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

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 boundary condition at an impedance wall in a non uniform duct with potential mean flow the boundary condition at an impedance wall in a duct with a steady mean flow requiring the specification of the normal component of acoustic particle velocity is examined it is found that when implemented in the weak formulation of the finite element method it can be considerably simplified the boundary condition would appear to require data which includes the tangential derivative of the tangential mean flow velocity the normal derivative of the normal component of mean flow velocity and the derivatives of the mean flow density and the boundary admittance along the boundary it is shown that with suitable rearrangement the normal and tangential velocity derivatives can be eliminated as can the derivatives of the mean flow density and admittance the boundary condition becomes only slightly more complicated than the corresponding boundary condition when mean flow is absent and is no more difficult to implement requiring only local values of tangential mean flow velocity density and admittance which are already required as data for the weak formulation of the field equation

three classes of minimal linear codes over the finite fields of odd characteristic minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two party computation in this paper we focus on constructing minimal linear codes with frac w min w max leq frac p 1 p for any odd prime p based on a generic construction of linear codes where w min and w max denote the minimum and maximum nonzero weights in a code respectively first we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal second we also present an infinite family of minimal linear codes by employing partial spreads which can be viewed as a generalization of the construction of ding et al in addition we determine the weight distributions of all these minimal linear codes

secret sharing schemes from binary linear codes in principle every linear code can be used to construct a secret sharing scheme however in general determining the access structure of the scheme is very hard on the other hand finding error correcting codes that produce secret sharing schemes with efficient access structures is also difficult in this paper we study a set of minimal codewords for certain classes of binary linear codes and then determine the access structure of secret sharing schemes based on these codes furthermore we prove that the secret sharing schemes obtained are democratic in the sense that every participant is involved in the same number of minimal access sets

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

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

minimal linear codes from characteristic functions minimal linear codes have interesting applications in secret sharing schemes and secure two party computation this paper uses characteristic functions of some subsets of mathbb f q to construct minimal linear codes by properties of characteristic functions we can obtain more minimal binary linear codes from known minimal binary linear codes which generalizes results of ding et al ieee trans inf theory vol 64 no 10 pp 6536 6545 2018 by characteristic functions corresponding to some subspaces of mathbb f q we obtain many minimal linear codes which generalizes results of ieee trans inf theory vol 64 no 10 pp 6536 6545 2018 and ieee trans inf theory vol 65 no 11 pp 7067 7078 2019 finally we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes

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

determination of the local weight distribution of binary linear block codes some methods to determine the local weight distribution of binary linear codes are presented two approaches are studied a computational approach and a theoretical approach for the computational approach an algorithm for computing the local weight distribution of codes using the automorphism group of the codes is devised in this algorithm a code is considered the set of cosets of a subcode and the set of cosets is partitioned into equivalence classes thus only the weight distributions of zero neighbors for each representative coset of equivalence classes are computed for the theoretical approach relations between the local weight distribution of a code its extended code and its even weight subcode are studied as a result the local weight distributions of some of the extended primitive bose chaudhuri hocquenghen bch codes reed muller codes primitive bch codes punctured reed muller codes and even weight subcodes of primitive bch codes and punctured reed muller codes are determined

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

galerkinlevin method for highly oscillatory integrals in this paper a new method namely galerkinlevin method is proposed to investigate the highly oscillatory integrals indeed we construct an algorithm based on the levin and galerkin methods using the levin method the given integral is transformed into a certain ordinary differential equation then a galerkin method is applied for solving the obtained ode the efficiency of the approach will be shown by applying the procedure on some prototype examples a new algorithm for solving the highly oscillatory integrals is proposed using the levin method a given integral is transformed into a certain ode error and convergence analysis of the proposed algorithm is performed

tightened upper bounds on the ml decoding error probability of binary linear block codes the performance of maximum likelihood ml decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability the upper bounds on the block and bit error probabilities are valid for any memoryless binary input and output symmetric communication channel and their effectiveness is exemplified for various ensembles of turbo like codes over the additive white gaussian noise awgn channel an expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds

two new families of two weight codes we construct two new infinite families of trace codes of dimension 2m over the ring mathbb f p u mathbb f p with u 2 u when p is an odd prime they have the algebraic structure of abelian codes their lee weight distribution is computed by using gauss sums by gray mapping we obtain two infinite families of linear p ary codes of respective lengths p m 1 2 and 2 p m 1 2 when m is singly even the first family gives five weight codes when m is odd and p equiv 3 pmod 4 the first family yields p ary two weight codes which are shown to be optimal by application of the griesmer bound the second family consists of two weight codes that are shown to be optimal by the griesmer bound whenever p 3 and m ge 3 or p ge 5 and m ge 4 applications to secret sharing schemes are given

