Nested quasicrystalline discretisations of the line

One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinatorial properties from the point of view of the theory of languages, and self-similarity with algebraic scaling factor $\theta$. We explain the relation of the cut-and-project sets to non-standard numeration systems based on $\theta$. We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences

Some Combinatorial Operators in Language Theory

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.Comment: 21 page

THE AMERICAN KOINE-ORIGIN, RISE, AND PLATEAU STAGE

The early dialect history of the British immigrants to the United States involved a leveling process, leading to the formation of a koine. Reports from British travelers in the eighteenth century indicate a "striking uniformity" in the English of the American colonists. Comparison with the early reports from Australia, and with general works on the theory of languages in migration, indicates that the development of a koine would be expected in a situation of the type pre.vailing, where dominant regional patterns of settling in the new country cannot be convincingly shown. Since observers in the late eighteenth century begin to be critical of American usage, it is concluded that the koine stage was drawing to a close by that time and individual American dialects were developing. It is concluded that the formation of those new dialects was traceable to influences other than the regional dialects of more recent immigrants from England.http://web.ku.edu/~starjrn

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions indexed by compositions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials.Comment: 30 page

Complexity of Problems of Commutative Grammars

We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v?) is P, and that the equivalence problem for context free grammars (do G_1 and G_2 generate the same language?) is in $\mathrm{\Pi_2^P}$