多元函数积分的简化计算

利用多元函数积分区域的对称性,可通过对被积函数以及积分区域的变换来简化多元函数积分的计算.     

Numerical computation of real or complex elliptic integrals

Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included.This work was supported by the Director of Energy Research, Office of Basic Energy Sciences. The Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract W-7405-ENG-82.     

Multiple Integral Inequalities for Schur Convex Functions on Symmetric and Convex Bodies

In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions defined on bodies $B⊂\mathbb{R}^n$ that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.     

The s-Perron,sap-Perron and ap-McShane Integrals

In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.     

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.     

对称性在重积分及曲面积分中的应用

在积分区域具有某种对称性时,给出重积分及曲面积分所具有的相应性质,并通过例题给出这些性质在重积分及曲线、曲面积分中的应用方法.     

凸体的弦幂积分公式

本文研究了高维欧氏空间中凸体的弦幂积分问题.利用积分的方法,获得了凸体弦幂积分的几个积分公式.     

On multiple integrals of special form

Multiple integrals generalizing the iterated kernels of integral operators are expressed as single integrals in the case of a special representation of the kernel (this is our theorem). Besides integral equations, Markov processes involve these integrals as well. As a consequence of the theorem, we obtain transition probability densities of certain Markov processes. As an illustration, we consider nine examples.     

Upper integral and its geometric meaning

The Hahn definition of the integral is recalled, the requirement of measurability of the integrand omitted. Both the upper and lower integrals comply with this definition and so does any measurable function between them. The outer product measure of the hypograph of a nonnegative bounded nonmeasurable function is equal to the upper integral which is equal to one of the Fan integrals. The outer measure of the graph of a bounded nonmeasurable function is equal to the difference between the upper and lower integrals. A norm for not necessarily measurable functions is defined with the upper integral. The linear space with this norm is complete. The convergence in this space implies the convergence in outer measure. The distance as an outer measure of the symmetric difference of two sets gives us a complete metric space of classes of subsets.      

Henstock-Kurzweil and McShane product integration; Descriptive definitions

The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function A is absolutely continuous. As a consequence we obtain that the McShane product integral of A over [a, b] exists and is invertible if and only if A is Bochner integrable on [a, b]. Supported by grant No. 201/04/0690 of the Grant Agency of the Czech Republic.     

函数粗集生成的粗积分(英文)

The concepts of the lower approximation integral,the upper approximation integral and rough integrals are given on the basis of function rough sets.Based on these concepts,the relation of the lower approximation integrals,the relation of the upper approximation integrals,the relation of rough integrals,and the double median theorem of rough integrals are discussed.Rough integrals have finite contraction characteristic and finite extension characteristic.     

Henstock-Kurzweil delta and nabla integrals

We will study the Henstock-Kurzweil delta and nabla integrals, which generalize the Henstock-Kurzweil integral. Many properties of these integrals will be obtained. These results will enable time scale researchers to study more general dynamic equations. The Henstock-Kurzweil delta (nabla) integral contains the Riemann delta (nabla) and Lebesque delta (nabla) integrals as special cases.     

The ap-Denjoy and ap-Henstock integrals

In this paper we define the ap-Denjoy integral and show that the ap-Denjoy integral is equivalent to the ap-Henstock integral and the integrals are equal.     

First integrals and families of symmetric periodic motions of a reversible mechanical system

A reversible mechanical system which allows of first integrals is studied. It is established that, for symmetric motions, the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles, are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case, the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case, when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia, all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents, of which two are simple and two form a Jordan Box. In typical situation, the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia.     

Hodge integrals with one λ-class

Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space.ELSV formula connects the Hodge integrals with Hurwitz numbers,and the generating function of Hurwitz numbers satisfies the cut-and-join equation.Therefore,it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula.In this paper,at first,we will review the method introduced in Goulden et al.’s paper to get the λ g conjecture for Hodge integral.Through some variables transformation,the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation.By comparing the coefficients of the lowest degree term of both sides in this equation,we can get the λ g conjecture.Then,in a similar way,we obtain our main result in this paper:a recursive formula for Hodge integral of type contains only one λ g 1-class.We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.     

Nonlinear stochastic integration with a non-smooth family of integrators

We consider nonlinear stochastic integrals of Itô-type w.r.t. a family of semimartingales which depend on a spatial parameter. These integrals were introduced by Carmona/Nualart, Kunita, and Le Jan. The extension of the elementary nonlinear integral is based on the condition that the semimartingale kernel has nice continuity properties in the spatial parameter. We investigate the case that continuity is not available and suggest different directions of generalization. This brings us beyond the case that any integral can be approximated by integrals with integrands taking only finitely many values.     

第一类曲面积分的一类特殊解法

对一些特殊曲面(如柱面、旋转曲面)上的第一类曲面积分,转化为二重积分的基本方法可能非常复杂,用类似计算三重积分的平行截面法能更简洁地解决此类问题.     

关于取值于局部凸空间中集值映射的积分

本文引进了一类集值映射的积分，研究了这种积分的若干性质．在此基础上，讨论了这种积分与抽象函数的Riemann积分、Bochner积分和Petits积分的关系．     

Differential equation for a functional integral

We propose a new method for calculating functional integrals in cases where the averaged (integrated) functional depends on functions of more than one variable. The method is analogous to that used by Feynman in the one-dimensional case (quantum mechanics). We consider the integration of functionals that depend on functions of two variables and are symmetric under rotations about a point in the plane. We assume that the functional integral is taken over functions defined in a finite spatial domain (in a disc of radius r). We obtain a differential equation describing change in the functional as the radius r increases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 184–188, August, 2008.     

Discrete Choquet integral and some of its symmetric extensions

Classical extensions of the Choquet integral (defined on [0,1]) to [−1,1] are the asymmetric and the symmetric Choquet integral, the second one being called also the Šipoš integral. Recently, the balancing Choquet integral was introduced as another kind of a symmetric extension of the discrete Choquet integral. We introduce and discuss a new type of such extension, the fusion Choquet integral, and discuss its properties and relationship to the balancing and the symmetric Choquet integral. The symmetric maximum introduced by Grabisch is shown to be a special case of the fusion and the balancing Choquet integral. Several extensions of OWA operators are also discussed.