Separating path systems of almost linear size

A separating path system for a graph G is a collection P of paths in G such that for every two edges e and f, there is a path in P that contains e but not f. We show that every n-vertex graph has a separating path system of size O(n log∗ n). This improves upon the previous best upper bound of O(n log n), and makes progress towards a conjecture of Falgas-Ravry– Kittipassorn–Kor´andi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluh´ar, according to which an O(n) bound should hold

Exploring the Mental Lexicon of the Multilingual: Vocabulary Size, Cognate Recognition and Lexical Access in the L1, L2 and L3

Recent empirical findings in the field of Multilingualism have shown that the mental lexicon of a language learner does not consist of separate entities, but rather of an intertwined system where languages can interact with each other (e.g. Cenoz, 2013; Szubko-Sitarek, 2015). Accordingly, multilingual language learners have been considered differently to second language learners in a growing number of studies, however studies on the variation in learners’ vocabulary size both in the L2 and L3 and the effect of cognates on the target languages have been relatively scarce. This paper, therefore, investigates the impact of prior lexical knowledge on additional language learning in the case of Hungarian native speakers, who use Romanian (a Romance language) as a second language (L2) and learn English as an L3. The study employs an adapted version of the widely used Vocabulary Size Test (Nation & Beglar, 2007), the Romanian Vocabulary Size Test (based on the Romanian Frequency List; Szabo, 2015) and a Hungarian test (based on a Hungarian frequency list; Varadi, 2002) in order to measure vocabulary sizes, cognate knowledge and response times in these languages. The findings, complemented by a self-rating language background questionnaire, indicate a strong link between Romanian and English lexical proficiency

Transitive simple subgroups of wreath products in product action

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of `Cartesian decompositions' of the permuted set, relating them to certain `Cartesian systemsof subgroups'. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.Comment: Submitte

Generalised sifting in black-box groups

We present a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. Our procedure is a Monte Carlo algorithm, and is presented and analysed in the context of black-box groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. Our major objective was that the procedures could be proved to be Monte Carlo algorithms, and their costs computed. In addition we explicitly determined suitable subset chains for six of the sporadic groups, and we implemented the algorithms involving these chains in the {\sf GAP} computational algebra system. It turns out that sample implementations perform well in practice. The implementations will be made available publicly in the form of a {\sf GAP} package