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).

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...

It is well known that beam is one of the basic structures in architecture. It is greatly used in the designing of bridge and construction. Recently, scientists bring forward the theory of combined beams. That is to say, we can bind up some stratified structure copings into one global combined beam with rock bolts. The deformations of an elastic beam in equilibrium state, whose two ends are simp...

In a sequence of independent Bernoulli trials the probability of success in the k:th trial is pk = a/(a+b+k−1). An explicit formula for the binomial moments of the number of two consecutive successes in the first n trials is obtained and some consequences of it are derived.

In an infinite sequence of independent Bernoulli trials with success probabilities pk = a/(a + b + k − 1) for k = 1, 2, 3, . . . , let Nr be the number of r ≥ 2 consecutive successes. Expressions for the first two moments of Nr are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is i...

The billiard in a polygon is not always ergodic and never K-mixing or Bernoulli. Here we consider billiard tables by attaching disks to each vertex of an arbitrary simply connected, convex polygon. We show that the billiard on such a table is ergodic, K-mixing and Bernoulli.

The chemical distance D(x, y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm μ depending on the dimension and the percolation parameter, the probability of the event