Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a\left(t\right)u_t$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions $a$: typically $a$ is equal to $1$ on $(0,T)$, equal to $0$ on $(T,qT)$ and is $qT$-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases,...

Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a\left(t\right)u_t$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0,T), equal to 0 on (T, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability....

Let ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ be the ring of Gaussian integers modulo $n$. We construct for ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ a cubic mapping graph $\Gamma \left(n\right)$ whose vertex set is all the elements of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ and for which there is a directed edge from $a\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ to $b\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ if $b=a^3$. This article investigates in detail the structure of $\Gamma \left(n\right)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in ${\Gamma }_1\left(n\right)$ is obtained and we also examine when a vertex lies on a $t$-cycle of ${\Gamma }_2\left(n\right)$, where ${\Gamma }_1\left(n\right)$ is induced by all the units of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ while ${\Gamma }_2\left(n\right)$ is induced by all the...

We prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.