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

Kolmogorov studied the problem of whether a function of the parameter p of the Bernoulli distribution Bernoulli[p] has an unbiased estimator based on a sample X1, X2, . . . , Xn of size n and proved that exactly the polynomial functions of degree at most n can be estimated. For the geometric distribution Geometric[p], we prove that exactly the functions that are analytic at p = 1 have unbiased ...

In this article, we study the generalized Bernoulli and Euler polynomials, and obtain relationships between them, based upon the technique of matrix representation.

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava–Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.