The problem of computing the exact value of ∑ 1/n is so classical (dating back to 1644) that it even has a name of its own: the Basel problem. The city of Basel was the hometown both of the famous brothers Jakob and Johann Bernoulli, who made serious but unsuccessful attempts to solve the problem, and to the prodigious mathematician Leonhard Euler, who found the first solution in 1735, see [3]....

We prove the hyperbolicity, ergodicity and thus the Bernoulli property of two hard balls in one of the following four polygons: the square, the equilateral triangle, the 45 − 45 − 90◦ triangle or the 30− 60− 90◦ triangle.

*Correspondence: [email protected] College of Science, China University of Petroleum (East China), Qingdao, Shandong 266555, People’s Republic of China Abstract In this paper, we reduce a linear d-dimensional beam equation with an x-periodic and t-quasi-periodic potential for most values of the frequency vector via the KAM theorem. We focus on the measure estimates of small divisor conditions and ...

We describe a new algorithm that computes the nth Bernoulli number in n4/3+o(1) bit operations. This improves on previous algorithms that had complexity n2+o(1).

The free vibration analysis of rotating functionally graded (FG) nano-beams under an in-plane thermal loading is provided for the first time in this paper. The formulation used is based on Euler-Bernoulli beam theory through Hamilton’s principle and the small scale effect has been formulated using the Eringen elasticity theory. Then, they are solved by a generalized differential quadrature meth...

This article deals with the mechanical analysis of a fixed-fixed nano-beam based on nonlocal elasticity theory. The nano-beam is sandwiched with two piezoelectric layers through it’s upper and lower sides. The electromechanical coupled equations governing the problem are derived based nonlocal theory considering to Euler-Bernoulli beam assumptions and based on the nonlocal piezoelectricity acco...

A triangular model base on an investigation which has done by Sinha et al. has been developed for evaluating embedded crack localization in beam-column structures. In the assessment of this member’s behavior, the effects of displacement slope are necessary. In order to propose a crack localization method for embedded crack, an efficient static data based indicator is proposed for this crack in ...

Modeling and control of a viscoelastic beam are considered. Piezoelectrical elements are bonded to the beam and used as actuators. The beam is also equipped with a strain gauge that serves as a sensing device. The beam system is described by the Euler-Bernoulli beam equation, which is Fourier transformed and numerically solved in the frequency domain. Then, two different approaches are evaluate...

Résumé. Cet article présente une méthode basée sur les cartes auto-organisatrices probabilistes dédiées à la classification non supervisée et la visualisation de données catégorielles et des données mixtes contenant des composantes quantitatives et binaires. Pour chacun de ces types de données, nous proposons un formalisme probabiliste dans lequel les unités de la carte topologique sont représe...