The k -regular sequences form a large class studied in number theory, combinatorics, and other parts of discrete mathematics. This class is known to be closed under many natural operations, such as term-by-term sum, product, running sum, and so forth, but it is not closed under running maximum. Proving the previously-known counterexample, involving the Stern sequence, required intricate arguments. In this note, we construct a significantly simpler example of a k -regular sequence whose running maximum is not k -regular.

In this paper, we study circular words arising in the development of equipment using shields in brachytherapy. This equipment has physical constraints that have to be taken into consideration. From an algorithmic point of view, the problem can be formulated as follows: Given a circular word, find a constrained circular word of the same length such that the Manhattan distance between these two words is minimal. We show that we can solve this problem in pseudo polynomial time (polynomial time in practice) using dynamic programming.

We compute the cardinality of the syntactic monoid of the language 0* repb (m$\mathbb{N}$) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integer