on the positive almost periodic type solutions for some nonlinear delay integral equations in this paper we discuss the existence of positive almost periodic type solutions for some nonlinear delay integral equations by constructing a new fixed point theorem in the cone some known results are extended

existence of a positive almost periodic solution for a nonlinear delay integral equation in this paper we discuss the existence of positive almost periodic solution for some nonlinear delay integral equation by constructing a new fixed point theorem in the cone some known results are extended

analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

convergence analysis of a high order nystrom integral equation method for surface scattering problems in this paper we present a convergence analysis for the nystrom method proposed in j comput phys 169 1 80 110 2001 for the solution of the combined boundary integral equation formulations of sound soft acoustic scattering problems in three dimensional space this fast and efficient scheme combines fft techniques and a polar change of variables that cancels out the kernel singularity we establish the stability of the algorithms in the l 2 norm and we derive convergence estimates in both the l 2 and l infty norms in particular our analysis establishes theoretically the previously observed super algebraic convergence of the method in cases in which the right hand side is smooth

a comparison of some methods for the evaluation of highly oscillatory integrals abstract in part i the extended clenshaw curtis method for finite fourier integrals is discussed and a number of timed comparisons are made between the various implementations which appear in the literature part ii deals with irregular oscillatory integrals and outlines the various methods which have been proposed for their evaluation their effectiveness is discussed in the light of a set of test examples

improvement of pufem for the numerical solution of high frequency elastic wave scattering on unstructured triangular mesh grids the purpose of this paper which builds on previous work int j numer meth engng 2009 77 1646 1669 is to improve a numerical scheme based on the partition of unity finite element method pufem for the solution of the time harmonic elastic wave equations the approach consists to approximate the displacement field by the standard finite element shape functions enriched locally by superimposing pressure p and shear s plane waves the aim is to accurately model two dimensional elastic wave problems on relatively coarse mesh grids capable of containing many wavelengths per nodal spacing for wide ranges of frequencies this allows us to relax the traditional requirement of about 10 nodal points per s wavelength in this work an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the pufem the main contribution here consists in developing an explicit closed form solution for two dimensional wave based integrals when the phase variation is linear in the local coordinate element system the evaluation of the element mass matrix is performed from appropriate edge integrals all other element matrices obtained by adequate splitting of the element stress tensor matrix are simply deduced from the element mass matrix entries the results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time numerical tests dealing with the scattering of s elastic plane waves by a circular rigid body show that for the same discretization level it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves however this increases the conditioning and the fill in of the global matrix at high frequency it is even possible to push the number of degrees of freedom per s wavelength under 2 and still achieve good accuracy finally some remarks on the choice of the numbers of p and s plane waves leading to better accuracy and conditioning are discussed copyright 2010 john wiley sons ltd

analysis of multiple scattering iterations for high frequency scattering problems ii the three dimensional scalar case in this paper we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology based on integral equations for the numerical solution of scattering problems at high frequencies in more detail here we extend the two dimensional results in part i of this work to fully three dimensional geometries as in the former case our concern here is the determination of the rate of convergence of the multiple scattering iterations for a collection of three dimensional convex obstacles that are inherent in the aforementioned high frequency schemes to this end we follow a similar strategy to that we devised in part i first we recast the iterated neumann multiple scattering series in the form of a sum of periodic orbits of increasing period corresponding to multiple reflections that periodically bounce off a series of scattering sub structures then we proceed to derive a high frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically and finally we analyze this recurrence to provide an explicit rate of convergence associated with each orbit while the procedure is analogous to its two dimensional counterpart the actual analysis is significantly more involved and perhaps more interestingly it uncovers new phenomena that cannot be distinguished in two dimensional configurations e g the further dependence of the convergence rate on the relative orientation of interacting structures as in the two dimensional case and beyond their intrinsic interest we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple scattering series and thus to provide significant savings in computational times

positive almost periodic solutions of some delay integral equations the delay integral equation x t tt f s x s ds which arises in models for the spread of epidemics is studied with the aim of establishing the existence of positive almost periodic solutions for large values of when f t x is uniformly almost periodic in t for x in compact subsets of r under reasonable assumptions on f it is shown that there exist two positive numbers 0 they do exist a priori bounds on the set of positive solutions and uniqueness results are also obtained

