rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

performance comparison of numerical inversion methods for laplace and hankel integral transforms in engineering problems abstract different methods for the numerical evaluations of the inverse laplace and inverse of joint laplace hankel integral transforms are applied to solve a wide range of initial boundary value problems often arising in engineering and applied mathematics the aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half space to concentrated impulse or those related to viscoelastic wave motion in layered media these classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations theoretical seismology soil structure interaction in time domain and so on therefore their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems some different methods are also investigated in detail and compared for the inversion of the joint hankel laplace transforms where more sophisticated integrand functions are encountered it is shown that durbin crump d amore fixed talbot gaver whyn rho gwr and direct integration methods have excellent performance and produce good results when applied to the same problems on the contrary gaver stehfest and piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for simple transformed functions particularly the performance of gwr algorithm is very good even for transformed functions with infinite number of singularities where the other methods fail in addition in case of double integral transforms only the fixed talbot durbin and weeks methods are recommended

global matric massey products and the compactified jacobian of the e6 singularity abstract in this paper we compute the compactified jacobian of the singularity e6 by g m greuel and h knorrer 1985 math ann 270 417 425 this singularity has only a finite number of isomorphism classes of rank 1 torsionfree modules using the theory of matric massey products in an earlier work we computed the local formal moduli with its local versal family for each local module and we studied the degeneracy of each local module here we prove a result showing how the local theory connects to the global theory i e we prove that the morphism from the local formal moduli of a local module to the local ring at the point corresponding to the module on the compactified jacobian is a smooth morphism in the case where m e6 i e the normalization this morphism is an isomorphism thus the degeneracy stratification diagram for the compactified jacobian can be found from the degeneracy of the normalization in the local case

a parameter method for computing highly oscillatory integrals this paper presents a new efficient parameter method for integration of the highly oscillatory integral 0 1f x e i w g x dx with an irregular oscillator the effectiveness and accuracy are tested by means of numerical examples for the case where g x has stationary points

the versal deformation space of a reflexive module on a rational cone abstract by an approach based on results of a ishii we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degree m for each component a resolution is given as the total space of a vector bundle on a grassmannian the vector bundle is a sum of copies of the cotangent bundle the canonical sub bundle the dual of the canonical quotient bundle and the trivial line bundle via an embedding in a trivial bundle we obtain the components by projection in particular we give equations for the minimal stratum in the chern class filtration of the versal deformation space we obtain a combinatorial description of the local deformation relation and a classification of the components in particular we give a formula for the number of components

on the implementation of discontinuous galerkin methods for volterra integral equations with highly oscillatory bessel kernels in this paper we consider implementation of discontinuous galerkin methods for weakly singular volterra integral equations of first and second kinds with highly oscillatory bessel kernels which cost the same operations independent of large values of frequencies preliminary numerical results show that the proposed method is efficient for verifying accuracy of approximate solutions

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

obstruction calculus for functors of artin rings i abstract in this paper we define and study obstruction theories for morphisms of functors of artin rings we prove the existence of a universal obstruction theory and we give explicit criteria for completeness and for linearity as applications we extend several results in the literature removing the finite dimensionality of the tangent space and the existence of a vector space of obstructions from the assumptions

parareal multiscale methods for highly oscillatory dynamical systems we introduce a new strategy for coupling the parallel in time parareal iterative methodology with multiscale integrators following the parareal framework the algorithm computes a low cost approximation of all slow variables in the system using an appropriate multiscale integrator which is refined using parallel fine scale integrations convergence is obtained using an alignment algorithm for fast phase like variables the method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables or to resolve all the state variables the numerical scheme does not require that the system is split into slow and fast coordinates moreover the dynamics may involve hidden slow variables for example due to resonances we propose an alignment algorithm for almost periodic solutions in which case convergence of the parareal iterations is proved the applicability of the method is demonstrated in numerical examples

a method for computing bessel function integrals abstract a method for numerical calculation of integrals containing bessel functions of integer or integer plus one half order is described the calculation involves first a one dimensional fourier sine or cosine transform followed by evaluation of the coefficient of the chebyshev series of the fourier transformed function in the case of the bessel function and evaluation of the legendre expansion coefficient in the case of the spherical bessel function a generalization of the method for the computation of an integral involving the bessel function of arbitrary real order v is presented as well

an efficient method for evaluating the integral of a class of highly oscillatory functions highly oscillatory integrals require special techniques for their effective evaluation various studies have been conducted to find computational methods for evaluating such integrals in this paper we present an efficient numerical method to evaluate a class of generalised fourier integrals on a line or a square with integrands of the form f x e i k g x under the assumption that in the domain of integration both f and g are sufficiently smooth and that g does not have any stationary critical points numerical analysis and results are given to illustrate the effectiveness of our method for computing generalised fourier integrals

