a fast algorithm for prandtl s integro differential equation collocation and quadrature methods for singular integro differential equations of prandtl s type are studied in weighted sobolev spaces a fast algorithm basing on the quadrature method is proposed convergence results and error estimates are given

on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

the numerical evaluation of a class of logarithmically singular integral transforms abstract we consider the numerical evaluation of several logarithmically singular integral transforms which occur in the numerical solution of various types of integral equations with logarithmically singular kernels when the equations involve cauchy singular kernels as well the method is a substantial improvement in speed accuracy and ease of programming over the method introduced by fromme golberg and werth 5 convergence of the rule is proved under suitable differentiability conditions on the smooth part of the kernel and numerical results are presented substantiating this result

the numerical evaluation of hadamard finite part integrals a quadrature rule is described for the numerical evaluation of hadamard finite part integrals with a double pole singularity within the range of integration the rule is based upon the observation that such an integral is the derivative of a cauchy principal value integral

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

calculating singular integrals as an ill posed problem applying numerical quadrature to singular integrals in a straightforward way leads to uncontrolled instability with respect to data errors in this paper we describe how to control this instability

birational automorphisms of fano hypersurfaces v v y v 0 between smooth three dimensional quartics v v4 v 0 v 0 4 in p is a projective isomorphism this is the original wording of their theorem in fact the arguments of im give a much stronger result namely they actually prove that smooth quartics are birationally superrigid the concept of birational rigidity is not visual being rather technical it makes little impression it is immediate implications of the property of being birationally super rigid that are really impressive for a projective variety x smooth in codimension 1 and a linear system jdj free from xed components we de ne the threshold of canonical adjunction or brie y just threshold of the pair x d setting

birational superrigidity and k stability of fano complete intersections of index 1 we prove that every smooth fano complete intersection of index 1 and codimension r in pn r is birationally superrigid and k stable if n 10r we also propose a generalization of tian s criterion of k stability and as an application prove the k stability of the complete intersection of a quadric and a cubic in p5 in the appendix written jointly with c stibitz we prove the conditional birational superrigidity of fano complete intersections of higher index in large dimension

collocation with chebyshev polynomials for a hypersingular integral equation on an interval abstract a collocation method for a first kind integral equation with a hypersingular kernel on an interval is analysed chebyshev polynomials of the second kind are used as the basis functions for the approximation and the collocation points are chosen to be chebyshev quadrature points in our analysis we introduce sobolev norms that reflect the singular structure of the exact solution at the endpoints of the interval numerical experiments are presented which underline the theoretical estimates

optimal convergence rates for some discrete projection methods using recent results of sloan on hyperinterpolation operators j approx theory 81 1995 238 we show how to obtain optimal convergence rates for a variety of discrete polynomially based galerkin and collocation methods for a variety of fredholm and singular integral equations these results improve on those given in golberg j integral equations appl 8 1996 307 and are analogous to those obtained by atkinson and bogomolny for spline based methods math comp 48 1987 596 these results are then applied to analyze double approximation methods where both numerical integration and data errors are considered here we show that the errors are dominated by those in the data hence it may not be possible in practice to obtain optimal convergence rates unless the data can be computed exactly

on the generic local langlands correspondence for gspin groups in the case of split gspin groups we prove an equality of l functions between automorphic local l functions defined by the langlands shahidi method and local artin l functions our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of levi subgroups of gspin by constructing langlands parameters for general generic irreducible admissible representations of gspin from the one for generic irreducible supercuspidal representations of its levi subgroups

local langlands correspondence for classical groups and affine hecke algebras using the results of colette moeglin on the representations of p adic classical groups based on methods of james arthur and its relation with representations of affine hecke algebras established by the author we show that the category of smooth complex representations of a quasi split p adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups a statement of this kind had been conjectured by g lusztig all classical groups general linear orthogonal symplectic and unitary groups appear in this context we get also parameterizations of representations of affine hecke algebras which seem not all to be in the literature yet all this sheds some light on what is known as the stable bernstein center

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

on the weights of certain quadratures for the numerical evaluation of cauchy principal value integrals and their derivatives we consider interpolatory quadrature rules of the type begin gathered frac d k d lambda k int limits 1 1 1 x alpha 1 x beta frac f x x lambda dx sum limits i 1 n w ni k lambda f x ni r n k f hfill k 0 1 alpha beta 0 hfill end gathered where the nodes x ni are the zeros of the jacobi polynomialp n x in particular we prove that sum limits i 1 n left w ni k lambda right o n k log n uniformly in 1 e 1 e furthermore we show that whenk 1 f x c p 1 1 p 1 andf p x h 1 1 0 1 we have r n 1 f o n p mu 1 delta 0 small as we like uniformly in 1 e 1 e

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

a sinc quadrature rule for hadamard finite part integrals a sinc quadrature rule is presented for the evaluation of hadamard finite part integrals of analytic functions integration over a general are in the complex plane is considered special treatment is given to integrals over the interval 1 1 theoretical error estimates are derived and numerical examples are included

hypersingular integral equations over a disc convergence of a spectral method and connection with tranter s method two dimensional hypersingular equations over a disc are considered a spectral method is developed using fourier series in the azimuthal direction and orthogonal polynomials in the radial direction the method is proved to be convergent then tranter s method is discussed this method was devised in the 1950s to solve certain pairs of dual integral equations it is shown that this method is also convergent because it leads to the same algebraic system as the spectral method

a chebyshev expansion of singular integrodifferential equations with a 2 in s t s t kernel singular integrodifferential equations of the first kind with a symmetric kernel having a 2 in s t s t singularity i e a derivative of a cauchy type kernel are studied an algorithm based on the expansion in chebyshev polynomials of the second kind is presented the method is applied to two dimensional wave scattering and convergence is demonstrated