a fully discrete galerkin method for high frequency exterior acoustic scattering in three dimensions standard galerkin discretization techniques with locally or globally supported basis functions for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension leading to large cpu time and memory requirements to set up the dense galerkin matrix with each entry requiring evaluation of multi dimensional highly oscillatory integrals in this work using globally supported basis functions we describe an efficient fully discrete galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional galerkin integrals with high order accuracy such high order order accuracy for various practically relevant frequencies email protected 1 100 000 substantially improves on approximations based on standard asymptotic techniques we demonstrate the efficiency of our algorithm for spherical and non spherical convex scattering for several wavenumbers 1

numerical evaluation of highly oscillatory bessel transforms abstract in this paper we mainly focus on numerical evaluation of highly oscillatory bessel transforms based on the multiple integral the schemes for computing this class of the transform are presented by means of the bounds of special functions we give the corresponding errors of the schemes to demonstrate the efficiency and accuracy of the proposed methods some test examples are shown

numerical methods for two classes of singularly oscillatory bessel transforms and their error analysis abstract this work introduces and analyzes both the complex integration method and the modified complex integration method for two classes of highly oscillatory bessel transforms with algebraic or algebraic logarithmic singularities we first transform the original integrals into the infinite integrals on 1 by the change of variable t x where 0 with the help of the integral representation of the bessel function the integral representation and the asymptotic series of the whittaker function we then use the cauchy s residue theorem and convert the above infinite integrals into the infinite integrals on 0 the integrands of the resulting infinite integrals are not oscillatory and decay exponentially fast fortunately they can be efficiently evaluated by using the gauss laguerre quadrature rule furthermore we conduct their error analysis in inverse powers of the frequency finally numerical experiments are provided to confirm our theoretical analysis we can see from theoretical analysis and numerical experiments that the accuracy of the quadrature increases both for the case of growing the number of nodes n and fixed and for the case of an increasing and fixed the number of nodes n the comparison of the proposed two methods and the filon type method presented in kang and ma 2017 is briefly discussed as well

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

frequency adapted galerkin boundary element methods for convex scattering problems we introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems in the exterior of two dimensional smooth convex obstacles to facilitate the applicability of our algorithms throughout the entire frequency spectrum we have enriched our galerkin approximation spaces through incorporation of oscillations in the incident field of radiation into the algebraic and trigonometric polynomial approximation spaces the resulting methodologies have three distinctive properties indeed from a theoretical point of view 1 they can be tuned to demand only an mathcal o k varepsilon o k increase for any varepsilon 0 0 in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumberk owing to the optimal adaptation of approximation spaces to asymptotic stretching shrinking of illuminated and deep shadow regions shadow boundaries and 2 they are convergent for each fixed wavenumberk thanks to the additional approximation spaces in the deep shadow region perhaps more importantly from a practical point of view 3 they give rise to linear systems with significantly enhanced condition numbers and this in turn allows for more accurate solutions if desired

relative group completions abstract we extend arbitrary group completions to the category of pairs g n where g is a group and n is a normal subgroup of g relative localizations are special cases our construction is a group theoretical analogue of fibrewise completion and fibrewise localization in homotopy theory and generalizes earlier work of hilton and others on relative localization at primes we use our approach to find conditions under which factoring out group radicals preserves exactness this has implications in the study of the effect of plus constructions on homotopy fibre sequences

on evaluation of bessel transforms with oscillatory and algebraic singular integrands in this paper we study efficient methods for computing the integrals of the form 0 1x a 1 x bf x j v wx dx where a b v w are the given constants and w 1 j v is the bessel function of the first kind and of order v f is a sufficiently smooth function on 0 1 firstly we express the moments in a closed form with the aid of special functions secondly we induce the filon type method based on the taylor interpolation polynomial at two endpoints and the hermite interpolation polynomial at clenshaw curtis points on evaluating the highly oscillatory bessel integrals with algebraic singularity theoretical results and numerical experiments perform that the methods are very efficient in obtaining very high precision approximations if w is sufficiently large