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

fast computation of a class of highly oscillatory integrals in this paper the problem of numerical evaluation of a class of highly oscillatory integrals a b f x x a b x k 1 n x k k e i x dx where 1 k 1 a k b and 1 has been discussed based on analytic continuation the integrals are transformed into the problems of integrating the forms on 0 with the integrand does not oscillate and decays exponentially fast which can be efficiently computed by using general gauss laguerre quadrature rule theoretical results and numerical experiments perform that the method is very efficient in obtaining very high precision approximations if is sufficiently large

on the convergence of the method for indefinite integration of oscillatory and singular functions we consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in keller 8 to this end we have analysed the properties of the difference operator related to the difference equation for the chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients properties of this operator were never investigated before the obtained results lead us to the conclusion that the studied method is always convergent we also give a rigorous proof of the error estimates

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

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

density of spherically embedded stiefel and grassmann codes the density of a code is the fraction of the coding space covered by packing balls centered around the codewords a high density indicates that a code performs well when used as a uniform point wise discretization of an ambient space this paper investigates the density of codes in the complex stiefel and grassmann manifolds equipped with the chordal distance arising from an euclidean embedding including the unitary group as a special case the choice of distance enables the treatment of the manifolds as subspaces of euclidean hyperspheres in this geometry the densest packings are not necessarily equivalent to maximum minimum distance codes computing a code s density follows from computing 1 the normalized volume of a metric ball and 2 the kissing radius the radius of the largest balls one can pack around the codewords without overlapping first the normalized volume of a metric ball is evaluated by asymptotic approximations the volume of a small ball can be well approximated by the volume of a locally equivalent tangential ball in order to properly normalize this approximation the precise volumes of the manifolds induced by their spherical embedding are computed for larger balls a hyperspherical cap approximation is used which is justified by a volume comparison theorem showing that the normalized volume of a ball in the stiefel or grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity then bounds on the kissing radius are derived alongside corresponding bounds on the density unlike spherical codes or codes in flat spaces the kissing radius of grassmann or stiefel codes cannot be exactly determined from its minimum distance it is nonetheless possible to derive bounds on density as functions of the minimum distance stiefel and grassmann codes have larger density than their image spherical codes when dimensions tend to infinity finally the bounds on density lead to refinements of the standard hamming bounds for stiefel and grassmann codes

asymptotic expansions of bessel anger and weber transformations in this paper we explore higher order numerical quadrature for the integrations of systems containing bessel anger and weber functions the method is constructed by finding the approximate solution of the differential equation and truncating the asymptotic series numerical examples based on theoretical results are presented to demonstrate the efficiency of the method

universal deformation rings of modules over frobenius algebras abstract let k be a field and let be a finite dimensional k algebra we prove that if is a self injective algebra then every finitely generated module v whose stable endomorphism ring is isomorphic to k has a universal deformation ring r v which is a complete local commutative noetherian k algebra with residue field k if is also a frobenius algebra we show that r v is stable under taking syzygies we investigate a particular frobenius algebra 0 of dihedral type as introduced by erdmann and we determine r 0 v for every finitely generated 0 module v whose stable endomorphism ring is isomorphic to k

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

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

on the base space of a semi universal deformation of rational quadruple points this paper is devoted to the study of a particular class of normal surface singularities the rational quadruple points we will determine the base space of a semi universal deformation of such a singularity the answer turns out to be unexpectedly simple the isomorphism type of the base space of a rational quadruple point is completely determined by two numbers s and n the base space then is isomorphic to s x b n where s is a smooth germ of dimension s and where b n is a certain universal space defined in 3 5 a rational quadruple point with the star shaped resolution graph shown in figure 1 has the factor b n in its space we call such a singularity an n star

the versal deformation of an isolated toric gorenstein singularity given a lattice polytope q n we define an affine scheme open image in new window that reflects the possibilities of splitting q into a minkowski sum denoting by y the toric gorenstein singularity induced by q we construct a flat family over open image in new window with y as special fiber in case y has an isolated singularity this family is versal

