1 First Lecture on Entropy Example 1.1 Consider the unilateral shifts f : {1, 2}N → {1, 2}N and g : {1, 2, 3}N → {1, 2, 3}N equipped with the Bernoulli measures μ and ν, respectively. Show that, for any set X ⊂ {1, 2}N satisfying f−1(X) = X and μ(X) = 1, there exists x ∈ X such that #(X ∩ f−1(x)) = 2. Conclude that if k 6= l then (f, μ) and (g, ν) can not be ergodically equivalent. Proof. Denot...

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...

Using the finite difference calculus and differentiation, we obtain several new identities for Bernoulli and Euler polynomials; some extend Miki’s and Matiyasevich’s identities, while others generalize a symmetric relation observed by Woodcock and some results due to Sun.

Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. The p-adic absolute value | |p on Cp is normalized so that |p|p 1/p. Let Z>0 be the set of natural numbers and Z≥0 Z>0 ∪ {0}. As is well known, the Bernoulli polynomials Bn x are defined by the ge...

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.

In this paper, the investigational results for a robust adaptive vibration control of a translating tensioned beam with a varying traveling speed are presented. The dynamics of the moving beam are modeled as an Euler-Bernoulli beam, in which the tension applied to the beam is given as a nonlinear spatiotemporally varying function. The moving beam span is divided into two parts, i.e., a controll...

The roots of Bernoulli polynomials, Bn(z), when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of C, the Bernoulli polynomials asymptotically behave like sine or cosine. Here we establish the asmptotic behavior of Bn(nz), compute the distribution of real roots of Bernoulli polynomials and show that, properly rescal...

In this note, we will point out, as a corollary of Popa’s rigidity theory, that the crossed product von Neumann algebras for Bernoulli shifts cannot have relative property T. This is an operator algebra analogue of the theorem shown by Neuhauser and Cherix-Martin-Valette for discrete groups. Our proof is different from that for groups.