a transformation approach for efficient evaluation of oscillatory surface integrals arising in three dimensional boundary element methods we propose a method for efficient evaluation of surface integrals arising in boundary element methods for three dimensional helmholtz problems with real positive wavenumber k modelling wave scattering and or radiation in homogeneous media to reduce the number of degrees of freedom required when k is large a common approach is to include in the approximation space oscillatory basis functions with support extending across many wavelengths a difficulty with this approach is that it leads to highly oscillatory surface integrals whose evaluation by standard quadrature would require at least o k2 quadrature points here we use equivalent contour integrals developed for aperture scattering in optics to reduce this requirement to o k and possible extensions to reduce it further to o 1 are identified the contour integral is derived for arbitrary shaped elements but its application is limited to planar elements in many cases in addition the transform regularises the singularity in the surface integrand caused by the green s function including for the hyper singular case under appropriate conditions an open source matlab code library is available to demonstrate our routines

filon clenshaw curtis rules for a class of highly oscillatory integrals with logarithmic singularities in this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities for these quadrature rules we derive error estimates in terms of n the number of nodes k the rate of oscillations and a sobolev like regularity of the function we prove that the method is not only robust but the error even decreases for fixed n as k increases practical issues about the implementation of the rule are also covered in this paper by a writing down ready to implement algorithms b analysing the numerical stability of the computations and c estimating the overall computational cost we finish by showing some numerical experiments which illustrate the theoretical results presented in this paper

fast integration of rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is presented the method is efficient for integrals involving bessel functions jv rx with large oscillation parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations

an o 1 integration scheme for three dimensional surface scattering problems we present an accurate method of o 1 complexity with respect to frequency i e a method that to achieve a prescribed error tolerance requires a bounded computational cost for arbitrarily high frequencies for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three dimensional space like the two dimensional counterpart of this algorithm which we introduced recently and which is applicable to scattering by curves in the plane the present method is based on a combination of two main elements 1 a high frequency ansatz for the unknown density in a boundary integral formulation of the problem and 2 an extension of the ideas of the method of stationary phase to allow for o 1 high order accurate integration of oscillatory functions the techniques we introduce to implement an efficient o 1 integrator in the present three dimensional context differ significantly from those used in the earlier two dimensional algorithm in particular in the present text we introduce an efficient canonical hybrid analytic numerical algorithm which in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two dimensional scattering surface we include numerical results illustrating the behavior of the integration algorithm on sound soft spheres with diameters of up to 5000 wavelengths in such cases for a single integral the algorithm yields accuracies of the order of three digits in computational times of less than two seconds in a preliminary full scattering simulation we present a solution with two digits of accuracy in the surface density was obtained in about three hours running time in a single 1 5ghz amd athlon processor for a sphere of 500 wavelengths in diameter

acceleration of an iterative method for the evaluation of high frequency multiple scattering effects high frequency integral equation methodologies display the capability of reproducing single scattering returns in frequency independent computational times and employ a neumann series formulation to handle multiple scattering effects this requires the solution of an enormously large number of single scattering problems to attain a reasonable numerical accuracy in geometrically challenging configurations here we propose a novel and effective krylov subspace method suitable for the use of high frequency integral equation techniques that significantly accelerates the convergence of neumann series we additionally complement this strategy utilizing a preconditioner based upon kirchhoff approximations that provides a further reduction in the overall computational cost

on the evaluation of bessel transformations with the oscillators via asymptotic series of whittaker functions in this paper we are concerned with numerical evaluation of bessel transformations with the oscillators first we rewrite the integrals by whittaker functions then based on the asymptotic series of whittaker functions the integrals are transformed into the problems involving the complex exponential functions which can be efficiently computed by using the complex integration method at the same time we obtain the entirely different asymptotic expansions from the references in particular the asymptotic expansions can be denoted by the incomplete gamma function the error analysis shows that the decay of the error drastically improves as the frequency grows the effectiveness and accuracy of the methods are tested for large arguments of bessel functions