asymptotic expansions and fast computation of oscillatory hilbert transforms in this paper we study the asymptotics and fast computation of the one sided oscillatory hilbert transforms of the form begin aligned h f t e i omega t x int nolimits 0 infty e i omega t frac f t t x dt quad omega 0 quad x ge 0 end aligned where the bar indicates the cauchy principal value and f is a real valued function with analytic continuation in the first quadrant except possibly a branch point of algebraic type at the origin when x 0 the integral is interpreted as a hadamard finite part integral provided it is divergent asymptotic expansions in inverse powers of omega are derived for each fixed x ge 0 which clarify the large omega behavior of this transform we then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms depending on the position of x we classify our discussion into three regimes namely x mathcal o 1 or x gg 1 0 x ll 1 and x 0 numerical experiments show that the convergence of the proposed methods greatly improve when the frequency omega increases some extensions to oscillatory hilbert transforms with bessel oscillators are briefly discussed as well

a comparative study of numerical integration based on haar wavelets and hybrid functions a quadrature rule based on uniform haar wavelets and hybrid functions is proposed to find approximate values of definite integrals the wavelet based algorithm can be easily extended to find numerical approximations for double triple and improper integrals the main advantage of this method is its efficiency and simple applicability error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method

method for numerical integration of rapidly oscillating functions in diffraction theory the quadrature of general highly oscillatory integrals is a relatively complicated computational problem that occurs in a wide range of practical applications e g in physics chemistry and image analysis it is often necessary to use a high number of nodal points with classical quadrature formulas in order to achieve a required accuracy of numerical integration of rapidly oscillating functions however an increase in integration points leads to a larger computational time our work describes and analyses a method for numerical integration of rapidly oscillating functions which enables to obtain the required accuracy with a smaller number of nodal points than classical integration rules and with a relatively simple integration formula the proposed method is generally suitable for integration of rapidly oscillating functions in various applications the method is demonstrated in examples of calculation of the diffraction integral in optics where the integrand is often a rapidly oscillatory function the results can be readily adapted to similar problems from other fields copyright 2009 john wiley sons ltd

kerdock codes for limited feedback precoded mimo systems codebook based limited feedback is a practical way to obtain partial channel state information at the transmitter in a precoded multiple input multiple output wireless system in this correspondence we propose a systematic codebook design the proposed kerdock codebook consists of multiple mutually unbiased unitary basis matrices with quaternary alphabet and the identity matrix we propose to derive the beamforming and precoding codebooks from this base codebook eliminating the requirement to store multiple codebooks furthermore we show that quaternary alphabet facilitates efficient codebook storage and codeword search we derive the chordal distance for two and four antenna codebooks showing that the proposed codebooks compare favorably with prior designs monte carlo simulations compare achievable rates and error rates for different codebooks sizes

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

algorithm for the computation of bessel function integrals abstract a fortran subroutine is given for the computation of integrals of the form c 0 f x j v x d x where v 0 1 10

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

on van der corput type lemmas for bessel and airy transforms and applications abstract this paper presents van der corput type lemmas for bessel and airy transforms with shifted jacobi weight functions these results on the asymptotic orders of the highly oscillatory integrals on the frequency are optimal furthermore from these estimates the convergence rates on filon type methods are easily derived

deformations of modules of maximal grade and the hilbert scheme at determinantal schemes let r be a polynomial ring and m a finitely generated graded r module of maximal grade which means that the ideal it a generated by the maximal minors of a homogeneous presentation matrix a of m has maximal codimension in r suppose x proj r it a is smooth in a sufficiently large open subset and dimx 1 then we prove that the local graded deformation functor of m is isomorphic to the local hilbert scheme functor at x proj r under a weak assumption which holds if dimx 2 under this assumption we get that the hilbert scheme is smooth at x and we give an explicit formula for the dimension of its local ring as a corollary we prove a conjecture of r m miro roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component v of the hilbert scheme also their conjecture on the dimension of v is proved for dimx 1 the cohomology h i nx of the normal sheaf of x in proj r is shown to vanish for 1 i dimx 2 finally the mentioned results slightly adapted remain true replacing r by any cohen macaulay quotient of a polynomial ring

computation of oscillatory integrals with an exponential kernel and jacobi type singularities abstract in this communication two rules are considered for computing oscillatory integrals with an exponential kernel and jacobi type singularities an asymptotic rule is presented with the aid of integration by parts an asymptotic filon type quadrature rule is constructed on the basis of a two point taylor interpolation polynomial error analysis on the proposed rules in inverse powers of large frequency is provided numerical examples are provided to illustrate that the two rules are very efficient in obtaining very high precision approximations if is sufficiently large

