an upgraded aca algorithm in complex field and its statistical analysis an upgraded adaptive cross approximation u aca algorithm in complex field is proposed including detailed discussion about both relative error estimator and pivoting threshold two new relative error estimators are constructed and the range of the pivoting threshold is determined theoretically besides a statistical analysis of the u aca algorithm for computational electromagnetics is provided to identify the credibility of existing relative error estimators and new ones the statistical experiments show that the u aca algorithm with either of the two new relative error estimators is a more robust algorithm at present for applications in computational electromagnetics compared with the previous aca algorithm

asymptotics of racah coefficients and polynomials ponzano and regge conjectured certain asymptotic formulae for racah coefficients with large angular momenta and gave convincing numerical evidence of their validity we provide rigorous proofs of their asymptotic formulae we use generating functions to derive an integral representation of racah polynomials and then apply complex analytic methods to determine the asymptotic behaviour of the racah coefficients

asymptotics of discrete orthogonal polynomials and the continuum limit of the toda lattice in a recent monograph deift and mclaughlin considered the continuum limit of the toda lattice and solved the problem using perturbation techniques such as the wentzel kramers brillouin wkb method we present an alternative approach based on the analytic theory of discrete orthogonal polynomials the method is based on some extremal problems in the theory of logarithmic potentials and relies on some results for the asymptotic theory of orthogonal polynomials with a discrete orthogonality measure

divergence runge phenomenon for least squares polynomial approximation on an equispaced grid and mock chebyshev subset interpolation runge showed more than a century ago that polynomial interpolation of a function f x using points evenly spaced on email protected 1 1 could diverge on parts of this interval even if f x was analytic everywhere on the interval least squares fitting of a polynomial of degree n to an evenly spaced grid with p points should improve accuracy if email protected n we show through numerical experiments that such an overdetermined fit reduces but does not eliminate the runge phenomenon more precisely define b n 1 p the least squares fit will fail to converge everywhere on 1 1 as n for fixed b if f x has a singularity inside a curve d b in the complex plane the width of the region enclosed by the convergence boundary d shrinks as b diminishes i e more points for a fixed polynomial degree but shrinks to the interval 1 1 only when b 0 we also show that the runge phenomenon can be completely defeated by interpolation on a mock chebyshev grid a subset of n 1 points from an equispaced grid with o n 2 points chosen to mimic the non uniform n 1 point chebyshev lobatto grid

krawtchouk polynomials and universal bounds for codes and designs in hamming spaces universal bounds for the cardinality of codes in the hamming space f sub r sup n with a given minimum distance d and or dual distance d are stated a self contained proof of optimality of these bounds in the framework of the linear programming method is given the necessary and sufficient conditions for attainability of the bounds are found the parameters of codes satisfying these conditions are presented in a table a new upper bound for the minimum distance of self dual codes and a new lower bound for the crosscorrelation of half linear codes are obtained

approximation on non tensor domains including squircles part iii single domain spectral methods have been largely restricted to tensor product bases on a tensor product grid to break the tensor barrier we study approximation in a domain bounded by a squircle the zero isoline of b x y x 2 y 2 1 the boundary varies smoothly from a circle 1 to the square polynomial least squares hyperinterpolation converges geometrically as long as the number of points p is at least double the number of basis functions n the polynomial grid was made denser near the boundaries chebyshev like by depositing grid points along wisely chosen contours of b gaussian radial basis functions rbfs were more robust in the sense that they too converged geometrically but hyperinterpolation p n and a chebyshevized grid were unnecessary a uniform grid truncated to include only those points within the squircle was satisfactory even without interpolation points on the boundary although boundary points are a cost effective improvement for a given number of points p however rbf interpolation was only slightly more accurate than polynomial hyperinterpolation and needed twice as many basis functions interpolation costs can be greatly reduced by exploiting the invariance of the squircle bounded domain to the eight elements d 4 dihedral group

finite part integral and boundary element method to solve three dimensional crack problems in piezoelectric materials using the hypersingular integral equation method based on body force method a planar crack in a three dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed this crack problem is reduced to solve a set of hypersingular integral equations compare with the crack problems in elastic isotropic materials it is shown that for the impermeable crack the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials based on the exact analytical solution of the singular stresses and electrical displacements near the crack front the numerical method of the hypersingular integral equation is proposed by the finite part integral method and boundary element method which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied finally the numerical solutions of the stress and electric field intensity factors of some examples are given