efficient method for the computation of oscillatory bessel transform and bessel hilbert transform abstract in this paper we study the numerical methods for the evaluation of two kinds of highly oscillatory bessel transforms firstly we rewrite both integrals as the sum of two integrals by rewriting the bessel function as a linear combination of whittaker w function we then transform one of integrals into the fourier type which can be transformed into the integrals on 0 and can be computed by some proper gaussian quadrature which take into account the asymptotic property of whittaker w function as x 0 the other can be efficiently computed based on the evaluation of special functions in addition error analysis for the presented methods is given moreover we also give an explicit formula for the integral 0 j x x d x in terms of meijer g function and then apply the method for the oscillatory bessel transforms to the computation of highly oscillatory bessel hilbert transforms theoretical results and numerical examples demonstrate the efficiency and accuracy of the proposed methods

a fast algorithm for chebyshev fourier and sinc interpolation onto an irregular grid a chebyshev or fourier series may be evaluated on the standard collocation grid by the fast fourier transform fft unfortunately the fft does not apply when one needs to sum a spectral series at n points which are spaced irregularly the cost becomes o n2 operations instead of the ffts o n log n this sort of off grid interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features it is even more crucial to semi lagrangian spectral algorithms for solving convection diffusion and navier stokes problems because off grid interpolation must be performed several times per time step in this work we describe an alternative algorithm the first step is to pad the set of spectral coefficients an with zeros and then take an fft of length 3n to interpolate the chebyshev series to a very fine grid the second step is to apply either the mth order euler sum acceleration or 2m 1 point lagrangian interpolation to approximate the sum of the series on the irregular grid we show that both methods yield full precision with m n allowing an order of magnitude reduction in cost with no loss of accuracy

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

the a core and a cover of a group abstract this paper provides a comprehensive investigation of the cellular approximation functor cell a g in the category of groups approximating a group g by a group a we also study related notions such as a injection a generation and a constructibility of a group g and we find several interesting connections with the schur multiplier h 2 g z our constructions are direct and are given in a slow and detailed manner

a contribution to scattering calculation for small wavelengths the high frequency panel method numerical treatment of electromagnetic scattering in the case of small wavelengths too small for reasonable matrix sizes when using discretization methods and too large for traditional asymptotic methods is considered an approach which takes advantage of the user friendly nature of matrix discretization methods while avoiding the explosive increase in computing effort as the wavelength decreases is suggested it appears that no sensible increase of matrix size is required as the wavelength decreases since this is an untried technique applied in an inaccessible computational domain the preliminary work described is confined to two dimensional conducting circular cylinders where the answers are well known and well understood the overall idea however appears to be quite general

a high frequency boundary element method for scattering by a class of nonconvex obstacles in this paper we propose and analyse a hybrid numerical asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound soft nonconvex polygons the approximation space is enriched with carefully chosen oscillatory basis functions these are selected via a study of the high frequency asymptotic behaviour of the solution we demonstrate via a rigorous error analysis supported by numerical examples that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases in contrast to the at least linear growth required by conventional methods this appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle our analysis is based on new frequency explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane

an efficient high order algorithm for acoustic scattering from penetrable thin structures in three dimensions this paper presents a high order accelerated algorithm for the solution of the integral equation formulation of volumetric scattering problems the scheme is particularly well suited to the analysis of thin structures as they arise in certain applications e g material coatings in addition it is also designed to be used in conjunction with existing low order fft based codes to upgrade their order of accuracy through a suitable treatment of material interfaces the high order convergence of the new procedure is attained through a combination of changes of parametric variables to resolve the singularities of the green function and partitions of unity to allow for a simple implementation of spectrally accurate quadratures away from singular points accelerated evaluations of the interaction between degrees of freedom on the other hand are accomplished by incorporating two face equivalent source approximations on cartesian grids a detailed account of the main algorithmic components of the scheme are presented together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

