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

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

minimum distance computation of ldpc codes using a branch and cut algorithm we give a branch and cut algorithm for finding the minimum distance of a binary linear block code we give two integer programming ip models and study the convex hull of the single constraint relaxation of these ip models we use the new inequalities as cuts in a branch and cut scheme finally we report computational results based on turbo and low density parity check ldpc codes that demonstrate the effectiveness of our cuts we demonstrate that our ip formulation and specific cuts are efficient tools for determining the minimum distance of moderate size linear block codes specifically they are very efficient for ldpc codes and provide us with an additional tool for solving this important problem

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

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

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

dual row modules and polyhedra of blocking group problems corresponding to every group problem is a row module duality for group problems is developed using the duality or orthogonality of the corresponding row modules the row module corresponding to a group problem is shown to include gomory s fractional cuts for the group polyhedron and all the vertices of the polyhedron of the blocking group problem the polyhedra corresponding to a pair of blocking group problems are shown to have a blocking nature i e the vertices of one include some of the facets of the other and mutatis mutandis the entire development is constructive the notions of contraction deletion expansion and extension are defined constructively and related to homomorphic liftings and suproblems in a dual setting roughly speaking a homomorphic lifting is dual to forming a subproblem a proof of the gastou johnson generalization of gomory s homomorphic lifting theorem is given and dual constructions are discussed a generalization of gomory s subadditive characterization to subproblems is given in the binary case it is closely related to the work of seymour on cones arising from binary matroids

integrability of centers perturbed by quasi homogeneous polynomials abstract the objective of this work is to give sufficient and necessary conditions for integrability of differential equation systems in the plane of the form y2p 1 x2 x y x2q 1 y2 x y wherex2 x y andy2 x y are p q quasi homogeneous polynomials of degree 2pq q 1 and 2pq p 1 respectively

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

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

generating limit cycles from a nilpotent critical point via normal forms abstract it is well known that the normal form theory can be applied to solve the center focus problem for monodromic planar nilpotent singularities in this paper we see how this theory can also be applied to generate limit cycles from this type of singularities

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

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

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

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

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

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

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

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

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

on the generality of the subadditive characterization of facets gomory s characterization of facets of the group problem and araoz s characterization for certain semigroups are shown to apply to a very general situation where only closure of the addition operation is assumed

the center conditions for a lienard system we obtain necessary and sufficient coefficient conditions for a center at the origin for a lienard system with nonlinearities of degree six the necessity of the conditions is derived from the first seven focus quantities and their sufficiency is proved by cherkas s method

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

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

binary clutter inequalities for integer programs we introduce a new class of valid inequalities for general integer linear programs called binary clutter bc inequalities they include the 0 frac 1 2 cuts of caprara and fischetti as a special case and have some interesting connections to binary matroids binary clutters and gomory corner polyhedra we show that the separation problem for bc cuts is strongly hard in general but polynomially solvable in certain special cases as a by product we also obtain new conditions under which 0 frac 1 2 cuts can be separated in polynomial time these ideas are then illustrated using the traveling salesman problem tsp as an example this leads to an interesting link between the tsp and two apparently unrelated problems the t join and max cut problems

cyclicity of a family of vector fields abstract we study the cyclicity and the center problem for a special family of planar differential equations we also relate this cyclicity with the maximum number of limit cycles that bifurcate from a hamiltonian vector field afterwards we apply the result obtained to several known types of differential equations

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

reversibility and quasi homogeneous normal forms of vector fields abstract this paper uses tools in quasi homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point our main result provides an algorithm via lie triangle that detects the non reversibility of vector fields that is it is possible to decide whether a planar center is not reversible some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields we find several families of nilpotent centers which are non reversible

generalized singular point quantity and integrability of degenerate resonant singular point in this paper integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied the concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied two methods of computing generalized singular point quantities were given furthermore the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time

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

small amplitude limit cycles in lienard systems via multiplicity abstract a recent paper of christopher and lloyd reduces the computation of the order of degeneracy of a weak focus for a polynomial lienard system x y f x y g x to the computation of the multiplicity of a polynomial map in this paper we first take advantage of that approach to obtain new lower and upper bounds for the maximum order of degeneracy of the origin in terms of the degrees of f and g later on we implement an algorithm to compute this maximum order for concrete values of these degrees as far as we know we enlarge the set of degrees for which this maximum order was known finally we extend christopher and lloyd s result to analytic degenerate or not lienard equations

