We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed ap...

We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69–91, 1999) for computing all the zeros of an analytic function that lie inside the unit circle. The algorithm uses only the function values and no (first or higher order) derivatives. Information about the location of the zeros is obtained from certain integrals along the unit circle. In numerical computati...

In 2007, Zeng and Li proposed a weighted triangular approximation of a fuzzy number. Unfortunately, this approximation may fail to be a fuzzy number. In this paper, we improve this approximation and propose a generalization by the name of weighted trapezoidal approximation. Their algorithms are also presented. Finally, some examples and relevant properties are discussed. © 2009 Elsevier B.V. Al...

In this paper, an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied, which is proposed by Grzegorzewski and Mrówka. A formula for computing the improved approximation is presented. Moreover, the nearest triangular approximation operator preserving the expected interval is also investigated. © 2007 Elsevier B.V. All rights reserved.

2 How to Create Instances of Fuzzy Numbers 3 2.1 Arbitrary Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Definition by Side Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Definition by α-cut Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Definition with Generating Functions Omitted: Shadowed Sets . . . . . . 6 2.2 Us...

Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting non-universality phenomenon. Though they have bounded variance, their fluctuations are asymptotically non-Gaussian but infinitely divisible. The fluctuations are asymptotically Gaussian for less smooth linear statistics for which the variance diverges. The degree of smooth...

In this note, we provide a new perspective on Euler–Maclaurin expansions of (offset) trapezoidal rule approximations of the finite-range integrals I [f ] = ∫ b a f (x) dx, where f ∈ C∞(a, b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f (x) has asymp...

High order approximations of an integral can be obtained by taking the linear combination of lower degree approximations in a systematic way. One of these approaches for 1-d integrals is known as Romberg Integration and is based upon the composite trapezoidal rule approximations and the well-known Euler-Maclaurin expansion of the error. Because of its theoretical nature, students in a classical...

An interesting property of the midpoint rule and trapezoidal rule, which is expressed by the so-called Hermite–Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite–Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. I...

Zipf’s law appears in many application areas but does not have a closed form expression, which may make its use cumbersome. Since it coincides with the truncated version of the Zeta distribution, in this paper we propose three approximate closed form expressions for the truncated Zeta distribution, which may be employed for Zipf’s law as well. The three approximations are based on the replaceme...