A celebrated unit distance conjecture due to Erdős says that the distances cannot arise more than C ϵ n 1 + times (for any > 0 ) among points in Euclidean plane (see e.g. [10] and references contained therein). In three dimensions, conjectured bound is 4 3 [8] [13] ). dimensions four higher, this problem, its general formulation, loses meaning because Lens example shows one can construct a set ...

We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms uniform distribution sequences. give a necessary condition number to be random expressed classical This extends result proved by Avigad sequences linear functions with integer coefficients wider class Koksma functions. And, requiring equidistribution respect every computably enumerable open set (respectively...

Focusing on nanocrystalline (nc) pure face-centered cubic metals, where systematic experimental data are available, this paper presents a brief overview of the recent progress made in improving mechanical properties of nc materials, and in quantitatively and mechanistically understanding the underlying mechanisms. The mechanical properties reviewed include strength, ductility, strain rate and t...