Let G ≤ Sn be a permutation group of degree n with a quotient isomorphic to Ak. We show that if n < (1/2 − o(1))k then the extension splits. Motivated by this result, Guralnick and Liebeck have shown that the conclusion fails to hold if we allow n to be n = 2k(k − 1), establishing a tight quadratic growth rate for the smallest degree of a faithful permutation representation of a non-split exten...

The paper is devoted to a concept of aggregation of binary fuzzy relations. Consider two n-ary vectors, when we know the degrees to which corresponding components are in relation we can measure the degree to which vectors are in relation. Tasks with such background justify the necessity of aggregation of binary fuzzy relations and we contribute to the topic in this paper. First we recall the ap...

We are investigating parts of the mathematical foundations of stemmatology, the science reconstructing the copying history of manuscripts. After Joseph Bédier in 1928 got suspicious about large amounts of root bifurcations he found in reconstructed stemmata, Paul Maas replied in 1937 using a mathematical argument that the proportion of root bifurcating stemmata among all possible stemmata is so...

Slaman and Woodin have developed and used set-theoretic methods to prove some remarkable theorems about automorphisms of, and de…nability in, the Turing degrees. Their methods apply to other coarser degree structures as well and, as they point out, give even stronger results for some of them. In particular, their methods can be used to show that the hyperarithmetic degrees are rigid and biinter...

The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size of maximum matchings in scale-free networks, where the size of maximum matchings is small and a perfect matching often does not exist. In this paper, we stud...