distributions of zeros of discrete and continuous polynomials from their recurrence relation the hypergeometric polynomials in a continuous or a discrete variable whose canonical forms are the so called classical orthogonal polynomial systems are objects which naturally appear in a broad range of physical and mathematical fields from quantum mechanics the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of sturm liouville differential and difference equations often they are encountered in the form of a three term recurrence relation ttrr which connects a polynomial of a given order with the polynomial of the contiguous orders this relation can be directly found in particular by use of lanczos type methods tight binding models or the application of the conventional discretization procedures to a given differential operator here the distribution of zeros and its asymptotic limit characterized by means of its moments around the origin is described for the continuous classical hermite laguerre jacobi bessel polynomials and for the discrete classical charlier meixner kravchuk hahn polynomials by means of a general procedure which i only requires the three term recurrence relation and ii avoids the often high brow subtleties of the potential theoretic considerations used in some recent approaches let us underline that the orthogonality condition is not required in our approach the moments are given in an explicit manner which at times allow us to recognize the analytical form of the corresponding distribution

a continuum limit of the relativistic toda lattice asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients we consider the continuum limit of the relativistic toda lattice in particular we propose a method in order to integrate this system of nonlinear partial differential equations for some particular initial data and boundary conditions before possible shocks first we recall the relation between the finite relativistic toda lattice and the theory of discrete laurent orthogonal polynomials our analysis is then based on some results for the asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients and the connection with a constrained and weighted extremal problem for logarithmic potentials

phase transitions and equilibrium measures in random matrix models the paper is devoted to a study of phase transitions in the hermitian random matrix models with a polynomial potential in an alternative equivalent language we study families of equilibrium measures on the real line in a polynomial external field the total mass of the measure is considered as the main parameter which may be interpreted also either as temperature or time our main tools are differentiation formulas with respect to the parameters of the problem and a representation of the equilibrium potential in terms of a hyperelliptic integral using this combination we introduce and investigate a dynamical system system of odes describing the evolution of families of equilibrium measures on this basis we are able to systematically derive a number of new results on phase transitions such as the local behavior of the system at all kinds of phase transitions as well as to review a number of known ones

on the constrained mock chebyshev least squares the algebraic polynomial interpolation on n 1 uniformly distributed nodes can be affected by the runge phenomenon also when the function f to be interpolated is analytic among all techniques that have been proposed to defeat this phenomenon there is the mock chebyshev interpolation which produces a polynomial p that interpolates f on a subset of m 1 of the given nodes whose elements mimic as well as possible the chebyshev lobatto points of order m in this work we use the simultaneous approximation theory to produce a polynomial p of degree r greater than m which still interpolates f on the m 1 mock chebyshev nodes minimizing at the same time the approximation error in a least squares sense on the other points of the sampling grid we give indications on how to select the degree r in order to obtain polynomial approximant good in the uniform norm furthermore we provide a sufficient condition under which the accuracy of the mock chebyshev interpolation in the uniform norm is improved numerical results are provided

weakly singular stress bem for 2d elastostatics a weakly singular stress bem is presented in which the linear state regularizing field is extended over the entire surface the algorithm employs standard conforming c0 elements with lagrangian interpolations and exclusively uses gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space the linear state stress bie on which the algorithm is based has no free term so that the bem treatment of external corners requires no special consideration other than to admit traction discontinuities the self regularizing nature of the somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary a key observation is the relation between bie density functions and successful interpolation orders numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations additionally the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions copyright 1999 john wiley sons ltd

approximate calculation of sums i bounds for the zeros of gram polynomials let n be a positive integer and x j be n equidistant points we propose an algorithmic approach for approximate calculation of sums of the form sum j 1 n f x j the method is based on the gaussian type quadrature formula for sums sum j 1 n f x j approx sum k 1 n b n k f g n k n n ll n where g n k n are the zeros of the so called gram polynomials this allows the calculation of sums with very large number of terms n to be reduced to sums with a much smaller number of summands n the first task in constructing such a formula is to calculate its nodes g n k n in this paper we obtain precise lower and upper bounds for g n k n numerical experiments show that the estimates for the zeros g n k n are very sharp and that the proposed method for calculation of sums is efficient

a mapped polynomial method for high accuracy approximations on arbitrary grids the focus of this paper is the approximation of analytic functions on compact intervals from their pointwise values on arbitrary grids we introduce a new method for this problem based on mapped polynomial approximation by careful selection of the mapping parameter we ensure both high accuracy of the approximation and an asymptotically optimal scaling of the polynomial degree with the grid spacing as we explain efficient implementation of this method can be achieved using nonuniform fast fourier transforms numerical results demonstrate the efficiency and accuracy of this approach

