tsai as outer iteration and the electric magnetic current combined field integral equation as inner iteration respectively in this case the new method offers the guarantee of high accuracy with faster convergence especially for electrically dielectric structures in large scale furthermore it presents an excellent compatibility with sparse approximate inverse preconditioning numerical results are given to demonstrate the accuracy efficiency and compatibility of the proposed method

evaluation of weakly singular integrals via generalized cartesian product rules based on the double exponential formula various weakly singular integrals over triangular and quadrangular domains arising in the mixed potential integral equation formulations are computed with the help of novel generalized cartesian product rules the proposed integration schemes utilize the so called double exponential quadrature rule originally developed for the integration of functions with singularities at the endpoints of the associated integration interval the final formulas can easily be incorporated in the context of singularity subtraction singularity cancellation and fully numerical methods often used for the evaluation of multidimensional singular integrals the performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods

numerical integration scheme for the near singular green function gradient on general triangles a new near singularity cancellation transformation quadrature scheme for triangle domains is presented it is tailored to the green function gradient kernel static and dynamic multiplied with higher order weighting functions on curvilinear triangles the transformation is analytically invertible attention is paid to ensuring accuracy in both tangential and normal components such integrals must be routinely evaluated in modern method of moments mom implementations the scheme follows the standard three subtriangle splitting approach of which the benefits are single parameter accuracy control a geometry independent total number of quadrature points and implementation simplicity extensive numerical results show superior efficiency and reliability to existing three subdomain schemes consistent dependable error convergence is demonstrated with very low variation in accuracy for fixed quadrature order and singularity height the scheme is suitable for practical use by mom developers

calculation of mom interaction integrals in highly conductive media the construction of the impedance matrix in the method of moments requires the calculation of interaction integrals between the expansion functions through the green s function and its derivatives the singular behavior of the green s function poses considerable problems for an accurate numerical evaluation of these integrals requiring techniques such as singularity extraction or cancellation in this contribution we will show why these methods fail when the medium is highly conductive a novel technique is proposed to handle these highly challenging integrals the complexity of the new method is independent of the conductivity

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

on the near interaction elements in integral equation solvers for electromagnetic scattering by three dimensional thin objects the accuracy of near interaction ni evaluation is investigated for hypersingular and strongly singular kernels in solving electromagnetic em integral equations we first define the ni between a field point and a source triangular patch using a nearness factor nf and show the accuracy of numerical integrations for the ni integrals with hypersingular or strongly singular kernels it is found that there exists a transition area or turning point before and after which the convergence behaviors of those integrations are greatly different we define the nf value at the 1 relative error as the turning point nf 0 and define nf les nf 0 as the ni range we find that nf 0 1 4 could be a bound of ni range for the triangles with a quality factor q 0 35 and analytical formulas should be used in evaluating ni elements within the bound numerical examples for em scattering by three dimensional 3d thin objects are used to demonstrate the significance of ni treatments for the accurate solutions

on the dual basis for solving electromagnetic surface integral equations a powerful technique for solving electromagnetic em surface integral equations sies for inhomogenous objects by the method of moments mom involves the well known rao wilton glisson rwg basis function to represent the electric current and for field orthogonality and numerical stability reasons a variation of the rwg basis known as the ntilde x rwg basis where ntilde is a unit normal vector at the object surface to represent the magnetic current though this combination provides a numerically efficient and effective solution that has been demonstrated on a variety of structures one cannot feel entirely comfortable because of the presence of fictitious magnetic current associated with the modified basis chen and wilton proposed a different smoother basis in 1990 that avoids the fictitious line charges but because of computational cost issues it has not been used beyond chen s dissertation recently this basis was rediscovered and has received considerable attention our work reexamines the dual basis exploring issues not addressed by chen and wilton and showing accurate solutions for a variety of em scattering structures

optimized numerical evaluation of singular and near singular potential integrals involving junction basis functions in analyzing conducting bodies composed of surfaces connected to wires via a method of moments discretization of the electric field integral equation junction basis functions are defined at connection points between wires and surfaces to correctly capture the physics of the current flux in the vicinity of the junction in evaluating the associated self term vector potential surface integrals two singularities are present one is related to the green s function and the other is due to the 1 behavior of the junction basis function that models the current near a junction here we present a simple and efficient numerical procedure for evaluating singular and near singular source vector potential integrals involving junction basis functions the approach is based on a double transformation that cancels both singularities and also overcomes some limitations of singularity subtraction methods the proposed scheme is then optimized to obtain at least four correct digits in the evaluation of these integrals essentially arbitrary accuracy is achievable by suitably increasing the quadrature order

integral equation based nonoverlapping ddm using the explicit boundary condition a new integral equation based nonoverlapping domain decomposition method nddm is proposed for analyzing electromagnetic em scattering from electrically large pec objects whose mlfma models are too large for the user s computer to accommodate the integral equation is first built on the entire pec surface and the entire surface is partitioned into some nonoverlapping subsurfaces the rwg functions are used to expand the surface current on the entire surface then each full rwg function across some boundary curve is split into two half rwg functions whose unknown coefficients are constrained by the boundary current continuity using the explicit boundary condition the convergence of the outer iterative scheme of the proposed nddm is investigated theoretically and demonstrated to be very good and hence a very large problem can be effectively solved in an ordinary computer some numerical examples are provided to demonstrate the correctness and robustness of the proposed method and to compare the proposed method with the existing overlapping ddms having similar attributes

kolmogorov s strong law of large numbers for fuzzy random variables in this paper kolmogorov s strong law of large numbers for sums of independent and level wise identically distributed fuzzy random variables is obtained

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

one way domain decomposition method with exact radiation condition and fast gmres solver for the solution of maxwell s equations for the solution of the time harmonic electromagnetic scattering problem by inhomogeneous 3 d objects a one way domain decomposition method ddm is considered the computational domain is partitioned into concentric subdomains on the interfaces of which robin type transmission conditions tcs are prescribed an integral representation of the electromagnetic fields on the outer boundary constitutes an exact radiation condition the global system obtained after discretization of the finite element fe formulations is solved via a krylov subspace iterative method gmres it is preconditioned in such a way that essentially only the solution of the fe subsystems in each subdomain is required this is made possible by a computationally cheap h curl h div transformation performed on the interfaces that separate the two outermost subdomains the eigenvalues of the preconditioned matrix of the system are bounded by two and optimized values of the coefficients involved in the local tcs on the interfaces are determined so as to maximize the minimum eigenvalue numerical experiments are presented that illustrate the numerical accuracy of this technique its fast convergence and legitimate the choices made for the optimized coefficients

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

iterative solutions for electromagnetic fields at perfectly reflective and transmissive interfaces using clifford algebra and the multidimensional cauchy integral a technique is presented for calculating the solution of maxwell s equations using a cfie based on the cauchy integral and formulated in the guise of clifford algebra this formulation has a geometric interpretation leading to an iterative method of solution which is easily proven as convergent and correct for both perfectly reflective and perfectly transmissive interfaces simple test cases involving a cubic boundary and plane wave and dipole sources are used to investigate the numerical performance of the method

asymptotic properties of linear filter for deterministic processes abstract it is known that kalman bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability in this paper we prove the stability of kalman bucy filter for the case of noise free dynamical system i e for deterministic processes the earlier stability results cannot be applied for this case as the system is not controllable we further show that the optimal linear filter for a general class of non gaussian initial conditions is asymptotically proximal to kalman bucy filter it is also shown that the filter corresponding to non zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of gaussian initial conditions