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

secret sharing schemes from three classes of linear codes secret sharing has been a subject of study for over 20 years and has had a number of real world applications there are several approaches to the construction of secret sharing schemes one of them is based on coding theory in principle every linear code can be used to construct secret sharing schemes but determining the access structure is very hard as this requires the complete characterization of the minimal codewords of the underlying linear code which is a difficult problem in general in this paper a sufficient condition for all nonzero codewords of a linear code to be minimal is derived from exponential sums some linear codes whose covering structure can be determined are constructed and then used to construct secret sharing schemes with nice access structures

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

on the complexity of minimum distance decoding of long linear codes we suggest a decoding algorithm of q ary linear codes which we call supercode decoding it ensures the error probability that approaches the error probability of minimum distance decoding as the length of the code grows for n spl rarr spl infin the algorithm has the maximum likelihood performance the asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known the algorithm develops the ideas of covering set decoding and split syndrome decoding

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

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

frequency domain prediction of turbofan noise radiation this paper describes a frequency domain numerical method for predicting noise radiation from ducted fans including acoustic treatment and non uniform background flow effects the method solves the euler equations linearized about a mean flow in the frequency domain a pseudo time derivative term is added to the frequency domain equations so that a time marching technique can be employed to drive the acoustic field to steady state explicitly this approach makes distributed parallel computing more viable for equations of this type and will allow for future use of well known convergence acceleration techniques such as multigrid to obtain the solutions efficiently simulations of the jt15d static test inlet are performed including the effects of liners and the results are compared with experimental data a generic engine geometry is used for demonstrating further the prediction capability of the code calculating the attenuation effects of different liner impedances and liner installation locations on the radiated sound fields

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

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

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

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

universally ideal secret sharing schemes given a set of parties 1 spl middot spl middot spl middot n an access structure is a monotone collection of subsets of the parties for a certain domain of secrets a secret sharing scheme for an access structure is a method for a dealer to distribute shares to the parties these shares enable subsets in the access structure to reconstruct the secret while subsets not in the access structure get no information about the secret a secret sharing scheme is ideal if the domains of the shares are the same as the domain of the secrets an access structure is universally ideal if there exists an ideal secret sharing scheme for it over every finite domain of secrets an obvious necessary condition for an access structure to be universally ideal is to be ideal over the binary and ternary domains of secrets the authors prove that this condition is also sufficient they also show that being ideal over just one of the two domains does not suffice for universally ideal access structures finally they give an exact characterization for each of these two conditions

on the covering structures of two classes of linear codes from perfect nonlinear functions in this paper the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions from fqm to itself are determined using a unified approach all the minimal codewords of these codes are characterized according to their weights which suggests that their covering structures are determined finally all the minimal access sets of the secret sharing schemes based on their dual codes are obtained

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

the partition of unity finite element method for short wave acoustic propagation on non uniform potential flows a novel numerical method is proposed for modelling time harmonic acoustic propagation of short wavelength disturbances on non uniform potential flows the method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space the basis functions are local solutions of the governing equations they are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non uniform many wavelengths can be included within a single element leading to ultra sparse meshes results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid based schemes numerical results for lined uniform two dimensional ducts and for non uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved numerical studies indicate that the pollution effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated

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

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

pseudo codeword analysis of tanner graphs from projective and euclidean planes we consider coded data transmission over a binary input output symmetric memoryless channel using a binary linear code in order to understand the performance of maximum likelihood ml decoding one studies the codewords in particular the minimal codewords and their hamming weights in the context of linear programming lp decoding one s attention needs to be shifted to the pseudo codewords in particular to the minimal pseudo codewords and their pseudo weights in this paper we investigate some families of codes that have good properties under lp decoding namely certain families of low density parity check ldpc codes that are derived from projective and euclidean planes we study the structure of their minimal pseudo codewords and give lower bounds on their pseudo weight besides this main focus we also present some results that hold for pseudo codewords and minimal pseudo codewords of any tanner graph and we highlight how the importance of minimal pseudo codewords under lp decoding varies depending on which binary input output symmetric memoryless channel is used

