Quadrature Rules for the Surface Integral of the Unit Sphere Based on Extremal Fundamental Systems

Quadrature rules for the surface integral of the unit Sphere S^r–1 based on an extremal fundamental system, i.e., a nodal system which provides fundamental Lagrange interpolatory polynomials with minimal uniform norm, are investigated. Such nodal systems always exist; their construction has been given in earlier work. Here the main results is that the corresponding interpolatory quadrature for the space of homogeneous polynomials of degree two is equally weighted for arbitrary r, and hence positive. For the full quadratic polynomial space we can prove positivity of the weights, only.     

Projective Modules over the Non-Commutative Sphere

The positive cone of the K₀-group of the non-commutative sphereB is explicitly determined by means of the four basic unboundedtrace functionals discovered by Bratteli, Elliott, Evans andKishimoto. The C*-algebra B is the crossed product A x Z₂ ofthe irrational rotation algebra A by the flip automorphism defined on the canonical unitary generators U, V by (U) = U*,(V) = V*, where VU = e²ⁱ UV and is an irrational real number.This result combined with Rieffel's cancellation techniquesis used to show that cancellation holds for all finitely generatedprojective modules over B. Subsequently, these modules are determinedup to isomorphism as finite direct sums of basic modules. Italso follows that two projections p and q in a matrix algebraover B are unitarily equivalent if, and only if, their vectortraces are equal: [p] = [q]. These results will have the following ramifications. They areused (elsewhere) to show that the flip automorphism on A isan inductive limit automorphism with respect to the basic buildingblock construction of Elliott and Evans for the irrational rotationalgebra. This will, in turn, yield a two-tower proof of thefact that B is approximately finite dimensional, first provedby Bratteli and Kishimoto.     

On the Existence and Construction of Tight Fourth Order Quadrature Rules for the Sphere

We study the so-called tight quadrature rules for polynomials of degree 4 on the unit sphere S ^D-1 and present precise formulae for the first 6 components of the nodes in terms of the parameter u := . In particular, we reobtain the well-known necessary condition for the existence of such rules saying that u has to be an odd integer and we sharpen it under an additional assumption.As a constructive application, two explicit tight fourth order quadrature rules for the case D = 7 are given.     

Asymptotic Error Formulae for Numerical Quadrature

A method, due to Fox, is used to derive asymptotic error formulaefor numerical procedures having the form (z+h, h).–(z,h)=(f,z,h).These procedures correspond to numerical quadrature for theintegrand (/h)(f,z,0)and compact expressions are given for determiningthe order of convergence as h 0, and the leading term in theerror. It is shown that a natural generalization of the Euler-Maclaurinexpansion is available. These results are applied to the particularcase where the Pt are polynomials in the differentiation operator.A related interpolation problem is also studied, and it is shownthat in certain cases higher order quadrature formulae are possiblewhen this interpolation problem is not poised.     

球面均匀分布的拟合优度检验

证明了基于惯量矩的d维单位球面上样本服从均匀分布的基本特征,得到球面均匀分布协差阵特征根估计的强相合性及渐近多元正态性.提出了检验球面上样本均匀性的渐近卡方统计量,证明了拟合优度检验的相合性并做检验功效的随机模拟.     

Numerical Quadrature of Periodic Singular Integral Equations

This paper presents quadrature formulae for the numerical integrationof a singular integral equation with Hilbert kernel. The formulaeare based on trigonometric interpolation. By integration a quadratureformula for an integral with a logarithmic singularity is obtained.Finally it is demonstrated how a singular integral equationwith infinite support can be solved by use of the precedingformulae.     

Distribution of Lattice Points over the Four-Dimensional Sphere

Let r_l(n) be the number of representations of n by a sum of l squares of integers and let 0 < A < 1 be a constant. It is proved that if (n,2)=1, then . Previously, the author obtained this asymptotics with a weaker error term O( . Bibliography: 12 titles.     

Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

Numerical Quadrature Computation of the Macdonald Function for Complex Orders

The use of Gaussian quadrature formulae is explored for the computation of the Macdonald function (modified Bessel function) of complex orders and positive arguments. It is shown that for arguments larger than one, Gaussian quadrature applied to the integral representation of this function is a viable approach, provided the (nonclassical) weight function is suitably chosen. In combination with Gauss–Legendre quadrature the approach works also for arguments smaller than one. For very small arguments, power series can be used. A Matlab routine is provided that implements this approach. AMS subject classification (2000) 33-04, 33C10, 65D15, 65D32     

数值积分的可选择误差界(英文)

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.     

Short Surface Waves due to a Heaving Sphere in a Hemispherical Lake

At the centre of a hemispherical lake, a sphere is making verticaloscillations about the half-immersed position. By constructingsuitable generalized Green's functions, a pair of dual integralequations is obtained for the velocity potential on the sphereand the lake boundary. After suitable modifications, these canbe solved iteratively for sufficiently large frequencies andthe asymptotic form of the virtual mass coefficient found.     

A Gaussian Quadrature Method for the Numerical Solution of the Goursat Problem

The author considers the numerical solution of the Goursat problemby using the cartesian product Gauss two point rule. Considerablesavings in time over other O(h⁵) methods is given. Three computationalexamples are considered.     

Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere S ^r⊂R ^r+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d _n =dim P _n points, where P _n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S ² of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

     

Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podle quantum sphere equal the winding numbers of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic K₀ of the standard quantum sphere.     

各向异性网格下具有数值积分的非协调有限元逼近

In this paper we mainly discuss the nonconforming finite element method for second order elliptic boundary value problems on anisotropic meshes.By changing the discretization form(i.e.,by use of numerical quadrature in the procedure of computing the left load),we obtain the optimal estimate O(h),which is as same as in the traditional finite element analysis when the load f∈H~1(Ω)∩C~0(Ω)which is weaker than the previous studies.The results obtained in this paper are also valid to the conforming triangular element and nonconforming Carey's element.     

Computational technique for singular boundary value problems by using Numerical Quadrature

In problems of physics and engineering we often come across singular boundary value problems that cannot be solved by the usual numerical methods. Special methods for solving such problems have been developed. These methods lead to banded systems, linear and nonlinear depending upon the nature of the boundary value problem. In this paper a difference method based on nonuniform mesh for a class of singular two-point boundary value problems of the form

Motion of a Force Source over an Elastic Surface

The motion of a variable point force P(t) over the free surface of an elastic half-space is investigated with a view toward studying wave processes in an elastic medium. Special attention is given to the determination of surface wavefronts, because the detection of the latter from seismogram data can be utilized to locate the position of a source in space and time. This type of planar problem has been investigated previously, for example, in [1], but the analytical solutions of the problem of a stationary source moving with a constant velocity have been obtained only in the limit t without regard for the transient process of wave propagation.     

Direct Numerical Simulation of Hypersonic Flow over a Cavity

An open rectangular cavity has served as a generic geometry in shock tunnel experiments with laminar flow being two‐dimensional upstream of the cavity. A high order central difference scheme with flux‐corrected transport has been employed to simulate the flow under the assumption that the inflow profile upstream of the cavity is laminar. The high order in space enables to reproduce well vortical structures and possible transition to turbulent flow.     

利用Legendre小波与Gauss-Legendre求积公式求解三重数值积分

提出利用Legendre小波和Gauss-Legendre求积公式求解几种积分区域的三重数值积分如长方体,四面体,圆柱体,圆锥和椭球体.通过某种线性或非线性变换将空间积分区域变换到空间长方体.利用Gauss-Legendre求积公式将三重积分转换成二重积分,然后利用Legendre小波对二重积分进行逼近.数值算例验证了方法的可行性和有效性.