Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from minimal $G$-flow. From this, deduce algebra generated by functions $\mathfrak{A}(G)$ proper subalgebra $\ell^\infty(G)$ and enveloping semigroup universal $M(G)$ quotient $\beta G$. When $G$ countable, also prove any metrizable, $G$-flow, there exists free, it exis...

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...

In particular, the values at x = 0 are called Bernoulli numbers of order k, that is, Bn (0) = Bn k) (see [1, 2, 4, 5, 9, 10, 14]). When k = 1, the polynomials or numbers are called ordinary. The polynomials Bn (x) and numbers Bn were first defined and studied by Norlund [9]. Also Carlitz [2] and others investigated their properties. Recently they have been studied by Adelberg [1], Howard [5], a...

Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as words. We give positive answer to their Open Problem 2: almost surely, all words are seen for site on $\mathbb{Z}^3$ with parameter $p = 1/2$. also extend this result various directions, proving same $\mathbb{Z}^d$, $d \geq 3$, any value \in (p\_c^{\tex...

The Vernam cipher (or one-time pad) has played an important rule in cryptography because it is a perfect secrecy system. For example, if an English text (presented in binary system) X1X2... is enciphered according to the formula Zi = (Xi + Yi) mod 2, where Y1Y2... is a key sequence generated by the Bernoulli source with equal probabilities of 0 and 1, anyone who knows Z1Z2... has no information...

In undirected graphical models, learning the graph structure and learning the functions that relate the predictive variables (features) to the responses given the structure are two topics that have been widely investigated in machine learning and statistics. Learning graphical models in two stages will have problems because graph structure may change after considering the features. The main con...

In this paper, we consider three types of functions given by products of Bernoulli and Genocchi functions and derive some new identities arising from Fourier series expansions associated with Bernoulli and Genocchi functions. Furthermore, we will express each of them in terms of Bernoulli functions.