The K-Degrees, Low for K Degrees,and Weakly Low for K Sets

The K-Degrees, Low for K Degrees, and Weakly Low for K Sets

We call A weakly low for K if there is a c such that KA(σ) ≥ K(σ)−c for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K iff Chaitin’s Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity o...

K-trivial, K-low and MLR-low Sequences: A Tutorial

A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of K-trivial, K-low and MartinLöf-random-low sets: three different definitions turns out to be equivalent for very non-trivial reasons. This paper, based on the course taught by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014, provides an exposition of t...

k-Sets and k-Facets

We survey problems, results, and methods concerning k-facets and k-sets of finite point sets in real affine d-space and the dual notion of levels in arrangements of hyperplanes.

Completely separating systems of k-sets for 7 ≤ k ≤ 10

Generalized Frames for B(H, K)

Frames play significant role in various areas of science and engineering. Motivated by the work of Chander Shekhar, S. K. Kaushik and Abas Askarizadeh, Mohammad Ali Dehghan, we introduce the concepts of $K$-frames for $B(mathcal{H, K})$ and  we establish some result. Also, we consider the relationships between $K$-Frames and $K$-Operator Frames for $B(mathcal{H})$.