On continuity of Guo Wuwen function

We show that the functions g and gs introduced by Guo Wuwen in [4] are continuous and semialgebraic. We use this fact to prove that the set Nn of ordered n-tuples of real numbers, realizable by nonnegative matrices, is a closed set

Group homomorphisms as error correcting codes

We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups $G$ and $H$. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance when $G$ is solvable or $H$ is nilpotent, in terms of the normal subgroup structure of $G$ as well as the prime divisors of $|G|$ and $|H|$. In particular, we show that in the above case, the distance is independent of the subgroup structure of $H$. We complement this by showing that, in general, the distance depends on the subgroup structure $G$.Comment: 13 page

Hankel Determinant Calculus for the Thue-Morse and related sequences

The Hankel determinants of certain automatic sequences $f$ are evaluated, based on a calculation modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as $LU$-decompo\-si\-tion and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence $g$ equal to $f$ modulo $p$ are equal to the Hankel determinants of $f$ modulo $p$. The clue then consists of finding a nice sequence $g$, whose Hankel determinants have closed-form expressions. Several examples are presented, including a result saying that the Hankel determinants of the Thue-Morse sequence are nonzero, first proved by Allouche, Peyri\`ere, Wen and Wen using determinant manipulation. The present approach shortens the proof of the latter result significantly. We also prove that the corresponding Hankel determinants do not vanish when the powers $2^n$ in the infinite product defining the $\pm 1$ Thue--Morse sequence are replaced by $3^n$