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

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

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

upper bounds to error probabilities of coded systems beyond the cutoff rate a family of upper bounds to error probabilities of coded systems was recently proposed by d divsalar see ieee communication theory workshop 1999 jpl tmo prog rep 42 139 1999 these bounds are valid for transmission over the additive white gaussian noise channel and require only the knowledge of the weight spectrum of the code words after illustrating these bounds we extend them to fading channels contrary to the union bound our bounds maintain their effectiveness below the signal to noise ratio snr at which the cutoff rate of the channel equals the rate of the code some applications are shown first we derive upper bounds to the minimum snr necessary to achieve zero error probability as the code block length increases to infinity next we use our bounds to predict the performance of turbo codes and low density parity check codes

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

on the voronoi neighbor ratio for binary linear block codes soft decision decoding of block codes is regarded as the geometrical problem of identifying the voronoi region within which a given input vector lies a measure called the neighbor ratio is proposed to characterize how many facets a voronoi region has theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes an asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1 2 or not for rates below this threshold all pairs of codewords tend to share a voronoi facet for higher rates a relatively small fraction of them do

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

a three dimensional hybrid finite element spectral analysis of noise radiation from turbofan inlets this paper describes a new three dimensional 3d analysis of tonal noise radiated from non axisymmetric turbofan inlets the novelty of the method is in combining a standard finite element discretisation of the acoustic field in the axial and radial coordinates with a fourier spectral representation in the circumferential direction the boundary conditions at the farfield fan face and acoustic liners are treated using the same spectral representation the resulting set of discrete acoustic equations are solved employing the well established bicgstab or qmr iterative algorithms and a very effective specialised preconditioner based on the axisymmetric mean geometry and flow field numerical examples demonstrate the suitability of the new method to engine configurations with realistic 3d features such as relatively large degrees of asymmetry and spliced acoustic liners the examples also illustrate the two advantages of the new method over a traditional 3d finite element approach the new method requires a significantly smaller number of unknowns as relatively few circumferential fourier modes in the spectral solution ensure an accurate field representation also due to the effective preconditioner the spectral linear solver benefits from stable iterations at a high rate of convergence

improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow we present an efficient multimodal method to describe the acoustic propagation in the presence of a uniform flow in a waveguide with locally a wall impedance treatment the method relies on a variational formulation of the problem which allows to derive a multimodal formulation within a rigorous mathematical framework notably to properly account for the boundary conditions on the walls being locally the myers condition and the neumann condition otherwise also the method uses an enriched basis with respect to the usual cosine basis able to absorb the less converging part of the modal series and thus to improve the convergence of the method using the cosine basis the modal method has a low convergence 1 n with n the order of truncation using the enriched basis the improvement in the convergence is shown to depend on the mach number from 1 n5 to roughly 1 n1 5 for m 0 to m close to unity the case of a continuously varying wall impedance is considered and we discuss the limiting case of piecewise constant impedance which defines pressure edge conditions at the impedance discontinuities

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

a reverse flow theorem and acoustic reciprocity in compressible potential flows in ducts a reverse flow theorem for acoustic propagation in compressible potential flow has been obtained directly from the field equations without recourse to energy conservation arguments a reciprocity theorem for the scattering matrix for the propagation of acoustic modes in a duct with either acoustically rigid walls or acoustically absorbing walls follows it is found that for a source at a specific end of the duct suitably scaled reflection matrices in direct and reverse flow have a reciprocal relationship scaled transmission matrices obtained for direct flow and reversed flow with simultaneous switching of source location from one end to the other also have a reciprocal relationship a related reverse flow theorem specialized to one dimensional acoustic propagation has also been obtained reciprocity relationships for the scattering coefficients for propagation are derived and are found to be similar though much simpler than in the case of multi mode propagation in one dimensional flow reciprocal relations and power conservation arguments are used to show that scaled power reflection and transmission coefficients are invariant to flow reversal and switching of source location from one end of the duct to the other numerical verification of the reciprocal relationships is given in a companion paper