efficient sampling of electromagnetic fields via the adaptive cross approximation the adaptive cross approximation method is applied to the problem of efficiently sampling an electromagnetic field across two dimensions the algorithm uses previously collected samples to determine which field samples to take next and reduces the number of samples required to accurately represent the field in this way the field may be sampled faster than it would have been if it had been raster scanned this is achieved without significant changes to the sensor hardware

an accurate stable and efficient domain type meshless method for the solution of mhd flow problems the aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to dirichlet neumann and robin boundary conditions toward this the meshless point collocation method mpcm is used for mhd flow problems in channels with fully insulating or partially insulating and partially conducting walls having rectangular circular elliptical or even arbitrary cross sections mpc is a truly meshless and computationally efficient method the maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently and the convergence proof for the numerical solution of the poisson problem with dirichlet boundary conditions have been attained also additionally in the present work convergence is attained for neumann and robin boundary conditions accordingly the shape functions are constructed using the moving least squares mls approximation the refinement procedure with meshless methods is obtained with an easily handled and fully automated manner we present results for hartmann number up to 10 5 the numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional fem and bem for the dirichlet neumann and robin boundary conditions of interior problems with simple or complex boundaries

uniform asymptotics and zeros of a system of orthogonal polynomials defined via a difference equation abstract we study a class of sieved pollaczek polynomials defined by a second order difference equation three term recurrence relation the measure of orthogonality is determined by using the markov theorem and the perron stieltjes inversion formula and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point n and a critical point n involving respectively the airy function ai and a 1 2 ai i 2 bi darboux s method the method of steepest descents and various uniform asymptotic techniques such as cubic transformations are used to derive the results asymptotic formulas for the least zeros the largest zeros and the zeros on both sides of n are also obtained several numerical examples are provided to compare the approximate zeros with the true values

symmetric weak form integral equation method for three dimensional fracture analysis a symmetric galerkin boundary element method is developed for the analysis of linearly elastic isotropic three dimensional solids containing fractures the formulation is based upon a weak form displacement integral equation and a weak form traction integral equation recently developed by li and mear 1997 these integral equations are only weakly singular and their validity requires only that the boundary displacement data be continuous hence allowing standard co elements to be employed as part of the numerical implementation a special crack tip element is developed which has a novel feature in that there exist degrees of freedom associated with the nodes at the crack front as a result a higher degree of approximation is achieved for the relevant displacement data on the crack and further the stress intensity factors are obtained directly in terms of the crack front nodal data various examples are treated for cracks in unbounded domains and for cracks in finite domains including both embedded and surface breaking cracks and it is demonstrated that highly accurate results can be achieved using relatively coarse meshes

on the convergence of the aca algorithm for radiation and scattering problems the adaptive cross approximation aca algorithm when used to accelerate the numerical solution of integral equations for radiation and scattering problems sometimes suffers from inaccuracies these inaccuracies occur when the aca convergence criterion which is based on an approximation of the residual relative error is prematurely satisfied this paper identifies the two sources of this problem and proposes adaptations of the algorithm that remedy them

spectral methods in non tensor geometry part ii single domain spectral methods have been largely restricted to tensor product bases on a tensor product grid to break the tensor barrier we studied approximation in two idealized families of domains one family is bounded by a squircle the zero isoline of b x y x 2 y 2 1 the boundary varies smoothly from a circle 1 to the square the other family is bounded by a perturbed quadrifolium the plane algebraic curve x 2 y 2 x 2 y 2 3 x 2 y 2 2 this varies smoothly from the singular self intersecting curve known as the quadrifolium to a circle as varies from zero to infinity we compared two different bivariate polynomial bases truncating by total degree zernike polynomials are natural for the disk tensor products of chebyshev polynomials are equally sensible for the square both yield an exponential rate of convergence for our non tensor neither disk nor square domains indeed the chebyshev basis worked well for the disk and the zernike polynomials were good for the square the expected differences due to numerical ill conditioning did not emerge much to our surprise the price for the nontensor domain was that hyperinterpolation was necessary that is least squares fitting with more interpolation constraints than unknowns denoting the number of interpolation points by p and the basis size by n a ratio of p n around two to three was optimum while p near one was very inacccurate a uniform grid truncated to include only those points within the squircle or other boundary curve was satisfactory even without interpolation points on the boundary although boundary points are a cost effective improvement interpolation costs were greatly reduced by exploiting the invariance of the squircle bounded and perturbed quadrifolium domains to the eight element d4 dihedral group

