Almost strong mixing group actions in topological dynamics

In ergodic theory, given sufficient conditions on the system, every weak mixing $mathbb{N}$-action is strong mixing along a density one subset of $mathbb{N}$. We ask if a similar statement holds in topological dynamics with density one replaced with thickness. We show that given sufficient initial conditions, a group action in topological dynamics is strong mixing on a thick subset of the group if and only if the system is $k$-transitive for all $k$, and conclude that an analogue of this statement from ergodic theory holds in topological dynamics when dealing with abelian groups.

On the Instability and Critical Damping Conditions, $ktau = 1/e$ and $ktau = pi/2$ of the equation $dot{theta} = -k theta(t-tau)$

In this note, I show that it is possible to use elementary mathematics, instead of the machinery of Lambert function, Laplace Transform, or numerics, to derive the instability condition, $k tau = pi/2$, and the critical damping condition, $ktau = 1/e$, for the time-delayed equation $dot{theta} = -k theta(t-tau)$. I hope it will be useful for the new comers to this equation, and perhaps even to the experts if this is a simpler method compared to othe