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 ...

An arbitrary-depth reduction theorem for the ‘convolution’ multiple L-values of Euler-Zagier type for a broad class of Dirichlet characters is proven by an analytic method. To this end, generalized polylogarithms associated to Dirichlet characters are defined. The proof uses the monodromies of these functions and the parities of generating functions of generalized Bernoulli numbers. Some genera...

In this paper we study the reliability of calculations of the structural reliability. It compares the exact reliability expression within the Bernoulli-Euler column theory with its counterpart obtained via the finite difference expression in the buckling context.

Techniques for modeling hysteresis and material nonlinearities in magnetostrictive materials are presented. Hys-teresis in these materials is due to both the driving magnetic eld and stress relations within the material and is signiicant throughout most of the drive range of the material. Two characterizations are presented here. The rst model is posed in terms of Preisach operators and is pure...

A possibly unknown approach to the problem of finding the common term of a power series is considered. A direct formula for evaluating this common term has been obtained. This formula provides useful expressions for direct evaluation of the number of partitions of a nonnegative integer and the partitions themselves. These expressions permit easily to work with power series, evaluate nth derivat...