On an algorithm for calculating diffraction integrals

An algorithm for calculating integrals of rapidly oscillating functions given on a smooth two-dimensional surface is proposed. The surface is approximated by a collection of flat triangles with the values of the integrand known at their vertices. These values are used as reference ones to extend the function to other points of a triangle. The integral of the extended function over the surface of a triangle is calculated exactly. The desired value of the full diffraction integral is determined as the sum of the integrals calculated over the surfaces of all triangles. The resulting formulas for integral calculation involve singularities (indeterminate forms). Much attention is given to representations of these formulas in such a way that the indeterminate forms are automatically evaluated. Numerical results are presented.     

Modified clenshaw-curtis method for the computation of Bessel function integrals

The numerical evaluation of Bessel function integrals may be difficult when the Bessel function is rapidly oscillating in the interval of integration. In the method presented here, the smooth factor of the integrand is replaced by a truncated Chebyshev series approximation and the resulting integral is computed exactly. The numerical aspects of this exact integration are discussed.     

Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation

The purpose of this paper is twofold. First, we present the existence theorem of an optimal trajectory in a nonconvex variational problem with recursive integral functionals by employing the norm-topology of a weighted Sobolev space. We show the continuity of the integral functional and the compactness of the set of admissible trajectories. Second, we show that a recursive integrand is represented by a normal integrand under the conditions guaranteeing the existence of optimal trajectories. We also demonstrate that if the recursive integrand satisfies the convexity conditions, then the normal integrand is a convex function. These results are achieved by the application of the representation theorem in Lp-spaces.     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.     

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.     

A localic theory of lower and upper integrals

An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non‐negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non‐negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined. Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A hybrid approach for the integration of a rational function

A hybrid integration algorithm obtaining an indefinite integral of a rational function (say q/r, q and r are polynomials) with floating-point but real coefficients is proposed. The algorithm consists of four steps and is based on combinations of symbolic and numeric computations (hybrid computation). The first step is a hybrid preprocessing stage. An integrand is decomposed into rational and logarithmic parts by using an approximate Horowitz' method which allows floating-point coefficients. Here, we replace the Euclidean GCD algorithm with an approximate-GCD algorithm which was proposed by Sasaki and Noda recently. It is easy to integrate the rational part. The logarithmic part is integrated numerically in the second step. Zeros of a denominator of it are computed by the numerical Durand-Kerner method which computes all zeros of a polynomial equation simultaneously. The integrand is then decomposed into partial fractions in the third step. Coefficients of partial fractions are determined by residue theory. Finally, in the fourth step, partial fractions are transformed into the resulting indefinite integral by using well-known rules of integrals. The hybrid algorithm proposed here gives both indefinite integrals and accurate values of definite integrals. Numerical errors in the hybrid algorithm depend only on errors in the second step. The algorithm evaluates some problems where numerical methods are inefficient or incapable, or a pure symbolic method is theoretically insufficient.     

A divergence theorem for Finsler metrics

By means of the general form of Stokes' theorem on manifolds a divergence theorem is derived for hypersurfaces which bound a compact region of ann-dimensional Finsler spaceF _n . In general the integrand of then-fold volume integral will depend on the covariant derivatives of an arbitrary vector field which defines the element of support; certain conditions under which this dependence may be circumvented are discussed. The scalar curvature ofF _n is expressed in terms of the divergence of a certain vector field: forn=2 this formula reduces to a particularly simple form, and its substitution into the aforementioned divergence theorem gives rise to a formula which represents a generalization of the classical Gauss-Bonnet Theorem.

     

Bounding the number of zeros of certain Abelian integrals

In this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n+k−1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way.     

From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus

In this article we give a new proof of Ito's formula inR ⁿ starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.     

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.     

Gaussian product-type quadratures

Ordinary N-term integral quadratures require the evaluation of the entire integrand at N points. However, m-by-n product type quadratures involve the evaluation of one factor of the integrand at m points and the reamaining factor at n points. The principal results of this paper include the generalization of the product-type quadrature concept to arbitrary weight functions and to infinite as well as finite intervals, the calculation of the mn coefficients of this product quadrature formula from the LU decoposition of one n-byn, and the extension of the precision of the formula. Nuerical examples are included to illustrate the application of Gaussian product-type quadratures and to compare them with the ordinary Gaussian quadratures.     

The integral geometry boundary determination problem for a pencil of straight lines

Under consideration is the problem of integrating finitely many functions over straight lines. Each function as well as the corresponding line is assumed unknown. The available information is the sum of integrals over all straight lines of a family of pencils in each of which the intersection of lines is a point of a given bounded open set in a finite-dimensional Euclidean space. Each integrand depends on a greater number of variables than the sum of the integrals. Hence, the conventional statement of the problem of determining the integrands becomes underspecified. In this situation we pose and study the problem of determining the discontinuity surfaces of the integrands. The uniqueness theorem is proven under the condition that these surfaces exist. The present article is a refinement of the previous studies of the authors and differs from them in [1–6] by not only some technical improvements but also the principally new fact that the integration is performed over an unknown set.     

Regularity for some scalar variational problems under general growth conditions

We consider integrals of the calculus of variations over a set Ω of ? ⁿ , and the related regularity result: are the minimizers smooth functions, say for example of classC ^∞(Ω)? Classically, the so-called natural growth conditions on the integrand have been the main sufficient assumptions for regularity. In recent years, motivated also by application, the interest in the study of this problem has increased under more general growth assumptions. In this paper, we propose some general growth conditions that guarantee regularity for a class of scalar variational problems.     

从高斯公式到格林公式和牛顿-莱布尼茨公式

本文讨论高等数学课程中,高斯公式、格林公式和牛顿-莱布尼兹公式之间的内在联系,指出格林公式和牛顿-莱布尼茨公式可以分别看作一维和二维欧氏空间中的高斯公式.实际上,n维欧氏空间中的高斯公式可以看作微积分基本定理在高维欧氏空间中的表述形式.利用高斯公式还可以导出定积分、二重积分和任意n重积分的分部积分公式.     

Computation of oscillating integrals

An adaptive quadrature method for the automatic computation of integrals with strongly oscillating integrand is presented. The integration method is based on a truncated Chebyshev series approximation. The algorithm uses a global subinterval division strategy. There is a protection against the influence of round-off errors. A Fortran implementation of the algorithm is given.     

On Balder’s Existence Theorem for Infinite-Horizon Optimal Control Problems

Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f₀, 0} of the utility function (integrand) f₀ is relaxed to the requirement that the integrals of f₀ over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.     

The indicator of contact boundaries for an integral geometry problem

We pose and study a rather particular integral geometry problem. In the two-dimensional space we consider all possible straight lines that cross some domain. The known data consist of the integrals over every line of this kind of an unknown piecewise smooth function that depends on both points of the domain and the variables characterizing the lines. The object we seek is the discontinuity curve of the integrand. This problem arose in the author’s previous research in X-ray tomography. In essence, it is a generalization of one mathematical aspect of flaw detection theory, but seems of interest in its own right. The main result of this article is the construction of a special function that can be unbounded only near the required curve. Precisely for this reason we call the function the indicator of contact boundaries. A uniqueness theorem for the solution follows rather easily from the property of indicators.     

对数三角函数的定积分

定义了三种积分表示的两元函数.这些两元函数有伽马函数表示,可以展开为幂级数.在积分符号内展开被积函数,先积分,再求和,也得到级数展开.对比展开系数,就得到一些对数三角函数定积分的值.选取合适的围道,得到其他两类对数三角函数定积分的值.