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

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

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

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

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

four problems on graphs with excluded minors the four problems we consider are the chinese postman odd cut co postman and odd circuit problems seymour s characterization of matroids having the max flow min cut property can be specialized to each of these four problems to show that the property holds whenever the graph has no certain excluded minor we develop a framework for characterizing graphs not having these excluded minors and use the excluded minor characterizations to solve each of the four optimization problems in this way a constructive proof of seymour s theorem is given for these special cases we also show how to solve the chinese postman problem on graphs having no four wheel minor where the max flow min cut property need not hold

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

bifurcation at the equator for a class of quintic polynomial differential system abstract the center problem and bifurcation of limit cycles for degenerate singular point and infinity are far from being solved in general in this paper we studied center conditions and bifurcation of limit cycles at the degenerate singular point and infinity the equator in a class of quintic polynomial differential system with two small parameters and eight normal parameters the method is based on two homeomorphic transformations of the infinity and degenerate singular point into linear singular point which allows us to compute the generalized lyapunov constants the singular point quantities for the origin and infinity the center conditions for the degenerate singular point and infinity are derived respectively the limit cycle configurations of 7 2 and 2 6 are obtained under simultaneous perturbation at the origin and infinity as far as we know it is the first time that the bifurcation of limit cycles from a degenerate singular point and infinity simultaneously is studied in planar differential system

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

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

polyhedra of regular p nary group problems the duality for group problems developed in 3 is restricted top nary group problems results for ternary group problems are obtained similar to those obtained by fulkerson and lehman for the binary case a complete facet description of the group polyhedron is available for a group problem having the fulkerson property a group problem has the fulkerson property if its vertices are the facets of the blocking group problem and if its facets are the vertices of the blocking group problem the fulkerson property is a generalization of the max flow min cut theorem of ford and fulkerson which is interpreted as a statement about the pair of row modules arising from a group problem we show that a group problem has the fulkerson property if the corresponding row module is regular

center conditions of a particular polynomial differential system with a nilpotent singularity abstract in this work we study the center conditions of a particular polynomial differential system with a nilpotent singularity using a new proposed algorithm this problem was initially studied in 19 where some center conditions were found using the cherkas method and its full characterization was established as an open problem

center problem for several differential equations via cherkas method cherkas method characterizes centers for analytic lienard differential equations we extend his method to degenerate lienard differential equations and we apply this extension to solve the center problem for several families of polynomial differential equations in particular we give all centers for some differential equations given by a vector field which is the sum of two quasi homogeneous ones

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

center problem for trigonometric lienard systems we give a complete algebraic characterization of the non degenerated centers for planar trigonometric lienard systems the main tools used in our proof are the classical results of cherkas on planar analytic lienard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials our results are also applied to some particular subfamilies of planar trigonometric lienard systems the results obtained are reminiscent of the ones for planar polynomial lienard systems but the proofs are different

binary group and chinese postman polyhedra a new proof of the characterization of the chinese postman polyhedra is given in developing this proof a theorem of gomory about homomorphic lifting of facets for group polyhedra is generalized to subproblems some results for the chinese postman problem are generalized to binary group problems in addition a connection is made between fulkerson s blocking polyhedra and a blocking pair of binary group problems a connection is also developed between minors and lifting of facets for group problems

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

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

the center problem for a family of systems of differential equations having a nilpotent singular point abstract we study the analytic system of differential equations in the plane x y t i 0 f q p 2 i s where p q n p q s n 1 p q 0 n n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we prove for this system the existence of a lyapunov function and we solve theoretically the center problem for such system finally as an application of the theoretical procedure we characterize the centers of several subfamilies

weierstrass integrability in lienard differential systems in this work we study the lienard differential systems that admit a weierstrass first integral or a weierstrass inverse integrating factor

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

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

local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations we study the analytic system of differential equations in the plane which can be written in a suitable coordinates system as email protected email protected email protected i 0 f q p 2 i s where p email protected n p 0 email protected n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we show the taylor expansion of the return map near the origin for this system which allow us to generate small amplitude limit cycles bifurcating from the critical point also as an application of the theoretical procedure we characterize the centers and we generate limit cycles of small amplitude from the origin of several families finally we give a new family integrable analytically which includes the centers of the systems studied