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. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).     

Profile empirical likelihood for parametric and semiparametric models

This paper introduces a profile empirical likelihood and a profile conditionally empirical likelihood to estimate the parameter of interest in the presence of nuisance parameters respectively for the parametric and semiparametric models. It is proven that these methods propose some efficient estimators of parameters of interest in the sense of least-favorable efficiency. Particularly, for the decomposable semiparametric models, an explicit representation for the estimator of parameter of interest is derived from the proposed nonparametric method. These new estimations are different from and more efficient than the existing estimations. Some examples and simulation studies are given to illustrate the theoretical results. The first author is supported by NNSF projects (10371059 and 10171051) of China. The second author is supported by a grant from The Research Grants Council of the Hong Kong Special Administrative Region, China (#HKU7060/04P). The third author is supported by the University Research Committee of the University of Hong Kong and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU7323/01M).     

Solvability of partial differential equations by meshless kernel methods

This paper first provides a common framework for partial differential equation problems in both strong and weak form by rewriting them as generalized interpolation problems. Then it is proven that any well-posed linear problem in strong or weak form can be solved by certain meshless kernel methods to any prescribed accuracy. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101205). Robert Schaback’s research in Hong Kong was sponsored by DFG and City University of Hong Kong.     

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.     

Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions

In this paper we consider a financial market model with frictions which include transaction costs, bid-ask spread and taxes. By using optimization, linear and nonlinear programming and convex programming techniques, several necessary and sufficient conditions are derived for the weak no-arbitrage. Some results on state prices are also provided. The results of this paper can provide at least some theoretical insight to the problem. This research was supported in part by a grant of the National Excellent Ph.D. Thesis Project of China (No. 200267), a grant of the National Natural Science Foundation of China (No. 10171115), a “Tenth Five-Year Plan” project of Ministry of Education of China (No. 01JA630009), a grant of the Natural Science Foundation of Guangdong Province (No. 011193), two grants from the Research Grants Council of Hong Kong (CityU 1081/02E, HKU 7139/01H), and a research grant of the University of Hong Kong.     

Multi-level spectral galerkin method for the navier-stokes problem I : spatial discretization

A multi-level spectral Galerkin method for the two-dimensional non-stationary Navier-Stokes equations is presented. The method proposed here is a multiscale method in which the fully nonlinear Navier-Stokes equations are solved only on a low-dimensional space subsequent approximations are generated on a succession of higher-dimensional spaces j=2, . . . ,J, by solving a linearized Navier-Stokes problem around the solution on the previous level. Error estimates depending on the kinematic viscosity 0<ν<1 are also presented for the J-level spectral Galerkin method. The optimal accuracy is achieved when We demonstrate theoretically that the J-level spectral Galerkin method is much more efficient than the standard one-level spectral Galerkin method on the highest-dimensional space . The work of this author was supported in part by the NSF of China 10371095, City University of Hong Kong Research Project 7001093 Hong Kong and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1084/02P)     

Regularity and well-posedness of a dual program for convex best <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-spline interpolation

An efficient approach to computing the convex best C ¹-spline interpolant to a given set of data is to solve an associated dual program by standard numerical methods (e.g., Newton’s method). We study regularity and well-posedness of the dual program: two important issues that have been not yet well-addressed in the literature. Our regularity results characterize the case when the generalized Hessian of the objective function is positive definite. We also give sufficient conditions for the coerciveness of the objective function. These results together specify conditions when the dual program is well-posed and hence justify why Newton’s method is likely to be successful in practice. Examples are given to illustrate the obtained results. The work was supported by EPSRC grant EP/D502535/1 for the first author and by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5141/01E) for the second author.     

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.     

Quadratic immersed finite element spaces and their approximation capabilities

This paper discusses a class of quadratic immersed finite element (IFE) spaces developed for solving second order elliptic interface problems. Unlike the linear IFE basis functions, the quadratic IFE local nodal basis functions cannot be uniquely defined by nodal values and interface jump conditions. Three types of one dimensional quadratic IFE basis functions are presented together with their extensions for forming the two dimensional IFE spaces based on rectangular partitions. Approximation capabilities of these IFE spaces are discussed. Finite element solutions based on these IFE for representative interface problems are presented to further illustrate capabilities of these IFE spaces. Dedicated to the 60th birthday of Charles A. Micchelli Mathematics subject classifications (2000) 65N15, 65N30, 65N50, 65Z05. Yanping Lin: Supported by NSERC. Weiwei Sun: This work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (project CityU 1141/01P).     

Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals

In this article, the general (composite) Newton-Cotes rules for evaluating Hadamard finite-part integrals with third-order singularity (which is also called “supersingular integrals”) are investigated and the emphasis is placed on their pointwise superconvergence and ultraconvergence. The main error of the general Newton-Cotes rules is derived, which is shown to be determined by a certain function . Based on the error expansion, the corresponding modified quadrature rules are also proposed. At last, some numerical experiments are carried out to validate the theoretical analysis.     

A Stochastic Model for Combined Activity/Destination/Route Choice Problems

In travel behavior modeling, an important topic is to investigate what drives people to travel. A systematic analysis should examine why, where and when various activities are engaged in, and how activity engagement is related to the spatial and institutional organization of an urban area. In view of this, this paper presents a stochastic model for solving the combined activity/destination/route choice problem. It is a time-dependent model for long-term transport planning such as travel demand forecasting. The activity/destination choices are based on multinomial logit formulae and, the route choice is governed by stochastic user equilibrium principle. The solution algorithm is proposed together with a numerical example for demonstration. It is shown that the proposed modeling approach provides a powerful tool for fully understanding and predicting the complex travel behavior at strategic level. The work described in this paper was substantially supported by the grants from the National Natural Science Foundation of China (Project No. 79825101), the Chinese Academy of Sciences (MADIS Research Project) and the Research Grants Council of the Hong Kong Special Administrative Region (Project No. PolyU5077/97E).     

The superconvergence of the Newton-Cotes rule for Cauchy principal value integrals

We consider the general (composite) Newton-Cotes method for the computation of Cauchy principal value integrals and focus on its pointwise superconvergence phenomenon, which means that the rate of convergence of the Newton-Cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point. The necessary and sufficient conditions satisfied by the superconvergence point are given. Moreover, the superconvergence estimate is obtained and the properties of the superconvergence points are investigated. Finally, some numerical examples are provided to validate the theoretical results.     

The numerical evaluation of Hadamard finite-part integrals

Summary 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.     

Explicit classes of permutation polynomials of $$ \mathbb{F}_{3^{3m} } $$

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over are presented. These polynomials have simple shapes, and they are related to planar functions. This work was supported by Australian Research Council (Grant No. DP0558773), National Natural Science Foundation of China (Grant No. 10571180) and the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No. 612405)     

The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals.     

Variance estimation for sample quantiles using them out ofn bootstrap

We consider the problem of estimating the variance of a sample quantile calculated from a random sample of sizen. Ther-th-order kernel-smoothed bootstrap estimator is known to yield an impressively small relative error of orderO(n ^−r/(2r+1) ). It nevertheless requires strong smoothness conditions on the underlying density function, and has a performance very sensitive to the precise choice of the bandwidth. The unsmoothed bootstrap has a poorer relative error of orderO(n ^−1/4), but works for less smooth density functions. We investigate a modified form of the bootstrap, known as them out ofn bootstrap, and show that it yields a relative error of order smaller thanO(n ^−1/4) under the same smoothness conditions required by the conventional unsmoothed bootstrap on the density function, provided that the bootstrap sample sizem is of an appropriate order. The estimator permits exact, simulation-free, computation and has accuracy fairly insensitive to the precise choice ofm. A simulation study is reported to provide empirical comparison of the various methods. Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7131/00P).     

On the Distributions of Two Classes of Multiple Dependent Aggregate Claims

In this paper we examine two classes of correlated aggregate claims distributions, with univariate claim counts and multivariate claim sizes. Firstly, we extend the results of Hesselager [ASTIN Bulletin, 24： 19-32（1994）] and Wang ＆ Sobrero＇s [ASTIN Bulletin, 24：161-166 （1994）] concerning recursions for compound distributions to a multivariate situation where each claim event generates a random vector. Then we give a multivariate continuous version of recursive algorithm for calculating a family of compound distribution. Especially, to some extent, we obtain a continuous version of the corresponding results in Sundt [ASTIN Bulletin, 29：29-45 （1999）] and Ambagaspitiya [Insurance： Mathematics and Economics, 24：301-308 （1999）]. Finally, we give an example and show how to use the algorithm for aggregate claim distribution of first class to compute recursively the compound distribution.     

A sinc quadrature rule for Hadamard finite-part integrals

Summary 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.     

A note on parameter estimations of panel vector autoregressive models with intercorrelation

This note considers parameter estimation for panel vector autoregressive models with intercorrelation. Conditional least squares estimators are derived and the asymptotic normality is established. A simulation is carried out for illustration.     

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method. This author was fully supported by Hong Kong RGC grant (Project No. 403403) This author acknowledges the support from a Direct Grant of CUHK during his visit at The Chinese University of Hong Kong.