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.

It has equilibrium (and in fact reversible) measure μ the product Bernoulli measure of density p on {0, 1}Z. It can also be constructed by attaching a Poisson process clock to every site in Z. Every time a clock rings, the site is refreshed to a Bernoulli variable of parameter p, but only if its right neighbour is empty. This constraint creates the ”bubbles” one can see on the picture. A natura...

In this study, we construct the two-variable Dirichlet q-L-function and the two-variable multiple Dirichlet-type Changhee q-L-function. These functions interpolate the q-Bernoulli polynomials and generalized Changhee q-Bernoulli polynomials. By using the Mellin transformation, we give an integral representation for the two-variable multiple Dirichlet-type Changhee q-L-function. We also obtain r...

We study the problem of entropy increase of the Bernoulli-shift map without recourse to the concept of trajectory and we discuss whether, and under which conditions if it does, the distribution density entropy coincides with the Kolmogorov-Sinai entropy, namely, with the trajectory entropy. 05.45.+b,03.65.Sq,05.20.-y Typeset using REVTEX 1

For d ≥ 2, we use results of Hochman and Meyerovitch to construct examples of Zd shifts of finite type of entropy logN , N ∈ N, which cannot factor topologically onto the Zd Bernoulli shift on N symbols.

1 Department of Mathematics and Computer Science, KonKuk University, Chungju 138-70, Republic of Korea 2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea 3 Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore 570# 005, India 4 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea 5 Department of ...

This paper describes the use of Bernoulli mixture models for extracting boolean rules from data. Bernoulli mixtures identify high data density areas on the corners of a hypercube. One corner represents a conjunction of literals in a boolean clause and the set of all identified corners, of the hypercube, indicates disjuncts of clauses to form a rule. Further class labels can be used to select fe...

A uniformly p to one endomorphism is a measure preserving map with entropy log p which is a.e. p to 1 and for which the conditional expectation of each preimage is precisely 1/p. The standard example of this is one sided p-shift with uniform i.i.d. Bernoulli measure. We give a characterization of those uniformly finite to one endomorphisms conjugate to this standard example by a condition on th...

A countable group Γ is called shift-minimal if every non-trivial measure preserving action of Γ weakly contained in the Bernoulli shift Γ y ([0, 1] , λ ) is free. We show that any group Γ whose reduced C∗-algebra Cr(Γ) admits a unique tracial state is shift-minimal, and that any group Γ admitting a free measure preserving action of cost> 1 contains a finite normal subgroup N such that Γ/N is sh...

It was recently shown that there exists a family Z 2 Markov random elds which are K but are not isomorphic to Bernoulli shifts 6]. In this paper we show that most distinct members of this family are not isomorphic. This implies that there is a two parameter family of Z 2 Markov random elds of the same entropy, no two of which are isomorphic.