letter to the editor on the filon and levin methods for highly oscillatory integral this paper shows that for any suitably smooth function f x and arbitrarily selected interpolation nodes c1 c2 cv in a b the filon method and the levin method for with the polynomial interpolation approach are identical when g x is a linear function based on this result a new efficient levin quadrature for is presented

bounds for codes in products of spaces grassmann and stiefel manifolds upper bounds are derived for codes in stiefel and grassmann manifolds with given minimum chordal distance they stem from upper bounds for codes in the product of unit spheres and projective spaces the new bounds are asymptotically better than the previously known ones

on the implementation of the asymptotic filon type method for infinite integrals with oscillatory bessel kernels in this paper we consider the implementation of the asymptotic filon type method for the semi infinite highly oscillatory bessel integrals of the form 1 f x c v wx dx where c v wx denotes bessel function j v wx of the first kind y v wx of the second kind h v 1 wx and h v 2 wx of the third kind and modified bessel function k v wx of the second kind respectively f is a smooth function on 1 lim x f k x 0 k 0 1 2 and w is large by approximating f by a linear combination of negative integer powers so that the moments can be expressed by some special functions we complete the implementation of the method furthermore we give the error analysis of the method for computing the integrals the method is very efficient in obtaining very high precision approximations if w is sufficiently large numerical examples are provided to confirm our analysis

on the levin iterative method for oscillatory integrals this paper considers and gives error analysis for levin iteration method to approximate bessel trigonometric transformation i f email protected a bf x cos r 1x j m rx dx for generalized fourier transformation i f email protected a bf x e i w g x dx under the condition that g x 0 for all email protected a b levin iteration method with the initial u 0 x 0 is identical to the asymptotic method

on generalized quadrature rules for fast oscillatory integrals abstract in this paper we present numerical analysis for the generalized quadrature rule for a b f x w r x d x where w r x is an oscillatory function and derive higher order generalized quadrature rules the results show that the generalized quadrature rules are efficient for bessel trigonometric transformations and the accuracy increases when oscillation becomes faster

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

numerical solutions to volterra integral equations of the second kind with oscillatory trigonometric kernels this paper explores quadrature methods for weakly singular volterra integral equation of the second kind with highly oscillatory trigonometric kernels and presents piecewise constant and linear collocation methods for approximation of the solution the evaluation of the highly oscillatory kernels of integral equation will give rise to the computation of oscillatory integrals which can be done by using efficient filon type methods moreover by using asymptotic analysis of the solution the convergence rates for piecewise constant and linear collocation methods which show the higher the frequency the more accurate the numerical results the effectiveness and accuracy of the proposed method are tested by numerical examples

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

analysis of a collocation method for integrating rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is analyzed the method is efficient for integrals involving bessel functions j rx with a large oscillation frequency parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations the analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases ko words oscillatory integrals collocation analysis

change of rings in deformation theory of modules given a b module m and any presentation b a j the obstruction theory of m as a b module is determined by the usual obstruction class o a for deforming m as an a module and a new obstruction class o j these two classes give the tool for constructing two obstruction maps which depend on each other and which characterise the hull of the deformation functor we obtain relations between the obstruction classes by studying a change of rings spectral sequence and by representing certain classes as elements in the yoneda complex calculation of the deformation functor of m as a b module including the generalised massey products is thus possible within any a free 2 presentation of m

efficient evaluation of oscillatory bessel hilbert transforms in this paper we study the asymptotics and evaluation of the oscillatory bessel hilbert transform 0 f x email protected n wx dx with 0 t the singularity of the hilbert transform is transferred to an individual oscillatory integral independent of f x for this singular integral we present two methods one is the combination of a filon type method and a complex integration method the other is the combination of a filon type method and an adaptive clenshaw curtis quadrature the remaining integral which is nonsingular can be well calculated with a combination of a filon type method and a gauss laguerre quadrature the efficiency and accuracy of the proposed methods are illustrated by numerical examples

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

bounds on packings of spheres in the grassmann manifold we derive the gilbert varshamov and hamming bounds for packings of spheres codes in the grassmann manifolds over r and c asymptotic expressions are obtained for the geodesic metric and projection frobenius chordal metric on the manifold