the superconvergence of the composite midpoint rule for the finite part integral the composite midpoint rule is probably the simplest one among the newton cotes rules for riemann integral however this rule is divergent in general for hadamard finite part integral in this paper we turn this rule to a useful one and apply it to evaluate hadamard finite part integral as well as to solve the relevant integral equation the key point is based on the investigation of its pointwise superconvergence phenomenon i e when the singular point coincides with some a priori known point the convergence rate of the midpoint rule is higher than what is globally possible we show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate by applying the midpoint rule to approximate the finite part integral and by choosing the superconvergence points as the collocation points we obtain a collocation scheme for solving the finite part integral equation more interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression by which an optimal error estimate is established some numerical examples are provided to validate the theoretical analysis

the convergence of several algorithms for solving integral equations with finite part integrals ii we establish the uniform convergence and obtain convergence rates for several algorithms for solving a class of hadamard singular integral equations these results improve on the mean square convergence shown in 2

numerical solution and its analysis for a system of hypersingular integral equations abstract a system of hypersingular integral equations occurs quite naturally in several branches of science and engineering during the formulation of many boundary value problems the analytical solution is known for the system of dominant equations however there are many real world problems such as crack problems occur in the field of fracture mechanics which may not be formulated as system of dominant equations therefore we propose a numerical method to find the approximate solution for such generalized form the convergence of the proposed method is proved in l n 2 space this convergence helps to derive theoretical error bound for the error between the exact and the approximate solution an application of the proposed method in finding numerical solution of hypersingular integral equation over the curves is shown finally the derived theoretical error bound is numerically calculated and validated with the help of numerical examples

a chebyshev expansion of singular integral equations with a logarithmic kernel abstract singular integral equations of the first kind with a symmetric kernel having a logarithmic singularity are studied an algorithm based on the expansion in chebyshev polynomials of the first kind is presented the method is applied to two dimensional wave scattering and convergence is demonstrated

gauss quadrature rules for finite part integrals we construct a set of polynomials n x which are orthogonal with respect to w x x 2 where wx is a weight function these polynomials can be used for the definition of a gauss quadrature formula for the finite part integral the process is exactly the same as the one used for the extraction of the classical gauss formula for the riemann integrals three different methods are derived the first and most accurate quadrature formula is successfully tested in some numerical examples the proposed quadrature formulas have many applications in problems of mathematical physics mechanics etc

the application of a boundary integral equation method to the prediction of ducted fan engine noise the prediction of ducted fan engine noise using a boundary integral equation method biem is considered governing equations for the biem are based on linearized acoustics and describe the scattering of incident sound by a thin finite length cylindrical duct in the presence of a uniform axial inflow a classical boundary value problem bvp is derived that includes an axisymmetric locally reacting liner on the duct interior using potential theory the bvp is recast as a system of hypersingular boundary integral equations with subsidiary conditions we describe the integral equation derivation and solution procedure in detail the development of the computationally efficient ducted fan noise prediction program tbiem3d which implements the biem and its utility in conducting parametric noise reduction studies are discussed unlike prediction methods based on spinning mode eigenfunction expansions the biem does not require the decomposition of the interior acoustic field into its radial and axial components which for the liner case avoids the solution of a difficult complex eigenvalue problem numerical spectral studies are presented to illustrate the nexus between the eigenfunction expansion representation and biem results we demonstrate biem liner capability by examining radiation patterns for several cases of practical interest

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

sigmoidal transformations and the euler maclaurin expansion for evaluating certain hadamard finite part integrals starting with some results of lyness concerning the euler maclaurin expansion of cauchy principal value integrals over left 0 1 right it is shown how by the use of sigmoidal transformations good approximations can be found for the hadamard finite part integral int scriptstyle 0 1 left phi left x right left x c right 2 right dx where 0 c 1 the analysis is illustrated by some numerical examples

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

numerical solution of integral equations with a logarithmic kernel by the method of arbitrary collocation points an integral equation whose kernel presents logarithmic singularity is numerically solved by the method of arbitrary collocation points acp as a first step a gaussian quadrature of order n hence of polynomial accuracy 2n 1 is employed for the numerical approximation of the integral until now the collocation which follows was performed on special points xk determined as roots of appropriate transcedental functions in order to retain the 2n 1 degree of polynomial accuracy of the gaussian quadrature in this paper an appropriate interpolatory technique is proposed so that xk may be arbitrary and yet the high 2n 1 accuracy of the gaussian quadrature is retained

the numerical evaluation of a 2 d cauchy principal value integral arising in boundary integral equation methods in this paper we consider the problem of computing 2 d cauchy principal value integrals of the form

birational geometry of fano hypersurfaces of index two we prove that every non trivial structure of a rationally connected fibre space on a generic in the sense of zariski topology hypersurface v of degree m in the m 1 dimensional projective space for m ge 16 is given by a pencil of hyperplane sections in particular the variety v is non rational and its group of birational self maps coincides with the group of biregular automorphisms and for that reason is trivial the proof is based on the techniques of the method of maximal singularities and inversion of adjunction

generalized mangler type principal value integrals with an application to fracture mechanics abstract a generalized interpretation of the classical mangler type principal value integrals is proposed this new interpretation in general is not equivalent to the classical one several formulae for this new class of integrals are proved concerning their relationship to cauchy type generalized cauchy type and ordinary mangler type principal value integrals moreover the plemelj formulae and quadrature rules are considered for the same class of integrals finally the present results are applied to the hypersingular integrodifferential equation of an interface crack in the classical theory of plane elasticity which takes now the form of a simple hypersingular integral equation

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed