a quadrature formula for integrals with nearby singularities the purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities in the boundary element method the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point especially field variables and their derivatives at a field point near a boundary cannot be computed accurately in the present paper a quadrature formula for l isolated singularities near the integration interval based on lagrange interpolatory polynomials is obtained the error estimation of the proposed formula is also given quadrature formulas for regular and singular integrals with conjugate poles are derived numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy

numerical computation of hypersingular integrals on the real semiaxis abstract in this paper we propose some different strategies to approximate hypersingular integrals 0 g x x t p 1 d x where p is a positive integer t 0 and the integral is understood in the hadamard finite part sense hadamard finite part integrals shortly fp integrals regarded as p th derivative of cauchy principal value integrals are of interest in the solution of hypersingular bie which model many different kind of physical and engineering problems see 1 and the references therein 2 3 4 the procedure we employ here is based on a simple tool like the truncated gaussian rule see 5 conveniently modified to remove numerical cancellation we will consider functions g having different decays at infinity the method is shown to be numerically stable and convergent and some error estimates in suitable zygmund type spaces are proved finally some numerical tests which confirm the efficiency of the proposed procedures are presented

hypersingular integral equations past present future abstract in this address attention is paid to solving hypersingular integral equations and also to its extension supersingular integral equations firstly a short discussion is given of the historical development of these mathematical tools secondly the author s own experience in solving such problems is discussed lastly a glimpse of the future is offered by looking at some of the present techniques for solving hypersingular integral equations

instrumental variable quantile regression a robust inference approach abstract in this paper we develop robust inference procedures for an instrumental variables model defined by y d u where d u is strictly increasing in u and u is a uniform variable that may depend on d but is independent of a set of instrumental variables z the proposed inferential procedures are computationally convenient in typical applications and can be carried out using software available for ordinary quantile regression our inferential procedure arises naturally from an estimation algorithm and has the important feature of being robust to weak and partial identification and remains valid even in cases where identification fails completely the use of the proposed procedures is illustrated through two empirical examples

generalized quadrature rules of gaussian type for numerical evaluation of singular integrals an efficient method for constructing a class of generalized quadrature formulae of gaussian type on 1 1 for integrands having logarithmic singularities is developed that kind of singular integrals are very common in the boundary element method several special cases for n point quadratures which are exact on both of the spaces p 2 n 2 1 1 1 the space of algebraic polynomials of degree at most 2 n 2 1 and l 2 1 1 1 span x k log x k 0 2 1 the logarithmic space where 1 n are presented regarding a direct connection of these 2 m point quadratures with m point quadratures of gaussian type with respect to the weight function t t 1 2 over 0 1 the method of construction is significantly simplified gaussian quadratures on 0 1 are exact for integrands of the form t p t q t log t where p and q are algebraic polynomials of degree at most 2 m 1 and 1 1 2 m respectively the obtained quadratures can be used in a software implementation of the boundary element method

a receding contact plane problem between a functionally graded layer and a homogeneous substrate abstract in this paper we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half space when the two bodies are pressed together the graded layer is modeled as a nonhomogeneous medium with an isotropic stress strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions since the contact between the two bodies is assumed to be frictionless then only compressive normal tractions can be transmitted in the contact area using integral transforms the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half length the global equilibrium condition of the layer is supplemented to solve the problem the singular integral equation is solved numerically using chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half length that satisfies the global equilibrium condition the main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact

instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings this paper reports estimates of the effects of jtpa training programs on the distribution of earnings the estimation uses a new instrumental variable iv method that measures program impacts on quantiles the quantile treatment effects qte estimator reduces to quantile regression when selection for treatment is exogenously determined qte can be computed as the solution to a convex linear programming problem although this requires first step estimation of a nuisance function we develop distribution theory for the case where the first step is estimated nonparametrically for women the empirical results show that the jtpa program had the largest proportional impact at low quantiles perhaps surprisingly however jtpa training raised the quantiles of earnings for men only in the upper half of the trainee earnings distribution

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

likelihood inference in some finite mixture models abstract parametric mixture models are commonly used in applied work especially empirical economics where these models are often employed to learn for example about the proportions of various types in a given population this paper examines the inference question on the proportions mixing probability in a simple mixture model in the presence of nuisance parameters when sample size is large it is well known that likelihood inference in mixture models is complicated due to 1 lack of point identification and 2 parameters for example mixing probabilities whose true value may lie on the boundary of the parameter space these issues cause the profiled likelihood ratio plr statistic to admit asymptotic limits that differ discontinuously depending on how the true density of the data approaches the regions of singularities where there is lack of point identification this lack of uniformity in the asymptotic distribution suggests that confidence intervals based on pointwise asymptotic approximations might lead to faulty inferences this paper examines this problem in details in a finite mixture model and provides possible fixes based on the parametric bootstrap we examine the performance of this parametric bootstrap in monte carlo experiments and apply it to data from beauty contest experiments we also examine small sample inferences and projection methods

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

likelihood estimation and inference in a class of nonregular econometric models we study inference in structural models with a jump in the conditional density where location and size of the jump are described by regression curves two prominent examples are auction models where the bid density jumps from zero to a positive value at the lowest cost and equilibrium job search models where the wage density jumps from one positive level to another at the reservation wage general inference in such models remained a long standing unresolved problem primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions this paper develops likelihood based estimation and inference methods for these models focusing on optimal bayes and maximum likelihood procedures we derive convergence rates and distribution theory and develop bayes and wald inference we show that bayes estimators and confidence intervals are attractive both theoretically and computationally and that bayes confidence intervals based on posterior quantiles provide a valid large sample inference method

application of entropy and energy measures of fuzziness to processing of ecg signal the paper deals with the application of entropy and energy measure of fuzziness to processing of ecg signal after formulation of the problem the notion of entropy measure of fuzziness is recalled next the idea of a fuzzy signal created from the original signal is proposed and the entropy measure of fuzziness determined for each sample is calculated the application of this measure to building a detection function of ecg signal is presented the theoretical considerations have been illustrated with digital processing of the real ecg signal

frictional contact problem for a rigid cylindrical stamp and an elastic layer resting on a half plane abstract the frictional contact problem for a layer resting on a homogeneous half plane is handled using linear elasticity theory in this study the layer is in contact with a rigid cylindrical stamp that is on the layer and applies a concentrated force in the normal and tangential directions friction between the component couples of layer stamp and layer half plane is taken into account the problem is reduced to a system of singular integral equations in which the contact pressures and the contact areas are the unknowns and it is treated using fourier transforms and the boundary conditions for the problem the system of singular integral equations is solved numerically using the gauss jacobi integration formula with equilibrium and consistency conditions numerical results for the contact pressures and the contact areas are given as a solution for both the frictional and the frictionless cases this work is the first study that investigates the effect of friction on the receding contact problem of a layer and a half plane with two contact areas

asymptotic efficiency in parametric structural models with parameter dependent support in certain auction search and related models the boundary of the support of the observed data depends on some of the parameters of interest for such nonregular models standard asymptotic distribution theory does not apply previous work has focused on characterizing the nonstandard limiting distributions of particular estimators in these models in contrast we study the problem of constructing efficient point estimators we show that the maximum likelihood estimator is generally inefficient but that the bayes estimator is efficient according to the local asymptotic minmax criterion for conventional loss functions we provide intuition for this result using le cam s limits of experiments framework

a time domain constrained fuzzy clustering method and its application to signal analysis this paper introduces a new fuzzy clustering method with time domain constraints which is used to signal analysis proposed method makes it possible to include natural constraints for signal analysis using fuzzy clustering that is the neighboring samples of signal belong to the same cluster this method can be called time domain constrained fuzzy clustering this paper introduces two approaches to include the above kind of constraints the first approach leads to the time domain constrained fuzzy c regression models method the second approach leads to the insensitive version of the above method which results in additional robustness for outliers and non gaussian noise finally simulations on synthetic as well as real life signals are realized to evaluate the performance of the time domain constrained fuzzy clustering methods a comparison with the traditional fuzzy c regression models is also made large scale simulations demonstrate the competitiveness of the proposed methods for signal analysis with respect to the traditional fuzzy clustering methods

measuring fuzzy uncertainty first this paper reviews several well known measures of fuzziness for discrete fuzzy sets then new multiplicative and additive classes are defined we show that each class satisfies five well known axioms for fuzziness measures and demonstrate that several existing measures are relatives of these classes the multiplicative class is based on nonnegative monotone increasing concave functions the additive class requires only nonnegative concave functions some relationships between several existing and the new measures are established and some new properties are derived the relative merits and drawbacks of different measures for applications are discussed a weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced finally we comment on the construction of measures that may assess all of the uncertainties associated with a physical system

application of sigmoidal transformations to weakly singular and near singular boundary element integrals accurate numerical determination of line integrals is fundamental to reliable implementation of the boundary element method for a source point distant from a particular element standard gaussian quadrature is adequate as well as being the technique of choice however when the integrals are weakly singular or nearly singular source point near the element this technique is no longer adequate here a co ordinate transformation technique based on sigmoidal transformations is introduced to evaluate weakly singular and near singular integrals a sigmoidal transformation has the effect of clustering the integration points towards the endpoints of the interval of integration the degree of clustering is governed by the order of the transformation comparison of this new method with existing co ordinate transformation techniques shows that more accurate evaluation of these integrals can be obtained based on observations of several integrals considered guidelines are suggested for the order of the sigmoidal transformations copyright 1999 john wiley sons ltd

on the estimation of the extreme value index and large quantile estimation l article se constitue de deux parties on donne une preuve simple de la convergence faible de l estimation de pickand pour le parametre principal d une loi de valeurs extremes on donne d autres conditions naturelles pour la convergence forte et pour la normalite asymptotique de l estimation ensuite une grande quantile d une loi est estimee par une combinaison de statistiques d ordre intermediaires ou extremes ceci mene a un intervalle de confiance asymptotique

efficient semiparametric estimation of quantile treatment effects this paper develops estimators for quantile treatment effects under the identifying restriction that selection to treatment is based on observable characteristics identification is achieved without requiring computation of the conditional quantiles of the potential outcomes instead the identification results for the marginal quantiles lead to an estimation procedure for the quantile treatment effect parameters that has two steps nonparametric estimation of the propensity score and computation of the difference between the solutions of two separate minimization problems root n consistency asymptotic normality and achievement of the semiparametric efficiency bound are shown for that estimator a consistent estimation procedure for the variance is also presented finally the method developed here is applied to evaluation of a job training program and to a monte carlo exercise results from the empirical application indicate that the method works relatively well even for a data set with limited overlap between treated and controls in the support of covariates the monte carlo study shows that for a relatively small sample size the method produces estimates with good precision and low bias especially for middle quantiles

green s function for a two dimensional exponentially graded elastic medium the freespace green function for a twodimensional exponentially graded elastic medium is derived the shear modulus is assumed to be an exponential function of the cartesian coordinates x y i e

superconvergence of the composite simpson s rule for a certain finite part integral and its applications the superconvergence phenomenon of the composite simpson s rule for the finite part integral with a third order singularity is studied the superconvergence points are located and the superconvergence estimate is obtained some applications of the superconvergence result including the evaluation of the finite part integrals and the solution of a certain finite part integral equation are also discussed and two algorithms are suggested numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms

evaluation of nearly singular integrals in boundary element contour and node methods for three dimensional linear elasticity this article addresses the issue of nearly singular integrals that arise in boundary element methods bems when one tries to evaluate displacements and stresses at points inside a body that lie close to its bounding surface an efficient new approach is proposed for the evaluation of these integrals the same issue is then discussed in the context of the boundary contour method bcm and the boundary node method bnm finally numerical results from both the bcm and the bnm are presented for an illustrative problem

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

gaussian approximation of suprema of empirical processes we develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of gaussian processes without taking the route of approximating empirical processes themselves in the sup norm we prove an abstract approximation theorem that is applicable to a wide variety of problems primarily in statistics especially the bound in the main approximation theorem is non asymptotic and the theorem does not require uniform boundedness of the class of functions the proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors the proof of which depends on an effective use of stein s method for normal approximation and some new empirical processes techniques we study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not donsker type importantly our new technique is able to prove the gaussian approximation for the supremum type statistics under considerably weak regularity conditions especially concerning the bandwidth and the number of series functions in those examples

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

evaluations of hypersingular integrals using gaussian quadrature a gaussian quadrature formula for hypersingular integrals with second order singularities is developed based on previous gaussian quadrature formulae for cauchy principal value integrals the formula uses classical orthonormal polynomials and the formula is then specialized to the case of legendre and chebyshev polynomials numerical experiments are carried out using the current formula and a previous formula developed by kutt it is found that the two methods generally give similar results and in some cases the current method works better it has also been shown that the current method allows the choice of an appropriate weight which can increase the convergence rate and the accuracy of the results copyright 1999 john wiley sons ltd

a composite transformation for numerical integration of singular integrals in the bem in this paper we present a composite non linear transformation of a polynomial transformation pt and a sigmoidal transformation st by the name of ct ps composite transformation with polynomial and sigmoidal transformations this new transformation can be used for both cauchy principal value cpv integrals and weakly singular integrals with interior singularities moreover it is implied that the numerical evaluation using ct ps would be more efficient than either pt or st used alone by numerical examples we show that the presented ct ps method with the gauss legendre quadrature rule improves the approximation errors of the existing methods effectively in particular it is demonstrated that the results of the present method are excellent for cpv integrals with near end point singularities copyright 2003 john wiley sons ltd

remarks on the gauss quadrature rule for a particular class of finite part integrals the method of finite part hadamard type or hypersingular integrals is already a standard approach in applied numerical methods in engineering and mainly in acoustics fluid dynamics elasticity and fracture mechanics here we study the convergence of the non classical possessing a negative node outside the integration interval as well as a negative weight corresponding to this node gauss legendre quadrature rule for finite part integrals on the interval 0 1 with the weight function 1 y and we prove its validity for integrands possessing a continuous first derivative the case of integrands possessing a higher order derivative is also considered and the rate of convergence is established these results are based on the derivation of a bound for the negative node in the above quadrature rule and further on the theory of convergence of the classical gauss quadrature rule numerical results verifying the results of this paper are also displayed the present results can be generalized to the gauss laguerre similarly non classical quadrature rule for finite part integrals on the interval 0 as well as to the two special but important in engineering applications non classical gaussian quadrature rules for cauchy type principal value integrals and mangler type finite part integrals have been proposed by tsamasphyros and dimou

semiparametric estimation of a binary response model with a change point due to a covariate threshold this paper is concerned with semiparametric estimation of a threshold binary response model the estimation method considered in the paper is semiparametric since the parameters for a regression function are finite dimensional while allowing for heteroskedasticity of unknown form in particular the paper considers manski 1975 1985 s maximum score estimator the model in this paper is irregular because of a change point due to an unknown threshold in a covariate this irregularity coupled with the discontinuity of the objective function of the maximum score estimator complicates the analysis of the asymptotic behavior of the estimator sufficient conditions for the identification of parameters are given and the consistency of the estimator is obtained it is shown that the estimator of the threshold parameter is n consistent and the estimator of the remaining regression parameters is cube root n consistent furthermore we obtain the asymptotic distribution of the estimators it turns out that a suitably normalized estimator of the regression parameters converges weakly to the distribution to which it would converge weakly if the true threshold value were known and likewise for the threshold estimator

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

indirect finite element evaluation of two dimensional finite part integral using fourier transformation a broad class of engineering problems in fracture mechanics thermal fluid transport and electromagnetic theory involve the evaluation of two dimensional finite part integrals of the form a method for evaluation of such integrals is developed by deriving an equivalent integral using fourier transformation this equivalent integral does not involve a kernel with singular behaviour consequently standard numerical integration methodologies with conventional analytical evaluation techniques can be used in the finite element computations the accuracy and convergence of the developed numerical procedure are successfully demonstrated by numerical examples for planar fracture geometries

