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.

We give an asymptotic expansion for the Taylor coefficients of L(P (z)) where L(z) is analytic in the open unit disc whose Taylor coefficients vary ‘smoothly’ and P (z) is a probability generating function. We show how this result applies to a variety of problems, amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal sequences.

We consider the Bernoulli first-passage percolation on Z (d ≥ 2). That is, the edge passage time is taken independently to be 1 with probability 1− p and 0 otherwise. Let μ(p) be the time constant. We prove in this paper that μ(p1)− μ(p2) ≥ μ(p2) 1− p2 (p2 − p1) for all 0 ≤ p1 < p2 < 1 by using Russo’s formula. AMS classification: 60K 35. 82B 43.

The boundary problem is considered for inhomogeneous increasing random walks on the square lattice Z2+ with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles.

We prove that if a countable discrete group Γ contains an infinite normal subgroup with the relative property (T) (e.g. Γ = SL(2,Z) ⋉Z, or Γ = H × H with H an infinite Kazhdan group and H arbitrary) and V is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a Bernoulli Γ-action ...

The physical nature of beamstrahlung during beam-beam interaction in linear colliders is reviewed. We first make the distinction between a dense beam and a dilute beam. We then review the characteristics of synchrotron radiation (SR) and bremsstrahlung, and argue that for a wide range of beam parameters beamstrahlung is SR in nature, even if the beam is dilute. Some issues concerning the specif...