a fast and well conditioned spectral method for singular integral equations abstract we develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing chebyshev and ultraspherical polynomials to reformulate the equations as almost banded infinite dimensional systems this is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels the resulting system can be solved in o m 2 n operations using an adaptive qr factorization where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution the complexity is reduced to o m n operations by pre caching the qr factorization when the same operator is used for multiple right hand sides stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity applications considered include the faraday cage and acoustic scattering for the helmholtz and gravity helmholtz equations including spectrally accurate numerical evaluation of the far and near field solution the julia software package singularintegralequations jl implements our method with a convenient user friendly interface

asymptotically derived boundary elements for the helmholtz equation in high frequencies we present an asymptotically derived boundary element method for the helmholtz equation in exterior domains each basis function is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase factor is determined a priori by using arguments from geometrical optics and the geometrical theory of diffraction while the smooth amplitude is represented by high order splines this yields a high order method in which the number of unknowns is virtually independent of the wavenumber k two types of diffracted basis functions are presented the first accounts for the dominant oscillatory behavior in the shadow region while the second also accounts for the decay of the amplitude there we show that the matrix a associated with the discrete problem has only o n significant entries as k where n is the number of basis functions hence it can be approximated with a matrix having o n terms and the relative error between a and rapidly converges to zero as k although the method is applicable to a variety of scatterers we focus our attention here on scattering from smooth closed convex bodies in two dimensions computations on a circular cylinder illustrate our results

the construction of cubature rules for multivariate highly oscillatory integrals we present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points a general method is presented to identify these points and to compute the weights of the corresponding rule the accuracy of the constructed rules increases with increasing frequency of the integrand for a fixed frequency the accuracy can be improved by incorporating more derivatives of the integrand the results are illustrated numerically for fourier integrals on a circle and on the unit sphere and for more general oscillators on a rectangular domain

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

a high order tangential basis algorithm for electromagnetic scattering by curved surfaces we describe analyze and demonstrate a high order spectrally accurate surface integral algorithm for simulating time harmonic electromagnetic waves scattered by a class of deterministic and stochastic perfectly conducting three dimensional obstacles a key feature of our method is spectrally accurate approximation of the tangential surface current using a new set of tangential basis functions the construction of spectrally accurate tangential basis functions allows a one third reduction in the number of unknowns required compared with algorithms using non tangential basis functions the spectral accuracy of the algorithm leads to discretized systems with substantially fewer unknowns than required by many industrial standard algorithms which use for example the method of moments combined with fast solvers based on the fast multipole method we demonstrate our algorithm by simulating electromagnetic waves scattered by medium sized obstacles diameter up to 50 times the incident wavelength using a direct solver in a small parallel cluster computing environment the ability to use a direct solver is a tremendous advantage for monostatic radar cross section computations where thousands of linear systems with one electromagnetic scattering matrix but many right hand sides induced by many transmitters must be solved

positive almost periodic solutions of some neutral integral equation abstract in this paper we investigate the existence of positive almost periodic solutions for some neutral integral equation by constructing a new fixed point theorem in the cone some new results are obtained

fast integration of highly oscillatory integrals with exotic oscillators in this paper we present an efficient filon type method for the integration of systems containing bessel functions with exotic oscillators based on a diffeomorphism transformation and give applications to airy transforms preliminary numerical results show the effectiveness and accuracy of the quadrature for large arguments of integral systems

exponentially fitted gauss laguerre quadrature rule for integrals over an unbounded interval new quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope the new formulae are derived by exponential fitting and they represent a generalization of the usual gauss laguerre formulae their weights and nodes depend on the frequency of oscillation in the integrand and thus the accuracy is massively increased rules with one up to six nodes are treated with details numerical illustrations are also presented

application of the integral equation asymptotic phase method to two dimensional scattering a hybrid procedure called the integral equation asymptotic phase ie ap method is investigated for scattering from perfectly conducting cylinders of arbitrary cross section shape the ie ap approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution to leave a slowly varying residual function that can be represented by a coarse density of unknowns in the present investigation the current density appearing within the combined field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function the resulting equation is discretized by the method of moments using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density even for scatterers with corners

on the calculation of highly oscillatory integrals with an algebraic singularity abstract in this paper we explore the evaluation of highly oscillatory integrals containing an algebraic singularity based on the change of variable t x r where r is a positive real number and the analytical continuation robust numerical results demonstrate the accuracy and effectiveness of the proposed approach