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

choice of specific interpolation points in quadrature formula for fast oscillating integrals integrals containing trigonometric functions with non linear argument having a large multiplier are treated with the help of a specific choice of interpolation points this allows excluding the contribution of internal points of interpolation and determining upper and low bounds of the considered integral

the method of computing formal moduli abstract in this paper we consider a method given earlier for computing the formal moduli of modules we start with a short resume of the method then we explain how this method can be implemented using a standard basis computation we list the formal moduli with versal families for all torsionfree rank 1 modules over the curve singularity e6 finally we discuss the degeneracy properties of these modules proving that the normalization deforms to all the other torsionfree rank 1 modules these examples are computed by implementing the algorithm in the singular language the program source is not included in this paper for obvious reasons but can be found in previous work by the author 1997 implementing the algorithm for computing formal moduli preprint series institute of mathematics university of oslo

global parametric solutions of scalar transport passive scalar transport involves complex interactions between advection and diffusion where the global transport rate depends upon scalar diffusivity and the values of the possibly large set of parameters controlling the advective flow although computation of a single solution of the advection diffusion equation ade is simple in general it is prohibitively expensive to compute the parametric variation of solutions over the full parameter space q even though this is crucial for e g optimization parameter estimation and elucidating the global structure of transport by decomposing the flows within q so as to exploit symmetries we derive a spectral method that solves the ade over q three orders of magnitude faster than other methods of similar accuracy solutions are expressed in terms of the exponentially decaying natural periodic patterns of the ade sometimes called strange eigenmodes we apply the method to the experimentally realisable rotated arc mixer chaotic flow both to establish numerical properties and to calculate the fine scale structure of the global solution space for transport in this chaotic flow over 10 5 solutions within q are resolved and spatial pattern locking a symmetry breaking transition to disordered spatial patterns and fractally distributed optima in transport rate are observed the method exhibits exponential convergence and efficiency increases with resolution of q

efficient numerical methods for bessel type of oscillatory integrals in this paper new quadrature rules are proposed for numerical evaluation of highly oscillatory integrals containing first kind of bessel functions j x meshless procedure with uniform and scattered nodes is used to cope with frequent irregular oscillations caused by bessel and bessel trigonometric functions in addition multi resolution quadrature rules based on hybrid functions and haar wavelets are used as supporting tools to handle the case of singularity of the meshless collocation method error bounds of the proposed methods are calculated and numerically verified by solving some benchmark test problems

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels based on the filon clenshaw curtis method for highly oscillatory integrals and together with the sommariva s result sommariva 2013 for clenshaw curtis quadrature rule we present a chebyshev collocation method for a class of fredholm integral equations with highly oscillatory kernels whose unknown function is assumed to be less oscillatory than the kernel in the proposed method the filon clenshaw curtis method is used to compute the involved oscillatory integrals which makes the proposed method very precise by solving only a small system of linear equations we can obtain a very satisfactory numerical solution the performance of the presented method is illustrated by several numerical examples compared with the method proposed by li et al 2012 this method enjoys a lower computational complexity furthermore numerical examples show that the presented method has a competitive advantage on the accuracy compared with the method in li et al 2012

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

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

efficient quadrature for highly oscillatory integrals involving critical points this paper based on the levin collocation method and levin type method together with composite two point gauss legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points the effectiveness and accuracy of the quadrature are tested

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

new quadrature rules for highly oscillatory integrals with stationary points in this paper new algorithms are being proposed for evaluation of highly oscillatory integrals hois with stationary point s the algorithms are based on modified levin quadrature mlq with multiquadric radial basis functions rbfs coupled with quadrature rules based on hybrid functions of order 8 hfq8 and haar wavelets quadrature hwq aziz et al 2011 part of the new procedure presented in this paper is comprised of transplanting monomials which are used in the conventional levin method by the rbfs the linear and hermite polynomials based quadratures xiang 2007 are being replaced by the new methods based on hwq and hfq8 respectively both the methods are merged with mlq to obtain the numerical solution of highly oscillatory integrals having stationary points the accuracy of the new methods is neither dampened by presence of the stationary point s nor by the large value of frequency parameter theoretical facts about the error analysis of the new methods are analyzed and proved numerical examples are included to show efficiency and accuracy of the new methods a new framework for the numerical solution of highly oscillatory integrals is proposed the integrals are bifurcated in the neighborhood of stationary point the integral on the smaller subinterval is solved by hybrid functions and haar wavelets the integral on the longer subinterval is solved by the meshless method with mq radial basis functions convergence analysis of the proposed methods is performed

