In the operator algebraic formulation of probability theory Markov processes typically appear as perturbations of Bernoulli processes. We develop a scattering theory for this situation. This theory applies to the isomorphism problem between Markov processes and Bernoulli shifts as well as to the description of open quantum systems.

The random β-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure forK for special values of β and use the projection to obtain results on the local dimension of the Bernoulli convolution.

In this paper, we give a short new proof of a recent result due to Schumacher concerning an extension of Faulhaber’s identity for the Bernoulli numbers. Our approach follows from basic manipulations involving the ordinary generating function for the Bernoulli polynomials in the context of the Zeon algebra.

The Rayleigh beam is a perturbation of the Bernoulli-Euler beam. We establish convergence of the solution of the Exact Controllability Problem for the Rayleigh beam to the correspondig solution of the Bernoulli-Euler beam.The problem of convergence is related to a Singular Perturbation Problem. The main tool in solving this problem is a weak version of a lower bound for hyperbolic polynomials.

We connect and generalize Matiyasevich’s identity #0102 with Bernoulli numbers and an identity of Candelpergher, Coppo and Delabaere on Ramanujan summation of the divergent series of the infinite sum of the harmonic numbers. The formulae are analytic continuation of Euler sums and lead to new recursion relations for derivatives of Bernoulli numbers. The techniques used are contour integration, ...

We use contour integrals and the Cauchy residue theorem in order to derive several summation formulas, in terms of the higher-order Bernoulli polynomials and the ordinary Bernoulli and Euler polynomials, for a remarkably general family of secant sums. Numerous (known or new) special cases are shown to follow readily from the summation formulas presented in this paper. 2007 Elsevier Inc. All rig...

We introduce a notion of Jacobi-Bernoulli cohomology associated to a semisimplicial Lie algebra (SELA). For an algebraic scheme X over C, we construct a tangent SELA TX and show that the Jacobi-Bernoulli cohomology of TX is related to infinitesimal deformations of X.

The purpose of this article is to present, in a simple way, an analytical approach to special numbers and polynomials. The approach is based on derivative polynomials. The paper is, to some extent, a review article, although it contains some new elements. In particular, it seems that some integral representations for Bernoulli numbers and Bernoulli polynomials are new.

ergodic theory : Bernoulli shifts. There is a class of transformations that play a central role in ergodic theory, namely the Bernoulli shifts. The reason for this is partly because they are in a certain sense the simplest examples of measure-preserving transformations and partly because of their role in applications. Definition of Bernoulli shifts. If we are given positive numbers Pu ' ' ' » P...