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

the entropy value for ecg of the sport horses in this study we compared the amplitude and the duration of p and t waves of sport horses before a training program with after a training program waves are transformed to the entropies by the same methods of engonul et al the entropy of the waves p and t in electrocardiogram and some comments int j math model comput vol 6 pp 159 173 2016 we have benefitted from the definition of fuzzy sets and have assigned an entropy for each p and t waves of sport horses then we have calculated cesaro entropies for each wave in a particular segment for ten beats in addition the graphical representations are obtained for healthy and diseased horses by formalization from the paper the entropy of the waves p and t in electrocardiogram and some comments by engonul et al

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

derivation of a new analytical solution for a general two dimensional finite part integral applicable in fracture mechanics an exact expression is derived for the general finite part integral over an inclined ellipticaldomain r denotes the distance of a point in to the singular point is a general function of the cartesian co ordinates x0 y0 the boundary of the region represents the equation z x0 y0 o these integrals appear during the numerical solution of plane crack problems in three dimensional elasticity where they are the dominant part of a hypersingular integral equation the availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and simultaneously lead to quicker numerical results the considered finite part integral can be expressed in closed form as function of complete elliptical integrals or gauss hypergeometric functions respectively formuias for special cases and some i j values and their numerical verification are given in appendices ii and iii

analysis and evaluation of singular integrals by the invariant imbedding approach the invariant imbedding formulae for evaluating singular integrals on flat domains are used to analyse the singular integrals which arise in the boundary element method although all orders of singularity are considered the primary objective of the paper is the evaluation of cauchy principal value integrals cpv conditions for boundedness of the cpv are derived and evaluation techniques are presented for situations in which the singular point lies on or is in close proximity to an element boundary specific cases analysed include straight and curved element boundaries and corners other numerical aspects of the formulae are also discussed by way of example the analysis techniques are applied to the volume integrals which arise in continuum inelastic problems

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

a gauss quadrature rule for hypersingular integrals abstract we construct a set of polynomials n x t which are orthogonal with respect to v x t w x x t p t a b where w x is a weight function and p is an integer as an application these polynomials can be used to extract gaussian quadrature rules for hypersingular integrals

computation of gauss type quadrature formulas gaussian formulas for a linear functional l such as a weighted integral are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to l in gauss type formulas one or more extraneous conditions such as pre assigning certain nodes replace some of the equations expressing exactness when applied to high order polynomials these extraneous conditions may be applied by modifying the same number of recursion coefficients we survey the methods of computing formulas from recursion coefficients methods of obtaining recursion coefficients and modifying them for gauss type formulas and questions of existence and numerical accuracy associated with those computations

extremal quantile treatment effects this paper establishes an asymptotic theory and inference method for quantile treatment effect estimators when the quantile index is close to or equal to zero such quantile treatment effects are of interest in many applications such as the effect of maternal smoking on an infant s adverse birth outcomes when the quantile index is close to zero the sparsity of data jeopardizes conventional asymptotic theory and bootstrap inference when the quantile index is zero there are no existing inference methods directly applicable in the treatment effect context this paper addresses both of these issues by proposing new inference methods that are shown to be asymptotically valid as well as having adequate finite sample properties

measures of entropy and fuzziness related to aggregation operators we suggest some new measures for the calculation of the entropy of a probability distribution one motivation for this is to provide tools for the application of the maximum entropy principle the first class of entropy measures is based on the ordered weighted averaging operator another class of these measures is based on the distance of the probability distribution from the uniform distribution a third class of entropy measures is based on the t conorms we then suggest some measures of fuzziness associated with a fuzzy set

nodal type collocation methods for hypersingular integral equations and nonlocal diffusion problems abstract in this paper we study nodal type quadrature rules for approximating hypersingular integrals and their applications to numerical solution of finite part integral equations and nonlocal diffusion problems we first derive explicit expressions for the quadrature coefficients and establish corresponding error estimates some collocation schemes are then constructed based on these rules to numerically solve certain type of finite part integral equations and nonlocal diffusion problems in one dimension and condition number and optimal error estimates for the proposed schemes are also rigorously obtained on uniform grids these schemes are of toeplitz structure which results in many advantages in developing fast linear solvers various numerical experiments are also performed to illustrate the theoretical results