searching globally optimal parameter sequence for defeating runge phenomenon by immunity genetic algorithm data interpolation is a fundamental data processing tool in scientific studies and engineering applications however when interpolating data points on an equidistant grid using polynomials the so called runge phenomenon may occur making polynomial interpolation unreliable although there are some methods proposed to defeat the runge phenomenon it is still an open problem which parameter sequence is the globally optimal for overcoming the runge phenomenon in this paper we develop an immunity genetic algorithm based method to solve this problem specifically we first model the runge phenomenon defeating problem as an optimization in which the objective function is the energy of the parametric curve an immunity genetic algorithm is then devised to determine the best iga parameter sequence which minimizes the objective function the resulting parametric curve overcomes the runge phenomenon by performing the proposed immunity genetic searching algorithm starting with some groups of randomly generated parameter sequences the resulted parameter sequences closely oscillate around the chebyshev parameter sequence therefore the chebyshev parameter sequence is most likely the globally optimal sequence conquering the runge phenomenon

extremal polynomials on discrete sets we study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets typical examples being the meixner polynomials and the charlier polynomials following ideas of rakhmanov dragnev and saff weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials which can be solved explicitly we give formulas for the contracted zero distributions the n th root asymptotics and the asymptotics of the largest zeros

a numerical solution of the constrained weighted energy problem a numerical algorithm is presented to solve the constrained weighted energy problem from potential theory as one of the possible applications of this algorithm we study the convergence properties of the rational lanczos iteration method for the symmetric eigenvalue problem the constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the ritz values the region depends on the distribution of the eigenvalues on the distribution of the poles and on the ratio between the size of the matrix and the number of iterations our algorithm gives the possibility of finding the boundary of this region in an effective way we give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational lanczos algorithm

the fourier extension method and discrete orthogonal polynomials on an arc of the circle abstract the fourier extension method also known as the fourier continuation method is a method for approximating non periodic functions on an interval using truncated fourier series with period larger than the interval on which the function is defined when the function being approximated is known at only finitely many points the approximation is constructed as a projection based on this discrete set of points in this paper we address the issue of estimating the absolute error in the approximation the error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle and these polynomials are then evaluated asymptotically using riemann hilbert methods

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

a stability barrier for reconstructions from fourier samples we prove that any stable method for resolving the gibbs phenomenon that is recovering high order accuracy from the first m fourier coefficients of an analytic and nonperiodic function can converge at best root exponentially fast in m any method with faster convergence must also be unstable and in particular exponential convergence implies exponential ill conditioning this result is analogous to a recent theorem of platte trefethen and kuijlaars concerning recovery from pointwise function values on an equispaced m grid the main step in our proof is an estimate for the maximal behavior of a polynomial of degree n with bounded m term fourier series this estimate is related to a conjecture of hrycak and grochenig in the final part of the paper we discuss the implications of our main theorem to polynomial based interpolation and least squares approaches for overcoming the gibbs phenomenon

fast analysis of large antenna arrays using the characteristic basis function method and the adaptive cross approximation algorithm the characteristic basis function method cbfm has been hybridized with the adaptive cross approximation aca algorithm to construct a reduced matrix equation in a time efficient manner and to solve electrically large antenna array problems in core with a solve time orders of magnitude less than those in the conventional methods various numerical examples are presented that demonstrate that the proposed method has a very good accuracy computational efficiency and reduced memory storage requirement specifically we analyze large 1 d and 2 d arrays of electrically interconnected tapered slot antennas tsas the entire computational domain is subdivided into many smaller subdomains each of which supports a set of characteristic basis functions cbfs we also present a novel scheme for generating the cbfs that do not conform to the edge condition at the truncated edge of each subdomain and provide a minor overlap between the cbfs in adjacent subdomains as a result the cbfs preserve the continuity of the surface current across the subdomain interfaces thereby circumventing the need to use separate ldquoconnectionrdquo basis functions

