Numerical integration over polygons using an eight-node quadrilateral spline finite element

In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures.     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

Surface integral methods for high-frequency electromagnetic scattering

Surface integral equation based methods are advantageous for simulation of electromagnetic waves scattered by three dimensional obstacles, because they efficiently reduce the dimension of the problem and are robust for high-frequency problems. However, the cost of setting up the associated discretized dense linear systems is prohibitive due to evaluation of highly oscillatory magnetic and electric dipole surface integral operators using standard cubatures. The computational complexity of evaluating such integrals depends on the incident wave frequency, and the size and shape of the obstacles. In this work we discuss a surface integral reformulation of the scattering problem that involves evaluation of surface integrals with a highly oscillatory physical density, and discuss methods for efficient evaluation of such integrals for a class of smooth three dimensional scatterers whose diameter is a large multiple of the incident wavelength. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gaussian cubature: A practitioner’s guide

Accurate modeling of management and economic processes often requires that researchers accurately approximate the expectations of functions of random variables. While commonly employed, Monte Carlo simulation techniques generally require large sample sizes to insure accuracy. For functions that are computationally burdensome, the Monte Carlo approach may be impractical. We propose a method to generate samples from multivariate distributions that contain far fewer points than reliable Monte Carlo samples, yet retain much of the original distributions’ information. Our method, Gaussian cubatures generated via linear programming, is designed to be feasible for joint, but independent distributions. While heuristic for joint, dependent distributions, this method appears to be very reliable and to accurately approximate expectations of an important class of functions.     

Product type multiple integration formulas

An integration formula of the type $$\int_a^b {f(x)g(x)dx \cong \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {a_{ij} f(xi)g(y_j ),} } } $$ referred to as a product quadrature, was first considered by R. Boland and C. Duris. In this paper, the author extends the concept of a product formula to multiple integrals. The definitions and some of the results for interpolatory, compound, and symmetric product quadratures have an analog for product cubatures and these are given.     

A redundant Klee–Minty construction with all the redundant constraints touching the feasible region

We have previously shown, by redundant Klee–Minty constructions, that the central path may arbitrarily closely visit every vertex of the Klee–Minty cube. In those constructions, the redundant constraints are far away from the feasible region. In this paper, we provide a construction in which all redundant constraints touch the feasible region.     

Finite-part integrals over polygons by an 8-node quadrilateral spline finite element

In this paper we consider the numerical integration on a polygonal domain Ω in ?² of a function F(x,y) which is integrable except at a point \(P_{0}=(x_{0},y_{0})\in{\stackrel{\circ}{\Omega}}\), where F becomes infinite of order two. We approximate either the finite-part or the two-dimensional Cauchy principal value of the integral by using a spline finite element method combined with a subdivision technique also of adaptive type. We prove the convergence of the obtained sequence of cubatures. Finally, to illustrate the behaviour of the proposed method, we present some numerical examples.     

Interpolation by local space curves

A scheme is proposed for the construction of a visually smooth and fair space curve passing through a prescribed sequence of irregularly spaced and non-coplanar points. A major feature is its locality of definition - in a given region the curve depends only of the nearby points. Also the construction is direct, compact and fast. Illustrative examples are briefly described.     

二维拟定常可压流Euler方程组的简单波

简单波是这样的流动,它在像空间中的像是一条曲线．“简单波理论是除基本流动结构以外构造流动问题的解的基础”,见Courant和Friedrichs的经典著作《超声速流与冲击波》．该文主要研究二维拟定常可压流Euler方程组的简单波的几何结构.根据这些几何诠释,还构造了绕一拟流线弯曲部的疏散和压缩的简单波流动结构．这种流动结构将作为一个局部流动结构出现在4个接触间断的Riemann问题的整体解中．     

Covering a polygonal region by rectangles

The problem of covering a compact polygonal region, called target region, with a finite family of rectangles is considered. Tools for mathematical modeling of the problem are provided. Especially, a function, called Γ-function, is introduced which indicates whether the rectangles with respect to their configuration form a cover of the target region or not. The construction of the Γ-function is similar to that of Φ-functions which have been proved to be an efficient tool for packing problems. A mathematical model of the covering problem based on the Γ-function is proposed as well as a solution strategy. The approach is illustrated by an example and some computational results are presented.     

Construction of new symmetric designs with parameters (70,24,8)

This article presents a construction of all symmetric designs with parameters (70,24,8) on which the group E₈ F₂₁ operates so that the automorphism of order 7 operates fixed-point-free. A method based on enumeration of cosets in group is used for the construction.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

京津冀协同应对自然灾害应急资源配置优化研究

京津冀协同发展已成为国家战略。京津冀区县毗邻,在灾害分布上具有临近相似性和区域差异性的特点。本文从整合京津冀地区应急资源、协同应对突发事件的角度出发,提出了京津冀地区统筹规划下区域协同应对自然灾害的新模式。本文以京津冀地区应急物资保障的经济性和物资分配的公平性为目标,建立了多阶段、多灾种的京津冀协同应对自然灾害应急资源配置的多目标规划模型,在此基础上设计了一种多目标免疫算法。结合京津冀地区自然灾害发生的情景概率,验证了模型的科学性与算法的有效性,为京津冀地区乃至世界城市群协同应对应急资源的优化配置提供理论指导与科学的决策。     

g-Quasikörper

A generalization of a construction for finite quasifields by Foulser is given, which yields all finite Dickson-nearfields. This constuction is then used to obtain locally finite quasifields as unions of chains of finite g-quasifields.     

On the question of representation of ultrafilters and their application in extension constructions

We study ultrafilters of broadly understood measurable spaces and possibilities of their application as generalized elements in the construction of attraction sets in abstract reachability problems with constraints of asymptotic nature. A class of measurable spaces is specified for which all ultrafilters including free ultrafilters (with empty intersection of all of their sets) are built constructively.     

A Markov Property for Set-Indexed Processes

We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.     

Solving boundary value problems of mathematical physics using radial basis function networks

A neural network method for solving boundary value problems of mathematical physics is developed. In particular, based on the trust region method, a method for learning radial basis function networks is proposed that significantly reduces the time needed for tuning their parameters. A method for solving coefficient inverse problems that does not require the construction and solution of adjoint problems is proposed.     

Multidimensional trigonometric interpolation

An investigation and construction of trigonometric interpolation polynomials in nonstandard regions, including multiply connected, is carried out. A method of boundary diffeomorphism of the initial region onto a canonical region is proposed and a case of interpolation on a nonuniform net in the initial region is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 568–571, April, 1990.     

Topological groups with compact proper invariant subgroups

A construction is given which lets one construct examples of locally compact inductively pronilpotent groups for which all proper invariant subgroups are compact and which do not contain proper open invariant subgroups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 708–710, May, 1990.