Dirichlet Uniformly Well-approximated Numbers

Fix an irrational number θ. For a real number τ > 0, consider the numbers y satisfying that for all large number Q, there exists an integer 1 ≤ n ≤ Q, such that ∥nθ − y∥ < Q−τ, where ∥⋅∥ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any τ > 0, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of θ. It is also proved that with respect to τ, the only possible discontinuous point of the Hausdorff dimension is τ = 1.

DISCONTINUOUS POINT PROCESSES FOR THE ANALYSIS OF REPAIRABLE UNITS

In this paper, a useful family of discontinuous point processes is proposed, which is able to analyze the failure pattern of repairable units subjected to repair actions that depart from the commonly assumed minimal repair policy. Maximum likelihood estimators of the parameters which index two special cases of the above family are derived, and procedures for testing the departure from the minimal repair assumption, based on asymptotic results, are discussed. An exact and unbiased testing procedure to be performed for small or moderate sample sizes is also proposed. Finally, numerical examples are given to illustrate the proposed estimation and testing procedures.