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

quadrature rules for finite element approximations of 1d nonlocal problems it is well known that calculations of the entries of the stiffness matrix in the finite element approximations of nonlocal diffusion and mechanics models are often very time consuming due to the double integration process over the domain and the singularities of the nonlocal kernel functions in this paper we propose some effective and accurate quadrature rules for computing these double integrals for one dimensional nonlocal problems in particular for problems with highly singular kernels the corresponding inner integrals can be first evaluated exactly in our method and the outer one then will be approximated by some popular quadrature rules with these quadrature rules the assembly of the stiffness matrix in the finite element method for the nonlocal problems becomes similar to that for the classical partial differential equations and is thus quite efficient

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

evaluation of hypersingular integrals in the boundary element method by complex variable techniques strongly singular and hypersingular integrals that occur in many two dimensional applications of the boundary element method e g potential theory and liner elasticity can be treated in a unified manner by complex analysis this paper presents an elegant approach for the evaluation of such hypersingular integrals the limit to the boundary method where a source point is moved from the inside or outside of a body to its boundary is carried out using complex variables the connection of this limit with the hadamard finite part of hypersingular integrals and principal value integrals is established many examples are worked out throughout the paper to illustrate the theory

quadrature formulas for functions with poles near the interval of integration on examine la construction des regles de quadrature de type interpolatoire utilisant uniquement des valeurs de fonctions reelles pour des fonctions ayant des paires complesex conjuguees de poles pres de l intervalle d integration

on subsampling estimators with unknown rate of convergence abstract politis and romano have put forth a general subsampling methodology for the construction of large sample confidence regions for a general unknown parameter associated with the probability distribution generating the stationary sequence x 1 x n the subsampling methodology hinges on approximating the large sample distribution of a statistic t n t n x 1 x n that is consistent for at some known rate n although subsampling has been shown to yield confidence regions for of asymptotically correct coverage under very weak assumptions the applicability of the methodology as it has been presented so far is limited if the rate of convergence n happens to be unknown or intractable in a particular setting in this article we show how it is possible to circumvent this limitation by a using the subsampling methodology to derive a consistent estimator of the rate n and b using the estimated rate to construct asymptotically correct confidence regions for based on subsampling

the superconvergence of newton cotes rules for the hadamard finite part integral on an interval we study the general composite newton cotes rules for the computation of hadamard finite part integral with the second order singularity and focus on their pointwise superconvergence phenomenon i e when the singular point coincides with some a priori known point the convergence rate is higher than what is globally possible we show that the superconvergence rate of the composite newton cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points several numerical examples are provided to validate the theoretical analysis

regularization of hypersingular and nearly singular integrals in the potential theory and elasticity both the hypersingular and nearly singular integrals which appear in the hypersingular boundary integralequations and integral representations of the secondary fields respectively are regularized by the application of the superposition principle two kinds of the non singular formulations namely those with the strongly singular and weakly singular kernels are presented in this paper the formulations are given in terms of the relevant boundary quantities and the collocation at element junctions is possible two and three dimensional problems are analysed simultaneously in a unique way for either internal or external problems of the potential theory and elasticity

two trigonometric quadrature formulae for evaluating hypersingular integrals two trigonometric quadrature formulae one of non interpolatory type and one of interpolatory type for computing the hypersingular integral are developed on the basis of trigonometric quadrature formulae for cauchy principal value integrals the formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae fast three term recurrence relations for evaluating the quadrature weights are derived numerical tests are carried out using the current formula as applications two simple crack problems are considered one is a semi infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions it is found that the present method generally gives superior results copyright 2002 john wiley sons ltd

a frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate abstract this paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space when they are pressed against each other the graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant poisson s ratio a segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction the entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction it is assumed that the contact region is under sliding contact conditions with the coulomb s law used to relate the tangential traction to the normal component employing fourier integral transforms and applying the necessary boundary conditions the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem the gauss chebyshev quadrature collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations this system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously the main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region

