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.      

A dynamical approach to accelerating numerical integration with equidistributed points

We show how ideas originating in the theory of dynamical systems inspire a new approach to numerical integration of functions. Any Lebesgue integral can be approximated by a sequence of integrals with respect to equidistributions, i.e. evenly weighted discrete probability measures concentrated on an equidistributed set. We prove that, in the case where the integrand is real analytic, suitable linear combinations of these equidistributions lead to a significant acceleration in the rate of convergence of the approximate integral. In particular, the rate of convergence is faster than that of any Newton-Cotes rule.     

On the estimation of multi-dimensional integrals with strongly oscillating integrands

The estimation of integrals with rapidly oscillating integrands is difficult, as positive and negative contributions will cancel nearly completely. The effect becomes more pronounced as the number of dimensions of the region of integration is increased. The article shows how an integral over ann-dimensional region can be reduced to integrals over itsn?1 dimensional surfaces in such a manner that the oscillating character of the integrand is taken into account. The method can be interpreted as the repeated application of integration by parts along lines normal to the wave fronts which are determined by the integrand. The remaining integral is estimated by the second mean value theorem of integral calculus. The integrations by parts appear only indirectly in the form of an application of Gauss' integral theorem to a vector field which is determined by the integrand.     

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.     

带Brown运动的随机奇异积分的存在性

在轨迹二阶导数具有Hlder连续的条件下,利用高阶奇异积分思想和概率极限的理论,研究了在Brown运动下的随机奇异积分.得到了以Brown运动为积分元的随机奇异积分是存在性定理.     

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.     

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.     

Curvilinear integrals of discontinuous functions over nonrectifiable paths and Riemann boundary‐value problem

We study a generalization of the concept of curvilinear integral for the case where path of integration is nonrectifiable, and integrand has discontinuities. Then we apply that integrals for solving of the Riemann boundary‐value problem for domains with nonrectifiable boundaries and discontinuous boundary data. The research is supported by RFBR (grant 18‐31‐00060) and is performed according to a special programme of the Russian government supporting research at Kazan Federal University. Mathematical Methods in the Applied Sciences journal encourages authors to share the data and other artifacts supporting the results in the paper by archiving it in an appropriate public repository. Authors may provide a data availability statement, including a link to the repository they have used, in order that this statement can be published in their paper. Shared data should be cited.     

无穷限反常积分收敛时被积函数的极限状态

根据无穷限反常积分∫a^＋∞f（x）dx收敛的柯西准则和定积分的性质,讨论被积函数f（x）当x→∞时。的极限状态,并得出当无穷限反常积分∫a^＋∞f（x）dx收敛且f（x）在[a,＋∞）上连续,或者无穷限反常积分∫a^＋∞f（x）dx绝对收敛时,存在数列{xn}∩[a,＋∞]且xn→＋∞（n→∞）,使limn→∞xnf（xn）=0.     

Numerical integration of functions with poles near the interval of integration

An automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole (or poles) of order 2, i.e., a double pole on the real axis or a complex conjugate pair of double poles, near the interval of integration. The present scheme is based on product integration rules of interpolatory type, using function values of the abscissae only in the interval of integration. The integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the Chebyshev polynomials. The fast Fourier transform (FFT) technique is used to generate efficiently the sequence of the finite Chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error. Numerical examples are included to illustrate the performance of the method.     

被积函数正负性周期变化的反常积分

从被积函数的正负性变化规律入手,借助交错级数的敛散性,给出并证明相应反常积分的敛散性,进而推广得出一类反常积分的敛散性判定定理．     

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.     

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 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.     

Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.     

On a class of Gauss-like quadrature rules

Summary. We consider a problem that arises in the evaluation of computer graphics illumination models. In particular, there is a need to find a finite set of wavelengths at which the illumination model should be evaluated. The result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene. These values are then used to determine the RGB coordinates of the light by evaluating three definite integrals, each with a common integrand (the SPD) and interval of integration but with distinct weight functions. We develop a method for selecting the sample wavelengths in an optimal manner. More abstractly, we examine the problem of numerically evaluating a set of definite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration. It is shown that when it is not efficient to use a set of Gauss rules because valuable information is wasted. We go on to extend the notions used in Gaussian quadrature to find an optimal set of shared abcissas that maximize precision in a well-defined sense. The classical Gauss rules come out as the special case and some analysis is given concerning the existence of these rules when . In particular, we give conditions on the weight functions that are sufficient to guarantee that the shared abcissas are real, distinct, and lie in the interval of integration. Finally, we examine some computational strategies for constructing these rules. Received July 15, 1991     

用参数展开法计算一类反常积分

对含参数反常积分I(t,s)=∫+∞0 x-1(1+x)-sdx,由贝塔函数的积分表示得到I(t,s)的伽马函数表示,再由伽马函数的级数展开,得到I(t,s)的参数级数展开.I(t,s)可在积分符号内按参数展开,参数系数是含对数函数的反常积分.对比同类参数的系数,可得一系列含对数函数反常积分的值.     

关于积分上限函数的周期性研究

探讨积分上限函数的周期性.得到积分上限函数是周期函数的两个充分必要条件.并得到当被积甬数是周期函数时.无穷限的反常积分发散性的一个结论.     

Fuzzy integrals and fuzzy measures with their values in complete lattices

Real-valued fuzzy measures and their associated fuzzy integrals were proposed by Sugeno. In this paper we consider fuzzy measures valued in complete lattices and their associated upper and lower fuzzy integrals. Moreover we define a class of functionals, called fuzzy integrals. In the setting of the complete lattices which are both Brouwerian and dually Brouwerian, the upper and lower fuzzy integrals are fuzzy integrals; while in that of the completely distributive complete lattices every fuzzy integral is both lower and upper fuzzy integral with respect to a fuzzy measure.     

Local and improper Daniell-Loomis integrals

In this paper we start from previous results obtained in [7] on the abstract space of Daniell-Loomis integrable functionsL, which is constructed like to the Daniell extension process, but without continuity assumptions on the elementary integral. The localized integral is used to prove thatL consists of those functions whose local upper and lower integrals are equal and finite, or thatL is closed with respect to improper integration. Our results are also holded in integration with respect to finitely additive measures.