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

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

numerical experiments on acoustic reciprocity in compressible potential flows in ducts a reciprocity theorem for the scattering matrix for the propagation of acoustic modes in a duct with acoustically hard walls or with acoustically absorbing walls has been given in a companion publication it was found that for a source at a specified end of the duct suitably scaled reflection matrices in direct and reverse flow have a reciprocal relationship scaled transmission matrices obtained for direct flow and reversed flow with simultaneous switching of source location from one end to the other also have a reciprocal relationship a reverse flow theorem for the equivalent one dimensional propagation model which is a good approximation to the three dimensional model at low frequencies was also obtained in this case using reciprocity and acoustic power conservation arguments it is additionally found that the acoustic power transmission coefficient is the same for a source at either end of the duct for a given flow direction this result leads to an invariance theorem which relates acoustic power propagated due to sources of equal pressure amplitude at the two ends of the duct a numerical verification of these reciprocal relationships is given here for propagation in axially symmetric circular and annular ducts with multi modal propagation and at low frequencies when a one dimensional model is appropriate

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

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

voronoi regions for binary linear block codes the voronoi regions of a block code govern many aspects of the code s performance on a gaussian channel and they are fundamental instruments in for example error probability analysis and soft decision decoding the article presents an efficient method for finding the boundaries of the voronoi regions for an arbitrary binary linear block code two theoretical results together lead to the voronoi regions first it is shown that the question of the voronoi neighborship can be reduced into testing a simpler relation called the gabriel neighborship second a fast method of recognising gabriel neighbors is proposed these results are finally employed to describe the voronoi regions for the golay codes and several bch codes including hamming codes

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

minimal vectors in linear codes minimal vectors in linear codes arise in numerous applications particularly in constructing decoding algorithms and studying linear secret sharing schemes however properties and structure of minimal vectors have been largely unknown we prove basic properties of minimal vectors in general linear codes then we characterize minimal vectors of a given weight and compute their number in several classes of codes including the hamming codes and second order reed muller codes further we extend the concept of minimal vectors to codes over rings and compute them for several examples turning to applications we introduce a general gradient like decoding algorithm of which minimal vectors decoding is an example the complexity of minimal vectors decoding for long codes is determined by the size of the set of minimal vectors therefore we compute this size for long randomly chosen codes another example of algorithms in this class is given by zero neighbors decoding we discuss relations between the two decoding methods in particular we show that for even codes the set of zero neighbors is strictly optimal in this class of algorithms this also implies that general asymptotic improvements of the zero neighbors algorithm in the frame of gradient like approach are impossible we also discuss a link to secret sharing schemes

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

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

seven classes of three weight cyclic codes cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms compared with linear block codes in this paper seven classes of three weight cyclic codes over gf p whose duals have two zeros are presented where p is an odd prime the weight distributions of the seven classes of cyclic codes are settled some of the cyclic codes are optimal in the sense that they meet certain bounds on linear codes the application of these cyclic codes in secret sharing is also considered

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

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 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 correctable errors of binary linear codes the error correction capability of binary linear codes with minimum distance decoding in particular the number of correctable uncorrectable errors is investigated for general linear codes and the first order reed muller codes for linear codes a lower bound on the number of uncorrectable errors is derived the bound for uncorrectable errors with a weight of half the minimum distance asymptotically coincides with the corresponding upper bound for reed muller codes and random linear codes for the first order reed muller codes the number of correctable uncorrectable errors with a weight of half the minimum distance plus one is determined this result is equivalent to deriving the number of boolean functions of m variables with nonlinearity 2 m 2 1 the monotone error structure and its related notions larger half and trial set which were introduced by helleseth klove and levenshtein are mainly used to derive the results

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

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

a mapped finite and infinite partition of unity method for convected acoustic radiation in axisymmetric domains a mapped infinite partition of unity method is developed to solve frequency domain acoustic applications for convected radiation in axisymmetric domains the paper presents the formulation and the discretization technique the partition of unity method is coupled with infinite elements based on the astley leis conjugated formulation the performances of the method are analysed on two applications convected propagation in an uniform duct and dipole radiation the aim of such a method compared to the classical finite element method is to solve shortwave applications with a respectable level of accuracy

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