study and solution of bem singular integral equation method in the case of concentrated loads abstract the boundary element method has been widely used in order to obtain numerical solutions in many problems in science and engineering on the other hand the singular integral equation method can treat successfully the singular fields appearing in engineering problems when the right hand side of it is a regular function in this paper we consider the case where the right hand side of a boundary integral equation singular integral equation is a distribution this problem arises in the case of concentrated forces and or moments our main scope is to propose appropriate methods for the numerical solution of singular integral equations when the loading is a concentrated force or a moment before that we are obliged to analyze the solvability of such equations the main key for the proof of the solvability is the proof of existence of cauchy integrals of distributions as a by product we find a very interesting result that the solution behaves as 1 r or 1 r2 at the point where the concentrated force or moment respectively is applied obviously the existence of essential singularities in the solution creates difficulties in the numerical treatment of integral equations thus two appropriate numerical methods and an interpolatory quadrature method are proposed for the numerical solution of bie sie numerical examples illustrate the importance of the proposed methods

a unified approach with spectral convergence for the evaluation of hypersingular and supersingular integrals with a periodic kernel in this paper a general class of methods is proposed for the evaluation of hypesingular supersingular integrals with a periodic integrand of singularity higher than or equal to 2 the method is based on regularizing the singular integrals into proper regular integrals when we can apply the classical trapezoidal rule error analysis is presented for hypesingular supersingular integrals with a singularity of order p 1 where p is a positive integer we prove that the convergence rate of the method is of o lnn n k a if the integrand u satisfies email protected email protected k p a 0 email protected for any positive integer k with exponential convergence rate if the integrand u is analytic several numerical examples are provided to support the theoretical error analysis

the superconvergence of the composite trapezoidal rule for hadamard finite part integrals the composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels however this quadrature rule has been less employed for hadamard finite part integrals due to the fact that its global convergence rate for hadamard finite part integrals with p 1 order singularity is p order lower than that for the riemann integrals in general in this paper we study the superconvergence of the composite trapezoidal rule for hadamard finite part integrals with the second order and the third order singularity respectively we obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for hadamard finite part integrals by noting the superconvergence phenomenon

extremal quantile regression quantile regression is an important tool for estimation of conditional quantiles of a response y given a vector of covariates x it can be used to measure the effect of covariates not only in the center of a distribution but also in the upper and lower tails this paper develops a theory of quantile regression in the tails specifically it obtains the large sample properties of extremal extreme order and intermediate order quantile regression estimators for the linear quantile regression model with the tails restricted to the domain of minimum attraction and closed under tail equivalence across regressor values this modeling setup combines restrictions of extreme value theory with leading homoscedastic and heteroscedastic linear specifications of regression analysis in large samples extreme order regression quantiles converge weakly to arg min functionals of stochastic integrals of poisson processes that depend on regressors while intermediate regression quantiles and their functionals converge to normal vectors with variance matrices dependent on the tail parameters and the regressor design

the use of rational functions in numerical quadrature quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree but also for rational functions having the same or the most important poles as the function to be integrated constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts the superiority of such rational polynomial methods is shown by an analysis of the remainder term and documented by numerical examples

a new algorithm for cauchy principal value and hadamard finite part integrals an algorithm for the approximate evaluation of integrals defined by cauchy principal value or by hadamard finite part has been proposed the convergence of the procedure is proved the stability of the algorithm is considered and some numerical examples are given

some numerical algorithms to evaluate hadamard finite part integrals abstract some algorithms are described for the numerical evaluation of hadamard finite part integrals of type 1 1 f x x t 2 v dx t where v is a jacobi weight convergence results and some numerical examples are given

fundamental formulation for frictional contact problems of coated systems abstract the plane problem about surface loading of an elastic layer perfectly bonded to an elastically dissimilar half plane is considered the fundamental solutions for concentrated forces acting perpendicular and parallel to the layer surface are obtained the stress and displacement fields in the coating layer and the substrate due to these concentrated forces are found on the basis of these expressions the fundamental integral equations are obtained which describe the frictional contact between an elastic body and a coated substrate from the fundamental integral equations a series of integral equations for special cases are deduced corresponding to practical contact situations finally the numerical results for a typical example are given to demonstrate the validity of the fundamental equations and numerical procedures given in this paper