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

efficient evaluation of the physical optics integrals for conducting surfaces using the uniform stationary phase method the computational time to evaluate the physical optics po expression by numerical integration increases rapidly with the increase of electrical size of scattering surfaces however the computational time of po integrals for electrically large object can be greatly reduced by using the stationary phase method which is independent of the wavenumber for this method the theory and numerical implementations for isolated critical points have been well developed however for cases of nearby critical points there are still a few issues to be considered especially in numerical implementations in this paper we mainly study the numerical implementations for several most common cases of nearby critical points in particular the cases of two nearby inner stationary phase points and complex inner stationary phase points are discussed in more details such cases occur frequently when the scattering surface includes convex concave parts but the numerical implementations to such cases have not been reported to our knowledge the difficulty lies in how to identify whether two inner stationary points and complex inner stationary points on the surfaces with arbitrary shapes are close to each other or not a strategy is designed to solve this difficulty by validation in some typical examples we find that the stationary phase method is robust enough to evaluate the po integrals accurately finally some interesting phenomena observed in numerical validations are interpreted

a sparse discretization for integral equation formulations of high frequency scattering problems we consider two dimensional scattering problems formulated as an integral equation defined on the boundary of the scattering obstacle the oscillatory nature of high frequency scattering problems necessitates a large number of unknowns in classical boundary element methods in addition the corresponding discretization matrix of the integral equation is dense we formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies the method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns but retains accuracy for lower frequencies new in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals as a result we obtain a sparse discretization matrix for the oscillatory problem moreover numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency the sparse discretization applies to problems where the phase of the solution can be predicted a priori for example in the case of smooth and convex scatterers

local solutions to high frequency 2d scattering problems we consider the solution of high frequency scattering problems in two dimensions modeled by an integral equation on the boundary of a smooth scattering object we devise a numerical method to obtain solutions on only parts of the boundary with little computational effort the method incorporates asymptotic properties of the solution and can therefore attain particularly good results for high frequencies we show that the integral equation in this approach reduces to an ordinary differential equation

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

extraction of uniformly accurate phase functions across smooth shadow boundaries in high frequency scattering problems several recent numerical schemes for high frequency scattering simulations are based on the extraction of known phase functions from an oscillatory solution the remaining function is typically no longer oscillatory and as such it can be approximated numerically with a number of degrees of freedom that does not depend on the frequency of the original problem knowledge of the phase of a solution typically comes from asymptotic analysis for example from geometrical optics we consider integral equation formulations of time harmonic scattering by a smooth and convex obstacle and focus on the so called shadow boundaries they are the points where the incoming waves are tangential to the boundary of the scatterer we devise a numerical method that incorporates advanced results from asymptotic analysis which describe the frequency dependent transitional behavior of the solution uniformly across these points we describe and resolve an apparent conflict between two theories that describe the asymptotic behav

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

using b spline frames to represent solutions of acoustics scattering problems abstract although frames which are a generalization of bases are important tools used in signal processing their potential in other fields of engineering and applied mathematics e g acoustics has not been fully explored yet gabor frames that are specific type of frames are very well adapted to oscillating functions and therefore have a great potential to efficiently represent functions in connection with the helmholtz operator in this paper representations of the solution of a scattering problem in 2d using gabor frames based on b splines as building blocks are investigated practical issues concerning the implementation of frames like the restriction of the frame elements to a finite interval and methods to determine the unknown coefficients for the representation i e by using dual frames or by solving a least squares problem are discussed numerical experiments are made comparing the different ways to determine the unknown expansion coefficients and the representations are compared in terms of their efficiency i e the number of used frame coefficients versus the accuracy of the approximation in all cases the coefficients calculated with a slightly modified orthogonal matching pursuit algorithm provide the best accuracy versus sparsity ratio

