On right conjugacy closed loops of twice prime order

The right conjugacy closed loops of order 2p, where p is an odd prime, are classified up to isomorphism.Comment: Clarified definitions, added some remarks and a tabl

Aperiodic order and pure point diffraction

We give a leisurely introduction into mathematical diffraction theory with a focus on pure point diffraction. In particular, we discuss various characterisations of pure point diffraction and common models arising from cut and project schemes. We finish with a list of open problems.Comment: 14 page

Solution of the coincidence problem in dimensions d ≤ 4

Abstract. Discrete point sets S such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries R such that S ∩ RS is a subset of S of finite density. These are the so-called coincidence isometries. They are important in understanding and classifying grain boundaries and twins in crystals and quasicrystals. It is the purpose of this contribution to introduce the corresponding coincidence problem in a mathematical setting and to demonstrate how it can be solved algebraically in dimensions 2, 3 and 4. Various examples both from crystals and quasicrystals are treated explicitly, in particular (hyper-)cubic lattices and quasicrystals with non-crystallographic point groups of type H2, H3 and H4. We derive parametrizations of all linear coincidence isometries, determine the corresponding coincidence index (the reciprocal of the density of coinciding points, also called Σ-factor), and finally encapsulate their statistics in suitable Dirichlet series generating functions. 1

Avoiding or limiting regularities in words

International audienceIt is commonly admitted that the origin of combinatorics on words goes back to the work of Axel Thue in the beginning of the twentieth century, with his results on repetition-free words. Thue showed that one can avoid cubes on infinite binary words and squares on ternary words. Up to now, a large part of the work on the theoretic part of combinatorics on words can be viewed as extensions or variations of Thue’s work, that is, showing the existence (or nonexistence) of infinite words avoiding, or limiting, a repetition-like pattern. The goal of this chapter is to present the state of the art in the domain and also to present general techniques used to prove a positive or a negative result. Given a repetition pattern P and an alphabet, we want to know if an infinite word without P exists. If it exists, we are also interested in the size of the language of words avoiding P, that is, the growth rate of the language. Otherwise, we are interested in the minimum number of factors P that a word must contain. We talk about limitation of usual, fractional, abelian, and k-abelian repetitions and other generalizations such as patterns and formulas. The last sections are dedicated to the presentation of general techniques to prove the existence or the nonexistence of an infinite word with a given property