Every in nite binary sequence is Turing reducible to a random one. This is a corollary of a result of P eter G acs stating that for every co-r.e. closed set with positive measure of in nite sequences there exists a computable mapping which maps a subset of the set onto the whole space of in nite sequences. Cristian Calude asked whether in this result one can replace the positive measure conditi...

Let Γ be a hypergraph with vertex set Ω, let p : Ω → [0, 1], and let Ωp be a random set formed by including every ω ∈ Ω independently with probability p(ω). We investigate the general question of deriving fine (asymptotic) estimates for the probability that Ωp is an independent set in Γ, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of lower...

We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy v...

Martin-Löf randomness was originally defined and studied in the context of the Cantor space 2ω . In [1] probability theoretic random closed sets (RACS) are used as the foundation for the study of Martin-Löf randomness in spaces of closed sets. Here we focus on the space of closed subsets of R and a particular family of measures on this space, the generalized Poisson processes. This gives a nove...

Enumerating closed sets that are frequent in a given database is a fundamental data mining technique that is used, e.g., in the context of market basket analysis, fraud detection, or Web personalization. There are two complementing reasons for the importance of closed sets— one semantical and one algorithmic: closed sets provide a condensed basis for non-redundant collections of interesting loc...

In this paper, by using random fixed point index theory, some new boundary conditions based on strictly convex or strictly concave functions are established and some new theorems for the solutions of a class of random semi-closed 1-set-contractive operator equations A(ω, x) = μx are obtained, which extend and generalize some corresponding results of Wang [S. Wang, Fixed Point Theory Appl., 2011...

The set of all m-tuples of compatible full conditional distributions on discrete random variables is an algebraic set whose defining ideal is a unimodular toric ideal. We identify the defining polynomials of these ideals with closed walks on a bipartite graph. Our algebraic characterization provides a natural generalization of the requirement that compatible conditionals have identical odds rat...

We study Markov interval maps with random holes. The holes are not necessarily elements of the Markov partition. Under a suitable, and physically relevant, assumption on the noise, we show that the transfer operator associated with the random open system can be reduced to a transfer operator associated with the closed deterministic system. Exploiting this fact, we show that the random open syst...

The main purpose of this paper is to study normed hypervector spaces. We generalize some definitions such as basis, convexity, operator norm, closed set, Cauchy sequences, and continuity in such spaces and prove some theorems about them.