a new boundary integral formulation for plane elastic bodies containing cracks and holes this paper presents a new boundary integral formulation for a plane elastic body containing an arbitrary number of cracks and holes the body is assumed to be linear elastic and isotropic but can be of either finite or infinite extend the cracks inside the body can be either internal or edge crack and either straight or curvilinear and the holes can be of arbitrary number and shape starting from somigliana formula we obtain a system of boundary integral equations by applying integration by parts in complex variables notation the stress and displacement components can be expressed in terms of muskhelishvili s analytic functions which are in turn written as functions of boundary traction and displacement data in the form of cauchy integral the complex boundary integral equations for traction involve only singularity of order 1 1 where r is the distance measured from the singular boundary points and no hypersingular terms appear this new boundary integral formulation provides an effective basis in solving problems both analytically and numerically to illustrate the validity of our new integral formulation a number of classical problems are re examined analytically using the present formulation i an infinite body containing a circular hole subject to far field biaxial stress internal pressure and a point force on the hole s boundary respectively and ii an infinite body containing a circular are crack under remote uniaxial tension to illustrate the applicability of the present formulation for boundary element method analysis two numerical examples for the interactions between two collinear cracks are considered and the results agree well with the existing solutions by chandra et al 1995 for the case of finite rectangular plates and with isida cited in p 195 of murakami 1987 for the case of infinite plates c 1999 elsevier science ltd all rights reserved

a fast algorithm for the convolution of functions with compact support using fourier extensions we present a new algorithm for computing the convolution of two compactly supported functions the algorithm approximates the functions to be convolved using fourier extensions and then uses the fast fourier transform to efficiently compute fourier extension approximations to the pieces of the result the complexity of the algorithm is mathcal o bigl n log n 2 bigr where n is the number of degrees of freedom used in each of the fourier extensions

superlinear convergence of conjugate gradients we give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other krylov subspace methods we present a new bound on the relative error after n iterations this bound is valid in an asymptotic sense when the size n of the system grows together with the number of iterations the bound depends on the asymptotic eigenvalue distribution and on the ratio n n under appropriate conditions we show that the bound is asymptotically sharp our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials an important tool in our investigations is a constrained energy problem in logarithmic potential theory the new asymptotic bounds for the rate of convergence are illustrated by discussing toeplitz systems as well as a model problem stemming from the discretization of the poisson equation

boundary element free method for fracture analysis of 2 d anisotropic piezoelectric solids this paper considers a 2 d fracture analysis of anisotropic piezoelectric solids by a boundary element free method a traction boundary integral equation bie that only involves the singular terms of order 1 r is first derived using integration by parts new variables namely the tangential derivative of the extended displacement the extended displacement density for the general boundary and the tangential derivative of the extended crack opening displacement the extended displacement dislocation density are introduced to the equation so that solution to curved crack problems is possible this resulted equation can be directly applied to general boundary and crack surface and no separate treatments are necessary for the upper and lower surfaces of the crack the extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions and the unknown weight functions are modelled using the moving least squares mls approximation the numerical scheme of the boundary element free method is established and an effective numerical procedure is adopted to evaluate the singular integrals the extended stress intensity factors sifs are computed for some selected example problems that contain straight or curved cracks and good numerical results are obtained copyright 2006 john wiley sons ltd

analyzing interaction between coplanar square cracks using an efficient boundary element free method summary this paper examines the interaction between coplanar square cracks by combining the moving least squares mls approximation and the derived boundary integral equation bie a new traction bie involving only the cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation the new traction bie can be directly applied to a crack surface and no displacement bie is necessary because all crack boundary conditions both upper and lower ones are incorporated a boundary element free method is then developed by combining the derived bie and mls approximation in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms and the unknown weight is approximated with the mls approximation the efficiency of the developed method is tested for isotropic and transversely isotropic media the interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed copyright 2012 john wiley sons ltd

the dual boundary element method effective implementation for crack problems summary the present paper is concerned with the effective numerical implementation of the two dimensional dual boundary element method for linear elastic crack problems the dual equations of the method are the displacement and the traction boundary integral equations when the displacement equation is applied on one of the crack surfaces and the traction equation on the other general mixed mode crack problems can be solved with a single region formulation both crack surfaces are discretized with discontinuous quadratic boundary elements this strategy not only automatically satisfies the necessary conditions for the existence of the finite part integrals which occur naturally but also circumvents the problem of collocation at crack tips crack kinks and crack edge corners examples of geometries with edge and embedded crack are analysed with the present method highly accurate results are obtained when the stress intensity factor is evaluated with the j integral technique the accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed mode conditions

fft based method with near matrix compression in this paper a novel grouping scheme of the basis functions within the framework of the fast fourier transform fft based method is proposed for creating a block sparse structure for the near matrix and then fft based method with near matrix compression is established for the efficient analysis of multiscale problems for a multiscale problem if an fft based method is required to maintain both higher efficiency and higher accuracy at calculating the far interactions then the near matrix will be inevitably very large compared with the traditional fft based method the proposed method can significantly reduce the near matrix memory requirement without increasing the time spent several numerical examples are provided to demonstrate the correctness and the efficiency of the proposed method

