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...

Let Nn = {(Tnk,Xnk), k ≥ 1} be a Bernoulli p.p. on Z = (0,∞) × [0,∞). We discuss weak limit theorems for Nn as well as for the associated sum and extremal processes Sn(t) = { ∑ Xnk : Tnk ≤ t} and Yn(t) = {∨Xnk : Tnk ≤ t} on an open subset of Z.

A flexible system described by Euler-Bernoulli beam equation is considered. The beam is clamped at one end, and is free at the other end. Boundary control force and torque inputs are applied at the free end of the beam. The transfer functions of the controllers are marginally stable and may contain some poles on the imaginary axis. Various stability results are shown and the application of the ...