numerical evaluation of a class of highly oscillatory integrals involving airy functions in this paper we consider the implementation of the clenshaw curtis filon type method for a class of highly oscillatory integrals 0 1 x 1 x f x ai x dx where ai x is an airy function and 1 1 by replacing f by its chebyshev interpolation polynomial at the clenshaw curtis points so that the modified moments can be computed by recursive formula based on special functions an efficient and stable method for this integral is presented error analysis for the presented method is given moreover the method shares the property that the larger the the higher the precision theoretical results and numerical examples show that the method is very efficient in obtaining very high precision approximations if is sufficiently large

linear programming bounds for codes in grassmannian spaces in this paper we develop the linear programming method to obtain bounds for the cardinality of grassmannian codes endowed with the chordal distance we obtain a bound and its asymptotic version that generalize the well known bound for codes in the real projective space obtained by kabatyanskiy and levenshtein and improve the hamming bound for sufficiently large minimal distances

a unified framework for asymptotic analysis and computation of finite hankel transform abstract in this paper we present a unified framework for asymptotic analysis of the finite hankel transform this framework enables us to derive asymptotic expansions of the transform including the cases where the oscillator has zeros and stationary points as a consequence two efficient and affordable methods for computing the transform numerically are discussed and a detailed analysis of their asymptotic error estimate is carried out numerical examples are provided to confirm our analysis

optimal by order quadrature formulae for fast oscillatory functions with inaccurately given a priori information abstract in this article the authors construct optimal by order quadrature formulae for integration of fast oscillatory functions in interpolational classes c 1 l n 1 and c 1 l n e 1 the construction of efficient formulae for numerical integration of fast oscillatory functions is based on the application of the residual method and the method of quasi solutions both cases weak and strong oscillations are considered results of numerical examples are presented

quadrature methods for multivariate highly oscillatory integrals using derivatives while there exist effective methods for univariate highly oscillatory quadrature this is not the case in a multivariate setting in this paper we embark on a project extending univariate theory to more variables inter alia we demonstrate that in the absence of critical points and subject to a nonresonance condition an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices a direct counterpart of the univariate case this provides a convenient avenue towards the generalization of asymptotic and filon type methods as formerly introduced by the authors in a single dimension to simplices and more generally to polytopes the nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth in effect its violation is equivalent to the presence of stationary points in a single dimension we further explore this issue and propose a technique that often can be used in this situation yet much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals

fast computation of seabed spherical wave reflection coefficients in geoacoustic inversion this paper develops a fast numerical approach to computing spherical wave reflection coefficients swrcs for layered seabeds which provides substantial savings in computation time when used as the forward model for geoacoustic inversion of broadband seabed reflectivity data the approach exploits the sommerfeld integral representation of swrcs as the hankel transform of a function proportional to the plane wave reflection coefficient pwrc and applies levin integration to the rapidly oscillating integrand cast as the product of a pre computed media independent matrix and a vector involving pwrcs at a sparse sampling of integration angles compared to conventional simpson s rule integration for computation of the swrc the levin integration yields speed up factors of an order of magnitude or more further it results in reduced memory requirements for storage of pre computed quantities a desirable property when a graphics processing unit gpu is used for parallel computation of swrcs the paper applies trans dimensional bayesian inversion to investigate the impact of forward modeling in terms of pwrcs and swrcs on the estimation of geoacoustic parameters and uncertainties model comparisons are quantified in simulated and measured data inversions by comparing the estimated geoacoustic parameters to the true parameters or core measurements respectively and by calculating the deviance information criterion for model selection

the linear kdv equation with an interface the interface problem for the linear korteweg de vries kdv equation in one dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain the location of the interface is known and a number of compatibility conditions at the boundary are imposed we provide an explicit characterization of sufficient interface conditions for the construction of a solution using fokas s unified transform method the problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives

on the numerical approximation for fourier type highly oscillatory integrals with gauss type quadrature rules in this paper we present an improved numerical steepest descent method for the approximation of fourier type highly oscillatory integrals based on the previous numerical steepest descent method the new method used the integrand information at endpoints and stationary points the asymptotic order is given that is improved both for the case of stationary points and stationary points free several numerical examples are presented which show the high efficiency of the proposed method numerical results support our theoretical analyses