A new approach to the 22-regularity of the ℓ\ell-abelian complexity of 22-automatic sequences

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.Comment: 44 pages, 2 figures; publication versio

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

Constant 2-labelling of weighted cycles

We introduce the concept of constant 2-labelling of a weighted graph and show how it can be used to obtain periodic sphere packing. Roughly speaking, a constant 2-labelling of a weighted graph is a 2-coloring (black and white) of its vertex set which preserves the sum of the weight of black vertices under some automorphisms. In this manuscript, we study this problem on weighted complete graphs and on weighted cycles. Our results on cycles allow us to determine (r,a,b)-codes in Z^2 whenever |a-b|>4 and r>1

Some properties of abelian return words (long abstract)

We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue–Morse word, the set of abelian derived sequences is infinite

Mathémagie - L'art de la divination

Cet exposé est basé sur la suite des exposés Mathémagie (I, II, III) de Michel Rigo. Nous présentons ici 10 tours de magie ne nécessitant aucune habileté particulière de la part de l'apprenti magicien : des tours de cartes, des tours de divination et le célèbre tour du ``barman aveugle avec des gants de boxe''. Contrairement au magicien qui ne dévoile jamais ses secrets, ici, nous expliquons que ces tours reposent sur diverses propriétés et constructions mathématiques, comme la théorie des graphes ou la combinatoire des mots

On a conjecture about regularity and l-abelian complexity

A natural generalization of automatic sequences over an infinite alphabet is given by the notion of k-regular sequences, introduced by Allouche and Shallit in 1992. The k-regularity of a sequence provides us with structural information about how the different terms are related to each other. We show that a sequence satisfying a certain symmetry property is 2-regular. We apply this theorem to develop a general approach for studying the l-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue–Morse word have 2-abelian complexity sequences that are 2-regular. The computation and arguments leading to these results fit into a quite general scheme that can be used to obtain additional regularity results. This supports the conjecture that the l-abelian complexity of a $k$-automatic sequence is a k-regular sequence

Biological evaluation of the Belgian beaches by means of terrestrial invertebrates

Beaches belong to the least studied ecosystems, although they contain typical habitats for a large amount of specialised terrestrial invertebrates. This specific beach fauna was quite diverse along the Belgian coast at the beginning of the twentieth century. Especially species bound to organic matter, washed up on the tide line, were well represented. As a result of the development of mass tourism, most of our beaches are subject to mechanical beach cleaning and the suppletion of sand. These activities are believed to be responsible for the degradation of the original habitat. However, documentation on this topic was scarce and evidence of negative effects on local biodiversity was lacking. Therefore, the main goal of this research was to make an inventory of the terrestrial arthropod fauna on Flemish beaches and analysing temporal and spatial variation, in function of abiotic components such as the degree of recreation and the intensity of mechanical beach cleaning. The main conclusion is that a high degree of recreation and mechanical beach cleaning indeed has a negative influence on the richness of the species bound to organic detritus and also induces a change in community structure of terrestrial invertebrates along the Flemish coast. Secondly, predators and even parasites are also indirectly influenced by these anthropogenic disturbances, as a result of the declining prey population. Excluding or at least reducing these impacts along certain sections of the Flemish coast, might ensure the preservation of the organic detritus on the tide line and hence its associated beach fauna