For any 1-1 measure preserving map T of a probability space we can form the [T, Id] and [T, T−1] automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T, Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T, Id] endomorphism is standar...

Based on congruences mod p and on properties of Bernoulli polynomials and Bernoulli numbers, several conditions are derived for x, k > 2 to satisfy the Diophantine equation 1k + 2k H-\(x 1 )* = xk . It is proved that ord2(x 3) = orà2k + 3 and that x cannot be divisible by any regular prime. Furthermore, by using the results of experiments with the above conditions on an SGI workstation it is pr...

An output measure is an image of a uniform Bernoulli measure on finitely many states. We discuss “generalized Bernoulli” measures of [F-J], and answer several questions posed in that paper, in particular we establish a condition when such a measure is a finitary output measure. Answering D. Goldstein’s question, we characterize output measures via block codes in terms of walks on labeled graphs...

We show that an expansive Z2 action on a compact abelian group is measurably isomorphic to a two–dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of the K property due to Lind, Schmidt and the author. As a corollary, we note that an expansive Z...

This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy. This answers a question of Ornstein and Weiss.

Recent results concerning the application of Lie transformation group methods to structural mechanics are presented. Focus is placed on the point Lie symmetries and conservation laws inherent to the Bernoulli–Euler and Timoshenko beam theories as well as to the Marguerrevon Kármán equations describing the large deflection of thin elastic shallow shells within the framework of the nonlinear Donn...

In this paper we investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. In particular, we present a new combinatorial formula for such sequences in terms of a ’shift by rank’ quasi-expansion based on ordered set partitions. As an application, we give a new proof of Dilcher’s ...

Abstract : We study a model of random electric networks with Bernoulli resistances. In the case of the lattice Z, we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most (log |v|) 23 whereas its expected value is of order log |v|, when v goes to infinity. When d 6= 2, expectation and variance are of the same order. Similar results are obtain...

We consider two aspects of generalized Bernoulli polynomials B n (z). One aspect is connected with deening functions instead of polynomials by making the degree n of the polynomial a complex variable. In the second problem we are concerned with the asymptotic behaviour of B n (z) when the degree n tends to innnity.