a rapid solution of a kind of 1d fredholm oscillatory integral equation how to solve oscillatory integral equations rapidly and accurately is an issue that attracts special attention in many engineering fields and theoretical studies in this paper a rapid solution method is put forward to solve a kind of special oscillatory integral equation whose unknown function is much less oscillatory than the kernel function in the method an improved levin quadrature method is adopted to solve the oscillatory integrals on the one hand the employment of this quadrature method makes the proposed method very accurate on the other hand only a small number of small scaled systems of linear equations are required to be solved so the computational complexity is also very small numerical examples confirm the advantages of the method

near peak scattering and the corrected kirchhoff approximation for a convex obstacle abstract the uniform asympotic behavior of the scattering amplitude near the forward peak in the case of classical scattering of waves by a convex obstacle is derived a microlocal model is obtained for the scattering operator this is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of fourier integral operators those with folding canonical relations a crucial ingredient consists of putting a fourier integral operator with folding canonical relation into a normal form the analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary thus providing a corrected version of the kirchhoff approximation

efficient computation of highly oscillatory integrals with hankel kernel in this paper we consider the evaluation of two kinds of oscillatory integrals with a hankel function as kernel we first rewrite these integrals as the integrals of fourier type by analytic continuation these fourier type integrals can be transformed into the integrals on 0 the integrands of which are not oscillatory and decay exponentially fast consequently the transformed integrals can be efficiently computed by using the generalized gauss laguerre quadrature rule moreover the error analysis for the presented methods is given the efficiency and accuracy of the methods have been demonstrated by both numerical experiments and theoretical results

multilevel fast multipole acceleration in the nystrom discretization of surface electromagnetic integral equations for composite objects the multilevel fast multipole algorithm mlfma based on the nystrom discretization of surface integral equations sies is developed for solving electromagnetic em scattering by large composite objects traditionally the mlfma is based on the method of moments mom discretization for the sies and it usually works well when the robust rao wilton glisson rwg basis function is enough to represent unknown currents however the rwg basis function may not represent both the electric and magnetic current in solving the electric field integral equation efie and magnetic field integral equation mfie for penetrable objects and how one represents another current could be a problem in the mom in this work we use the nystrom method as a tool to discretize the sies and incorporate the mlfma to accelerate the solutions for electrically large problems the advantages of the nystrom discretization include the simple mechanism of implementation lower requirements on mesh quality and no use of basis and testing functions these benefits are particularly desired in the mlfma because the solved problems are very large and complex in general numerical examples are presented to demonstrate the effectiveness of the proposed scheme

central extensions and generalized plus constructions we describe the effect of homological plus constructions on the homotopy groups of eilenberg maclane spaces in terms of universal central extensions

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

pseudo almost periodic solutions of some differential equations ii we first present some results about pseudo almost periodic functions then we use these results to show some theorems about existence and uniqueness of the pseudo almost periodic solutions of the linear and the quasi linear nonautonomous equations finally we set up some theorems of the existence of pseudo almost periodic solutions implying the existence of almost periodic solutions of general ordinary differential equation systems

numerical evaluation of the two dimensional partition of unity boundary integrals for helmholtz problems there has been considerable attention given in recent years to the problem of extending finite and boundary element based analysis of helmholtz problems to higher frequencies one approach is the partition of unity method which has been applied successfully to boundary integral solutions of helmholtz problems providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy these benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions in this paper we adapt the numerical steepest descent method to evaluate these integrals for two dimensional problems the approach is successful in reducing the computational effort for most integrals encountered the paper includes some numerical features that are important for successful practical implementation of the algorithm

high frequency scattering by convex curvilinear polygons we consider the scattering of a time harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with lipschitz boundary for standard boundary or finite element methods with a piecewise polynomial approximation space the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave here we propose a novel galerkin boundary element method with a hybrid approximation space consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes a uniform mesh on illuminated sides and graded meshes refined towards the corners of the polygon on illuminated and shadow sides numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases

idempotent transformations of finite groups we describe the action of idempotent transformations on finite groups we show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group this is done in terms of the first two homology groups we prove for example that except special linear groups such an orbit can have at most 7 elements we also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