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

multilevel fast adaptive cross approximation algorithm with characteristic basis functions this paper presents a multilevel fast adaptive cross approximation mlfaca algorithm for accelerated iterative solution of the method of moments mom matrix equation for electrically large targets the mlfaca compresses the impedance submatrices between well separated blocks into products of sparse matrices constructed with the aid of the fast adaptive cross sampling facs scheme and the butterfly algorithm as a result the mlfaca can reduce both the computational time and the storage of the mom to mathbf o mathbf n mathbf log mathbf 2 text n where mathbf n is the number of the rao wilton glisson rwg basis functions in the analyzed target meanwhile the mlfaca maintains the adaptive and kernel independent properties furthermore the characteristic basis function method cbfm is employed to decrease the size of the outer matrices of the mlfaca to further reduce the storage and iteration time numerical results are presented to demonstrate the advantages of the proposed method including a successful solution of a scattering problem involving 10 861 668 rwg basis functions

bounds for polynomials with a unit discrete norm let e be the set of n equidistant points in 1 1 and fn e be the set of all polynomials p of degree n with max p c c e 1 we prove that 7t kn n x max p x clog p e arctan ky s x r v 1 n2 jv2 where n n and c is an absolute constant the result is essentially sharp bounds for kn z z g c uniform for n n are also obtained the method of proof of those results is a general one it allows one to obtain sharp or sharp up to a log n factor bounds for kn under rather general assumptions on e j5 n a model result is announced for a class of sets e main components of the method are discussed in some detail in the process of investigating the case of equally spaced points

constrained leja points and the numerical solution of the constrained energy problem a numerical method for solving the constrained energy problem from potential theory is proposed the method is based on the use of constrained leja points several numerical examples related to significant problems are presented and the results are discussed

zeros of classical orthogonal polynomials of a discrete variable in this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable considered as functions of a parameter by using a theorem of a markov and the so called hellmannfeynman theorem comparisons with previous results for zeros of hahn meixner kravchuk and charlier polynomials are also presented

derivatives of the three dimensional green s functions for anisotropic materials abstract a concise formulation is presented for the derivatives of green s functions of three dimensional generally anisotropic elastic materials direct calculation for derivatives of the green s function on the cartesian coordinate system is a common practice which however usually leads to a complicated course in this paper the green s function derived by ting and lee ting t c t lee v g 1997 the three dimensional elastostatic green s function for general anisotropic linear elastic solids the quarterly journal of mechanics and applied mathematics 50 3 407 426 is extended to obtain the derivatives using a spherical coordinate system the green s function can be shown as the composition of two independent functions one depends only on the radial distance of the field point to the origin and the other is in spherical angles the method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly with the application of cauchy residue theorem the contour integral can be evaluated in terms of the stroh eigenvalues of a sextic equation for the degenerate case evaluation of residues at multiple poles is also given applications of the present result are made to examine the green s functions and stress components for isotropic and transversely isotropic materials the results are in exact agreement with existing solutions

stochastic estimation of the frobenius norm in the aca convergence criterion the adaptive cross approximation aca algorithm has been used in many fast integral equation solvers for electromagnetic radiation and scattering problems it efficiently computes a low rank approximation to the interaction matrix between mutually distant parts of a scattering object the aca is an iterative algorithm that needs an accurate and efficient convergence criterion the evaluation of this criterion may consume a considerable part of the computational resources this communication presents an efficient new way to evaluate the convergence criterion using a stochastic approach

multiscale compressed block decomposition for fast direct solution of method of moments linear system the multiscale compressed block decomposition algorithm ms cbd is presented for highly accelerated direct non iterative solution of electromagnetic scattering and radiation problems with the method of moments mom the algorithm is demonstrated to exhibit n2 computational complexity and storage requirements scaling with n1 5 for electrically large objects several numerical examples illustrate the efficiency of the method in particular for problems with multiple excitation vectors the largest problem presented in this paper is the monostatic rcs of the nasa almond at 50 ghz for one thousand incidence angles discretized using 442 089 rwg basis functions being entirely algebraic ms cbd is independent of the greens function of the problem