on generalized gaussian quadrature rules for singular and nearly singular integrals we construct and analyze generalized gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities the rules have quadrature points inside the interval of integration and the weights are all strictly positive such rules date back to the study of chebyshev sets but their use in applications has only recently been appreciated we provide error estimates and we show that the convergence rate is unaffected by the singularity of the integrand we characterize the quadrature rules in terms of two families of functions that share many properties with orthogonal polynomials but that are orthogonal with respect to a discrete scalar product that in most cases is not known a priori

positive solutions of nonlinear integral equations arising in infectious diseases on considere l equation integrale non lineaire x t t t f s x s ds on utilise la theorie d indice du point fixe et la technique monotone pour etablir trois theoremes sur les solutions le premier affirme l existence de deux solutions non negatives non triviales de periode les deux autres concernent l existence et l unicite des solutions positives non triviales sous certaines conditions

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

the numerical steepest descent path method for calculating physical optics integrals on smooth conducting quadratic surfaces in this paper we use the numerical steepest descent path nsdp method to analyze the highly oscillatory physical optics po integral on smooth conducting parabolic surfaces including both monostatic and bistatic cases quadratic variations of the amplitude and phase functions are used to approximate the integrand of po integral then the surface po integral is reduced into several highly oscillatory line integrals by invoking the nsdp method these highly oscillatory po line integrals are defined on the corresponding nsdps furthermore the critical point contributions for the po integral are exactly extracted and represented based on the nsdps the proposed nsdp method for calculating the po integral on the smooth conducting surfaces is frequency independent and error controllable compared with the traditional asymptotic expansion approach the nsdp method significantly improves the po integral accuracy by around two digits when the working wave frequencies are not extremely large numerical results are given to validate the nsdp method

towards a full bandwidth numerical acoustic model prediction models are at the heart of modern acoustic engineering current commercial room acoustic simulation software almost exclusively approximates the propagation of sound geometrically as rays or beams these assumptions yield efficient algorithms but the maximum accuracy they can achieve is limited by how well the geometric assumption represents sound propagation in a given space this comprises their accuracy at low frequencies in particular methods that directly model wave effects are more accurate but they have a computational cost that scales with problem size and frequency effectively limiting them to small or low frequency scenarios this paper will report the results of initial research into a new full bandwidth model which aims to be accurate and efficient for all frequencies the name proposed for this is the wave matching method this builds on the boundary element method with the premise that if an appropriate interpolation scheme is designed then the model will become geometrically dominated at high frequencies other propagation modes may then be removed without significant error yielding an algorithm which is accurate and efficient this paper will present the general concepts of wave matching and the results from some numerical test cases

a high frequency hp boundary element method for scattering by convex polygons in this paper we propose and analyze a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound soft convex polygons in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution we demonstrate both theoretically and via numerical examples exponential convergence with respect to the order of the polynomials moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern in which the dependence on the frequency of all constants is explicit importantly these estimates prove that to achieve any desired accuracy in the computation of these quantities it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases in contrast to the at least linear growth required by conventional methods

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

fast non polynomial interpolation and integration for functions with logarithmic singularities abstract a fast non polynomial interpolation is proposed in this paper for functions with logarithmic singularities it can be executed fast with the discrete cosine transform based on this interpolation a new quadrature is proposed for a kind of logarithmically singular integrals the interpolation and integration errors are also analyzed numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature

approximation of highly oscillatory integrals containing special functions abstract a modified form of levin s method based on multi quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory bessel function of the first kind of order and oscillatory airy function the method converts the oscillatory integrals into a system of coupled ordinary differential equations odes and subsequently numerical solution of the coupled odes is obtained by meshless collocation procedure a multi resolution quadrature is used to tackle singularity of the proposed method theoretical error analysis of the proposed methods is performed in asymptotic sense numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory airy and bessel integrals

a galerkin boundary element method for high frequency scattering by convex polygons in this paper we consider the problem of time harmonic acoustic scattering in two dimensions by convex polygons standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave here we present a novel galerkin boundary element method which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon we prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency numerical results demonstrate the same logarithmic dependence on the frequency for the galerkin method solution our boundary element method is a discretization of a well known second kind combined layer potential integral equation we provide a proof that this equation and its adjoint are well posed and equivalent to the boundary value problem in a sobolev space setting for general lipschitz domains